THE QUANTUM THEORY THE QUANTUM THEORY BY FRITZ REICHE PROFESSOR OF PHYSICS IN THE UNIVERSITY OF BRESLAU TRANSLATED BY H. S. HATFIELD, B.Sc., Ph.D.. AND HENRY L. BROSE. M.A. WITH FIFTEEN DIAGRAMS NEW YORK E. P. BUTTON AND COMPANY PUBLISHERS 66038 PRINTED IN GREAT BRITAIN BY THE ABERDEEN UNIVERSITY PRESS c o CONTENTS ^ CHAP. PAGE ^ INTRODUCTION 1 j, I. THE ORIGIN OF THE QUANTUM HYPOTHESIS .... 2 k\ II. THE FAILURE OF CLASSICAL STATISTICS .... 13 III. THE DEVELOPMENT AND THE RAMIFICATIONS OF THE QUANTUM THEORY. * 16 IV. THE EXTENSION OF THE DOCTRINE OF QUANTA TO THE Mo- ^ LECULAR THEORY OF SOLID BODIES .... 29 V. THE INTRUSION OF QUANTA INTO THE THEORY OF GASES . 68 VI. THE QUANTUM THEORY OF THE OPTICAL SERIES. THE DE- VELOPMENT OF THE QUANTUM THEORY FOR SEVERAL DEGREES OF FREEDOM 84 VII. THE QUANTUM THEORY OF RONTGEN SPECTRA . . . 109 VIII. PHENOMENA OF MOLECULAR MODELS i ..... 117 ) IX. THE FUTURE . . . ' 125 W MATHEMATICAL NOTES AND REFERENCES .... 127 A INDEX . . 181 THE QUANTUM THEORY INTRODUCTION r ~r" v HE old saying that small causes give rise to great effects J[ has been confirmed more than once in the history of physics. For, very frequently, inconspicuous differences be- tween theory and experiment (which did not, however, escape the vigilant eye of the investigator) have become starting- points of new and important researches. Out of the well-known Michelson-Morley experiment, which, in spite of the application of the most powerful methods of exact optical measurement, failed to show an influence of the earth's movement on the propagation of light as was predicted by classical theory, there arose the great structure of Einstein's Theory of Kelativity. In the same way the trifling difference between the measured and calculated values of black-body radiation gave rise to the Quantum Theory which, formulated by Max Planck, was destined to revolutionise in the course of time almost all departments of physics. The quantum theory is yet comparatively young. It is therefore not surprising that we are confronted with an unfinished theory still in process of development which, changing constantly in many directions, must often destroy what it has built up a short time before. But under such circumstances as these, in which the theory is continually deriving new nourishment from a fresh stream of ideas and suggestions, there is a peculiar fascination in attempting to review the life-history of the quantum theory to the present time and in disclosing the kernel which will certainly out- last changes of form. 1 CHAPTEE I The Origin of the Quantum Hypothesis i. Black-Body Radiation and its Realisation in Practice THE Quantum Theory first saw light in 1900. When, in the years immediately preceding (1897-1899), Lummer and Pringsheim made their fundamental measurements 1 of black-body radiation at the ReichsanstaU, they could have had no premonition that their careful experiments would become the starting-point of a revolution such as has seldom occurred in physics. In the field of heat radiation chief interest at that time was centred in the radiation of a black body (briefly called " black- body " radiation), that is, of a body which absorbs completely all radiation which falls on it and which thus reflects, trans- mits, and scatters 2 none. We may shortly call to mind the following facts. It is known that any body at a given temperature sends out energy in the form of radiation into the surrounding space. This radiation is not energy in a single simple form but is made up of a number of single radiations of different colours, i.e. of different wave-lengths A or of different frequencies 3 v. In other words, it forms in general a spectrum in which radiations of all frequencies between v = and v = oo are represented. Further, these radiations are present in varying " intensities." We define this term thus. Consider the radiation emitted from unit surface of the body per_second in a certain direction ; break it up spectrally and cut out of the spectrum a small frequency interval dv such that it contains all frequencies between v and v + dv. The energy of radiation E v thus sliced out (namely, the emissivity of the body for the frequency v) may be defined in the following terms : * E v = ^K v d v , . . (1) 2 BLACK-BODY RADIATION 3 provided that as we shall assume for the sake of simplicity the surface. of the body emits uniform and unpolarised radiation in" all directions. The magnitude K,, thus defined is called the intensity of radiation of the body for the frequency v. It is in general a more or less complicated function of the frequency v, of the absolute temperature of the body T, and of the inherent properties of the body. The black body alone is unique in this respect. For its radiation and therefore its K v is, as G. Kirchhoff* was the first to point out, dependent only on the frequency .v and the absolute temperature T, that is, mathematically, K v =f(v,T) .... (2) This formula which gives the relation between the intensity of radiation from a black body, the temperature, and the " colour " is called the radiation formula or the law of radia- tion of a black body. To calculate this- relationship on the one hand and to measure it on the other were unsolved problems at that time. Unimpeachable measurements were of course possible only if one could succeed in constructing a black body which approached sufficiently near the theoretical ideal. This im- portant step, the realisation of the black body, was taken by 0. Lummer and W. Wien* on the basis of KirchhofTs 1 Law of Cavity Eadiation, which states : In an enclosure or a cavity which is enclosed on all sides by reflecting walls, externally protected from, exchanging heat with its surroundings, and evacuated, the condition of " black radiation " is auto- matically set up if all the emitting and absorbing bodies at the walls or in the enclosure are at the same temperature. In a space, therefore, which is hermetically surrounded by bodies at the same temperature T and which is prevented from ex- changing heat with its surroundings, every beam of radiation is identical in quality and intensity with that which would be emitted by a black body at the temperature T. Lummer and Wien, therefore, had only to construct a uniformly heated enclosure wi.th blackened walls having a small opening. The radiation emitted from this opening was then " black " to an approximation which was the closer 4 THE QUANTUM THEORY the smaller the opening, that is, the less the completeness of the enclosure was disturbed. The manner in which the intensity Kv of the black radiation thus realised depended on the frequency v and the temperature T had next to be determined. The above-mentioned investigation of Lummer and Pringsheim was devoted to this purpose. 2. The Stefan-Boltzmann Law of Radiation and Wien's Displacement Law While experimental research was proceeding on its way, theory was not idle, for valuable pioneer work was being done inasmuch as two fundamental laws were set up. In the first place, L. Boltzmann* proved, with the help of thermodynamics, the law previously enunciated by Stefan, 9 that the sum-total of the radiation from a black body, taking all the frequencies together, namely, the quantity K = \ K. v dv, is proportional to the fourth power of its absolute Jo temperature : 10 K = y . T 4 (y = const.) . . . (3) The laws proposed by Wien n entered more deeply into the question. Wien imagines the black radiation enclosed in a closed space with a perfectly reflecting piston as one wall, and then supposes the radiation to be compressed adiabatically, as in the case of gases (that is, no passage of heat to or from the cavity is allowed during the process), by infinitely slow movements of the piston. Now, if we express the change which this process causes in the energy of a definite colour interval dv in two ways, and if we take into consideration that the waves reflected at the piston undergo a change of colour according to Doppler's principle, we succeed in limiting very considerably the unknown functional dependence of the quantity K v on v and T. There is thus obtained a re- lation of the form 12 in which c is the velocity of light in vaciio, the function F being left undetermined. From this, Wien's Displacement Law, the conclusion 13 may be drawn that the frequency WIEN'S LAW OF RADIATION 5 I'max for which 1C (plotted as a function of v) is a maximum is displaced towards higher values proportional to T as the temperature increases : Vmax = COnst. . T . . . (4ft) If, as is usual in physical measurement, we use the wave- length A = - instead of the frequency as the variable, Wieris Law assumes a somewhat different form. For if we consider the radiant energy of a narrow range of wave-length d\ cor- responding to the frequency range dv, and write it in the form E^dX, then EidK = K^dv, that is : E^ - K*> . ^. In A place of (4) and (4a) we then get the relations : >-max . T . const. = S . . (5a) 3. Wien's Law of Radiation To formulate the law of radiation it was therefore neces- sary only to evaluate the unknown function F in (4) or (5). But this was just the central point of the whole question, and the most difficult part of the problem. Here, too, Wien made the first successful attack. On the basis of not entirely unobjectionable calculations, which were founded on Maxwell's law of distribution of velocities among gas molecules, he arrived at the following specialised form 14 of the function F : p a e -pf ( a an d ft are two constants). Thus the law of radiation (4) assumes the form K,a.-4 . . . . (6) which is called Wien's Law of Kadiation. How far did experiment confirm these theoretical results ? While the Stefan-Boltzmann Law and Wien's Displacement Law were confirmed to a large extent by the observations of Lunimer and Pringsheim, both experimenters found Wien's 6 THE QUANTUM THEORY Law of Eadiation confirmed only for high frequencies, that is, for short wave-lengths (more precisely, for large values of m), and detected, on the other hand, systematic dis for small frequencies, that is, for long wave-lengths. They maintained with unswerving persistence that these discrepan- cies were real in spite of objections from authoritative quarters. For while F. Paschen 17 imagined that he had proved by his work that Wien's Law of Radiation was universally valid, Max Planck, in his detailed theory of irreversible processes of radiation, 18 had arrived again at Wien's radiation formula by a more rigorous method. Starting from Kirchhoff's Law of Cavity Radiation, according to which the presence of any emitting or radiating substance whatsoever in a uniformly heated enclosure produces and ensures the maintenance of the condition of black-body radiation, Planck chose as the simplest schematic model of such a substance a system of linear electromagnetic oscillators, and investigated the equilibrium of the radiation set up between them and the radiation of the enclosure. This is to be understood as fol- lows : Each of the Planck oscillators as such we may, for example, assume bound electrons capable of vibration pos- sesses a fixed natural frequency v and responds, on account of its weak damping, only to those waves of the radiation in the enclosure whose frequencies lie in the immediate neigh- bourhood of v, while all other waves pass over it without effect. The oscillator thus acts selectively, as a resonator, in just the same way as a tuning-fork of definite pitch com- mences to sound only when its own " proper" tone, or one very near it, is contained in the volume of sound which strikes it. In this process of resonance, however, the oscillator ex- changes energy with the radiation inasmuch as, on the one hand, it acts as a resonator in abstracting energy from the external radiation, and, on the other, it acts as an oscillator and radiates energy by its own vibration. Hence a dynamic equilibrium is set up between the oscillator and the radiation of the enclosure, and, indeed, between just those waves of the radiation which have the frequency v. In this state of equilibrium the radiation of frequency v acquires an intensity K t . which, according to Kirchhoff's Law, is equal to the intensity THE QUANTUM HYPOTHESIS 7 of black-body radiation at this temperature. Secondly, the energy U of the oscillator passes in the course of time through all possible values, the mean value 19 U of which is found to be proportional to the intensity K v , a result which seems im- mediately plausible since the excitation of the oscillator will be greater the more intense the radiation that falls on it. The exact calculation of this relationship between S v and U on the basis of classical electrodynamics this is the first part of Planck's calculations leads to the fundamental formula : *,=*? - - ' (7) In the second part Planck * determined U, although by a method that is not free from ambiguity, as a function of v and T on the basis of the second law of thermodynamics. He obtained ff-*-4 .... (8) The combination of (7) and (8) gives us Wien's Law of Radia- tion (6). 4. The Quantum Hypothesis. Planck's Law of Radiation Lummer and Pringsheim, however, refused to surrender. In a fresh investigation 21 in 1900 they showed that in the region of long waves Wien's radiation formula undoubtedly did not agree with the results of observation. As a result of this, Planck, in an important paper 22 which must be regarded as marking the creation of the quantum theory, decided to modify his method of deducing the law of radiation, namely, by altering the expression (8) which gives the mean energy of the oscillator, but which is not unique. He proceeded as follows. 23 In order to distribute the whole available energy among the oscillators, he imagined this energy divided into a discrete number of finite " elements of energy " (energy quanta) of magnitude e, and supposed these energy quanta to be distributed at random among the individual oscillators exactly as a given number of balls, say 5, may be distributed among a certain number of boxes, say 3. Each such distri- bution (of 5 balls among 3 boxes) may obviously be carried 8 THE QUANTUM THEORY out in a number of different ways, whereby, however, we are not concerned with which particular balls lie in which par- ticular boxes, but with the number contained in each. 2 * Now since each such " distribution " corresponds to a definite state of the system, it follows from what has just been said that each condition may be realised in a number of different ways, that is, each condition is characterised by a certain number of possibilities of realisation. This number is called by Planck the thermodynamic probability W of the condition in question. For it is obvious that the probability of a con- dition or state is the greater, i.e. it will occur the more fre- quently, the greater the number of ways in which it may be realised. By means of the usual formulae of permuta- tions and combinations, of which the latter alone come into consideration here, it was possible to calculate the probability of any given distribution of the elements of energy among the oscillators, and thus also the probability of a given energetic condition of the system of oscillators as a func- tion of the mean energy U of an oscillator and of the energy quantum. Now, L. BoUzmann has given an extremely fertile rule, which connects the probability of state W of a system with its entropy S, a magnitude which, as is well known, plays a similar role in the second law of thermodyna- mics to that played by energy in the first. Thus S was ob- tained as a function of U and c. If now, on the other hand, one applied the second law itself, which expresses the en- tropy S as a function of the mean energy U and the absolute temperature T, the following result was obtained by this cir- cuitous process : the entropy, as an auxiliary magnitude, was eliminated, and a relation between U, T, and e was gained. This fundamental result, first obtained by Planck, is as follows : U = _! - (k being a constant) . . (9) But from (7) and TPtenVDisplacement Law (4) it follows that for the mean energy U of an oscillator, a relationship of the following form exists : THE QUANTUM HYPOTHESIS 9 A comparison of (9) and (10) shows that U assumes the form required by (10) only when e is set proportional to v, the frequency. This is an essential point of Planck's Theory : if we are to remain in agreement with Wien's Displacement Law, the energy element c must be set equal to hv t = hv . . . . (11) The constant h, which, on account of its dimensions (energy x time), is called Planck's Quantum of Action, has played, as we shall see, a r61e of undreamed-of importance in the further development of the quantum theory. By combining the formulae (7), (9), and (11) the renowned radiation law of Planck follows at once : - (12) - 1 which Planck first deduced in the year 1900 in the manner above described, that is, by the hypothesis of energy quanta. In the same year as well as in the following year this Law of Kadiation was confirmed very satisfactorily by H. Rubens and F. Kurlbaum 2 for long waves, and by F. Paschen Z1 for short waves. The later measurements of radiation emitted by black bodies,28 particularly the exact work carried out by E, Warburg and his collaborators at the Eeichsanstalt, have also demonstrated the validity of Planck's formula. In opposition to this, W. Nernst and Th. Wulf& as the result of a critical review of the whole experimental material available up to that date, have recently shown the existence of deviations (up to 7 per cent) between the measured and the calculated values according to Planck's formula, and hence feel themselves constrained to decide against the exact validity of Planck's formula. Whatever view is taken of this criticism, it is at any rate a powerful incentive to take up anew the measurement of the radiation emitted by black bodies with all the finesse and precautions of modern experimental science, and thereby to decide finally the important question whether Planck's Law is exactly valid or not. For short wave-lengths, i.e. high frequencies (more exactly, 10 THE QUANTUM THEORY for high values of =-^\, Planck's formula assumes the form and thus passes over into Wien's Law (cf. formula (6), which, as we have seen, was confirmed by experiment for these frequencies). In the other limiting case, i.e. for long waves, low frequencies (more exactly for small values of ?-=, Planck's formula assumes the form *.- .... (H) as is easily found by developing the exponential function efk as a series. This limiting law, which has been confirmed in the region of long wave-lengths, had been given pre- viously by Lord Rayleigh. 30 Planck's formula thus contains Wien's Law and Eayleigh's Law as limiting cases. If we use the wave-length X instead of the frequency v, Planck's Law takes the form . . . (15) To make this clear, the intensity of radiation E^ is plotted in Pig. 1 as a function of A. for various values of T. The curves which exhibit K. v as a function of v have a quite similar appearance. The maximum of the ^-curves lies at the point at which _A has the value 4-9651. It follows that Xinax ' T = TaSr-T = 6>042 x 1 9 ? = 8 (16) 4'yoOl . K K a relation, which is identical in form with Wien's Displace- ment Law (5a). For the total radiation we get from (12) or (15) CONSEQUENCES OF PLANCK'S THEORY 11 an equation which gives expression to the Stefan- Boltzmann Law 31 (3). From (16) and (17) we recognise that the measurement (a) of the total radiation (K) and (6) of the wave-length of the maximum (A. max ), at a fixed known temperature, allows us to calculate the two constants h and k of the radiation formula. 32 From Kurlbaum's measurements of the Stefan-Boltzmann constant y, which were available at that time, and from the constant 8 of Wien's Displacement Law (measured by Lummer and Pringsheim) Planck 33 found the fol- lowing values : h = 6-548 x 10- 27 [erg.sec.] k = 1-346. 10 -"fl^l Ldeg.J (18) Corresponding to the varying values which have been found in the course of time for the constants y and 8, the values h and k have undergone changes which are not worth while recording here. For particularly _/l the measurement of p IG i_ the total radiation as we see from the strongly varying values given in note 15 has not yet reached a sufficient degree of certainty, to allow a very ac- curate calculation of the two radiation constants h and k to be based on the Stefan-Boltzmann constant. Methods which allow h to be determined with undoubtedly much greater accuracy will be described later. 5. Consequences of Planck's Theory The deduction of the radiation formula and the determina- tion of its constants did not, however, exhaust the successes of Planck's new theory ; on the contrary, important relation- ships of this theory to other departments of physics became 12 THE QUANTUM THEORY immediately revealed. For it was found 3* that the constant k of the radiation formula is nothing other than the quotient of the absolute gas constant B (which appears in the equa- tion of state of an ideal gas) and the so-called Avogadro number N, i.e. the number of molecules in a grammolecule. *-* '. . . . (19) As the value of R is sufficiently accurately known from thermodynamics Planck,** by making use of the radiation measurements, was able to calculate the value of N. By using (18) he found N = 6-175 x 10 23 . \. . (20) The agreement of this value with the values deduced by quite different methods is very striking. 86 Avogadro s Law forms the bridge to the electron theory. For it is known that the electric charge which travels in electrolysis with 1 gramme-ion, that is, with N-ions, is a fundamental con- stant of nature, which is called the Faraday. Its value was, according to the position of measurements at that time, 9658 . 3 . 10 10 electrostatic units (the value nowadays ac- cepted 37 is 9649-4 . 2-999 . 10 10 ). If now each monovalent- ion carries the charge e, of the electron, the equation Ne = 9658 . 3 . 10 10 . . . (21) must hold. From this, by using (20), we get e = 4-69 x 10- 10 electrostatic units .. (22) The value of the electron charge thus calculated by Planck from the theory of radiation differs only by about 2 per cent from the latest and most exact measurements of R. A. Millikan** who found the value e = 4-774 . 10 - 10 electrostatic units. . (23) A truly astonishing result. CHAPTEE II The Failure of Classical Statistics I. The Equipartition Law and Rayleigh's Law of Radiation IF these great successes had justified faith in Planck's Theory, it was also soon recognised as had already been emphasised by Planck in his first papers that the central point of the theory lay in the Quantum Hypothesis, i.e. in the novel and repulsive conception, that the energy of the oscilla- tors of natural period v was not a continuously variable magnitude, but always an integral multiple of the element of energy, that is e = hv. The recognition of the necessity of this hypothesis has forced itself upon us more and more in the course of time, and has become established, more especially through indirect evidence, inasmuch as every attempt to work with the classical theory has led logically to a false law of radiation. For when Planck turned the radiation problem into a problem of probability for a definite amount of energy was to be divided among the oscillators according to chance, and the mean value U of the energy of an oscillator was to be calculated it became possible to apply the methods of the statistical mechanics founded by Clerk Maxwell, L. Boltzmann, and Willard Gibbs. And the application of these methods to the case in question appeared to be demanded from the start, if the standpoint, self-evident in classical physics, that the energy of the oscillator could assume in continuous sequence all values between and CD were adopted. What, then, did statistical mechanics require? One of its chief laws is the law of the equipartition of kinetic energy, * according to which in a state of statistical equilibrium at absolute temperature T every degree of freedom of a mechan- ical system, however complicated, possesses the mean kinetic 13 14 THE QUANTUM THEORY energy ^TcT. In this expression the constant k is defined by (19), and is thus the same constant as that which appears in the Law of Eadiation. A system of/ degrees of freedom, therefore, possesses at a temperature Ta mean kinetic energy / . $kT. For example, the atom of a monatomic gas is a configuration which possesses three degrees of freedom, if we regard it from the point of view of mechanics as a mass- point. Its kinetic energy at the temperature T has therefore a mean value* %kT, independent of its mass, a result which has been known in the kinetic theory of gases since the time of Maxwell, and which is deduced as a consequence of his law of distribution of velocities. Planck's linear oscillator, which is essentially identical with an electron vibrating in a straight line, possesses one degree of freedom ; its kinetic energy at the temperature T has therefore the mean value -^kT. Now the mean potential energy of the oscillator is equal to its mean kinetic energy.* 1 As a result, its mean total energy (kinetic plus potential) has the value C7= kT . . . . (24) This result of classical statistics, when combined with the relation (7) deduced from classical electrodynamics, gives Rayleigh's Law of Radiation K v = ^kT . . ... (25) which, as we saw (cf. (14)), is contained in Planck's Law of Radiation as a limiting case for small values of pL that is, for long waves or high temperatures. This Law of Radiation of Bayleigh which, deduced as it is from the fundamental principles of classical statistics and electrodynamics, should be able to claim general validity for all frequencies and all temperatures, stands none the less in glaring contradiction to observation. For while all observed curves of distribution of energy of a black body (i.e. K v plotted as a function of v, T being constant) always show a maximum, the curve expressed by (25) rises without limit for rising values of v, and therefore gives for the sum K 2 / Kvdv an infinitely large value. FRUITLESS ATTEMPTS AT IMPROVEMENT 15 2. Fruitless Attempts at Improvement From very different quarters and in the most varied ways attempts were made, as time went on, to escape from Rayleigh's Law without discarding classical statistical mechanics. All in vain. Thus 7. H. Jeans,* 2 without making use of a "material" oscillator, considered only the radiation as such in an enclosure, and distributed the whole energy of radiation according to the Law of Equipartition over the individual " degrees of freedom of radiation " (which are here the individual vibrations that are possible in an en- closure). Further, H. A. Lorentz** deduced in a penetrating investigation the thermal radiation of the metals, starting from the conception that the free " conduction electrons," which carry the current, produce the radiation by their collisions with the atoms, and applying the Law of Equipartition to the motion of these electrons. The problem was attacked in a somewhat different fashion by A. Einstein and L. Hop/.** They imagined the Planck oscillator firmly attached to a molecule, and then considered this complex exposed to the radiation and the impacts of other molecules. The Law of Radiation could then be deduced from the condition that the impulse, which the impacts of the molecules give to the com- plex, must not on the average be changed by the impulses, which the radiation gives to the oscillator. We may also mention a paper of A. D. Fokker** which was supplemented by M. Planck.* 6 In this, by the aid of a general law due to Einstein, the statistical equilibrium between the radiation and a large number of oscillators was examined on the basis of the classical theories. All these different ways ended, however, at the same point ; they all led to Rayleigh's Law. And finally, at the Solvay Congress in Brussels in 1911, H. A. Lorentz showed, in the most general manner imaginable, that we arrive of necessity at this wrong law, if we assume the validity of Hamilton's Principle and of the Principle of Equipartition for the totality of the pheno- mena (of mechanical and electromagnetic nature) which take place in an enclosure containing radiation, matter, and electrons. Only in the limiting case of high temperatures or small frequencies do the results of the classical theory agree with the results of observation.* 8 CHAPTEE III The Development and the Ramifications of the Quantum Theory i. The Absorption and Emission of Quanta A S stated above, the conviction was bound to establish Xl^tself that every attempt to deduce the laws of radiation on the basis of classical statistics and electrodynamics was doomed from the outset to failure, and it was necessary to introduce a hitherto unknown discontinuity into the theory. It was, of course, clear that this " atomising of energy " would conflict sharply with existing and apparently well-founded theories. For if the energy of the Planck oscillator was only to amount to integral multiples of e = hv, and therefore was only to be able to have the values 0, e, 2e, 3e . . . then, since the oscillator only changes its energy by emission and ab- sorption, the conclusion was inevitable that oscillators cannot absorb and emit amounts of energy of any magnitude but only whole multiples of e. (Quantum emission and quantum absorption.