,transcriber's ,notes3 ,underscores 80 before and after a word or phrase indicate italics in the original text4 ,equal signs 80 before and after a word or phrase indicate bold in the original text4 ,small capitals have been converted to ,,solid capitals4 ,illustrations have been moved so they do not break up paragraphs4 ,typographical and punctuation errors have been silently corrected4 ,page #da3 8#a1jjj1jjj feet or #cjj1jjj kilometres per second0 changed to 8#a1jjj1jjj1jjj feet or #cjj1jjj kilometres per second0 ,,the ,,atom ,,and ,,the ,,bohr ,,theory ,,of ,,its ,,structure ,original ,title3 8,bohr's ,atomteori1 almenfatteligt fremstillet0 ,translated from the ,danish by #a ,fellow of the ,american-,scandinavian ,foundation1 #aibc1 and ,,rachel ,t4 ,,lindsay ,illustration3 ,niels ,bohr ,,the ,,atom ,,and ,,the ,,bohr ,,theory ,,of ,,its ,,structure ,an ,elementary ,presentation ,,by ,h4 ,a4 ,,kramers ,,lecturer ,,at ,,the ,,institute ,,of ,,theoretical ,,physics ,,in ,,the ,,university ,,of ,,copenhagen ,,and ,,helge ,,holst ,,librarian ,,at ,,the ,,royal ,,technical ,,college ,,of ,,copenhagen ,,with ,a ,,foreword ,,by ,,sir ,,ernest ,,rutherford1 ,f4,r4,s4 ,illustration ,,new ,,york ,,alfred ,a4 ,,knopf #aibc ,,printed ,,in ,,great ,,britain ,,by ,,morrison ,,and ,,gibb ,,ltd41 ,,edinburgh ,,preface ,at the close of the nineteenth century and the beginning of the twentieth1 our knowledge of the activities in the interior of matter experienced a development which surpassed the boldest hopes that could have been entertained by the chemists and physicists of the nineteenth century4 ,the smallest particles of chemistry1 the atoms of the elements1 which hitherto had been approached merely by inductive thought1 now became tangible realities1 so to speak1 which could be counted and whose tracks could be photographed4 ,a series of remarkable experimental investigations1 stimulated largely by the ,english physicist1 ,j4 ,j4 ,thomson1 had disclosed the existence of negatively charged particles1 the so-called electrons1 _/ the mass of the smallest atom of the known elements4 ,a theory of electrons1 based on ,maxwell's classical electrodynamical theory and developed mainly through the labours of ,lorentz in ,holland and ,larmor in ,england1 had brought the problem of atomic structure into close connection with the theory of radiation4 ,the experiments of ,rutherford proved1 beyond a doubt1 that atoms were composed simply of light1 negative electric particles1 and small heavy1 positive electric particles4 ,the new 8quantum theory0 of ,planck was proving itself very powerful in overcoming grave difficulties in the theory of radiation4 ,the time thus seemed ripe for a comprehensive investigation of the fundamental problem of physics--the constitution of matter1 and an explanation in terms of simple general laws of the physical and chemical properties of the atoms of the elements4 ,during the first ten years of the new century the problem was attacked with great zeal by many scientists1 and many interesting atomic models were developed and studied4 ,but most of these had more significance for chemistry than for physics1 and it was not until #aiac that the work of the ,danish physicist1 ,niels ,bohr1 paved the way for a really physical investigation of the problem in a remarkable series of papers on the spectrum and atomic structure of hydrogen4 ,the ideas of ,bohr1 founded as they were on the quantum theory1 were startling and revolutionary1 but their immense success in explaining the facts of experience after a time won for them the wide recognition of the scientific world1 and stimulated work by other investigators along similar lines4 ,the past decade has witnessed an enormous development at the hands of scientists in all parts of the world of ,bohr's original conceptions2 but through it all ,bohr has remained the leading spirit1 and the theory which1 at the present time1 gives the most comprehensive view of atomic structure may1 therefore1 most properly bear the name of ,bohr4 ,it is the object of this book to give the reader a glimpse of the fundamental conceptions of this theory1 together with some of the most significant results it has attained4 ,the book is designed to meet the needs of those who wish to keep abreast of modern developments in science1 but have neither time nor inclination to delve into the highly mathematical abstract literature in which the developments are usually concealed4 ,it is with this in mind that the first four chapters have been devoted to a general survey of those parts of physics and chemistry which have close connection with atomic theory4 ,no attempt has been made at a mathematical development1 and the physical meaning of such mathematical formul as do occur has been clearly emphasized in the text4 ,it is hoped1 however1 that even those readers whose acquaintance with atomic theory is more than casual1 will find the book a stimulus to further study of the ,bohr theory4 ,here we wish to record our best thanks to ,mr4 and ,mrs4 ,lindsay for the ability and the great care with which they have carried out the translation from the ,danish original4 ,,foreword ,during the last decade there has been a great advance in our knowledge of the structure of the atom and of the relation between the atoms of the chemical elements4 ,in the later stages1 science owes much to the remarkable achievements of ,professor ,niels ,bohr and his co-workers in ,copenhagen4 ,for the first time1 we have been given a consistent theory to explain the arrangement and motion of the electrons in the outer atom4 ,the theory of ,bohr is not only able to account in considerable detail for the variation in the properties of the elements exemplified by the periodic law1 but also for the main features of the spectra1 both ,x-ray and optical1 shown by all elements4 ,this volume1 written by ,dr4 ,kramers and ,mr4 ,holst1 gives a simple and interesting account of our knowledge of atomic structure1 with special reference to the work of ,professor ,bohr4 ,dr4 ,kramers is in an especially fortunate position to give a first-hand account of this subject1 for he has been a valued assistant to ,professor ,bohr in developing his theories1 and has himself made important original contributions to our knowledge in this branch of inquiry4 ,i can confidently recommend this book to ,english readers as a clearly written and accurate account of the development of our ideas on atomic structure4 ,it is written in simple language1 and the essential ideas are explained without mathematical calculations4 ,this book should prove attractive not only to the general scientific reader1 but also to the student who wishes to gain a broad general idea of this subject before entering into the details of the mathematical theory4 ,e4 ,,rutherford4 ,,cavendish ,,laboratory1 ,,cambridge1 #hth ,october #aibc4 ,,contents ,,page ,,preface vii ,,foreword xi ,,chap4 ,i4 ,,atoms ,,and ,,molecules #a ,,ii4 ,,light ,,waves ,,and ,,the ,,spectrum #cd ,,iii4 ,,ions ,,and ,,electrons #fa ,,iv4 ,,the ,,nuclear ,,atom #hc ,v4 ,,the ,,bohr ,,theory ,,of ,,the ,,hydrogen ,,spectrum #aje ,,vi4 ,,various ,,applications ,,of ,,the ,,bohr ,,theory #aec ,,vii4 ,,atomic ,,structure ,,and ,,the ,,chemical ,,properties ,,of ,,the ,,elements #ahj ,,interpretation ,,of ,,symbols ,,and ,,physical ,,constants #bji ,,coloured ,,plates ,,plate ,i4 ,,spectrum ,,plates ,,according ,,to ,,the ,,original ,,drawings ,,of ,,bunsen ,,and ,,kirchhoff ,at end ,,ii4 ,,principal ,,features ,,of ,,atomic ,,structure ,,in ,,some ,,of ,,the ,,elements--,,atomic ,,structure ,,of ,,radium ,at end ,,the ,,atom ,,and ,,the ,,bohr ,,theory ,,of ,,its ,,structure ,,chapter ,i ,,atoms ,,and ,,molecules ,introduction4 ,as early as #djj ,b4,c4 the ,greek philosopher1 ,democritus1 taught that the world consisted of empty space and an infinite number of small invisible particles4 ,these particles1 differing in form and magnitude1 by their arrangements and movements1 by their unions and disunions1 caused the existence of physical bodies with different characteristics1 and also produced the observed variations in these bodies4 ,this theory1 which no doubt antedated ,democritus1 later became known as the ,atomic ,theory1 since the particles were called atoms1 i4e4 the 8indivisible40 ,but the atomic conception was not the generally accepted one in antiquity4 ,aristotle 7#chd-cbb ,b4,c47 was not an atomist1 and denied the existence of discontinuous matter2 his philosophy had a tremendous influence upon the ideas of the ancients1 and even upon the beliefs of the ,middle ,ages4 ,it must be confessed that his conception of the continuity of matter seemed to agree best with experiment1 because of the apparent homogeneity of physical substances such as metal1 glass1 water and air4 ,but even this apparent homogeneity could not be considered entirely inconsistent with the atomic theory1 for1 according to the latter1 the atoms were so small as to be invisible4 ,moreover1 the atomic theory left the way open for a more complete understanding of the properties of matter4 ,thus when air was compressed and thereafter allowed to expand1 or when salt was dissolved in water producing an apparently new homogeneous liquid1 salt water1 or when silver was melted by heat1 or light changed colour on passing through wine1 it was clear that something had happened in the interior of the substances in question4 ,but complete homogeneity is synonymous with inactivity4 ,how is it possible to obtain a definite idea of the inner activity lying at the bottom of these changes of state1 if we do not think of the phenomenon as an interplay between the different parts of the apparently homogeneous matter8 ,thus1 in the examples above1 the decrease in the volume of the air might be considered as due to the particles drawing nearer to each other2 the dissolving of salt in water might be looked upon as the movement of the salt particles in between the water particles and the combination of the two kinds2 the melting of silver might naturally appear to be due to the loosening of bonds between the individual silver particles4 ,the atomic theory had thus a sound physical basis1 and proved particularly attractive to those philosophers who tried to explain the mysterious activity of matter in terms of exact measurements4 ,the atomic hypothesis was never completely overthrown1 being supported after the time of ,aristotle by ,epicurus 7c4 #cjj ,b4,c471 who introduced the term 8atom10 and by the ,latin poet1 ,lucretius 7c4 #ge ,b4,c47 in his ,de ,rerum ,natura4 ,even in the ,middle ,ages it was supported by men of independent thought1 such as ,nicholas of ,autrucia1 who assumed that all natural activities were due to unions or disunions of atoms4 ,it is interesting to note that in #acdh he was forced to retract this heresy4 ,with the impetus given to the new physics by ,galileo 7#afjj7 the atomic view gradually spread1 sometimes explicitly stated as atomic theory1 sometimes as a background for the ideas of individual philosophers4 ,various investigators developed comprehensive atomic theories in which they attempted to explain nearly everything from purely arbitrary hypotheses2 they occasionally arrived at very curious and amusing conceptions4 ,for example1 about #afej the ,frenchman1 ,pierre ,gassendi1 following some of the ancient atomists1 explained the solidity of bodies by assuming a hook-like form of atom so that the various atoms in a solid body could be hooked together4 ,he thought of frost as an element with tetrahedral atoms1 that is1 atoms with four plane faces and with four vertices each2 the vertices produced the characteristic pricking sensation in the skin4 ,a much more thorough treatment of the atomic theory was given by ,boscovich 7#aggj74 ,he saw that it was unnecessary to conceive of the atoms as spheres1 cubes1 or other sharply defined physical bodies2 he considered them simply as points in space1 mathematical points with the additional property of being centres of force4 ,he assumed that any two atoms influenced each other with a force which varied1 according to a complicated formula1 with the distance between the centres4 ,but the time was hardly ripe for such a theory1 inspired as it evidently was by ,newton's teachings about the gravitational forces between the bodies of the universe4 ,indeed there were no physical experiments whose results could1 with certainty1 be assumed to express the properties of the individual atoms4 ,the ,atomic ,theory and ,chemistry4 ,illustration3 ,,fig4 #a4--,the four elements and the four fundamental characteristics4 ,in the meantime atomic investigations of a very different nature had been influencing the new science of chemistry1 in which the atomic theory was later to prove itself extraordinarily fruitful4 ,it was particularly unfortunate that in chemistry1 concerned as it is with the inner activities of the elements1 ,aristotle's philosophy was long the prevailing one4 ,he adopted and developed the famous theory of the four 8elements10 namely1 the dry and cold earth1 the cold and damp water1 the damp and warm air1 the warm and dry fire4 ,these 8elements0 must not be confused with the chemical elements known at the present day2 they were merely representatives of the different consistent combinations of the four fundamental qualities1 dryness and wetness1 heat and cold4 ,from the symmetry in the system these were supposed to be the principles by means of which all the properties of matter could be explained4 ,neither the four 8elements0 nor the four fundamental qualities could be clearly defined2 they were vague ideas to be discussed in long dialectic treatises1 but were founded upon no physical quantities which could be measured4 ,a system of chemistry which had its theoretical foundations in the ,greek elemental conceptions naturally had to work in the dark4 ,undoubtedly this uncertainty contributed to the relatively insignificant results of all the labour expended in the ,middle ,ages on chemical experiments1 many of which had to do with the attempt to transmute the base metals into gold4 ,naturally there were many important contributions to chemistry1 and the theories were changed and developed in many ways in the course of time4 ,the alchemists of the ,middle ,ages thought that metal consisted only of sulphur and quicksilver2 but the interpretation of this idea was influenced by the ,greek elemental theory which was maintained at the same time2 thus these new metal 8elements0 were considered by many merely as the expressions of certain aspects of the metallic characteristics1 rather than as definite substances1 identical with the elements bearing these names4 ,it is1 however1 necessary to guard against attributing to a single conception too great influence on the historical development of the chemical and physical sciences4 ,that the growth of the latter was hindered for so long a time was due more to the uncritical faith in authority and to the whole characteristic psychological point of view which governed ,western thought in the centuries preceding the ,renaissance4 ,robert ,boyle 7#afbg-afia7 is one of the men to whom great honour is due for brushing aside the old ideas about the elements which had originated in obscure philosophical meditations4 ,to him an element was simply a substance which by no method could be separated into other substances1 but which could unite with other elements to form chemical compounds possessing very different characteristics1 including that of being decomposable into their constituent elements4 ,undoubtedly ,boyle's clear conception of this matter was connected with his representation of matter as of an atomic nature4 ,according to the atomic conception1 the chemical processes do not depend upon changes within the element itself1 but rather in the union or disunion of the constituent atoms4 ,thus when iron sulphide is produced by heating iron and sulphur together1 according to this conception1 the iron atoms and the sulphur atoms combine in such a way that each iron atom links itself with a sulphur atom4 ,there is then a definite meaning in the statement that iron sulphide consists of iron and sulphur1 and that these two substances are both represented in the new substance4 ,there is also a definite meaning1 for instance1 in the statement that iron is an element1 namely1 that by no known means can the iron be broken down into different kinds of atoms which can be reunited to produce a substance different from iron4 ,the clarity which the atomic interpretation gave to the conception of chemical elements and compounds was surely most useful to chemical research in the following years2 but before the atomic theory could play a really great rle in chemistry1 it had to undergo considerable development4 ,in the time of ,boyle1 and even later1 there was still uncertainty as to which substances were the elements4 ,thus1 water was generally considered as an element4 ,according to the so-called phlogiston theory developed by the ,german ,stahl 7#affj-agcd71 a theory which prevailed in chemistry for many years1 the metals were chemical compounds consisting of a gaseous substance1 phlogiston1 which was driven off when the metals were heated in air1 and the metallic oxide which was left behind4 ,it was not until the latter half of the eighteenth century that the foundation was laid for the new chemical science by a series of discoveries and researches carried on by the ,swedish scientist ,scheele1 the ,englishmen ,priestley and ,cavendish1 and particularly by the ,frenchman ,lavoisier4 ,it was then discovered that water is a chemical compound of the gaseous elements oxygen and hydrogen1 that air is principally a mixture 7not a compound7 of oxygen and nitrogen1 that combustion is a chemical process in which some substance is united with oxygen1 that metals are elements1 while metallic oxides1 on the other hand1 are compounds of metal and oxygen1 etc4 ,of special significance for the atomic theory was the fact that ,lavoisier made weighing one of the most powerful tools of scientific chemistry4 ,weighing had indeed been used previously in chemical experiments1 but the experimenters had been satisfied with very crude precision1 and the results had little influence on chemical theory4 ,for example1 the phlogiston theory was maintained in spite of the fact that it was well known that metallic oxide weighed more than the metal from which it was obtained4 ,lavoisier now showed1 by very careful weighings1 that chemical combinations or decompositions can never change the total weight of the substances involved2 a given quantity of metallic oxide weighs just as much as the metal and the oxygen taken individually1 or vice versa4 ,from the point of view of the atomic theory1 this obviously means that the weight of individual atoms is not changed in the combinations of atoms which occur in the chemical processes4 ,in other words1 the weight of an atom is an invariable quantity4 ,here1 then1 we have the first property of the atom itself to be established by experiment--a property1 indeed1 which most atomists had already tacitly assumed4 ,moreover1 by the practice of weighing it was determined that to every chemical combination there corresponds a definite weight ratio among the constituent parts4 ,this also had been previously accepted by most chemists as highly probable2 but it must be admitted that the law at one time was assailed from several sides4 ,in comparing the weight ratios in different chemical compounds certain rules were1 in the meantime1 obtained4 ,in many ways the most important of these1 the so-called law of multiple proportions1 was enunciated in the beginning of the last century by the ,englishman1 ,john ,dalton4 ,as an example of this law we may take two compounds of carbon and hydrogen called methane or marsh gas and ethylene1 in which the quantities of hydrogen compounded with the same quantity of carbon are as two is to one4 ,another example may be seen in the compounds of carbon and oxygen4 ,in the two compounds of carbon and oxygen1 carbon monoxide and carbon dioxide1 the weight ratios between the carbon and oxygen are respectively as three to four and three to eight4 ,a definite quantity of carbon has thus in carbon dioxide combined with just twice as much oxygen as in carbon monoxide4 ,no less than five oxygen compounds with nitrogen are known1 where with a given quantity of nitrogen the oxygen is combined in ratios of one1 two1 three1 four and five4 ,these simple number relations can be explained very easily by the atomic theory1 by assuming1 first1 that all atoms of the same element have the same weight2 and second1 that in a chemical combination between two elements the atoms combine to form an atomic group characteristic of the compound in question--a compound atom1 as ,dalton called it1 or a molecule1 as the atomic group is now called4 ,these molecules consist of comparatively few atoms1 as1 for example1 one of each kind1 or one of one kind and two1 three or four of another1 or two of one kind and three or four of another1 etc4 ,when three elements are involved in a chemical compound the molecule must contain at least three atoms1 but there may be four1 five1 six or more4 ,the law of multiple proportions thus takes on a more complicated character1 but it remains apparent even in this case4 ,when ,dalton in the beginning of last century formulated the theory of the formation of chemical compounds from the atoms of the elements1 he at once turned atomic theory into the path of more practical research1 and it was soon evident that chemical research had then obtained a valuable tool4 ,it may be said that ,dalton's atomic theory is the firm foundation upon which modern chemistry is built4 ,illustration ,illustration ,illustration ,illustration3 ,,fig4 #b4--,representation of a part of ,dalton's atomic table 7of #ahjh7 where the atom of each element has its own symbol1 and chemical compounds are indicated by the union of the atoms of the elements into groups by #b1 #c1 #d 444 7binary1 ternary1 quaternary 444 atoms74 ,below are given the designations of the different atoms1 and in parentheses the atomic weight given by ,dalton with that of hydrogen as unity and the designations of the indicated atomic groups4 ,atoms of the ,elements4--#a4 ,hydrogen 7#a72 #b4 ,azote 7#e72 #c4 ,carbon 7#e72 #d4 ,oxygen 7#g72 #e4 ,phosphorus 7#i72 #f4 ,sulphur 7#ac72 #g4 ,magnesia 7#bj72 #h4 ,lime 7#bc72 #i4 ,soda 7#bh72 #aj4 ,potash 7#db72 #aa4 ,strontites 7#df72 #ab4 ,barytes 7#fh72 #ac4 ,iron 7#ch72 #ad4 ,zinc 7#ef72 #ae4 ,copper 7#ef72 #af4 ,lead 7#ie72 #ag4 ,silver 7#ajj72 #ah4 ,platina 7#ajj72 #ai4 ,gold 7#adj72 #bj4 ,mercury 7#afg74 ,chemical ,compounds4--#ba4 ,water2 #bb4 ,ammonia2 #bc4 #bf4 #bg4 and #cj4 ,oxygen compounds of ,azote2 #bd4 #bi4 and #cc4 ,hydrogen compounds of ,carbon2 #be4 ,carbon monoxide2 #bh4 ,carbon dioxide2 #ca4 ,sulphuric acid2 #cb4 ,hydro-sulphuric acid4 ,while ,dalton's theory could not give information about the absolute weights in grams of the atoms of various elements1 it could say something about the relative atomic weights1 i4e41 the ratios of the weights of the different kinds of atoms1 although it is true that these ratios could not always be determined with certainty4 ,if1 for example1 the ratio between the oxygen and hydrogen in water is found to be as eight to one1 then the weight ratio between the oxygen atom and the hydrogen atom will be as eight to one1 if the water molecule is composed of one oxygen atom and one hydrogen atom 7as ,dalton supposed1 see ,fig4 #b74 ,but it will be as sixteen to one1 if the water molecule is composed of one oxygen and two hydrogen atoms 7as we now know to be the case74 ,on the other hand1 a ratio of seven to one will be compatible with the experimental ratio of eight to one only if we assume that the water molecule consists of fifteen atoms1 eight of oxygen and seven of hydrogen1 a very improbable hypothesis4 ,in another case let us compare the quantities of oxygen and of hydrogen which are compounded with the same quantities of carbon in the two substances1 carbon monoxide and methane respectively4 ,on the assumption that the molecules in question have a simple structure1 we can draw conclusions about the ratio of the atomic weights of hydrogen and oxygen4 ,now1 if a ratio such as seven to one or fourteen to one is obtained while the analysis of water gives eight to one or sixteen to one1 then either the structure of the molecule is more complicated than was assumed1 or the analyses must be improved by more careful experiments4 ,we can thus understand that the atomic theory can serve as a controlling influence on the analysis of chemical compounds4 ,in order to choose between the different possible ratios of atomic weights1 for example1 the eight to one or the sixteen to one in the case of oxygen and hydrogen1 ,dalton had to make certain arbitrary assumptions4 ,the first of these is that two elements of which only one compound is known appear with but one atom each in a molecule4 ,partly on account of this assumption and partly on account of the incompleteness of his analyses1 ,dalton's values of the ratios of the atomic weights of the atoms and his pictures of the structure of molecules differ from those of the present day1 as is obvious from ,fig4 #b4 ,a much firmer foundation for the choice made appears later in the ,avogadro ,law2 starting with the fact that different gases show great similarity in their physical conduct--for instance1 all expand by an increase of of their volume with an increase in temperature from #j ,c4 to #a ,c4--the ,italian1 ,avogadro1 in #ahaa1 put forward the hypothesis that equal volumes of all gases at the same temperature and pressure contain equal numbers of molecules4 ,a few examples suffice to show the usefulness of this rule4 ,when one volume of the gas chlorine unites with one volume of hydrogen there result two volumes of the gas1 hydrogen chloride1 at the same temperature and pressure4 ,according to ,avogadro's ,law one molecule of chlorine and one molecule of hydrogen unite to become two molecules of hydrogen chloride1 and since each of these two molecules must contain at least one atom of hydrogen and one atom of chlorine1 it follows that one molecule of chlorine must contain two atoms of chlorine and that one molecule of hydrogen must contain two atoms of hydrogen4 ,from this one can see that even in the elements the atoms are united into molecules4 ,it is now well established that most elements have diatomic molecules1 though some1 including mercury and many other metals1 are monatomic4 ,when oxygen and hydrogen unite to form water1 one litre of oxygen and two litres of hydrogen produce two litres of water vapour at same temperature and pressure4 ,accordingly1 one molecule of oxygen and two molecules of hydrogen form two molecules of water4 ,if the oxygen molecule is diatomic like the hydrogen1 then one molecule of water contains one atom of oxygen and two atoms of hydrogen4 ,since the weight ratio between the oxygen and hydrogen in water is eight to one1 the atomic weight of oxygen must be sixteen times that of hydrogen4 ,through such considerations1 supported by certain other rules1 it has gradually proved possible to obtain reliable figures for the ratios between the atomic weights of all known elements and the atomic weight of hydrogen4 ,for convenience it was customary to assign the number #a to the latter and to call the ratio between the weight of the atom of a given element and the weight of the hydrogen atom the atomic weight of the element in question4 ,thus the atomic weight of oxygen is #af1 that of carbon #ab1 because one carbon atom weighs as much as #ab hydrogen atoms4 ,nitrogen has the atomic weight #ad1 sulphur #cb1 copper #fc4f1 etc4 ,a summary of the chemical properties and chemical compounds was greatly facilitated by the symbolic system initiated by the ,swedish chemist1 ,berzelius4 ,in this system the initial of the ,latin name of the element 7sometimes with one other letter from the ,latin name7 is made to indicate the element itself1 an atom of the element1 and its atomic weight with respect to hydrogen as unity1 while a small subscript to the initial designates the number of atoms to be used4 ,for example1 in the chemical formula for sulphuric acid1 ,h,,so1 the symbolic formula means that this substance is a chemical compound of hydrogen1 sulphur and oxygen1 that the acid molecule consists of two atoms of hydrogen1 one atom of sulphur and four atoms of oxygen1 and that the weight ratios between the three constituent parts is as #ba #b to #cb to #daf #fd1 or as #a3af3cb4 ,to say that the chemical formula of zinc chloride is ,zn,cl means that the zinc chloride molecule consists of one atom of zinc and two atoms of chlorine4 ,furthermore the changes which take place in a chemical process may be indicated in a very simple way4 ,thus the decomposition of water into hydrogen and oxygen may be represented by the so-called chemical 8equation0 #b;,h,o #b;,h,o1 where ,h and ,o signify the molecules of hydrogen and oxygen respectively4 ,conversely1 the combination of hydrogen and oxygen to form water will be given by the equation #b;,h,o #b;,h,o4 ,as a consequence of the development of the atomic theory the atoms of the elements became1 so to speak1 the building stones of which the elements and the chemical compounds are built4 ,it can also be said that the atoms are the smallest particles which the chemists reckon with in the chemical processes1 but it does not follow from the theory that these building stones in themselves are indivisible4 ,the theory leaves the way open to the idea that they are composed of smaller parts4 ,a belief founded on such an idea was indeed enunciated by the ,englishman1 ,prout1 a short time after ,dalton had developed his atomic theory4 ,prout assumed that the hydrogen atoms were the fundamental ones1 and that the atoms of the other elements consisted of a smaller or larger number of the atoms of hydrogen4 ,this might explain the fact that within the limits of experimental error1 many atomic weights seemed to be integral multiples of that of hydrogen--#af for oxygen1 #ad for nitrogen1 and #ab for carbon1 etc4 ,this led to the possibility that the same might hold for all elements1 and this hypothesis gave impetus to very careful determinations of atomic weights4 ,these1 however1 showed that the assumption of the integral multiples could not be verified4 ,it therefore seemed as if ,prout's hypothesis would have to be given up4 ,it has1 however1 recently come into its own again1 although the situation is more complicated than ,prout had imagined 7see p4 #ig74 ,dalton's atomic theory gave no information about the atoms except that the atoms of each element had a definite constant weight1 and that they could combine to form molecules in certain simple ratios4 ,what the forces are which unite them into such combinations1 and why they prefer certain unions to others1 were very perplexing problems1 which could only be solved when chemical and physical research had collected a great mass of information as a surer source of speculation4 ,from the knowledge of atomic weights it was easy to calculate what weight ratios might be found to exist in chemical compounds1 the molecules of which consisted of simple atomic combinations4 ,thus many compounds which were later produced in the laboratory were first predicted theoretically1 but only a small part of the total number of possible compounds 7corresponding to simple atomic combinations7 could actually be produced4 ,clearly it was one of the greatest problems of chemistry to find the laws governing these cases4 ,it had early been known that the elements seemed to fall into two groups1 characterized by certain fundamental differences1 the metals and the metalloids4 ,in addition1 there were recognized two very important groups of chemical compounds1 i4e4 acids and bases1 possessing the property of neutralizing each other to form a third group of compounds1 the so-called salts4 ,the phenomenon called electrolysis1 in which an electric current separates a dissolved salt or an acid into two parts which are carried respectively with and against the direction of the current1 indicates strongly that the forces holding the atoms together in the molecule are of an electrical nature1 i4e4 of the same nature as those forces which bring together bodies of opposite electrical charges4 ,one is led to denote all metals as electropositive and all metalloids as electronegative1 which means that in a compound consisting of a metal and a metalloid the metal appears with a positive charge and the metalloid with a negative charge4 ,the chemist ,berzelius did a great deal to develop electrical theories for chemical processes4 ,great difficulties1 however1 were encountered1 some proving for the time being insurmountable4 ,such a difficulty1 for example1 is the circumstance that two atoms of the same kind 7like two hydrogen atoms7 can unite into a diatomic molecule1 although one might expect them to be similarly electrified and to repel rather than attract each other4 ,another circumstance playing a very important part in determining the chemical compounds which are possible1 is the consideration of what is called valence4 ,as mentioned above1 one atom of oxygen combines with two atoms of hydrogen to form water1 while one atom of chlorine combines with but one atom of hydrogen to form hydrogen chloride4 ,the oxygen atom thus seems to be 8equivalent0 to two hydrogen atoms or two chlorine atoms1 while one chlorine atom is 8equivalent0 to one hydrogen atom4 ,the atoms of hydrogen and chlorine are for this reason called monovalent1 while that of oxygen is called divalent4 ,again an acid is a chemical compound containing hydrogen1 in which the hydrogen can be replaced by a metal to produce a metallic salt4 ,thus1 when zinc is dissolved by sulphuric acid to form hydrogen and the salt zinc sulphate1 the hydrogen of the acid is replaced by the zinc and the chemical change may be expressed by the formula ,zn,h,,so ,h,zn,s,o ,in this1 one atom of zinc changes place with two atoms of hydrogen4 ,the zinc atom is therefore divalent4 ,this is consistent with the fact that one zinc atom will combine with one oxygen atom to form zinc oxide4 ,to take another example1 if silver is dissolved in nitric acid1 one atom of silver is exchanged for one atom of hydrogen4 ,silver1 therefore1 is monovalent1 and we should expect that one atom of oxygen would unite with two atoms of silver4 ,some elements are trivalent1 as1 for example1 nitrogen1 which combines with hydrogen to form ammonia1 ,,nh2 others1 again1 are tetravalent1 such as carbon1 which unites with hydrogen to form marsh gas ,,ch1 and with oxygen to form carbon dioxide ,,co4 ,a valence greater than seven or eight has not been found in any element4 ,illustration3 ,,fig4 #c4--,rough illustrations of the valences of the elements4 ,a4 ,hydrogen chloride 7,h,cl72 ,b4 ,water 7,h,o72 ,c4 ,methane 7,,ch72 ,d4 ,ethylene 7,c,h74 ,if we consider the matter rather roughly and more or less as ,gassendi did1 we can explain the concept of valence by assuming that the atoms possess hooks2 thus hydrogen and chlorine are each furnished with one hook1 oxygen and zinc with two hooks1 nitrogen with three hooks1 etc4 ,when a hydrogen atom and a chlorine atom are hooked together1 there are no free hooks left1 and consequently the compound is said to be saturated4 ,when one hydrogen atom is hooked into each of the hooks of an oxygen or carbon atom the saturation is also complete 7see ,fig4 #c1 ,a1 ,b1 ,c74 ,the matter is not so simple as this1 however1 since the same element can often appear with different valences4 ,iron may be divalent1 trivalent or hexavalent in different compounds4 ,in many cases1 however1 where an examination of the weight ratios seems to show that an element has changed its valence1 this is not really true4 ,it was mentioned previously that carbon forms another compound with hydrogen in addition to ,,ch1 namely1 ethylene1 containing half as much hydrogen in proportion to the same amount of carbon4 ,with the aid of ,avogadro's ,law1 it is found that the ethylene molecule is not ,,ch but ,c,h4 ,thus we can say that the two carbon atoms in the molecule are held together by two pairs of hooks1 and consequently the compound can be expressed by the formula ,h-,c-,h ,h-,c-,h where the dashes correspond to hooks 7cf4 ,fig4 #c1 ,d74 ,such a formula is called a structural formula4 ,even if we are not allowed to think of the atoms in the molecules as held together by hooks1 it is well to have some sort of concrete picture of molecular structure4 ,it is possible to represent the tetravalent carbon atom in the form of a tetrahedron1 and to consider the united atoms or atomic groups as placed at the four vertices4 ,with such a spatial representation we can get an idea about many chemical questions which otherwise would be difficult to explain4 ,we know1 for example1 that two compound molecules having the same kind and number of atoms and the same bonds 7and hence the same structural formul71 may yet be different in that they are images of each other like a pair of gloves4 ,substances whose molecules are symmetrical in this way can be distinguished from each other by their different action on the passage of light4 ,this molecular chemistry of space1 or stereo-chemistry as it is called1 has proved of great importance in explaining difficult problems in organic chemistry1 i4e4 the chemistry of carbon4 ,although there have never been many chemists who really have believed the carbon atom to be a rigid tetrahedron1 we must admit that in this way it has been possible to get on the track of the secrets of atomic structure4 ,in comparing the properties of the elements with their atomic weights1 there has been discovered a peculiar relation which remained for a long time without explanation1 but which later suggested a certain connection between the inner structure of an atom and its chemical properties4 ,we refer to the natural or periodic system of the elements which was enunciated in #ahfi by the ,russian chemist1 ,mendelejeff1 and about the same time and independently by the ,german1 ,lothar ,meyer4 ,this system will be understood most clearly by examining the table on p4 #bc1 where the elements with their respective atomic weights and chemical symbols are arranged in numbered columns so that the atomic weights increase upon reading the table from left to right or from top to bottom4 ,it will be seen that in each of the nine columns there are collected elements with related properties1 forming what may be called chemical families4 ,the table as here given is of a recent date and differs from the old table of ,mendelejeff1 both in the greater number of elements and in the particulars of the arrangement4 ,with each element there is associated a number which indicates its position in the series with respect to increasing atomic weight4 ,thus hydrogen has the number #a1 helium #b1 etc41 up to uranium1 the atom of which is the heaviest of any known element1 and to which the number #ib is given4 ,in each of the columns the elements fall naturally into two sub-groups1 and this division is indicated in the table by placing the chemical symbols to the right or left in the column4 ,on close examination it becomes evident that the regularity in the system is not entirely simple4 ,first of all some cases will be found where the atomic weight of one element is greater than that of the following element4 7,the cases of argon and potassium on the one hand and cobalt and nickel on the other are examples47 ,such an interchange is absolutely necessary if the elements which belong to the same chemical family are to be placed in the same column4 ,as a second instance of irregularity1 attention must be called to ,column ,,viii4 in the table4 ,while in the first score or so of elements it is always found that two successive elements have different properties and clearly belong to distinct chemical families1 in the so-called iron group 7iron1 cobalt and nickel7 we meet with a case where successive elements resemble each other in many respects 7for instance1 in their magnetic properties74 ,since there are two more such 8triads0 in the periodic system1 however1 we cannot properly call this an irregularity4 ,but in addition to these difficulties there is what we may even call a kind of inelegance presented by the so-called 8rare earths0 group4 ,in this group there follow after lanthanum thirteen elements whose properties are rather similar1 so that it is very difficult to separate them from each other in the mixtures in which they occur in the minerals of nature4 7,in the table these elements are enclosed in a frame47 ,on the other hand1 the apparent absence of an element in certain places in the table 7indicated by a dash7 cannot by any means be looked upon as irregular4 ,in ,mendelejeff's first system there were many vacant spaces4 ,with the help of his table ,mendelejeff was1 to some extent1 able to predict the properties of the missing elements4 ,an example of this is the case of the element between gallium and arsenic4 ,this is called germanium1 and was discovered to have precisely the properties which had been predicted for it--a discovery which was one of the greatest triumphs in favour of the reality of the periodic system4 ,on the whole1 the elements discovered since the time of ,mendelejeff have found their natural positions in the table4 ,this is seen1 for example1 in the case of the so-called 8inactive gases0 of the atmosphere1 helium1 argon1 neon1 xenon1 krypton and niton1 which have the common property of being able to form no chemical combinations whatever4 ,their valence is therefore zero1 and in the table they are placed by themselves in a separate column headed with zero4 ,to explain the mystery of the periodic system1 it was necessary to make clear not only the regularity of it1 but also the apparent irregularities which seemed to be arbitrary individual peculiarities of certain elements or groups4 ,in the periodic system1 chemistry laid down some rather searching tests for future theories of atomic structure4 ,,the ,,periodic ,,or ,,natural ,,system ,,of ,,the ,,elements ---------------------------------------- ---------------------------------------- ---------------------------------------- -------- #j ,i4 ,,ii4 ,,iii4 ,,iv4 ,v4 ,,vi4 ,,vii4 ,,viii4 #a ,hydrogen ,h #ajjh ---------------------------------------- ---------------------------------------- ---------------------------------------- -------- #b ,helium #c ,lithium #d ,beryllium #e ,boron #f ,carbon #g ,nitrogen #h ,oxygen #i ,fluorine ,he #djj ,li #fid ,be #ia ,b #aaj ,c #abj ,n #adj ,o #af ,f #aij ---------------------------------------- ---------------------------------------- ---------------------------------------- ------- #aj ,neon #aa ,sodium #ab ,magnesium #ac ,aluminium #ad ,silicon #e ,phosphorus #af ,sulphur #ag ,chlorine ,ne #bjb ,na #bcj ,mg #bdc ,al #bga ,si #bhc ,p #caj ,s #cba ,cl #cee ---------------------------------------- ---------------------------------------- ---------------------------------------- ------- #ah ,argon #ai ,potassium #bj ,calcium #ba ,scandium #bb ,titanium #bc ,vanadium #bd ,chromium #be ,manganese #bf ,iron #bg ,cobalt ,a #cii ,k #cia ,ca #dja ,sc #dda ,ti #dha ,v #eaj ,cr #ebj ,mn #edi ,fe #eeh ,co #eij #bh ,nickel ,ni #eh4g ---------------------------------------- ---------------------------------------- ---------------------------------------- ------- #bi ,copper #cj ,zinc #ca ,gallium #cb ,germanium #cc ,arsenic #cd ,selenium #ce ,bromine ,cu #fcf ,zn #fed ,ga #fii ,ge #gbe ,as #gej ,se #gib ,br #gii ---------------------------------------- ---------------------------------------- ---------------------------------------- ------- #cf ,krypton#cg ,rubidium #ch ,strontium #ci ,yttrium #dj ,zirconium #da ,niobium #db ,molybdenum #dc -- #dd ,ruthenium #de ,rhodium ,kr #hbi ,rb #hed ,sr #hgf ,y #hhg ,zr #ijf ,nb #ice ,mo #ifj ,ru #ajag ,rh #ajbi #df ,palladium ,pd #ajfg ---------------------------------------- ---------------------------------------- ---------------------------------------- ------- #dg ,silver #dh ,cadmium #di ,indium #ej ,tin #ea ,antimony #eb ,tellurium #ec ,jodine ,ag #ajgi ,cd #aabd ,in #aadh ,sn #aahg ,sb #abjb ,te #abge ,j #abfi ---------------------------------------- -------- #ed ,xenon #ee ,caesium #ef ,barium #eg ,lanthanum #eh ,cerium #ei ,praseodymium #fj ,neodymium #fa -- #fb ,samarium ,x #acjb ,cs #acbh ,ba #acgc ,la #acij ,ce #adjb ,pr #adjf ,nd #addc ,sm #aejd #fc ,europium #fd ,gadolinium #fe ,terbium #ff ,dysprosium ,eu #aebj ,gd #aegc ,tb #aeib ,dy #afbe #fg ,holmium #fh ,erbium #fi ,thulium #gj ,ytterbium #ga ,cassiopeium ,ho #afce ,er #afgg ,tm #afhe ,yb #agcj ,cp #agej ---------------------------------------- -------- #gb ,hafnium #gc ,tantalum #gd ,tungsten #ge -- #gf ,osmium #gg ,iridium ,hf #agi ,ta #ahae ,w #ahdj ,os #aiji ,ir #aiba #gh ,platinum ,pt #aieb ---------------------------------------- ---------------------------------------- ---------------------------------------- ------- #gi ,gold #hj ,mercury #ha ,thallium #hb ,lead #hc ,bismuth #hd ,polonium #he -- ,au #aigb ,hg #bjjf ,tl #bjdj ,pb #bjgb ,bi #bjij ,po #bajj ---------------------------------------- ---------------------------------------- ---------------------------------------- ------- #hf ,niton #hg -- #hh ,radium #hi ,actinium #ij ,thorium#ia ,protactinium #ib ,uranium ,ni #bbbj ,ra #bbfj ,ac8 ,th #bcba ,pa8 ,u #bch ---------------------------------------- ---------------------------------------- ---------------------------------------- ------- ,the ,molecular ,theory of ,physics4 ,from a consideration of the chemical properties of the elements we shall now turn to an examination of the physical characteristics1 although in a certain sense chemistry itself is but one special phase of physics4 ,if matter is really constructed of independently existing particles--atoms and molecules--the interplay of the individual parts must determine not only the chemical activities1 but also the other properties of matter4 ,since most of these properties are different for different substances1 or in other words are 8molecular properties10 it is reasonable to suppose that in many cases explanations can be more readily given by considering the molecules as the fundamental parts4 ,it is natural that the first attempts to develop a molecular theory concerned gases1 for their physical properties are much simpler than those of liquids or solids4 ,this simplicity is indeed easily understood on the molecular theory4 ,when a liquid by evaporation is transformed into a gas1 the same weight of the element has a volume several hundred times greater than before4 ,the molecules1 packed together tightly in the liquid1 in the gas are separated from each other and can move freely without influencing each other appreciably4 ,when two of them come very close to each other1 mutually repulsive forces will arise to prevent collision4 ,since it must be assumed that in such a 8collision0 the individual molecules do not change1 they can then to a certain extent be considered as elastic bodies1 spheres for instance4 ,from considerations of this nature the kinetic theory of gases developed4 ,according to this a mass of gas consists of an immense number of very small molecules4 ,each molecule travels with great velocity in a straight line until it meets an obstruction1 such as another molecule or the wall of the containing vessel2 after such an encounter the molecule travels in a second direction until it collides again1 and so on4 ,the pressure of the gas on the wall of the container is the result of the very many collisions which each little piece of wall receives in a short interval of time4 ,the magnitude of the pressure depends upon the number1 mass and velocity of the molecules4 ,the velocity will be different for the individual molecules in a gas1 even if all the molecules are of the same kind1 but at a given temperature an average velocity can be determined and used4 ,if the temperature is increased1 this average molecular velocity will be increased1 and if at the same time the volume is kept constant1 the pressure of the gas on the walls will be increased4 ,if the temperature and the average velocity remain constant while the volume is halved1 there will be twice as many molecules per cubic centimetre as before4 ,therefore1 on each square centimetre of the containing wall there will be twice as many collisions1 and consequently the pressure will be doubled4 ,boyle's ,law1 that the pressure of a gas at a given temperature is inversely proportional to its volume1 is thus an immediate result of the molecular theory4 ,the molecular theory also throws new light upon the correspondence between heat and mechanical work and upon the law of the conservation of energy1 which about the middle of the nineteenth century was enunciated by the ,englishman1 ,joule1 the ,germans1 ,mayer and ,helmholtz1 and the ,dane1 ,colding4 ,a brief discussion of heat and energy will be given here1 since some conception of these phenomena is necessary in understanding what follows4 ,to lift a stone of #e pounds through a distance of #aj feet demands an expenditure of work amounting to #e #aj #ej foot-pounds2 but the stone is now enabled to perform an equally large amount of work in falling back these #aj feet4 ,the stone1 by its height above the earth and by the attraction of the earth1 now has in its elevated position what is called 8potential0 energy to the amount of #ej foot-pounds4 ,if the stone as it falls lifts another weight by some such device as a block and tackle1 the potential energy lost by the falling stone will be transferred to the lifted one4 ,if the apparatus is frictionless1 the falling stone can lift #e pounds #aj feet or #aj pounds #e feet1 etc41 so that all the #ej foot-pounds of potential energy will be stored in the second stone4 ,if instead of being used to lift the second stone1 the original stone is allowed to fall freely or to roll down an inclined plane without friction1 the velocity will increase as the stone falls1 and1 as the potential energy is lost1 another form of energy1 known as energy of motion or kinetic energy1 is gained4 ,conversely1 a body when it loses its velocity can do work1 such as stretching a spring or setting another body in motion4 ,let us suppose that the stone is fastened to a cord and is swinging like a pendulum in a vacuum where there is no resistance to its motion4 ,the pendulum will alternately sink and rise again to the same height4 ,as the pendulum sinks1 the potential energy will be changed into kinetic energy1 but as it rises again the kinetic will be exchanged for potential4 ,thus there is no loss of energy1 but merely a continuous exchange between the two forms4 ,if a moving body meets resistance1 or if its free fall is halted by a fixed body1 it might seem as if1 at last1 the energy were lost4 ,this1 however1 is not the case1 for another transformation occurs4 ,every one knows that heat is developed by friction1 and that heat can produce work1 as in a steam-engine4 ,careful investigations have shown that a given amount of mechanical work will always produce a certain definite amount of heat1 that is1 #djj foot-pounds of work1 if converted into heat1 will always produce #a ,b4,t4,u4 of heat1 which is the amount necessary to raise the temperature of #a pound of water #a ,f4 ,conversely1 when heat is converted into work1 #a ,b4,t4,u4 of heat 8vanishes0 every time #djj foot-pounds of work are produced4 ,heat then is just a special form of energy1 and the development of heat by friction or collision is merely a transformation of energy from one form to another4 ,with the assistance of the molecular theory it becomes possible to interpret as purely mechanical the transformation of mechanical work into heat energy4 ,let us suppose that a falling body strikes a piston at the top of a gas-filled cylinder1 closed at the bottom4 ,if the piston is driven down1 the gas will be compressed and therefore heated1 for the speed of the molecules will be increased by collisions with the piston in its downward motion4 ,in this example the kinetic energy given to the piston by the exterior falling body is used to increase the kinetic energy of the molecules of the gas4 ,when the molecules contain more than one atom1 attention must also be given to the rotations of the atoms in a molecule about each other4 ,a part of any added kinetic energy in the gas will be used to increase the energy of the atomic rotations4 ,the next step is to assume that1 in solids and liquids1 heat is purely a molecular motion4 ,here1 too1 the development of heat after collision with a moving body should be treated as a transformation of the kinetic energy of an individual1 visible body into an inner kinetic energy1 divided among the innumerable invisible molecules of the heated solid or liquid4 ,in considering the internal conduct of gases it is unnecessary 7at least in the main7 to consider any inner forces except the repulsions in the collisions of the molecules4 ,in solids and liquids1 however1 the attractions of the tightly packed molecules for each other must not be neglected4 ,indeed the situation is too complicated to be explained by any simple molecular theory4 ,not all energy transformations can be considered as purely mechanical4 ,for instance1 heat can be produced in a body by rays from the sun or from a hot fire1 and1 conversely1 a hot body can lose its heat by radiation4 ,here1 also1 we are concerned with transformations of energy2 therefore the law for the conservation of energy still holds1 i4e4 the total amount of energy can neither be increased nor decreased by transformations from one form to another4 ,for the production of #a ,b4,t4,u4 of heat a definite amount of radiation energy is required2 conversely1 the same amount of radiation energy is produced when #a ,b4,t4,u4 of heat is transformed into radiation4 ,this change cannot1 however1 be explained as the result of mechanical interplay between bodies in motion4 ,the mechanical theory of heat is very useful when we restrict ourselves to the transfer of heat from one body to another1 which is in contact with it4 ,when applied to gases the theory leads directly to ,avogadro's ,law4 ,if two masses of gas have the same temperature1 i4e41 if no exchange of heat between them takes place even if they are in contact with each other1 then the average value of the kinetic energy of the molecules must be the same in both gases4 ,if one gas is hydrogen and the other oxygen1 the lighter hydrogen molecules must have a greater velocity than the heavier oxygen molecules2 otherwise they cannot have the same kinetic energy 7the kinetic energy of a body is one-half the product of the mass and the square of the velocity74 ,since the pressure of a gas depends upon the kinetic energy of the molecules and upon their number per cubic centimetre1 at the same temperature and pressure equal volumes must contain equal numbers of oxygen and of hydrogen molecules4 ,as ,joule showed in #ahea1 from the mass of a gas per cubic centimetre and from its pressure per square centimetre1 the average velocity of the molecules can be calculated4 ,for hydrogen at #j ,c4 and atmospheric pressure the average velocity is about #eejj feet per second2 for oxygen under the same conditions it is something over #acjj feet per second4 ,all these results of the atomic and molecular theory1 however1 gave no information about the absolute weight of the individual atoms and molecules1 nor about their magnitude nor the number of molecules in a cubic centimetre at a given temperature and pressure4 ,as long as such questions were unsolved there was a suggestion of unreality in the theory4 ,the suspicion was easily aroused that the theory was merely a convenient scheme for picturing a series of observations1 and that atoms and molecules were merely creations of the imagination4 ,the theory would seem more plausible if its supporters could say how large and how heavy the atoms and molecules were4 ,the molecular theory of gases showed how to solve these problems which chemistry had been powerless to solve4 ,let us assume that the temperature of a mass of gas is #ajj ,c4 at a certain altitude1 and #j ,c4 one metre lower1 i4e41 the molecules have different average velocities in the two places4 ,the difference between the velocities will gradually decrease and disappear on account of molecular collisions4 ,we might expect this 8levelling out0 process or equilibration to proceed very rapidly because of the great velocity of the molecules1 but we must consider the fact that the molecules are not entirely free in their movements4 ,in reality they will travel but very short distances before meeting other molecules1 and consequently their directions of motion will change4 ,it is easy to understand that the difference between the velocities of the molecules of the gas will not disappear so quickly when the molecules move in zigzag lines with very short straight stretches4 ,the greater velocity in one part of the gas will then influence the velocity in the other part only through many intermediate steps4 ,gases are therefore poor conductors of heat4 ,when the molecular velocity of a gas and its conductivity of heat are known1 the average length of the small straight pieces of the zigzag lines can be calculated--in other words1 the length of the mean free path4 ,this length is very short2 for oxygen at standard temperature and pressure it is about one ten-thousandth of a millimetre1 or #ja 1 where is #jjja millimetre or one micron4 ,in addition to the velocity of the molecules1 the length of the mean free path depends upon the average distance between the centres of two neighbouring molecules 7in other words1 upon the number of molecules per cubic centimetre7 and upon their size4 ,there is difficulty in defining the size of molecules because1 as a rule1 each contains at least two atoms2 but it is helpful to consider the molecules1 temporarily1 as elastic spheres4 ,even with this assumption we cannot yet determine their dimensions from the mean free path1 since there are two unknowns1 the dimensions of the molecules and their number per cubic centimetre4 ,upon these two quantities depends1 however1 also the volume which will contain this number of molecules1 if they are packed closely together4 ,if we assume that we meet such a packing when the substance is condensed in liquid form1 this volume can be calculated from a knowledge of the ratio between the volume in liquid form and the volume of the same mass in gaseous form 7at #j ,c4 and atmospheric pressure74 ,then from this result and the length of the mean free path the two unknowns can be determined4 ,although the assumptions are imperfect1 they serve to give an idea about the dimensions of the molecules2 the results found in this way are of the same order of magnitude as those derived later by more perfect methods of an electrical nature4 ,the radius of a molecule1 considered as a sphere1 is of the order of magnitude #ja 1 where means #aj millimetre or #jjja micron4 ,even if a molecule is by no means a rigid sphere1 the value given shows that the molecule is almost unbelievably small1 or1 in other words1 that it can produce appreciable attraction and repulsion in only a very small region in space4 ,the number of molecules in a cubic centimetre of gas at #j ,c4 and atmospheric pressure has been calculated with fair accuracy as approximately #bg #aj4 ,from this number and from the weight of a cubic centimetre of a given gas the weight of one molecule can be found4 ,one hydrogen molecule weighs about #afe #aj grams1 and one gram of hydrogen contains about #f #aj atoms and #c #aj molecules4 ,the weight of the atoms of the other elements can be found by multiplying the weight of the hydrogen atom by the relative atomic weight of the element in question--#af for oxygen1 #ad for nitrogen1 etc4 ,if the pressure on the gas is reduced as much as possible 7to about one ten-millionth of an atmosphere7 there will still be #c #aj molecules in a cubic centimetre1 and the average distance between molecules will be about one micron4 ,the mean free path between two collisions will be considerable1 about two metres1 for instance1 in the case of hydrogen4 ,the values found for the number1 weight and dimensions of molecules are either so very large or so extremely small that many people1 instead of having more faith in the atomic and molecular theory1 perhaps may be more than ever inclined to suppose the atoms and molecules to be mere creations of the imagination4 ,in fact1 it is only two or three decades ago that some physicists and chemists--led by the celebrated ,german scientist1 ,wilhelm ,ostwald--denied the existence of atoms and molecules1 and even went so far as to try to remove the atomic theory from science4 ,when these sceptics1 in defence of their views1 said that the atoms and molecules were1 and for ever would be1 completely inaccessible to observation1 it had to be admitted at that time that they were seemingly sure of their argument1 in this one objection at any rate4 ,a series of remarkable discoveries at the close of the nineteenth century so increased our knowledge of the atoms and improved the methods of studying them that all doubts about their existence had to be silenced4 ,however incredible it may sound1 we are now in a position to examine many of the activities of a single atom1 and even to count atoms1 one by one1 and to photograph the path of an individual atom4 ,all these discoveries depend upon the behaviour of atoms as electrically charged1 moving under the influence of electrical forces4 ,this subject will be developed in another section after a discussion of some phenomena of light1 an understanding of which is necessary for the appreciation of the theory of atomic structure proposed by ,niels ,bohr4 ,in the molecular theory of gases1 where we have to do with neutral molecules1 much progress has in the last years been made by the ,dane1 ,martin ,knudsen1 in his experiments at a very low pressure1 when the molecules can travel relatively far without colliding with other molecules4 ,while his researches give information on many interesting and important details1 his work gives at the same time evidence of a very direct nature concerning the existence of atoms and molecules4 ,,chapter ,,ii ,,light ,,waves ,,and ,,the ,,spectrum ,the ,wave ,theory of ,light4 ,there have been several theories about the nature of light4 ,the great ,english physicist1 ,isaac ,newton 7#afdb-agbg71 who was a pioneer in the study of light as well as in that of mechanics1 favoured an atomic explanation of light2 i4e41 he thought that it consisted of particles or light corpuscules which were emitted from luminous bodies like projectiles from a cannon4 ,in contrast to this 8emission0 theory was the wave theory of ,newton's contemporary1 the ,dutch scientist1 ,huygens4 ,according to him1 light was a wave motion passing from luminous bodies into a substance called the ether1 which filled the otherwise empty universe and permeated all bodies1 at least all transparent ones4 ,in the nineteenth century the wave theory1 particularly through the work of the ,englishman1 ,young1 and the ,frenchman1 ,fresnel1 came to prevail over the emission theory4 ,since the wave theory plays an important part in the following chapters1 a discussion of the general characteristics of all wave motions is appropriate here4 ,the examples will include water waves on the surface of a body of water1 and sound waves in air4 ,illustration3 ,,fig4 #d4--,photograph of the interference between two similar wave systems4 ,illustration3 ,,fig4 #e4--;,a section of the same picture enlarged4 7,from ,grimsehl1 ,lehrbuch der ,physik47 ,let us suppose that we are in a boat which is anchored on a body of water and let us watch the regular waves which pass us4 ,if there is neither wind nor current1 a light body like a cork1 lying on the surface1 rises with the wave crests and sinks with the troughs1 going forward slightly with the former and backward with the latter1 but remaining1 on the whole1 in the same spot4 ,since the cork follows the surrounding water particles1 it shows their movements1 and we thus see that the individual particles are in oscillation1 or more accurately1 in circulation1 one circulation being completed during the time in which the wave motion advances a wave-length1 i4e41 the distance from one crest to the next4 ,this interval of time is called the time of oscillation1 or the period4 ,if the number of crests passed in a given time is counted1 the oscillations of the individual particles in the same time can be determined4 ,the number of oscillations in the unit of time1 which we here may take to be one minute1 is called the frequency4 ,if the frequency is forty and the wave-length is three metres1 the wave progresses #c #dj #abj metres in one minute4 ,the velocity with which the wave motion advances1 or in other words its velocity of propagation1 is then #abj metres per minute4 ,we thus have the rule that velocity of propagation is equal to the product of frequency and wave-length 7cf4 ,fig4 #h74 ,on the surface of a body of water there may exist at the same time several wave systems2 large waves created by winds which have themselves perhaps died down1 small ripples produced by breezes and running over the larger waves1 and waves from ships1 etc4 ,the form of the surface and the changes of form may thus be very complicated2 but the problem is simplified by combining the motions of the individual wave systems at any given point4 ,if one system at a given time gives a crest and another at the same instant also gives a crest at the same point1 the two together produce a higher crest4 ,similarly1 the resultant of two simultaneous troughs is a deeper trough2 a crest from one system and a simultaneous trough from the other partially destroy or neutralize each other4 ,a very interesting yet simple case of such 8interference0 of two wave systems is obtained when the systems have equal wave-lengths and equal amplitudes4 ,such an interference can be produced by throwing two stones1 as much alike as possible1 into the water at the same time1 at a short distance from each other4 ,when the two sets of wave rings meet there is created a network of crests and troughs4 ,figs4 #d and #e show photographs of such an interference1 produced by setting in oscillation two spheres which were suspended over a body of water4 ,illustration3 ,,fig4 #f4--,schematic representation of an interference4 ,in ,fig4 #f there is a schematic representation of an interference of the same nature4 ,let us examine the situation at points along the lower boundary line4 ,at #j1 which is equidistant from the two wave centres1 there is evidently a wave crest in each system2 therefore there is a resultant crest of double the amplitude of a single crest if the two systems have the same amplitude4 ,half a period later there is a trough in each system with a resultant trough of twice the amplitude of a single trough4 ,thus higher crests and deeper troughs alternate4 ,the same situation is found at point #b1 a wave-length farther from the left than from the right wave centre2 in fact1 these results are found at all points such as #b1 #b1 #d and #d1 where the difference in distance from the two wave centres is an even number of wave-lengths4 ,at the point #a1 on the other hand1 where the difference between the distance from the centres is one-half a wave-length1 a crest from one system meets a trough from the other1 and the resultant is neither crest nor trough but zero4 ,there is the same result at points #a1 #c1 #c1 #e1 #e1 etc41 where the difference between the distances from the two wave centres is an odd number of half wave-lengths4 ,by throwing a stone into the water in front of a smooth wall an interference is obtained1 similar to the one described above4 ,the waves are reflected from the wall as if they came from a centre at a point behind the wall and symmetrically placed with respect to the point where the stone actually falls4 ,illustration3 ,,fig4 #g4--,waves which are reflected by a board and pass through a hole in it4 ,when a wave system meets a wall in which there is a small hole1 this opening acts as a new wave centre1 from which1 on the other side of the wall1 there spread half-rings of crests and troughs4 ,but if the waves are small and the opening is large in proportion to the wave-length1 the case is essentially different4 ,let us suppose that wave rings originate at every point of the opening4 ,as a result of the co-operation of all these wave systems the crests and troughs will advance1 just as before1 in the original direction of propagation1 i4e41 along straight lines drawn from the original wave centre through the opening2 lines of radiation1 we may call them4 ,it can be shown1 however1 that as these lines of radiation deviate more and more from the normal to the wall1 the interference between wave systems weakens the resultant wave motion4 ,if the deviation from the normal to the wall is increased1 the weakening varies in magnitude1 provided that the waves are sufficiently small2 but even if the wave motions at times may thus 8flare up0 somewhat1 still on the whole they will decrease as the deviation from the normal to the wall is increased4 ,the smaller the waves in comparison to the opening1 the more marked is the decrease of the wave motions as the distance from the normal to the wall is increased1 and the more nearly the waves will move on in straight lines4 ,that light moves in straight lines1 so that opaque objects cast sharp shadows1 is therefore consistent with the wave theory1 provided the light waves are very small2 though it is reasonable to expect that on the passage of light through narrow openings there will be produced an appreciable bending in the direction of the rays4 ,this supposition agrees entirely with experiment4 ,as early as the middle of the seventeenth century1 the ,italian ,grimaldi discovered such a diffraction of light which passes through a narrow opening into a dark room4 ,illustration3 ,,fig4 #h4--,schematic representation of a wave4 ,a and ,b denote crests2 ,c denotes a trough4 wave-length4 amplitude of wave4 ,if ,t denotes the time the wave takes to travel from ,a to ,b1 and #a_/,t the frequency1 the wave velocity v will be equal to _/,t 4 ,points ,p and ,p are points in the same phase4 ,in both light and sound the use of such terms as wave and wave motion is figurative1 for crests and troughs are lacking4 ,but this choice of terms is commendable1 because sound and light possess an essential property similar to one possessed by water waves4 ,what happens when a tuning-fork emits sound waves into the surrounding air1 is that the air particles are set in oscillation in the direction of the propagation of sound4 ,all the particles of air have the same period as the tuning-fork1 and the number of oscillations per second determines the pitch of the note produced2 but the air particles at different distances from the tuning-fork are not all simultaneously in the same phase or condition of oscillation4 ,if one particle1 at a certain distance from the source of sound and at a given time1 is moving most rapidly away from the source1 then at the same time there is another particle1 somewhat farther along the direction of propagation1 which is moving towards the source most rapidly4 ,this alternation of direction will exist all along the path of the sound4 ,where the particles are approaching each other1 the air is in a state of condensation1 and where the particles are drawing apart1 the air is in a state of rarefaction4 ,while the individual particles are oscillating in approximately the same place1 the condensations and rarefactions1 like troughs and crests in water1 advance with a velocity which is called the velocity of sound4 ,if we call the distance between two consecutive points in the same phase a wave-length1 and the number of oscillations in a period of time the frequency1 then1 as in the case of water waves1 the velocity of propagation will be equal to the product of frequency and wave-length4 ,light1 like sound1 is a periodic change of the conditions in the different points of space4 ,these changes which emanate from the source of light1 in the course of one period advance one wave-length1 i4e41 the distance between two successive points in the same phase and lying in the direction of propagation4 ,as in the cases of sound and water waves1 the velocity of propagation or the velocity of light is equal to the product of frequency and wave-length4 ,if this velocity is indicated by the letter c1 the frequency by and the wave-length by 1 then c c c or ---- or ---- 4 ,the velocity of light in free space is a constant1 the same for all wave-lengths4 ,it was first determined by the ,danish astronomer ,ole ,rmer 7#afgf7 by observations of the moons of ,jupiter4 ,according to the measurements of the present day the velocity of light is about #a1jjj1jjj1jjj feet or #cjj1jjj kilometres per second4 ,in centimetres it is thus about #c #aj4 ,efforts have been made to consider light waves1 like sound waves1 as produced by the oscillations of particles1 not of the air1 but of a particular substance1 the 8ether10 filling and permeating everything2 but all attempts to form definite representations of the material properties of the ether and of the movements of its particles have been unsuccessful4 ,the electromagnetic theory of light1 enunciated about fifty years ago by the ,scottish physicist1 ,maxwell1 has furnished information of an essentially different character concerning the nature of light waves4 ,let us suppose that electricity is oscillating in a conductor connecting two metal spheres1 for instance4 ,the spheres1 therefore1 have1 alternately1 positive and negative charges4 ,then according to ,maxwell's theory we shall expect that in the surrounding space there will spread a kind of electromagnetic wave with a velocity equal to that of light4 ,wherever these waves are1 there should arise electric and magnetic forces at right angles to each other and to the direction of propagation of the waves2 the forces should change direction in rhythm with the movements of electricity in the emitting conductor4 ,by way of illustration let us assume that we have somewhere in space an immensely small and light body or particle with an electric charge4 ,if1 in the region in question1 an electromagnetic wave motion takes place1 then the charged particle will oscillate as a result of the periodically changing electrical forces4 ,the particle here plays the same rle as the cork on the surface of the water 7cf4 p4 #ce72 the charged body thus makes the electrical oscillations in space apparent just as the cork shows the oscillations of the water4 ,in addition to the electrical forces there are also magnetic forces in an electromagnetic wave4 ,we can imagine that they are made apparent by using a very small steel magnet instead of the charged body4 ,according to ,maxwell's theory1 the magnet exposed to the electromagnetic wave will perform rapid oscillations4 ,maxwell came to the conclusion that light consisted of electromagnetic waves of a similar nature1 but much more delicate than could possibly be produced and made visible directly by electrical means4 ,in the latter part of the nineteenth century the ,german physicist1 ,h4 ,hertz1 succeeded in producing electromagnetic waves with oscillations of the order of magnitude of #ajj1jjj1jjj per second1 corresponding to wave-lengths of the order of magnitude of several metres4 c #c #aj 7 ---- -------- #cjj cm474 #aj ,moreover1 he proved the existence of the oscillating electric forces by producing electric sparks in a circle of wire held in the path of the waves4 ,he showed also that these electromagnetic waves were reflected and interfered with each other according to the same laws as in the case of light waves4 ,after these discoveries there could be no reasonable doubt that light waves were actually electromagnetic waves1 but so small that it would be totally impossible to examine the oscillations directly with the assistance of electric instruments4 ,but there was in this work of ,hertz no solution of the problems about the nature of ether and the processes underlying the oscillations4 ,more and more1 scientists have been inclined to rest satisfied with the electromagnetic description of light waves and to give up speculation on the nature of the ether4 ,indeed1 within the last few years1 specially through the influence of ,einstein's theory of relativity1 many doubts have arisen as to the actual existence of the ether4 ,the disagreement about its existence is1 however1 more a matter of definition than of reality4 ,we can still talk about light as consisting of ether waves if we abandon the old conception of the ether as a rigid elastic body with definite material properties1 such as specific gravity1 hardness and elasticity4 ,the ,dispersion of ,light4 ,it has been said that the wave-length of light is much shorter than that of the ,hertzian waves4 ,this does not mean that all light waves have the same wave-length and frequency4 ,the light which comes to us from the sun is composed of waves of many different wave-lengths and frequencies1 to each of which corresponds a particular colour4 ,in this respect also light may be compared with sound4 ,in whatever way a sound is produced1 it is in general of a complicated nature1 composed of many distinct notes1 each with its characteristic wave-length and frequency4 ,naturally the air particles cannot oscillate in several different ways simultaneously4 ,at a given time1 however1 we can think of the condensation and rarefactions of the air or the oscillations of the particles corresponding to different tones1 as compounded with each other in a way similar to that in which the resultant crests and troughs are produced on a body of water with several coexistent wave systems4 ,when we say that the complicated wave-movement emitted from some sound-producing instrument consists of different tones1 this does not only mean that we may imagine it purely mathematically as resolved into a series of simpler wave systems4 ,the resolution may also take place in a more physical way4 ,let us assume that we have a collection of strings each of which will produce a note of particular pitch4 ,now1 if sound waves meet this collection of strings1 each string is set in oscillation by the one wave in the compound sound wave which corresponds to it4 ,each string is then said to act as a resonator for the note in question4 ,the notes which set the resonator strings in oscillation sound more loudly in the neighbourhood of the resonators2 but1 as the wave train continues on its journey the tones taken out by the strings will become weak in contrast to those notes which found no corresponding strings4 ,the resonator is said to absorb the notes with which it is in pitch4 ,light which is composed of different colours1 i4e41 of wave systems with different wave-lengths1 can also be resolved or dispersed1 but by a method different from that in the case of sound4 ,when light passes from one medium to another1 as from air to glass or vice versa1 it is refracted1 i4e41 the direction of the light rays is changed2 but if the light is composed of different colours the refraction is accompanied by a 8spreading0 of the colours which is called dispersion4 ,if we look through a glass prism so that the light from the object examined must pass in and out through two faces of the prism which make not too great an angle with each other1 the light-producing object is not only displaced by the refraction1 but has coloured edges4 ,newton was the first to explain the relation of the production of the colours to refraction4 ,he made an experiment with sunlight1 which he sent through a narrow opening into a dark room4 ,the sunlight was then by a glass prism transformed or dispersed into a band of colour1 a spectrum consisting of all the colours of the rainbow1 red1 yellow1 green1 blue and violet1 in the order named1 and with continuous transition stages between neighbouring colours4 ,illustration3 ,,fig4 #i4--,prism spectroscope4 ,to the right is seen the collimator1 to the left the telescope1 in the foreground a scheme for illuminating the cross-wire4 7,from an old print47 ,in ,newton's original experiment the different wave-lengths were but imperfectly separated4 ,a spectrum with pure wave-lengths can be obtained with a spectroscope 7cf4 ,fig4 #i74 ,the light to be investigated illuminates an adjustable vertical slit in one end of a long tube1 called the collimator1 with a lens in the other end4 ,if the slit is in the focal plane of the lens1 the light at any point in the slit goes in parallel rays after meeting the lens4 ,it then meets a prism1 with vertical edges1 placed on a little revolving platform4 ,the rays1 refracted by the prism1 go in a new direction into a telescope whose objective lens gives in its focal plane1 for every colour1 a clear vertical image of the slit4 ,these images can be examined through the ocular of the telescope2 but since the different colours are not refracted equally1 each coloured image of the slit has its own place4 ,the totality of the slit images then forms a horizontal spectrum of the same height as the individual images4 ,by revolving the collimator different parts of the spectrum can be put in the middle of the field of view4 ,to facilitate measurements in the spectrum there is in the focal plane of the collimator a sliding cross-wire with an adjusting screw or a vertical strand of spider web4 ,illustration3 ,,fig4 #aj4--,the mode of operation of a grating4 ,a1 grating2 ,c1 ,d1 ,e 444 ,h1 slits2 ,m ,m1 incident rays4 ,when ,d ,d1 ,e ,e 444 are a whole number of wave-lengths1 the light waves which move in the direction indicated by ,c ,n and are collected by a lens1 at the focal point will all be in the same phase and therefore will reinforce each other4 ,in other directions the light action from one slit is compensated by that from another4 ,instead of using the refraction of light in a prism to separate the wave-lengths1 we can use the interference which arises when a bundle of parallel light waves passes through a ruled grating1 consisting of a great many very fine parallel lines1 equidistant from each other2 such a grating can be made by ruling lines with a diamond point on the metal coating of a silvered plate of glass4 ,from each line there are sent out light waves in all directions2 but if we are considering light of one definite colour 7a given wave-length1 monochromatic light71 the interference among the waves from all the slits practically destroys all waves except in the direction of the original rays and in the directions making certain angles with the former1 dependent upon the wave-length and the distance between two successive lines 7the grating space74 ,monochromatic light can be obtained by using as the source of light a spirit flame1 coloured yellow with common salt 7sodium chloride74 ,if the slit in a spectroscope is lighted with a yellow light from such a flame1 and if a grating normal to the direction of the rays is substituted for the prism1 then in the telescope there is seen a yellow image of the slit1 and on each side of it one1 two1 three or more yellow images4 ,if sunlight is used the central image is white1 since all the colours are here assembled4 ,the other images become spectra because the different colours are unequally refracted4 ,in these grating spectra1 which according to their distance from the central line are called spectra of the first1 second or third order1 the violet part lies nearest to the central line1 the red part farthest away4 ,since the deflection is the greater the greater the wave-length1 then violet light must have the shortest wave-length and red the greatest4 ,from the amount of the refraction and the size of the grating space the wave-length of the light under investigation can be calculated4 ,for the yellow light from our spirit flame the wave-length is about #jjjjehi mm4 or #jehi or #ehi 4 ,in centimetres the wave-length is #jjjjjehi cm42 from the formula c_/1 #ebf #aj4 ,the frequency is thus almost inconceivably large4 ,for the most distant red and violet in the spectrum the wave-lengths are respectively about #hjj and #djj 1 and the frequencies #cge #aj and #gej #aj oscillations per second4 ,in scientific experiments a grating of specular metal with parallel rulings is substituted for the transparent grating4 ,the spectrum is then given by the reflected light from the parts between the rulings4 ,specular