^^^^m,M 
 
CORNELL 
 
 UNIVERSITY 
 
 LIBRARY 
 
 GIFT OF 
 
 Bimund L. Murdnck "55 
 Franklin ¥. Murdock '52 
 
Cornell University Library 
 QC 475.J43 
 
 Report on radiation and tiie quantum-tlieo 
 
 3 1924 012 330 407 
 
Cornell University 
 Library 
 
 The original of tliis bool< is in 
 tine Cornell University Library. 
 
 There are no known copyright restrictions in 
 the United States on the use of the text. 
 
 http://www.archive.org/details/cu31924012330407 
 
THE 
 
 PHYSICAL SOCIETY 
 
 OF 
 
 LONDON. 
 
 REPOKT ON RADIATION 
 
 AND THE 
 
 QUANTUM-THEORY. 
 
 BT 
 
 J. H. JEANS, M.A., F.R.S. 
 
 Price to Non-Fellows, 6s net, post free 6s. 3cl. 
 Bound in cloth, 8s. 6cl., post free 8s. Sdi 
 
 LONDON: 
 
 "THE ELECTRICIAN" PRINTING & PUBLISHING 00., LTD., 
 
 1, 2 AND 3, Salisbury Court, Fleet Stbbbi. 
 
 1914. 
 
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CONTENTS. 
 
 Chapter Pagb 
 
 L — Introductory : On the Need foe a Quantqm- 
 
 Theoey ', ] 
 
 II. — General Discussion op the Radiation Prob- 
 lem, ACCOEDING to THE CLASSICAL MECHANICS fi 
 
 III. — The Development of the Quantum-Theory. ... 30 
 
 IV.^The Line Speotea of the Elements 43 
 
 V. — The Photo-electeic Effect , 58 
 
 VI. — ^The Specific Heat of Solids 66 
 
 VII. — On the Physical Basis for the Quantum- 
 Theory 79 
 
IV. 
 
 PRINCIPAL EEFERENCES OTHER THAN THE DE- 
 TAILED PAPERS REFERRED TO IN THE TEXT. 
 
 " La Theorie du Rayonnement et les Quanta." (Gauthier- 
 Villars, Paris, 1912.) Reports presented to the first Solway 
 Congress of Physics at Brussels (October, 1911), with the dis- 
 cussion thereon. 
 
 [Reports presented by Lorentz, Jeans, Warburg, Rubens, 
 Planck, Knudsen, Perrin, Nernst, Kamerlingh-Onnes, Sommer- 
 feld, Langevin and Einstein, nearly all dealing with the 
 Quantum-Theory. Comments and discussion by the foregoing, 
 and also by Poincare, Lord Rayleigh, Wien, Rutherford, 
 Brillouin, Mme. Curie, Hasenohrl, De Broghe and Lindemann.] 
 
 " Vortrage ilber die Kinetische Theorie der Materie und der 
 Elcktrizitat " ; Mathematische Vorlesungen an der Universitat 
 Gottingon. (Teubner, Leipzig, 1914.) 
 
 [Reports by Planck, Debye, Nernst, v. Smoluchowski, 
 Sommci'fcid, Lorentz and others, largely concerned with the 
 developments of the Quantum-Theory.] 
 
 " Report of the British Association (1913) at Birmingham." 
 Section A. — Discussion on Radiation ; pp. 376-386. 
 
 [Discussion by Lorentz, Pringsheim, Love and Larmor. 
 Opening by Jeans.] 
 
 Max Planck : " Acht Vorlesungen liber Theoretische Physik." 
 (S. Hirzel, Leipzig, 1910.) 
 
 [The fifth and sixth lectures deal with Planck's form of the 
 Quantum-Theory.] 
 
 Jlenri Poincare: " Dernieres pensees." (Flammarion, 
 Paris, 1913.) [An essay, entitled " L'hypothese des Quanta," 
 gives in a non-technical form Poincare's final judgment on the 
 Quantum-Theory.] 
 
RADIiTION AND THE QUANTUM-THEORY. 
 