} This conclusion is in absolute contradiction to classical electrodynamics. For, according to the electron theory, an electromagnetic oscillator, for instance a vibrat- ing electron, emits and absorbs in a field of radiation perfectly continuously, that is to say, in sufficiently short times it emits or absorbs indefinitely small amounts of energy. 2. Einstein's Light-quanta ; Phenomena of Fluctuation in a Field of Radiation Thus at the very entrance into the new country there yawned a gulf, which had either, in view of the previous success of the classical theory, to be bridged over by a com- promise ; or, failing this, tradition would have to be discarded and the gap would be relentlessly enlarged. Einstein felt him- 16 EINSTEIN'S LIGHT-QUANTA 17 self compelled to take the latter radical course. On the basis of very original considerations, 49 he set up the hypothesis that the energy quanta not only played a part, as Planck held, in the interaction between radiation and matter (resonators or oscillators), but that radiation, when propagated through a vacuum or any medium, possesses a quantum-like structure (Light- quantum hypothesis). Accordingly, all radiation was to consist of indivisible " radiation quanta " ; when the energy is being propagated from the exciting centre, it is not divided evenly in the form of spherical waves over ever-increasing volumes of space, but remains concentrated in a finite numbe of energy complexes, which move like material structures, and can only be emitted and absorbed as whole individuals. Einstein believed himself forced to this strange conception, which breaks with all the observations that appear to support the undulatory theory, by several investigations, all of which led to the same conclusion. He was per- suaded to this view by the result of calculations dealing with certain phenomena of fluctuation, phenomena which are familiar to us in statistics and particularly in the kinetic theory of gases. It is well known that in a gas which contains n molecules in a volume v , the spatial distribution of these molecules is far from constant, being subject to vari- ation on account of the motion of the molecules. Indeed, in principle, extreme cases are possible as that, for example, in which all n molecules are collected at a given moment in a fractional part v( hv p , i.e. v e > v p . And this is Stokes' Law. Further, another fact in the realm of phosphorescence phenomena speaks against the undulation hypothesis and in favour of that of light-quanta. According to the classical undulatory theory, all molecules of a phosphorescent body on which a light- wave impinges, should absorb energy from the wave, and thus all simultaneously become able to emit phosphorescent light. In reality, relatively only very few molecules are excited to phosphorescence at the same time, and only gradually, in the course of time, does the number of molecules excited increase. It would thus appear as if the light-wave falling on the phosphorescent body has not equal intensity along its whole front as the classical theory assumes but rather as if it consists of single energy-com- plexes thrown out by the source of light, so that the wave-point possesses, as it were, a "beady" structure, in which active portions (light-quanta) alternate with inactive gaps. This conception of the " beady " wave-front had played a part before the advent of Einstein's hypothesis of light-quanta. J". /. Thomson 8B had tried to make use of it to explain the 20 THE QUANTUM THEORY fact that, when a gas is ionised by ultra-violet light or Kontgen rays, only a relatively extremely small number of gas-mole- cules are ionised. This is a phenomenon which is quite analogous to the above-named phenomenon of phosphor- escence ; for these, too, according to Lenard's view, the exci- tation consists in the disjunction, through the agency of the radiation, of electrons from the molecules of the phosphor- escent body, and these electrons attach themselves to " storage atoms." On the return of these electrons to the parent molecules, energy is set free and sent out as phosphorescent light. The ionisation of gases by ultra-violet light or Ront- gen rays M is also capable of being explained naturally by the light-quantum hypothesis. If we suppose with Einstein, that one light-quantum hv is used up in ionising one mole- cule, then hv > /, where / is the work required to ionise one molecule, that is to say, to remove an electron from it. We have under consideration here a phenomenon which be- longs to the great branch of photo-electric phenomena,* 1 i.e. the liberation of electrons from gases, metals, and other sub- stances by the action of light. According to the hypothesis of light-quanta, in all these processes light-quanta are changed into kinetic energy of the electrons hurled off from the body. If we again adopt Einstein 's standpoint, according to which one light-quantum hv is transformed into the kinetic energy of one projected electron, we must have the following re- lation M for the energy of emission of the emitted electrons, each having a mass ra : fyn V 2 = hv - P . . . . (29) This is called Einstein's Law of the Photo-electric Effect. In this, P is the work that has to be done to tear the electron away from the atom, and to project it from the point at which it is torn from the atom up to the point at which it leaves the surface of the body. For the energy of the emitted electrons we thus obtain a linear increase with the periodicity of the light which releases them. This law, which many in- vestigators have attempted to prove, with varying success, has recently been verified by B. A. Millikan* 9 for the normal photo-electric effect w of the metals Na and Li with such a degree of accuracy that we can actually use this method for ELECTRONIC ENERGY 21 the exact determination of h. The value found by Millikan, h = 6-57 x 10 ~ 27 , is in good agreement with the value h = 6'548 x 10' 27 found by Planck from radiation measure- ments. In an entirely similar manner as was used for the phenomena of phosphorescence, the phenomena of fluores- cence in the regions of the Rontgen and visible radiations may be explained by the hypothesis of light-quanta. The in- vestigations of Ch. Barkla, Sadler, M. de Broglie, and E. Wagner 61 have shown the following : if a body is inun- dated with Eontgen rays, and if the absorption of these rays by the body is measured whilst the hardness (i.e. the frequency v e ) of the rays is varied, the absorption, as we pass from lower to higher v e , suddenly increases to a high value for a certain value of v e . At the same moment the body begins, at the expense of the energy absorbed, to emit a secondary Rontgen radiation characteristic of the body itself in the form of a line spectrum. It further appears that all lines emitted have a lower v than that of the exciting radiation. As a matter of fact, the hypothesis of light-quanta requires that the radiation-quantum hv of all rays emitted as secondary radia- tion should be smaller than the quantum hv of the primary exciting rays. For example, the region of frequencies which serves to excite the " ^-series " stretches from a sharply defined limit v k (the so called " edge of the absorption band ") upwards towards higher frequencies; whereby v k is some- what larger than the hardest known line (y) of the ^-series. In other words, the excitation of secondary Rontgen radiation by primary Rontgen rays also obeys Stokes' Law. 4. The Transformation of Electronic Energy into Light-quanta It is very significant, that the transformation of light- quanta into kinetic enei'gy of electrons is also, as it were, "reversible," that is, the opposite process also occurs in nature, by which light-quanta result from the kinetic energy of charged particles. A good example of processes of this kind is afforded by the generation of Rontgen rays by the impact of quickly-moving electrons (cathode rays) on matter. If, say, the characteristic X"-series of a certain element is to 22 THE QUANTUM THEORY be generated by the impact of cathode rays upon an anti- cathode formed of the said element, then the kinetic energy E of an impinging electron must exceed a critical value E K . For if we imagine E changed into a light-quantum hv e , then v e must fall within the region of excitation of the JT-series, and must thus be ^ V K (V K being the frequency of the edge of the absorption band). It follows that E ^> hv K ( = E K ). From this there follows an important relation between the frequency V K of the edge of the absorption band and the critical value EK of the electronic energy, i.e. the smallest value of the energy at which the electron is just able to generate the required secondary radiation. This quantum-relation E K = hv K has proved quite correct according to measurements carried out by D. L. Webster & and E. Wagner w and conversely presents, when E K and V K are sufficiently accurately known, a method for the determination of /i. 6 * Now, it is known that the cathode rays, on striking the anti-cathode, do not merely excite the characteristic Eontgen radiation, that is a line spectrum, but excite a continuous spectrum as well, the so-called " impulse radiation " (Brems- strahlung). If we therefore select any frequency v of this continuous spectrum, the ideas of the hypothesis of light- quanta immediately suggest the conclusion that a definite minimum energy E m of the impinging electrons is necessary to excite this frequency v, and that we must have E m = hv. The investigations of D. L. Webster & W. Duane and F. L. Hunt**, A. W. Hull and M. Bice^E. Wagner, M F. Dessauer and E. Back 66 have confirmed these formulae with the greatest accuracy, and thus form the foundation of one of the most trustworthy methods for the precise measurement of the magnitude h. The following values were obtained : h = 6-50 x 10-27 (Duane-Hunt) ; h = 6-53 x 10 - 2 ~ (Webster) ; h = 6-49 x 10 ~ 27 (Wagner). We also meet with similar phenomena in the visible and neighbouring regions of the spectrum. Thus /. Franck and Cr. Hertz m showed that the impact of electrons upon mercury vapour molecules can be used to excite a definite characteristic fluorescence line of mercury of wave-length A = 25364 (i.e. v = 1-183 . 10 16 ), if the kinetic energy of the HYPOTHESIS OF LIGHT-QUANTA 28 electron exceeds a certain critical value E Q . In this con- nexion they found that the relation E = hv was again fulfilled with great accuracy. 70 We shall return to these experiments and others connected with them later, since they play an important part in confirming the most recent model of the atom. 5. Other Applications of the Hypothesis of Light-quanta In a considerable number of other cases, which shall only be noticed shortly at this point, the hypothesis of light-quanta has proved of value, especially in the hands of /. Stark 71 and Einstein. Thus Stark ra has made use of this hypothesis to interpret the fact that the canal-ray particles emit their " kinetic radiation " only when their speed exceeds a certain value. He has also propounded general laws for the position of band-spectra of chemical compounds by arguing on the basis of the hypothesis of light-quanta. 78 Finally, Einstein 7 * and Stark 78 have considered photo-chemical reactions from the standpoint of the hypothesis of light-quanta and have enun- ciated a fundamental law, which has been verified, at least partially, by the detailed investigations of E. Warburg. 6. Planck's Second Theory In spite of all the successes which the quantum hypothesis of light is able to show, we must not leave out of consideration that this radical view, at least in its existing form, is very difficult to bring into agreement with the classical undulatory theory. Since on the one hand the phenomena of interference and diffraction, in all their observed minutiae, are excellently described by the wave-theory, but offer almost insuperable difficulties to the quantum theory of light, it is easy to under- stand that few scientists could make up their minds to ap- prove of such a far-reaching change in the old and well-tested conception of the propagation of light, a change that entailed perhaps its complete abandonment. This more cautious and conservative standpoint was taken up by M. Planck, who retains it to this day, inasmuch as he preferred to locate the quantum property in matter (the oscillators) or at least to confine it to the process of interaction between matter and 24 THE QUANTUM THEORY radiation while endeavouring to retain the classical wave- theory for the propagation of radiation in space. None the less, serious hindrances had already intruded themselves in the development of his first quantum hypothesis (quantum emission and quantum absorption). For H. A. Lorentz 1 ? 1 pointed out quite rightly that the conception, especially of quantum absorption, leads to peculiar difficulties. He showed that the time which an oscillator requires for the absorption of a quantum of energy turns out to belong to an improbable degree when the external field of radiation is sufficiently weak. Moreover, it would be possible to interrupt the radiation at will before the oscillator had absorbed a whole quantum. As a result of these objections Planck determined to modify the quantum hypothesis as follows. 78 Absorption proceeds con- tinuously and according to the laws of classical electrodynamics : the energy of the oscillators is therefore continuously variable, and can assume any value between and oo . On the other hand, emission occurs in quanta, and the oscillator can emit only when its energy amounts to just a whole multiple of = hv. Whether it then emits or not is determined by a law of prob- ability. But if it does emit, then it loses its whole momentary energy, and therefore emits quanta. Between two emissions its energy -content grows by absorption continuously and in pro- portion to the time. According to this second theory of Planck, which is called the theory of quantum emission, the mean energy U of a linear oscillator is ~ greater than in the first theory. 79 While in the former case the mean energy of the oscillator at abso- lute zero was equal to zero (see equation (9) from which, when T = 0, U 0), in the case of this second theory it is equal to . The oscillators retain therefore at the zero- point a zero-point energy of value - as a mean, inasmuch 2 as they assume, when T = 0, all possible energies between and hv. Nevertheless, this theory also, when the relation (7) is correspondingly modified, leads to Planck's Law of Kadiation. In the course of time Planck has made several further ZERO-POINT ENERGY 25 attempts 80 to enlarge and modify this second theory too. For example, he has temporarily assumed the emission also to be continuous, and relegated the quantum element to the excita- tion of the oscillators by molecular or electronic impacts. He has, however, repeatedly returned in essentials to the second form of his theory (continuous absorption, quantum emission). 7. Zero-point Energy In more than one direction, this theory has had further results. The appearance of the mean zero-point energy, which is peculiar to this second theory of Planck, became the starting-point of a series of researches, in which certain physicists, going beyond Planck, postulated the existence of a true (not mean] zero-point energy equal for all oscillators. On this basis, Einstein and 0. Stern 81 have given a deduction of Planck's Law which avoids all discontinuities other than the existence of this zero-point energy. In the year 1916, Nernst 82 took a still more radical step in postulating the existence of a " zero-point radiation " which was also to be present at the absolute zero of temperature and was to exist independently of heat radiation, filling the whole of space, and such that the oscillators, as well as all molecular structures, set themselves in equilibrium with it by taking up the zero-point energy. Even if we regard these views more or less sceptically, one thing cannot be ignored : many facts undoubtedly support the conception that at the absolute zero by no means all motion has ceased. We need only draw attention to the fact, that, according to the view of F. Richarz*z P. Langevin& and according to the experiments of Einstein, W. J. de Haas * and E. Beck** Para- and Dia- magnetism are produced by rotating electrons and that this magnetism remains in existence down to the lowest tempera- tures. 8. Theory of the Quantum of Action In yet another respect has Planck's theory proved stimu- lating, in virtue of a special formulation which Planck gave it 87 at the Solvay Congress in Brussels during 1911. For here Planck gave expression for the first time to the idea that the appearance of energy-quanta is only a secondary 26 THE QUANTUM THEORY matter, being only the consequence of a deeper and more general law. This law, which is to be regarded as the pre- cursor of the latest development of the doctrine of quanta, may be formulated as follows : Suppose the momentary state of a Planck oscillator, say a linearly vibrating electron, to be defined according to Gibb's method by its displacement q from its position of rest and by its impulse or momentum p, and suppose it to be represented in a q-p plane (the state- or phase-plane). Every point of the q-p plane, that is, every phase-point, corresponds to a definite momentary condition of the oscillator. The postulate is then made that not all points of this plane of states are equivalent. On the contrary, there FIG. 2. are certain states of the oscillator which are distinguished by a peculiarity. The totality of the phase-points that cor- respond to these peculiar states form a family of discrete curves which surround one another. In the case of the Planck oscillator these curves are concentric ellipses (see Fig. 2) which divide the phase-plane into ring-like strips. The postulate of the quantum theory now consists in this, that these ring strips all possess the same area h. If we calculate on this basis the energy possessed by an oscillator in one of these unique states, we find 80 that it is a whole multiple of hv. These special states (represented in the phase-plane by the points of the discrete ellipses) are, there- THEORY OF THE QUANTUM OF ACTION 27 fore, according to Planck's first theory, the only dynamically possible and stable states of the oscillator. If an oscillator emits or absorbs, its phase-point jumps from one ellipse to another. The state of affairs is different if we accept Planck's second theory. According to this, all conditions of the oscil- lator, that is all points on the phase-plane, are dynamically possible. On the other hand, emission takes place only in the states specially distinguished by the ellipses. Seen from this new point of view, the energy-quanta are, therefore, only a result of the partitioning of the phase-plane. Mathe- matically, we may express this " structure of the phase-plane " thus : the ?ith unique curve encloses a surface of area nh, or, in symbolic language, I" [dqdp = \ nh . '. (30) The double integral is taken over the surface ; the single integral is taken around the boundary curve of the nth ellipse. On this basis for systems of one degree of freedom, which is called Planck's theory of " the action-quantum " for h has the dimensions of an action the modern extension of the quantum theory for several degrees of freedom has, as we shall see, been erected. Further, a line of argument proposed and developed by A. Sommerfeld takes its origin here. Starting from the fact just mentioned, that Planck's constant h possesses the dimen- sions of action (energy-time), Sommerfeld set up the hypo- thesis w that for every purely molecular process, say the release of an electron in the photo-electric effect, or the stopping of an electron by the anticathode in the generation of Eontgen rays, the quantity called action (L - V)dt, known to us V J from Hamilton's Principle, has the value - . Here L and V Air are the kinetic and potential energies of the electron respec- tively, T is the duration of the molecular process 4 ! say, for example, the time which is required for the release of the electron from the atomic complex during the photo-electric effect, or the stopping of the electron by the anti-cathode. 28 THE QUANTUM THEORY This formulation of the quantum hypothesis is, as it were, an expression of the well-known fact that large amounts of energy are absorbed or given up in short times, whereas small amounts are absorbed or emitted in longer times by the molecules, so that on the whole the product of the energy transferred and the duration of the time of exchange is a constant. In fact, fast cathode rays, for example, are stopped by matter in a shorter time and therefore generate harder Kontgen rays than slow cathode rays. Sommerfeld has applied his theory successfully to the mechanism of the generation of Rontgen rays and y-rays. 90 Sommerfeld and P. Debye w have worked out on the same basis a theory of the photo-electric effect, which, like the hypothesis of light- quanta, also leads to Einstein's Law (29). CHAPTEK IV The Extension of the Doctrine of Quanta to the Molecular Theory of Solid Bodies 92 i. Dulong and Petit's Law IT was a particularly fortunate circumstance for the con- solidation of the doctrine of quanta that the failure of classical statistics was not confined to the theory of radiation, but, as appears later, extended to the molecular theory of solid bodies. Thus there arose in quite another field a strong sup- port for the quantum hypothesis, namely, in the field of Atomic Heats. The Atomic Heat of a substance (in the case of poly- atomic bodies we say the " Molecular Heat ") is defined as the product of its specific heat and its atomic weight (or molec- ular weight) ; or, otherwise expressed, it is that amount of heat which must be communicated to a " gramme-atom " m (or gramme-molecule) of the body, in order that its tempera- ture may be raised by one degree. According to our present conceptions, the thermal content of a monatomic solid, say a crystal, is nothing more than the energy of the elastic vibrations of its atoms, which are arranged in the form of a space-lattice, about their positions of equilibrium. If we apply classical statistics to these vibrations, and particularly the law of equipartition of kinetic energy, we arrive at the following conclusion : The mean kinetic energy of an atom 01L/TT vibrating in space, i.e. with three degrees of freedom, is , and its mean potential energy is equal to the same amount, 94 so that its total energy is therefore 3kT. If we now consider 1 gramme-atom of the body, that is, a system of N atoms (where N is the Avogadro number, approximately 6 x 10 23 ), we get for the mean energy of the body, remembering (19), E = 3kTN = 3RT . . . (31) 29 30 THE QUANTUM THEORY where R is the absolute gas-constant. It follows that the atomic heat of the body at constant volume becomes : 0,.^-3*-fr94[] . . (32) This is the law of Dulong and Petit, 9 * according to which the atomic heat (at constant volume) of monatomic solid bodies has the value 5'94 jj , independently of the temperature. 96 This law is actually obeyed by many elements more or less closely. 97 On the other hand, elements have long been known which are far from following this rule, and which show systematic differences, especially at low temperatures. Thus, as early as the year 1875, F. H. Weber & found that the atomic heat of diamond at - 50 C. is about 0'75 ?^. The deg. atomic heats of other elements as well (boron, beryllium, silicon) have also been shown to be much too small at ordinary temperatures. And altogether it appeared that the defect from Dulong and Petit' 's normal value occurs quite generally at low temperatures, and becomes the more pro- nounced, the lower the temperature. The classical theory offered no solution of these low values of the atomic heat. 99 2. Einstein's Theory of Atomic Heats Einstein was the first to recognise 10 that in this case, too, the quantum theory was destined to solve the difficulty. Precisely as in the theory of radiation, the method of classical statistics leads of necessity to a wrong law in the field of atomic heats. Hence, here also, we must abandon the laiv of the equipartilion of energy. In fact, we need only imagine electric charges distributed among the atoms 101 and then we see that, exactly like the Planck oscillators, they must set themselves in equilibrium with the heat-radiation which is always present in the body. This means, however, that the relation (7), according to which U = ~a K,., must be set up between the mean energy U of an atom vibrating linearly with frequency v, and the intensity of radiation K,,. If we now take Planck's radiation formula (12) as empirically EINSTEIN'S THEORY OF ATOMIC HEATS 31 given, it follows immediately that the mean energy U of the linearly vibrating atom must possess, not the value kT given by classical statistics, but the value given by the quantum theory, namely, U = hv For the atom which vibrates in space we get, therefore by an obvious generalisation in place of the classical value 3kT, the quantum value : ^j ^- p-- .. X / / I / 1 / . o 4 1 j 0.3 a, * ) 2-Lv aJ according as to whether we are considering transverse or longitudinal frequencies. The series of overtones therefore extends without limit to infinity. In reality, however, as the body consists of N atoms (mass-points), it may not possess more than 3N natural frequencies. In order to attain this, Debye helps himself out by means of the following bold supposition. Instead of calculating strictly the elastic spec- trum of the real body consisting of N atoms, he replaces it by that of the continuum as an approximation, but breaks it off arbitrarily at the 3Nth natural period. Debye thus gets the greatest frequency v m which occurs, that is, the upper limit of the elastic spectrum, from the condition : therefore [9JV I "(H) . (44) The atomic heat of the body, which follows from (42), is 40 THE QUANTUM THEORY a result which can easily be brought into the following more simple form : 1M The atomic heat is therefore only a function of the magnitude x m , that is, it depends only on the ratio ^ : here ^ = ^ This result may be expressed in Debye's terms thus : reckon- ing the temperature T as a multiple of a temperature which is characteristic of the particular body, then the atomic heat is represented for all monatomic bodies by the same curve. Hence we must be able to bring the C curves of all monatomic bodies into coincidence, if only the scale of temperature be suitably chosen for each substance. 128 For high tempera- tures, the Debye formula passes over, as it must do, into the classical value of Dulong and Petit, C v = 3.R, 128 just as do the Einstein and Nernst-Lindemann formulae. On the other hand, it differs from these latter in falling much more slowly at low temperatures. For while the atomic heats, according to both Einstein and Nernst-Lindemann, fall exponentially / 1 constN (with 7- e ~~T~) at low temperatures, Debye's formula leads to the fundamental law, 127 that the atomic heats of all bodies at low temperatures are proportional to the third power of the absolute temperature. It is further remarkable, that we may write formula (44) for the maximum natural frequency in a form such that only measurable magnitudes occur in it. For if we express the two velocities of sound Ct and c\ in terms of the elastic constants of the body, and replace the volume Fof the gramme-atom by ,. . atomic weight (A) . the quotient -- density g (p) ( >, * follows that ,5-28.10*.^) where " ' DEBYE'S THEORY OF ATOMIC HEATS 41 In it K is again the compressibility of the body, where JL i (53) r ( > = kv ; r<*> = kv These still very complicated formulae may, according to Born, be brought into a very simple and comprehensive form LATTICE THEORY OF ATOMIC HEATS 51 by limiting our considerations to low temperatures and in- troducing certain approximations. As we have already re- cognised, at low temperatures only the long waves contribute to the energy-content. Hence we shall apply in formula (53) all those approximations which are introduced by confining ourselves to long waves. Let us consider first r ( ;p. Here we set in place of the v{S of (50) the constant values v., which are independent of the wave-length A. and of the wave- direction. If we do this, we can place the constant factors in front of both integration signs, and write !