gratings can be made by ruling on a concave mirror1 which focuses the rays so that a glass lens is unnecessary4 ,gratings with several hundred lines or rulings to the millimetre give excellent spectra1 with strength of light and marked dispersion4 ,the preparation of the first really good gratings is due to the experimental skill of the ,american1 ,rowland1 who in #ahgj built a dividing engine from which the greater part of the good gratings now in use originate4 ,the contribution which ,rowland thereby made to physical science can hardly be over-estimated4 ,spectral ,lines4 ,in the early part of the nineteenth century ,wollaston1 in ,england1 and later ,fraunhofer in ,germany1 discovered dark lines in the solar spectrum1 a discovery which meant that certain colours were missing4 ,the most noticeable of these so-called 8,fraunhofer ,lines0 were named with the letters ,a1 ,b1 ,c1 ,d1 ,e1 ,f1 ,g1 ,h1 from red to violet4 ,it was later discovered that some of the lines were double1 that the ,d-line1 for instance1 can be resolved into ,d and ,d2 other letters1 such as b and h1 were introduced to denote new lines4 ,with improvements in the methods of experiment and research the number of lines has increased to hundreds and even thousands4 ,the light from a glowing solid or liquid element forms1 on the other hand1 a continuous spectrum1 i4e4 a spectrum which has no dark lines4 ,an illustration of the solar spectrum with the strongest ,fraunhofer lines is given at the end of the book4 ,in contrast to the solar spectrum with dark lines on a bright background are the so-called line spectra1 which consist of bright lines on a dark background4 ,the first known line spectrum was the one given by light from the spirit flame coloured with common salt1 mentioned in connection with monochromatic light4 ,as has been said1 this spectrum had just one yellow line which was later found to consist of two lines close to each other4 ,it is sodium chloride which colours the flame yellow4 ,the colour is due1 not to the chlorine1 but to the sodium1 for the same double yellow line can be produced by using other sodium salts not compounded with chlorine4 ,the yellow light is therefore called sodium light4 ,no4 #g in the table of spectra at the end of the book shows the spectrum produced by sodium vapour in a flame4 7,on account of the small scale in the figure it is not shown that the yellow line is double47 ,another interesting discovery was soon made1 namely1 that the sodium line has exactly the same wave-length as the light lacking in the solar spectrum1 where the double ,d-line is located4 ,about #ahfj ,kirchhoff and ,bunsen explained this remarkable coincidence as well as others of the same nature4 ,they showed by direct experiment that if sodium vapour is at a high temperature it can not only send out the yellow light1 but also absorb light of the same wave-length when rays from a still warmer glowing body pass through the vapour4 ,this phenomenon is something like that in the case of sound waves where a resonator absorbs the pitch which it can emit itself4 ,the existence of the dark ,d-line in the solar spectrum must then mean that in the outer layer of the sun there is sodium vapour present of lower temperature than the white-hot interior of the sun1 and that the light corresponding to the ,d-line is absorbed by the vapour4 ,several ingenious experiments1 which cannot be described here1 have given further evidence in favour of this explanation4 ,in the other line spectra1 just as in that from the common salt flame1 definite lines correspond to definite elements and not to chemical compounds4 ,the emission of these lines is then not a molecular characteristic1 but an atomic one4 ,the line spectra of metals can often be produced by vaporizing a metallic salt in a spirit flame or in a hot1 colourless gas flame 7from a ,bunsen burner74 ,it is even better to use an electric arc or strong electric sparks4 ,the atoms from which gaseous molecules are formed can also be made to emit light which by means of the spectroscope is shown to consist of a line spectrum4 ,these results are obtained by means of electric discharges of various kinds1 arcs1 and spark discharges through tubes where the gas is in a rarefied state4 ,the other ,fraunhofer lines in the solar spectrum correspond to bright lines in the line spectra of certain elements which exist here on earth4 ,these ,fraunhofer lines must then be assumed to be caused by the absorption of light by the elements in question4 ,this may be explained by the presence of these elements as gases in the solar atmosphere1 through which passes the light from the inner layer4 ,this inner surface would in itself emit a continuous spectrum4 ,the work of ,kirchhoff and ,bunsen put at the disposal of science became a new tool of incalculable scope4 ,first and foremost1 spectrum examinations were taken into the service of chemistry as spectrum analysis4 ,it has thus become possible to analyse quantities of matter so small that the general methods of chemistry would be quite powerless to detect them4 ,it is also possible by spectrum analysis to detect minute traces of an element2 several elements were in this way first discovered by the spectroscope4 ,moreover1 chemical analysis has been extended to the study of the sun and stars4 ,the spectral lines have given us answers to many problems of physics--problems which formerly seemed insoluble4 ,last but not least spectrum analysis has given us a key to the deepest secrets of the atom1 a key which ,niels ,bohr has taught us how to use4 ,in the discussion of the spectrum we have hitherto restricted ourselves to the visible spectrum limited on the one side by red and on the other by violet4 ,but these boundaries are in reality fortuitous1 determined by the human eye4 ,the spectrum can be studied by other methods than those of direct observation4 ,the more indirect methods include the effect of the rays on photographic plates and their heating effect on fine conducting wires for electricity1 held in various parts of the spectrum4 ,it has thus been discovered that beyond the visible violet end of the spectrum there is an ultra-violet region with strong photographic activity and an infra-red region producing marked heat effects4 ,there are both dark and light spectral lines in these new parts of the spectrum4 ,the fact that glass is not transparent to ultra-violet or infra-red rays has been an obstacle in the experiments1 but the difficulty can be overcome by using other substances1 such as quartz or rock salt1 for the prisms and lenses1 or by substituting concave gratings4 ,by special means it has been possible to detect rays with wave-lengths as great as #cjj and as small as about #jjb 1 corresponding to frequencies between #aj1 and #ae #aj vibrations per second1 while the wave-lengths of the luminous rays lie between #jh and #jd 4 ,the term 8light wave0 is often used to refer to the ultra-violet and infra-red rays which can be shown in the spectra produced by prisms or gratings4 ,illustration3 ,,fig4 #aa4--,photographic effect of ,x-rays1 which are passed through the atom grating in a magnesia crystal4 ,the electrically produced electromagnetic waves1 as already mentioned1 have wave-lengths much greater than #cjj 4 ,in wireless telegraphy there are generally used wave-lengths of one kilometre or more1 corresponding to frequencies of #cjj1jjj vibrations per second or less4 ,by direct electrical methods it has1 however1 not been possible to obtain wave-lengths less than about one-half a centimetre1 a length differing considerably from the #jc millimetre wave of the longest infra-red rays4 ,wave-lengths much less than #jjb or #bj exist in the so-called ,rntgen rays or ,x-rays with wave-lengths as small as #jja corresponding to a frequency of #cj #aj4 ,these rays cannot possibly be studied even with the finest artificially made gratings1 but crystals1 on account of the regular arrangement of the atoms1 give a kind of natural grating of extraordinary fineness4 ,with the use of crystal gratings success has been attained in decomposing the ,rntgen rays into a kind of spectrum1 in measuring the wave-lengths of the ,x-rays and in studying the interior structure of the crystals4 ,the ,german ,laue1 the discoverer of the peculiar action of crystals on ,x-rays 7#aiab71 let the ,x-rays beams pass through the crystal1 obtaining thereby photographs of the kind illustrated in ,fig4 #aa4 ,later on essential progress was due to the ,englishmen1 ,w4 ,h4 and ,w4 ,l4 ,bragg1 who worked out a method of investigation by which beams of ,x-rays are reflected from crystal faces4 ,the greatest wave-length which it has been possible to measure for ,x-rays is about #ae 1 which is still a long way from the #bj of the furthermost ultra-violet rays4 ,it may be said that the spectrum since ,fraunhofer has been made not only longer but also finer1 for the accuracy of measuring wave-lengths has been much increased4 ,it is now possible to determine the wave-length of a line in the spectrum to about #jjja or even less1 and to measure extraordinarily small changes in wave-lengths1 caused by different physical influences4 ,in addition to the continuous spectra emitted by glowing solids or liquids1 and to the line spectra emitted by gases1 and to the absorption spectra with dark lines1 there are spectra of still another kind4 ,these are the absorption spectra which are produced by the passage of white light through coloured glass or coloured fluids4 ,here instead of fine dark lines there are broader dark absorption bands1 the spectrum being limited to the individual bright parts4 ,there are also the band spectra proper1 which1 like the line spectra1 are purely emission spectra1 given by the light from gases under particular conditions2 these seem to consist of a series of bright bands which follow each other with a certain regularity 7cf4 ,fig4 #ab74 ,with stronger dispersion the bands are shown to consist of groups of bright lines4 ,illustration3 ,,fig4 #ab4--,spectra produced by discharges of different character through a glass tube containing nitrogen at a pressure of _/ that of the atmosphere4 ,above1 a band spectrum2 below1 a line spectrum4 ,since the line spectra are most important in the atomic theory1 we shall examine them here more carefully4 ,the line spectra of the various elements differ very much from each other with respect to their complexity4 ,while many metals give a great number of lines 7iron1 for instance1 gives more than five thousand71 others give only a few1 at least in a simple spectroscope4 ,with a more powerful spectroscope the simplicity of structure is lost1 since weaker lines appear and other lines which had seemed single are now seen to be double or triple4 ,moreover1 the number of lines is increased by extending the investigation to the ultra-violet and infra-red regions of the spectrum4 ,the sodium spectrum1 at first1 seemed to consist of one single yellow line1 but later this was shown to be a double line1 and still later several pairs of weaker double lines were discovered4 ,the kind and number of lines obtained depends not only upon the efficiency of the spectroscope1 but also upon the physical conditions under which the spectrum is obtained4 ,the eager attempts of the physicists to find laws governing the distribution of the lines have been successful in some spectra4 ,for instance1 the line spectra of lithium1 sodium1 potassium and other metals can be arranged into three rows1 each consisting of double lines4 ,the difference between the frequencies of the two 8components0 of the double lines was found to be exactly the same for most of the lines in one of these spectra1 and for the spectra of different elements there was discovered a simple relationship between this difference in frequency and the atomic weight of the element in question4 ,but this regularity was but a scrap1 so to speak2 scientists were still very far from a law which could exactly account for the distribution of lines in a single series1 not to mention the lines in an entire spectrum or in all the spectra4 ,the first important step in this direction was made about #ahhe by the ,swiss physicist1 ,balmer1 in his investigations with the hydrogen spectrum1 the simplest of all the spectra4 ,in the visible part there are just three lines1 one red1 one green-blue and one violet1 corresponding to the ,fraunhofer lines ,c1 ,f and h4 ,these hydrogen lines are now generally known by the letters ,h1 ,h and ,h4 ,in the ultra-violet region there are many lines also4 ,balmer discovered that wave-lengths of the red and of the green hydrogen line are to each other exactly as two integers1 namely1 as #bg to #bj1 and that the wave-lengths of the green and violet lines are to each other as #bh to #be4 ,continued reflection on this correspondence led him to enunciate a rule which can be expressed by a simple formula4 ,when frequency is substituted for wave-length ,balmer's formula is written as 7 #a #a 7 ,k7--- - -------71 7 #d n 7 where is the frequency of a hydrogen line1 ,k a constant equal to #cbi #aj and n an integer4 ,if n takes on different values1 becomes the frequency for the different hydrogen lines4 ,if n #a is negative1 for n #b is zero4 ,these values of n therefore have no meaning with regard to 4 ,but if n #c1 then gives the frequency for the red hydrogen line ,h2 n #d gives the frequency of the green line ,h and n #e that of the violet line ,h4 ,gradually more than thirty hydrogen lines have been found1 agreeing accurately with the formula for different values of n4 ,some of these lines were not found in experiment1 but were discovered in the spectrum of certain stars2 the exact agreement of these lines with ,balmer's formula was strong evidence for the belief that they are due to hydrogen4 ,the formula thus proved itself valuable in revealing the secrets of the heavens4 ,as n increases #a_/n approaches zero1 and can be made as close to zero as desired by letting n increase indefinitely4 ,in mathematical terminology1 as n 1 #a_/n #j and ,k_/#d #hbc #aj1 corresponding to a wave-length of #cfe 4 ,physically this means that the line spectrum of hydrogen in the ultra-violet is limited by a line corresponding to that frequency4 ,near this limit the hydrogen lines corresponding to ,balmer's formula are tightly packed together4 ,for n #bj differs but little from ,k_/#d1 and the distance between two successive lines corresponding to an increase of #a in n becomes more and more insignificant4 ,fig4 #ac1 where the numbers indicate the wave-lengths in the ngstrm unit 7#ja 71 shows the crowding of the hydrogen lines towards a definite boundary4 ,the following table1 where ,k has the accurate value of #cbijcfd #aj1 shows how exactly the values calculated from the formula agree with experiment4 ,illustration3 ,,fig4 #ac4--,lines in the hydrogen spectrum corresponding to the ,balmer series4 ,,table ,,of ,,some ,,of ,,the ,,lines ,,of ,,the ,,balmer ,,series ---------------------------------------- ----------------------- ,k7 - #a_/n7 7calculated74 7found74 7found74 ---------------------------------------- ---------------------- n #c,k7 - _/ 7 #def1iie bills#def1iif bills#fefdfj ,h n #d,k7 - _/ 7 #faf1idc 8 #faf1idc 8 #dhfbfh 8 ,h n #e,k7 - _/ 7 #fij1igf 8 #fij1igf 8 #dcdafh 8 ,h n #f,k7 - _/ 7 #gca1aib 8 #gca1aic 8 #dajbhh 8 ,h n #g,k7 - _/ 7 #gee1ddj 8 #gee1dda 8 #cigaai 8 ,h ---------------------------------------- ---------------------- n #bj,k7 - _/7 #had1cfe 8 #had1cfa 8 #cfhcjg 8 ---------------------------------------- ---------------------- ,from arguments in connection with the work of the ,swedish scientist1 ,rydberg1 in the spectra of other elements1 ,ritz1 a fellow countryman of ,balmer's1 has made it seem probable that the hydrogen spectrum contains other lines besides those corresponding to ,balmer's formula4 ,he assumed that the hydrogen spectrum1 like other spectra1 contains several series of lines and that ,balmer's formula corresponds to only one series4 ,ritz then enunciated a more comprehensive formula1 the ,balmer-,ritz formula3 7 #a #a 7 ,k7 ------- - -----71 7 n n7 where ,k has the same value as before1 and both n and n are integers which can pass through a series of different values4 ,for n #b1 the ,balmer series is given2 to n #a1 and n #b1 #c 444 there corresponds a second series which lies entirely in the ultra-violet region1 and to n #c1 n #d1 #e 444 a series lying entirely in the infra-red4 ,lines have actually been found belonging to these series4 ,formul1 similar to the ,ritz one1 have been set up for the line spectra of other elements1 and represent pretty accurately the distribution of the lines4 ,the frequencies are each represented by the difference between two terms1 each of which contains an integer1 which can pass through a series of values4 ,but while the hydrogen formula1 except for the ns1 depends only upon one constant quantity ,k and its terms have the simple form ,k_/n1 the formula is more complicated with the other elements4 ,the term can often be written1 with a high degree of exactness1 as ,k_/7n 71 where ,k is1 with considerable accuracy1 the same constant as in the hydrogen formula4 ,for a given element can assume several different values2 therefore the number of series is greater and the spectrum is even more complicated than that of hydrogen4 ,all these formul are1 however1 purely empirical1 derived from the values of wave-lengths and frequencies found in spectrum measurements4 ,they represent certain more or less simple bookkeeping rules1 by which we can register both old and new lines1 enter them in rows1 arrange them according to a definite system4 ,but from the beginning there could be no doubt that these rules had a deeper physical meaning which it was not yet possible to know4 ,there was no visible correspondence between the spectral line formul and the other physical characteristics of the elements which emitted the spectra2 not even in their form did the formul show any resemblance to formul obtained in other physical branches4 ,,chapter ,,iii ,,ions ,,and ,,electrons ,early ,theories and ,laws of ,electricity4 ,the fundamental phenomena of electricity1 which were first made the subject of careful study about two centuries ago1 are that certain substances can be electrified by friction so that somehow they can attract light bodies1 and that the charges of electricity may be either 8positive0 or 8negative40 ,bodies with like charges repel each other1 while those with unlike charges attract each other1 and either partially or entirely neutralize each other when they are brought close together4 ,moreover1 it had long ago been discovered that in some substances electricity can move freely from place to place1 while in others there is resistance to the movement4 ,the former bodies are now called conductors and include metals1 while the latter are called insulators1 glass1 porcelain and air being members of this class4 ,in order to explain the phenomena some imagined that there were two kinds of 8electric substances0 or 8fluids02 and since no change in weight could be discovered in a body when it was electrified1 it was1 in general1 assumed that the electric fluids were weightless4 ,in the normal1 neutral body it was believed that these fluids were mixed in equal quantities1 thereby neutralizing each other2 on this account they were supposed to be of opposite characteristics1 so one was called positive and the other negative4 ,according to a second theory1 there was assumed to be just one kind of electricity1 which was present in a normal amount in neutral bodies2 positive electricity was caused by a superfluity of the fluid2 negative1 by a deficit4 ,in both theories it was possible to talk of the amount of positive or negative electricity which a body contained or with which it was 8charged10 because the supporters of the one-fluid idea understood by the terms positive and negative a superfluity and a deficit1 respectively1 of the one fluid4 ,in both theories it was possible to talk about the direction of the electric current in a conductor1 since the supporters of the two-fluid theory understood by 8direction0 that in which the electric forces sent the positive electricity1 or the opposite to that in which the negative would be sent4 ,it could not be decided whether positive electricity went in the one direction or the negative in the other1 or whether each simultaneously moved in its own direction4 ,both theories were quite arbitrary in designating the electric charge in glass1 which was rubbed with woollen cloth1 as positive4 ,on the whole1 neither theory seemed to have any essential advantage over the other2 the difference between them seemed to lie more in phraseology than in actual fact4 ,that the positive and negative states of electricity could not be taken as 8symmetric0 seemed1 however1 to follow from the so-called discharge phenomena1 in which electricity1 with the emission of light1 streams out into the air from strongly charged 7positive or negative7 bodies1 or passes through the air between positive and negative bodies in sparks1 electric arcs or in some other way4 ,in a discharge in air between a metal point and a metal plate1 for instance1 a bush-shaped glow is seen to extend from the point when the charge there is positive1 while only a little star appears when the charge is negative4 ,naturally1 we cannot discuss here the many electric phenomena and laws1 and must be satisfied with a brief description of those which are of importance in the atomic theory4 ,in this latter category belongs ,coulomb's ,law1 formulated about #aghe4 ,according to this law1 the repulsions or attractions between two electrically charged bodies are directly as the product of the charges and inversely as the square of the distance between them 7as in the case of the gravitational attraction between two neutral bodies1 according to ,newton's ,law74 ,the unit in measuring electric charges can be taken as that amount which will repel an equal amount of electricity of the same kind at unit distance with unit force4 ,if we use the scientific or 8absolute0 system1 in which the unit of length is one centimetre1 that of time one second and that of mass one gram1 then the unit of force is one dyne1 which is a little greater than the earth's attraction on a milligram weight4 ,let us suppose that two small bodies with equal charges of positive 7or negative7 electricity are at a distance of one centimetre from each other4 ,if they repel each other with a charge of one dyne1 then the amount of electricity with which each is charged is called the absolute electrostatic unit of electricity4 ,if one body has a charge three times as great and the other has a charge four times as great1 the repulsion is #c #d #ab times greater4 ,if the distance between the bodies is increased from one to five1 the repulsion is twenty-five times as small1 since #e #be4 ,if the charge of one body is substituted by a negative one of same magnitude the repulsion becomes an attraction of the same magnitude4 ,in the early part of the nineteenth century methods were found for producing a steady electric current in metal wires4 ,in #ahbj1 the ,danish physicist1 ,h4 ,c4 rsted1 discovered that an electric current influences a magnet in a characteristic way1 and that1 conversely1 the current is affected by the forces emanating from the magnet1 by a magnetic field in other words4 ,the ,french scientist1 ,ampre1 soon afterwards formulated exact laws for the electromagnetic forces between magnets and currents4 ,in #ahca1 the ,english physicist1 ,faraday1 discovered that an electric current is induced in a wire when currents or magnets in its neighbourhood are moved or change strength4 ,faraday's views on electric and magnetic fields of force around currents and magnets were further of fundamental importance to the electromagnetic wave theory as developed by ,maxwell4 ,the branch of physics dealing with all these phenomena is now generally known as electrodynamics4 ,illustration3 ,,fig4 #ad4--,picture of electrolysis of hydrogen chloride4 ,a1 anode2 ,k1 cathode2 ,h1 hydrogen atoms2 ,cl1 chlorine atoms4 ,electrolysis4 ,faraday also studied the chemical effects which an electric current produces upon being conducted between two metal plates1 called electrodes1 which are immersed in a solution of salts or acids4 ,the current separates the salt or acid into two parts which are carried by the electric forces in two opposite directions4 ,this separation is called electrolysis4 ,if the liquid is dilute hydrochloric acid 7,h,cl71 the hydrogen goes with the current to the negative electrode1 the cathode1 and takes the positive electricity with it1 while the chlorine goes against the current and takes the negative electricity to the positive electrode1 the anode4 ,we must then assume with the ,swedish scientist1 ,arrhenius1 that1 under the influence of the water1 the molecules of hydrogen chloride always are separated into positive hydrogen atoms and negative chlorine atoms1 and that the electric forces from the anode and the cathode carry these atoms respectively with and against the current4 ,the electrically charged wandering atoms are called ions1 i4e4 wanderers4 ,the positive electricity taken by the hydrogen atoms to the cathode goes into the metal conductor1 while the anode must receive from the metal conductor an equal amount of positive electricity to be given to the chlorine atoms to neutralize them4 ,the negative charge of a chlorine atom must then be as large as the positive charge of a hydrogen atom4 ,these assumptions imply that equal numbers of the two kinds of atoms are present in the whole quantity of atoms transferred in any period of time4 ,faraday found that the quantity of hydrogen which in the above experiment is transferred to the cathode in a given time is proportional to the quantity of electricity transferred in the same time4 ,a gram of hydrogen always takes the same amount of electricity with it4 ,by experiment this amount of electricity can be determined1 and1 since the weight in grams of the hydrogen atom is known1 it is possible to calculate the amount of one atom4 ,in electrostatic units it is #dgg #aj1 i4e41 #dgg billionth#a parts4 ,a chlorine atom then carries with it #dgg #aj electrostatic units of negative electricity4 ,since its atomic weight is #cee1 then #cee grams of chlorine will take as much electricity as #a gram of hydrogen4 ,the ratio e_/m between the charge e and the mass m is then #cee times as great for hydrogen as for chlorine4 #a ,billion used here to mean one million million1 and trillion to mean one million billion4 ,we have temporarily restricted ourselves to the electrolysis of hydrogen chloride4 ,let us now assume that we have chloride of zinc 7,zn,cl71 which1 by electrolysis1 is separated into chlorine and zinc4 ,each atom of chlorine will1 as before1 carry #dgg #aj units of negative electricity to the anode2 but since zinc is divalent 7cf4 p4 #ag7 and one atom of zinc is joined to two of chlorine1 therefore one atom of zinc must carry a charge of #b #dgg #aj units of positive electricity to the cathode4 ,an atom or a group of atoms1 with valence of three1 in electrolysis carries #c #dgg #aj units1 etc4 ,we see then1 that the quantity of electricity which accompanies the atoms in electrolysis is always #dgg #aj electrostatic units or an integral multiple thereof4 ,this suggests the thought that electricity is atomic and that the quantity #dgg #aj units is the smallest amount of electricity which can exist independently1 i4e41 the elementary quantum of electricity or the 8atom of electricity40 ,the atom of a monovalent element1 when charged or ionized1 should have one atom of electricity2 a divalent1 two1 etc4 ,on the two-fluid theory it was most reasonable to assume that there were two kinds of atoms of electricity representing1 respectively1 positive and negative electricity4 ,in ,fig4 #ae there is given1 in accordance with the two-fluid theory1 a rough picture of a chlorine ion and a hydrogen ion and their union into a molecule4 ,illustration3 ,,fig4 #ae4--,provisional representation 7according to the two-fluid theory7 of ,a1 a hydrogen ion2 ,b1 a chlorine ion2 and ,c1 a molecule of hydrogen chloride4 ,the atoms of electricity seemed to differ essentially from the usual atoms of the elements in their apparent inability to live independently2 they seemed to exist only in connection with the atoms of the elements4 ,they would seem much more real if they could exist independently4 ,that such existence really is possible1 has been discovered by the study of the motion of electricity in gases4 ,vacuum ,tube ,phenomena4 ,illustration3 ,,fig4 #af4--,vacuum tube with cathode rays and a shadow-producing cross4 ,p and ,n1 conducting wires for the electric current2 a1 cathode2 b1 anode and shadow-producer2 c1 d1 the shadow4 ,illustration3 ,,fig4 #ag4--,vacuum tube1 where a bundle of cathode rays are deviated by electric forces4 ,a1 anode2 ,k1 cathode4 ,it has previously been said that air is an insulator for electricity1 a statement which is1 in general1 true2 however1 as has also been said1 electric sparks and arcs can pass through air4 ,moreover1 it has been discovered that exhausted air is a very good conductor1 so that a strong current can pass between two metal electrodes in a glass tube where the air is exhausted1 if the electrodes are connected to an outer conductor by metal wires fused into the glass4 ,in these vacuum tubes there are produced remarkable light effects1 at first inexplicable4 ,when the air is very much exhausted1 to a hundred thousandth of the atmospheric pressure or less1 strong electric forces 7large difference of potential between the electrodes7 are needed to produce an electric discharge4 ,such a discharge assumes an entirely new character2 in the interior of the glass tube there is hardly any light to be seen1 but the glass wall opposite the negative electrode 7the cathode7 glows with a greenish tint 7fluorescence74 ,if a small metal plate is put in the tube between the cathode and the glass wall1 a shadow is cast on the wall1 just as if light were produced by rays1 emitted from the cathode at right angles to its surface 7cf4 ,fig4 #af74 ,the ,english physicist1 ,crookes1 was one of the first to study these cathode rays4 ,he assumed that they are not ether waves like the light rays1 but that they consist of particles which are hurled from the cathode with great velocity in straight lines2 they light the wall by their collisions with it4 ,there was soon no doubt as to the correctness of ,crookes' theory4 ,the cathode rays are evidently particles of negative electricity1 which by repulsions are driven from the cathode 7the negative electrode74 ,a metal plate bombarded by the rays becomes charged negatively4 ,let us suppose that we have a small bundle of cathode rays1 obtained by passing the rays from the cathode ,k 7cf4 ,fig4 #ag7 through two narrow openings ,s and ,s4 ,it can then be shown that the bundle of rays is deviated not only by electric forces1 but also by magnetic action from a magnet which is held near the glass4 ,in the figure there is shown a deviation of the kind mentioned1 caused by making the plates at ,b and ,c respectively positive and negative2 since ,b attracts the negative particles and ,c repels them1 the light spot produced by the bundle of rays is moved from ,m to ,m4 ,the magnetic deviation is in agreement with rsted's rules for the reciprocal actions between currents and magnets1 if we consider the bundle of rays produced by moving electric particles as an electric current4 7,since the electric particles travelling in the direction of the rays are negative1 and since it is customary by the expression 8direction of current0 to understand the direction opposite to that in which the negative electricity moves1 then1 in the case of the cathode rays just mentioned1 the direction of the current must be opposite to that of the rays47 ,from measurements of the magnetic and electric deviations it is possible to find not only the velocity of the particles1 but also the ratio e_/m between the charge e of the particle and its mass m4 ,the velocity varies with the potential at the cathode1 and may be very great1 #ej1jjj km4 per second1 for instance 7about one-sixth the speed of light71 or more4 ,it has been found that e_/m always has the same value1 regardless of the metal of the cathode and of the gas in the tube2 this means that the particles are not atoms of the elements1 but something quite new4 ,it has also been found that e_/m is about two thousand times as large as the ratio between the charge and the mass of the hydrogen atom in electrolysis4 ,if we now assume that e is just the elementary quantum of electricity #dgg #aj1 which in magnitude amounts to the charge of the hydrogen atom in electrolysis 7but is negative71 then m must have about _/ the mass of the hydrogen atom4 ,this assumption as to the size of e has been justified by experiments of more direct nature4 ,the experiments with charge and mass of electrons which have in particular been carried out by the ,english physicist1 ,j4 ,j4 ,thomson1 give reason then to suppose these quite new and unknown particles to be free atoms of negative electricity2 they have been given the name of electrons4 ,gradually more information about them has been acquired4 ,thus it has been possible in various ways to determine directly the charge on the electron1 independently of its mass4 ,special mention must be made of the brilliant investigations of the ,american1 ,millikan1 on the motion of very small electrified oil-drops through air under the influence of an electric force4 ,to ,millikan is due the above-mentioned value of e1 which is accurate to one part in five hundred4 ,further1 the mass of the electron has been more exactly calculated as about _/ that of the hydrogen atom4 ,their magnitude has also been learned2 the radius of the electron is estimated as #ae #aj cm4 or #ae #aj 1 an order of magnitude one ten-thousandth that of the molecule or atom4 ,after the atom of negative electricity had been isolated1 in the form of cathode rays1 the next suggestion was that corresponding positive electric particles might be discharged from the anode in a vacuum tube4 ,by special methods success has been attained in showing and studying rays of positive particles4 ,in order to separate them from the negative cathode ray particles the ,german scientist1 ,goldstein1 let the positive particles pass through canals in the cathode2 they are therefore called canal rays4 ,the velocity of the particles is much less than that of the cathode rays1 and the ratio e_/m between charge and mass is much smaller and varies according to the gas in the tube4 ,in experiments where the tube contains hydrogen1 rays are always found for which e_/m1 as in electrolysis1 is about _/ of the ratio in the cathode rays4 ,therefore there can be scarcely any doubt that these canal rays are made up of charged hydrogen atoms or hydrogen ions4 ,the values found with other gases indicate that the particles are atoms 7or molecules sometimes7 of the elements in question1 with charges one or more times the elementary quantum of electricity 7#dgg #aj electrostatic units74 ,research in this field has also been due in particular to ,j4 ,j4 ,thomson4 ,from his results1 as well as from those obtained by other methods1 it follows that positive electricity1 unlike negative1 cannot appear of its own accord1 but is inextricably connected to the atoms of the elements4 ,the ,nature of ,electricity4 ,the earlier conceptions of a one or two-fluid explanation of the phenomena of electricity appear now in a new light4 ,we are led to think of a neutral atom as consisting of one mass charged with positive electricity together with as many electrons negatively charged as are sufficient to neutralize the positive4 ,if the atom loses one1 two or three electrons1 it becomes positive with a charge of one1 two or three elementary quanta of electricity1 or for the sake of simplicity and brevity we say that the atom has one1 two or three 8charges40 ,if1 on the other hand1 it takes up one1 two or three extra electrons it has one1 two or three negative charges4 ,fig4 #ah can give help in understanding these ideas1 but it must not be thought that the electrons are arranged in the way indicated4 ,the substances1 which appear as electropositive in electrolysis--i4e4 hydrogen and metals--should then be such that their atoms easily lose one or more electrons1 while the electronegative elements should1 on the other hand1 easily take up extra electrons4 ,elements should be monovalent or divalent according as their atoms are apt to lose or to take up one or two electrons4 ,from investigations with the vacuum tube it appears1 however1 that the atoms of the same element can in this respect behave in more ways than would be expected from electrolysis or chemical valence4 ,illustration3 ,,fig4 #ah4--,provisional representation 7according to the electron theory7 of ,a1 a neutral atom2 ,b1 the same atom with two positive charges 7a divalent positive ion7 and ,c1 the same atom with two negative charges 7a divalent negative ion74 ,when an electric current passes through a metal wire1 it must be assumed that the atoms of the metal remain in place1 while the electrical forces carry the electrons in a direction opposite to that which usually is considered as the direction of the current 7cf4 p4 #gj74 ,the motion of the electrons must not be supposed to proceed without hindrance1 but rather as the result of a complicated interplay1 by no means completely understood1 whereby the electrons are freed from and caught by the atoms and travel backwards and forwards1 in such a way that through every section of the metal wire a surplus of electrons is steadily passing in the direction opposite to the so-called direction of the current4 ,the number of surplus electrons which in every second passes through a section of the thin metal wire in an ordinary twenty-five candle incandescent light1 at #bbj volts1 amounts to about one trillion 7#aj71 or #ajjj million 7#aj7 in #jjjj1jjj1jja of a second4 ,if the metal conducting wire ends in the cathode of a vacuum tube1 the electrons carried through the wire pass freely into the tube as cathode rays from the cathode4 ,this motion of electricity agrees best with the one-fluid theory1 since the electrons1 which here alone accomplish the passage of the electricity1 may be considered as the fundamental parts of electricity4 ,in this respect the choice of the terms positive and negative is very unfortunate1 since a body with a negative charge actually has a surplus of electrons4 ,moreover1 the electrons really have mass2 but since the mass of a single electron is only _/ that of the atom of the lightest element1 hydrogen1 and since in an electrified body which can be weighed by scale there is always but an infinitesimal number of charged atoms1 it is easy to understand that1 formerly1 electricity seemed to be without weight4 ,in electrolysis1 where the motion of electricity is accomplished by positive and negative ions1 we have a closer connection with the two-fluid theory4 ,in motions of electricity through air the situation suggests both the one-fluid and the two-fluid theories1 since the passage of electricity is sometimes carried on exclusively by the electrons1 and sometimes partly by them and partly by larger positive and negative ions1 i4e41 atoms or molecules with positive and negative charges4 ,the ,electron ,theory4 ,proceeding on the assumption that the electric and optical properties of the elements are determined by the activity of the electric particles1 the ,dutch physicist ,lorentz and the ,english physicist ,larmor succeeded in formulating an extraordinarily comprehensive 8electron theory10 by which the electrodynamic laws for the variations in state of the ether were adapted to the doctrine of ions and electrons4 ,this ,lorentz theory must be recognized as one of the finest and most significant results of nineteenth century physical research4 ,it was one of the most suggestive problems of this theory to account for the emission of light waves from the atom4 ,from the previously described electromagnetic theory of light 7cf4 p4 #db7 it follows that an electron oscillating in an atom will emit light waves in the ether1 and that the frequency of these waves will naturally be equal to the number of oscillations of the electron in a second4 ,if this last quantity is designated as 1 then ,it may then be supposed that the electrons in the undisturbed atom are in a state of rest1 comparable with that of a ball in the bottom of a bowl4 ,when the atom in some way is 8shaken10 one or more of the electrons in the atom begins to oscillate with a definite frequency1 just as the ball might roll back and forth in the bowl if the bowl was shaken4 ,this means that the atom is emitting light waves1 which1 for each individual electron have a definite wave-length corresponding to the frequency of the oscillations1 and that1 in the spectrum of the emitted light1 the observed spectral lines correspond to these wave-lengths4 ,strong support for this view was afforded by ,zeeman's discovery of the influence of a magnetic field upon spectral lines4 ,zeeman1 a ,dutch physicist1 discovered1 about twenty-five years ago1 that when a glowing vacuum tube is placed between the poles of a strong electromagnet1 the spectral lines in the emitted light are split so that each line is divided into three components with very little distance between them4 ,it was one of the great triumphs of the electron theory that ,lorentz was able to show that such an effect was to be expected if it was assumed that the oscillations of light were produced by small oscillating electric particles within the atom4 ,from the experiments and from the known laws concerning the reciprocal actions of a magnet and an electric current 7here the moving particle71 the theory enabled ,lorentz to find not only the ratio e_/m between the electric charge of each of these particles and its mass1 but also the nature of the charge4 ,he could conclude from ,zeeman's experiment that the charge is negative and that the ratio e_/m is the same as that found for the cathode rays4 ,after this there could not well be doubt that the electrons in the atoms were the origin of the light which gives the lines of the