 CHAPTER I. 
 
 ON THE NEED FOR A QUANTUM-THEORY. 
 
 Introductory. 
 
 1. The quantum-theory, whicli is now found to be applic- 
 able in many departments of physical science, had its origin 
 in an attempt to account for the spectrum of black-body 
 radiation. It is therefore appropriate, as well as convenient, 
 that the present report should approach the theory through 
 the radiation problem. The arrangement adopted will be as 
 follows : We shall first explain how the quantum-theory is 
 demanded, and indeed made inevitable, by the known facts of 
 black-body radiation (Chapters I. and II.) ; we shall next 
 discuss the radiation problem with the help of the quantum- 
 theory (Chapter III.), and shall then proceed to consider the 
 bearing of the quantum-theory on certain other problems of 
 physics, namely, the line spectra of the elements (Chapter IV.), 
 the photo-electric effect (Chapter V.), and the specific heats of 
 soHds (Chapter VI.) A final chapter is devoted to discussing 
 whether a physical basis can be found for the conceptions of 
 the theory. 
 
 2. The quantum-theory, it will be understood from the 
 outset, represents a complete departure from the old Newtonian 
 system of mechanics. Until the end of the last century, the 
 Newtonian mechanics had shown a capacity of explaining and 
 interpreting practically all the phenomena to which it had been 
 applied. It was natural that physical science should attempt 
 to progress by explaining more and more phenomena in terms 
 of the Newtonian mechanics. It was hoped that whatever 
 phenomena still defied explanation would, with the acquisition 
 of more intimate knowledge, be ultimately found to be simply 
 further illustrations of the truth of the Newtonian system of 
 laws. 
 
 Gradually certain phenomena have emerged from the general 
 
 B 
 
2 RADIATION AND THE QUANTUM-THEORY. 
 
 mass as resisting explanation in terms of these laws, and the 
 •eonviction has more and more developed that something out- 
 side the Newtonian laws is needed for their explanation. Of 
 course, the mere discovery that a phenomenon is difficult to 
 explain in the Newtonian way is no adequate reason for 
 abandoning a system of laws which is known to hold through- 
 out vast regions of natural phenomena ; what we have to 
 discuss is whether there are not certain phenomena which are 
 in flat contradiction to the Newtonian laws. From demon- 
 strating that a matter is difficult to proving that it is impossible 
 is a long step, but if this step can be taken with respect to the 
 explanation of even one well-estabhshed phenomenon of Nature, 
 then the logical necessity of rejecting the impossibility becomes 
 vmanswerable. 
 
 The phenomenon which is believed to provide the crucial 
 test as to the universal validity of the Newtonian mechanics 
 is the following : the total radiant energy per unit volume of ether 
 in temferature-equilihrium with matter is finite, and not infinite. 
 It is a matter merely of mathematical demonstration that this 
 fact is incompatible with Newtonian mechanics. Of course, 
 it cannot be denied that the phenomenon itself is in some ways 
 open to question. It may be argued that there are at least the 
 possibilities that radiation never can be in temperature-equiK- 
 brium with matter, or that ether as a seat of energy is a figment 
 of the imagination. These possibiUties must, of course, be 
 discussed. It may be argued that the mathematical proof is 
 open to question : the proof in its finished form was given by 
 Poincare, and the validity of his mathematical reasoning has 
 never been challenged, but this cannot carry absolute con- 
 viction. Like every other scientific judgment the matter has 
 to be one of balancing probabilities. But a search for the 
 considerations to be put in the other side of the balance reveals, 
 to many minds at least, a total deficiency of real logical argu- 
 ments. We, as physical machines, are built on a scale which 
 is large compared with the scale of light waves and electrons, 
 from which it has resulted that our first physical experiments, 
 as a race, have been concerned with matter also on a scale very 
 large in comparison with its ultimate structure. The New- 
 tonian laws have imdoubtedly been found adequate to explain 
 the whole series of what we may call large-scale phenomena, 
 but no adequate reasons have, so far, been given for asserting 
 that they must also be the laws which govern small-scale 
 phenomena. The fact seems to be that the old laws are not. 
 