-*> V^,. FU>3 , where,, The factor in square brackets has, however, a simple meaning. From the law of distribution of the natural periods we see, namely, that this factor gives the sum-total of all natural frequencies that occur in one of the 3s branches of the spectrum ; it therefore has the value N, which as has already been said, is the number of basic groups which go to make up the crystal. If we choose the piece of crystal under con- sideration such that its size is so that N is equal to the Avogadro number, then if we remember that Nk = R for I* 2 ), the expression 3 follows. If we compare this result with (34) we see that rop excepting for the missing factor 3 consists of 3(s - 1) Einstein functions. We write the expression in the form M where ^ =_^ . (55) in which the abbreviation is obvious. The fact that, in using 52 THE QUANTUM THEORY these approximations, we come across Einstein factors, i.e. that we encounter the " monochromatic " theory, might have been anticipated. For since we treated the v/s here as con- stants that are quite independent of wave-length and wave- direction, these vibrations represent processes which have nothing to do with the propagation of elastic waves in the crystal as a whole : and this means that the individual particles, uncoupled as it were, perform 3(s -- 1) mono- chromatic vibrations. The approximate evaluation of the first part rW is quite different. For here we have to use for the frequencies v i V 2> V 3 * ne relations (49), which connect the three acoustic natural frequencies with wave-length and wave-direction. Here we have therefore to deal with three real elastic oscilla- tions, which are propagated in the crystal with the three different acoustic velocities q-^ri), q*(ii), q s (n), each of which depends on the direction (n). The crystal acts here as a dynamic whole, exactly as in Debye's point of view. Hence we may conjecture that r ( J> allows itself to be brought into the form of three Debye functions (45). The more exact calculation confirms this supposition, and gives us 1M (56) which, taking Debye's formula (45) into consideration, we may write in the following immediately intelligible form : I^-IDft . . (57) i=l The three magnitudes Xi here play the part of three upper limits of frequency. Their values are where the three magnitudes qt represent certain mean direc- tions of the acoustic velocities, which therefore no longer LATTICE THEORY OF ATOMIC HEATS 53 depend on the wave-direction. From (55) and (57) we get for the thermal capacity of the piece of crystal considered - (59) Now, since JV particles of each of the s different kinds of particles are present, that is one gramme-atom of each kind of particle exactly for N is the Avogadro number the piece of crystal contains s gramme-atoms of different sorts of particles. If, therefore, we cut the crystal into s equal N pieces in such a manner, that each piece comprises only - basic groups, then each of these pieces contains a so-called "mean" gramme-atom. Hence if we now consider only a p single one of these pieces, its thermal capacity is ; we call S it the " mean atomic heat " C v , and we may write 3 D + (* . . (60) Here the #/s have the same meaning as in (58). For the N piece of crystal now under consideration consists of basic s groups, and has therefore the volume . Formula (58), however, obviously remains unchanged when we replace in it N and V by and . The quantity -, the volume of a s s s mean gramme-atom, is also called the mean atomic volume. In the case of chemical compounds, in which several sorts of atoms occur in the basic group, and also in the case of polyatomic elements, in which the basic group contains several particles of a like sort, we frequently speak of the molecular heat. In doing so, we follow the usual chemical conception, inasmuch as we imagine the s particles of the basic group divided into one or several sub-groups, and regard each sub- group, taken alone, as a molecule. If then the molecule 54 THE QUANTUM THEORY contains q atoms, then qC,, is the mean molecular heat; for example, the basic group of rock-salt (NaCl) contains one sodium ion and one chlorine ion. The whole piece of crystal, which, by definition, contains - = basic groups, comprises S A therefore sodium ions and the same number of chlorine ions, that is to say - " NaCl-molecules." q is in this special case equal to 2. Hence 2C,, represents the thermal capacity of N " NaCl-molecules,'' that is, the mean molecular heat of rock-salt. If among the s particles of the basic group there are p atomic residues and s p electrons, the number of Einstein factors in (59) reduces to 3(p - 1), since the 3(s - p) ultra- violet frequencies arising from the s - p electrons contribute only in a vanishingly small degree to the atomic heat as com- pared with the infra-red. We thus arrive at the law : the mean molecular heat of a crystal wJiose basic group includes p (similar or different) atomic residues, is made up, at a suf- ficiently low temperature, of three Debye terms (with, in general, three different upper limits of frequency) and 3{p - 1) Einstein terms (in which the 3(p-I) infra-red natural frequencies for long waves appear as frequency numbers). When we descend to the lowest temperatures, the Einstein terms disappear exponentially, and only the three Debye terms remain, for these, as we know, decrease much more slowly. In them we can further replace all the upper limits of the three integrals (see (56)) by GO , so that the integrals thereby become numerical constants. Eemembering (58) we get the fundamental law, that the molecular heat of every crystal at the loivest temperatures is proportional to the third power of the absolute temperature. So the general lattice theory con- firms Debye's result. The formula obtained has the following simple form : 141 where VA is the " mean atomic volume " mean atomic weights - _ - . - i* - 1 TESTS OF THE BORN-KARMAN THEORY 55 and q represents a quantity which, if suitably defined, may be called the mean acoustic velocity, introduced in place of the three different acoustic velocities q lt q 2 , q s . Also in the other extreme case, for high temperatures, a very useful formula can be obtained, as H. Thirring iK showed. He started from (52) and developed the exponential functions in series. The following value is then obtained for the mean atomic heat : where the coefficients J v J 2 , J 3 , . . . depend in a complicated manner on the elastic constants of the crystals, the atomic masses, and the atomic distances. 10. Tests of the Born-Karman Theory How do matters stand with regard to the testing of the Born-Kdrmdn Theory? We see at once that it is incom- parably more difficult than in the case of Debye's Theory : for even in simple cases, the calculation of the mean atomic heat of a crystal is very complicated, and requires above all a more exact knowledge of its elastic behaviour than we at present possess. Only by restricting our attention to low and very low temperatures on the one hand, where the formulae (60) and (61) may be applied, and, on the other, to the region of high temperatures, within the limits of applicability of Thirring's formula (62), are we enabled to carry our calculations for a number of simple substances to the point of comparison with experimental results. Born and Kdrmdn themselves, in one of their first publications 14a tested the formula (61), valid for the lowest temperatures (Debye's TMaw), by comparing its results with those of experiment. They limited themselves in this case to metals (Al, Cu, Ag, Pb) which, however at any rate in the usual form are not proper crystals, but irregular crystalline aggregates. For this reason, they proceeded as if the metal were an isotropic body, and obtained the mean acoustic velocity the only quantity in (61) which in general requires 56 THE QUANTUM THEORY extensive calculation from the following relation which holds for isotropic bodies : 1H Q 1 O N+! < 68) Here q l and q t are the velocities of propagation of the longitudinal and transverse elastic waves, magnitudes, there- fore, which may be simply calculated from the two elastic constants of the isotropic body and its density. 1 * 9 The agreement of the values of C v thus found with the experi- mental data is, especially in the case of Al and Cu (and also Pb), quite good. A. Eucken w has, however, pointed out rightly, that no weight should be attached to this agreement. For the values of the elastic constants which Born and Kdrmdn used for calculating q t and q t are those which are correct at the ordinary room temperature. If we take their dependence on temperature into account, the good agreement between theory and experiment disappears. Metals are, indeed, not isotropic bodies, and hence it is not permissible to use the observable elastic constants, which depend upon temperature, in calculating q. Matters are much more favourable in the case of real crystals, in which, as experiments by E. Madelung i show, the elastic constants vary very little with temperature. But here the calculation of the mean acoustic velocity q gives rise in general to notable difficulties, 1 * 8 which may, however, be cleared away in simple cases by a very practical method due to L. Hopf and G. Lechner. 9 Eopf and Leclmer were thus enabled successfully to carry out the calculations for sylvin (KC1), rock-salt (NaCl) fluor-spar (CaF 2 ) and pyrites (FeS 2 ). They proceeded to calculate the quantity q from the observed values of C v , assuming the correctness of formula (61), and they then compared these with the value of q calculated from elastic data. The result showed very satisfac- tory agreement. 1 o It is of particular interest to test the very clear formula (60) which gives the mean atomic heat as a sum of three Debi/e, functions and 3(s - 1) Einstein functions. Here the three infra-red natural frequencies v, vj?, v coincide, and the TESTS OF THE BORN-KlRMAN THEORY 57 three Einstein functions become equal to one another. If we introduce the further approximation of replacing the three different quantities Xi in the Debye formula by a mean value x, it follows that C, = \{D(x) + E(x)} . . . (64) In this we use the value of x deduced from formula (58) by merely replacing JT, I must decrease with =f a hypothesis which, as we shall see, has latterly been upheld by several investigators. Now, although the agreement between theory and experi- ment could thus be compelled by special assumptions at high temperatures, the region of low temperatures revealed itself 64 THE QUANTUM THEORY as the vulnerable point of the theory. For experiments by H. Kamerlingli-Onnes m in the laboratory for low tempera- tures at Leyden had shown that the resistance of metals at very low temperatures (the experiments extended as far down as 1'6 abs.) falls away to a quite extraordinary degree, and practically disappears before the zero-point is reached. At any rate, the resistance cannot, as follows in view of what has just been said from formula (67&), sink proportionately to only the first power of the temperature ; on the contrary the fall is without doubt proportional to a higher power. That the Wiedemann- Franz Law also ceases to be valid in this region, has been proved by experiments of C. H. Lees 174 and W. Meissner. In order to escape from all these difficulties the quantum theory was appealed to, and attempts were made, in the most varied ways, to make it harmonise with the existing theory. A first attack was ventured by W. Nernst 176 and Kamerlingh- Onnes, 111 who gave for the resistance of the metals empirical formulae which linked up directly with the form of Planck's energy equation (9) and which gave the change in the resist- ance with temperature satisfactorily. F. A. Lindemann m and W. Wien 119 conceived more detailed theories. Linde- mann accepts in his first paper J. J. Thomson's hypothesis, according to which N is proportional to >JT, and retains the equipartition law for the motion of the electrons, so that q also becomes proportional to JT. Then, according to (676), the variation of the resistance - with the temperature depend* , then the corresponding momentum or impulse p is known to be none other than the moment of momentum. 193 It follows from this, since p is independent of , that 2irp* = rih . . . (82) in agreement with (81). HEAT OF ROTATION OF DIATOMIC GASES IS In the same way, on the basis of Planck's first theory, namely, the conception that the special quantum rates of rotation v n are the only possible ones, and using the dumb-bell model, the author 196 has recently carried out the strict calcula- tion for structures with two degrees of freedom (free axes of rotation), making use of the later ideas of the quantum theory. This stricter method likewise gives us curves for the rotational heat which are useless, for they also have a maximum and a subsequent minimum, as in Ehrenfest's case. Only by making special subsidiary assumptions, such as excluding certain quantum states, can we get curves which rise steadily with increasing temperature, and which agree, at least to a certain extent, with observation. 197 . "* . f-r E, E r E' r E r FIG. 7. Not much more satisfactory results were obtained in those investigations which, again with the use of the dumb-bell model, were based on Planck's second theory. According to this theory, the discrete values v n of the rotational speeds are not the only possible ones ; on the contrary, the molecule can rotate with all rotational speeds between and oo , and hence can assume all values of rotational energy between and oo, exactly like the Planck oscillators in Planck's second theory. The peculiarity of the special quantum values (80) for the energy here consists in the following : imagine the energies E r plotted as abscissae (Fig. 7) and the corresponding prob- abilities w as ordinates ; then a step-ladder curve results, 74 THE QUANTUM THEORY the steps of which lie exactly at the values Ety. The prob- ability that a given value E r of the rotational energy will appear is therefore constant within the range of energy between E (r ? and JB,"* 1 * but changes suddenly at the ends of this range. According to Planck's first theory, which allows only she quantum values E%\ the encircled points alone have a meaning. Only at those points is the probability other taan zero, while all intermediate values of the energy possess the probability zero, that is, do not occur. In this case, too, the problem was first solved for one degree of freedom (fixed axis of rotation). E. Holm lK and J. v. Weyssenhoff 199 found, in agreement with one another, a steadily rising curve for the rotational heat, which fitted the observations well at low temperatures, but undoubtedly went too high at higher temperatures (from about 140 abs. up- wards). But when the modern development of the quantum hypo- thesis for several degrees of freedom, to which we shall be introduced later, was available, a stricter calculation for free axes of rotation, i.e. for two degrees of freedom, could be carried out. This problem was attacked on the one hand by M. Planck,* 10 on the other by Frau S. Rotszayn but was treated differently in each case. Planck started with the premise that this problem belongs to the category of so-called " degenerate " problems. This term is to convey the follow- ing: the molecule rotates, when no external forces act on it, according to the doctrines of mechanics, with con- stant speed in a spatially-fixed plane. The position of this plane in space must, so Planck argues, be of no im- portance for the statistical state of the molecule. Hence the condition of rotation of the molecule in the sense of the quantum theory is determined by a single quantity, namely, the rotational energy. In spite of the fact, therefore, that the problem is originally and naturally a problem of two degrees of freedom for the position of the molecule in space is determined by two angles we must, according to Planck, treat it in the quantum theory as a problem of only one degree of freedom. The two degrees of freedom coalesce, as it were ; they are " coherent" In contrast to this, Frau Rotszayn proceeds to turn the HEAT OF ROTATION OF DIATOMIC GASES 75 problem into a non-degenerate one by the addition of an ex- ternal field, and after solving this problem, reduces the field of force till it vanishes. This method which was also used by the author in the paper above cited, appears to be par- ticularly advantageous, when the calculation is based on Planck's first theory, for peculiar difficulties arise in " degen- erate " cases. Success here decides in favour of the second method. For while Planck finds a curve 2 02 which rises above the classical value to a maximum, and then descends asymp- totically towards the value R and is therefore of no use the calculation of Frau Rotszayn gives a steadily rising curve, which agrees well with the measurements for lower and higher temperatures ; only the value observed at T = 197 abs. lies about 10 per cent too low. 203 While all the above-mentioned investigations are based on the dumb-bell model, which can only be regarded as a pro- visional, schematic construction, P. S. Epstein* in 1916 carried out the corresponding calculations for another mole- cular model proposed by N. Bohr.w* This model of the hydrogen molecule, to which we shall return later, is built up of two positive hydrogen atoms, each of which carry a single positive charge, and around the connecting line of which two electrons, diametrically opposite, rotate in a fixed circle at a fixed rate (see Fig. 8). Since the equilibrium in this purely electrical system is determined by the play of the Coulomb attractions and the centrifugal forces, and since the radius of the electron is determined by a quantum condition, this model possesses the advantage that all its dimensions are completely 76 THE QUANTUM THEORY fixed, so that there is no longer any question of the arbitrari- ness of the moment of inertia. The " dumb-bell knobs " are represented here by the two positively-charged hydrogen atoms; the rotations of the molecule hitherto considered would therefore correspond to those motions in which the molecule rotates with a moment of inertia J" about an axis at right angles to the line joining the atoms. But to this there must very plainly be added the rotation of the system about the axis of symmetry (i.e. the line joining the atoms), which results from the extremely rapid rotation of the elec- trons. The moment of inertia corresponding to this axis is, in consequence of the extremely small mass of the electrons, very small compared with J. The whole system obviously possesses, if we regard it approximately as rigid, the properties of a symmetrical top. Its motion is therefore, in consequence of its own rotation about the axis of symmetry, not a rotation, but instead the well-known motion, "regular precession." 206 Epstein treated the problem from this point of view but could not obtain agreement at low temperatures with the moment of inertia calculated from the model itself, 207 namely, / = 2 -82 x 10 ~ 41 . Presumably, this failure depends on the fact that the model does not correspond with reality, and in fact we shall see later, that well-founded doubts have arisen as to the correctness of the Bohr model. We must therefore admit, unfortunately as one of a number of instances in the quantum theory, that the important problem of the rotational heat of hydrogen still awaits solution. 2. The Bjerrum Infra-red Rotation-spectrum N. BjerrumM* has applied the relation (79) in a very interesting manner to the infra-red absorption of polyatomic gaseous compounds. These gases (for example HC1, HBr, CO, H 2 in the form of steam, but on the other hand not the elementary gases H 2 , 2 , N 2 , C1 2 ) show, according to the investigations of S. P. Langley, F, Paschen, H. Eubens, 211 H. Bubens and E. Aschkinass, H. Eubens and G. Hettner, W. Burmeisterpu Eva v. Bahr,* u extensive absorption bands in the short- and long- wave infra-red. While in the long- wave infra-red we account for the absorption by the rotating molecule, which, composed of positively and negatively INFRA-RED ROTATION-SPECTRUM 77 charged atoms, act like electric double poles and hence in turning emit and absorb radiation, Bjerrum was the first to point out that the molecular rotation must also make itself noticeable in the short-wave infra-red. For if there exists in this region a linear vibration V Q of the ions in the molecule relatively to one another and hence an absorption at this point and if, in addition, the whole molecule rotates at the speed v r , then it is known that there will be produced as a result of the composition of the vibration with the rotation 218 two new vibrations (and, correspondingly, two new regions of absorption) having the periods v e + v r and VQ - v r , sym- metrically disposed on both sides of the ionic vibration VQ. On the whole, then, we have three points of absorption : v r , VQ ~ VT, vo + v r , to which we must add the non-rotational state VQ as a fourth. But if now, according to Planck's first theory, the molecule can only rotate with discrete speeds of rotation v n [see (79)], we get symmetrically to the original position of absorption v = VQ and, on both sides of it, a series of further discrete equidistant positions of absorption : v = i/o + v n = v + n~-} * \(n - I, 2, 3 . . .) . (83) v = v - v n = v - n These discrete equidistant positions of absorption have actually been found by Eva v. Bahr in the case of water vapour and gaseous hydrochloric acid, and were measured later with still greater accuracy by H. Rubens and G. Hettner for water vapour. In an examination carried out on an extensive scale E. S. Imes zl1 has once more thoroughly investigated the hydrogen halides (HC1, HBr, HF) and con- firmed the law (83) for the position of the absorption lines. It was thereby found that the middle line VQ was always missing. From the standpoint of the theory here described this would mean that the non-rotational state does not exist, that is, that the molecules always rotate (zero-point rotation). A. Eucken? 1 * who discussed the results of E, v. Bahr, which were at that time the only ones known, deduced from the good agreement between observation and calculation that Planck's second theory is not valid, for the experiments 78 THE QUANTUM THEORY seemed so obviously to prove that the molecule can actually only rotate with the discrete speeds v n . This conclusion, however, is not inevitable, as Planck 219 showed in a pene- trating investigation. On the contrary, the observations may after all be explained, surprising as it may seem, on the basis of his second theory (continuous " classical " absorp- tion ; all speeds of rotation possible). This curious result is explained as follows : Let w(E r )dE r be the probability that a molecule possesses exactly the rotational energy E r ; hence for N molecules Nw(E r )dE r will be the number that will possess exactly the rotational energy E r . These molecules rotate therefore according to (78) with the speed _ 1 l2E r "" \ ular weight M = mN, and set k >: then C = C + a log M, where C = log ^7 = 10'17 If we finally use the base 10 instead of the natural base e for our logarithms, and measure the vapour pressure not in absolute measure but in atmospheres, we get the chemical constant C' used by Nernst, which is related to Sterns vai-ie for C thus : C' = xJL* = 6-0057 For it we finally get the simple expression C' = C' + 4 Iogi M, where C' = - 1-59 . (91) This formula has been brilliantly verified by experiment. The hitherto most trustworthy measurements of vapour pressure and chemical states of equilibrium give in the case of hydro- gen, argon, and mercury the values - 1-69 0-15, - 1-65 0-06, - 1-62 0'03 We are therefore justified in saying with Stern that the expression (90) for the chemical constant of the monatomic gases is theoretically and experimentally one of the best founded results of the Quantum Theory. CHAPTEE VI The Quantum Theory of the Optical Series. The Development of the Quantum Theory for several Degrees of Freedom 23 i. The Thomson and the Rutherford Atomic Models THE greatest advance since M. v. Lane's discovery of the method of Eontgen-spectroscopy for determining crystal structure was made in the realm of atomic theory in 1913, when the Danish physicist Niels Bohr placed the atomic models in the service of the quantum theory. Bohr's labours have in their turn reacted on the quantum theory and fertil- ised it, and thus a marvellous abundance of notable successes have been achieved in recent years through the interaction be- tween the dynamics of the atom and "the quantum hypothesis. Among serviceable atomic models, the one proposed by J. J. Thomson long occupied a much favoured position ; accord- ing to it, the electropositive part of an atom, which constitutes the most important part of its mass, is supposed to be a sphere of "atomic dimensions" (radius about 10 ~ 8 cms.) filled with a positive space charge in the interior of which the negative parts, the electrons, rest in a stable position of equi- librium. This model has the great advantage of explaining on purely electrical grounds the possibility of " quasi-elastic- ally bound " electrons, i.e. such electrons as, being displaced -*(?-*} < 96 > where N is again the Rydberg number as defined in (98). If we here set s = 3, n = 4, 5, 6 . . . we get the so-called " principal series of hydrogen " which was observed by Foivler 2fl2 and very recently measured with great care by F. Paschen. 3 For s = 4, n = 5, 6, 7 . . . we get the so-called " second subsidiary series of hydrogen," which was observed by Pickering 28* and Evans. m Both series were, before the advent of Bohr's Theory, falsely ascribed to hydrogen. A new and extremely noteworthy result of Bohr's Theory is revealed, if we allow for the movement of the nucleus in our calculations. For, in reality, the nucleus is not stationary, but nucleus and electron revolve about their common centre of gravity. By taking this fact into account we are led to a slightly altered expression for the Rydberg constant. In place of (95) we get the formula in which M denotes the mass of the nucleus. It follows from this that for different elements, for instance, hydrogen and helium, the Rydberg constant differs somewhat and is smaller for hydrogen than for helium (since M H < Mff e ). In general, the value of the Rydberg constant increases with increase of atomic weight tending towards a limiting value. All this is in perfect agreement with the results of many years of spectroscopic research. In the same way as emission, absorption has a quantum- like character, according to Bohr's model. If light, say of the first Balmer line (E a ), falls upon a hydrogen atom, a quantum hv of this external H a radiation is used to "raise" the electron into the third quantum orbit. An amount of energy hvff a is taken from the external radiation, that is, light from the line H* is absorbed. STRUCTURE OF THE PERIODIC SYSTEM 91 4. The Structure of the Periodic System Even in his earliest papers Bohr endeavoured to construct for the higher elements as well (Li, Be, B, C, etc.), in con- nexion with the Periodic System, suitable atomic models with several rings of electrons, each occupied by several electrons, in which, for example, the well-known octaves of the system are reproduced by a regular arrangement of the external electrons which recurs at every eighth element, while the number of the electrons revolving in the outermost ring is equal to the valency of the element in question. W. Kossel 286 arrived at a similar structure of the atoms as a result of a profound investigation of the formation of mole- cules from atoms. Also, L. Vegard, 251 A. Sommerfeld* and B. Ladenbnrg 2 s9 have constructed analogous atomic models, particularly taking into account the well-known up-and-down curve of atomic volumes, and using them to explain other periodically varying properties (paramagnetism, ionic colour). These considerations, although they are tending indisputably along the right lines as far as the general principles are con- cerned, are not yet firmly established in detail. 5. The Quantum Hypothesis for Several Degrees of Freedom While the quantum hypothesis in its most primitive form demonstrated in this way its innate power by entering the field of atomic dynamics, it had, in doing so, gained little as far as its own development was concerned. But the fruits of Bohr's Theory ripened more rapidly than could have been divined. Already the year 1915 brought a decisive develop- ment : almost simultaneously, Planck and Sommerfeld inde- pendently found the solution of a problem that had long been a burning question, namely, the extension of tlie quantum theory to several degrees of freedom. Sommerfeld^ retained a close connexion with Bohr's Theory in attacking the problem. The first main condition of this theory related to the choice of " allowable " stationary orbits among all those mechanically possible. According to this, as we saw, only those orbits were allowed for which the moment of momentum (Impuls- moment) p is a whole multiple of -. This may also be 92 THE QUANTUM THEORY expressed according to (81) and (82) thus : among all mechan- ically possible paths, only those are allowable and stationary for which the pJuise-integral fulfils the condition : nh . (98) In this quantum condition we are to replace according to (82) the general co-ordinate q by the angle of rotation (the " azimuth ") , the impulse ^> by the " impulse (or momentum) corresponding to ," namely, p^ (the moment of momentum). The integration is thereby to be extended over the whole range of values of the variable q, that is, in the present case, from to 2*-. In the case of the original Bohr Theory, which considers only circular orbits, there naturally exists only a single quantum condition, namely, that for the case q = <, since the angle of rotation is the only variable of the path. Matters are otherwise, when we reject the limitation to circular orbits, and hence take .STe^/cr-ellipses into account. Then each point of the path is determined by two variables, namely, by the distance r of the electron from the nucleus, which is at the focus of the ellipse, and by the angle (the " azimuth ") which r makes with a fixed direction (say with the straight line, which joins the nucleus to the perihelion). In this case we are presented with a problem of two degrees of freedom, with two generalised co-ordinates, r and $ (polar co-ordinates). The simple extension of the quantum hypo- thesis by Sommerfeld now consists in setting up in this case' two quantum conditions of the form (98), one for the co- ordinate <, which agrees with the single quantum condition of Bohr's Theory, and a new one for the co-ordinate r, so that the selection of the stationary orbits is here determined by the two following equations : nh. . . (99) n'h . . . . (100) n and n are here whole numbers, p$ and p,. are the impulses (momenta) corresponding to the co-ordinates and r. 261 The THE QUANTUM HYPOTHESIS 93 integration in (100) is to be taken over the full range of values of r, that is, from the smallest value r m i n (perihelion) to the greatest value r max (aphelion) and back to the smallest fmin. (99) is called the azimuthal quantum condition, n being the azimuthal quantum number ; (100) is the radial quantum condition, ri the radial quantum number. In a corresponding manner the extension may be carried out for more than two degrees of freedom. If the system has / degrees of freedom, and if it is therefore characterised by the / generalised co-ordinates q v q.