spectrum4 ,it seemed1 however1 quite unfeasible for the theory to explain the details in a spectrum--to derive1 for instance1 ,balmer's formula1 or to show why hydrogen has these lines1 copper those1 etc4 ,these difficulties1 combined with the great number of lines in the different spectra1 seemed to mean that there were many electrons in an atom and that the structure of an atom was exceedingly complicated4 ,ionization by ,x-rays and ,rays from ,radium4 ,radioactivity4 ,as has been said1 the electrons in a vacuum tube cause its wall to emit a greenish light when they strike it4 ,upon meeting the glass wall or a piece of metal 7the anticathode7 placed in the tube the electrons cause also the emission of the peculiar1 penetrating rays called ,rntgen rays in honour of their discoverer1 or more commonly ,x-rays4 ,they may be described as ultra-violet rays with exceedingly small wave-lengths 7cf4 p4 #ed74 ,when1 further1 the electrons meet gas molecules in the tube they break them to pieces1 separating them into positive and negative ions 7ionization74 ,the positive ions are the ones which appear in the canal rays4 ,the ions set in motion by electrical forces can break other gas molecules to pieces1 thus assisting in the ionization process4 ,at the same time the gas molecules and atoms are made to produce disturbances in the ether1 and thus to cause the light phenomena which arise in a tube which is not too strongly exhausted4 ,the free air can be ionized in various ways2 this ionization can be detected because the air becomes more or less conducting4 ,in fact1 electric forces will drive the positive and negative ions through the air in opposite directions1 thus giving rise to an electric current4 ,if the ionization process is not steadily continued1 the air gradually loses its conductivity1 since the positive and negative ions recombine into neutral atoms or molecules4 ,ionization can be produced by flames1 since the air rising from a flame contains ions4 ,a strong ionization can also be brought about by ,x-rays and by ultra-violet rays4 ,in the higher strata of the atmosphere the ultra-violet rays of the sun exercise an ionizing influence4 ,most of all1 however1 the air is ionized by rays from the so-called radioactive substances which in very small quantities are distributed about the world4 ,the characteristic radiation from these substances was discovered in the last decade of the nineteenth century by the ,french physicist1 ,becquerel1 and afterwards studied by ,m4 and ,mme4 ,curie4 ,from the radioactive uranium mineral1 pitchblende1 the latter separated the many times more strongly radioactive element radium4 ,the proper nature of the rays was later explained1 particularly through the investigations of the ,english physicists1 ,rutherford1 ,soddy and ,ramsay4 ,these rays1 which can produce heat effects1 photographic effects and ionization1 are of three quite different classes1 and accordingly are known as -rays1 -rays1 and -rays4 ,the last named1 like the ,x-rays1 are ultra-violet rays1 but they have often even shorter wave-lengths and a much greater power of penetration than the usual ,x-rays4 ,the -rays are electrons which are ejected with much greater velocity than the cathode rays2 in some cases their velocity goes up to #iih per cent4 that of light4 ,the -rays are positive atomic ions1 which move with a velocity varying according to the emitting radioactive element from _/ to almost _/ that of light4 ,it has further been proved that the -particles are atoms of the element helium1 which has the atomic weight #d1 and that they possess two positive charges1 i4e41 they must take up two electrons to produce a neutral helium atom4 ,there is no doubt that the process which takes place in the emission of radiation from the radioactive elements is a transformation of the element1 an explosion of the atoms accompanied by the emission either of double-charged helium atoms or of electrons1 and the forming of the atoms of a new element4 ,the energy of the rays is an internal atomic energy1 freed by these transformations4 ,the element uranium1 with the greatest of all known atomic weights 7#bch71 passes1 by several intermediate steps1 into radium with atomic weight #bbf2 from radium there comes1 after a series of steps1 lead1 or1 in any case1 an element which1 in all its chemical properties1 behaves like lead4 ,we shall go no further into this subject1 merely remarking that the transformations are quite independent of the chemical combinations into which the radioactive elements have entered1 and of all external influences4 ,when -particles from radium are sent against a screen with a coating of especially prepared zinc sulphide1 on this screen1 in the dark1 there can be seen a characteristic light phenomenon1 the so-called scintillation1 which consists of many flashes of light4 ,each individual flash means that an -particle1 a helium atom1 has hit the screen4 ,in this bombardment by atoms the individual atom-projectiles are made visible in a manner similar to that in which the individual raindrops which fall on the surface of a body of water are made visible by the wave rings which spread from the places where the drops meet the water4 ,this flash of light was the first effect of the individual atom to be available for investigation and observation4 ,the incredibility of anything so small as an atom producing a visible effect is lessened when1 instead of paying attention merely to the small size or mass of the atom1 its kinetic energy is considered2 this energy is proportional to the square of the velocity1 which is here of overwhelming magnitude4 ,for the most rapid -particles the velocity is #bbf #aj cm4 per second2 their kinetic energy is then about _/ of the kinetic energy of a weight of one milligram of a substance at a velocity of one centimetre per second4 ,this energy may seem very small1 but1 at least1 it is not a magnitude of 8inconceivable minuteness10 and it is sufficient under the conditions given above to produce a visible light effect4 ,we must here also consider the extreme sensitiveness of the eye4 ,illustration3 ,,fig4 #ai4--,photograph of paths described by -particles 7positive helium ions7 emitted from a radioactive substance4 ,more practical methods of revealing the effects of the individual -particles and of counting them are founded on their very strong ionization power4 ,by strengthening the ionization power of -particles1 ,rutherford and ,geiger were able to make the air in a so-called ionization chamber so good a conductor that an individual -particle caused a deflection in an electrical apparatus1 an electrometre4 ,illustration3 ,,fig4 #bj4--,photograph of the path of a -particle 7an electron74 7,both #ai and #bj are photographs by ,c4 ,t4 ,r4 ,wilson47 ,with a more direct method the ,english scientist ,c4 ,t4 ,r4 ,wilson has shown the paths of the -particles by making use of the characteristic property of ions1 that in damp air they attract the neutral water molecules which then form drops of water with the ions as nuclei4 ,in air which is completely free of dust and ions the water vapour is not condensed1 even if the temperature is decreased so as to give rise to supersaturation1 but as soon as the air is ionized the vapour condenses into a fog4 ,when ,wilson sent -particles through air1 supersaturated with water vapour1 the vapour condensed into small drops on the ions produced by the particles2 the streaks of fog thus obtained could be photographed4 ,fig4 #ai shows such a photograph of the paths of a number of atoms4 ,when a streak of fog ends abruptly it does not mean that the -particles have suddenly halted1 but that their velocity has decreased so that they can no longer break the molecules of air to pieces1 producing ions4 ,the paths of the -particles have been photographed in the same way1 although an electron of the -particles has a mass about #gjjj times smaller than that of a helium atom2 the electron has1 however1 a far greater velocity than the helium atom4 ,this velocity causes the ions to be farther apart1 so that each drop of water formed around the individual ions can appear in the photograph by itself 7cf4 ,fig4 #bj74 ,,chapter ,,iv ,,the ,,nuclear ,,atom ,introduction4 ,we are now brought face to face with the fundamental question1 hardly touched upon at all in the previous part of this work1 namely1 that of the construction and mode of operation of the atomic mechanism itself4 ,in the first place we must ask3 ,what is the 8architecture0 of the atom1 that is1 what positions do the positive and negative particles take up with respect to each other1 and how many are there of each kind8 ,in the second place1 of what sort are the processes which take place in an atom1 and how can we make them interpret the physical and chemical properties of the elements8 ,in this chapter we shall keep essentially to the first question1 and consider especially the great contribution which ,rutherford made in #aiaa to its answer in his discovery of the positive atomic nucleus and in the development of what is known as the ,rutherford atomic model or nuclear atom4 ,rutherford's ,atom ,model4 ,rutherford's discovery was the result of an investigation which1 in its main outlines1 was carried out as follows3 a dense stream of -particles from a powerful radium preparation was sent into a highly exhausted chamber through a little opening4 ,on a zinc sulphide screen1 placed a little distance behind the opening1 there was then produced by this bombardment of atomic projectiles1 a small1 sharply defined spot of light4 ,the opening was next covered by a thin metal plate1 which can be considered as a piece of chain mail formed of densely-packed atoms4 ,the -particles1 working their way through the atoms1 easily traversed this 8piece of mail0 because of their great velocity4 ,but now it was seen that the spot of light broadened out a little and was no longer sharply limited4 ,from this fact one could conclude that the -particles in passing among the many atoms in the metal plate suffered countless1 very small deflections1 thus producing a slight spreading of the rays4 ,it could also be seen that some1 though comparatively few1 of the -particles broke utterly away from the stream1 and travelled farther in new directions1 some1 indeed1 glancing back from the metal plate in the direction in which they had come 7cf4 ,fig4 #ba74 ,the situation was approximately as if one had discharged a quantity of small shot through a wall of butter1 and nearly all the pellets had gone through the wall in an almost unchanged direction1 but that one or two individual ones had in some apparently uncalled for fashion come travelling back from the interior of the butter4 ,one might naturally conclude from this circumstance that here and there in the butter were located some small1 hard1 heavy objects1 for example1 some small pellets with which some of the projectiles by chance had collided4 ,accordingly1 it seemed as if there were located in the metal sheet some small hard objects4 ,these could hardly be the electrons of the metal atoms1 because -particles1 as has been stated before1 are helium atoms with a mass over seven thousand times that of a single electron2 and if such an atom collided with an electron1 it would easily push the electron aside without itself being deviated materially in its path4 ,hardly any other possibility remained than to assume that what the -particles had collided with was the positive part of the atom1 whose mass is of the same order of magnitude as the mass of the helium atom 7cf4 ,fig4 #ba74 ,a mathematical investigation showed that the large deflections were produced because the -particles in question had passed1 on their way1 through a tremendously strong electric field of the kind which will exist about an electric charge concentrated into a very small space and acting on other charges according to ,coulomb's ,law4 ,when1 in the foregoing1 the word 8collision0 is used1 it must not be taken to mean simply a collision of elastic spheres2 rather the two particles 7the -particle and the positive particle of the metal atom7 come so near to each other in the flight of the former that the very great electrical forces brought into play cause a significant deflection of the -particles from their original course4 ,illustration3 ,,fig4 #ba4--,tracks of -particles in the interior of matter4 ,while #a and #c undergo small deflections by collisions with electrons1 #b is sharply deflected by a positive nucleus4 ,rutherford was thus led to the hypothesis that nearly all of the mass of the atom is concentrated into a positively charged nucleus1 which1 like the electrons1 is very small in comparison with the size of the whole atom2 while the rest of the mass is apportioned among a number of negative electrons which must be assumed to rotate about the nucleus under the attraction of the latter1 just as the planets rotate about the sun4 ,under this hypothesis the outer limits of the atom must be regarded as given by the outermost electron orbits4 ,the assumption of an atom of this structure makes it at once intelligible why1 in general1 the -particles can travel through the atom without being deflected materially by the nuclear repulsion1 and why the very great deflections occur as seldom as is indicated by experiment4 ,this latter circumstance has1 on the other hand1 no explanation in the atomic model previously suggested by ,lord ,kelvin and amplified by ,j4 ,j4 ,thomson1 in which the positive electricity was assumed to be distributed over the whole volume of the atom1 while the electrons were supposed to move in rings at varying distances from the centre of the atom4 ,illustration3 ,,fig4 #bb4--,photograph of the paths of two -particles 7positive helium ions74 ,one collides with an atomic nucleus4 ,the same characteristic phenomenon made evident in the passage of -particles through substances by the investigations of ,rutherford appears in a more direct way in ,wilson's researches discussed on p4 #ha4 ,his photographs of the paths of -particles through air supersaturated with water vapour 7see ,fig4 #bb7 show pronounced kinks in the paths of individual particles4 ,thus in the figure referred to1 there are shown the paths of two -particles4 ,one of these is almost a straight line 7with a very slight curvature71 while the other shows a very perceptible deflection as it approaches the immediate neighbourhood of the nucleus of an atom1 and finally a very abrupt kink2 at the latter place it is clear that the -particle has penetrated very close to the nucleus4 ,if one examines the picture more closely1 there will be seen a very small fork at the place where the kink is located4 ,here the path seems to have divided into two branches1 a shorter and a longer4 ,this leads one at once to suspect that a collision between two bodies has taken place1 and that after the collision each body has travelled its own path1 just as if1 to return to the analogy of the bombardment of the butter wall1 one had been able to drive two pellets out of the butter by shooting in only one4 ,or1 to take perhaps a more familiar example1 when a moving billiard ball collides at random with a stationary one1 after the collision they both move off in different directions4 ,so1 when the -particle hits at random the atomic nucleus1 both particle and nucleus move off in different directions2 though in this case1 since the nucleus has the much greater mass of the two1 it moves more slowly1 after the collision1 than the -particle1 and has1 therefore1 a much shorter range in the air than the lighter1 swifter -particle4 ,had the gas in which the collisions took place been hydrogen1 for example1 the recoil paths of the hydrogen nuclei would have been longer than those of the -particles1 because the mass of the hydrogen nucleus is but one quarter the mass of the -particle 7helium atom74 ,the collision experiments on which ,rutherford's theory is founded are of so direct and decisive a character that one can hardly call it a theory1 but rather a fact1 founded on observation1 showing conclusively that the atom is built after the fashion indicated4 ,continued researches have amassed a quantity of important facts about atoms4 ,thus1 ,rutherford was able to show that the radius of the nucleus is of the order of magnitude #aj to #aj;cm4 ,this means really that it is only when an -particle approaches so near the centre of an atom that forces come into play which no longer follow ,coulomb's ,law for the repulsion between two point charges of the same sign 7in contrast to the case in the ordinary deflections of -particles74 ,it should be remarked1 however1 that in the case of the hydrogen nucleus theoretical considerations give foundation for the assumption that its radius is really many times smaller than the radius of the electron1 which is some #bjjj times lighter2 experiments by which this assumption can be tested are not at hand at present4 ,the ,nuclear ,charge2 ,atomic ,number2 ,atomic ,weight4 ,it is not necessary to have recourse to a new research to determine the masses of the nuclei of various atoms1 because the mass of the nucleus is for all practical purposes the mass of the atom4 ,accordingly1 if the mass of the hydrogen nucleus is taken as unity1 the atomic mass is equal to the atomic weight as previously defined4 ,the individual electrons which accompany the nucleus are so light that their mass has relatively little influence 7within the limits of experimental accuracy7 on the total mass of the atom4 ,on the other hand1 a problem of the greatest importance which immediately suggests itself is to determine the magnitude of the positive charge of the nucleus4 ,this naturally must be an integral multiple of the fundamental quantum of negative electricity1 namely1 #dgg #aj electrostatic units1 or if we prefer to call this simply the 8unit0 charge1 then the nuclear charge must be an integer4 ,otherwise a neutral atom could not be formed of a nucleus and electrons1 for in a neutral atom the number of negative electrons which move about the nucleus must be equal to the number of positive charges in the nucleus4 ,the determination of this number is1 accordingly1 equivalent to the settling of the important question1 how many electrons surround the nucleus in the normal neutral state of the atom of the element in question4 ,the answer to the question is easiest in the case of the helium atom4 ,for when this is expelled as an -particle1 it carries1 as ,rutherford was able to show1 a positive charge of two units--in other words1 two electrons are necessary to change the positive ion into a neutral atom4 ,at the same time there is every reason to suppose that the -particle is simply a helium nucleus deprived of its electrons2 it follows1 therefore1 that the electron system of the neutral helium atom consists of two electrons4 ,since the atomic weight of helium is four1 the number of electrons is consequently one-half the atomic weight4 ,rutherford's investigation of the deflections of -particles in passing through various media had already led him to believe that for many other elements1 to a considerable approximation1 the nuclear charge and hence the number of electrons was equal to half the atomic weight4 ,hydrogen1 of course1 must form an exception1 since its atomic weight is unity4 ,the positive charge on the hydrogen nucleus is one elementary quantum1 and in the neutral state of the atom1 only one electron rotates about it4 ,fig4 #bc gives a representation of the structure of the hydrogen atom1 and the structures of the two types of hydrogen ions formed respectively by the loss and gain of an electron4 ,in the picture1 the position of the electron is1 of course1 arbitrary1 and for the sake of simplicity its path is supposed to be circular4 ,illustration3 ,,fig4 #bc4--,schematic representation of the nuclear atom4 ,a1 a neutral hydrogen atom2 ,b1 a positive1 and ,c1 a negative hydrogen ion2 ,k1 atomic nuclei2 ,e1 electrons4 ,as has just been indicated1 ,rutherford's rule for the number of electrons is only an approximation4 ,a ,dutch physicist1 van den ,broek1 conceived in the meantime the idea that the number of electrons in the atom of an element is equal to its order number in the periodic table 7its 8atomic number10 as it is now called74 ,especially through a systematic investigation of the ,x-ray spectra characteristic of the different elements this has proved to be the correct rule4 ,in fact1 using ,bragg's reflection method of ,x-rays from crystal surfaces 7cf4 p4 #ed71 the ,englishman1 ,moseley1 made in #aiad the far-reaching discovery that these spectra possess an exceptionally simple structure1 which made it possible in a simple way to attach an order number to each element 7given on p4 #bc74 ,on the basis of ,bohr's theory1 established a year before1 it could be directly proved that this order number must be identical with the number of positive elementary charges on the nucleus4 ,the number which formerly indicated simply the position of an element in the periodic system has thus obtained a profound physical significance1 and in comparison the atomic weight has come to have but a secondary meaning4 ,the inversion of argon and potassium in the periodic system 7mentioned on p4 #ba71 which seemed to be an exception to the regularity displayed by the system as a whole1 obtains an easy explanation on the van den ,broek rule2 for to explain the inversion we need only assume that potassium has one electron more than argon1 though its atomic weight is less than that of argon4 ,we see at once that the atomic weight and number of electrons 7or what is the same thing--the nuclear charge7 are not directly correlated to each other4 ,and since the periodic system based on the atomic number represents the correct arrangement of the elements according to their respective properties 7especially their chemical properties71 we are led naturally to the conclusion that it is the atomic number and not the atomic weight that determines chemical characteristics4 ,the conception of the relatively great importance of the atomic number as compared with the atomic weight has in recent years received overwhelming support from the researches of ,soddy1 ,fajans1 ,russell1 ,hevesy and others who have discovered the existence of so-called isotope elements 7from the ,greek isos same1 and topos place71 substances with different nuclear masses 7atomic weights7 and different radioactive properties 7if there are any71 but with the same nuclear charge1 the same number of electrons and1 consequently1 occupying the same place in the periodic system4 ,two such isotopes are practically equivalent in all their chemical properties as well as in most of their physical characteristics4 ,one of the oldest examples of isotopes is provided by ordinary lead with the atomic weight #bjgb and the substance found in pitchblende with the atomic weight #bjf1 but identical1 chemically1 with ordinary lead4 ,this latter form of lead has already been referred to on p4 #gi as the end product of radioactive disintegrations1 and hence it is sometimes called radium lead4 ,by his investigations of canal rays the ,english physicist ,aston has just recently shown that many substances which have always been assumed to be simple elements1 are in reality mixtures of isotopes4 ,the atomic weight of chlorine determined in the usual way is #cee1 but in the discharge tube two kinds of chlorine atoms appear1 having atomic weights #ce and #cg respectively2 and it must be assumed that these two kinds of chlorine are present in all the compounds of chlorine known on the earth in the ratio of1 roughly1 three to one4 ,to separate such mixtures into their constituent parts is extremely difficult1 precisely because the constituents have identical properties apart from a small difference in density1 which stands in direct connection with the atomic weight4 ,such a separation was first carried out successfully by the ,danish chemist1 ,brnsted1 in collaboration with the ,hungarian chemist1 ,hevesy 7#aiba74 ,these two scientists were able to separate a large quantity of mercury of density #aceiee into two portions of slightly different densities4 ,all the different isotopes of which mercury is a mixture were1 indeed1 not wholly separated2 they were represented in the two portions in different proportions4 ,thus1 in one of the first attempts1 the density of the one part was #aceihf and of the other #aceibj 7at #j ,c74 ,it is a perfectly reasonable supposition that it is the electron system which determines the external properties of the atom1 that is1 those properties which depend on the interplay of two or more atoms4 ,for the electron1 rotating about the nucleus at a considerable distance1 separates1 so to speak1 the nucleus from the surrounding space1 and must therefore be assumed to be the organ which connects the atom with the rest of the universe4 ,one might also expect the structure of the electron system to depend wholly on the nuclear charge1 i4e4 on the atomic number and not on the mass of the nucleus1 since it is the nuclear electrical attraction which holds the electrons in their orbits and not the relatively insignificant gravitational attraction4 ,it thus becomes intelligible that the properties of the elements can be divided into two sharply defined classes1 namely3 7#a7 properties of the nucleus1 and 7#b7 properties of the electron system in the atom4 ,the credit for first recognizing the sharp distinction between these two classes1 a distinction fundamental for a detailed study of the atom1 is due to ,niels ,bohr4 ,the properties of the nucleus determine--7a7 the radioactive processes1 or explosions of the nucleus1 and related processes2 7b7 collisions1 where two nuclei approach extremely near to each other2 and 7c7 weight which1 as mentioned above1 stands in direct connection with atomic weight4 ,the properties of the electron system are1 on the other hand1 the determining factors in all other physical and chemical activities1 and1 as has been stated1 are functions1 we may say1 of the atomic number of the given element4 ,the ,bohr theory may be said to concern itself with the chemical and physical properties of the atom with the exception of those which have to do with the nucleus4 ,we shall consequently devote our attention in the next chapters to the electron system4 ,but before turning to this we shall dwell a little further upon the atomic nucleus4 ,the ,structure of the ,nucleus4 ,that the nucleus is not an elementary indivisible particle but a system of particles1 is clearly shown by the radioactive processes in which -particles and -particles 7electrons7 are shot out of the nuclei of radioactive elements4 ,bohr was the first to see clearly that not only the -particles emitted in such cases come from the nucleus1 but that the -particles also have their source there4 ,there is now no doubt that1 in addition to the outer electrons of the atom1 which are the determining factor in the atomic number1 there must also be1 in the radioactive substances at any rate1 special nuclear electrons which lead a more hidden existence in the interior of the nucleus4 ,one can easily understand that isotopes may result as products of radioactive disintegration4 ,for example1 let us suppose that a nucleus emits first an -particle 7i4e4 a helium nucleus with two positive charges71 and thereafter sends out two electrons1 each with its negative charge1 in two new disintegrations4 ,the nuclear charge in the resultant atom will then obviously be the same as before1 because the loss of the two electrons exactly neutralizes that of the -particle4 ,but the atomic weight will be diminished by four units 7i4e4 the weight of the helium nucleus1 remembering also that the electrons have but very negligible masses74 ,among the radioactive substances are recognized many examples of isotope elements1 with atomic weights differing precisely by four4 ,the radioactive element uranium is the element with the greatest atomic weight 7#bch71 and atomic number 7#ib71 and consequently with the greatest nuclear charge4 ,almost all the other radioactive substances are those with high atomic numbers in the periodic system4 ,the cause of radioactivity must be sought in the hypothesis that the nuclei of the radioactive elements are very complicated systems with small stability1 and therefore break down rather easily into less complicated and more stable systems with the emission of some of their constituent particles2 the corpuscular rays thus produced possess a considerable amount of kinetic energy4 ,accordingly1 by analogy1 the nuclei of the non-radioactive elements may be assumed to be composed of nuclear electrons and positive particles2 hydrogen alone excepted4 ,the simplest assumption is that the hydrogen nucleus is the real quantum or atom of positive electricity1 just as the electron is the atom of negative electricity4 ,on this theory all substances are built up of two kinds only of fundamental particles1 namely1 hydrogen nuclei and electrons4 ,that these particles may themselves consist of constituent parts is1 of course1 an open possibility1 but such speculation is beyond our experience up to the present4 ,in every nucleus there are more positive hydrogen nuclei than there are negative electrons1 so that the nucleus has a residual positive charge of a magnitude equal to the difference between the number of hydrogen nuclei and nuclear electrons4 ,if we now pass from hydrogen which has the atomic weight1 atomic number and nuclear charge of unity1 we next encounter helium with the atomic weight #d1 atomic number and nuclear charge #b4 ,the helium nucleus should therefore consist of #d hydrogen nuclei1 which would together account for the atomic weight of #d4 ,but since these represent #d positive charges1 there must also be present in the nucleus #b negative electrons to make the resultant nuclear charge equal to #b4 ,we could indeed hardly conceive of a system composed of #d positive hydrogen nuclei alone2 for the forces of repulsion would soon drive the separate parts asunder4 ,the two electrons can1 so to speak1 serve to hold the system together4 ,fig4 #bd gives a rough representation of the helium atom4 ,it must be carefully noted that the picture is purely schematic and the distances arbitrary4 ,the helium nucleus1 composed of #d hydrogen nuclei and #b electrons1 seems to possess extreme stability1 and it is not improbable that helium nuclei occur as higher units in the structure of the nuclei of not only the radioactive substances but also the other elements4 ,we shall perhaps be very near the truth in saying that all nuclei are built up of combinations of hydrogen nuclei1 helium nuclei and electrons4 ,illustration3 ,,fig4 #bd4--,schematic representation of a helium atom4 ,k1 nuclear system with four hydrogen nuclei and two nuclear electrons2 ,e1 electrons in the outer electron system4 ,in nitrogen1 with the atomic weight #ad and atomic number #g1 the nucleus should consist of #ad hydrogen nuclei 7with #ab of them compounded1 perhaps1 into #c helium nuclei7 and #g nuclear electrons1 reducing the resultant positive nuclear charge from #ad to #g4 ,uranium1 with atomic number #ib and atomic weight #bch1 should have a nucleus composed of #bch hydrogen nuclei and #adf electrons1 and so on for the others4 ,we see at once that the conception of the nucleus here propounded leads us back to the old hypothesis of ,prout 7see p4 #ae7 that all atomic weights should be integral multiples of that of hydrogen4 ,this hypothesis apparently disagreed with atomic weight measurements1 but the isotope researches have vanquished this difficulty2 thus it has been mentioned before that chlorine with an atomic weight of #cee appears to be a mixture of isotopes with atomic weights #ce and #cg1 and other cases have a similar explanation4 ,yet the rule cannot be wholly and completely exact4 ,for1 in the first place1 the mass of the electrons must contribute something1 though this contribution is far too small to be measured4 ,but there is also a second matter which plays a part here4 ,this is the law enunciated by ,einstein in his relativity theory1 that every increase or decrease in the energy of a body is correlated with an increase or decrease in the mass of the body1 proportional to the energy change4 ,we must1 therefore1 expect that the masses of the various atomic nuclei will depend not only on the number of hydrogen nuclei 7and electrons71 but also on the energy represented in the attractions and repulsions between the particles of the system1 and in their mutual motions1 or the energy which comes into play in the formation and disintegration of nuclear systems4 ,this is presumably closely connected1 although in a way which is not clearly understood1 with the fact1 that if the atomic weights of the elements are to come out integers1 that of hydrogen must not be taken as #a but as #ajjh2 that is1 the atomic weight unit must be chosen a little smaller than the atomic weight of hydrogen 7cf4 table1 p4 #bc74 ,transformation of ,elements and ,liberation of ,atomic ,energy4 ,we shall now treat very briefly two questions which have profoundly interested many people1 because they are concerned with possible practical applications of our new knowledge of atoms4 ,the first question is this3 ,can one not1 from this knowledge1 bring about the transformation of one element into another8 ,in answering this1 it can1 of course1 be said immediately that among the radioactive substances such transformations are constantly taking place without human interference1 and we certainly have no right to state offhand that it will be impossible for man ever to bring about such a transformation artificially4 ,for example1 if we could succeed in getting one hydrogen nucleus loose from the nucleus of mercury1 the latter would thereby be changed into a gold nucleus4 ,such a thing is not only conceivable1 but in the last few years it has become a reality1 though1 to be sure1 not with the substances here mentioned4 ,in #aiai ,rutherford1 by bombarding nitrogen 7,n #ad7 with -particles1 was able to knock loose some hydrogen nuclei from the nitrogen nucleus2 perhaps he succeeded thereby in changing the nitrogen nuclei into carbon nuclei 7,c #ab7 by the breaking off of two hydrogen nuclei from each nitrogen nucleus4 ,but to disintegrate very few nitrogen nuclei1 ,rutherford had to employ a formidable bombardment with hundreds of thousands of projectiles 7-particles72 and even if he had ended with gold instead of carbon1 this would have been1 from the economic point of view1 a very foolish way of making gold2 and at the present time we know of no other artificial method for the transformation of elements4 ,that ,rutherford's investigation has1 in any case1 extraordinarily great interest and scientific value is another matter4 ,the second question is whether one cannot liberate and utilize the energy latent in the interior of the atom4 ,this question1 which was suggested in the first instance by the discovery of radium1 has recently attracted considerable attention because of reports that1 according to ,einstein's relativity theory1 one gram of any substance by virtue of its mass alone must contain a quantity of energy equal to that produced by the burning of #cjjj tons of coal4 ,the meaning of this statement is this3 it has already been mentioned that according to the relativity theory a decrease in the energy of a body brings about a decrease in its mass2 it is immaterial in what form the energy is given up1 whether as heat1 elastic oscillations1 or the like2 all that is said is1 that to a certain decrease in mass1 will correspond a perfectly definite emission of energy in some form4 ,if we now could imagine the whole mass of one gram of a substance to be 8destroyed0 7i4e4 caused to disappear utterly as a physical substance71 and to reappear as heat energy1 for example1 then we could compute from the known relation between mass and energy1 that the heat energy thus brought about would be equivalent to that obtained by the burning of #cjjj tons of coal4 ,but in order that all this energy should be developed1 even the hydrogen nuclei and the electrons would have to be 8destroyed10 and no phenomenon is known1 supporting the supposition that such a 8destruction0 of the fundamental particles of a substance is possible1 or that it is possible to transform these particles into other types of energy4 ,a thought like this must rather be stamped as fantasy1 the origin of which is to be found in a misunderstanding of a purely scientific mode of expression4 ,the case is essentially different with those quantities of energy which must be assumed to be freed or absorbed in the transformation of one nuclear system into another1 that is1 in elemental transformations4 ,though these are far smaller in amount1 the radioactive processes indicate that they are not wholly to be despised4 ,for one gram of radium will upon complete disintegration to non-radioactive material give off as much energy as is equivalent to #dfj kg4 of coal4 ,but even here we must confess that it will take about #agjj years for only half of the radium to be transformed4 ,it is not at all impossible that other elemental transformations might lead to just as great energy developments as appear in the disintegration of radioactive substances4 ,let us imagine that four hydrogen nuclei1 which together have a mass of #d #ajjh #djcb1 and two electrons could join together to form a helium nucleus with atomic weight very close to #d4 ,this process would thus result in a loss of mass which must be assumed to appear in another form of energy4 ,the amount of energy obtainable in this way from one gram of hydrogen would be considerably more than that given off by the disintegration of one gram of radium4 ,there can hardly exist any doubt that in nature there occur not only disintegrations1 but also 7perhaps in the interior of the stars7 building-up processes in which compound nuclei result from simple ones4 ,it is therefore natural to suppose that by exerting on hydrogen exceptional conditions of temperature1 pressure1 electrical changes1 etc41 we could succeed by experiments here on earth in forming helium from it with the development of considerable energy4 ,but at the same time it is very likely that even under favourable circumstances such a process would take place with very great slowness1 because the formation of a helium nucleus might well be a very infrequent occurrence2 it would probably be the result of a certain succession of collisions between hydrogen nuclei and electrons1 a combination whose probability of occurrence in a certain number of collisions is infinitely less than the probability of winning the largest prize in a lottery with the same number of chances4 ,nature has time enough to wait for 8wins10 while mankind unfortunately has not4 ,we know concerning the disintegration of the radioactive substances that it is of the character here indicated2 of the great number of atoms to be found in a very small mass of a radioactive substance1 now one explodes and now another4 ,but why fortune should pick out one particular atom is as difficult to understand as why in a lottery one particular number should prove to be the lucky one rather than any other4 ,our only understanding of the whole matter rests on the law of averages1 or probability as we may call it4 ,we know that of a billion radium atoms 7#aj7 on the average thirteen explode every second2 and even if in any single collection of a billion a few more or a few less may explode1 the average of thirteen per second per billion will always be maintained in dealing with larger and larger numbers of atoms1 as1 for example1 with a thousand billion or a million billion4 ,for other radioactive substances we get wholly different averages for the number of atoms disintegrating per second2 but in no case are we able to penetrate into the inner character of the process of disintegration itself4 ,and what holds true of the radioactive substances will also hold true probably for elemental changes of all kinds2 ,rutherford with his hundreds of thousands of -particle projectiles was able to make sure of but a few lucky 8shots40 ,the whole matter must at this stage be looked upon as governed wholly by chance4 ,one interested in speculating on what would happen if it were possible to bring about artificially a transformation of elements propagating itself from atom to atom with the liberation of energy1 would find food for serious thought in the fact that the quantities of energy which would be liberated in this way would be many1 many times greater than those which we now know of in connection with chemical processes4 ,there is then offered the possibility of explosions more extensive and more violent than any which the mind can now conceive4 ,the idea has been suggested that the world catastrophes represented in the heavens by the sudden appearance of very bright stars may be the result of such a release of sub-atomic energy1 brought about perhaps by the 8super-wisdom0 of the unlucky inhabitants themselves4 ,but this is1 of course1 mere fanciful conjecture4 ,it seems clear1 however1 that we need have no fear that in investigating the problem of atomic energy we are releasing forces which we cannot control1 because we can at present see no way to liberate the energy of atomic nuclei beyond that which ,nature herself provides1 to say nothing of a practical solution of the energy problem4 ,the time has certainly not yet come for the technician to follow in the theoretical investigator's footsteps in this branch of science4 ,one hesitates1 however1 to predict what the future may bring forth4 ,interesting and significant as is the insight which ,rutherford and others have opened up into the inner workings of the nucleus1 the study of the electron system of the atom bears more intimately upon the various branches of physical and chemical science1 and hence presents greater possibilities of attaining1 in a less remote future1 to discoveries of practical significance4 ,,chapter ,v ,,the ,,bohr ,,theory ,,of ,,the ,,hydrogen ,,spectrum ,the ,nuclear ,atom and ,electrodynamics4 ,even if ,rutherford had not yet succeeded in giving a complete answer to the first of the questions propounded in the previous chapter1 namely1 that concerning the positions of the positive and negative particles of the atom1 one