THE NEED FOR A QUANTUM-THEORY. 3 
 
 SO to speak, fine-grained enough to supply the whole truth with 
 regard to small-scale phenomena. 
 
 In view of the fact that our preconceived ideas must neces- 
 sarily form the starting point in every train of thought, and 
 that these preconceived ideas are very intimately bound up 
 with the Newtonian mechanics, it seems advisable to begin the 
 present report with an explanation of the reasons which are 
 believed to compel the abandonment of this system. In the 
 present chapter these reasons are discussed in a purely physical 
 way, without mathematical analysis and therefore, of course, 
 without rigorous proof. In Chapters II. and III., which are 
 largely mathematical, an account is given of the analysis 
 which is needed to work out the physical ideas discussed in the 
 first chapter. The report is so arranged that the mathematical 
 parts of Chapters II. and III. may conveniently be omitted by 
 those readers whose interest centres mainly in physical ideas 
 rather than in abstract reasoning. The contents of the sub- 
 sequent chapters have been already explained and are sufficiently 
 indicated by their titles. 
 
 The Crucial Phenomenon. 
 
 3. As has already been stated the phenomenon which it is 
 convenient to consider first, as providing a crucial test of the 
 Newtonian mechanics, is the phenomenon that the total radiant 
 energy per unit volume of ether in temperature-equilibrium with 
 matter is finite, and not infinite. 
 
 Assuming for the moment the truth of this as an experi- 
 mental fact, we may try to understand its physical bearings. 
 To make the question as definite and as simple as possible, let 
 us fix our attention on an enclosure with perfectly reflecting 
 walls in which there is a mass of, say, iron at 0°C., and let us 
 suppose that there is a state of equilibrium inside this enclosure. 
 The iron is continually radiating energy out from its surface 
 into the surrounding ether inside the enclosure, and is also 
 absorbing energy from the ether. From the condition of 
 equilibrium, the rates of exchange must just balance. If we 
 assume, for additional simplicity, that the iron is coated with a 
 perfectly .ibsorbing paint, then in point of fact each square 
 centimetre of surface emits 3x 10^ ergs of radiation per second 
 into the ether, and also absorbs SxlO'^ ergs per second of 
 radiation falling on to it from the ether. The energy in the 
 ether is of density 4x10-^ ergs per cubic centimetre ; the heat 
 energy in the iron is of the order of 8x 10^ ergs per cubic centi- 
 
 B2 
 
4 RAPT.VTJO.V AND THE QUAN'TTJM-THEORY. 
 
 metre. The iicat energy of the iron resides in the oscillations 
 of its atoms, each atom moving with an average velocity of 
 about 30,000 cms. per second. 
 
 4. A very little consideration will show that this state of 
 things is different from what might be expected by analogy 
 from other systems which are known to obey the ordinary 
 dynamical laws. Consider, for instance, a tank of water (to 
 Teprescnt the ether) in which is floated a system of corks (to 
 ffcpresent atoms of matter) connected by light springs or 
 -elastics so that they can oscillate relatively to one another. 
 
 Suppose that initially the surface of the water is at rest. Let 
 'the system of corks be set into violent oscillation and placed on 
 the surface of the water. The motion of the corks will set up 
 waves in the water, and these waves will spread all over the 
 surface of the water, undergoing reflection when they meet the 
 walls of the tank. We know that ultimately the corks will be 
 reduced to rest ; the energy of their motion will be transformed 
 'first into the energy of waves and ripples on the surface of the 
 water, and then, owing to the viscosity of the water, into heat 
 energy in the water. A final state in which the corks continue 
 to oscillate with extreme vigour while the water has almost no 
 energy is unthinkable ; we expect a final state in which practic- 
 ally all the energy has found its way into the water. 
 