^, q s . . . and the corre- sponding impulses p v p. 2 , p 3 . . ., then the " allowable " movements of the system are limited by the / quantum conditions : \Pi d( li = n i h > p-A = nji, - - \Pfdqj- = njh . (101) (n v n. 2 . . . HJ- are positive whole numbers). In every one of the / phase-integrals the integration is to be performed over the full range of values of the co-ordinate in question. A difficulty, which arose here from the outset, was the question as to which co-ordinates ought to be chosen for the application of the quantum rule (101), or whether the choice is immaterial. In general, we may characterise a system of several degrees of freedom by various types of co-ordinates ; for instance, we may describe the Kepler movement of the electron either by polar co-ordinates r and , or by Cartesian co-ordinates x and y. This question is the more urgent, when one considers that the separate phase-integrals Ip^; do not really become constants for every choice of co-ordinates, as is required by the quantum rule (101) , 262 P. S. Epstein 263 and K. Schwarzschild 2 ^ have solved, independently of one another, this problem of the " correct choice of co-ordinates " to a certain extent. Incidentally, an interesting and sur- prising relation of the quantum rules (101) to a long-known theorem of classical dynamics was revealed, which had been propounded by Jacobi and Hamilton, and had hitherto been successfully applied in celestial mechanics. Finally, quite lately, A. Einstein, 26 * by modifying the expression (101), has 94 THE QUANTUM THEORY put forward a quantum hypothesis which has the advantage of being independent of the choice of co-ordinates. But a closer discussion of these abstract investigations would lead us too far here. The second formulation of the quantum hypothesis for several degrees of freedom is due, as already mentioned, to M. Planck.* It is, as it were, more cautious in its nature than the more radical attack of Sommerfeld. Planck, con- tinuing directly from the division of the phase-plane of linear oscillators already discussed, starts from the so-called Gibbs phase-space to deal with more complicated systems. For a system of /degrees of freedom, which is characterised by the co-ordinates q v q. 2 . . . qy and the impulses p lt p. 2 . . . pf, the Gibbs phase-space is that 2/ dimensional space, the points of which possess the 2/ 1 co-ordinates q 1 . . . p/. Each point of the phase-space (phase-points) represents, therefore, a definite momentary state of the system in question. Planck now gives this phase-space, in exact analogy to the phase- plane, a cellular structure, by bringing into prominence certain specially distinguished boundary surfaces. At the same time the size of the cells is proportional to h f . The points of intersection of those boundary surfaces then repre- sent the distinctive quantum states or phases of the system (that is, according to Planck's first theory the only possible, the "allowable" conditions). In contrast with Sommerfeld' s Theory, in which the motion of a system of / degrees of freedom is always determined by / quantum conditions, in Planck's, under certain circumstances, the case may occur that fewer quantum conditions than degrees of freedom exist, so that several (" coherent ") degrees of freedom are limited by a single quantum condition. 6. Sommerfeld's Theory of Relativistic Fine-structure That these theories had found the kernel of the matter was soon to be shown by applying them to Bohr's atomic model. According to them from among all the mechanically possible paths, which the electron can describe about the -fold positively charged nucleus, the allowable, stationary paths must be determined by the two quantum conditions (99) and (100). This gives, in place of the discrete, quantised circles RELATIVISTIC FINE-STRUCTURE 95 of Bohr, discretely quantised Kepler ellipses, among which also the Bohr circles are included, as special cases. And further, the ellipses are quantum-determined, both with re- ference to their sizes (i.e. to their major axes), and to their form (i.e. the relation of the axes to one another), so that here every orbit, as compared with Bohr, is characterised by two quantum numbers n and w'. 267 In place of formula (93) for the hydrogen type of series, we get the general formula : a68 v = Nzf, _ J_ _ 1 "I (102) L(s + s'Y (n + nj] Here again N, the Rydberg constant, is given by (95), or more exactly (the motion of the nucleus being taken into account) by (97) ; s and s' are the two quantum numbers (azimuthal and radial) of the final orbit of the electron ; n and n' are the quantum numbers of its initial orbit. Since also, as a result of this more complete view of Sommerfeld, the number of allowable orbits is greatly increased, as com- pared with those arising from Bohr's Theory (owing to the addition of the ellipses), the electrons have a great many more possibilities in passing from one orbit to another, that is, the chances of generating spectral lines are multiplied. But we easily recognise the following fact : if we choose as the final orbit of the electron any one of those orbits, for which the sum of the quantum numbers s + s' has a definite value, say s + s' = 2, and as initial orbit, any one of those paths, for which n + n' has a definite value, say n + n' = 3, then all the different transitions of the electrons from any one of these initial orbits to any one of these final orbits generate always the same line (in the case of the figures above chosen it will be the first Balmer line) ; for according to (102) the frequency of the line emitted depends only upon the sum s + s', and the sum n + n', and on the other hand not on the separate values of s, s', n, n'. It would thus appear as if nothing is gained physically by Sommerfeld' s elaboration of the theory as compared with Bohr's original theory. How- ever, as Bohr had already pointed out, the calculations are incomplete in one important respect, which become of funda- mental importance when consistently taken into account, and which represents the main achievement of Sommerf eld's 96 THE QUANTUM THEORY theory of spectral lines. Namely, the velocities of the electrons, which appear in these problems, cannot be con- sidered negligibly small compared with the velocity of light. In this case, however, we cannot, as we know, calculate by the methods of classical mechanics, which regards the mass of the electron as constant, but must take our stand upon the theory of relativity, and hence take into account the variations of the mass of the electron with its speed. Sommerfeld com- pleted the calculation in this respect. The paths of the electron and the nucleus differ, in this refinement of the theory, from the ordinary Kepler ellipse in that the perihelion of the orbit advances in the course of time, and that the path loses its closed character. This has the effect that the energy of the electron in the stationary quantum-chosen orbits which here also are determined by (99) and (100) are no longer solely dependent on the sum of the quantum numbers as in the case of the non-relativistic Kepler motion, but that the quantum numbers n and n also enter, separately, into the expression for the energy. Only as a first approximation, therefore, i.e. when the relativity correction is neglected, will the frequency v of the spectral line emitted depend on the quantum sums s + s' and n + n' alone, as (102) shows. If we take into account the relativistic change of mass of the electron, on the other hand, v will also depend on the individual values of s, s', n, w'. 269 It follows, therefore, that the various possibilities, above considered, of the generation of a definite spectral line, that is, the passage of an electron from any one of the initial orbits s + s' = constant to any one of the final orbits n + n' = constant, no longer produce exactly the same line, but give rise to slightly different lines, which, how- ever, on account of the smallness of the relativity effect, lie very close together. This is Sommerfeld' s explanation of the fine-structure of the spectral lines in the case of the hydrogen type of spectra. For example, according to Sommerfeld, the first line of the Balmer series (the red hydrogen line H a ) must consist of five components, which are arranged in two chief groups (of two and three each). The mean distance of these two groups from one another should amount, according to the theory, 270 to about 0'126A ; the best measurements of the hydrogen doublet gave the value 0'124A (Paschen, Mciasner). HIGHER ELEMENTS 97 If this agreement already speaks strongly in favour of Sommer- f eld's Theory, the exact measurements, by F. Paschen, of the fine-structure of the lines of positive helium (Fowler series) have given a still more convincing proof of its correctness ; almost without an exception, all the components required by the theory of the fine-structure appeared on the photographic plate, and thus proved strikingly the existence of the stationary paths of the electron and its relativistic change of mass. Two interesting consequences may yet be mentioned here ; they are directly connected with Sommerfeld's Theory and Paschen's observations. First of all they have rendered possible the use of the fine-structure measurements for a direct " spectroscopic " determination of the three funda- mental constants e, m (mass of the electron at infinitely low speeds), and h.^ 2 Secondly, K. Glitscher was able to show that we only find the spectroscopic observations, for example, the size of the hydrogen doublet, in agreement with the theory, when we use for the variation in the mass of the electron the formula given by the theory of relativity. On the other hand, Abraham's Theory of the rigid electron leads to formulae which do not agree with experiment. 7. Higher Elements We thus see that Rutherford's atomic model as further developed by Bohr and Sommerfield far exceeded the ex- pectations which it could reasonably be expected to fulfil. At any rate, it has revealed to us the optical series of hydrogen and helium with undreamed-of precision as far as the finest details. But beyond these primary gains, it has undertaken a further series of successful attacks. Thus Landew* was successful in calculating the two series-systems of neutral helium (helium and parhelium) by taking, in contra- distinction to Bohr, a model of the neutral helium atom in which the two electrons circle around the double positive nucleus in two different orbits, either co-planar or else inclined at an angle to one another. In this case then, the external electron, the leaps of which generate the radiation, moves in a field in which the simple Coulomb Law no longer holds, on account of the disturbing influence of the inner electron. Examples of this type which differ from that of 7 98 THE QUANTUM THEORY hydrogen have been generally investigated by Sommp.rfeld, who has shown 273 that by giving up the Coulomb field we arrive, to a first and second approximation, at the Bydberg and Eitz forms of the series laws. A very promising beginning in setting up a quantum theory of the spectral lines was thus made. 8. The Stark Effect and the Zeeman Effect in Bohr's Theory of the Atom Under the circumstances the question forces itself upon us, whether the atomic model in its present state of development is able to account for the Stark effect, that is, the splitting up of the spectral lines as a result of the action of an external electric field on the electrons emitting the lines. For, as we may remember, the original TJwmson model had completely failed just at this point. And how do matters stand as regards the Zeeman effect, the splitting up of spectral lines as a result of an external magnetic field? Could the new model explain these phenomena as well as the old ? Both questions have fortunately been answered in the affirmative. As regards the Stark effect, P. S. Epstein, in an important paper, succeeded in demonstrating the following : if we calculate the motion of the electron under the influence of the nucleus and the external field, according to the methods usual in celestial mechanics, and then choose from among all mechanically possible motions the allowable stationary orbits by applying the modern quantum rules for several degrees of freedom, and if, thirdly, we allow the electron to leap from one of these stationary paths into another (whereby we limit the infinite number of possible passages by a "principle of selection" presently to be discussed), then the Bohr frequency formula '(92) gives with the most admirable accuracy and completeness, both as regards position and number, all the components of the resolved lines as observed by Stark in the cases of hydrogen and positive helium. This astonishing result must be re- garded as a further strong support of the correctness of Bohr's model and its system of quanta. The theory of the explanation of the Zeeman effect has up to the present not been quite so successful, It is true that Debyc and SELECTION OF RUBINOWICZ AND BOHR 99 Sommerfield 278 have been able to derive the normal Zeeman effect (division of the original line into a triplet when the line of observation is perpendicular to the lines of force) by calculation from the model. The explanation, however, of two important phenomena in this field has not yet been accomplished : firstly, the anomalous Zeeman effect and its laws (Runge-Preston rule), and secondly, the fact, discovered by Paschen and Back, 1219 that even in the case of lines with a complicated fine-structure, the normal triplet is formed as the magnetic field grows. Further investigation will, it may be hoped, unravel those difficulties. 9. The Principles of Selection of Rubinowicz and Bohr Inasmuch as the foregoing considerations deal only with the position of lines in the spectrum, i.e. with their frequency, we are still confronted with the problem of their form of vibration, i.e. their intensity and polarisation. Moreover, the important question had yet to be answered, whether all leaps of the electron from any one stationary path to any other are possible, or whether the number of allowable transitions must be limited by some " principle of selection." This also is, fundamentally, a question of intensity, for the position may be regarded as follows : the forbidden transitions corre- spond to zero intensity. The solution of this whole complex of problems has been greatly advanced quite recently. In the first place, A. Rubinowicz, & by applying the law of the conservation of the moment of momentum (impuls-moment) to the system atom + radiated wave, arrived at a principle of selection and a rule of polarisation of the following form : in atoms of the hydrogen type, which are removed from the influence of external fields of force, the azimuthal quantum number n of the electron [see formula (99)] can only alter by 0, +1, or - 1, when emission takes place. In the first case, the light radiated is linearly polarised, in the two other cases circularly. The position of the plane of the orbit remains unchanged during the process of emission. In the case of atoms differing from the hydrogen type, and of more complicated structure, the position is less simple; if we set the total moment of momentum of all the masses forming part of the system (we know that this 100 THE QUANTUM THEORY impulse remains constant during the motion), equal to a whole number, n*, times ^, it is just the changes in this number n* during the emission which must be limited by the principle of selection in the same manner, as, in the case above, the alterations in the azimuthal quantum number of the individual electron in its leaps were limited. Here also, zero change in the azimuthal quantum number gives linear polarisation, changes by + 1, on the other hand, lead to circular polarisation. In place of the orbital plane we get the " invariable plane " (at right angles to the total moments of momentum or impulse-moments), the position of which in space remains unaltered. If, finally, the atom is exposed to an external field, say a homogeneous electric field (Stark effect) or a homogeneous magnetic field (Zeeman effect), then, as we know, only that component of the total turning impulse remains constant during the motion of the masses forming parts of the atom which is parallel to the external field. If we set these components of impulse = n^, then only the alteration of this number n will be limited by the principle of selection (that is, the alterations must be Q l 1). The principle of selection is thus clearly weakened in its action by the external field, and can, if fields of irregular strength and direction act on the atom, become completely illusory, as, for example, in the case of electric discharges. By means of entirely different considerations, N. Bohr 281 arrived at results which coincide, in essentials, with those of Eubinowicz, but exceed them greatly in range. Bohr started from the fact that in the limit for large quantum numbers, when the successive stationary states of the atom differ very little in the energy they involve, the frequency that the electron emits in its passage between neighbouring states becomes identical with the rate of revolution in the stationary orbit. 282 The electron therefore emits, according to Bohr's frequency condition, the same line that it sends out accord- ing to the classical theory of electrons. In other words, for very high quantum numbers, the quantum theory passes over into the classical theory. (Bohr's "Principle of Correspon- dence or Analogy.") Arguing from this principle, Bohr pro- SELECTION OF RUBINOWICZ AND BOHR 101 ceeds as follows : according to classical mechanics, the motion of the electron in Bohr's atom may be represented as the super- position of component harmonic vibrations of the frequency : "kl = Tl 2 + . . . + TjWf . . (103) Here, T I . . . T/ are whole numbers which in general may have all values between oo and + oo ; the o^ . . . ay are certain constants which depend on the character of the motion : / is the number of degrees of freedom. Let the amplitude of the partial vibration characterised by the numbers TJ to T/ be A T i . . . A r f. Then, according to classical electrodynamics, vki is the frequency of the radiated partial wave (T! . . . T,) and A^ . . . A*f is a measure of its in- tensity. On the other hand, the following result is derived from the quantum theory (Bohr's frequency formula) for high quantum numbers : in the transition from an initial state characterised by the quantum numbers m v m 2 . . . w/ into a final state corresponding to the quantum numbers n^ . . . Hf, a line of frequency VQU = (% - n^ + (ra 2 - w 2 )w 2 + . . . + (m f - n f )u>f . . . (104) is emitted. Here the quantities T 2 = m 2 ~ n 2 > . . . T/ = ra/ - w/ . . . (105) i.e. the " classical " partial vibration (r { . . . T/) corresponds to that quantum transition, in which the quantum numbers alter by exactly TJ . . . T/. The polarisation and intensity of the wave emitted during this q^lantum transition may be calculated from the form of vibration and amplitude of the " corresponding classical" partial oscillation. This principle which has been derived for high quantum numbers is extrapolated by Bohr with great boldness over the region of all quantum numbers. Thus the important " principle of correspondence " is obtained. If in the development of the electronic motion in terms of partial vibrations the term (r lt r 2 . . . ?/) is missing, then the corresponding transition 102 THE QUANTUM THEORY is not present. Hence there follows, for example, for atoms of the hydrogen type in a field free from force, the law that the azimuthal quantum number can in all emissions only change by + 1 or 1, both of which lead to circularly polarised radiation. This law is somewhat more limited in form than that of Rubinowicz. Both the principles of selection and the rules for the polarisation and the intensity have stood the test of compari- son with experiment. Bubinowicz himself showed that his principle of selection and the rule of polarisation are in agree- ment with Paschen's measurements of the fine-structure of the helium lines, and further with the observations of the Stark effect and the normal Zeeman effect. P. S. Epstein 283 and H. A. Kramers* went still further, and were able to prove by profound investigations, based on Bohr's Theory, that the calculations of intensity along the lines sketched above were also in surprising agreement with observation. Finally, Sommerfeld and Kossel 283 in an interesting study have applied the Rubinowicz principle of selection to spectra differing from the hydrogen type as well, and have shown that it is able to explain why certain series appear more readily and are more favoured than others, as it were, and that, by the selection of the " possible " transitions, it sets a limit to the multiplicity of possible combinations in a manner which, so it appears, entirely agrees with experience. 10. Collision of Electrons on the Basis of the Bohr Atom While in this way, through the interpretation and unravell- ing of the universe and the almost bewildering abundance of spectroscopic observations, the conviction of the correctness of Bohr's atomic model deepened more and more, a series of observations of quite another kind became known and contri- buted considerably to the consolidation of Bohr's Theory. These were the investigations already mentioned earlier in connexion with the light-quantum hypothesis, which dealt with the collision of free electrons with gas molecules and atoms. These researches were conducted particularly by J. Franck and G. Hertz 28fl and, in succession, by a considerable number of American investigators in a systematic manner. The manifold results of these interesting researches may be COLLISION OF ELECTRONS 103 sketched here schematically by a simple example. What have we to expect when electrons collide with a Bohr Atom ? As a simple type of Bohr atom, let us choose a model in which z electrons revolve around a 2-fold positively charged nucleus in stationary quantum paths. The nature and spatial arrange- ment of these paths, as well as the distribution of the electrons among the individual paths will be left open, and we shall FIG. 10. make only the simplifying assumption that one electron the so-called valency electron revolves alone in the outermost orbit (1) (see Fig. 10). Let this be the " normal," unexcited state of the atom. The hydrogen atom (z = 1) is, as we know, constituted in this way, and, of the neutral complicated atoms, the atoms of the vapours of the alkali metals (Li, Na, K, Kb, Cs) very probably also fall under this scheme. If by any addition of energy the electron is " raised " from its normal 104 THE QUANTUM THEORY orbit (1) to a higher orbit (that is, one having more energy), say into the orbit (2), (3), (4) and so forth, and if it "falls" from these back into orbit (1), then the 1, 2, 3 . . . line of the so-called " Absorption-series of the unexcited atom " (principal series) is emitted. The frequencies of the lines emitted are regulated by Bohr's frequency condition (92), i.e. that the loss of energy W n W l incurred in passing from the nib. to the first orbit is equal to a quantum hv n>l of the line emitted : Wn-W^hv^. . . . (106) The additional energy required to " raise " the electron to the higher energy level can be obtained in two ways : firstly by absorption of external radiation ; secondly (and that is the case we are dealing with here) by electronic impact. If external radiation of frequency v n 1 = 8-14 - 10' 12 , in striking agreement with the value of E^ . For mercury vapour, the limit in question of the principal series A.^ = 1-188 10 - 5 . From this follows, according to (108), FOQ = 10-4 volts while the measurements of various workers gave the value 10*2 to 10'3 volts (Tate, Bergen, Davis and Goucher ; Hughes and Dixon ; Bishop 6 ). From all these examples, which could be considerably multiplied, the conclusion may be drawn with convincing clearness that the Bohr conceptions have laid bare the nature of the con- struction and the mode of action of the atom with un- precedented lucidity. 11. Einstein's Deduction of Planck's Law of Radiation on the Basis of the Bohr Atom Under these circumstances the suggestion naturally arises to refound the law of black-body radiation by taking as the ele- mentary absorbing and emitting structure Bohr's model in place of the linear oscillator used by Planck. Einstein 289 has taken this step. In a highly important study he investigated PLANCK'S LAW OF RADIATION 107 the equilibrium of energy and momentum between black-body radiation and a generalised Bohr model, which, stripped of all special properties, has only to fulfil the quantum condition of being able to assume a discrete series of different states. For the interaction between the radiation and the atom absorption (Einstrahhing) and emission (Ausstrahlung) Einstein intro- duces the following simple hypotheses : the frequency of the emissions, i.e. the transitions, accompanied by loss of energy, of the atom from a condition (2) of higher energy, E%, to a condition (1) of lower energy, E v shall follow the same statistical law as that which governs the disintegration of radioactive bodies, i.e. the number of transitions 2 -> 1 in the time dt, or, as we may say, the number of atoms (2) that " dis- integrate " in this time is proportional to dt N v where N% denotes the number of atoms momentarily in the state (2). But, according to Einstein, a different law regulates the processes called into existence by the effect of external radi- ation. Under the influence of external radiation two things may happen : either an atom may pass from state (1) to state (2) by taking up energy, this is the " proper positive absorp- tion." Or the case may also occur, that, as a result of the phase-relation between the field of the external radiation and the atom, the atom loses energy through the action of the im- pinging radiation, and hence passes from state (2) to state (1) ("negative absorption"). The rate at which both kinds of transition are repeated is then proportional to the intensity K v of the external radiation : the number of transitions 1 -> 2 associated with positive absorption in the time dt is therefore proportional to N-^dtK.,, ; the number of transitions 2 - 1 as- sociated with negative absorption is proportional to N 2 dtK. v . Here N^ is the number of atoms momentarily in the state (1). Nj_ and N 2 are determined by the laws of distribution known from the theory of gases and statistical mathematics and en- larged in conformity with the quantum theory. There follows from the energy equilibrium between in-coming and out-going radiation at the temperature T . . . (109) 108 THE QUANTUM THEORY where k is Boltzmann's constant, and A is a constant inde- pendent of the temperature. From Wieris Displacement Law (4) it follows, firstly that A is proportional to v 3 and secondly that E 2 - E! is proportional to v. If, therefore, we write E 2 - E l = hv . . . . (110) we recognise in this expression Bohr's frequency condition (92). In this way K,, assumes the form of Planck's Law of Eadia- tion, arising in a surprisingly simple and elegant manner from a minimum of hypotheses of a general character. Einstein, in pursuing and deepening these conceptions by writing down the expression for the equilibrium of the momenta in addition to the energies of the in-coming and out-going radiation, was led to the remarkable conclusion that the radiation of Bohr atoms cannot take place in spherical waves, as the classical theory of electrons requires, but that the process of emission must have a particular direction like the shot from a cannon. We cannot fail to recognise that this brings the conception that radiation has a quantum-like structure (light-quantum hypo- thesis) within realisable bounds. CHAPTEK VII The Quantum Theory of Rontgen Spectra i. The Analysis of Rontgen Spectra T) AEALLEL with the development of the science of optical [ spectra, a theory of Eontgen spectra has been developed of late years upon the same basis. This theory has already shed much light on the structure of atoms and thus forms a desirable extension of the theory of optical spectra. The investigations of Ch. Barkla, W. H. and W. L. Bragg, Moseley and Darwin, Siegbahn and Friman,' 290 among others, have shown that by the impact of cathode rays upon the anti- cathode of a Eontgen tube two kinds of Eontgen rays arise : first, the so-called " impact radiation " (Bremsstrahlung) con- sisting of an extensive and continuous range of wave-lengths (similar to the continuous background of visible spectra) ; secondly, the " characteristic radiation," a typical line- spectrum, the structure of which depends so essentially on the material of the anti-cathode that a glance at this spectrum suffices us to deduce immediately and unmistakably the nature of the material of which the anti-cathode is composed. Thus along- side the optical spectrum analysis of Bunsen and Kirchhoff a Eontgen- or X-ray analysis presents itself. It has further been shown that the characteristic X-ray spectrum is a purely atomic property, and, indeed, an additive one. If we examine, for example, the X-ray spectrum, which is emitted by an anti-cathode of brass (copper + zinc), we find the lines of both copper and zinc unaltered and occupying the same positions as if only one metal were present in turn. No new lines appear. Accordingly we are led to suppose that the line-spectrum arises in the atoms of the anti-cathode, and is generated there by the impinging electrons of the cathode 109 110 THE QUANTUM THEORY rays. The further important fact appeared that the lines of the characteristic spectrum may be arranged in series, just like those of the optical spectrum. Thus we have discovered up to the present a short-wave ^-series, a long- wave .L-series, and a still longer-wave M-series. The most curious feature of these spectra is their connexion, by a definite law, with the atomic number of their element in the periodic system. If we plot the position of a certain line (say the first line K a of the ^-series) for the successive elements of the periodic system, a perfectly regular progres- sive shift is revealed : the line advances with increasing atomic number steadily towards the shorter waves. The re- gularity of this advance is such that we can recognise gaps or false positions of elements in the periodic system immediately by an excessive jump. Now, according to the hypothesis, already mentioned, of Eiitherford, v. d. Broek, and Bohr, the atomic number of an element is nothing other than the number of its nuclear charge, that is, the number of elemen- tary positive charges of its nucleus. If to this we add the phenomenon just discussed, according to which the steady advance of the nuclear charge in the series of the elements is reflected in the steady displacement of the X-ray lines, then we are forced to the view that the origin of tlie X-ray spectra must be localised in the immediate neighbourhood of the nucleus, that is, in the inmost part of the atom. For in this region the nucleus clearly has the greatest power and is least disturbed by external electrons, and hence it is here, too, that the growth of the nuclear charge will make itself most felt. The connexion between the position of the X-ray lines and the atomic number z was first formulated by G. Moseley. He found for the frequency of K a (first line of the JfT-series) and L a (first line of the Z/-series) the empirical relation (Ill) where N is the Eydberg number. The similarity of these relations, which are only approxj- THE ANALYSIS OF RONTGEN SPECTRA 111 mately valid, with Bohr's formula (93) for the series of the hydrogen type is so striking, that it was an obvious step to seek to find the explanation of the Eontgen series by arguing on the basis of Bohr's model. This problem was attacked chiefly by W, Kossel, 2 A. Sommerfeld, L. Vegard, P. Debye, J. Kroo,&* and A. Smekal. 