might at any rate hope that his general explanation of the structure of the atom--that is1 the division into the nucleus and surrounding electrons1 and the determination of the number of electrons in the atoms of the various elements--would furnish a good foundation for the answer to the second question about the connection between the atomic processes and the physical and chemical properties of matter4 ,but in the beginning this seemed so far from being true that it appeared almost hopeless to find a solution of the problem of the atom in this way4 ,we shall best understand the meaning of this if we consider the simplest of the elemental atoms1 namely1 the atom of hydrogen with its positive nucleus and its one electron revolving about the nucleus4 ,how could it be possible to explain from such a simple structure the many sharp spectral lines given by the ,balmer-,ritz formula 7p4 #eg78 ,as has previously been mentioned1 the classical electron theory seemed to demand a very complicated atomic structure for the explanation of these lines4 ,according to the electron theory1 the atoms may be likened to stringed instruments which are capable of emitting a great number of tones1 and in these atoms the electrons are naturally supposed to correspond to the 8strings40 ,but the hydrogen atom has only one electron1 and it hardly seems credible that in a mass of hydrogen the individual atoms would be tuned for different 8tones10 with definite frequencies of vibration4 ,now1 it certainly cannot be concluded from the analogy with the stringed instrument that a single electron can emit light of only a single frequency at one time1 corresponding to a single spectral line4 ,for a plucked string will1 as we know1 give rise to a simple tone only if it vibrates in a very definite and particularly simple way2 in general it will emit a compound sound which may be conceived as made up of a 8fundamental0 and its so-called 8overtones10 or 8harmonies0 whose frequencies are #b1 #c1 444 times that of the fundamental 7i4e4 integral multiples of the latter74 ,these overtones may arise even separately because the string1 instead of vibrating as a whole1 may be divided into #b1 #c1 444 equally long vibrating parts1 giving respectively #b1 #c1 444 times as great frequencies of vibration4 ,we call such vibrations 8harmonic oscillations40 ,the simultaneous existence of these different modes of oscillation of the string may be thought of in the same way as the simultaneous existence of wave systems of different wave-lengths on the surface of water4 ,corresponding to the possibility of resolving the motion of the string into its 8harmonic components10 the compound sound waves produced by the string can be resolved by resonators 7cf4 p4 #dd7 into tones possessing the frequencies of these components4 ,according to the laws of electrodynamics the situation with the electron revolving about the hydrogen nucleus might be expected to be somewhat similar to that described above in connection with the vibrating string4 ,if the orbit of the electron were a circle1 it should emit into the ether electromagnetic waves of a single definite wave-length and corresponding frequency1 1 equal to 1 the frequency of rotation of the electron in its orbit2 that is1 the number of revolutions per second4 ,but just as a planet under the attraction of the sun1 varying inversely as the square of the distance1 moves in an ellipse with the sun at one focus1 so the electron1 under the attraction of the positive nucleus1 which also follows the inverse square law1 will in general be able to move in an ellipse with the nucleus at one focus4 ,the electromagnetic waves which are emitted from such a moving electron may on the electron theory be considered as composed of light waves corresponding to a series of harmonic oscillations with the frequencies3 1 #b1 #c 444 and so on1 where 1 as before1 is the frequency of revolution of the electron4 ,according as the actual orbit deviates more or less from a circle1 the frequencies 1 444 will appear stronger or weaker in the compound light waves emitted4 ,but the actual distribution of spectral lines in the real hydrogen spectrum presents no likeness whatever to this distribution of frequencies4 ,from this it is evident that no agreement can be reached between the classical electron theory on the one hand and the ,rutherford atom model on the other4 ,indeed1 the disagreement between the two is really far more fundamental than has just been indicated4 ,according to ,lorentz's explanation of the emission of light waves1 the electrons in a substance 7see again p4 #ge7 should have certain equilibrium positions1 and should oscillate about these when pushed out of them by some external impulse4 ,the energy which is given to the electron by such an impulse is expended in the emission of the light waves and is thus transformed into radiation energy in the emitted light1 while the electrons fall to rest again unless they receive in the meantime a new impulse4 ,we can get an understanding of what these impulses in various cases may be by thinking of them1 in the case of a glowing solid1 for example1 as due to the collisions of the molecules2 or in the case of the glowing gas in a discharge tube1 from the collisions of electrons and ions4 ,the oscillating system represented by the electron 7the 8oscillator07 will possess under these circumstances great analogy with a string which after being set into vibration by a stroke gradually comes back to rest1 while the energy expended in the stroke is emitted in the form of sound waves4 ,although the vibrations of the string become weaker after a while1 the period of the vibrations will remain unchanged2 the string vibrations like pendulum oscillations have an invariable period1 and the same will be the case with the frequency of the electron if the force which pulls it back into its equilibrium position is directly proportional to the displacement from this position 7the 8harmonic motion0 force74 ,rutherford's atomic model is1 however1 a system of a kind wholly different from the 8oscillator0 of the electron theory4 ,the one revolving hydrogen electron will find a position of 8rest0 or equilibrium only in the nucleus itself1 and if it once becomes united with the latter it will not easily escape2 it will then probably become a nuclear electron1 and such a process would be nothing less than a transformation of elements 7see p4 #gi74 ,on the other hand1 it follows necessarily from the fundamental laws of electrodynamics that the revolving electron must emit radiation energy1 and1 because of the resultant loss of energy1 must gradually shrink its path and approach nearer the nucleus4 ,but since the nuclear attraction on the electron is inversely proportional to the square of the distance1 the period of revolution will be gradually decreased and hence the frequency of revolution 1 and the frequency of the emitted light will gradually increase4 ,the spectral lines emitted from a great number of atoms should1 accordingly1 be distributed evenly from the red end of the spectrum to the violet1 or in other words there should be no line spectrum at all4 ,it is thus clear that ,rutherford's model was not only unable to account for the number and distribution of the spectral lines2 but that with the application of the ordinary electrodynamic laws it was quite impossible to account for the existence even of spectral lines4 ,indeed1 it had to be admitted that an electrodynamic system of the kind indicated was mechanically unstable and therefore an impossible system2 and this would apply not merely to the hydrogen atom1 but to all nuclear atoms with positive nuclei and systems of revolving electrons4 ,however one looks at the matter1 there thus appears to be an irremediable disagreement between the ,rutherford theory of atomic structure and the fundamental electrodynamic assumptions of ,lorentz's theory of electrons4 ,as has been emphasized1 however1 ,rutherford founded his atomic model on such a direct and clear-cut investigation that any other interpretation of his experiments is hardly possible4 ,if the result to which he attained could not be reconciled with the theory of electrodynamics1 then1 as has been said1 this was so much the worse for the theory4 ,it could1 however1 hardly be expected that physicists in general would be very willing to give up the conceptions of electrodynamics1 even if its basis was being seriously damaged by ,rutherford's atomic projectiles4 ,surmounted by its crowning glory--the ,lorentz electron theory--the classical electrodynamics stood at the beginning of the present century a structure both solid and spacious1 uniting in its construction nearly all the physical knowledge accumulated during the centuries1 optics as well as electricity1 thermodynamics as well as mechanics4 ,with the collapse of such a structure one might well feel that physics had suddenly become homeless4 ,the ,quantum ,theory4 ,in a field completely different from the above the conclusion had also been reached that there was something wrong with the classical electrodynamics4 ,through his very extended speculations on thermodynamic equilibrium in the radiation process1 ,planck 7#aijj7 had reached the point of view expressed in his quantum theory1 which was just as irreconcilable with the fundamental electrodynamic laws as the ,rutherford atom4 ,a complete representation of this theory would lead us too far2 we shall merely give a short account of the foundations on which it rests4 ,by a black body is generally understood a body which absorbs all the light falling upon it1 and1 accordingly1 can reflect none4 ,physicists1 however1 denote by the term 8perfect black body0 in an extended sense1 a body which at all temperatures absorbs all the radiation falling upon it1 whether this be in the form of visible light1 or ultra-violet1 or infra-red radiation4 ,from considerations which were developed some sixty years ago by ,kirchhoff1 it can be stated that the radiation which is emitted by such a body when heated does not depend on the nature of the body but merely on its temperature1 and that it is greater than that emitted by any other body whatever at the same temperature4 ,such radiation is called temperature radiation or sometimes 8black0 radiation1 though the latter term is apt to be misleading1 since a 8perfect black body0 emitting black radiation may glow at white heat4 ,it may be of interest to note here the fundamental law deduced by ,kirchhoff1 which may best be illustrated by saying that good absorbers of radiation are also good radiators4 ,an instructive experiment illustrating this is performed by painting a figure in lampblack on a piece of white porcelain4 ,the lampblack surface is clearly a better absorber of radiant energy than the white porcelain4 ,when the whole is heated in a blast flame1 the lampblack figure glows much more brightly than the surrounding porcelain1 thus showing that at the same temperature it is also the better radiator4 ,following the same law we conclude that highly reflecting bodies are not good radiators1 a fact that has practical significance in house heating4 ,the perfect black body1 then1 being the best absorber of radiation1 is also the best radiator4 ,in actual practice no body is absolutely black4 ,even a body coated with lampblack reflects about #aj per cent4 of the light waves incident on it4 ,the ,danish physicist ,christiansen remarked long ago that a real black body could be produced only if an arrangement could be made whereby the incident waves could be reflected several times in succession before finally being emitted4 ,to take the case of lampblack1 three such reflections would reduce the re-emission from the lampblacked body to only _/ of the radiation initially falling on it4 ,this type of black body was finally realized by making a cavity in an oven having as its only opening a very small peep-hole1 and keeping the temperature of the wall of the oven uniform4 ,if a ray is sent into the cavity through the peep-hole1 it will become1 so to speak1 captured1 because1 when once inside it will suffer countless reflections from the walls of the cavity1 having more and more of its energy absorbed at each reflection4 ,very little of the radiation thus entering will ever get out again1 and consequently such a body will act very much like a 8perfect black body0 according to the theoretical definition above4 ,accordingly1 the radiation which is emitted from such a glowing cavity through the peep-hole will be black or practically black radiation4 ,the cavity itself will be criss-crossed in all directions by radiation emitted from one part of the inner surface of the cavity and absorbed by 7and partially reflected from7 other parts of the surface4 ,when the walls of the cavity are kept at a fixed and uniform temperature there will automatically be produced a state of equilibrium in which every cubic centimetre of the cavity will contain a definite quantity of radiation energy1 dependent only on the temperature of the walls4 ,further1 in the equilibrium state the radiation energy will be distributed in a perfectly definite way 7dependent as before on the temperature only7 among the various types of radiation corresponding to different wave-lengths and frequencies4 ,if there is too much radiation of one kind and too little of another1 the walls of the cavity will absorb more of the first kind than they emit1 and emit more of the second kind than they absorb1 and so the state will vary until the right proportion for equilibrium is attained4 ,this distribution of the radiation over widely differing wave-lengths can be investigated by examining spectroscopically the light emitted from the peep-hole in the cavity4 ,then1 by means of a bolometer or some other instrument1 the heat development in the different portions of the spectrum can be measured4 ,at a temperature of #aejj ,c41 for example1 one will find that the maximum energy is represented for rays of wave-length in the close vicinity of #ah 1 i4e4 in the extreme infra-red4 ,if the temperature is raised1 the energy maximum travels off in the direction of the violet end of the spectrum2 if the temperature is lowered1 it will move farther down into the infra-red4 ,it is also possible to make a theoretical calculation of the distribution of energy in the spectrum of the black-body radiation at a given temperature4 ,but the results obtained do not agree with experiment4 ,the ,english physicists1 ,rayleigh and ,jeans1 developed on the basis of the classical electrodynamic laws and by apparently convincing arguments a distribution law according to which actual radiation equilibrium becomes impossible1 since if it were true the energy in the radiation would tend more and more to go over to the region of short wave-lengths and high frequencies1 and this shifting would apparently go on indefinitely4 ,the theory thus leads to results which are not only in disagreement with experiment1 but which must be looked upon as extremely unreasonable in themselves4 ,planck1 however1 had vanquished these difficulties and had obtained a radiation law in agreement with experiment by introducing an extremely curious hypothesis4 ,like ,lorentz1 he thought of radiation as produced through the medium of small vibrating systems or oscillators1 which could emit or absorb rays of a definite frequency 4 ,but while1 according to the ,lorentz theory and the classical electrodynamics1 radiation can be emitted in infinitely small quantities 7i4e4 small without limit71 ,planck assumed that an oscillator can emit and absorb energy only in certain definite quantities called quanta1 where the fundamental quantum of radiation is dependent on the frequency of the oscillator1 varying directly with the latter4 ,if thus we denote the smallest quantity of energy which an oscillator of frequency can emit or absorb by ,e1 then we can write ,e h1 where h is a definite constant fixed for all frequencies4 ,accordingly the cavity can receive radiation energy of frequency from the radiating oscillators in its wall1 or transfer energy to these in no smaller quantity than h4 ,the total energy of that kind emitted or absorbed at any given time will always be an integral multiple of h4 ,oscillators with a frequency #a times as great will emit energy in quanta which are #a times larger1 and so on4 ,the quantity h is independent1 not only of the wave-length1 but also of the temperature and nature of the emitting body4 ,this constant1 the so-called ,planck constant1 is thus a universal constant4 ,if one uses the 8absolute0 units of length1 mass and time 7see table1 p4 #baj71 its value comes out as #fed #aj4 ,for the frequency #gej #aj vibrations per second1 corresponding to the extreme violet in the visible spectrum1 the ,planck energy quantum thus becomes about #e #aj erg1 or #cfi #aj foot-pounds 7note that the 8erg0 is the 8absolute0 unit of work1 or the amount of work done when a body is moved through a distance of #a cm4 by a force of one dyne acting in the direction of the motion1 while the 8foot-pound0 is the work done when a force of #a pound moves a body #a foot in the direction of the force74 ,for light belonging to the red end of the spectrum1 the energy is about half as great4 ,if we pass1 however1 to the highest frequencies and the shortest wave-lengths which are known1 namely those corresponding to the 8hardest0 7i4e4 most penetrating7 -rays 7see p4 #gh71 we meet with energy quanta which are a million times larger1 i4e4 #b #aj erg1 although they are still very small compared to any amount of energy measurable mechanically4 ,this remarkable theory of quanta1 which in the hands of ,planck still possessed a rather abstract character1 proved under ,einstein's ingenious treatment to have the greatest significance in many problems which1 like heat radiation1 had provided physicists with many difficulties4 ,for by assuming that energy in general could only be given up and taken in in quanta1 certain facts about the specific heats of bodies could be accounted for--facts which the older physics had proved powerless to explain4 ,the ,planck energy quanta1 as ,einstein showed1 could also explain in a very direct and satisfactory way the photoelectric effect1 as it is called4 ,this effect consists in the freeing of electrons from a metal plate which ultra-violet rays are allowed to strike4 ,the maximum velocity with which these electrons are propelled from the plate is found to be independent of the intensity of the incident light1 but dependent simply on the frequency of the radiation4 ,careful measurements have indeed shown1 as ,einstein predicted1 that the incident light really does utilize an energy quantum h to free each electron and give it velocity 7cf4 p4 #agb74 ,of the different methods which nowadays are at hand1 the photoelectric effect constitutes one of the best means of determining the value of h4 ,it has been applied for that purpose by ,millikan1 to whose ingenious experiments the most accurate direct determination of h is actually due4 ,all this lay completely outside the laws of electrodynamics1 and pointed to the existence of unknown and more fundamental laws4 ,but1 for the time being1 physicists had to be satisfied merely with a recognition of the fact that these mysterious energy quanta play a very significant part in many phenomena4 ,a decade ago physics1 as regards radiation problems1 was in a very unsettled state2 with four separate branches of knowledge1 each of which seemed firm and well-founded enough in itself1 but which had no common connecting link1 indeed1 were even to some extent inconsistent with each other4 ,the first of these was the classical electrodynamics surmounted by the felicitous electron theory of ,lorentz and ,larmor4 ,the second was the empirical knowledge of the spectra resting on the work of ,balmer1 ,ritz and ,rydberg4 ,the third was ,rutherford's nuclear atom model4 ,and the fourth was ,planck's quantum theory of heat radiation4 ,it was quite evident that progress in the theory of radiation and the structure of the atom was hopeless as long as these four points of view remained uncorrelated4 ,main ,outlines of the ,bohr ,theory4 ,such was the situation when1 in #aiac1 ,bohr published his atomic theory1 in which he was able with great ingenuity to unite the nuclear atom1 the ,balmer-,ritz formula and the quantum theory4 ,as far as electrodynamics is concerned1 the impossibility of retaining that in its classical form was presented in a much clearer way than ever before4 ,but1 as will presently be evident1 the ,bohr theory has a very definite connection with the classical theory1 and ,bohr's attempts to preserve and develop this connection have proved to be of the greatest significance for his theory4 ,in spite of the fundamental rupture with the old ideas1 the ,bohr theory strives to absorb all that is useful in the classical point of view4 ,at the head of the theory appear the two fundamental hypotheses or postulates on the properties of the atom4 ,the first postulate states that for each atom or atomic system there exists a number of definite states of motion1 called 8stationary states10 in which the atom 7or atomic system7 can exist without radiating energy4 ,a finite change in the energy content of the atom can take place only in a process in which the atom passes completely from one stationary state to another4 ,the second postulate states that if such a transition takes place with the emission or absorption of electromagnetic light waves1 these waves will have a definite frequency1 the magnitude of which is determined by the change in the energy content of the atom4 ,if we denote the change in energy by ,e and the frequency by we may write ,e ,e h1 or ----- h where h is the ,planck constant4 ,in consequence of the second postulate the emission as well as the absorption of energy by the atom always takes place in quanta4 ,the two postulates say nothing concerning the nature of the motion in the stationary states4 ,in the applications1 however1 a connection with the ,rutherford atomic model is established4 ,confining our attention first to the hydrogen atom1 the system with which we are concerned consists1 accordingly1 of a positive nucleus and one electron revolving about it4 ,the various states of motion which the electron can assume in virtue of the first postulate are a series of orbits at different distances from the nucleus4 ,in each of these 8stationary orbits0 the electron follows the general mechanical laws of motion2 i4e4 under the nuclear attraction which is inversely proportional to the square of the distance1 the electron describes an ellipse with the nucleus at one focus1 as has previously been stated2 but in contradiction with the classical electrodynamics it will emit no radiation while moving in this orbit4 ,fig4 #be shows a series of these orbits1 to which the numbers #a1 #b1 #c1 #d have been attached1 and which for simplicity are represented as circular4 ,illustration3 ,,fig4 #be4--,the ,bohr model of the hydrogen atom in the simplified form 7with circles instead of ellipses74 ,if the electron passes from an outer orbit to an inner one2 for example1 if it goes from number #d to number #b1 or from number #b to number #a1 the electric force which attracts it to the nucleus will do work just as the force of gravity does work when a stone falls to the ground4 ,a part of this work is used to increase the kinetic energy of the electron1 making its velocity in the inner orbit greater than in the outer1 but the rest of the work is transformed into radiation energy which is emitted from the atom in the form of monochromatic light4 ,in consequence of the second postulate the frequency of the emitted radiation is proportional to the energy loss4 ,when the electron has reached the innermost orbit 7the one denoted by #a in the figure71 it cannot get any nearer the nucleus and hence cannot emit any more radiation unless it first is impelled to pass from its inner orbit to an outer orbit again by the absorption of external energy sufficient to bring about this change4 ,once in the outer orbit again1 it is in a state to produce radiation by falling in a second time4 ,the innermost orbit represents thus the electron's equilibrium state1 and corresponds to the normal state of the atom4 ,if we try to illustrate the matter with an analogy from the theory of sound1 we can do so by comparing the atom not with a stringed instrument1 but with a hypothetical musical instrument of a wholly different kind4 ,let us imagine that we have placed one over another and concentrically a series of circular discs of progressively smaller radii1 and let us suppose that a small sphere can move around any one of these without friction and without emitting sound4 ,in such a motion the system may be said to be in a 8stationary state40 ,sooner or later the sphere may fall from the first disc on to one lower down and continue to roll around on the second1 having emitted a sound1 let us assume1 by its fall4 ,by passing thus from one stationary state to another it loses a quantity of energy equal to the work which would be necessary to raise it again to the disc previously occupied1 and to bring it back to the original state of motion4 ,we can assume that the energy which is lost in the fall reappears in a sound wave emitted by the instrument1 and that the pitch of the sound emitted is proportional to the energy sent out4 ,if1 moreover1 we imagine that the lowermost disc is grooved in such a way that the sphere cannot fall farther1 then this fanciful instrument can provide a very rough analogy with the ,bohr atom4 ,we must beware1 however1 of stretching the analogy farther than is here indicated4 ,it must be specially emphasized here that the frequency of the sound emitted in the above example has no connection with the frequency of revolution of the sphere4 ,in the ,bohr atom1 likewise1 the frequency of revolution of the electron in its stationary orbit has no direct connection with the frequency of the radiation emitted when the electron passes from this orbit to another4 ,this is a very surprising break with all previous views on radiation1 a break whose revolutionary character should not be under-estimated4 ,but1 however unreasonable it might seem to give up the direct connection between the revolutional frequency and the radiation frequency1 it was absolutely necessary if the ,rutherford atomic model was to be preserved4 ,and as we shall now see1 the new point of view of the ,bohr theory leads naturally to an interpretation of the ,balmer-,ritz formula1 which had previously not been connected with any other physical theory4 ,the quantity of energy ,e1 which the atom gives up when the electron passes from an outer to an inner orbit1 or which1 conversely1 is taken in when the electron passes from an inner to an outer orbit1 may1 as has been indicated1 be regarded as the difference between the energy contents of the atom in the two stationary states4 ,this difference may be expressed in the following way4 ,let us imagine that we eject the electron from a given orbit 7e4g4 ,no4 #b in the diagram7 so that it is sent to 8infinity10 or1 in other words1 is sent so far away from the nucleus that the attraction of the latter becomes negligible4 ,to bring about this removal of the electron from the atom demands a certain amount of energy1 which we can call the ionizing work corresponding to the stationary orbit in question4 ,we may here designate it as ,a4 ,to eject the electron from the orbit ,no4 #d will demand a smaller amount of ionizing work1 ,a4 ,the difference ,a - ,a is accordingly the work which must be done to transfer the electron from the orbit ,no4 #b to the orbit ,no4 #d4 ,this is1 however1 exactly equal to the quantity ,e of energy which will be emitted as light when the electron passes from orbit ,no4 #d to orbit ,no4 #b4 ,if we call the frequency of this light 1 then from the relations ,e h and ,e ,a - ,a1 we have h ,a - ,a ,if1 now1 in place of this specific example using the stationary orbits #b and #d we take any two orbits designated by the numbers n 7for the inner7 and n 7for the outer71 we can write for the frequency of the radiation emitted for a transition between these arbitrary states h ,a - ,a or ,a ,a ----- - ----- h h ,we have now reached the point where we ought to bring in the ,balmer-,ritz formula for the distribution of the lines in the hydrogen spectrum4 ,this formula may be written 7see p4 #ei7 ,k ,k ------ - ----- n n ,we can now see very clearly the similarity between the formula derived from the spectrum investigations and that derived from the two ,bohr postulates4 ,in both formul the frequency appears as the difference between two terms which are characterized in both cases by two integral numbers1 in the first formula1 numbers denoting two stationary orbits in the ,bohr model for hydrogen1 and in the second the two numbers which in the ,balmer-,ritz formula for the hydrogen spectrum characterize1 respectively1 a series and one of the lines of the series4 ,to obtain complete agreement we have merely to equate the corresponding terms in the two formul4 ,thus we have for any arbitrary integer n ,a ,k ----- ---- or h n h,k ,a ------ n ,for the innermost stationary orbit1 for which n #a1 the ionizing work ,a will accordingly be equal to the product of the constants h and ,k of ,planck and ,balmer respectively2 and for the orbits ,no4 #b1 ,no4 #c1 ,no4 #d1 etc41 the values will be respectively 1 _/1 _/1 etc41 of this product4 ,from the charges on the nucleus and the electron1 which are both equal to the elementary quantum e of electricity 7see p4 #ij71 and from the ionizing energy for a given orbit we can now find by the use of simple mechanical considerations the radius of the orbit4 ,if we denote the radii of the orbits #a1 #b1 #c 444 by a1 a1 a 4441 we then obtain for the diameters #b;a1 #b;a1 #b;a 444 the values #b;a #ajef #aj cm4 7or approximately #b;a #aj cm471 #b;a #d #aj cm41 #b;a #i #aj cm41 etc4 ,it is seen that the radii of the orbits are in the proportion #a1 #d1 #i 4441 or in other words the squares of the integers which determine the orbit numbers4 ,it is in this proportion that the circles in ,fig4 #be are drawn4 ,we must remember1 however1 that we have here for the moment been thinking of the orbits as circles1 while in reality they must in general be assumed to be ellipses4 ,the foregoing considerations will1 however1 still hold with the single change that #b;a will now mean1 instead of the diameter of a circle1 the major axis of an ellipse4 ,let us return to the formul ,a ,a ----- - ---- h h ,k ,k and ------ - ----- n n ,here n denotes in the first formula the index number for the inner of the two orbits between which the transition is supposed to take place1 while in the second formula n denotes a definite series in the hydrogen spectrum4 ,if n is #b while n takes on the values #c1 #d1 #e 444 then in the ,bohr model of the hydrogen atom this corresponds to a series of transitions to the orbit ,no4 #b from the orbits #c1 #d1 #e 4441 while in the hydrogen spectrum this corresponds to the lines in the ,balmer series1 namely1 the red line 7,h7 corresponding to the transition #c-b1 the blue-green line 7,h7 to #d-b1 the violet line 7,h7 to #e-b and so on4 ,if we now put n #a while n takes the values #b1 #c1 #d 4441 we get in the atom transitions to the orbit ,no4 #a from the orbits ,no4 #b1 #c1 #d 4441 corresponding in the spectrum to what is called the ,lyman series in the ultra-violet 7named after the ,american physicist ,lyman1 who has carried on extensive researches in the ultra-violet region of the spectrum74 ,thus every line in the hydrogen spectrum is represented by a transition between two definite stationary states in the hydrogen atom1 since this transition will give the frequency corresponding to the line in question4 ,at first sight this would seem perhaps to be such an extraordinary satisfactory result that it would prove an overwhelming witness in favour of the ,bohr theory4 ,a little more careful thought1 on the other hand1 would perhaps cause a complete reversion from enthusiasm and lead some to say that the whole thing has not the slightest value1 because the stationary states were so chosen that agreement might be made with the ,balmer-,ritz formula4 ,this last consideration1 indeed1 states the truth in so far that the agreement between the formula and the theory1 at least as developed here up to this point1 is of a purely formal nature4 ,in the ,bohr postulates the frequencies of the emitted radiation are determined by a difference between two of a series of energy quantities1 characterizing the stationary states1 just as in the ,balmer-,ritz formula they appear as a difference between two of a series of terms 7,k1 ,k_/#d1 ,k_/#i1 4447 each characterized by its integer4 ,now by characterizing the quantities of energy in the stationary states by a series of integers 7in itself a wholly arbitrary procedure7 complete agreement between the ,bohr stationary states idea and the spectral formul can be attained4 ,it is not even necessary to introduce the ,rutherford atomic model to attain this end4 ,by bringing in this specific model1 one might join the new theory to the knowledge already gained of the atomic structure1 and1 so to speak1 crystallize the hitherto undefined or only vaguely defined stationary states into more definite form as revolution in certain concrete orbits4 ,this would then lead to a more comprehensive conception of atomic structure4 ,but the theory unfortunately would still be rather arbitrary1 since there would seem to be no justification for picking out certain fixed orbits with definite diameters or major axes to play a special rle4 ,one cannot wonder then that many scientists considered the ,bohr theory unacceptable1 or at any rate were inclined to look upon it simply as an arbitrary1 unreasonable conception which really explained nothing4 ,naturally1 ,bohr himself clearly recognized the formal nature of the agreement between the ,balmer-,ritz formula and his postulates4 ,but ,bohr was the first to see that the quantum theory afforded the possibility of bringing about such an agreement1 and he saw1 moreover1 that the agreement was not merely fortuitous1 but contained within it something really fundamental1 on which one could build further4 ,that atomic processes on his theory took on an unreasonable character 7compared with the classical theory7 was nothing to worry about1 for ,bohr had come to the clear recognition that it was completely impossible to understand from known laws the ,planck-,einstein 8quantum radiation10 or to deduce the properties of the spectrum from the ,rutherford atom alone4 ,he therefore saw that his theory was really not introducing new improbabilities1 but was only causing the fundamental nature of the contradictions which had previously hindered development in this field to appear in a clearer light4 ,but in addition to this the choice of the dimensions of the stationary states was by no means so arbitrary as might appear in the foregoing4 ,in his first presentation of the theory of the hydrogen spectrum1 ,bohr had derived his results from certain considerations connected with the quantum theory--considerations of a purely formal nature1 indeed1 just as those developed in the preceding1 but leading to agreement with the spectral formul4 ,he1 moreover1 called attention to the fact that the values obtained for the orbital dimensions were of the same order of magnitude as those which could be expected on wholly different grounds4 ,the diameter of the innermost orbit1 i4e41 that which defines the outer limit of the atom in the normal state1 was found to be1 as has been noted above1 about #aj cm41 i4e41 of the same order of magnitude as the values obtained for the diameters of molecules on the kinetic theory of gases 7see p4 #bg74 ,the stationary states corresponding to very high quantum numbers one could expect to meet only when hydrogen was very attenuated1 for otherwise there could be no room for the large orbits4 ,we note that the #cbnd orbit must have a diameter #cb 7or over #ajjj times7 as great as the innermost orbit4 ,since1 now1 lines with high number in a hydrogen series correspond on the ,bohr theory to transitions from orbits of high number to an inner orbit1 it became understandable why only comparatively few lines of the ,balmer series are ordinarily observed in the discharge tube1 while many more lines are observed in the spectra of certain stars4 ,for in such stars the possibility is left open for hydrogen to exist in a very attenuated state1 and yet in such large masses that the lines in question can become strong enough for observation4 ,in fact1 one must assume that in a great mass of hydrogen a very large number of atoms send out simultaneously light of the wave-length corresponding to one line4 ,for the ionizing work1 i4e41 the work necessary to eject the electron completely from the normal state and thus make the atom into a positive ion1 the ,bohr theory gives a value of the same order of magnitude as the so-called 8ionization potentials0 which have been found by experiment for various gases4 ,an exact correspondence between theory and experiment could for hydrogen not be attained with certainty1 because the hydrogen atoms in hydrogen gas under ordinary conditions always appear united in molecules4 ,in his very first paper1 however1 ,bohr had studied ,balmer's formula also from another point of view1 and had derived in this way an expression for the ,rydberg constant ,k which agreed with experiment4 ,these considerations have reference to the above-mentioned connection of the theory with the classical theory of electrodynamics4 ,such a connection had previously been known to exist in the fact that1 for long wave-lengths1 the radiation formula of ,planck reduces practically to the ,rayleigh ,jeans ,law which can be derived from electrodynamics4 ,this is related to the fact that when is small 7long wave-lengths71 the energy quantum h is very small1 and hence the character of the radiation emitted will approach more and more nearly to a continuous 8unquantized0 radiation4 ,one might then expect that the ,bohr theory also should lead in the limit of long wave-lengths and small frequencies to results resembling those of the ordinary electrodynamic theory of the radiation process4 ,on the ,bohr theory we get the long wave-lengths for transitions between two stationary states of high numbers 7numbers which also differ little from each other74 ,thus suppose n is a very large number4 ,then the transition from the orbit n to the orbit n - #a will give rise to radiation of great wave-length4 ,for in this case ,a and ,a differ very little1 and accordingly h is very small1 as must be also4 ,according to the electrodynamic theory of radiation1 the revolving electron should emit radiation whose frequency is equal to the electron's frequency of revolution4 ,according to the ,bohr theory it is impossible to fulfil this condition exactly1 since radiation results from a transition between two stationary orbits in each of which the electron has a distinct revolutional frequency4 ,but if n is a large number1 the difference between the frequencies of revolution and for the two orbits n and n - #a1 respectively1 becomes very small2 for example1 for n #ajj1 it is only #c per cent4 ,for a certain high value of n1 then1 the frequency of the emitted radiation can therefore be approximately equal to the frequency of revolution of the electron in both the two orbits1 between which the transition takes place4 ,but even if this proved correct for values of n about #ajj1 one could not be sure beforehand whether it would work out right for still larger values of n1 for example1 #ajjj4 ,in order to investigate this latter point we must look into the formul for the revolutional frequency in a stationary orbit and for the radiation frequency 4 ,since1 according to the ,bohr theory1 we can apply the usual laws of mechanics to revolution in a stationary orbit1 it is an easy matter to find an expression for 4 ,from a short mathematical calculation we can deduce that ,r_/n1 where ,r is the frequency of revolution for the first orbit 7n #a74 ,we find 1 on the other hand1 by substituting in the ,balmer-,ritz formula the numbers n and n - #a1 and a simple calculation shows that for great values of n1 the expression for will approach in the limit the simple form #b,k_/nsup#c4 ,for large orbit numbers1 accordingly varies as 1 i4e41 inversely proportional to the third power of n1 and by equating ,r and #b,k1 we find that the values for and tend more and more to become equal4 ,in this way the value of ,k1 the ,balmer constant1 may be computed4 ,it is found that m ,k #b;e----- h where e is the charge on the electron1 m the mass of the electron1 and h is ,planck's constant4 ,upon the substitution of the experimental values for these quantities1 a value of ,k is determined which agrees with the experimental value 7from the spectral lines investigation7 of #cbi #aj within the accuracy to which e1 m and h are obtainable4 ,this agreement has from the very first been a significant support for the ,bohr theory4 ,one might now object that we have here considered radiation due to a transition between two successive stationary states1 e4g41 ,no4 #ajj and ,no4 #ii1 or the like 7a 8single jump0 we might call it74 ,on the other hand1 for transitions between states whose numbers differ by #b1 #c1 #d or more 7as in a double jump1 or a triple jump7 the agreement found above will wholly disappear1 and doubt be cast on its value4 ,for in such cases of high orbit numbers the frequency of revolution will remain approximately the same even for a difference of #b1 #c1 #d or more in orbit number2 but the radiation frequency for a double jump will be nearly twice that for a single jump1 while that for a triple jump will be nearly three times1 etc4 ,accordingly1 for approximately the same revolutional frequency we shall have in these cases for the radiation frequency very nearly 1 #b1 #c1 etc4 ,we must1 however1 remember that when the orbit in the stationary states is not a circle1 but an ellipse 7as must in general be assumed to be the case71 the classical electrodynamics require that the electron emits besides the 8fundamental0 radiation of frequency 1 the overtones of frequencies #b1 #c 4444 ,we then also here see the outward similarity between the ,bohr theory and the classical electrodynamics4 ,we may say that the radiation of frequency 1 produced by a single jump1 corresponds to the fundamental harmonic component in the motion of the electron1 while the radiation of frequency 1 emitted by a double jump1 corresponds to the first overtone1 etc4 ,the similarity is1 however1 only of a formal nature1 since the processes of radiation1 according to the ,bohr theory1 are of quite different nature than would be expected from the laws of electrodynamics4 ,in order to show how fundamental is the difference1 even where the similarity seems greatest1 let us assume that we have a mass of hydrogen with a very large number of atoms in orbits1 corresponding to very high numbers1 and that the revolutional frequency can practically be set equal to the same quantity 4 ,there may take place transitions between orbits with the difference #a1 #b1 #c 444 in number1 and as the result of these different transitions we shall find1 by spectrum analysis1 in the emitted radiation frequencies which are practically 1 #b1 #c1 etc4 ,according to the radiation theory of electrodynamics we should also get these frequencies and the spectral lines corresponding to them4 ,it must1 however1 be assumed that they are produced by the simultaneous emission from every individual radiating atom of a fundamental and a series of overtones4 ,according to the ,bohr theory1 on the other hand1 each individual radiating atom at a given time emits only one definite line corresponding to a definite frequency 7monochromatic radiation74 ,we can now realize that the ,bohr theory takes us into unknown regions1 that it points towards fundamental laws of nature about which we previously had no ideas4 ,the fundamental postulates of electrodynamics1 which for a long time seemed to be the fundamental laws of the physical world itself1 by which there was hope of explaining the laws of mechanics and of light and of everything else1 were disclosed by the ,bohr theory as merely superficial and only applicable to large-scale phenomena4 ,the apparently exact account of the activities of nature1 obtained by the formul of electrodynamics1 often veiled processes of a nature entirely different from those the formul were supposed to describe4 ,one might then express some surprise that the laws of electrodynamics could have been obtained at all and interpreted as the most fundamental of all laws4 ,it must1 however1 be remembered that the ,bohr theory for large wave-lengths1 i4e41 the slow oscillations1 leads to a formal agreement with electrodynamics4 ,it must1 moreover1 be remembered that the laws of electrodynamics are established on the basis of large-scale electric and magnetic processes which do not refer to the activities of separate atoms1 but in which very great numbers of electrons are carried in a certain direction in the electric conductors or vibrate in oscillations which are extremely slow compared with light oscillations4 ,moreover1 the observed laws1 even if they can account for many phenomena in light1 early showed their inability to explain the nature of the spectrum and many other problems connected with the detailed structure of matter4 ,indeed the more this structure was studied1 the greater became the difficulties1 the stronger the evidence that the solution cannot be obtained in the classical way4 ,if we ask whether ,bohr has succeeded in setting up new fundamental laws1 which can be quantitatively formulated1 to replace the laws of electrodynamics and to be used in the derivation of everything that happens in the atom and so in all nature1 this question must receive a negative answer4 ,the motion of the electron in a given stationary state may1 at any rate to a considerable extent1 be calculated by the laws of mechanics4 ,we do not know1 however1 why certain orbits are1 in this way1 preferred over others1 nor why the electrons jump from outer to inner orbits1 nor why they sometimes go from one stationary orbit to the next and sometimes jump over one or more orbits1 nor why they cannot come any closer to the nucleus than the innermost orbit1 nor why1 in these transitions1 they emit radiation of a frequency determined according to the rules mentioned4 ,it must not be forgotten that in science we must always be patient with the question 8,why80 ,we can never get to the bottom of things4 ,on account of the nature of the problem1 answers cannot be given to the questions why the smallest material particles 7for the time being hydrogen nuclei and electrons7--the elementary physical individuals--exist1 or why the fundamental laws for their mutual relationships--the most elementary relationships existing between them--are of this or that nature2 a satisfactory answer would necessarily refer to something even more elementary4 ,we cannot claim more than a complete description of the relative positions and motions of the fundamental particles and of the laws governing their mutual action and their interplay with the ether4 ,if we examine our knowledge of the atomic processes in the light of this ideal we are tempted1 however1 to consider it as boundless ignorance4 ,we are inconceivably far from being able to give a description of the atomic mechanism1 such as would enable us to follow1 for example1 an electron from place to place during its entire motion1 or to consider the stationary states as links in the whole instead of isolated 8gifts from above40 ,during the transition from one stationary state to another we have no knowledge at all of the existence of the electron1 indeed we do not even know whether it exists at that time or whether it perhaps is dissolved in the ether to be re-formed in a new stationary state4 ,but even if we turn aside from such a paradoxical consideration1 it must be recognized that we do not know what path the electron follows between two stationary orbits nor how long a time the transition takes4 ,as has been done in this book1 the transition is often denoted as a jump1 and many are inclined to believe that the electron in its entire journey from a distant outer orbit to the innermost spends the greatest part of the time in the stationary orbits1 while each transition takes but an infinitesimally short time4 ,this1 however1 in itself does not follow from the theory1 nor is it implied in the expression 8the stationary states40 ,these states may in a certain sense be considered as way stations2 but when we ask whether an electron stays long in the station1 or whether the stationary state is simply a transfer point where the electron changes its method of travelling so that the frequency of its radiation is changed1 these are other matters1 and we cannot here go into the considerations connected with them4 ,to get an idea of some of the difficulties inherent in the attempt to make concrete pictures of the nature of the processes1 let us again consider the analogy between the ,bohr atom of hydrogen and a special kind of musical instrument in which sounds are produced by the fall of a small sphere between discs at various heights 7see p4 #abj74 ,it will be most natural here to think of the sounds as developed by the sphere when it hits the lower disc1 and to think of the tones of higher pitch as given by the harder blows1 corresponding to the larger energy 7determinative of the pitch7 released by the fall4 ,we can1 however1 by no means transfer such a picture to the atomic model4 ,for in the latter we cannot think of the stationary state as a material thing which the electron can hit1 and it is also unreasonable to imagine that the radiation is not emitted until the moment when the transition is over and the electron has arrived in its new stationary state4 ,we must1 on the contrary1 assume that the emission of radiation takes place during the whole transition1 whether the latter consumes a shorter or longer time4 ,if it were the case that a transition always took place between two successive stationary states1 it would then be possible to use the musical instrument to illustrate the matter4 ,let us denote the discs from the lowest one up with the numbers #a1 #b1 #c1 444 corresponding to the stationary states #a1 #b1 #c1 444 and for the moment consider a fall from disc #f to disc #e4 ,we can now imagine that the space between the two discs is in some way tuned for a definite note4 ,thus we might place between the discs a series of sheets of paper having such intervals between them that the sphere in its fall strikes their edges at equal intervals of time1 e4g41 _/ second4 ,the disturbance then set up will produce a sound with the frequency #ajj vibrations per second4 ,if the distance between the discs #e and #d is double that between #f and #e1 the sphere in the fall from #e to #d will lose double the energy lost in the descent from #f to #e1 and will therefore emit a note of double frequency4 ,the sheets of paper in the space between #e and #d must then be packed more tightly than between #f and #e4 ,and so the space between any two discs may thus be said to have its own particular classification or 8tuning40 ,in analogy with this we might think of the space about a hydrogen nucleus divided by the stationary states into sections each with its own 8tuning40 ,but apart from the intrinsic peculiarity of such an arrangement and the particular difficulties it will meet in trying to explain the more complicated phenomena to be mentioned later1 the one fact that the electron in a transition from one stationary state to another can jump over one or more intervening stationary orbits1 makes such a representation impossible4 ,if the sphere in the given example could fall from disc #f to disc #d1 it should during the whole descent emit a note of higher pitch than in the descent from #f to #e4 ,but this could not possibly take place1 if the space from #f to #e1 which must be traversed en route to #d1 is tuned for a lower note4 ,the same consideration applies to the hydrogen atom4 ,naturally it is not impossible to continue the effort to illustrate the matter in some concrete manner 7one might1 for example1 imagine separate channels each with its own particular tuning between the same two discs74 ,but in all these attempts the situation must become more and more complicated rather than more simple4 ,on the whole it is very difficult to understand how a hydrogen atom1 where the electron makes a transition from orbit #f to orbit #d1 can during the entire transition emit a radiation with a frequency different from that when the electron goes from orbit #f to orbit #e4 ,although it seems as if the two electrons in making the transition are at first under identical conditions1 still1 nevertheless1 the one which is going to orbit #d emits from the first a radiation different from that emitted by the one going to orbit #e4 ,even from the very beginning the electron seems to arrange its conduct according to the goal of its motion and also according to future events4 ,but such a gift is wont to be the privilege of thinking beings that can anticipate certain future occurrences4 ,the inanimate objects of physics should observe causal laws in a more direct manner1 i4e41 allow their conduct to be determined by their previous states and the contemporaneous influences on them4 ,there is a difficulty of a similar nature in the fact that from the same stationary orbit the electron sometimes starts for a single jump1 another time for a double jump1 and so on4 ,from certain considerations it is often possible to propound laws for the probability of the different jumps1 so that for a great quantity of atoms it is possible to calculate the strengths 7intensities7 of the corresponding spectral lines4 ,but we can no more give the reason why one given electron at a given time determines to make a double jump while another decides to make a single jump or not to jump at all1 than we can say why a certain radium nucleus among many explodes at a given moment 7cf4 p4 #ajb74 ,this similarity between the occurrence of radiation processes on the ,bohr theory and of the radioactive processes has especially been emphasized by ,einstein4 ,it must1 by no means1 be said that the causal laws do not hold for the atomic processes1 but the hints given here indicate how difficult it will be to reach an understanding--in the usual sense--of these processes and consequently of the processes of physics in general4 ,there is much that might indicate that1 on the whole1 it is impossible to obtain a consistent picture of atomic processes in space and time with the help of the motions of the nuclei and the electrons and the variations in the state of the ether1 and with the application of such fundamental conceptions of physics as mass1 electric charge and energy4 ,even if this were the case1 it does not follow that a comprehensive description in time and space of the physical processes is impossible in principle2 but the hope of attaining such a description must perhaps be allied to the representation of 8physical individuals0 or material particles of an even lower order of magnitude than the smallest particles now known--electrons and hydrogen nuclei--and to ideas of more fundamental nature than those now known2 we are here outside our present sphere of experience4 ,from all the above remarks it would be very easy to get the impression that the ,bohr theory1 while it gives us a glimpse into depths previously unsuspected1 at the same time leads us into a fog1 where it is impossible to find the way4 ,this is very far from being the case4 ,on the contrary1 it has thrown new light on a host of physical phenomena of different kinds so that they now appear in a coherence previously unattainable4 ,that this light is not deceptive follows from the fact that the theory1 which has been gradually developed by ,bohr and many other investigators1 has made it possible to predict and to account for many phenomena with remarkable accuracy and in complete agreement with experimental observation4 ,the fundamental concepts are1 on the one hand1 the stationary states1 where the usual laws of mechanics can be applied 7although only within certain limits71 and1 on the other1 the 8quantum rule0 for transitions between the states4 ,but at the very beginning it has been necessary in many respects to grope in the dark1 guided in part by the experimental results and in part by various assumptions1 often very arbitrary4 ,for ,bohr himself1 a most important guide has been the so-called correspondence principle1 which expresses the previously mentioned connection with the classical electrodynamics4 ,it is difficult to explain in what it consists1 because it cannot be expressed in exact quantitative laws1 and it is1 on this account1 also difficult to apply4 ,in ,bohr's hands it has been extraordinarily fruitful in the most varied fields2 while other more definite and more easily applicable rules of guidance have indeed given important results in individual cases1 they have shown their limitations by failing in other cases4 ,we can here merely indicate what the correspondence principle is4 ,as has been said 7cf4 p4 #acj71 it has been found that in the limiting region 7sufficiently low frequencies7 where the ,bohr theory and the classical electrodynamics are merged in their outward features1 a series of frequencies 1 1 for monochromatic radiation1 emitted by different atoms in the single jumps1 double jumps1 etc41 of the electrons1 are equal to the frequencies 1 #b1 #c 444 which1 according to the laws of electrodynamics1 are contained in each of these atoms respectively as fundamental and the first1 second 444 overtones in the motion of the electron4 ,farther away from this region the two sets of frequencies are no longer equally large1 but it is easy to understand1 from the foregoing1 the meaning of the statement that1 for example1 the radiation of a triple jump with the frequency 8corresponds0 to the second overtone #c in the revolution of the electron4 ,it is this correspondence which ,bohr traces back to the regions where there is even a great difference in two successive orbits and where the frequency produced by a transition between these orbits is very different from the frequencies of revolution in the two orbits or their overtones4 ,he expresses himself as follows3 8,the probability for the occurrence of single1 double1 triple jumps1 etc41 is conditioned by the presence in the motion of the atom of the different constituent harmonic vibrations having the frequency of the fundamental1 first overtone1 second overtone1 etc41 respectively40 ,in order to understand how this 8correspondence10 apparently so indefinite1 can be used to derive important results1 we shall give an illustration4 ,let us assume that the mechanical theory for the revolution of an electron in the hydrogen atom had led to the result that the orbits of the electrons always had to be circles4 ,according to the laws of electrodynamics1 the motion of the electron would in this case never give any overtones1 and1 according to the correspondence principle1 there could not appear among the frequencies emitted by hydrogen any which would correspond to the overtones1 i4e41 there would not be any double jumps1 triple jumps1 etc41 produced1 but the only transitions would be those between successive stationary orbits4 ,the investigation of the spectrum shows1 however1 that multiple jumps occur as well as single jumps1 and this fact may be taken as evidence that the orbits in the hydrogen atom are not usually circles4 ,let us next assume that1 instead1 we had obtained the result that the orbits of the electrons are always ellipses of a certain quite definite eccentricity1 corresponding to certain definite ratios in intensity between the overtones and the fundamentals2 that1 for instance1 the intensity of the classical radiation due to the first overtone is in all states of motion always one-half that due to the fundamental1 the intensity due to the second overtone always one-third that due to the fundamental1 etc4 ,then the radiation actually emitted should1 according to the correspondence principle1 be such that the intensities of the lines corresponding to the double and triple jumps1 which start from a given stationary state1 are respectively one-half and one-third of the intensity corresponding to a single jump from the same state4 ,by these examples we can obtain an idea of how the correspondence principle may in certain cases account for various facts1 as to what spectral lines cannot be expected to appear at all1 although they would be given by a particular transition1 and concerning the distribution of intensities in those which really appear4 ,the illustration given above1 however1 has really nothing much to do with actual problems1 and objections may be raised to the rough way in which the illustration has been handled4 ,the correspondence principle has its particular province in more complicated electron motions than those which appear in the unperturbed hydrogen atom--motions which1 unlike the simple elliptical motion1 are not composed of a series of harmonic oscillations 71 #b1 #c 4447 but may be considered as compounded of oscillations whose frequencies have other ratios4 ,the correspondence principle has1 in such cases1 given rise to important discoveries and predictions which agree completely with the observations4 ,we have dwelt thus long upon the difficult correspondence principle1 because it is one of ,bohr's deepest thoughts and chief guides4 ,it has made possible a more consistent presentation of the whole theory1 and it bids fair to remain the keystone of its future development4 ,but from these general considerations we shall now proceed to more special phases of the problem and examine one of the first great triumphs in which the theory showed its ability to lead the way where previously there had been no path4 ,the ,false ,hydrogen ,spectrum4 ,in #ahig the ,american astronomer1 ,pickering1 discovered in the spectrum of a star1 in addition to the usual lines given by the ,balmer series1 a series of lines each of which lay about midway between two lines of the ,balmer series2 the frequencies of these lines could be represented by a formula which was very similar to the ,balmer formula2 it was necessary merely to substitute n #c1 #d1 #e1 etc41 in the formula on p4 #eg instead of n #c1 #d1 #e1 etc4 ,it was later discovered that in many stars there was a line corresponding to n _/ or n #b in the usual ,balmer-,ritz formula 7p4 #ei74 ,it was considered that these must be hydrogen lines1 and that the spectral formula for this element should properly be written #a #a ,k ---------- - --------- 7 n_/#b 7 7 n_/#b 7 where n and n can assume integral values4 ,this was done since it was not to be believed that the spectral properties of chemically different elements could be so similar4 ,this view was very much strengthened when ,fowler1 in #aiab1 discovered the ,pickering lines in the light from a vacuum tube containing a mixture of hydrogen and helium4 ,it could not quite be understood1 however1 why the new lines did not in general appear in the hydrogen spectrum4 ,according to the ,bohr theory for the hydrogen spectrum it was impossible--except by giving up the agreement 7cf4 p4 #abi7 with electrodynamics in the region of high orbit numbers--to attribute to the hydrogen atom the emission of lines corresponding to a formula where the whole numbers were halved4 ,the formula given above might1 however1 also be written as 7 #a #a 7 #d,k7------ - ------74 7 n n 7 ,if the earlier calculations had been carried out a little more generally1 i4e41 if instead of equating the nuclear charge with #a elementary electric quantum e1 as in hydrogen1 it had been equated with ,ne where ,n is an integer1 then the frequency might have been written as 7 #a #a 7 ,n,k7------ - ------74 7 n n 7 ,this formula is evidently the same as that just given when ,n equals #b4 ,now we know that helium has the atomic number and nuclear charge #b 7cf4 p4 #ij72 a normal neutral helium atom has two electrons and it is1 therefore1 very different from a hydrogen atom4 ,if1 however1 a helium atom has lost one electron and therefore has become a positive ion with one charge1 it is a system like the hydrogen atom with only one single electron moving about the nucleus4 ,it differs in its 8outer0 characteristics from the hydrogen atom only in having a nuclear charge twice as great1 i4e4 its spectral formula must be given with ,n #b1 or ,n #d4 ,the formula for the supposed hydrogen lines would consequently fit the case of a helium atom which has lost an electron4 ,bohr was aware of this1 and he therefore suggested that the lines in question were due1 not to hydrogen1 but to helium4 ,at first all the authorities in the field of spectroscopy were against this view2 but most of the doubt was dispelled when ,evans showed that the lines could be produced in a vacuum tube where there was only helium with not a trace of hydrogen4 ,in a letter to ,nature in ,september #aiac1 ,fowler objected to the ,bohr theory on the ground that the disputed line-formula did not exactly correspond to the formula with #d,k1 but that there was a slight disagreement4 ,bohr's answer was immediate4 ,he called attention to the fact that--since temporarily he had sought only a first approximation--in his calculations he had taken the mass of the nucleus to be infinite in comparison to the mass of the electron1 so that the nucleus could be considered exactly at the focus of the ellipse described by the electron4 ,in reality1 he said1 it must be assumed that nucleus and electron move about their common centre of gravity1 just as in the motion in the solar system it must be assumed that not the centre of the sun1 but the centre of gravity of the entire system remains fixed4 ,this motion of the nucleus leads to the introduction of a factor ,m_/7,m m7 in the expression for the constant ,k given on p4 #abi1 where ,m is the mass of the nucleus and m that of the electron1 which in hydrogen is that of the nucleus4 ,in helium1 ,m is four times as large as in hydrogen1 so that the given factor here has a slightly different value4 ,the difference in the values for ,k for the hydrogen and for the helium spectrum which was found by ,fowler1 is #jjd per cent41 which agrees exactly with the theoretical difference4 ,bohr thus turned ,fowler's objection into a strong argument in favour of the theory4 ,the ,introduction of more than one ,quantum ,number4 ,during the first years after #aiac1 ,bohr was practically alone in working out his theory1 at that time still assailed by many1 and in showing its application to many problems4 ,in #aiaf1 however1 the theorists in other countries1 led by the well-known ,munich professor1 ,sommerfeld1 began to associate themselves with the ,bohr theory1 and their investigations gave rise to much essential progress4 ,we shall here mention some of the most important contributions4 ,in the theory for the hydrogen spectrum propounded above1 it was assumed that we had to do with a single series of stationary orbits1 each characterized by its quantum number4 ,but as shown by theoretical investigations each of the stationary orbits must1 when more detail is asked for1 also be indicated by an additional quantum number4 ,illustration3 ,,fig4 #bf4--;,a compound electron motion produced by the very rapid rotation of an elliptical orbit4 ,this is closely connected with the fact that the motion of the electron is not quite so simple as previously assumed4 ,we have assumed that the electron moves about the nucleus just as a planet moves about the sun 7according to ,kepler's ,laws71 in an ellipse with the sun at one focus1 since the electron is influenced by an attraction inversely proportional to the square of the distance1 just as the planets are attracted by the sun according to ,newton's ,law4 ,we must1 however1 remember that we are here concerned with the electric attraction which at a given distance is determined1 not by mass1 but by the electric charges in question4 ,if the latter remain unchanged1 while the mass of the electron varies1 the motion will be changed1 because the same force has less effect upon a greater mass4 ,according to the ,einstein principle of relativity1 the mass of the electrons1 in accordance with ideas expounded long ago by ,j4 ,j4 ,thomson1 will not be constant1 but to a certain extent depend upon the velocity1 which will vary from place to place1 when the orbit is an ellipse4 ,as a result of this1 the motion becomes a central motion of more general nature than a ,kepler ellipse4 ,since the influence of the change of mass is very small1 the orbit can still be considered as an approximate ,kepler ellipse2 but the major axis will slowly rotate in the plane of the orbit4 ,in reality1 the orbit will not therefore be closed1 but will have the character which is shown in ,fig4 #bf2 this1 however1 corresponds to a much more rapid rotation of the major axis than that which actually takes place in the hydrogen atom1 where--even in case of the swiftest rotation--the electron will revolve about #dj1jjj times round the nucleus at the same time as the major axis turns round once4 ,if the electron moves in a fixed ,kepler ellipse1 the energy content of the atom will be determined by the major axis of the ellipse only4 ,if these axes for the stationary states with quantum numbers #a1 #b1 #c 444 are respectively denoted by #b;a1 #b;a1 #b;a 444 the frequency1 for instance1 in the transition from ,no4 #c to ,no4 #b--since it is determined by the loss of energy--will be the same whether the orbits are circles or ellipses4 ,if1 on the other hand1 the electron moves in an ellipse which itself rotates slowly1 the energy content1 as can be shown mathematically1 will depend not only upon the major axis of the ellipse1 but also to a slight degree upon its eccentricity1 or1 in other words1 on its minor axis4 ,then in the transition #c-b we shall get different energy losses and consequently different frequencies1 according as the ellipse is more or less elongated4 ,if it were the case that the eccentricity of the ellipses for a certain quantum number could take arbitrary values1 then in the transition between two numbers we could get frequencies which may take any value within a certain small interval1 i4e41 a mass of hydrogen with its great quantity of atoms would give diffuse spectral lines1 i4e4 lines which are broadened over a small continuous spectral interval4 ,this is1 however1 not the case2 but long before the appearance of the ,bohr theory it had been discovered that the hydrogen lines1 which we hitherto have considered as single1 possess what is called a fine structure4 ,with a spectroscopic apparatus of high resolving power each line can be separated into two lying very close to each other4 ,this fine structure can now be explained by the fact that in a stationary state with quantum number #c and major axis of the orbit #b;a1 for instance1 the eccentricity of the orbit has neither one single definite value nor all possible values1 but1 on the contrary1 it has several discrete values of definite magnitude1 to which there correspond slightly different but definite values of the energy content of the atom4 ,it is now possible to designate the series of stationary orbits1 which have the major axis #b;a with the principal quantum number #c1 with subscripts giving the auxiliary quantum number for stationary orbits corresponding to the different eccentricities1 so that the series is known as #c1 #c1 #c4 ,instead of a single line corresponding to the transition #c-b1 there are then obtained several spectral lines lying closely together and corresponding to transitions such as #c-b1 #c-b1 etc4 ,by theoretical considerations1 requiring considerable mathematical qualifications1 but of essentially the same formal nature as those ,bohr had originally applied to the determination of the stationary orbits in hydrogen1 ,sommerfeld was led to certain formal quantum rules which permit the fixing of the stationary states of the hydrogen atom corresponding to such a double set of quantum numbers4 ,the results he obtained as regards the fine structure of the hydrogen lines agree with observation inside the limit of experimental error4 ,although ,sommerfeld's methods have also been very fruitful when applied to the spectra of other elements1 they were still of a purely formal and rather arbitrary nature2 it is1 therefore1 of great importance that the ,leiden professor1 ,ehrenfest1 and ,bohr succeeded later in handling the problem from a more fundamental point of view1 ,bohr making use of the correspondence principle previously mentioned4 ,it should be said here1 by way of suggestion1 that ,bohr used the fact that the motions of the electrons are not simple periodic but 8multiple periodic40 ,we see this most simply if we think of the revolution of the electrons in the elliptical orbit as representing one period1 and the rotation for the major axis of the ellipse as representing a second period4 ,illustration3 ,,fig4 #bg4--,the model of the hydrogen atom with stationary orbits corresponding to principal quantum numbers and auxiliary quantum numbers4 ,fig4 #bg shows a number of the possible stationary orbits in the hydrogen atom according to ,sommerfeld's theory2 for the sake of simplicity the orbits are drawn as completely closed ellipses4 ,if we examine1 for instance1 the orbits with principal quantum number #d1 we have here three more or less elongated ellipses1 #d1 #d1 #d1 and the circle #d2 in all of them the major axis has the same length1 and the length of the major axis is to that of the minor axis as the principal quantum number is to the auxiliary quantum number 7for the circle #d 3 #d #a74 ,on the whole1 to a principal quantum number n there correspond the auxiliary quantum numbers #a1 #b 444 n1 and the orbit for which the auxiliary quantum number equals the principal quantum number is a circle4 ,we see that in the more complicated hydrogen atom model there is possibility for a much greater number of different transitions than in the simple model 7,fig4 #be1 p4 #aai74 ,some of the transitions are indicated by arrows4 ,since the energy content of the atom is almost the same for orbits with the same principal quantum number and different auxiliary quantum numbers1 three transitions like #c-b1 #c-b and #c-b will give about the same frequency1 and therefore spectral lines which lie very close together4 ,in a transition like #d-d the emitted energy quantum h1 and also 1 will be so extremely small that the corresponding line will be too far out in the infra-red for any possibility of observing it4 ,it must be pointed out that the above considerations only hold if the hydrogen atoms1 strictly speaking1 are undisturbed4 ,thus1 very small external forces1 which may be due to the neighbourhood of other atoms1 etc41 will be sufficient to cause changes in the eccentricity of the stationary orbits4 ,in such a case the above definition of the auxiliary quantum number becomes obviously illusory1 and the original character of the fine structure disappears4 ,this is in agreement with the experiments1 since the ,sommerfeld fine structure can be found only when the conditions in the discharge tube are especially quiet and favourable4 ,influence of ,magnetic and ,electric ,fields on the ,hydrogen ,lines4 ,illustration3 ,,fig4 #bh4--,the splitting of three hydrogen lines under the influence of a strong electric field4 ,as previously mentioned 7p4 #gf71 the spectral lines are split into three components when the atoms emitting lights are exposed to magnetic forces4 ,the agreement found here between observation and the ,lorentz electron theory was considered as strong evidence of the correctness of the latter4 ,according to the ,bohr theory1 the picture upon which this explanation rested must be abandoned entirely2 but fortunately it has been shown that the ,bohr theory leads to the same results2 and1 moreover1 ,bohr1 with the assistance of the correspondence principle1 has been able to set forth the more fundamental reason for this agreement4 ,the ,german scientist1 ,stark1 showed1 in #aiab1 that hydrogen lines are also split by electric fields of force4 ,in ,fig4 #bh it is shown how very complicated this phenomenon is2 here the classical electron theory could not at all explain what happened4 ,this phenomenon could also be accounted for by the extended ,bohr theory 7with the introduction of more than one quantum number71 as it was shown independently by ,epstein and by ,schwarzschild in #aiaf2 further1 the correspondence principle has again shown its superiority1 since it makes possible an approximate determination of the different intensities of the different lines4 ,a calculation carried out by ,h4 ,a4 ,kramers has shown that the theory gives a remarkably good agreement with the experiments4 ,not until we think of the extraordinary accuracy of the measurements which are obtained by spectrum analysis1 can we thoroughly appreciate the importance of the quantitative agreement between theory and observation in the hydrogen spectrum that has just been mentioned4 ,moreover1 we must remember how completely helpless we previously were in the strange puzzles offered even by the simplest of all spectra1 that of hydrogen4 ,,chapter ,,vi ,,various ,,applications ,,of ,,the ,,bohr ,,theory ,introduction4 ,we have dwelt at length upon the theory of the hydrogen spectrum because it was particularly in this relatively simple spectrum that the ,bohr theory first showed its fertility4 ,moreover1 by studying the case of the hydrogen atom with its one electron1 it is easier to gain insight into the fundamental ideas of the ,bohr theory and its revolutionary character4 ,naturally1 the theory is limited neither to the hydrogen atom nor to spectral phenomena1 but has a much more general application4 ,as has already been said1 it takes1 as its problem1 the explanation of every one of the physical and chemical properties of all the elements1 with the exception of those properties known to be nuclear 7cf4 p4 #id74 ,this very comprehensive problem can naturally1 even in its main outlines1 be solved but gradually and by the co-operation of many scientists1 and it is quite impossible to go very deeply into the great work which has already been accomplished1 and into the difficulties which ,bohr and the others working on the problem have overcome4 ,we must be content with showing some especially significant features4 ,different ,emission ,spectra4 ,while the neutral hydrogen atom consists simply of a positive nucleus and one electron revolving about the nucleus1 the other elements1 in the neutral state1 have from two up to #ib electrons in the system of electrons revolving around the nucleus4 ,even #b electrons1 as in the helium atom1 make the situation far more complicated1 since we have in this case a system of #c bodies which mutually attract or repel each other4 ,we are thus confronted with what1 in astronomy1 is known as the three-body problem1 a problem considered with respect by all mathematicians on account of its difficulties4 ,in astronomy1 the difficulties are restricted very much when the mass of one body is many times greater than that of the others1 as in the case of the mass of the sun in relation to that of the other planets4 ,here1 by comparatively simple methods1 it is possible to calculate the motions inside a finite time-interval with a high degree of approximation even when there are not two but many planets involved4 ,we might now be tempted to believe that in the atom we had to deal with comparatively simple systems--solar systems on small scale--since the mass of the nucelus is many times greater than that of the electrons4 ,but even if the suggested comparison illustrates the position of the nucleus as the central body which holds the electrons together by its power of attraction1 the comparison in other respects is misleading4 ,while the orbits of the planets in the solar system may be at any distance whatsoever from the sun1 and the motions of the planets are everywhere governed by the laws of mechanics1 the atomic processes1 according to the ,bohr theory1 are characterized by certain stationary states1 and only in these can the laws of mechanics possibly be applied4 ,but in addition1 the forces between nucleus and electrons are determined not at all by the masses1 but rather by the electric charges4 ,in the helium atom the nuclear charge is only double that of an electron1 and the attraction of the nucleus for an electron will therefore be only twice as large as the repulsions between two electrons at the same distance apart4 ,this repulsion under these circumstances will1 therefore1 also have great influence on the ensuing motion4 ,in elements with higher atomic numbers the nuclear charge has greater predominance over the electron charges2 but1 on the other hand1 there are then more electrons4 ,the situation is in each case more complicated than in the hydrogen atom4 ,nevertheless1 the line spectra of the elements of higher atomic number show how the lines1 as in the hydrogen spectrum1 are arranged in series although in a more complicated manner 7cf4 p4 #ei72 in any case in many instances there is great similarity between the radiation from the hydrogen atom and that from the more complicated atoms4 ,thus in the line spectra of many elements1 just as in that of hydrogen1 the frequency of every line can be expressed as a difference between two terms1 involving certain integers which can pass through a series of values4 ,from the combinations of terms1 two at a time1 the values of corresponding to the different spectral lines can be derived4 ,this so-called combination principle enunciated by the ,swiss physicist1 ,ritz1 can evidently be directly interpreted on the basis of ,bohr's postulates1 since the different combinations may be assumed to correspond to definite atomic processes1 in which there is a transition between two stationary states1 each of which corresponds to a spectral term4 ,moreover1 the terms 7cf4 p4 #ei7 may often be approximately given by the ,rydberg formula ,k ---------- 7n 7 where ,k has about the same value as in hydrogen1 and can take on a series of values 1 444 1 while n takes on integer values4 ,since we thus determine the different lines by assigning values to the two integers n and k in each term1 we have in this respect something like the fine structure in the hydrogen spectrum1 where the stationary states are determined by a principal quantum number and an auxiliary quantum number4 ,the spectra of which we are speaking here1 and for which the terms have the form given above1 are often called arc spectra1 because they are emitted particularly in the light from the electric arc or from the vacuum tube4 ,we must expect that the similarity which exists in the law for the distribution of spectral lines will correspond to a similarity in the atomic processes of hydrogen and the other elements4 ,the hydrogen atom emits radiation corresponding to the different spectral lines when an electron from an outer stationary orbit jumps1 with a spring of varying size1 to an orbit with lower number1 and at last finds rest in the innermost orbit in a normal state1 where the energy of the atom is as small as possible4 ,similarly1 we must assume that the electrons in other atoms1 during processes of radiation1 may proceed in towards the nucleus until they are collected as tightly as possible about the nucleus1 corresponding to the normal state of the atom1 where its energy content is as small as possible3 8capture0 of electrons by the nucleus4 ,the region in space which1 in the normal state1 includes the entire electron system1 must be assumed to be of the same order of magnitude as the dimensions of the atom and molecule which are derived from the kinetic theory of gases4 ,this normal state may be called a 8quiescent0 state1 since the atom cannot emit radiation until it has been excited by the introduction of energy from without4 ,this excitation process consists of freeing one 7or more7 electrons1 in some way or other1 from the normal state and either removing it out to a stationary orbit farther away from the nucleus or ejecting it completely from the atom4 ,not all electrons can be equally easily removed from the quiescent state4 ,those moving in small orbits near the nucleus will be tighter bound than those moving in larger orbits