1 And thus, in addition to the theory of the optical spectra which take their origin at the periphery of the atom, a theory of the Kontgen spectra has arisen which leads us FIG. 11. into the inmost regions of the atom. According to this theory we may picture to ourselves, in general terms, the emission of the Rontgen spectra as follows : we consider a neutral Bohr atom, consisting of a 2-fold nucleus, around which z electrons revolve. These z electrons may be arranged in different rings. The innermost, single-quantum ring, the so- called -ST-ring, carries, let us say, p l electrons in its normal state ; let the second ring, the ir-ring, be a two-quantum ring occupied by p. 2 electrons, the third, three-quantum, the .M-ring with^ 3 electrons, and so on (Fig. 11). The question whether 112 THE QUANTUM THEORY we can reach our goal with this conception of the ring by assuming the quantum numbers to increase as we go outwards, and whether we are to take the rings as co-planar or inclined to one another will be left open. The preparation for the emission of the -BT-series consists in this, that by the addition of energy whether by absorption of external radiation or by electronic impact an electron of the K-ring is removed to in- finity, that is, the atom is, so to speak, ionised " inside," i.e. in the .BT-ring. If the energy of the atom before this inner ionisation = W , and after the ionisation = W K , then the amount W K - W of energy must be provided. Hence every radiation, the energy quantum of which satisfies the condition hv ^ W K - W , can on being absorbed effect the tearing of the electron out of the K-r'mg. If we allow the v of the external radiation to grow slowly from small values, then, at the point "TT7" \JU V K = *_ 9, a sudden increase of the absorption occurs, because from this point onwards the external radiant energy is used for the " ionisation of the K-ring." Thus an absorp- tion-band extends from v = V K towards higher frequencies, the edge of the band lying at V K . This phenomenon of the " edge of the absorption-band " has already been interpreted above in the sense of the hypothesis of light-quanta. If the addition of energy is provided by the impact of a strange electron, coming from without, then its energy must be E> W K - W , that is, E ^> ~hv K , a relation, which we have already deduced earlier from the standpoint of the quantum hypothesis of light. By ionisation of the .ST-ring the atom is now prepared for ^-emission. If now an electron falls from the 2-quantum Zv-ring into the 1-quantum J5"-ring, filling up, so to speak, the gap produced there, then the first line of the K- series, K a , will be emitted. If on the other hand the gap in the .ST-ring is filled by an electron of the 3 -quantum If -ring, or the 4-quantum .W-ring, Kp or K y result respectively. The position is quite analogous as regards the L- and .M-series. If, by the addition of energy (absorption or electron-impact), an electron of the Zi-ring is battered off, that is if the L-ring is ionised, then the atom is prepared for the emission of the ZJ-series. If, now, the gap in the 2-quantum L-ring is filled by an electron of the 3-quantum M-ring, the first line of the Z/-series, FINE-STRUCTURE OF RONTGEN LINES 118 L a , results; if it is filled by an electron of the N-riug, the second line of the .L-series, Ly, results (the notation is not quite consistent but will serve the present purpose), and so forth. The converse phenomenon to line emission, viz. line absorp- tion, with which we are acquainted in visible spectra, appears at first sight to be missing here. That is, however, as W. Kossel 298 recently showed, an error. It is true that the ejected electron of the .ST-ring, for example, cannot in general be caught upon the L-, M-, or .N-ring, because all places on them are already occupied. An absorption of the lines K a , Kp, K y , is therefore in this case impossible. But the electron of the K-ring can certainly come to rest on an unoccupied quantum orbit outside the occupied rings, that is, outside the surface of the atom. In this process a " line " is actually absorbed, namely, that line of which the hv is equal to the energy- difference between the K-rmg and the final orbit of the ejected electron. This refinement of our considerations shows, then, that the electron from the .fiT-ring does not need to be raised immediately to infinity, but that line absorptions may occur before the edge of the band of absorption is reached. 2. The Fine-structure of Rontgen Lines It is particularly noteworthy that Sommer/eld succeeded also in the field of X-ray spectra in explaining the fine- structure of the lines by calling in the aid of the theory of relativity. Thus, for example, the 2-quantum .L-orbit is "double"; it can occur as a circle (n = 0, n = 2) or as an ellipse 2" (n = 1, n = 1). Hence the line which is emitted by the electron of which the L-rmg is the initial orbit, namely, K a , is a doublet (K a and K a -). In just the same way, those lines for which the Iv-orbit is the final orbit of the electron are doublets, namely, the line L a (more exactly L a >) to which Lp is added to make a doublet ; further, L y which forms a doublet with L&, and so forth. The distance between the components of the doublets (expressed in frequencies) comes out, according to Sommerfeld's Theory, as approximately pro- portional to the fourth power of the atomic number z. Hence here, in the X-ray region, where we are dealing for the most part with elements having fairly high atomic numbers, the 8 114 THE QUANTUM THEORY doublets appear microscopically enlarged as compared with the microscopic hydrogen-doublet (z = 1). During the emis- sion of X-rays the electron approaches very near to the highly-charged nucleus, and hence the relativistic effects of the resolution of the lines are much greater than in the case of the optical spectra, in which the electron is moving at the surface of the atom, where it is almost entirely screened from the action of the strong nucleus by the remaining electrons. With the help of the following relation deduced theoretically and adapted to experimental evidence, < - fr")' -.< 112 > Sommerfeld was able to calculate the hydrogen-doublet from the observed L-doublets, and compare it with the results of experiment. The agreement is very satisfactory. 3. The Distribution of Electrons among the Rings. Objections to the Ring-arrangement of Electrons The quantitative calculation of the simplest case, namely, the emission of K a , led Debye to the conclusion that the .ST-ring in the normal state consists of three electrons. To this Kroo, by elaborating the calculation, adds the con- clusion that the L-r'mg contains in its normal state nine electrons. With these two distribution numbers, p l = 3, p 2 = 9, the position of K a could be represented as a function of the atomic number z for all elements. The emission of K a takes place according to the following obvious scheme : I K-ring L-ring Normal state | 3 9 . _ . . , , _ . Initial state [ 2 | 9 > I ni8atlon of * Z-nng. KEafBtate | 8 | 8 > EmiSS1On f *" The two distribution numbers (Besetzungszahleri) thus found for the two innermost rings excite our attention. For on the basis of the Periodic System with its periods of eight we ought to expect, according to Kossel, the numbers 2 and 8. DISTRIBUTION OF ELECTRONS 115 The strange occurrence of the numbers 3 and 9 becomes an objection, when we consider the case of sodium (z = 11). Here, according to Kossel, we should expect the numbers 2, 8, 1, since in all probability an electron (the valency electron) revolves alone, as in the case of all alkali metals, around the outside quantum orbit (M-ring). In any case it is impossible that the two innermost rings together should, in the normal state, contain 12 (= 3 + 9) electrons. If we attempt to go a step further still on the basis of Kroo's numbers 3 and 9, and to set up a formula which represents for all 2*8 the position of L a in conformity with observation, and thereby to determine the number of electrons p 3 on the .M-ring, we find, as A. Smekal 300 showed, that this mode of representation is impossible with any combination 3, 9, p y Nor do we fare better if we incline the various rings to one another, and take their interaction into account. The suspicion is forced upon us, that perhaps the whole conception of the arrangement into plane rings does not correspond with fact, but that, rather, the electrons in the atom form spatially symmetrical figures. This suspicion is very much strengthened by a series of pro- found investigations carried out by M.. Born and A. Landd. 901 Following on M. Bom's investigations of the dynamics of the crystal-lattice, which we discussed in detail earlier hi connection with the atomic heat of solids, the two in- vestigators asked themselves the question, whether it is possible to build up the cubic crystal-lattice of the alkaline halides (NaCl, NaBr, Nal; KC1, KBr, KI, etc.) from ions of Bohr atoms, by taking into account only the mutual electro- static forces; and whether this method, if possible, would enable them to prophesy the crystal properties (lattice-con- stant, compressibility) from the atomic models of the two constituent ions. The answer to this question has been, on the whole, in the affirmative. But when the calculation of the compressibility of these crystals was carried out, the remarkable result manifested itself that crystals are found to be too soft, that is, insufficiently rigid, if the conception of the ring-arrangement of electrons in the atom is maintained. On the other hand, we get good agreement with the observations if, following Born, we introduce the hypothesis that the electrons are arranged spatially. A complex of eight electrons, 116 THE QUANTUM THEORY as occurs in sodium, potassium, etc., does not therefore occupy a plane 8-ring ; the eight electrons describe paths of ciibical symmetry. Into the still obscure region of these " spatial " electron paths, A. Lande 302 has made some successful in- cursions. From all that has been said it would appear to be certain that in dealing with Rontgen spectra, too, we can no longer be content with the arrangement of the electron rings in planes, and that the whole quantitative theory of the Eontgen series, including Sommerfeld's fine-structure of the K- and the L- doublets, must be built up on a fresh foundation. CHAPTEB VIII Phenomena of Molecular Models i. Dispersion and Magneto-rotation of the H 2 Molecule WHILE the X-ray spectra and the spectra of the optical series arise from the atoms of the elements (and hence their theory links up with the atomic models), there is a series of phenomena which, in the case of polyatomic substances, are peculiar to the molecules, and the theory of which, therefore, is founded on the molecular models. Chief among these are the normal dispersion, the rotation of the plane of polarisation in the magnetic field (magneto-rotation), and, further, the great and complicated subject of band-spectra. Up till a few years ago, dispersion and magneto-rotation had been exclusively treated from the standpoint of the Thomson model, that is, with the help of quasi-elastically bound electrons, and this explanation had served in turn as a powerful support for this model. Nevertheless, discrepancies in these theories had long been known. For example, measurements calculated upon the basis of the dispersion theories of Drude, Voigi, or Planck led to values for the ratio of the charge to the mass of the electron ( ) which, in com- parison with the direct measurements of this quantity (based upon the deflection of the cathode- or /3-rays in the electric and magnetic fields) which were much too small. When, however, the Thomson model became displaced by the Rutherford-Bohr model, and the successes of the Bohr atomic model increased at an undreamed-of rate, the question arose whether an unobjectionable theory of dispersion and magneto- rotation could not be founded upon these new views. The difficult position, into which we are brought by this problem, 117 118 THE QUANTUM THEORY arises from the fact that we do not actually know a single instance of the exact manner in which a polyatomic Bohr molecule is built up from its nuclei and electrons. The exact knowledge of this structure, and the motion of all the electrons is absolutely necessary, if we desire to know how the molecule reacts upon external waves (dispersion). It is true that W. Kosseiw* has, in a detailed study already referred to above, pointed out the general guiding lines along which, from the chemical point of view, the building-up of the atom from molecules must be carried out, but the details of this construction remain open. Only in a few of the simplest cases have detailed molecular pictures been con- structed and closely tested. Thus Bohr, as we remarked in discussing the atomic heat of gases, has already proposed a model of the diatomic hydrogen molecule. It has the follow- ing construction (see Fig. 8) : two singly-positive nuclei (that is, each consisting of only a single positive charge) are separated by the distance 26. In the vertical plane which bisects the line joining the nuclei, two electrons rotate, diametrally opposite one another, on a circle of diameter 2a. The equilibrium of the Coulomb and the centrifugal forces requires that a = b^/3. By means of this relation, and by the quantum condition that each electron must have the moment of momentum , the model is completely determined in all 2ir its dimensions and speeds. It was this model which was the first to be proposed : it was examined by P. Debye 3* with reference to its dispersion. On account of its sym- metrical structure the molecule possesses no electrical mo- ment in its normal state. If, on the other hand, it is struck by an external light wave, the motion of its electrons is periodically disturbed ; they depart from the normal quantum path, fall into forced vibration, and thus generate an electric moment which changes periodically in step with the external wave. Thus the original motion of the primary wave is changed, and dispersion results. We may conceive this as follows : Let c be the velocity of the primary wave in vacuo. The oscillations of the electrons generate a secondary wave which spreads out from the molecules. All these secondary waves combine with the primary wave to a form new wave OBJECTIONS TO BOHR'S MODEL 119 which moves with the altered velocity q, the value of which depends on the frequency of the primary wave. But just this is the phenomenon of dispersion. The electronic vibra- tions which occur here are not oscillations about positions of equilibrium, as in the case of the quasi-elastic model, but oscillations about stationary paths. Moreover, here, the force holding the electrons, as opposed to the usual classical theories of dispersion, is anisotropic (that is, the electron is held by different forces in different directions) ; above all, by means of this anisotropy, it was possible to explain away the disagreement in the value of , which had previously been WIC found to be too small ; and Debye succeeded, on the basis of the normal value of , in deducing from the theory the observed dispersion curve of hydrogen, that is, the curve which shows how its coefficient of refraction depends on the wave-length. It should be noted that in the formula for the coefficient of refraction, no single constant is arbitrary, but that the dispersion formula is made up entirely of universal constants. Using the same method (calculus of disturbances), P. Scherrer 30 * has calculated the rotation of the plane of polarisation which linearly polarised light undergoes in its passage through hydrogen under the influence of a magnetic field. His efforts were equally successful. 2. Objections to Bohr's Model of the Hydrogen Molecule In spite of the successes which the Bohr model of the hydrogen molecule has won, a list of weighty objections to it has accumulated in the course of time. That the con- tribution which the rotation (more accurately, the regular precession) of this molecule makes to the molecular heat at low temperatures, does not correspond with the observations of Eucken, has been shown by P. S. Epstein, as we have already mentioned. Also at high temperatures, when the oscillations of the two nuclei relatively to one .another con- tribute to the molecular heat, no agreement between theory and observation has been found in the case of the Bohr model, as G. Laski 306 recently showed, 120 THE QUANTUM THEORY Further, the model must possess, in consequence of the revolving electrons, an almost fixed magnetic moment parallel to the axis of the nucleus, that is to say, it must be equivalent to a molecular elementary magnet, which endeavours to set itself, in an external magnetic field, parallel to the lines of force. Hydrogen ought, therefore, to be paramagnetic, whereas it is diamagnetic. Another very important objection, to which Nernst in particular drew attention, is the following : if we calculate the work which is necessary to separate the molecule into its two atoms, the so-called heat of dissociation, we get s 07 the value, 61,000 calories. On the other hand, Langmuir 3 08 found 84,000 cals., Isnardi** 95,000 cals., /. FrancJc, P. Knipping and Thea Kriiger 81,000 ( 5700) cals. In any case, the calculated heat of dissociation comes out 25 per cent, too small.sioa Finally, W. Lenz 311 has recently increased the objections to the hydrogen model by an important one based on a theory of band-spectra, which we shall discuss below. He proved that the band-lines of hydrogen and nitrogen can exhibit the observed Zeeman effect, only if these molecules possess no moment of momentum around the nuclear axis. The fact that the two electrons in Bohr's molecular model revolve in the same sense, however, endows it with just such a moment of momentum. On the whole, the Bohr model does not seem to correspond to reality ; the arrangement of the two nuclei and electrons must plainly be quite different. No satisfactory model, however, has yet been found. 3. Models of Higher Molecules Matters are no better in the case of models of the more complicated molecules. It is true that Sommerfeld 312 and F. Pawer 313 have also worked out the theories of dispersion and magneto-rotation in the case of the more general Bohr models (N 2 and O 2 ) which are constructed on the lines of the hydrogen model. According to Sommerfeld, four electrons revolve about the line joining the two nuclei in the case of oxygen, each of which acts with an effective charge + 2e ; in the case of nitrogen, a ring of six electrons rotates about the nuclear axis, while the nuclei carry triple effective charges. QUANTUM THEORY OF BAND-SPECTRA 121 Sommerfeld was able to obtain agreement with observation only by setting up for each electron of a valency ring of 2s-electrons the unaccountably strange quantum condition : moment of momentum = ^ ^ s > undoubtedly a most unsatisfactory result. Gerda Laski 31 * obtained better results with some- what different models, which she chose in such a way that the specific heat of the two gases at high temperatures agreed with the observations of Pier. 318 According to her ideas, the nitrogen molecule must consist of two seven-fold positive nuclei, each of which is closely surrounded by a 1-quantum ring of two (or three) electrons. The " valency ring " in the central vertical plane is 2-quantum and contains ten (or eight) electrons. Analogously, the oxygen molecule consists of two eight-fold positive nuclei, each encircled by a 1-quantum ring of two (or three) electrons, whereas the 2-quantum valency ring contains twelve (or ten) electrons. The same objections apply to some extent to these models of Sommerfeld and Laski as to the hydrogen model. For example, they give no account of why oxygen should be paramagnetic, and nitrogen, on the other hand, diamagnetic. Moreover, the above-mentioned objection of Lenz applies in full force to these models ; for they all possess moments of momentum around the nuclear axis. In conclusion, we feel bound to admit that the exact constitution of even the simplest models is at present unknown to us. 4. The Quantum Theory of Band-spectra To conclude this chapter, we shall turn our attention to the band- spectra, and collect together shortly what the quantum theory has been able to assert about them up to the present time. That they belong to molecules and compounds may nowadays be regarded as certain. The first attempt to con- struct a logical quantum theory of band-spectra was under- taken by K. Schwarzschild 316 who clearly recognised the importance of the rotation of the molecule in the production of these spectra. His conceptions may be defined as follows : a system of electrons revolves at a definite quantum distance around a molecule which itself rotates according to quantum conditions, the assumption being made for the sake of 122 THE QUANTUM THEORY simplicity that the motion of the electrons is not influenced by the motion of the molecule. If E is the quantum energy of the electrons, E r the quantised rotational energy of the molecule, then E + E r = E is the total energy of the system. If the three chief moments of inertia of the molecule / are equal to one another, then it follows, just as in (80), that where n denotes the rotational quantum number. Therefore If, now, the system passes from one quantum state having the electronic energy E Q and the rotational quantum number n into another quantum state having the electronic energy E' and the rotational quantum number n', then it follows from Bohr's frequency formula (92) that the frequency of the line radiated is given by _ E -E' (n - n> 7 T o 2 T ' ' V A / If we keep all the quantum numbers which occur here, except- ing n, constant, and allow n to vary, then we get a series of lines progressing towards the violet and having the frequencies v = a + bn 2 (a and b are constants) . (115) This is a formula which had already been given empirically by Deslandres,* 11 and which is approximately true for the lines of many bands. Following Schwarzschild, T. Heurlinger 318 and W. Lenz* 19 in particular, have further developed and refined the quantum theory of band-spectra. For example, Lenz has pictured the molecule as a symmetrical top having two moments of inertia and a rotational rigidity (moment of momentum) around the axis of the figure, and hence deals from the outset with a regular precession of the molecule in place of a rotation. Using Bohr's frequency formula, and applying the principles of selection, he obtained the following general foi-mula for the lines of a band : v a + bn + en 2 (a, b t c are constants) . (116) QUANTUM THEORY OF BAND-SPECTRA 123 which is obeyed, according to Heurlinger, in the case of the so-called " cyanogen " lines of nitrogen, for example. In addition to the lines given by (116), Lenz's Theory requires the occurrence of the series given by the formula + + 2 . . . (117) for the case that the molecule really possesses a finite moment of momentum about its axis of figure. A series which follows this law does not, however, exist in the cyanogen bands, ac- cording to Heurlinger. Lenz deduces from this the conclusion already mentioned, that the nitrogen model does not possess a rotational rigidity about its axis. By calculating the Zeeman effects of the band lines, and comparing them with observation, Lenz was able to confirm this, and to extend it to the hydrogen molecule. The infra-red Bjerrum absorption bands of the diatomic and polyatomic gas compounds, which we had discussed at length in Chapter Y, belong to the general type of band-spectra. If we are to deduce them from a theory consistently founded on quanta and not, as we did earlier, half according to the quantum, half according to the classical theory we must follow closely the course pursued above, with the difference that, in place of the energy of the electronic system there will appear the energy of the atoms, 920 with which the rotational energy of the molecule is combined, as a first approximation, additively. The logical carrying out of this calculation (in which Bohr's frequency formula and the principle of corre- spondence are applied), which was undertaken by Heurlinger & l and the author, 322 gives for the structure of the " fluted " ab- sorption bands an arrangement of lines which at first sight does not appear to agree with the beautiful and exact measure- ments of Imes. 323 The theory gives for the position of the absorption lines a formula l/ = Vo(n + i) JL, (n=l,2,3...) (118) and therefore requires that all neighbouring lines be equi- distant, including the two in the middle (n = 0). On the other hand, Imes' observations show with indubitable clearness that the interval between the two middle lines is twice as great as 124 THE QUANTUM THEORY the interval between all neighbouring lines. This apparent contradiction is explained, as A. Kratzer 32 * recently showed, in a surprising fashion, if we take into account the intensity of the absorption lines according to Bohr's Principle of Analogy. For it then appears that the first absorption line to the right of the middle v -line, namely, the line h " = "o + SW (which is derived from formula (118) by setting n = and using the positive sign for the second term) is of vanishingly small intensity. This line is generated when the molecule passes over from an initial rotationless and vibrationless state into the final state in which the two ions oscillate relatively to one another with one quantum, and in which, at the same time, the molecule rotates as a whole with one quantum. The rotationless and vibrationless state has, however, a vanishingly small probability ; the number of transitions from this initial state per second, and therefore the intensity of the correspond- ing absorption line, is hence vanishingly small. By the dis- appearance of the first line to the right of the middle position v , the structure of the lines as observed by Imes is actually reproduced, as one may easily recognise ; in the formula, the " middle" of the line structure is displaced from the point v to the right by the amount oZFr The absorption lines group themselves equidistantly and symmetrically on both sides of the missing " middle," v = v + Q-^J- This state of affairs O7T J may be expressed by writing, in formal agreement with (83), " = "''4^7 (*-:i,8,8...)