farther from the nucleus4 ,the arc spectrum is now caused by driving one of the most loosely bound electrons out into an orbit farther from the nucleus or removing it completely from the atom4 ,in the latter case the rest of the atom1 which with the loss of the negative electron becomes a positive ion1 easily binds another electron1 which1 with the emission of radiation1 corresponding to lines of the series spectrum1 can approach closer to the nucleus4 ,let us now assume1 first1 that this radiating electron moves at so great a distance from the nucleus and the other electrons that the entire inner system can be considered as concentrated in one point2 then the situation is quite as if we had to deal with a hydrogen atom4 ,if the atomic number is as high as #bi 7copper71 for instance1 the nuclear charge will consist of twenty-nine elementary quanta of positive electricity2 but since there are twenty-eight electrons in the inner system1 the resultant effect is that of only one elementary quantum of positive electricity1 as in the case of a hydrogen nucleus4 ,the spectral lines which are emitted in the jumps between the more distant paths will be practically the same as hydrogen lines4 ,but1 since in the jumps between these distant orbits1 very small energy quanta will be emitted1 the frequencies are very small1 the wave-lengths very great1 i4e41 the lines in question lie far out in the infra-red4 ,illustration3 ,,fig4 #bi4--;,different stationary orbits which the outermost 7#aath7 electron of sodium may describe4 ,when the electron has come in so close to the nucleus that the distances in the inner system cannot be assumed to be small in comparison to the distance of the outer electron from the nucleus1 the situation is changed4 ,the force with which the nucleus and the inner electrons together will work upon the outer electron will be appreciably different from the inverse square law of attraction of a point charge4 ,the consequence of this difference is that the major axis in the ellipse of the electron rotates slowly in the plane of the orbit as described in case of the theory of the fine structure of the hydrogen lines 7cf4 p4 #adf71 and even if the cause is different the result is the same2 the orbit of the outer electron in the stationary states will be characterized by a quantum number n and an auxiliary quantum number k4 ,if the electron comes still closer to the nucleus1 its motion is even more complicated4 ,when the electron in its revolution is nearest the nucleus it will be able to dive into the region of the inner electrons1 and we can get motions like those shown in ,fig4 #bi for one of the eleven sodium electrons4 ,the inner dotted circle is the boundary of the inner system which is given by the nucleus and the ten electrons remaining in the 8quiescent0 state--little disturbed by the restless ,no4 #aa4 ,in the figure we can see greater or smaller parts of ,no4 #aa's different stationary orbits with principal quantum numbers #c and #d4 ,we shall not account further for the different orbits and the spectral lines produced by the transitions between orbits1 but shall merely remark that the yellow sodium line1 which corresponds to the ,fraunhofer ,d-line 7cf4 p4 #di71 is produced by the transition #c-c1 between two orbits with the same principal quantum number4 ,the sketch shows to a certain degree how fully many details of the atomic processes can already be explained4 ,the theory can even give a natural explanation of why the ,d-line is double4 ,we have restricted ourselves to the case where only one electron is removed from the normal state of the neutral atom4 ,it may1 however1 happen that two electrons are ejected from the atom so that it becomes a positive ion with two charges4 ,when an electron from the outside is approaching this doubly charged ion it will1 at a distance1 be acted upon as if the ion were a helium nucleus with two positive charges4 ,the situation1 in other words1 will be as in the case of the false hydrogen spectrum 7cf4 p4 #adb71 where the constant ,k in the formula for the hydrogen spectrum is replaced by another which is very close to #d,k4 ,but if the atom is not one of helium1 but one with a higher atomic number1 the stationary orbits of the outer electron which approach closely to the nucleus will not coincide exactly with those in the ionized helium atom1 corresponding to the fact that the terms in the formula for the spectrum1 instead of the simple form #d,k_/n1 have the more complicated form #d,k_/7n 74 ,spectra of this nature are often called spark spectra1 since they appear especially strong in electric sparks2 they appear also in light from vacuum tubes1 when an interruptor is placed in the circuit1 making the discharge intermittent and more intense4 ,an atom with several electrons can1 however1 be much more violently excited from its quiescent state when an electron in the inner region of the atom is ejected by a swiftly moving electron 7a cathode ray particle or a -particle from radium7 which travels through the atom4 ,such an invasion produces a serious disturbance in the stability of the electron system2 a reconstruction follows1 in which one of the outer1 more loosely bound electrons takes the vacant position4 ,in the transitions1 in which these outer electrons come in1 rather large energy quanta are emitted4 ,the emitted radiation has therefore a very high frequency2 monochromatic ,x-rays are thus emitted4 ,since these have their origin in processes far within the atom1 it can be understood that the different elements have different characteristic ,x-ray spectra1 which can give very valuable information about the structure of the electron system 7cf4 p4 #ia74 ,between these ,x-ray spectra and the series spectra previously mentioned there lie1 as connecting links1 those spectra which are produced when electrons are ejected from a group in the atom which does not belong to the innermost group1 but does not1 on the other hand1 belong in the outermost group in the normal atom4 ,we have very little experimental knowledge about such spectra1 because the spectral lines involved have wave-lengths lying between about #ae and #ajj 4 ,rays with these wave-lengths are absorbed very easily by all possible substances2 they have very little effect on photographic plates1 where they are absorbed by the gelatine coating before they have an opportunity to influence the molecules susceptible to light4 ,but there can be scarcely any doubt that1 in the course of a few years1 experimental technique will have reached such efficiency that this domain of the spectrum1 so important for the atomic theory1 will also become accessible to experiment4 ,in individual cases1 wave-lengths as small as #bj have already been obtained by ,millikan4 ,of entirely different character from these spectra are the band spectra4 ,they are in general produced by electric discharges through gases which are not very highly attenuated 7cf4 p4 #ee72 they are not due to purely atomic processes1 but can be designated as molecular spectra4 ,their special character is due to motions in the molecule1 not only motions of the electrons1 but also oscillations and rotations of the nuclei about each other4 ,we shall not go into these problems here2 in what follows we shall investigate a certain type of band spectra somewhat more closely in connection with the absorption of radiation4 ,while the band spectra with a spectroscope of high resolving power can be more or less completely resolved into lines1 this is not the case with the continuous spectra4 ,they are emitted not only by glowing solids 7cf4 p4 #ed71 but also by many gaseous substances4 ,when such gases are exposed to electric discharges they emit1 in addition to the line spectra and band spectra1 continuous spectra which in certain parts of the spectrum furnish a background for bright lines which come out more strongly4 ,it might seem impossible to correlate these with the ,bohr theory2 but in reality a spectrum does not always have to consist of sharp lines4 ,this can at once be seen from the correspondence principle4 ,if the motions in the stationary states are of such nature that they can be resolved into a number of discrete harmonic oscillations each with its own period 7for instance the orbit of an electron in a rotating ellipse2 cf4 p4 #adi71 then1 according to the correspondence principle1 in the transition between two such stationary states there are produced sharp spectral lines 8corresponding0 to these harmonic components4 ,but not all motions of atomic systems can be thus resolved into a number of definite harmonic oscillations4 ,when this cannot be done1 the stationary states cannot be expected to be such that transitions between them produce radiation which can be resolved into sharp lines4 ,a simple example1 where it is easily intelligible that the ,bohr theory will not lead to sharp lines1 is obtained in a simple consideration of the hydrogen atom4 ,let us examine the lines belonging to the ,balmer series which are produced when an electron passes to the ,no4 #b orbit from an orbit with higher orbit number1 which is farther from the nucleus4 ,as has been said1 we obtain here an upper limit for the frequency corresponding to a value of the outer orbit number which is infinite2 this means1 in reality1 that the electron in one jump comes in from a distance so great that the attraction of the nucleus is infinitely small4 ,the energy released by such a jump is the same as the ionizing energy ,a which is required to eject the electron from the orbit ,no4 #b and drive it from the atom4 ,it is here assumed1 however1 that the electron out in the distance was practically at rest4 ,if the captured electron has a certain initial velocity outside1 it will have a corresponding kinetic energy ,a4 ,when in one jump this electron comes from the outside into orbit ,no4 #b1 the energy lost by the electron and emitted in the form of radiation will be the sum of the ionizing energy ,a and the original kinetic energy ,a4 ,the frequency will then become greater than that corresponding to ,a2 and since the velocity of the electron before it is captured is not restricted to certain definite values1 neither is the value of 4 ,the radiation from a great quantity of hydrogen atoms which are binding electrons in this way will1 in the spectrum1 not be concentrated in certain lines1 but will be distributed over a region in the ultra-violet which lies outside of the limit calculated from the ,balmer formula2 still in a certain sense this continuous spectrum is correlated with the ,balmer series4 ,in the spectra from certain stars there has actually been discovered a continuous spectrum1 which lies beyond the limits of the ,balmer series and may be said to continue it4 ,also the ,x-rays1 which are generally used in medicine1 have varying frequencies2 this is caused by the fact that some of the electrons which1 in an ,x-ray tube1 strike the atoms of the anticathode and travel far into it at a high speed1 lose a part or all of their velocity without ejecting inner electrons4 ,the lost kinetic energy then appears directly as radiation4 ,these remarks ought to be sufficient to show that the radiation1 for instance1 from a glowing body1 where the interplay of atoms and molecules is very complicated1 can give a continuous spectrum4 ,electron ,collisions4 ,the excitation of an atom in the normal state 7cf4 p4 #aeg71 by which one of its electrons is removed to an outer stationary orbit1 may be caused by a foreign electron which strikes the atom4 ,a study of collisions between atoms and free electrons is therefore of the greatest importance when investigating more closely the conditions by which series spectra are produced4 ,these investigations can be carried out by giving free electrons definite velocities by letting them pass through an electric field1 where the 8difference of potential0 is known in the path traversed by the electrons4 ,when an electron moves through a region with a difference of potential of one volt 7the usual technical unit71 the kinetic energy of the electron will be increased by a definite amount 7of #af #aj erg74 ,if its initial velocity is zero1 its passage through this field will make the velocity #fjj km4 per second2 if the potential difference were #d volts1 #i volts1 etc41 the velocity obtained by the electron would be #b1 #c1 etc41 times larger4 ,for the sake of brevity we shall say that the kinetic energy of an electron is1 for instance1 #ae volts1 when we mean that the kinetic energy is as great as would be given by a difference of potential of #ae volts4 ,in #aiac the ,german physicist ,franck began a series of experiments by methods which made it possible to regulate accurately the velocity of the electrons1 and to determine the kinetic energy before and after collisions with atoms4 ,he first applied the methods to mercury vapour1 where the conditions are particularly simple1 since the mercury molecules consist of only one atom4 ,franck bombarded mercury vapour with electrons all of which had the same velocity4 ,he then showed that if the kinetic energy of the electrons was less than #di volts the collisions with the atoms were completely 8elastic10 i4e41 the direction of the electron could be changed by the collision1 but not its velocity4 ,if1 however1 the velocity of the impinging electrons was increased so much that it was somewhat larger than #di volts1 there was an abrupt change in the situation1 since many of the collisions became completely inelastic1 i4e41 the colliding electron lost its entire velocity and gave up its entire kinetic energy to the atom4 ,if the initial velocity was even greater1 so that the kinetic energy of the colliding electron was #f volts1 for instance1 then when the collision took place there would always be lost a kinetic energy of #di volts1 since the electrons would either preserve their kinetic energy intact or have it reduced to #aa volt 7cf4 ,fig4 #cj74 ,illustration3 ,,fig4 #cj4--,schematic drawing of ,franck's experiment with electron collisions4 ,g is a glowing metal wire which emits electrons4 ,if between ,g and the wire net ,t there is a difference of potential of #f volts1 the electrons will pass through the holes of the net with great velocity out into the space ,r1 where there is mercury vapour4 a represents a free electron ,f and a mercury atom ,hg before the collision1 while b represents them after the collision2 with the collision ,f loses a kinetic energy corresponding to #di volts2 at the same time a bound electron ,b in the atom goes over to a larger stationary orbit4 ,this remarkable phenomenon can be understood from the ,bohr theory if we assume that to send the most loosely bound electron in the mercury atom out to the nearest outer stationary orbit there is required an energy of #di volts1 since in that case1 according to the first postulate1 an energy of less than this magnitude cannot be absorbed by the atom4 ,the use of the word 8understanding0 must here be qualified2 if the forces which influence the free electron as it comes into the electron system of the mercury atom are no other than the usual repulsion from the electrons and the attraction from the nucleus1 the conduct of the colliding electron can in no way be explained by the laws of mechanics4 ,but what happens is in agreement with the characteristic stability of the stationary states1 and ,bohr had prophesied how it would happen4 ,curiously enough ,franck believed in the beginning that his experiment disagreed with the ,bohr theory because he made the mistake of supposing that what happened was merely ionization1 i4e41 complete disruption of a bound electron from a mercury atom4 ,franck's experiments showed1 moreover1 that mercury vapour1 as soon as the inelastic collisions appeared1 began to emit ultra-violet light of a definite wave-length1 namely1 #becg 4 ,the product of the frequency of this light and ,planck's constant h agrees exactly with the energy quantum possessed by an electron which has passed a potential difference of #di volts2 but this also agrees with what might be expected1 according to the ,bohr theory1 from the radiation the removed electron would emit upon returning to the normal state4 ,the energy which is respectively absorbed and emitted in the two transitions must be indeed h4 ,since an electron can not only be driven out to the next stationary orbit1 but also to an even more distant one 7or entirely ejected7 and thence can come in again in one or more jumps1 it is evident that a far more complicated situation may arise4 ,the ,franck experiment1 which now has been extended to many other elements1 clearly gives extraordinarily valuable information in such cases4 ,in mercury it has been found that the energy a free electron must have in order to eject an electron from an atom and turn the atom into a positive ion1 corresponds to a difference of potential of #ajh volts1 a value which ,bohr had predicted4 ,at the same time that ,franck's experiments1 in this respect and in others1 have strengthened the ,bohr theory in the most satisfactory way1 they have also advanced its development very much4 ,indeed it may be said that they have been of the greatest help in atomic research4 ,even if the spectroscope has greater importance1 the investigations on electron collisions make the realities in the ,bohr theory accessible to study in a more direct and palpable manner4 ,illustration3 ,,fig4 #ca4--,stratification of light in a vacuum tube4 ,the peculiarities in the electron collisions appear most clearly in an old and well-known phenomenon of light1 namely1 the stratification of the light in a vacuum tube 7,fig4 #ca74 ,this stratification1 which previously seemed so incomprehensible1 agrees exactly with the feature so fundamental in the atomic theory that a free electron cannot give energy under a certain quantum to an atom4 ,we can imagine that1 in the non-illuminated central space between the bright strata1 the electrons each time under the influence of the outer electric field obtain the amount of kinetic energy which must serve to excite the atoms of the attenuated vapour4 ,as has been said 7p4 #afa71 electron collisions may cause the emission of characteristic ,x-rays2 but to produce them very great energy is required4 ,therefore the electrons which are to produce this effect must have an opportunity to pass freely through a certain region under the influence of a proportionately strong electric field 7with potential of from #ajjj to #ajj1jjj volts and more74 ,the electrons find such a field in a highly exhausted ,x-ray tube1 where the electrons under strong potential are driven from the cathode against the anticathode1 into which they penetrate deeply4 ,absorption4 ,in the experiments previously described it was the electron collisions which furnished the energy required to excite the atoms1 i4e41 to carry them from the normal state over into a stationary state with greater energy4 ,this 8excitation energy0 may1 however1 also be furnished to the atoms in the form of radiation energy2 we shall now examine this case more closely4 ,let us assume that to transfer an atom from the normal state to another stationary state1 or1 in other words1 to transfer one of the electrons to an outer stationary orbit1 a certain quantity of energy ,e is demanded2 then the radiation emitted by the atom when it returns to the normal state will have a frequency depending upon the relation ,e h or ,e_/h1 where h1 as usual1 is the ,planck constant4 ,but just as the atom in the transition from the stationary state to the normal state can emit radiation only with the definite frequency 1 then the opposite transition can only be performed by absorption of radiation with the same frequency2 when this happens the absorbed radiation energy has exactly the value ,eh4 ,this reciprocity1 which may be considered as a direct consequence of the ,bohr postulates1 agrees with what has been said 7cf4 p4 #ej7 about the correspondence between the lines in the line spectrum of an element and the dark absorption lines of that element--e4g41 the ,fraunhofer lines in the solar spectrum4 ,let us examine1 as an example1 the yellow sodium line1 the ,d-line4 ,light with the corresponding frequency1 #ebf #aj vibrations per second1 is emitted by a sodium atom1 when the loosest bound electron goes over from a stationary orbit with quantum numbers #c to the orbit #c1 which belongs to the normal state of the sodium atom4 ,the transition in the opposite direction1 #c to #c1 can take place under absorption of radiation only when in the light from some other source of light1 which passes the sodium atoms1 there are found rays with the frequency #ebf #aj4 ,even if there is present radiation energy with some other frequency1 the sodium atoms take no notice of this energy2 they absorb only rays with the frequency stated1 and every time an atom absorbs energy from a ray the energy taken is always an energy quantum of the magnitude h1 i4e4 about #fed #aj #ebf #aj #cdd #aj ergs 7#a erg is the unit of energy used in the determination of h74 ,when there are present a large number of sodium atoms 7as1 for instance1 in the previously mentioned common salt flame71 the transition #c to #c can take place in some atoms1 the transition #c to #c in others2 therefore1 at the same time there can be absorption and radiation of the light in question4 ,whether absorption or radiation at any given time has the upper hand depends upon various conditions 7temperature1 etc474 ,for the sake of simplicity we have here tacitly understood that there can be but one definite transition 7from the normal state7 corresponding to the assumption that the sodium spectrum had no other lines than the ,d-line4 ,in reality this is not the case1 and there can equally occur absorption of rays with larger frequencies belonging to other spectral lines in the sodium atom and corresponding to other possible transitions between stationary states in the sodium atom4 ,if the temperature of the sodium vapour is sufficiently low1 in which case almost all the atoms are in the normal state1 it is evident that in the absorption only those lines will appear which correspond to transitions from the normal state1 and which therefore form only a part of all the lines of the sodium spectrum4 ,we thus obtain an explanation of the previously enigmatical circumstance that not all spectral lines which can appear in emission will be found in absorption4 ,at the same time we get1 in absorption experiments1 valuable information about the structure of the atom beyond what the observations in the emission spectra are able to give4 ,interesting phenomena may arise owing to the fact that the jumps between the stationary states of the atom sometimes1 as we know1 take place in single jumps1 sometimes in double or multiple jumps1 so that the intermediate stationary states are jumped over4 ,there is then evidently a possibility that absorption can take place1 for instance1 with a double jump of an electron1 which may later return to the original stationary orbit in two single jumps4 ,the absorbed radiation energy will then appear in emission with two frequencies which are entirely different from the frequency of the absorbed rays 7this latter in this case will be the sum of the other two74 ,when an element is illuminated with a certain kind of rays1 it can1 in other words1 emit in return rays of a different nature4 ,such changes of frequencies have also been observed in experiment2 they contain1 in principle1 an explanation of the characteristic phenomenon called fluorescence4 ,we shall not go further into this problem1 but dwell for a time on the characteristic phenomenon of absorption which is known as the photoelectric effect4 ,in this phenomenon 7cf4 p4 #aaf7 a metal plate1 by illumination with ultra-violet light1 is made to send out electrons with velocities the maximum value of which is independent of the strength of the illumination1 but depends only on the frequency of the rays4 ,what happens is that some of the electrons in the metal which otherwise have1 as their function1 the conduction of the electric current1 by absorbing radiation energy1 free themselves from the metal and leave it with a certain velocity4 ,the reason why the rays for most metals must be ultra-violet 7i4e4 have a high frequency and consequently correspond to a proportionately large energy quantum7 depends upon the fact that the energy quantum absorbed by the electrons must be large enough to carry out the work of freeing the electrons4 ,but as long as the frequency of the rays 7and therefore their energy quantum7 is no less than what is needed for the freeing process1 it does not need to have certain fixed values4 ,if the energy quantum h which the rays can give off is greater than is required to free the electrons1 the surplus becomes kinetic energy in the electrons1 which thus acquire a velocity which is the greater the greater the frequency 1 and which coincides with the maximum velocity observed in the experiments4 ,what happens here is evidently something which can be considered as the reverse of the process which leads to the production of the continuous hydrogen spectrum 7described on p4 #afc74 ,in the latter case1 electrons with different velocities are bound by the hydrogen atoms1 which thus emit rays with frequencies increasing with increasing velocity1 while1 vice versa1 in the photoelectric effect rays with different frequencies free the electrons and give them velocities increasing with increasing frequencies4 ,it must be acknowledged that there is something very curious in this effect4 ,if the electromagnetic waves1 as has been assumed1 are distributed evenly over the field of radiation1 it is not easy to understand why they give energy to some atoms and not to others1 and why the selected ones always--with a given frequency--acquire a definite energy quantum1 independent of the intensity of the radiation4 ,for small intensities of the incident radiation1 the atom1 in order to acquire the proper quantum1 must absorb energy from a greater part of the field of radiation 7or for a longer time7 than for large intensities4 ,when the atoms acquire energy in electron collisions1 the situation is apparently easier to understand1 since in this case the colliding electrons give their kinetic energy to definite atoms1 namely1 those which they strike4 ,einstein1 in #aije1 when there was not yet any talk of the nuclear atom or the ,bohr theory1 enunciated his theory of light quanta1 according to which the energy of radiation is not only emitted and absorbed by the atoms in certain quanta1 with magnitudes determined by the frequency 1 but is also present in the field of radiation in such quanta4 ,when an atom emits an energy quantum h1 this energy will not spread out in waves on all sides1 but will travel in a definite direction--like a little lump of energy1 we might say4 ,these light quanta1 as they are called1 can1 like the electrons1 hit certain atoms4 ,but even if in this theory the difficulties mentioned are1 apparently1 overcome1 far greater difficulties are introduced2 indeed it may be said that the whole wave theory becomes shrouded in darkness4 ,the very number which characterizes the different kinds of rays loses its significance as a frequency and the phenomena of interference--reflection1 dispersion1 diffraction1 and so on--which are so fundamental in the wave theory of the propagation of light1 and on which1 for instance1 the mechanism of the human eye is based1 receive no explanation in the theory of light quanta4 ,for instance1 in order to understand that grating spectra can be produced at all1 we must think of a co-operation of the light from all the rulings 7cf4 ,fig4 #aj1 p4 #dg71 and this co-operation cannot arise if all the slits at a given moment do not receive light emitted from the same atom4 ,in a bundle of rays which comes in at right angles to a grating1 we must1 in order to explain the interference1 assume that the state of oscillation at a given moment is the same in all slits1 that1 for instance1 there are wave crests in all at the same time1 if we borrow a picture from the representation of water waves4 ,only in this case there can behind the grating at certain fixed places--for which the difference in the wave-length of the distances from successive slits is a whole number of wave-lengths--steadily come wave crests from all the slits at one moment and wave troughs from all at another moment 7the classical explanation of the 8mechanism0 of a grating74 ,if we imagine1 however1 that some slits are hit by light quanta from one atom and others from a second atom1 it is pure chance if there are wave crests simultaneously in all slits1 because the different atoms in a source of light emit light at different times1 depending purely on chance4 ,an understanding of the observed effect of a grating on light seems then out of question4 ,the theory of light quanta may thus be compared with medicine which will cause the disease to vanish but kill the patient4 ,when ,einstein1 who has made so many essential contributions in the field of the quantum theory1 advocated these remarkable representations about the propagation of radiation energy he was naturally not blind to the great difficulties just indicated4 ,his apprehension of the mysterious light in which the phenomena of interference appear on his theory is shown in the fact that in his considerations he introduces something which he calls a 8ghost0 field of radiation to help to account for the observed facts4 ,but he has evidently wished to follow the paradoxical in the phenomena of radiation out to the end in the hope of making some advance in our knowledge4 ,this matter is introduced here because the ,einstein light quanta have played an important part in discussions about the quantum theory1 and some readers may have heard about them without being clear as to the real standing of the theory of light quanta4 ,the fact must be emphasized that this theory in no way has sprung from the ,bohr theory1 to say nothing of its being a necessary consequence of it4 ,in the ,bohr theory1 absorption and radiation must be said to be completely reciprocal processes1 i4e4 processes of essentially the same nature1 but proceeding in opposite directions4 ,in itself it cannot be said to be more incomprehensible that an atom absorbs energy from a field of radiation in agreement with the ,bohr postulates than that it emits energy into the field2 but in both cases we naturally encounter the great difficulties mentioned in ,chap4 ,v4 ,we have hitherto restricted ourselves to the purely atomic processes4 ,but just as in the emission of radiation we meet spectra which owe their characteristics to molecular processes 7band spectra1 cf4 p4 #afb71 we have also absorption spectra with characteristics depending essentially upon motions of the atomic nuclei in the molecules4 ,a particularly interesting and instructive example of this nature is met with in the infra-red region of the spectrum in certain broad absorption lines or absorption bands1 which are due to gases having molecules containing several atoms4 ,in hydrogen chloride1 for instance1 there is found1 in the region of the spectrum which corresponds to a wave-length of about #ce 1 such an absorption band1 which by more accurate investigation has been shown to consist of a great number of absorption lines4 ,the explanation of this collection of lines must be sought in the motions which the hydrogen nucleus and the chlorine nucleus perform1 as they in part vibrate with respect to each other and in part rotate about their common centre of gravity4 ,just as in the case of the motions of the electrons in the atom1 there are also certain stationary states for the nuclear motions4 ,when the molecule absorbs radiation energy it will go from one of these states to another1 where the energy content is greater4 ,this absorption of energy proceeds according to the quantum rule1 i4e41 the product of the ,planck constant h and the frequency for the absorbed radiation must be equal to the difference in energy between the two stationary states2 only those rays which have frequencies fulfilling this condition are absorbed4 ,illustration3 ,,fig4 #cb4--,schematic representation of possible motions in a molecule of hydrogen chloride4 ,o is the centre of gravity of the molecule4 ,the black circles give the states of equilibrium of the nuclei1 the circles s their outer positions in oscillating1 and the circles r positions during the rotation of the nuclei4 ,in hydrogen chloride1 at standard temperature1 the molecules will be in different stationary states of rotation 7cf4 the remarks on p4 #bg71 corresponding to different definite values of the rotation frequency1 while the nuclei1 on the other hand1 must be assumed to be at rest with reference to each other1 i4e41 they preserve their mutual distance4 ,in ,fig4 #cb1 ,h and ,cl indicate the circles which the two nuclei will describe about the centre of gravity2 here1 however1 it must be remarked that the hydrogen circle is drawn too small in comparison with that of chlorine4 ,if heat rays with all possible wave-lengths around #ce are sent through the hydrogen chloride1 that radiation energy will be absorbed which can in part set the nuclei in oscillation and in part change the state of rotation4 ,let us for a moment assume that only the former change could happen4 ,then a ray with wave-length #cdf would be absorbed1 this frequency corresponding to the energy in the stationary state of oscillation into which the molecule goes2 this frequency is very nearly equal to the frequency with which the nuclei vibrate relatively to each other4 ,in reality1 at the same time that the nucleus is set in oscillation1 there will always be a change in the state of rotation--consisting either in an increase or in a decrease in the velocity of rotation4 ,the energy absorbed1 and therefore the frequency for the radiation absorbed1 is thereby changed a little1 so that in the spectrum of the rays sent through we do not obtain an absorption line corresponding to #cdf 1 but a line somewhat removed from that4 ,since there are1 however1 many stationary states of rotation to start from1 and since in some molecules there is one transition1 in others another1 we get many absorption lines on each side of #cdf 4 ,even before ,bohr propounded his theory1 at a time when the quantum theory did not yet have a clarified form1 the ,danish chemist1 ,niels ,bjerrum1 had predicted that the infra-red absorption lines ought to have such a structure4 ,this structure must be interpreted in the above way which differs somewhat from ,bjerrum's ideas1 but his prediction was essentially strengthened by investigations1 and it was one of the most significant features in the development of the quantum theory prior to #aiac4 ,the first to detect the structure of the infra-red absorption bands was the ,swedish physicist1 ,eva von ,bahr4 ,her experiments were later extended in a most significant way by the work of ,imes and other ,american investigators4 ,they enable us to calculate exactly the distance between the two nuclei in the molecule4 ,it may be asked what becomes of the energy which the hydrogen chloride molecule thus absorbs1 and whether it necessarily after a longer or shorter time must be re-emitted as radiation4 ,the latter is not the case4 ,in a collision between molecules or atoms1 the energy which one molecule 7or atom7 has absorbed by radiation can undoubtedly be transferred to another molecule1 the velocity of which is thereby increased4 ,the theoretical necessity of the occurrence of such collisions was clearly shown for the first time in a very significant investigation by two of ,bohr's students1 ,klein and ,rosseland4 ,without collisions of this nature the radiation energy absorbed could never be transformed into heat energy4 ,here we come to a very great and important field1 which has a very close connection with the theory of the chemical processes and to a better explanation of which the more recent experiments of ,franck and his co-workers have made important contributions4 ,,chapter ,,vii ,,the ,,structure ,,of ,,the ,,atom ,,and ,,the ,,chemical ,,properties ,,of ,,the ,,elements ,introduction4 ,we have hitherto restricted ourselves mainly to those applications of the ,bohr theory which have a direct connection with the processes of radiation4 ,we have shown how fertile the theory has proved to be1 how many problems1 previously inexplicable1 have been solved1 and what exact agreement has been established between experiment and theory in this comprehensive field4 ,we may now ask how the theory accounts for the chemical behaviour of the different elements4 ,as early as #aiac1 ,bohr1 in connection with his researches on spectral phenomena1 had considered the chemical properties of the elements and had pointed out interesting possibilities4 ,illustration3 ,,fig4 #cc4--;,early representation of the formation of a hydrogen molecule 7,bohr1 #aiac74 ,the ,combination of ,atoms into ,molecules4 ,in his discussion of hydrogen1 ,bohr suggested a model for the structure of its molecule1 which we shall give here1 because1 by a simple example1 it illustrates how two neutral atoms may form a molecule 7cf4 p4 #ac74 ,in ,fig4 #cc1 a1 ,k,e and ,k,e are two neutral hydrogen atoms which are approaching each other with the orbits of the two electrons parallel4 ,the nucleus ,k and the electron ,e then attract each other as do the nucleus ,k and the electron ,e4 ,the two electrons repel each other as do the two nuclei2 but when the electrons are in opposite positions of their orbits1 the forces of attraction outweigh the effect of the forces of repulsion4 ,calculation shows that when the atoms are allowed to approach each other the positions of the atoms will be as is shown in ,fig4 #cc1 b1 where the orbits are closer to each other than the nuclei4 ,finally1 for small distances of the two nuclei1 the two orbits will be merged into one1 as is shown in ,fig4 #cc1 c4 ,this orbit will be slightly larger than the original ones4 ,the two hydrogen atoms may1 in this way1 combine into one molecule4 ,in fact1 an equilibrium position can be found for which the nuclei are held together1 in spite of their mutual repulsion1 by the attractive forces existing between them and the electrons which are moving in their common orbit4 ,it must1 however1 be assumed1 for reasons which cannot be given here1 that the hydrogen molecule is1 in reality1 constructed somewhat differently2 probably the orbits of the electrons make an angle with each other4 ,the formation of a hydrogen molecule may also be supposed to occur when a positive hydrogen ion1 i4e41 a hydrogen nucleus1 and a negative hydrogen ion1 i4e41 a hydrogen nucleus with two electrons1 are drawn together by their mutual attractions4 ,the forces of attraction would be much stronger than in the first example given1 and the formation of a neutral molecule would not take place in the same way4 ,more energy would also be released1 but the final result will be the same4 ,just as in the case of the hydrogen molecule1 other molecules may be formed from atoms belonging to the same or to different elements4 ,the method of formation of molecules varies according as it is a union of neutral molecules in the normal state1 or a union of positive and negative ions4 ,conversely1 by chemical decomposition a molecule can be separated either into neutral parts or into ions4 ,if1 for instance1 common salt 7sodium chloride1 ,na,cl7 is dissolved in water1 the salt molecules are1 under the influence of the water molecules1 decomposed in ,na-ions with one positive charge and ,cl-ions with one negative charge1 corresponding to the monovalent electropositive character of sodium and the monovalent electronegative character of chlorine4 ,the possibilities are1 however1 far from being exhausted by these two methods of composition and decomposition4 ,an atom may exist not only in the normal state where it has its complete number of electrons collected as tightly as possible about the nucleus1 and in the ionized state with one or more too many or too few electrons2 but in a neutral atom one or more of the outer electrons may be in a stationary orbit at a greater distance from the nucleus than corresponds to the normal stationary state4 ,it is easy to understand that an atom in such an 8excited0 or 7as it is called in chemistry7 active state often finds it easier to act in concert with other atoms than when it is in the normal state2 in this latter state the atom is often more like a little compact lump of neutral substance than in the active state4 ,it will1 in any case1 be understood that the interplay between the atoms1 which reveals itself in the chemical processes or reactions between different elements1 offers many opportunities for the ,bohr theory to give in the future a more detailed explanation than was possible to the earlier theories of chemistry4 ,we must also mention the fact that it has become possible to elucidate in main features the phenomena1 hitherto unexplained1 of the chemical effects of light1 as on a photographic plate 7photochemistry7 and of catalysis1 which consists in bringing about1 or accelerating1 the chemical interaction between two substances by the presence of a third substance which does not itself enter in the compound1 and often needs only to be present in very small quantities4 ,it must1 however1 be emphasized that at present we do not yet possess a detailed theory of molecular constitution comparable with our knowledge of the structure of the atoms4 ,the ,periodic ,system4 ,instead of inquiring how the chemical processes may take place1 we shall now study the general correlation between the chemical properties and the atomic numbers of the elements1 a correlation which has found its empirical expression in the natural or periodic system of the elements 7cf4 p4 #bc74 ,the explanation of the puzzles of this system must be said to be one of the finest results which ,bohr has obtained1 and it constitutes a striking evidence in favour of the quantum theory of atoms4 ,there is nothing new in the idea of connecting the arrangement of the elements in the periodic system with an arrangement of particles in the atom in regular groups1 the character of which varies1 so to say1 periodically with increasing number of particles4 ,in the atom model