| where . (119) From the constant interval between neighbouring lines, namely . (120) the moment of inertia of the rotating molecule can be cal- culated with great accuracy. 328 CHAPTEE IX The Future IN the preceding pages the author has attempted to give in broad outline the most important features of the doctrine of quanta, its origin, its development, and its ramifications. If we now survey the whole structure, as it stands before us, from its foundations to the highest story, we cannot avoid a feeling of admiration ; admiration for the few who clear-sightedly recognised the necessity for the pew doctrine and fought against tradition, thus laying the founda- tions for the astonishing successes which have sprung from the quantum theory in so short a time. None the less, no one who studies the quantum theory will be spared bitter disappointment. For we must admit that, in spite of a comprehensive formulation of quantum rules, we have not come one step nearer to understanding the heart of the matter. That there are discrete mechanical and electrical systems, characterised by quantum conditions and marked out from the infinite continuity of " classically " possible states, appears certain. But where does the deeper cause lie, which brings about this discontinuity in nature? Will a knowledge of the nature of electricity and of the con- stitution of the electromagnetic field serve to read the riddle ? And even if we do not set ourselves so distant a goal, there remains an abundance of unanswered questions. The decision has not yet been made, as to whether, as Planck's first theory requires, only quantum-allowed states exist (or are stable), or whether, according to Planck's second formula- tion, the intermediate states are also possible. We are still completely in the dark about the details of the absorption and emission process, and do not in the least understand 125 126 THE QUANTUM THEORY why the energy quanta ejected explosively as radiation should form themselves into the trains of waves which we observe far away from the atom. Is radiation really pro- pagated in the manner claimed by the classical theory, or has it also a quantum character ? Over all these problems there hovers at the present time a mysterious obscurity. In spite of the enormous empirical and theoretical material which lies before us, the flame of thought which shall illumine the obscurity is still wanting. Let us hope that the day is not far distant when the mighty labours of our generation will be brought to a successful conclusion. Mathematical Notes and References 1 0. Lummer and E. Pringsheim, Wiedem. Ann. 63, 395 (1897) ; Verhandl. d. deutsch. physikal. Ges. 1899, pp. 23, 215; ibid., 1900, p. 163. Of. also O. Lummer and E. JahnJee, Drudes Ann. 3, 283 (1900), and O. Lummer, E. Jahnke and E. Pringsheim, Drudes Ann. 4, 225 (1901). 2 Of. M. Planck, Vorlesungen iiber die Theorie der Warmestrahlung (Leipzig 1906), 10. 3 Frequency () = velocity oflight in vacuo (c) ^ wave-length in vacuo (A.) iCf., for example, M. Planck, Vorlesungen iiber Warmestrahlung (1906), 17. 3 O. Kirchhoff, Gesammelte Abhandlungen (J. A. Earth, Leipzig 1882), pp. 573 et seq. ; Berliner Akademieberichte, 1859, p. 216 ; Poggend. Ann. 109, 275 (1860). 6O. Lummer and W. Wien, Wiedem. Ann. 56, 451 (1895). Cf. also O. Lummer and F. Kurlbaum, Verhandl. d. deutsch. physikal. Ges. 17, 106 (1898). 7 Of. Note 5. 8 L. Boltzmann, Wiedem. Ann. 22, 291 (1884). 9J. Stefan, Wiener Ber. 79, 391 (1879). 10 The Stefan-Boltemann Law is deduced as follows : Let the energy of black-body radiation at the temperature T, which is enclosed in a space of volume V having a movable piston, be U = Vu, where u is the " spatial " density of the radiant energy. The pressure, equal in all directions, which the radiation exerts upon the piston and walls is, according to electrodynamics, p = $u. If we supply to this system at the temperature T (that is, isothermally) an amount of heat d'Q, then its energy increases by dU, and the radiation does work pdV in push- ing back the piston. Therefore, according to the first law of thermo- dynamics, and owing to the two relations above : d'Q = dU + pdV = udV + Vdu + dV = | udV + Vdu. According to the second law of thermodynamics, -Q must be a com- plete differential. Hence the following relation holds : \_ 3 m 4 d u\dT _1 ,'ldt* u)dT_l 4 fl dw _ u] 3\TdT Z^J 127 128 THE QUANTUM THEORY i.e. -^ = 4^, which, integrated, gives u = a!" where a is a constant. Now, as we can easily see, the total radiation 00 K 2 / TH v dv is distinguished from the density of radiation u only by a constant factor (see M. Planck, Lectures in Radiation (1906), 22), hence the total radiation is proportional to the fourth power of the absolute temperature and this is the Stefan-Boltzmann Law. 11 W. Wien, Sitzungsber. d. Akad. d. Wissensch. Berlin, 9 Feb. 1893, p. 55 ; Wiedem. Ann. 52, 132 (1894). Of. also Max Abraham, Theorie der Elektrizitat II, 43 (1914); M. Planck, Vorlesungen iiber die Theorie der Warmestrahlung (Leipzig 1906), pp. 68 etseq. ; W. Westphal, Verhandl. d. deutsch. physikal. Ges. 1914, p. 93; H. A. Lorentz, Akad. d. Wissensch. Amsterdam, 18 May 1901, p. 607. 12 Formula (4) of the text (Wieris Law of Displacement) may be obtained by means of a simple dimensional calculation, as L. Hopf recently showed in the " Naturwissenschaften " (8, 109, 110 (1920)). We assume that Kx depends only on v, T, and the velocity of light c. The dimensions of Kx are obtained from the fact that, according to (1), energy surface x time From this it follows that [K,] - [-']. If we set Kx = const. v x T y c z then, remembering that T has the dimensions of energy, we get [m- 2 ] = const. [-* m y - V* r *" f t~ Z 1 = const. \m y ?V+* r*~ -*] Hence x = 2; # = 1 ; 2 = -2 which gives us, Kx = const. . ?1 . T. This relation is not, however, as we shall see, generally valid. In fact, oo it would give no finite value for K = 2 \"K. v dv. But, according to the 6 Stefan-Boltzmann Law (3), K = y T 4 . Hence the constant of Kx may still depend on a dimensionless combination of the four variables y, v, T, c. If, therefore, we set const. =f(y^T n c^y' a ) then the argument of the function / must have the dimension 0. If, further, we remember that [ BfKf* surface x time H = . NOTES AND REFERENCES 129 it then follows that = [t-t .mri.fr, t-*i .K.t - . t ] = [< -*--?+. Hence const. = /[(?)". c- - 7 ,] = < Therefore K , = Jr.* (j) = *. J. or, finally, Kl> = 3 F (j} 13 If we plot K, - *(} function of v, keeping T constant, the maximum of this curve if one is present lies at that point at which = 0. This gives where F 1 is the differential coefficient of F with respect to the argument. This equation, in which only occurs as unknown, gives a definite value for ^. In other words, for v = max, it follows that ^^ = const. II W. Wien, Wied. Ann 58, 662 (1896). 13 0. Lummer and E. Pringslieim, Wied. Ann. 63, 395 (197) ; Drude's Ann. 3, 159 (1900) : Veih. d. deutsch. phys. Ges. 1, 23 and *15 (1899). The total radiation emitted per second from 1 cm. 2 in one direction is, by formula (1) ( s = According to the Stefan-Boltzmann Law, S is nroportional to T 4 , there- fore S = ffT*. (The constant of proportionality a is related to the constant y occurring in (3) by the equation v = iry.) The absolute measurement of S gave the following values for a, in chronological order : according to Ch. Ffry [Bull. Soc. Franc. Phys. 4 (1909)]. = 6-51. 10 ~ 12 > according to Ch. F&ry and M. Drecq [Journ. de Phys. (5) 1, 551 (1911)]. = 5*67 . 10 ~ 12 i> according to G. A. Shakespear [Proc. Roy. Soc. (A) 86, 180 (1911)]. ==5-54. 10 ~ 12 M according to W. H. Westphal [Verhandl. d. deutech. physikal. Ges. 14, 987 (1912)]. - 6-05. 10 " 12 >, according to L. Puccianti [Cim. (6) 4, 31 (1912)]. = 5-89. 10 ~ 12 ,, according to Keene [Proc. Roy. Soc. (A) 88, 49 (1913)]. = 5-57.10-12 according to W. H. Westphal [Verhandl, d. deutsch. physikal. Ges. 15, 897 (1913)]. -= 5-85 . 10 ~ 12 , according to W. Gerlach [Phys. Zeitschr. 17, 150 (1916)]. As regards Wieris Law of Displacement, the relation (5a) was tested and found to be confirmed. From Fig. 1, in which E*. is plotted as a function of A. for different values of A., we see clearly how the maximum of the curve becomes displaced towards shorter wave-lengths as the temperature rises. For the constant on the right-hand side of relation (5o) the measure- ments gave the following values : const. = 0-294 [cm. deg.] according to O, Lummer and E. Pringsfoim [Verhandl. d. deutsch. physikal. Ges. 1, 23 and 215 (1899)]. = 0-292 according to F. Paschen [Drude's Ann. 6, G57 (1901)]. = 0-2911 according to Coblentz [Bull. Bur. of Stand. 10, 1 (1914)]. 16 O. Lummer and E. Pringsheim, Verhandl. d. deutsch. physikal. Ges. 1, 215 (1899). 17 F. Paschen, Berliner Ber. 1899, pp. 405, 959. 18 M. Planck, Absorption und Emission elektr. Wellen durch Resonant. Sitzungsber. d. Berl. Akad. d. Wiss. 21 March 1895, pp. 289-301 ; Wiedem. Ann. 57, 1-14 (1896). tJber elektr. Schwingungen, welche durch Re- sonanz erregt und durch Strahlung gedampft werden. Sitzungsber. d. Berl. Akad. d. Wiss. 20 Febr. 1896, pp. 151-170 ; Wiedem. Ann. 60, 577-599 (1897). tTber irreversible Strahlungsvorgange. (1. Mitteilung.) Sitzungs- ber. d. Berl. Akad. d. Wiss., 4 Febr. 1897, pp. 57-68. (2. Mitteilung) ibid., 8 July 1897, pp. 715-717. (3. Mitteilung) ibid., 16 Dec. 1897, pp. 1122- 1145. (4 Mitteilung) ibid., 1 July 1898, pp. 449-476. (5. Mitteilung) ibid., 18 May 1899, pp. 440-480. (Supplement.) ibid., 9 May 1901, pp. 544-555 ; NOTES AND REFERENCES 181 Drudes Ann. 1, 69-122 (1900). (Supplement.) Drudes Ann. 6, 818-831 (1901). Entropie und Temperatur strahlender Warme. Drudes Ann. 1, 719-737 (1900). 19 In place of the mean value, with respect to time, of the energy of a single oscillator, we may use the spatial mean value of the momentary energy of a whole system consisting of very many oscillators. 20 In this second, more difficult part of the calculation, Planck takes his stand upon the second law of thermodynamics, and seeks, from this view, to determine a phase-quantity S of the oscillator, which possesses the well-known property of the entropy, that it increases in all irreversible processes. He arrived at the solution : This function possessed, as Planck showed, the required property of en- tropy, but it was not the only function with this property. And in fact it appeared later, that in the deduction of the above expression, a readily suggested but unjustified supposition had been made. The expression given in the text, formula (8), for the mean energy U follows from S by applying the second law in the form : MO. Lummer and E. Pringsheim, Verhandl. d. deutach. physikal. Ges. 1900, p. 163. 22 M . Planck, Verb. d. deutsch. phys. Ges. 1900, p. 237. It is of historic interest to note that Planck had already, in a somewhat earb'er paper (Verh. d. deutsch. phys. Ges. 1900, p. 202), arrived at the true law of radiation by a purely formal alteration of Wien's formula, which was not further explained. Cf. also Ann. d. Phys. 4, 553 (1901) ; 4, 561 (1901) ; 6, 818 (1901) ; 9, 629 (1902). 23 Let N oscillators be present. Let the total energy to be divided among them be UN = NU. The " state " or phase of the oscillator- system, the probability of which is to be calculated, is then defined by the fact that N oscillators possess the energy Ujt. We divide UN into P energy elements , so that UN = N . U = P*. The number of possible ways of distributing P balls among N boxes is, however, (N+P- 1)1 (N - 1) 1 P ! ' This is therefore the probability of the state, which corresponds to the distribution of P energy elements among N oscillators. P. Ehrenfest aud H. Kamerlingh-Onnes give a very simple deduction of this formula in Ann. d. Phys. 46, 1021 (1915). The rule mentioned in the text, which is due to Boltzmann, states 182 THE QUANTUM THEORY that the entropy Sjt of the oscillator system is connected with the prob- ability TFby the fundamental relation Sir = k log W where k is a constant. In this theorem of Boltzmann the following law of the growth of en- tropy (second law of thermodynamics) is contained : if a system passes from an improbable condition into a more probable one, then by this transition W, and therefore the entropy S, increases. If we here insert the value of W, and, since N and P are very large numbers, use Stirling's approximation formula \oge(Nl)=N(logeN- 1) then, if we set for P, N , we get by an easy calculation and hence the entropy S of one oscillator becomes : But according to the Second Law (see note 20) dS 1 If we carry out the differentiation on the left-hand side, and solve the re- sulting relation between U, T, and e, with respect to U, we get the ex- pression (9) of the text. 2< Of. the paper by Ehrenfest and Katnerlingh-Onnes cited in the previous note. 2S This law is essentially identical with Boltzmann' 's H- Theorem. Of. L. Boltztnann, Vorlesungen iiber Gastheorie Bd. I, p. 38 (1896) ; Sit- zungsber. d. Wiener Akad. d. Wiss. (II) 76, 373 (1877). Of. also P. Ehrenfest, Phys. Zeitschr. 15, 657 (1914). 36 H. Rubens and F. Kurlhaum, Sitzungsber. d. Berl. Akad. d. Wiss. 1900, p. 929 ; Ann. d. Phys. 4, 649 (1901). 27 F. Paschen, Ann. d. Phys. 4, 277 (1901). 28 L. Holborn and 8. Valentiner, Ann. d. Phys. 22, 1 (1907) ; Coblentz, Physical Review, 31, 317 (1910) ; E. Baisch, Ann. d. Phys. 35, 543 (1911) ; E. Warburg, O. Leithttuser, E. Hupka and C. Milller, Ann. d. Phys. 40, 609 (1913) ; E. Warburg and C. Milller, Ann. d. Phys. 48, 410 (1915). W. Nernst and Th. Wulf, Ber. d. deutsch. phys. Ges. 21, 294 (1919). 80 Lord RayUigh, Phil. Mag. 49, 539 (1900). 31 The " Stefan-Boltzmann constant of total radiation" 0-29) or both, are seriously affected by experimental error, or whether after all as Nernst and Wulf maintain Planck's formula is not right, must be left for the future to decide. 38 M. Planck, Ann. d. Phys. 4, 553 (1901). 84 If we apply Boltzmann's relation S = k log W (quoted in note 15), which connects the entropy S with the probability of state W, to one gramme-molecule of an ideal gas, then by calculating the probability of a certain state, i.e. a certain distribution of velocities among the molecules, we arrive at the following value for the entropy of the gas S kN(l log* U + log V) + const. 184 THE QUANTUM THEORY (Of., for example, M. Planck, Lectures on the Theory of Radiation (1906), 143.) Here N is the number of molecules in a gramme- molecule (Avogadro's number), U the energy, V the volume of the gas. Now, according to the Second Law of Thermodynamics, jo dU + must be a complete differential, where p and T denote pressure and temperature of the gas. Hence the relation \dVju~ T must hold. This gives kV = P ie p _kNT If we compare this with the equation of state of an ideal gas in thermo- dynamics, p = ~, we get for the absolute gas constant R the value B = kN from which formula (19) of the text follows. 88 M. Planck, Ann. d. Phys. 4, 564-566 (1901). 38 Compare, for example, the table of the values of Avogadro's number given in the report of J. Perrin at the Solvay Congress in Brussels (1911). [A. Eucken, Die Theorie der Strahlung und der Quanten. Abhandlungen der Bunsen-Gesellschaft Nr. 7, Wilh. Knapp, Halle 1914.] 37 R. A. Millikan, Phil. Mag. (6) 34, 13 (1917). Mlbid., from the values given by Millikan for the electronic charge e = 4-774 x 10- 10 (electrostatic units) and from the electrochemical constant F = 969-4 . 2'999 . 1010 electrostatic units, there follows for Avogadro's number the value N = 6-0617 . 1023. 39 Of., for example, W. Gibbs' Elements of Statistical Mechanics, Chapter V. WThe term "mean value" may be taken as referring to time or to space. If we select a definite atom, and follow it a long time upon its zig-zag path, and from the mean of the values which its kinetic energy assumes in the course of time, we get the " time-mean." If, on the other hand, we select a large number of identical atoms of the gas at a particular instant and again form the mean of the values of the kinetic energies which these atoms possess at the instant in question, we get the " space-mean." *1 If x is the elongation of the oscillator (electron) vibrating with the natural frequency, then x = A sin (2mrf), where A is the amplitude and t the time ; the mean kinetic energy becomes NOTES AND REFERENCES 135 The mean potential energy is : V Hence, as stated, L = V: i.e. the mean kinetic energy = the mean potential energy. 42 /. H. Jeans, Phil. Mag. 10, 91 (1905). MH. A. Lorentz, Proc. Kon. Akad. v. Wet., Amsterdam 1903, p. 666. The theory of electrons (Teubner, Leipzig 1909), Oh. II. 444. Einstein and L. Hopf, Ann. d. Phys. 33, 1105 (1910). UA. D. Fokker, Ann. d. Phys. 43, 810 (1914). 46 M. Planck, Ber. d. Berl. Akad. d. Wiss., 8 July 1915, p. 512. 47 H. A. Lorentz. Die Theorie d. Strahlung u. d. Quanten ; Abhand- lungen der Deutschen Bunsen-Gesellschaft. Nr. 7. v. A. Eitcken. Halle, W. Knapp 1914 pp. 10 et seq. 48 By a suitable modification of classical statistics in the sense of the quantum theory, we can obtain the expression (9) for the mean energy of an oscillator in the following manner which is worthy of notice. Let a number N of similar oscillators with the most varied values for the energy be given. We require to find how great is the probability w, that an oscillator possess a certain energy value U; or, otherwise expressed, how many of the N oscillators possess the energy U. In order to answer this question, we find it best to take first of all the standpoint of Oibbs' statistical mechanics, that is, of " classical " statistics. In place of the special case in question, namely, that of the linear oscillator, let us consider at once quite generally a system of / degrees of freedom, and characterise it by / generalised co-ordinates 2i2z ^ and by the corresponding impulses or momenta 2*1 .Pa . . Pf. (Here, the impulse pi is thus defined : form the kinetic energy of the system as a function of the generalised velocities qi J-, then OT- . <** pi = -~r. \ In particular, the linear oscillator (vibrating electron) will qi' be described by a co-ordinate q, namely, the elongation of the electron, and the impulse p = m -%. In general, therefore, 2/ quantities are necessary in order to define completely the momentary state of a system. Hence we can represent this momentary state by a point (" phase-point ") in the 2/- dimensional space in which 9i . . . Pf (of the " phase-space ") are co-ordinates. We now consider a number N of similar systems of this kind, which are in thermodynamic equilibrium with a very large reservoir at the temperature T. Then the probability that the co-ordinatea and impulses lie in the small intervals q 1 . . . g t + dq v etc., and Pi . . . Pi + rfpj, etc., that is, that the "phase-point" of the system lie in the element dn = dq^dq^ . . . dqj, dp^p t . . . dpf of the phase- space is, according to Oibbs, 136 THE QUANTUM THEORY Here E is the energy of the system, and k is the constant defined in (19). The integration in the denominator is to be taken over all possible values of the 2/ quantities q 1 . . . pf, or, as we may say, over all possible " phases," or over the whole region of the phase-space concerned. Among the N systems there are then Nw, whose phase-points lie in the element dfi of the phase-space. This is therefore a " distribution " of the N systems over the phase-space. This distribution is called Canonical ; it represents a generalisation of Maxwell's familiar law of distribution of velocities which may be deduced from it by special- ising it for the case of the gas atom, that is, by setting/ =3. The sum of all probabilities is naturally 1. Indeed, it is at once clear that L-> For the mean value of the energy E we get J-M. -/*'**. /."*> If we apply this equation to the linear oscillator we get / f - TT JjUe Wgqdp ffe-^dp N W ' <7 -!' + -0MV If we introduce the auxiliary variables { and n, defined by j = TW2m j n = ,JL, and hence dqdp = ddn * fj'2m TV we get U = f 2 + T? and, therefore, it suggests itself to us to write f|= \rj= where is a parametric angle. If we interpret { and rj as Cartesian co- ordinates of a point in the plane, then JU and

= in agreement with (24). This is the standpoint of classical statistics. The quantum statistics of the oscillator may be immediately deduced from this, if we elaborate the canonical law of distribution i - Je kT dqdp in a suitable manner If we here again introduce dqdp = -dUdf, and integrate with respect to $, we get _ wu= e * r dU Je'^dU as the probability that the energy of the oscillators lies between U and U H- dU. Now the quantum theory demands that the energy U shall assume only the discrete values U , U lt U , . . . Un- The transition may best be effected by laying down the condition : E shall only be able to assume the values contained in the narrow intervals between U and U + a, U^ and I7j + o, and generally Un and U n + o. Then dU = a, and the integral in the denominator changes into a sum. Thus it follows that n n thus a is eliminated ; if we now proceed to the limit a = 0, w remains 138 THE QUANTUM THEORY unaltered. Hence w n is the canonical distribution function generalised for quantum conditions, and hence, among N oscillators, Nw n have an energy of the value U n . We now get for the mean energy V * 2ft* v Now, according to the first form of the quantum theory, U n = tie = nh v (n = 0, 1, 2, 8 . . . oo )". Therefore u = 2 V o S, o If we set fc5>, for convenience, = x, then Further, from which we get in agreement with (9) The canonical distribution may be still further generalised by the intro- duction of certain "weight factors," which are intended to express the fact that the individual quantum states of the system considered have, a priori, different probabilities. This happens, for example, if each quantum state may be realised in different ways, and if the number of these possi- bilities of realisation is different for the different quantum states. Then, the different states will have different "weights," and a "weight factor" v n p n has to be included in the exper mental function e ~*r so that the can- onical distribution function assumes the form - = C . f^ff' Here C depends on the temperature ; p n , on the other hand, doe not. NOTES AND REFERENCES 189 194. Einstein, Ann. d. Phys. 17, 132 (1905) ; 20, 199 (1906) ; Verhandl. d. deutsch. physikal. Ges. 11, 482 (1909) ; Bericht Einstein auf dem Solvay-Kongress in Brussels 1911 ; cf. A. Eucken, Die Theorie der Strahlung und der Quanten ; Abhandl. d. deutsch. Bunsen-Gesellschaft, Nr. 7 (Halle, W. Knapp 1914), pp. 330 et seq. Cf. also W. Wien, Vorlesungen iiber neuere Probleme der theoretischen Physik (Teubner, Leipzig and Berlin 1913), 4. Vorlesung. H. A. Lorentz, Les theories statistiques en thermodynamique (Teubner, Leipzig and Berlin 1916), 42 et seq. SO A. Einstein, Ann. d. Phys. 17, 132 (1905). 31 A. Einstein, Phys. Zeitschr. 10, 185 (1909). 52 This formula may be deduced as follows : Firstly, from e = E - E the frequently used relation ? = W - 2E . IS + (E)* = W - (IT) 2 follows. In order now to calculate the two quantities E* (mean of the squares of the energy) and (E)* (square of the mean energy), which are known to differ from each other in general, we do best to take the standpoint of Gibbs' statistical mechanics (see note 48). According to this, the probability that the co-ordinates and impulses lie in the small intervals q l . . . q l + dq lt etc., p^ . . . p 1 + dp lt etc, that is, that the "phase-point" lies in the element dq t dq 2 . . . dqfdp^dp^ . . . dp, = dn of the " phase-space " : Then the mean of the energy follows in the usual way : Likewise, We then form dE 140 THE QUANTUM THEORY Therefore, dE F-Wjy We also arrive at the same formula, if instead of the classical canonical distribution function, we start from the quantum distribution function 68 The mean energy of radiation of frequency v in the volume v is = vu v dv, where the monochromatic density of radiation is if Planck's Law is taken as the basis. (Of., for example, M. Planck, Lectures on the Theory of Radiation, Engl. Transl.) According to formula (28) deduced in the previous note, it therefore follows that If we eliminate T on the right-hand side by substituting for e** its value 1 + 8vhvS , it follows that C'Uy u v vdv , The second term on the right is required by the Undulatory Theory for at each point of the volume v the most varied trains of waves of radiation cross one another's paths with every possible amplitude and phase. The interference of all these waves thus generates at the point considered an intensity, which varies continually, and hence the energy of the volume v also varies. If we calculate the mean of the square of the energy, i.e. a , wo find precisely the second term of the above formula. (Of., for example, H. A. Lorents, Les theories statistiques en thermodynamique (Teubner, Leipzig and Berlin), 1916, pp. 114 et seq,) The first term is not, however, explained by the classical undulatory theory. On the other hand, it becomes endowed with meaning if we suppose that the radiant energy consists of a certain whole number NOTES AND REFERENCES 141 (n) of finite energy complexes of the value hv. For then E = n hy, and therefore E = n hv, where n is the mean about which the number n varies. If 5 = n -_ n be the variation of the number n, then it follows that e = E - E = Shy^ where 2 = S 2 feV. But, according to a well-known law of statistics, S 2 = n. (Cf., for example, H. A, Lorentz, loc. cit., 26 and 27.) Hence ? = nfeV = ~E'hv. This is exactly the first term in the above formula. MA. Einstein, Ann. d. Phys. 17, 144 (1905). 95 J. J. Thomson, Conduction of Electricity through Gases. 96.4. Einstein, Ann. d. Phys. 17, 147 (1905). 97 Of. R. Pohl and P. Pringsheim, Die lichtelektrischen Erschein- ungen. Sammlung Vieweg Heft 1 (Braunschweig 1914). 98 A. Einstein, Ann. d. Phys. 17, 145 (1905). 99 R. A. Millikan, Phys. Zeitschr. 17, 217 (1916). 60 According to Pohl and Pringsheim, we have to distinguish between the normal and the selective photo-effect : in the case of the normal effect the number of electrons torn off (per calorie of the Jight-energy absorbed) is independent of the orientation of the electrical vector of the light-wave, and increases, starting from an upper limit of the wave- length, in general uniformly as the wave-length decreases. In the case of the selective effect, on the other hand, which only appears when the electrical vector of the light-wave possesses a component vertical to the metallic surface, the number of electrons torn off (per calorie of light- energy absorbed) shows a decided maximum at a definite wave-length. 61 Oh. Barkla, Phil. Mag. 7, 543, 812; 15, 218. Jahrb. d. Radioak- tivitat u. Elektronik, 5, p. 239, 1908. Ch. Barkla and Sadler, Phil. Mag. 17, 739. Ch. Barkla, Jahrb. d. Radioaktivitat u. Elektronik, 1910, p. 12. if. de Broglie, G. R. 25 May and 15 June 1914, p. 1785. Ch. Barkla, Phil. Mag. 16, 550. E. Wagner. Ann. d. Phys. 46, 868 (1915); Sit- zungsber. d. bayer. Akad. 1916, p. 39. 62 D. L. Webster, Proc. Americ. Acad. 2, 90(1916); Physic. Review, 7, 587 (1916). 63 E. Wagner, Ann. d. Phys. 46, 868 (1915). 64 Of., for example, E. Wagner, Phys. Zeitschr. 18, 443 (1917). The value that Wagner calculates for h is : h = 6-62 .10-27. 65 W. Duane and F. L. Hunt, Physic. Review, 6, 166 (1915). 86.4. W. Hull and If. Rice, Proc. Americ. Acad. 2, 265 (1916). 67 E. Wagner, Phys. Zeitschr. 18, 440 et seq. (1917) ; Ann. d. Phys. 57, 401 (1918). 68 F. Dessauer and E. Back, Ber. d. deutsch. physikal. Qes. 21, 168 (1919). 69 J. Franck and G. Hertz, Verhandl. d. deutsch. physikal. Ges. 16, 512 (1914). 70 The critical potential measured by Franck and Hertz amounted to V= 4-9 volts = electrostatic units, and therefore the critical energy 300 of the electron is v _ 4-774. 10 -10. 4-9 300 142 THE QUANTUM THEORY The wave-length A of the mercury line emitted is \ = 25364 = 2-536.10-5. Hence we must get 17 j.e i. V* 4-774. 10 -10. 4-9. 2-536. 10 -5 ,7= fc-, ,.e. h = = - 3 . 102-3 ^r = 6-59.10-27 and this is in good agreement with the results of other measurements. 71 Cf., for example, J. Stark, Prinzipien der Atomdynamik II. (S. Hirzel, Leipzig 1911), Chs. IV and V. 72 J. Stark, Ber. d. deutsch. phys. Ges. 10, 713 (1908) ; Phys. Zeitschr. 8, 913 (1907) ; 9, 767 (1908). Canal-rays are positively charged particles of matter, which move in a vacuum tube in the direction : anode to cathode ; the latter is pierced with holes through which the canal-rays pass into the space behind the cathode. If we generate such canal-rays in a vacuum tube rilled with hydrogen, we find that the series lines of hydrogen are emitted. Now, if we observe this emission spectroscopically " from the front," that is, so that the canal-rays are moving towards the observer, we see, firstly, at its usual place in the spectrum, the sharp series line (line of rest, "in- tensity of rest ") ; secondly, we see displaced towards the violet, a broadened strip (line of motion, "intensity of motion" or "dynamic intensity"). These lines represent the series line emitted by the moving canal-ray particles, which is displaced towards the region of higher frequencies on account of the Doppler effect. Since the canal-rays do not possess a single uniform velocity, and since particles with all possible velocities occur, the displaced strip is not sharp, but softened and broadened. The " intensity at rest " is therefore emitted when the quickly moving canal particles strike " resting " molecules, i.e. gas- molecules which are moving comparatively slowly and irregularly, and excite these to emit the series lines. The " intensity of motion," on the other hand, is excited by the unidirectionally moving canal particles themselves, when they hit gas-molecules. Now, it is very remarkable that the interval between the intensity of rest and that of motion is not filled in, but that the emission of the in- tensity of motion becomes observable only above a certain velocity. Stark interpreted this fact in terms of the light-quantum hypothesis thus : If %mv z is the kinetic energy of a canal-ray particle, and if the fraction arau 2 (a > 1) is transformed into a light-quantum hv upon collision with a gas-molecule, then we must have h < -me 2 ; that is, the spectral line of frequency v can be generated only by canal-rays, the Telocity of which >A/?5 \ can The proportionality between the critical velocity and *Jv has been fairly well borne out. It should be remarked here that J. Stark has lately abandoned the theory of light-quanta. (Cf. J. Stark, Verh. d. deutsch. physik. Ges. 16, 304 (1904) ; 18, 42 (1916).) NOTES AND REFERENCES 148 73 J. Stark, Phys. Zeitschr. 9, 85, 356 (1908). J. Stark and W. Steubing, Phys. Zeitschr. 9, 481 (1908). J. Stark, Phys. Zeitschr. 9, 889 (1908). In these papers J. Stark defends the view that the band-spectra are emitted when a "valency electron" belonging to the atom or molecule is pushed out of its normal position and then returns again to its initial position, counterbalancing the work done in displacement. If the energy of deformation (valency energy) E is changed into a light- quantum, then we must have hv = E, i.e. v ^- All lines of the ^ 7t ET band must therefore lie below the edge v = _ . If the valency energy E is changed by chemical processes, the band-spectrum must be dis- placed accordingly. 74,1 Einstein, Ann. d. Phys. 17, 148 (1905). 75 J. Stark, Phys. Zeitschr. 9, 889 (1908) ; Ann. d. Phys. 38, 467 (1912). The fundamental law of photochemical decomposition enunciated by Stark and Einstein states : If a molecule dissociates at all owing to j,he absorption of radiation of frequency v, then it will absorb an amount of energy hv when it dissociates. This energy, therefore, represents the heat of reaction, which will be set free upon recombination of the products of decomposition. This law was later deduced by A. Einstein for the range of validity of Wierfs Law of Radiation without the assistance of the light-quantum hypothesis, by purely thermodynamical methods. (Of. Ann. d. Phys. 37, 832 (1912), and 38, 881 (1912).) nE. Warburg, Ber. d. Berl. Akad. d. Wiss. 1911, p. 746; 1913, p. 644 ; 1914, p. 872 ; 1915, p. 230 ; 1916, p. 314 ; 1918, pp. 300, 1228. Cf. also " Naturwissenschaften," 5, 489 (1917). 77 H. A. Lorentz, Phys. Zeitschr. 11, 1250 (1910). 78 M. Planck, Ber. d. deutsch. physikal. Ges. 13, 138 (1911); Ann. d. Phys. 37, 6i2 (1912). 79 On account of the continuous (classical) absorption, all energy values of the oscillator in an elementary region, say between n and (n + l)e, are equally probable. The mean energy in the nth elementary region is, therefore, Prom the canonical law of distribution extended in the sense of the quantum theory, it then follows that 144 THE QUANTUM THEORY (cf. note 48). If we further set e = hy it follows that In place of relation (7) of the text we get here and this leads to Planck's Law of Radiation. 80 M. Planck, Sitzungsber. d. Kgl. Preuss. Akad. d. Wiss. 3 April, 1913, p. 350; ibid., 30 July 1914, p. 918; ibid., 8 July 1915, p. 512. 81 A. Einstein and O. Stern, Ann. d. Phys. 40, 551 (1913). 82 W. Nernst, Verhandl. d. deutsch. physikal. Ges. 18, 83 (1916). 83 F. Richarz, Wiedem. Ann. 52, 410 (1894J. 84 Report by P. Langevin at the Solvay Congress in Brussels, 1911. Cf. A. Eucken, Die Theorie der Strahlung und der Quanten. Abhandl. d. deutsch. Bunsen-Ges., Nr. 7 (W. Knapp, Halle 1914), pp. 318 et seq. 8M. Einstein and W. J. de Haas, Verhandl. d. deutsch. physikal. Ges. 17, 152, 203, 420 (1915). A. Einstein ibid., 18, 173 (1916). W. J. de Haas, ibid., 18, 423 (1916). 86 E. Beck, Ann. d. Phys. 60, 109 (1919). 87 Report by Planck at the Solvay Congress in Brussels, 1911. See A. Eucken, Die Theorie der Strahlung und der Quanten. Abhandl. d. deutsoh. Bunsen-Ges., Nr. 7 (W. Knapp, Halle 1914), p. 77. 88 If q is the elongation of a linearly vibrating electron of mass ra (os- cillator) and v its period of oscillation, then the energy of this configur- ation is The first term represents the kinetic and the second the potential energy. Now the impulse (the momentum) is p = m ~Ti- Therefore, we may write i.e. NOTES AND REFERENCES 145 The curves U = const., that is, those curves in the phase-plane, which correspond to the states of constant energy of the oscillator, are therefore ellipses with the semi-axes For a definite value of U we get a completely definite ellipse. The " phase-point " of the oscillators would continually revolve in this ellipse, if the electron, without emitting or absorbing, were to execute pure har- monic oscillations : for then its energy would remain permanently constant. If we allow U to vary continuously, i.e. if we give it other and again other values in continuous succession, we get an unlimited manifold of concentric ellipses. The quantum theory, as formulated in (30) in the text, selects from this infinite manifold a discrete set of ellipses, and distinguishes them as the "quantised" ellipses which correspond to the " characteristic states " of the oscillator. To these belong the "quantum energy- values " U , U lt U a . . . Dn. Now the nth ellipse encloses an area nh. The area of the nth ellipse is, however, hence we must have = nh i.e. U n = that is, in the nth quantum state the oscillator possesses an amount of energy nt = nhv. tOA. Sommerfeld, Phys. Zeitschr. 12, 1057 (1911). Report by A. Sommerfeld at the Solvay Congress in Brussels, 1911. Cf. A. Eucken, Die Theorie der Strahlung und der Quanten. Abhandl. d. deutsch. Bunsen-Ges., Nr. 7 (W. Knapp, Halle 1914), p. 252. 80 Report by Sommerfeld at the Solvay Congress, 1911. 91 A. Sommerfeld and P. Debye, Ann. d. Phys. 41, 873 (1913). 92 Cf., for example, the recent summary by E. Schrddinger, Der Energieinhalt der Festkorper im Lichte der neueren Forschung. Phys. Zeitschr. 20, 420, 450, 474 (1919). A complete set of references accom- panies this account. 93 One gramme-atom of a substance, the atomic weight of which is a, is defined as the quantity a grammes of the substance. For example, one gramme-atom of copper is equal to 63'57 grammes of copper, since 63-57 is the atomic weight of copper. Exactly analogous is the definition of the gramme- molecule (also called " mol"). One gramme- molecule of oxygen is 32 grammes of oxygen, for the molecular weight of oxygen (diatomic) is 32. If c is the specific heat of a substance of atomic weight a, it signifies that one gramme of the substance requires an amount of heat c to raise its temperature by 1 C. Hence we must communicate to a gramme-atom 10 146 THE QUANTUM THEORY of the substance, i.e. to a grammeB of it, an amount of heat C = ea in order to raise its temperature by 1 C. C is then called the atomic heat. 94 The equality of the mean potential and the mean kinetic energies is true here as in the case of the linear Planck oscillator (vibrating electron), of. note 41. This equality is, in general, always present when the forces which act upon the atoms and restore them to their positions of rest (zero positions) are Ivnea/r functions of the relative displacements of the atoms, that is, when the force is "quasi-elastic," that is, proportional to the displacement from the zero position. Cf. in this connexion L. Boltzmann, Wiener Ber. 63 (11), 731 (1871), and F. Richarz, Wied. Ann. 67, 702 (1899). WDulong and Petit, Ann. de chim. et de phys. 10, 395 (1819). 96 The quantity usually obtained by measurement is not the atomic heat at constant volume C,,, but the atomic heat at constant pressure C f . For this we get values which in general fluctuate about the value 6'4 cal./deg. The calculation of C c from C p is based on the thenno- dynamically deduced formula C p - C, = a * VT where a is the cubical coefficient of thermal expansion, K the (isothermal) cubical compressibility, and V the atomic volume = atomic wei g ht . density 97 E.g. we find for silver at C C p = 6-00 aluminium 58 C p = 5-82 copper 17 C C p = 5-79 lead 17 C p = 6'33 iodine ,, 25 C C p = 6-64 zinc 17 C C p = 6-03 98 F. E. Weber, Poggend. Ann. 147, 311 (1872) ; 154, 367, 553 (1875). 99 As a possible way out, the "agglomeration hypothesis," supported by F. Richarz [Marburger Ber. 1904, p. 1], C. Benedicks [Ann. d. Phys. 42, 183 (1913)] and others, has been put forward. According to this, as the temperature falls the number of degrees of freedom of the system diminishes by " freezing-in," as it were, in that certain linkages become completely rigid. According to this, however, the compressibility should decrease greatly as the temperature falls, which, according to E. Orii- neisen's measurements is not the case [Verb. d. deutsch. phys. Ges. 13, 491 (1911)]. Compare also in this connexion the report of E. Schro- dinger quoted in note 92. 100.4. Einstein, Ann. d. Phys. 22, 180, 800 (1907). 101 Cf. A. Einstein, Ann. d. Phys. 35, 683 ff. (1911), also the report by Einstein at the Solvay Congress in Brussels, 1911 ; see A. Eucken, Die Theorie der Strahlung und der Quanten. Abhandl. d. deutsch. Bunsen-Ges., Nr. 7 (W. Knapp, Halle 1914), pp. 330 et seq. 1024. Einstein, Ann. d. Phys. 34, 170, 590 (1911) ; 35, 679 (1911). IWThe nature of the dependence of the frequency v on the three NOTES AND REFERENCES 147 quantities A, p, K may, according to Einstein (loc. cit.), be obtained by a simple dimensional calculation. If we assume that v depends only on the mass in of the atoms, their distance apart d, and the compressibility K of the body, then an equation of the following form must hold v = C . m x . & . K z . C is here a numerical constant ; x, y and z are numbers which remain to be determined. The dimensions of the frequency \y] are \t - 1] ; the dimensions of m and d are \ni] and [i], and the dimensions of the compressibility K follow from its definition : _ _ increase in volume _ increase in pressure x original volume K has therefore the dimensions I! = [-surface "I ->"-" ' L force J L We thus get the following dimensional equation t~ l Hence x - 2 = 0; y -f z = ; "2z = - 1 from which we get x- -J; y- +4; 2- -4 We have therefore, Let N be Avogadro's number, i.e. the number of atoms in the gramme- atom. Then the atomic weight of the body is numerically equal to the mass of the gramme-atom, i.e. A = mN. If we imagine the atoms arranged upon a cubical space-lattice with sides d, then the density must satisfy the equation from this it follows that and hence i. from which, it follows that 148 THE QUANTUM THEORY Einstein determines the factor C by assuming simply that only the twenty-six neighbouring atoms act upon the displaced atom. IMF. A. Lindemann, Phys. Zeitschr. 11, 609 (1910). Lindemann's formula may be shortly deduced thus : Let r = a sin (2-irvt) be the elonga- tion of an atom which is vibrating with the amplitude a and the frequency y. The mean energy of this atom is E ' f (ar)" + f ' At the melting-point, according to Lindemann's conception, a is of the same order as d (distance apart of atoms). On the other hand, the mean energy of the atoms at high temperatures = 'SkT, or, at the melting-point 3kT s . (The melting-point, as a rule, is high.) From this it follows that = SkT, But we have (see note 103) m =^j- d = m*,-i = AlN-*f-*> Hence v = const. I 7 ,* . 4~*JSM-*V = const. T,* . 4~* . />*. 105 E. Qrilneisen, Ann. d. Phys. 39, 291 et seq. (1912). 106 E. Madelung, Nachr. d. kgl. Ges. d. Wiss. zu Qottingen, mathem.- physikal. Klasse 1909, p. 100, and 1910, p. 1. 107 W. Sutherland, Phil. Mag. (6), 20, 657 (1910). 108 If n and K are the coefficients of refraction and extinction of a substance respectively, then, according to Maxwell's Theory, its reflect- ing power is n - If we require the point of maximum reflection, we have to form the equation ^^ = 0, which gives after reduction the following relation : From this we see that the position of maximum reflection does not coincide exactly with the position of maximum absorption (^ = Oj, \ov / but that it lies the nearer to it, the less the coefficient of refraction varies with the frequency. On the other hand, the point of maximum -absorption lies, according to the dispersion theory, in the immediate ^neighbourhood of the natural frequency v,.. 109 H. Rvtens and E. F. Nichuh, Wiedem. Ann. 60, 418 (1897). Also NOTES AND REFERENCES 149 77. Rubens and H. Hollnagel, Ber. d. kgl. preuss. Akad. d. Wiss. 1910, p. 45; //. Hollnagel, Dissert. Berlin 1910; H. Rubens, Ber. d. kgl. preuss. Akad. d. Wiss. 1913, p. 513 ; H. Rubens and H. v. Wartenberg, ibid., 1914, p. 169. As an example we give here the following small table in which \ denotes the wave-length of the " residual " rays, as given by the above investigators. A. A. NaCl KC1 AgCl HgCl 52 M 63-4 M 81 -5 M 98-8/i T1C1 KBr AgBr TIBr 91-6 M 82-6 M 112-7 M 117 M HO Of., however, note 108. 111 W. Nernst and F. A. Lindemann, Sitzungsber. d. kgl. preuss. Akad. d. Wiss. 1911, p. 494 ; W. Nernst, Ann. d. Phys. 36, 426 (1911). 112 The following short table gives the values for v which are calculated from Einstein's formula (35), Lindemann's formula (36), from the "residual rays" (see note 109), and from the observed atomic heat according to an empirical formula (40) proposed by Nernst and Lindemann. For more detailed data with, in part, corrected numerical factors see C. E. Blom, Ann. d. Phys. 42, 1397 (1913). Substance V E "L ''residual rays ''atomic heat (Nerntt-Lindemann) Al 6-7 . 10 12 7-6 . 10" 8-3 . 10" Cu 5-7 . 10 12 6-8 . 10 12 6-7 . 10 12 Zn 4-4 . 10 12 4-8 . 10 12 Ag 4-1 . 10 12 4-4 . 10 12 4-5.10 12 Pb 2-2 . 10 12 1-8. 10 12 1-5 . 10 12 Diamond 32-5 . 10 12 40 . 10 12 NaCl 7-2. 10 12 5-8 . 10 12 5-9 . 10 12 KC1 5-6 . 10 12 4-7 . 10 12 4-5 . 10 12 113 W. Nernst, F. Koref, F. A. Lindemann, Untersuchungen iiber die spezifische Warme bei tiefen Temperaturen. I. u. II. Sitzungsber. d. kgl. preuss. Akad. d. Wiss. 1910, 3 March. W. Nernst, idem III., ibid., 1911, 9 March. F. A. Lindemann, idem IV., ibid., 1911, 9 March. W. Nernst and F. A. Lindemann, idem V., ibid., 1911, 27 April. W. Nernst and F. A. Lindemann, idem VI., ibid., 1912, 12 Dec. W. Nernst, idem Vn., ibid., 1912, 12 Dec. W. Nernst and F. Schwers, idem VIII., loc. tit., 1914. W. Nernst, Der Energieinhalt fester Stoffe. Ann. d. Phys. 36, 395 (1911). HI W. Nernst, Die theoretischen und experimentellen Grundlagen des neuen Warmesatzes. (W. Knapp, Halle 1918.) 115 The First Law states : If d'Q is the heat supplied to a system, d'A 150 THE QUANTUM THEORY the work done on the system from outside, then the increase of energy U of the system is given by dU = d'Q + d'A. The Second Law states : if d'Q is supplied reversibly at the temperature T, then ^ is the complete differential of the entropy S, hence Let us follow Helmholtz and introduce the " free energy" F denned by F= U- T- S. Then it follows that dF = dU - T.dS - S.dT=d'Q + d'A - T.dS - S.dT i.e. dF = d'A - S . dT for every reversible process. If the process is isothermal (dT = 0) then it follows that dF = d'A or, for a finite change of state, F. A - F l = A. If we set A' = - A, so that A' is the work gained, we get Fj - F, = A'. That is, the work gained in the isothermal reversible process which is, as may be shown, the maximum obtainable is equal to the decrease of free energy. Further, it follows, since at constant volume F the work d'A = 0, that Therefore, formulating these expressions for two states, we get /pjOffii - "a)~| _ ^ _ p^ _ ^J7 } _ {/ 2 j or, finally, if we write for short t7j - U z = U' an equation much used in physical chemistry. Since, now, according to Nernst's heat theorem, ('),. =o \OT Jhm T=Q (A' - U') vanishes for T = 0, being above the first order. Hence lim ?L4' ~ . ^') = Q and hence also NOTES AND REFERENCES 151 This is equation (89) of the text. From -= - S, it follows further that ^ F ] ' " F ^ - S 2 - 8 lt or Q _ O _ 9-4' and hence Nernst's Theorem may be formulated thus lim (S 2 - S,) = r=o that is, in /; neighbourhood of the absolute zero all processes proceed without change of entropy. 116 Cf., for example, M. Planck, Lectures on Thermodynamics. Planck goes further than Nernst inasmuch as he postulates that not only the difference of the entropies S 2 - S x is zero at absolute zero (see previous note) but also that the individual values themselves become zero. Hence, according to Planck, at the absolute zero of temperature the entropy of every chemically homogeneous body is equal to zero. From this the con- clusion given in the text, lim^^,j = may be deduced immediately. It follows from the relation (occurring in the last note) F - U = - TS and from Planck's version of Nernst's Theorem, that F - 0" vanishes for T = 0, being of higher order than the first. Hence = or limf^-H S) T=o\oT ) or, finally, 117 For low temperatures, that is, for high values of x = ^ Einstein's formula (34) takes the following form : C v = 3Rx*e~ x . The falling-off at low temperatures therefore follows an exponential law ; more exactly, it varies as i const 118 W. Nernst and F. A. Lindemann, Sitzungsber. d. kgl. preuss. Akad. d. Wiss. 1911, p. 494 ; Zeitschr. f. Elektrochemie, 17, 817 (1911). 119 A. Einstein, Ann. d. Phys. 35, 679 (1911). 120 For if we regard the atoms as mass-points, then each atom hao three degrees of freedom ; the whole body has therefore '3N degrees of freedom. As is proved in mechanics, however (cf. R. H. Weber and R. Oans, 152 THE QUANTUM THEORY Repertoriuin der Physik Bd. I. pp. 175 et seq.), a mechanical system of 3N degrees of freedom has 3N natural frequencies, and the moit general small motion of each atom consists in a superposition of these 3N natural frequencies. 121 P. Debye, Ann. d. Phys. 39, 789 (1912). 122 M. Born and Th. v. Kdrmdn, Phys. Zeitschr. 13, 297 (1912); 14, 15, 65 (1913). Of. also M. Born, Ann. d. Phys. 44, 605 (1914) ; M. Born, Dynamik der Kristallgitter (Teubner, Leipzig and Berlin 1915). 123 Of., for example, R. Ortvay, tjber die Abzahlung der Eigenschwin- gungen fester Korper. Ann. d. Phys. 42, 745 (1913). Ortvay considers the natural frequencies of an elastic cube, each side of which has the length L. There are found to be three groups of natural frequencies. The first two groups are the transversal frequencies, the third group is the group of the longitudinal frequencies. That the trans- versal frequencies form two groups (moreover identical) is easily seen. For in the case of a transversal vibration, which is propagated in, say, the direction of the aj-axis, two equal alternatives are probable, namely, that the particles vibrate parallel to the y- or to the 2-axis. In the case of the longitudinal oscillations, however, there is naturally only one group ; for in the case of propagation along the z-axis there is only one possibility, namely, that the particles vibrate parallel to the x-axis. The frequencies of the first two groups are characterised by the values the third group by Here c t and c t are the velocities of propagation of transversal and longitu- dinal waves in the body, whereas a, b, c are arbitrary positive whole numbers. If therefore we give a, b, c all possible values in all possible com- binations, we get all the possible transversal and longitudinal natural fre- quencies, which together form the elastic spectrum of the cube. If now we inquire how many transversal natural frequencies of the first group fall below v, this means nothing else than inquiring how many trios of values (a, b, c) fulfil the condition Imagine a, b, c as co-ordinates of a point in space. Then all possible trios (a, b, c) of values are represented by the total "lattice-points" of the positive space octant, and the above question is answered by counting how many lattice-points are at a distance less than ''from the origin (0, 0,0). c NOTES AND REFERENCES 158 All these lattice-points lie within the positive octant of the sphere whose radius is -. Since now one lattice-point is assigned to every volume of c t magnitude 1 namely, every elementary cube the required number of lattice-points, provided that it is sufficiently large, is equal to the volume of the positive spherical octant of radius ", i.e. is equal to If V = L 3 is the volume of the given cubical body, then the number of the transverse natural frequencies below v belonging to the first group is Zl = T F cJ The number belonging to the second group is the same, that is rr r? 4ir T r V Finally, the number of the longitudinal frequencies corresponding to these is 4ir v"' We thus get for the total of all natural frequencies below v The total of natural frequencies in the interval v . . . v + dv follows by differentiation with respect to v and this is just formula (43) of the text. 124 In formula (43) for Z(v)d v let us replace, according to formula (44) of the text, the factor Then it follows that 154 THE QUANTUM THEORY If we now aet - and - m , we get 128 A table showing how the Debye function C depends on x m is given by Nernst (Die theoretischen und experimentellen Grundlagen des neuen Warmesatzes. W. Knapp, Halle 1918, p. 201). In it the simple Einstein function [formula (34) of the text] is also tabulated. 126 If T is great, then x m is small compared with 1 ; then we may replace in the integral of (45) e x by 1 in the numerator, and e x - 1 by x in the denominator. It then follows that 127 If Tia small, then x m is large, and we may replace the upper limit of the integral as a first approximation by oo . The integral will thus become a numerical constant independent of x m , and it follows that C. - j* . const. - 9 - . T3 . const. 128 From the theory of elasticity it follows that and c = J HE \H-o where K is the compressibility, p the density, and a the ratio transverse contraction longitudinal dilatation* If we insert these values in (44) and note further that V = -, formula (46) of the text follows. 129 As the number of frequencies below v is proportional to v*, we get, for example, the following picture : if we divide the interval from to v m into 10 parts, and if only one natural frequency lies in the first division, then in the following divisions there will be 7, 19, 37, 61, 91, 127, 169, 217, 271 natural frequencies ; i.e. the natural frequencies crowd continually closer together. 180 P. Debye, Ann. d. Phys. 39, 789 (1912) ; W. Nernst and F. A. Lind&mann, Sitzungsber. d. Berl. Akad. d. Wiss. 1912, p. 1160. 131 A. Eucken and F. Schwers, Verhandl. d. deutsch. physikal. Ges. 15, 578 (1913) ; W. Nernst and F. Schwers, Sitzungsber. d. Berl. Akad. d. Wiss. 1914, p. 355 ; P. Grttnther, Ann. d. Phys. 51, 828 (1916) ; W. H. NOTES AND REFERENCES 155 Keesom and Kmnerlingh-Onnes, Amsterdam Proc. 17, 894 (1915). Cf. also the graphic tables by E. SchrOdinger, Phys. Zeitschr. 20, 498 (1919). 182 If we introduce into equation (44) of the text, a " mean acoustic velocity " c, by the obvious definition then for the order of magnitude of the smallest wave-length \ m i n , there follows c It now the atoms in the cubical space-lattice, for example, are arranged so as to be a distance a apart, then Na* = V, and hence VI- Amin \ 3 133 For references see note 122. 131 Of. Born, Dynamik der Kristallgitter, 19. 133 .F. Haber, Verb. d. deutsch. phys. Ges. 13, 1117 (1911). For if the atomic residue (mass m) and the electron (mass /i) are held to their zero positions by forces of the same order of magnitude, and if they vibrate independently of one another (a simplifying supposition) the equation of vibration of the atom is nix + a?x = 0, the solution of which is x = A sin( ~~F^t ). The infra-red frequency of the atom is, therefore, a y r = 2a- / , and correspondingly, the ultra-violet frequency of the electron Hence Habeas Law follows : v r : v v = and - = A sin 2irvt - (n + p) In order to find the relation between v and \ (that is, the " law of dis- persion"), let us insert the above formula in the equation of motion. Then it follows that - (n + 1) 1 + sin [znt - (n - 1) 1 * J L A J -n^l\ * Ji - 2*1 sin - *] . (l - cos *2\ \ J \ A / That is, = * J~sin ()- , m sin (Y if we set LJt = , m . W \w \A/ \A/ ir Vro 187 Of. .Bom, Dynamik der Kristallgitter, p. 51. From the special case treated in the previous note, we also recognise the truth of law (49) ; for if A is much greater than a, the dispersion law takes the form v = n , where q = v m ita, represents the velocity of A A propagation of the wave, and this is independent of the wave-length. 138 The statement that a given direction lies in the element of solid angle dti is intended to convey the following sense : about an arbitrary origin O describe a " unit sphere," i.e. a sphere of radius 1. Now let a cone of infinitely small angle be constructed of rays passing through 0, the point of the cone lying at O. Let this cone cut out of the surface of the unit sphere a small element of surface dn. Now let the parallel ray to the "given direction" be drawn through O (here, for example, the wave- normals). If this ray lies in the cone just constructed, then we say that the " given direction " lies in the elementary solid angle dfl. 139 The capacity for heat of a certain finite body is that amount of heat which must be imparted to the whole body in order that its temperature be raised by 1 C. If M is the mass of the body, and c its specific heat, then its capacity for heat is NOTES AND REFERENCES 157 From the mean energy content E of the whole body, r follows by differentiation with respect to the temperature r-* 1 140 This somewhat complicated calculation runs as follows : we start from the formula and first replace \ by g . Thus we get and the integral with respect to A. is transformed into one with respect to vi. The limits of this integral are vi = g*'( n ) [corresponding to \ = A^n)] and vi = (corresponding to A. = oo ). If we further set we get 3 47T *<> fe'T 3 ^ C dn ;'x 4 e*dx In place of the quantities gi(i) and x(n) which still depend essenti- ally on the direction, let certain mean values be introduced. Firstly, let us set 4ir In this way three mean acoustic velocities g lf 2 2 , g 3 , independent of the direction, are defmed. We further introduce in place of Am(n) a mean value independent of the direction, in the following manner. In deduc- ing formula (55) we saw that . Am(n) 158 THE QUANTUM THEORY If we carry out the integration with respect to x, we get to v r da x f o u, 47T dn Now, in a way analogous to that used for the acoustic velocities g, we set Hence Into ^ n ) = \ we introduce in place of qi(n) and \ m (n) the mean values qi and \ m , which are independent of direction ; thereby xi(n) also becomes independent of direction, and is transformed into It follows that '- 1 * 1 ""' 141 At the lowest temperatures . 32^2 Ji- - } i=i Xl b 3 a NOTES AND REFERENCES 159 Now the value of the integrals = ^w 4 . If we further set R = Nk, and for z~J the value (59), we get 15** ' Zfe l If we introduce_in place of the three acoustic velocities^", 2j, 3s a mean acoustic velocity q by means of the definition it follows that Finally for we can write V^ (mean atomic volume) and thus get the formula 148 H. Thirring, Phys. Zeitsohr. U, 867 (1913) ; 15, 127, 180 (1914). 143 M. Born and Th. v. Kdrmdn, Phys. Zeitschr. 14, 15 (1913). 144 Cf. note 182. 148 Cf. note 128. 1484. Eucken, Verhandl. d. deutsch. physikal. Ges. 15, 571 (1913). Cf. also A. E'ucken, Die Theorie der Strahlung und der Quanten (W. Knapp, Halle 1914), pp. 386 et seq., Appendix. 147 Cf. A. Eucken, Die Theorie der Strahlung und der Quanten (W. Knapp, Halle 1914), p. 387. 148 To calculate the mean acoustic velocity q, the relation given in note 141 is used 3 4ir We have therefore to obtain from the "dispersion equation" of the crystal in question (for long waves) the values of the three acoustic velocities q^n), q 2 (n), q s (n) as functions of the wave-direction ; q is then obtained from the above formula by integration over all directions and finally summation. 149 L. Hopf and G. Lechner, Verhandl. d. deutsch. physikal. Gee. 16, 643 (1914). 180 The following short table is taken from the paper of Hopf and Lechner cited in note 149 : 160 THE QUANTUM THEORY Crystal 2 calc. from C y 5 calc. from elastic data Sylvin . . . Rock salt . Fluor-spar . Pyrites 2-36 . 10 B 2-82 . 10 8 4-02 . 10 8 5-43 . 105 2-03 . 10 s 2-72 . 105 3-82 . 10 8 5-12 . 105 181 W. Nernst, Vortrage iiber die kinetische Theorie der Materia und der Elektrizitat. Wolfskehl-Kongress 1913 in Gottingen (Teubner, Leipzig and Berlin 1914), pp. 63 et seq. 182 W. Nernst, ibid., pp. 81 et seg_. 153 E. Schrodinger, Phys. Zeitschr. 20, 503 (1919). Schrodinger correctly points out that apart from the substitution of one single mean x for the three quantities x> in the Debye terms the approximation above all in the second part of C v (i.e. the replacement of the 3(s - 1) frequencies v t . . . v 3 by the constants ^ . . . i>,) may not be permissible in many cases : namely, in those cases in which the masses of the various kinds of atoms are not very different from one another. If we were to allow so he argues the masses of the different kinds of atoms and the forces acting upon them gradually to become equal to one another, a simple atomic lattice would result, and during this process the 3(s - 1) branches of the spectrum, which correspond to the second type of motion, would merge into the three first branches. " They cannot therefore even be approximately monochromatic if the masses differ only slightly." 