of ,lord ,kelvin and ,j4 ,j4 ,thomson 7cf4 p4 #hf71 with the positive electricity distributed over the volume of the whole atom1 ,thomson tried to explain certain leading characteristics of the periodic system by imagining the electrons as arranged in several circular rings about the centre of the atom4 ,he pointed out that the stability of the electronic configurations of this type varied in a remarkable periodic way with the number of electrons in the atom4 ,by considerations of this nature ,thomson was able to enunciate a series of analogies to the behaviour of the elements in the periodic system as regards the tendency of the neutral atoms to lose one or more electrons 7electropositive elements7 or to take up one or more electrons 7electronegative elements74 ,but1 setting aside possible objections to his considerations and calculations1 the connection with the system was very loose and general1 and his theory lost its fundamental support when his atomic model had to give way to ,rutherford's4 ,with ,bohr's theory the demand for a stable system of electrons was placed in an entirely new light4 ,in his treatise of #aiac1 ,bohr tried to give an explanation of the structure of the atom1 by thinking of the electrons as moving in a larger or smaller number of circular rings about the nucleus4 ,his theory did not exclude the possibility of orbits of electrons having different directions in space instead of lying in one plane or being parallel4 ,the tendency of the considerations was to attain a definite1 unique determination of the structure of the atom1 as is demanded by the pronounced stability of the chemical and physical properties of the elements4 ,the results were1 however1 rather unsatisfactory1 and it became more and more clear that the bases of the quantum theory were not sufficiently developed to lead in an unambiguous way to a definite picture of the atom4 ,nowadays the simple conception of the electrons moving in circular rings in the field of the nucleus is definitely abandoned1 and replaced by a picture of atomic constitution of which we shall speak presently4 ,in the following years the general conception of the group distribution of the electrons in the atom formed the basis of many theoretical investigations1 which in various respects have led to a closer understanding of chemical and physical facts4 ,the ,german physicist1 ,kossel1 showed that the characteristic ,x-ray spectra of the elements1 which are due to a process of reconstruction of the atom subsequent to the removal of one or more of the innermost electrons 7cf4 p4 #afa71 give a most striking support to the assumption that the electrons are distributed in different groups in which they are bound with different strength to the atom4 ,the connection between the electron groups and the chemical valence properties of the atoms1 to which ,thomson had first drawn attention and which also played an important part in ,bohr's early considerations1 was further developed in a significant way by ,kossel1 as well as by ,lewis and by ,langmuir in ,america4 ,these chemical theories had1 however1 little or no connection with the quantum theory of atomic processes2 even the special features of the ,rutherford atom1 which are of essential importance in the theory of the hydrogen spectrum and of other spectra1 played only a subordinate part4 ,in #aibj ,bohr showed how1 by the development of the quantum theory which had taken place in the meantime1 and the main features of which consisted in the introduction of more than one quantum number for the determination of the stationary states and in the establishment of the correspondence principle1 the problem of the structure of the atom had appeared in a new light4 ,in fact1 he outlined a general picture of atomic constitution1 based on the quantum theory1 which in a remarkable way accounted for the properties of the elements4 ,in order to decide doubtful questions1 he has often had to call to his aid the observed properties of elements1 and it must be readily admitted that the finishing touches of the theory are still lacking4 ,but from his general starting-point he has been able to outline the architecture of even the most complex atomic structures and to explain1 not only the known regularities1 but also the apparent irregularities of the periodic system of the elements4 ,the method ,bohr used in his attempt to solve the problem was to study how a neutral normal atom may gradually be formed by the successive capturing and binding of the individual electrons in the field of force about the nucleus of the atom4 ,he began by assuming that he had a solitary nucleus with a positive charge of a given magnitude4 ,to this nucleus free electrons are now added1 one after the other1 until the nucleus has taken on the number sufficient to neutralize the nuclear charge4 ,each individual electron undergoes a 8binding0 process1 i4e4 it can move in different possible stationary orbits about the nucleus and the electrons already bound4 ,with the emission of radiation it can go from stationary states with greater energy to others with less energy1 ending its journey by remaining in the orbit which corresponds to the least possible energy4 ,we may designate this state of least energy as the normal state of the system1 which1 however1 is only a positive atomic ion1 so long as all the electrons needed for neutralization are not yet captured4 ,from the exposition in the preceding chapter it will be seen that the ordinary series spectra 7arc spectra7 may be considered as corresponding to the last stage in this formation process1 since the emission of each line in such a spectrum is due to a transition between two stationary states in each of which ,n-#a electrons are bound in their normal state1 i4e4 as tight as possible1 by the nucleus1 while the ,nth electron moves in an orbit mainly outside the region of the other electrons4 ,in the same way the spark spectra give witness of the last stage but one of the formation process of the atom1 since here ,n-#b electrons are bound in their normal state while an ,n-#ath electron moves in an orbit large compared with the dimensions of the orbits of the inner electrons4 ,from these remarks it will be clear that the study of the series spectra is of great importance for the closer investigation of the process of formation of the atom outlined above4 ,furthermore1 the general ideas of the correspondence principle1 which directly connects the possibility of transition from one stationary state to another with the motion of the electron1 has been very useful in throwing light on the individual capturing processes and on the stability of the electronic configurations formed by these4 ,in what follows we cannot1 however1 reproduce ,bohr's arguments at length2 we must satisfy ourselves with some hints here and there1 and for the rest restrict ourselves to giving some of the principal results4 ,before going farther we shall recall what has previously been said about the quantum numbers4 ,in the undisturbed hydrogen atom1 the stationary orbits can be numbered with the principal quantum numbers #a1 #b1 #c 444 n4 ,but to each principal quantum number there corresponds not one but several states1 each with its auxiliary quantum number #a1 #b1 #c 444 k1 k at the most being equal to the principal quantum number4 ,in a similar way1 the stationary orbits of the electrons in an atom containing several electrons can be indicated by two quantum numbers1 the #c orbit1 for instance1 being that with principal quantum number #c and auxiliary quantum number #b4 ,but while in the hydrogen atom the principal quantum number n1 in the stationary orbits which are slowly rotating ellipses1 is very simply connected with the length of the major axis of the ellipse1 and k3 n is the ratio between the minor and major axes1 still in other atoms with complex systems of electrons the significance of the principal quantum number is not so simple and the orbit of an electron consists of a sequence of loops of more complicated form 7cf4 ,fig4 #bi74 ,we must satisfy ourselves with the statement that a definition of their significance can be given1 but only by mathematical-physical considerations which we cannot enter into here4 ,it may1 however1 be stated that1 if we restrict ourselves to a definite atom1 the rule will hold that1 among a series of orbits with the same auxiliary quantum number but different principal quantum numbers1 that orbit in which the electron attains a greater distance from the nucleus has the higher number4 ,another rule which holds is1 that an orbit with a small auxiliary quantum number in comparison with its principal quantum number 7as #d for instance1 cf4 ,fig4 #bi71 will consist of very oblong loops with a very great difference between the greatest and least distances of the electron from the nucleus1 while the orbit will be a circle when the two quantum numbers are the same as for #a1 #b1 #c4 ,although each orbit has two quantum numbers1 we often speak simply of the #a-1 #b-1 #c- 444 n-quantum orbits1 meaning here the orbits with the principal quantum numbers #a1 #b1 #c 444 n4 ,the one electron of hydrogen will1 upon being captured1 first be at 8rest0 when it reaches the #a-path1 and we might perhaps be led to expect that in the atoms with greater nuclear charges the electrons in the normal state also would be in the one quantum orbit #a1 because to this corresponds the least energy in hydrogen4 ,this assumption formed actually the basis of ,bohr's work of #aiac on the structure of the heavier atoms4 ,it cannot be maintained1 however4 ,considerations of theoretical and empirical nature lead to the assumption that the electrons which already are gathered about the nucleus can make room only to a certain extent for new ones1 moving in orbits of the same principal quantum number4 ,those electrons which are captured later are kept at an appropriate distance2 they are1 for instance1 prevented from passing from a #c-quantum orbit to a #b-quantum one1 if the number of electrons moving in #b-quantum orbits has reached a certain maximum value4 ,when it is said that the captured electrons end in the stationary state which corresponds to the least energy1 it must1 therefore1 mean1 not the #a-quantum orbit1 but the innermost possible under the existing circumstances4 ,the final result will be that the electrons are distributed in groups1 which are characterized each by their quantum numbers in such a way that passing from the nucleus to the surface of the atom1 the successive groups correspond to successive integer values of the quantum number1 the innermost group being characterized by the quantum number one4 ,moreover1 each group is subdivided into sub-groups corresponding to the different values which the auxiliary quantum number may take4 ,that the electrons first collected keep the latecomers at an appropriate distance must be understood with reservations2 a new electron moving in an elongated orbit can very well come into the territory already occupied2 in fact1 it may come closer to the nucleus than some of the innermost groups of electrons4 ,in case an outer electron thus dives into the inner groups1 it makes a very short visit1 travelling about the nucleus like a comet which at one time on its elongated orbit comes in among the planets and perhaps draws closer to the sun than the innermost planet1 but during the greater part of its travelling time moves in distant regions beyond the boundaries of the planetary system4 ,it is a very important characteristic of the ,bohr theory of atomic architecture that the outer electrons thus penetrate far into the interior of the atom and thus chain the whole system together4 ,such a 8comet electron0 has1 however1 a motion of a very different nature from that of a comet in the solar system4 ,let us suppose that the nuclear charge is #ee 7,caesium71 that there already are fifty-four electrons gathered tightly about the nucleus1 and that ,no4 #ee in an orbit consisting of oblong loops moves far away from the nucleus1 but at certain times comes in close to it4 ,then1 for the greater part of its orbit1 this electron will be subject to approximately the same attraction as the attraction towards one single charge1 as a hydrogen nucleus2 but when ,no4 #ee comes within the fifty-four electrons it will for a very short time be influenced by the entire nuclear charge #ee4 ,together with the nearness of the nucleus1 this will cause ,no4 #ee to acquire a remarkably high velocity and to move in an orbit quite different from the elliptical one it followed outside4 ,moreover1 the great velocity of the electron during its short visit to the nucleus is in a considerable degree determinative of its principal quantum number2 this will be higher than would be expected from the dimensions of the outer part of the orbit if we supposed the motion to take place about a hydrogen nucleus 7cf4 ,figs4 #bg and #bi74 ,after these general remarks we shall try in a few lines to sketch the ,bohr theory of the structure of the atomic systems from the simplest to the most complicated4 ,we shall not examine the entire periodic system with its ninety-two elements1 but here and there we shall bring to light a trait which will illustrate the problem--partly in connection with the schematic representations in the atomic diagrams at the end of the book4 ,description of the ,atomic ,diagrams4 ,the curves drawn represent parts of the orbital loops of the electrons in the neutral atoms of different elements4 ,although the attempt has been made to give a true picture of these orbits as regards their dimensions1 the drawings must still be considered as largely symbolic4 ,thus in reality the orbits do not lie in the same plane1 but are oriented in different ways in space4 ,it would have been impracticable to show the different planes of the orbits in the figure4 ,moreover1 there is still a good deal of uncertainty as to the relative positions of these planes4 ,on this account the orbits belonging to the same sub-group1 i4e41 designated by the same quantum numbers1 are placed in a symmetric scheme in the sketch4 ,for groups of circular orbits the rule has been followed to draw only one of them as a circle1 while the others in the simpler atoms are drawn in projection as ellipses within the circle1 and in the more complicated atoms are omitted entirely4 ,the two circular orbits of the helium atom are both drawn in projection as ellipses4 ,further1 for the sake of clearness1 no attempt has been made to draw the inner loops of the non-circular orbits of electrons which dive into the interior of the atom4 ,in lithium only1 the inner loop of the orbit of the #b electron has been shown by dotted lines4 ,in order to distinguish the groups of orbits with different principal quantum numbers two colours have been used1 red and black1 the red indicating the orbits with uneven quantum numbers1 as #a1 #c1 #e1 the black those with even quantum numbers1 as #b1 #d1 #f4 ,wherever possible the nucleus is indicated by a black dot2 but in the sketches of atoms with higher atomic numbers the #a-quantum orbits are merged into one little cross and the nucleus has been omitted4 ,it should be noticed that the radium atom is drawn on a scale twice as great as that for the other atoms4 ,we shall begin with the capture of the first electron4 ,if the nucleus is a hydrogen nucleus the hydrogen atom is completed when the electron has come into the #a orbit1 a circle with diameter of about #aj cm4 7cf4 the diagram74 ,if the nucleus had had a greater nuclear charge the ,no4 #a electron would have behaved in the same way1 but the radius of its orbit would have been less in the same ratio as the nuclear charge was greater4 ,for a lead nucleus1 with charge 7atomic number7 #hb1 the radius of the #a orbit is _/ that of the hydrogen #a orbit4 ,since atoms with high atomic numbers thus collect the electrons more tightly about them it is understandable that1 in spite of their greater number of electrons1 they can be of the same order of magnitude as the simpler atoms4 ,let us now examine the helium atom4 ,the first electron1 which its nucleus 7charge #b7 catches1 moves as shown in a circle #a1 but with a smaller radius than in the case of the hydrogen atom4 ,electron ,no4 #b can be caught in different ways1 and the closer study of the conditions prevailing here1 which are still comparatively simple since there are only two electrons1 has been of greatest importance in the further development of the whole theory4 ,we cannot go into it here1 but must content ourselves with saying that the stable final result of the binding of the second electron consists in the two electrons moving in circular #a-quantum orbits of the same size with their planes making an angle with each other 7cf4 the diagram74 ,this state has a very stable character1 and the helium atom is therefore very disinclined to interplay with other atoms1 with other helium atoms as well as with those of other elements4 ,helium is therefore monatomic and a chemically inactive gas4 ,in all atomic nuclei with higher charges than the helium nucleus the first two orbits are also bound into two #a-quantum circular orbits at an angle with each other2 this group cannot take up any new electron having the same principal quantum number4 ,it takes on an independent existence and forms the innermost electron group in all atoms of atomic number higher than #b4 ,electron ,no4 #c will accordingly not be bound in the same group with #a and #b4 ,it must be satisfied with a #b-quantum orbit1 #b1 which consists of oblong loops1 and1 when nearest the nucleus1 comes into the territory of the #a-quantum orbits4 ,it is but loosely bound compared to the first two electrons1 and the lithium atom1 which has only three electrons1 can therefore easily let ,no4 #c loose so that the atom becomes a positive ion4 ,lithium is therefore a strongly electropositive monovalent metal4 ,the element beryllium 7,no4 #d7 will probably have two electrons in the orbits #b2 it will therefore be divalent4 ,but during the short visit of these electrons to the nucleus they are subject to a greater nuclear charge than in lithium4 ,the #b electrons are therefore1 in beryllium1 more firmly bound than in lithium1 and the electropositive character of beryllium is therefore less marked4 ,we have something essentially new in the boron atom 7atomic ,no4 #e7 where the two electrons ,no4 #c and ,no4 #d are taken into #b orbits1 but where ,no4 #e will very probably be bound in a circular #b orbit4 ,how the conditions will be in the normal state of the following atoms preceding neon is not known with certainty4 ,we only know that the electrons coming after the first two will be captured in #b-quantum orbits1 the dimensions of which get smaller1 according as the atomic number increases4 ,the neon atom 7compare the diagram7 has a particularly stable structure1 both 8closed0 and symmetric1 which besides two #a-orbits contains four electrons in #b orbits and four electrons in #b orbits4 ,as regards the four electrons in #b orbits1 they do not have symmetrical positions at every moment or move simultaneously either towards or away from the nucleus2 on the contrary1 it must be assumed that the electrons come closest to the nucleus at different moments at equal intervals of time4 ,the name of inert or inactive gases is given to the entire series of helium 7#b71 neon 7#aj71 argon 7#ah71 krypton 7#cf71 xenon 7#ed7 and niton 7#hf71 the ,o-column in the periodic system given in the table on p4 #bc4 ,all these elements are monatomic and quite unwilling to enter into chemical compounds with other elements 7although there is about #a per cent4 of argon in the air about us this element has1 on this account1 escaped the observation of chemists until about #ahie1 when it was discovered by the ,english chemist1 ,ramsay74 ,this complete chemical inactivity is explained by the fact that the atoms of all these elements have a nicely finished 8architecture0 with all the electrons firmly bound in symmetrical configurations4 ,they may be said to form the mile posts of the periodic system1 and to be the ideals towards which the other atoms aspire4 ,the table shows how the electrons in the atoms of these gases are divided among the types of orbits corresponding to the different quantum numbers4 ,,table ,,showing ,,the ,,distribution ,,of ,,the ,,electrons ,,of ,,different ,,orbital ,,types ,,in ,,the ,,neutral ,,atoms ,,of ,,the ,,inactive ,,gases4 ---------------------------------------- ------------ ,quantum ,numbers4 ,atomic------------------------------ ,number #a #b #b #c #c #c #d #d #d #d ---------------------------------------- --- ,helium #b #b ,neon #aj #b #d #d ,argon #ah #b #d #d #d #d - ,krypton #cf #b #d #d #f #f #f #d #d - - ,xenon #ed #b #d #d #f #f #f #f #f #f - ,niton #hf #b #d #d #f #f #f #h #h #h #h 8 #aah #b #d #d #f #f #f #h #h #h #h ---------------------------------------- --- ---------------------------------------- --------------------------- ,quantum ,numbers4 ,atomic--------------------------------- -------- ,number #e #e #e #e #e #f #f #f #f #f #f #g #g #g ---------------------------------------- -------------- ,helium #b ,neon #aj ,argon #ah ,krypton #cf ,xenon #ed #d #d - - - ,niton #hf #f #f #f - - #d #d - - - - 8 #aah #h #h #h #h - #f #f #f - - - #d #d - ---------------------------------------- -------------- ,the elements fluorine1 oxygen and nitrogen can attain the ideal neon-architecture by binding respectively one1 two and three additional electrons4 ,naturally they do not become neon atoms1 but merely negative atomic ions with single1 double or triple charge2 and their tendency in this direction appears in their character of monovalent1 divalent and trivalent electronegative elements respectively4 ,if we return to carbon it can probably not become a tetravalent negative ion by binding four free electrons2 but in the typical carbon compound1 methane 7,,ch71 the neon ideal is realized in another manner4 ,in fact1 it is reasonable to assume that the four electrons of the hydrogen atoms together with the six of the carbon atom give approximately a neon-architecture4 ,the four hydrogen nuclei naturally cannot be combined with the carbon nucleus2 the mutual repulsions keep them at a distance4 ,they will probably assume very symmetrical positions within the electron system which holds them together4 ,the nitrogen atom may in a similar way find completion in a neutral molecule with neon-architecture1 if it unites with three hydrogen atoms to form ammonia ,,nh2 but the three hydrogen nuclei1 although having symmetrical positions1 will not lie in the same plane as the nitrogen nucleus4 ,the electric centre of gravity for the positive nuclei will therefore not coincide with the centre of gravity for the negative electron system4 ,the molecule obtains thus what might be called a positive and a negative pole1 and this dipolar character appears in the electrical action of ammonia 7its dielectric constant74 ,something similar holds true for the water molecule1 where1 in a neon-architecture of electrons1 in addition to the oxygen nucleus in the centre there are two hydrogen nuclei which are not co-linear with the oxygen nucleus4 ,if we go on from neon in the periodic system we come to sodium 7#aa74 ,when the sodium nucleus captures electron ,no4 #aa1 this cannot find room in the neon-architecture formed by the first ten electrons4 ,since the eleventh electron thus cannot find a place in either a #b or a #b orbit1 it is bound in a #c orbit 7cf4 ,fig4 #bi and diagram at the end74 ,the atom then has a character like that of the lithium atom1 and we can therefore understand the chemical relationship between the two elements1 which are both monovalent electropositive metals4 ,we shall not dwell longer upon the individual elements of the atomic series4 ,if we pass from neon through sodium 7#aa71 magnesium 7#ab71 aluminium 7#ac71 etc41 to argon 7#ah71 we get what is essentially a repetition of the situation in the series from lithium to neon4 ,we first get two orbits of the #c type in magnesium1 a #c orbit is for the first time added in aluminium1 and for the atomic number #ah1 eight #c-quantum orbits1 together with the eight orbits of the inner #b-quantum group and the two of the innermost #a-quantum group1 give the symmetric architecture of argon 7cf4 table on p4 #aif1 and diagram at the end74 ,the architecture of the argon atom is in a certain sense less complete than that of the neon atom4 ,in argon there are indeed four orbits of the #c type and four of the #c type1 but the third kind of #c-quantum orbit1 the circular #c one1 is missing4 ,nor does it appear in the next element1 potassium 7#ai74 ,the electron ,no4 #ai prefers1 instead of the #c orbit1 the #d orbit1 which consists of oblong loops and which gives a firmer binding because it dives in among the electrons bound earlier1 while the circle #c would lie outside them all4 ,we thus obtain an atom of type similar to the lithium and sodium atoms4 ,but the slighted #c path lies1 so to speak1 on the watch to steal a place for itself in the neutral atom1 and this has grave results for the subsequent development4 ,even in calcium 7#bj71 after the first eighteen electrons are bound in the argon architecture1 both the nineteenth and the twentieth go into a #d orbit1 and the behaviour of calcium is like that of magnesium4 ,but since the increasing nuclear charge means for the electron ,no4 #ai a decrease in the dimensions and an increase in the binding of the orbits corresponding to the quantum number #c1 a point will finally be attained where the #c orbit of the nineteenth electron lies within the boundaries of what may be called the argon system1 i4e41 the architecture corresponding to the first eighteen electrons1 and corresponds to a stronger binding than a #d orbit would do4 ,in scandium 7#ba7 the #c-type orbit occurs for the first time in the neutral atom and will not only come into competition with the #d type1 but will also cause a disturbance in the #c-quantum groups1 which in the following elements must undergo reconstruction4 ,as long as this lasts the situation is very complicated and uncertain4 ,when the reorganization is almost completed1 we come to the blotting out of chemical differences1 particularly known from the triad1 iron1 cobalt and nickel4 ,moreover1 there comes a fluctuation in the valency of the elements4 ,iron can1 as has been said1 be divalent1 trivalent or hexavalent4 ,this oscillation in valency begins in titanium4 ,we should perhaps expect that the reconstruction would be completed long before nickel 7#bh7 is reached1 because even with twenty-two electrons we could get four orbits of each of the #c-quantum types 7#c1 #c and #c72 but from the chemical facts we are led to assume that in a completed group of #c-quantum orbits there can be room for six electrons in each sub-group4 ,at first sight we should1 then1 expect the end of the reconstruction with nickel1 which has indeed eighteen electrons more than neon where the group of #b-quantum orbits was completed4 ,we might expect that nickel would be an inert element in the series with helium1 neon1 and argon4 ,on the contrary1 nickel merely imitates cobalt4 ,this is explained by the fact that the group of eighteen #c-quantum orbits1 although it has a symmetric architecture1 is weakly constructed if the nuclear charge is not sufficiently large4 ,the binding of this group is too weak for it to exist as the outer group in a neutral atom4 ,in nickel the electrons1 in a less symmetrical manner1 will probably arrange themselves with seventeen #c-quantum orbits and one #d-quantum orbit4 ,the group of eighteen #c-quantum orbits becomes stable1 however1 when the nuclear charge is equal to or larger than #bi1 in which case it can become the outer group in a positive ion4 ,in this we find the explanation of the properties of the atom of copper4 ,the neutral copper atom has its twenty-ninth electron bound in a #d orbit consisting of oblong loops 7cf4 diagram at the end72 this electron can easily be freed and leaves a positive monovalent copper ion with a symmetrical architecture4 ,even under these circumstances1 although possessing a certain stability1 the ion is not very firmly constructed4 ,thus the fact that copper can be both monovalent and divalent1 must be explained by the assumption that for a nuclear charge #bi1 the #c-quantum group still easily loses an electron4 ,when we come to zinc 7#cj7 the group of eighteen is more firmly bound2 zinc is a pronounced divalent metal which in its properties reminds us of calcium and magnesium4 ,from zinc 7#cj7 to krypton 7#cf7 we have a series of elements which in a certain way repeat the series from magnesium 7#ab7 to argon 7#ah74 ,in ,fig4 #cd is shown ,bohr's arrangement of the periodic system in which the systematic correlation of the properties of the element appears somewhat clearer than in the usual plan 7cf4 p4 #bc74 ,it shows great similarity with an arrangement proposed nearly thirty years ago by the ,danish chemist1 ,julius ,thomsen4 ,the elements from scandium to nickel1 where1 in the neutral atom1 the electron group of #c-quantum orbits is in a state of reconstruction1 are placed in a frame2 the neutral oblique lines connect elements which are 8homologous10 i4e41 similar in chemical and physical 7spectral7 respect4 ,illustration3 ,,fig4 #cd4--,the periodic system of the elements4 ,the elements where an inner group of orbits is in a stage of reconstruction are framed4 ,the oblique lines connect elements which in physical and chemical respects have similar properties4 ,in krypton 7#cf7 we again have a stable architecture with an outer group of eight electrons1 four in #d orbits and four in #d orbits4 ,owing to the appearance of #d orbits in the normal state of the atoms of elements with atomic number higher than #ch1 there follows in the fifth period of the natural system a reconstruction and provisional completion of the #d-quantum orbits to a group of eighteen electrons1 which shows a great simplicity with the completion of the #c-quantum group in the fourth period4 ,in ,fig4 #cd the elements where the #d-quantum group is in a state of reconstruction are framed4 ,the #d-quantum group with eighteen electrons is of more stable construction than the group of eighteen #c-quantum orbits in the elements with an atomic number lower by eighteen4 ,this is due to the fact that all the orbits in the first-mentioned group are oblong and therefore moored1 so to say1 in the inner groups1 while in the complete group of #c-quantum orbits there are six circular orbits4 ,this is the reason why silver1 in contrast to copper1 is monovalent4 ,the next inactive gas is xenon 7#ed71 which outside of the #d-quantum group has a group of eight electrons in #e-quantum orbits1 four in #e orbits and four in #e orbits4 ,we notice that in xenon the group of #d-quantum orbits still lacks the #d orbits4 ,on the theory we must1 therefore1 expect to meet a new process of completion and reconstruction when proceeding in the system of the elements4 ,the theoretical argument is similar to that which applies in the case of the completion of the #c-quantum group which takes place in the fourth period of the natural system4 ,in fact1 in the formation of the normal atoms of the elements next after xenon1 caesium1 #ee1 and barium1 #ef1 the fifty-four electrons first captured will form a xenon configuration1 while the fifty-fifth electron will be bound in a #f orbit1 consisting of very oblong loops1 which represents a much stronger binding than a circular #d orbit4 ,calculation shows1 however1 that with increasing nuclear charge there must soon appear an element for which a #d orbit will represent a stronger binding than any other orbit4 ,this is actually the case in cerium 7#eh71 and starting from this element we meet a series of elements where1 in the normal neutral atom1 the #d-quantum group is in a state of reconstruction4 ,this reconstruction must occur far within the atom1 since the group of eighteen #d-quantum orbits in xenon is already covered by an outer group of eight #e-quantum orbits4 ,the result is a whole series of elements with very slight outward differences between their neutral atoms1 and therefore with very similar properties4 ,this is the rare earths group1 which in such a strange way seemed to break down the order of the natural system 7cf4 p4 #ba71 but which thus finds its natural explanation in the quantum theory of the structure of the atom4 ,the elements in which the #d-quantum group is in a state of reconstruction are1 in ,fig4 #cd1 enclosed in the inner frame in the sixth period4 ,moreover1 in the outer frame all elements are enclosed where the group of #e-quantum orbits is in a state of reconstruction1 which started1 even before cerium in lanthanum 7#eg71 where the fifty-fifth electron in the normal state is bound in a #e orbit4 ,the element cassiopeium1 with atomic number #ga1 which is the last of the rare earths1 stands just outside the inner frame1 because in the normal neutral atom of this element the #d-quantum group is just completed2 this group1 instead of eighteen electrons with six electrons in each sub-group1 consists now of thirty-two electrons with eight electrons in each sub-group4 ,the theory was able to predict that the element with atomic number #gb1 which until a short time ago had never been found1 and the properties of which had been the subject of some discussion1 must in its chemical properties differ considerably from the trivalent rare earths and show a resemblance to the tetravalent elements zirconium 7#dj71 and thorium 7#ij74 ,this expectation has recently been confirmed by the work of ,hevesy and ,coster in ,copenhagen1 who have observed1 by means of ,x-ray investigations1 that most zircon minerals contain considerable quantities 7#a to #aj per cent47 of an element of atomic number #gb1 which has chemical properties resembling very much those of zirconium1 and which on this account had hitherto not been detected by chemical investigation4 ,a preliminary investigation of the atomic weight of this new element1 for which its discoverers have proposed the name hafnium 7,hafnia ,copenhagen71 gave values lying between #agh-ahj1 in accordance with what might be expected from the atomic weight of the elements 7#ga7 and 7#gc74 7,cf4 p4 #bc47 ,the further completion of the groups of #e- and #f-quantum orbits1 which in the rare earths had temporarily come to a standstill1 is resumed in hafnium and goes on in a way very similar to that in which the #d- and #e-quantum groups in the fifth1 and the #c- and #d-quantum groups in the fourth period underwent completion4 ,thus the reconstruction of the #e-quantum group which began in lanthanum1 and which receives a characteristic expression in the triad of the platinum metals1 has come to a provisional conclusion in gold 7#gi71 gold being the first element outside the two frames which1 in ,fig4 #cd1 appear in the sixth period4 ,the neutral gold atom possesses1 in its normal state--besides two #a-quantum orbits1 eight #b-quantum orbits1 eighteen #c-quantum orbits1 thirty-two #d-quantum orbits and eighteen #e-quantum orbits--one loosely bound electron in a #f orbit4 ,in niton 7#hf71 finally1 we meet again an inactive gas1 the structure of the atom of which is indicated in the table on p4 #aif4 ,in this element the difference between nuclear and electron properties appears very conspicuously1 since the structure of the electron system is particularly stable1 while that of the nucleus is unstable4 ,niton1 in fact1 is a radioactive element which is known in three isotopic forms2 one of these is the disintegration product of radium1 the so-called radium emanation2 it then has a very brief life4 ,in the course of four days over half of the nuclei in a given quantity of radium emanation will explode4 ,in the diagram at the end of the book1 as an example of an atom with very complicated structure1 there is given a schematic representation of the atom of the famous element radium1 on a scale twice as large as the one used in the other atoms4 ,it follows clearly enough1 from what has been said in ,chap4 ,,iv41 that the structure of the electron system has nothing to do with the radioactivity4 ,all the remarkable radiation activities are due to the nucleus itself4 ,there has not even been room in the figure to draw the nucleus2 the #a-quantum orbits consist only of a small cross1 and in the other groups we have contented ourselves with summary indications4 ,the electron system1 with its eighty-eight electrons1 is1 however1 in itself very interesting1 with its symmetry in the number of electrons in the different groups4 ,in the different quantum groups from #a to #g there are found respectively two1 eight1 eighteen1 thirty-two1 eighteen1 eight and two electrons4 ,the last group is naturally of a very different nature from the first2 they are 8valence electrons10 which easily get loose and leave behind a positive radium ion with stable niton architecture4 ,radium then belongs to the family of the divalent metals1 magnesium1 calcium1 strontium and barium4 ,four places from radium is uranium 7#ib7 and the end of the journey1 if we restrict ourselves to the elements which are known to exist4 ,one could very well continue the building-up process still further and discuss what structure would have to be assumed for the atoms of the elements with higher atomic numbers4 ,that they cannot exist is not the fault of the electron system but of the nuclei1 which would become too complicated and too large to be stable4 ,in the table on p4 #aif there is shown the probable structure of the atom of the inert gas following niton2 it must be assumed to have one hundred and eighteen electrons distributed in groups of two1 eight1 eighteen1 thirty-two1 thirty-two1 eighteen and eight among the quantum groups from #a to #g4 ,as has been said1 in all this symmetrical structure of the atoms of the elements1 ,bohr has in many cases had to rely upon general considerations of the information that observation gives about the properties of the individual elements4 ,it must1 however1 not be forgotten that the backbone of the theory is and remains the general laws of the quantum theory1 applied to the nucleus atom in the same way as they originally were applied to the hydrogen atom1 leading thereby to the interpretation of the hydrogen spectrum4 ,we have1 further1 a most striking evidence as to the correctness of ,bohr's ideas in the fact that not only do the pictures of the atoms which he has drawn agree with the known chemical facts about the elements1 but they are also able to explain in the most satisfactory manner possible the most essential features of the characteristic ,x-ray spectra of the different elements1 a field we shall not enter upon here4 ,in all that has been said above we have been considering the ,bohr theory simply as a means of gaining a deeper understanding of the laws which determine activities in the atomic world4 ,perhaps we shall now be asked if we can 8utilize0 the theory1 or1 in other words1 if it can be put to practical use4 ,to this natural and not unwarranted question we may first give the very general answer1 that progress in our knowledge of the laws of nature always contributes sooner or later1 directly or indirectly1 to increase our mastery over nature4 ,but the connection between science and practical application may be more or less conspicuous1 the path from science to practical application more or less smooth4 ,it must be admitted that the ,bohr theory1 in its present state of development1 hardly leads to results of direct practical application4 ,but since it shows the way to a more thorough understanding of the details in a great number of physical and chemical processes1 where the peculiar properties of the different elements play parts of decided importance1 then in reality it offers a wealth of possibilities for making predictions about the course of the processes--predictions which undoubtedly in the course of time will be of practical use in many ways4 ,in this connection the discovery of the element hafnium1 discussed on p4 #bjd1 may be mentioned4 ,it must be for the future to show what the ,bohr theory can do for technical practice4 ,below is given an explanation of the different symbols which occur at various places in the book2 also the values of important physical constants4 #a m4 #a metre2 #a cm4 #a centimetre #jcid inches4 #a #a micron #a_/ajjj of a millimetre #jjjja cm4 #aj cm4 #a #a_/a1jjj1jjj of a millimetre #aj cm4 #a cm4 #a cubic centimetre4 #a g4 #a gram2 #a kg4 #a kilogram #bb pounds4 #a kgm4 #a kilogrammetre 7the work or the energy required to lift #a kg4 #a m474 #a erg #ajb #aj kgm4 #gdh #aj foot-pounds4 represents wave-length4 represents frequency 7number of oscillations in #a second74 represents frequency of rotation 7number of rotations in #a second74 n represents an integer 7particularly the ,bohr quantum numbers74 ,the velocity of light is c #c #aj cm4 per second #ii #aj feet per second4 ,the wave-length of yellow sodium light is #jehi #ehi #bcb #aj inches4 ,the frequency of yellow sodium light is #ebf #aj vibrations per second4 ,the number of molecules per cm4 at #j ,c4 and atmospheric pressure is about #bg #aj4 ,the number of hydrogen atoms in #a g4 is about #faj4 ,the mass of a hydrogen atom is #afe #aj g4 ,the elementary quantum of electricity is #dgg #aj 8electrostatic units40 ,the negative electric charge of an electron is #a elementary quantum 7#a negative charge74 ,the positive electric charge of a hydrogen nucleus is #a elementary quantum 7#a positive charge74 ,the mass of an electron is that of the hydrogen atom4 ,the diameter of an electron is estimated to be about #c #aj cm4 ,the diameter of the atomic nucleus is of the order of magnitude #aj to #aj cm4 ,the diameter of a hydrogen atom in the normal state 7the diameter of the first stationary orbit in ,bohr's model7 is #ajef #aj cm4 ,the ,balmer constant ,k #cbi #aj4 ,the ,planck constant h #fed #aj4 ,an energy quantum is ,e h4 ,the ,balmer-,ritz formula for the frequencies of the lines in the hydrogen spectrum is 7 #a #a 7 ,k 7 ------ - -------7 7 n n 7 ,illustration3 ,,diagrams ,,of ,,spectra ,,after ,,bunsen ,,and ,,kirchhoff ,illustration3 ,,structure ,,of ,,the ,,radium ,,atom ,illustration3 ,,main ,,lines ,,of ,,the ,,atomic ,,structure ,,of ,,some ,,elements