184 H. Thirring, Phys. Zeitschr. 15, 127, 180 (1914). 188 M. Born, Ann. d. Phys. 44, 605 (1914). 186 E. Oriineisen, Ann. d. Phys. 39, 257 (1912). 187 S. Ratnowski, Verhandl. d. deutsch. physikal. Ges. 15, 75 (1913). 188 Vortrage iiber die kinetische Theorie der Materie und der Elek- trizitat. Wolfskehl-Kongresz zu Gottingen, 1913. (Teubner, Leipzig and Berlin 1914), Vortrag P. Debye. 189 If U is the energy, and S the entropy of the system, then the " free energy " is defined according to Helmholtz by the relation F = U - S . T. It then follows from note 115 that dF = d'A - S . dT where d'A is the work done from without. If we set in the usual way d'A = - pdV (p = pressure, V = volume) then dF=- pdV- SdT. From this we get immediately the equation (66) in the text NOTES AND REFERENCES 161 Similarly, fdF\ _ _ s and hence 160 P. Debye, loc. cit., note 158. 161 E. Oriineisen, Ann. d. Phys. 26, 211 (1908) ; 33, 65 (1910) ; 39, 285 (1912). 162 P. Debye, loc. cit., note 158. 1634. Eucken, Ann. d. Phys. 34, 185 (1911) ; Verhandl. d. deutsch. physikal. Ges. 13, 829 (1911). IMP. Drude, Ann. d. Phys. 1, 566 (1900). 163 #. Riecke, Wiedem. Ann. 66, 353, 545 (1898). 166 Of., for example, H. A. Lorentz, The Theory of Electrons (Teubner, Leipzig, and Berlin 1909). 167 Let g be the average velocity of the electrons along the free path I. Then the electron takes the time T = - to pass over this free path. During this time it is exposed to the electrical force E of the external field. Its increase in velocity due to this force is at the commencement of the free path = 0, at the end of it = ?=, where e and m are the fli charge and mass of the electron respectively. In the mean, therefore, the small additional velocity generated by the field is Ag = -^^- = |^-- The electrons stream unidirectionally with this velocity against the field. If N is the number of electrons per unit volume, then through unit area of the surface there streams per second a quantity of electricity This is, however, the "current density" I which is known to be connected with the field E by the relation I = ffE- The expression (67) for the conductivity the number of electrons per unit of volume, is small compared with N, say If we insert this value, then we get for the free path 200 5-4 . 10 17 (in electrostatic units) A = 63-57 k = 1-4 . 10- 16 T= 273 m = 0-9 . 10- 27 N* = 6-1 . 10 23 P = 8-9 e = 4-77 . 10- 10 . With these values we get I is of the order 5-7 . 10-". 164 THE QUANTUM THEORY Since the atomic distance is of the order of magnitude 2 . 10~ 8 , the electrons would therefore only suffer collision after passing many thou- sands of atoms. This is unacceptable, since the "radius of molecular action " of the atoms itself has dimensions which fall within the order of magnitude of about 10 ~ 8 . 171 JET. A. Lorentz, loc. cit., note 43. 172 J. J. Thomson, The Corpuscular Theory of Matter. 173 H. Kammerlingh-Onnes, Leiden Communicat. 1913, 133. 171 C. H. Lees, Phil. Trans. (A) 208, 381-443 (1908). 175 W. Meissner, Ann. d. Phys. 47, 1001 (1915). 176 W. Nernst, Berl. Ber. 1911, p. 310. 177 H. Kammerlingh-Onnes, Leiden Communicat. 119, 22 (1911). 178 F. A. Lindemann, Berl. Ber. 1911, p. 316. 179 W. Wien, Berl. Ber. 1913, p. 184. Of. also Vorlesungen iiber neuere Probleme der theoretischen Physik. (Teubner, Leipzig and Berlin 1913.) 3. Vorlesung. 180 If s is the radius of atomic action, N the number of stationary atoms per unit of volume, then, according to a well-known result of the kinetic theory of gases, the mean free paths of the electrons Let us set where s is the radius of atomic action for T = 0, that is, when the atoms are at rest ; let a be the amplitude of atomic vibration. Now the mean energy E of this vibration (frequency i>), on the one hand, = (2irv)*a z (M is the atomic mass) ; on the other hand, it is, according to Planck- Einstein, From this it follows that Now, according to formula (67) of the text, the resistance w * 2m 2 r *r-I* If we here set for q the value,*/ (cf. note 170), and for N, according to J. J. Thomson's supposition, a^T, and for =- the value *Ni* = *N(a' l + 2as + si) NOTES AND REFERENCES 165 it follows that an expression, which contains only a and s as unknown constants. If we set then W assumes the form given in the formula (70). 181 F. A. Lindemann, Phil. Mag. 29, 127 (1915). 181a F. Haber, Berl. Akad. Ber. 1919, pp. 506 and 990. 182 J. -Stark, Jahrb. d. Radioakt. u. Elektronik 9, 188 (1912). 183 G. Borelius, Ann. d. Phys. 57, 278 (1918). 184 K. Herzfeld, Ann. d. Phys. 41, 27 (1913). 185 If we set %mq* = E, therefore q = \K, the first of the two for- mulae (72) follows from (67). If we further take into account that in Drude's 9 /7 77 1 Theory E = f kT, that is, that & - | ^ then from (68) the second for- mula (72) follows. 186 F. v. Hauer, Ann. d. Phys. 51, 189 (1916). 187 W. Nernst, Berl. Ber. 1911, p. 65. 1884. Eucken, Berl. Ber. 1912, p. 141. 189 K. Scheel and W. Heuse, Ann. d. Phys. 40, 473 (1913). Of. also L. Holborn, K. Scheel and F. Henning, Warmetabellen der physikal.- techn. Reichsanstalt (Vieweg 1919). 1904. Einstein aud O. Stem, Ann, d. Phys. 40, 551 (1918). 191 The quantum formulae (76) and (77) properly correspond to the Planck oscillator, that is, to a system of one degree of freedom, while here, in. the case of rotation, we have to do with two degrees of freedom. But the energy of the Planck oscillator is composed of two equal parts, a kinetic and a potential part, while in the case of rotation only kinetic energy comes into question. This is often expressed thus : the Planck oscillator possesses one potential and one kinetic degree of freedom, while the rotating molecule possesses two kinetic degrees of freedom. 192 P. Ehrenfest, Verhandl. d. deutsch. physikal. Ges. 15, 451 (1918). 193 According to note 48, the quantum canonical distribution function is 166 THE QUANTUM THEORY and the mean energy is ~kT If we here set all p n 's = 1, and if for E n we substitute the value E (n from (80), there follows for the mean rotational energy of a molecule o and for the heat of rotation of hydrogen we get the expression 19* The turning impulse (moment of momentum) of a system, the mass-points of which possess the mass mi, the velocities vi, and the dis- tances n from a fixed point (say the origin of co-ordinates), is a vector of the value In the present case, the system consists only of the two atoms (mass M) , which rotate around a circle of radius r with the constant velocity v = r 2irv. Hence here |U| =p = 2Mr a . 2 = J. 2w, where J = 2Mr* is the moment of inertia. 198 The impulse (or momentum) pi corresponding to a generalised co- (3L ordinate qi is, according to note 48, defined by the relation pi = ^ . .' where qi = -jr, and L is the kinetic energy of the system. Now here the angle of rotation $ is chosen as a generalised co-ordinate. But the kinetic energy of a body rotating about a fixed axis is known to be = J ' (moment of inertia) x (angular velocity) 2 , hence J/d\*_J- L= 2\-Tt) ~2*- Hence ])Q = ov = J

s ^ uce > following a proposal of H. A. Lorentz, he set the rota- tional energy E^ equal to nhv n , in contrast to Ehrenfest's formulation (78), which rests on a sounder basis. 209 S. P. Langley, Annals of the Astrophysical Observatory of the Smithsonian Institution, Vol. I, p. 127, Plate XX (1900). 210 F. Paschen, Wiedem. Ann. 51, 1 5 52, 209 ; 53, 335 (1894). 211 H. Rubens, Berl. Ber. 1913, p. 513. 212 H Rubens and E. Aschkinass, Wiedem. Ann. 64, 584 (1898). 213 If. Rubens and G. Hettner, Berl. Ber. 1916, p. 167. See also G. Hettner, Ann. d. Phys. 55, 476 (1918). 214 W. Burmeister, Ber. d. deutsch. physikal. Ges. 15, 589 (1913). 215 Eva v. Bahr, Ber. d. deutsch. physikal. Ges. 15, 710, 731, 1150 (1 M6 3 <3f . Lord RayUigh, Phil. Mag. 34, 410 (1892). Let an HC1 mole- cule, for example, be considered, which consists of a positively charged 168 THE QUANTUM THEORY hydrogen atom H+ and a negatively chlorine atom Cl~ (see Fig. 13). Let its centre of gravity be S, and let a be the distance of the H+ atom from S. Let the line joining the two atoms be the axis of x', and let this axis turn in the positive direction about S at the rate of v r revolutions per second with respect to the fixed x-7/-system. If, now, the two atoms vibrate relatively to one another with the frequency r fl and the amplitude A, then the x' co-ordinate of the H+ atoms may be represented thus x' = a + A sin (2irv t). If we project this vibration upon the fixed co-ordinate system, it follows that (x = x' cos (2*vJ) = o cos (2irv r t) + A sin (2trv t) cos (2*V) \y = y' sin (2irv r t) = a sin (torvj) + A sin (2*V) sin (2W) FIG. 13. for which we may also write fa = a cos (2y r O + 4 sin 2(r + , r )t + 4 sin 2*( v , - , r )t |T/ = a sin (2w^) - ^cos 2 1 r(^ + , r )i + ^ cos 2(ir - Vf )t . From the point of view of the system at rest we have thus three oscillations : (a) the left-circular oscillation x = a cos (2^)|^ h ^ {requenc T/ = oin(2x^)J (6) the left-circular oscillation with the frequency v u + v r NOTES AND REFERENCES (c) the right-circular oscillation I with the frequency v n - 1 169 ~ sin 2 217 E. S. Imes, Astroph. Journ. 50, 251 (1919). 218 A. Eucken, Ber. d. deutsch. phys. Ges. 15, 1159 (1913). Eucken has here, on account of the asymmetrical form of the hydrogen molecule, assumed two different moments of inertia the angular velocity (frequency of rotation) of the electron in the circular orbit, then the condition for equilibrium between the attraction of the nucleus and the centrifugal force is e _ = maw 2 or ma s w 2 = eE = e*z. According to Bohr's second hypothesis the moment of momentum p( = mva=ma? we get for the discrete radii of the permissible quantum orbits and the corresponding frequencies of rotation ^BvWfm The energy (kinetic + potential) is (aT7 t \ 2tf -T) -*"*-*- therefore the discrete quantum values of the energy are * Wn? ' If, in this expression, we set we recognise, that W is a function of , and hence of v = -. The energy of the electron in the Rutherford model therefore depends, as stated in the text, on its frequency of rotation v. If the electron passes from the ntb to the sth quantum path, then, ac- cording to Bohr's third hypothesis, a homogeneous spectral line is emitted of frequency W n - W s 2irVw.2 2 /l 1 172 THE QUANTUM THEORY where N = Wm 2*8 Cf . note 237. 2*9 It is of historical interest to note that, before Bohr, A. E. Hems in 1910 (Sitzungsber. d. Wiener Akad. 10 March, 1910) succeeded in repre- senting Rydberg's number in terms of the universal constants e, h, m ; his result differed from that of Bohr only by a factor 8. He deduced his result as follows. Starting from /. J. Thomson's atomic model, which was generally accepted at that time, he calculated the maximum oscilla- tion-frequency (no. of revolutions) /max of the electron in the simplest atom (hydrogen atom) for the case when this atom, provided with one energy-quantum, was circling just on the surface of the positive sphere. He obtained 4Vw "max = p This maximum frequency was next identified by Haas with the series limit (n = oo ) in Balmer's formula Then it follows that which is a value 8 times greater than JVjjohr- Haas used this relation to calculate from the three quantities, the Bydberg number N, Planck's constant h, and the ratio ~ , all of which he assumed known, the charge m e of the electron. In consequence of the factor 8 he obtained the value e = 3'18 . 10~ 10 , a value that is too small according to our present know- ledge, but which agreed well with the measurements of J. J. Thomson and H. A. Wilson, which were available at that time. 230 Th. Lyman, Phil Mag. 29, 284 (1915). 251 F. Paschen, Ann. d. Phys. 27, 565 (1908). 252.4. Fowler, Month. Not. Roy. Astron. Soc. 73, Dec. 1912. 233 F. Paschen, Ann. d. Phys. 27, 565 (1908). 23* E. C. Pickering, Astroph. Journ. 4, 369 <1896) ; 5, 92 (1897). 238 E. J. Evans, Nature, 93, 241 (1914). 236 W. Kossel, Ann. d. Phys. 49, 229 (1916) ; Die Naturwissenschaften 7, 339, 360 (1919). 237 L. Vegard, Verhandl. d. deutsch. physikal. Ges. 19, 344 (1917). 238 A. Sommerfeld, Atombau und Spektrallinien. (An English edition translated from the 3rd German edition (1922) is being prepared by Messrs. Methuen & Co., Ltd.) 259 R. Ladenburg, Die Naturwissenschaften 8, 5 (1920). 280 .4. Sommerfeld, Ann. d. Phys. 51, 1 (1916). NOTES AND REFERENCES 173 281 Expressed in terms of polar co-ordinates the kinetic energy L as- sumes the well-known form : L = (^ + rV). In it, m denotes the mass of the electron, the dots represent differentia- tion with respect to the time. The impulses p r and p^ are then defined as follows (see note 48) : p r =?>L = mr ; p. = !%L = mr^. 3r' ^ 262 Only when each impulse _p f depends solely on the corresponding 2j (or when it is a constant), and when, in addition, the limits of the phase-integral are independent of the g/s, does the phase-integral work out to a constant. This is by no means the case for any arbitrary choice of the co-ordinate-system. 263 P. S. Epstein, Ann. d. Phys. 50, 489 ; 51, 168 (1916). 264 K. Schwarzschdd, Sitzungsber. d. Berl. Akad. d. Wiss. 4. Map 1916. * 263 A. Einstein, Verhandl. d. deutsch. physikal. Ges. 19, 82 (1917). 266 M. Planck, Verhandl. d. deutsch. physikal. Ges. 17, 407, 438 (1915) ; Ann. d. Phys. 50, 385 (1916). 267 The semi-major axis of the ellipse, which is characterised by the values n and n', here has the value The ratio of the axis is b _ n a~ n + n'' We see that n' = corresponds to the case of Bohr's circular orbits. 268 The energy of the electron moving in the Kepler ellipse (n, n') here has the value = _ 2TrVs a w _ _ Nhz* h\n + n'Y ~ (n + n') a ' The series formula (102) of the text then follows from Bohr's Law of Frequency 269 If account is taken of the influence of relativity, the series formula for the spectra of the hydrogen type become to a first approximation V = VQ + V l where 174 THE QUANTUM THEORY In these expressions the symbols N and a have the following meaning : N= 2 f e * m o a = ^f; a 2 is of the order 5-3.10-5 m is the mass of the electron at vanishingly small velocities. Hence whereas the first term i/ gives the old formula, which was obtained by neglecting the influence of relativity, the small additional term v l represents the influence of relativity. As we observe, v l does not only depend on the quantum sums s + s' and n + n', but also on the individual values s, s', n, n'. This member, v v is thus responsible for the fine-structure. 270 If we apply the formula of the preceding note to H a , we have to set z = 1, s + s' = 2, n + n' = 3. We then get l s' 1 n'~\ + r i + * 24 84 J n AVjj j 1 [- -i 1 r 1 1 t? *> * i i n r c I 1 i 1 "i * Pia. 15. Corresponding to the possibilities of partition and 2 + 01 circle 1 + I/ ellipse 2 ^^ orbits n + ri = 3 = 3 + 0^ circle \ = 2 + 1 L ellipse L 3 initial orbits = 1 + 2] ellipse ) (for dynamical reasons the azimuthal quantum number n cannot under normal conditions assume a zero value), we should expect 2.3 = 6 possibilities of production and hence 6 components of the fine-structure of Ha.. One of these components, however, namely, the one correspond- ing to the transition of the electron from the circle (n = 3, n' = 0) to the ellipse (s = 1, s' = 1) does not present itself under normal conditions, as follows from the " Principle of Selection " enunciated by Rubinowicz NOTES AND REFERENCES 175 and Sommerfeld (see Chapter VI, 9). Hence 5 components of the fine-structure remain ; their position is exhibited in Fig. 15. As we see, the 5 components arrange themselves into two main groups, containing 3 and 2 members, respectively. The ' missing " line Ila is dotted in. The distance AVH between I a and II a , Ib and lib, Ic and II C is called the " theoretical hydrogen doublet." According to the above formula the frequency-number of the line la (3, 0-2, 0) is The frequency-number of the line Ha (3, >1, 1) is Thus A "H = */J,, - "/a = ^=1-095.1010 corresponding to AA.H = 0'157A. The hydrogen-doublet actually observed is measured from about the middle of I a and It, to the middle of lib and II C , owing to the absence of IIa. This leads to the value 0'8AA. H , that is, to 0-126A. According to a principle of correspondence enunciated by Bohr (see Chapter VI, 9), as a result of which the azimuthal quantum number can only vary by + 1, the components Ib and II C are also absent. 271 F. Paschen, Ann. d. Phys. 50, 901 (1916). 272 From formula (97) of the text we get for the two Rydberg constants for hydrogen and helium : Moreover, according to note 269, we get the third formula giving the value of the constant for the fine-structure : From the first two relations, by using MHO = &Ma, we get m a , __M^ ~ 176 THE QUANTUM THEORY and hence _L = JL N H-* N H* m c MHC ' Nne - Na ' The two Rydberg numbers NH and NHC have been measured by Pastfien with great accuracy : Nil = (109677-691 + 0'06) . c Nue = (109722-144 0'04) . c. Moreover, = F is the electrochemical equivalent (Faraday'* num- MH . c ber), that is, the charge which, in electrolysis, accompanies one gramme- atom (i.e. N = - atoms). This number has the value F = 9649-4 electromagnetic units. If we insert the three values of N a , NH* and * - in the relation above MH.C deduced, we get JL = 1-7686 . 10 7 electromagnetic units, m c a value which agrees very well with those values of this quantity which were obtained by direct methods (deflection of the cathode- and /5-rays in the electric and magnetic field). Let us now write or, using the value of - given above, MH ^m^ = 3 h 4 The right-hand side of this equation is known. If we combine with it m c O /2 = = 7-290. 10- 3 which follows from Paschen's measurements of the fine-structure in the case of helium, we have three equations in three unknowns e, m g , h. From them we get e = (4-766 0-088). 10 - 10 h = (6-526 + 0-200) . 10 - 27 . According to Sommerfeld it is more advantageous to use Millikan's value for e. We then get fe =(4-774 0-004). 10 - 10 { h = (6-545 0-009) . 1Q- 87 U = (7-295 0-005) . 10-. NOTES AND REFERENCES 177 273 K. Glitscher, Ann. d. Phys. 52, 608 (1917). 274 A. Lande, Phys. Zeitschr. 20, 228 (1919) ; 21, 114 (1920). 278 Cf. A. Sommerfeld, Atombau und Spektrallinien. Ch. IV, 6. 276 P. S. Epstein, Ann. d. Phys. 50, 489 (1916). 277 P. Debye, Gottinger Nachr. 3 June, 1916. 278/1. Sommerfeld, Phys. Zeitschr. 17, 491 (1916). Cf. also Atombau und Spektrallinien. Ch. VI, 5. 279 F. Paschen and E. Back, Ann. d. Phys. 39, 897 (1912) ; 40, 960 (1913). 2804. Rubinowicz, Phys. Zeitschr. 19, 441, 465 (1918). 281 N. Bohr, On the Quantum Theory of Line-spectra. Parts I and II. D. Kgl. Danske Vidensk. Seisk. Skrifter, Naturvidensk. og Mathem. Afd. 8, Baekke IV, 1. Kopenhagen 1918. 282 The number of revolutions of the electron per second in the sth quantum circle of Bohr is, in the case of hydrogen, according to note 247 : On the other hand, it follows from formula (93) of the text, if we take s considerably greater than 1 (high quantum numbers), and n = s + 1 (transition between neighbouring circles), that 283 P. S. Epstein, Ann. d. Phys. 58, 553 (1919). 284 H. A. Kramers, Intensities of Spectral Lines. D. Kgl. Danske Vidensk. Selsk. Skrifter, Naturvidensk. og Mathem. Afd. 8, Raekke III, 3. Kopenhagen 1919. 283.4. Sommerfeld and W. Kossel, Ber. d. deutsch. physikal. Ges. 21, 240 (1919). 286 J. Franck and G. Hertz, Phys. Zeitschr. 20, 132 (1919) ; in which references are also given. Cf . also J. Franck and P. Knipping, Phys. Zeitschr. 20, 481 (1919) ; J. Franck, P. Knipping and Thea KrOger, Ber. d. deutsch. physikal. Ges. 21, 728 (1919). 287 J. Tate and Foote, Phil. Mag. July, 1918. 288 References are given in the report by J. Franck and G. Hertz, men- tioned in note 286. 289 A. Einstein, Phys. Zeitschr. 18, 121 (1917). Let us consider the two quantum states (1) and (2) of the atom, with the energies E l and E z (.E 2 > -EJ). The number of transitions 2 - 1 which take place in the time dt owing to radiation is then, according to Einstein, NiA 2l dt, in which N 9 is the number of atoms in the state 2, and, therefore, accord- ing to note 48 ~ &. N,, = Nw, N being the total number of atoms. 4 al is a factor of proportionality. 12 178 THE QUANTUM THEORY The introduction of external monochromatic radiation of frequency / and intensity K,, firstly brings about positive absorption, that is, transi- tions l->2. The number of these in the time dt is, according to Einstein, JVj.B 12 K,,, in which .B 12 is a factor of proportionality, N l is the number of atoms in the state 1, and hence 5 jY, = NCpje M. Secondly, the external radiation also effects transitions 2 - 1 (nega- tive absorption). The number of these that occur in the time dt = -ZV-j-B^Kj/, where jB 2] is a factor of proportionality. When the energy exchange is in equilibrium the number of transitions 2 > 1 must be equal to the number of transitions 1 > 2, hence ~ M' i.e. When the temperature increases indefinitely, K* must also increase to infinitely great values ; from this it follows that = 1. Finally, if we set -^p- = A for shortness, we get the relation given in the text: ^ A 290 Of. the resume* by E. Wagner, Phys. Zeitgchr. 18, 405, 432, 461, 488 (1917). 291 O. Moseley, Phil. Mag. 26, 1024 (1913) ; 27, 703 (1914). 292 W. Kossel, Verhandl. d. deutsch. physikal. Ges. 16, 898, 953 (1914) ; 18, 339 (1916). 293 A. Sommerfeld, Ann. d. Phys. 51, 125 (1916) ; Phys. Zeitschr. 19, 297 (1918). Of. also Atombau und Spektrallinien. Oh. Ill, Ch. IV 4, Ch. V 5. 29* L. Vegard, Verhandl. d. deutsch. physikal. Ges. 1917, pp. 328, 344 ; Phys. Zeitschr. 20, 97, 121 (1919). 298 P. Debye, Phys. Zeitschr. 18, 276 (1917). 296 J. Kro6, Phys. Zeitschr. 19, 307 (1918). 297 A. Smekal, Wiener Ber. Ila 127, 1229 (1918) ; 128, 639 (1919) ; Verhandl. d. deutsch. physikal. Ges. 21, 149 (1919). Of. also A. Smekal and F. Reiche, Ann. d. Phys. 57, 124 (1918). NOTES AND REFERENCES 179 298 W. Kossel, Zeitschr. f. Physik 1, 119 (1920). 299 Since the L-ring consists of several electrons, we take the expression " elliptic motion " to mean the following type of motion : each electron independently describes an elliptic path about the nucleus, whereby the electrons are at each moment situated at the corners of a regular polygon which shares in the motion of the electrons, alternately contracting and expanding during this motion (" elliptical associates"), cf. Sommerfeld. Atombau and Spektrallinien. 3004. Stnekal, Wiener Ber. Ha 128, 639 (1919). 301 M. Born and A. Lande, Berl. Akad. Ber. 1918, p. 1048 ; Verhandl, d. deutsch. physikal. Ges. 20, 202, 210 (1918) ; M. Born, ibid., 20, 230 (1918) ; Ann. d. Phys. 61, 87 (1920). 302 A. Lande, Verhandl. d. deutsch. physikal. Ges. 21, 2, 644, 653 (1919) ; Zeitschr. f. Phys. 2, 83 (1920). Cf. also A. Lande and E. Madelung, Zeitschr. f. Phys. 2, 230 (1920). 303 W. Kossel, Ann. d. Phys. 49, 229 (1916). 3M P. Debye, Munch. Akad. Ber. 9 Jan. 1915. 305 P.. Scherrer, Die Botationsdispersion des Wasserstoffs. Dissertation, Gottingen, 1916. 306 G. Laski, Phys. Zeitschr. 20, 269, 550 (1919). 307 Cf., for example, A. Sommerfeld, Atombau und Spektrallinien, Ch. IV, 6. 308 Langmuir, Journ. Amer. Chem. Soc. 34, 860 (1912) ; Zeitsohr. f. Electrochemie 23, 217 (1917). 309 Isnardi, Zeitschr. f. Elektrochemie 21, 405 (1915). 310 /. Franck, P. Knipping and Thea KriLger, Ber. d. deutsch. physikal. Ges. 21, 728 (1919). 310a Planck has made an attempt to alter Bohr's model in such a way that the right heat of dissociation results. See M. Planck, Berl. Akad. Ber. 1919, p. 914. Cf. also H. Kallmann, Dissertation, Berlin 1920. 311 W. Lenz, Ber. d. deutsch. physikal. Ges. 21, 632 (1919). 3124. Sommerfeld, Ann. d. Phys. 53, 497 (1917). 313 F. Pauer, Ann. d. Phys. 56, 261 (1918). 314 G. Laski, see note 314. 315 M. Pier, Zeitschr. f. Elektrochemie 16, 897 (1910). 316 K. Schwarzschild, Berl. Akad. Ber. 1916, p. 548. 317 H. Deslandres, Compt. Rend. 138, 317 (1904). 318 T. Heurlinger, Phys. Zeitschr. 20, 188 (1919) ; Zeitschr. f. Physik 1, 82 (1920). 319 W. Lenz, see note 311. 320 A different view is upheld by /. Burgers (Versl. K. Ak. van Wet. Amsterdam 26, 115, 1917), in which, also, jumping electrons produce the middle line in the infra-red of band. In contrast with Schwartschild and Lenz, Burger assumes that the motion of the electrons is influenced by the rotation of the molecule. The energy of the system is then not com- posed additively of the energy of the electrons and the rotational energy of the molecule, but a third term has to be added, which is due to the Coriolis force of the rotating system. 180 THE QUANTUM THEORY 321 T. Heurlinger, see note 318. 322 F. Reiche, Zeitschr. f. Physik 1, Heft 4, 283 (1920). 323 E. S. Imes, Astrophys. Journ. 50, 251 (1919). 324 A. Kratzer, Dissertation, Munchen 1920. 323 In the case of the gases investigated by Imes, namely HC1, HBr, and HP, the following moments of inertia were found : /HOI = 2 ' 6 * 10- 40 ; J H Br = 3-27 . 10 - 40 ; J HF = 1'37 . 10- 40 . INDEX Absorption band, edge of, 21, 112. continuous, 78. Acoustic vibrations, 157. Action, quantum of, 9, 27. Atomic heat, 29. numbers, 86, 110. Avogadro's constant, 12, 29, 163. Law, 12. Azimuthal quantum number, 92. B Back, 22. Balmer, 90. Barkla, 20, 109. Beck, 25. Benedicks, 146. Bergen, 106. Bergmann series, 89. Bishop, 106. Bjerrum, 75, 123. Bohr, 75. Bohr's model of the atom, 86, 119. principle of analogy or corre- spondence, 100. selection, 98. Boltzmann, 4, 13. Born, 38, 42. Bragg, W. H. and W. L., 109. Bravais, 37. Bremsstrahlung, 22, 109. Broek, van den, 86. Broglie, de, 21. Bunsen, 109. Burgers, 179. Canal rays, 142. Cavity radiation, law of, 3. Characteristic Bontgen radiation, 109. Chemical constant, 81. Compressibility, 33. Conductivity, thermal, 60. Coulomb, 12. Davis, 106. Debye, 38, 39, 58. Debye's formula, 40. Degeneration of gases, 79. Deslandres, 122. Dessauer, 22. Diamagnetism, 25. Diamond, 33. Dispersion, 47. Displacement Law, Wien's, 4. Distribution numbers, 114. of velocities, Maxwell's Law of, 5. Dix&n, 100. Doppler, 4. Drude, 61, 84, 17. Duane, 22. Dulong and Petit's Law, 29, 30. Ehrenfest, 71. Einstein, 1, 15, 16, 30, 107. functions, 31, 32, 51. Einstein's hypothesis of light- quanta, 16, 17, 107. Law of the photo-electric effect, 20. Elastic collisions, 104. spectrum, 42. Electron theory of metals, 61. Emissivity, 2. Entropy, 8, 151. Epstein, 75, 98. Equipartition, 13. Eucken, 56, 61, 69. 181 182 THE QUANTUM THEORY P Fine-structure, Sommerf eld's theory of, 94. of the Rontgen rays, 113. Fokker, 15. Foote, 105. Franck, 22, 102. Friman, 109. Oeiger, 85. Gibbs, 13, 26. Glitscher, 97. Ooiicher, 106. Gramme-molecule, 145. Grilneisen, 58. Haas de, 25. Haber, 46, 66, 155. Hamilton, 15. Hamilton's principle, 27. Hauer, von, 67. Heat theorem, Nernsfs, 35. Helium, 97. Helmholtz, 160. .ZTerte, 22, 102. Herzfeld, 66. Heurlinger, 122. Iftmse, 69. floo&e's Law, 59. flop/, 15, 56. Hughes, 106. fltiK, 22. Hunt, 22. Hydrogen type of series, 88. Imes, 123. Impulse or momentum, 26. radiation, 22, 109. Infra-red dispersion frequencies, 4i Intensity, 2. Isnardi, 120. Jeans, 15. K Kammerlingh-Onnes, 64. Karmdn, 38. Kepler ellipses, 95. Kinetic radiation, 23. Kirchhoff, 3. .KbsseZ, 91, 102, 105. Kramers, 102. Kratzer, 124. K-rings, 111. roo, 111, 115. K-series, 21. Kurlbaum, 9. Ladenburg, 91. Langmuir, 120. Losfci, 119. Lattice theory of atomic heats, 42. Lowe, 35 Law, Avogadro's, 12. equipartition, 13. of cavity radiation, 3. of radiation, Planck's, 10. --- Bayleigh's, 10, 14. --- Wieris, 5, 7. Stefan and Boltzmann, 5. L-doublets, 114. Lechner, 56. Lees, 64. Lenard, 19. Lews, 120, 122. Light-quanta, Einstein's, 16. Lindemann, 33, 64, 66. Lorentz, 15, 24, 61, 161. Lorenz, 62. L-rings, 111. Lummer, 2, 3, 4. M , 33. Marsden, 85. Maxwell, 5, 13. Mean atomic volume, 53. Meissner, 64. Metals, electron theory of, 61. Michelson-Morley, 1. Millikan, 12, 20. Molecular heat, 53. Momentum, 26, 166. moment of, 72, 166. Moseley, 110. INDEX 188 N Nernst, 9. and Lindemann's formula, 37. Nernst's heat theorem, 35. Nichols, 33. Orbits, allowable, 87. Ortvay, 152. Paramagnetism, 25. Parhelium, 97. Paschen, 6, 9, 89. Pauer, 120. Phase-integral, 92. Phase-space, 26, 92, 94, 136. Photo -electrons, 20, 141. Pier, 121. Planck, 6, 15, 24, 27, 91, 117. Poisson's ratio, 41. Polarisation, 168. Precession, 167. Pringsheim, 2, 4. Probability, thermodynamic, 8. Q Quanta, energy-, 7. Quantum of action, 9. Radial quantum number, 92. Ratnowski, 58. Bayleigh, 10. Reflection, metallic, 45. Relativity, 95, 96. Residual rays, 35, 149. Resonance lines, 104. potentials, 105. Resonators, Planck's, 6. Bice, 22. Richards, 58. Richarz, 25. Riecke, 61. Rontgen radiation, 21. Rotation spectra, 76. Rotszayn, 75. Rubens, 9, 33. Rubinowicz' principle of selection, 98, 99, 174. Rutherford, 84, 171. Rydberg, 90, 110. Sackur, 80. Sadler, 21. Scheel, 69. Scherrer, 79, 119. Schrodinger, 57. Schwarzschild, 93, 121. Selection, principle of, 98. Sommerfeld, 27, 91, 94, 102, 120. Stark, 23, 66. effect, 98. Stefan, 4. Stefan-Boltzmann, Law, 5. Stern, 25, 70, 81. Stokes, 19. Sutherland, 33. Thirring, 55. Thomson, J. J., 19, 63, 84. atom, 169. Vegard, 91. 117. w Wagner, 21, 22. Warburg, 9, 23. Weber, 30. Webster, 22. Weyssenhof, von, 74. Wien, 3, 4. Wien's displacement Law, 4. Law of radiation, 5, 7. WW/, 9. Zeeman, 84. effect, 98, 120. PRINTED IN GBBAT BRITAIN AT THE UNIVERSITY PRESS, ABERDEEN This book is DUE on the last date stamped below 301945 MAR 19 1946 aim MAR 2 7 1953 JUL 27 !95b Form L-9-35m-8,'28 ,2s* UC SOUTHERN REGIONAL LIBRARY FACIL t,