 5. Let us examine another analogy. Let the corks be 
 replaced by spherical lead shot, again connected by light 
 springs, and let the system be suspended in a closed chamber 
 ■containing air. After the system of shot has been started in 
 
 violent oscillation, let the chamber be closed up. The motion 
 -of the shot will set up waves in the air, and these will again be 
 dissipated by viscosity. A final steady state in which the 
 spheres continue to oscillate with high velocities for ever is 
 again unthinkable. In point of fact, we know that the final 
 state will be one in which the spheres are all at rest in their 
 positions of equihbrium, or rather, to be quite precise, in which 
 they oscillate with the infinitesimally small velocities of the 
 Brownian movements appropriate to particles of their size; 
 practically all the energy will have passed from the spheres 
 ■ into the surrounding air. 
 
 The Kinetic Theory of Gases enables us to trace out the 
 transformation of the energy which originally resided in the 
 oscillations of the spheres. In the final steady state we know 
 .that this energy will be the heat energy of the air. Let us 
 
THE NEED FOR A QUANTUM-THEORY. 5> 
 
 simplify tlie discussion by assuming that the air was originally 
 
 at or close to the absolute zero of temperature, and suppose 
 
 that in the steady state the temperature of the air is T. If 
 
 there are N molecules of air, their energy in the steady state 
 
 5 
 will be - NRT, and if there are n spheres, the energy of their 
 
 . 3 
 
 Brownian movements will be -nRT. where R is 'the gas con- 
 
 stant. The total energy in the steady state will be ( hN-I- -v JRT,,, 
 
 and T will be determined from the condition that this quancity 
 must be equal to the total energy of the original oscillations of 
 the spheres. Since N, the number of molecules, will be enor- 
 mously large compared with n, the number of spheres, it is 
 clear that practically all the energy will be contributed by the 
 
 5 
 term -^-NRT. By the time the steady state is reached the 
 
 energy is almost entirely transferred from the spheres to the gas.. 
 The molecules of the gas move with velocities determined 
 statistically by the well-known law of Maxwell. Suppose that 
 the position and velocity of every molecule were known at any- 
 instant, then, by an apphcation of Fourier analysis, it would be- 
 possible to regard this motion as made up of trains of waves — 
 i.e., any molecular motion, however complex, can be regarded 
 as coinciding at each instant with the motion produced by a 
 certain system of sound waves in the air, and the energy of the 
 molecular motion will be exactly the same as the energy of 
 these trains of waves (c/. Chapter VI. below). If the mole- 
 cular velocities at any instant obey Maxwell's law, it is found* 
 that the energy of these trains of waves is distributed between, 
 the different wave-lengths according to the law 
 
 ijiRTtm {!> 
 
 That is to say, if we could devise some system of resonators 
 which would pick up and sort out the molecular motions in the 
 way in which a spectroscope sorts out light, then the energy 
 per unit volume of the waves of wave-length intermediate 
 between A and A+c?A would be found to be that expressed 
 in formula (1). One reservation has to be made with respect 
 to this formula, namely, that it does not apply to waves of 
 wave-length as short as, or comparable with, the distance 
 
 *Cf. " Temperature-radiation and the Partition of Energy in Continuous 
 Media," Phil. Mag. 17, p. 229 (Feb., 1909). 
 
6 RADIATION AND THE QUANTUM-THEORY. 
 
 between adjacent molecules. But a good approximation can 
 be obtained by supposing that the formula holds accurately 
 from A=co to a limiting wave-length /lm,and that there are no 
 waves at all of wave-length shorter than l^. The total energy 
 is then 
 
 =Am 4 T!T 
 
 47rRT;rW.=- ~3, (2) 
 
 j: 
 
 and A„ can be fomid by equating this to the total energy 
 of the molecules. 
 
 Of the energy imder consideration, analysed by the integral 
 in equation (2), it will be seen that only one-eighth of the total 
 energy will be of wave-length greater than 2)^, while seven- 
 eights is of wave-length intermediate between this and X^. In 
 a chamber filled with ordinary air 2™ will be of the order of 
 10"' cms. Only one millionth part of the total energy will be 
 of wave length greater than 10"^ cms. Thus we may say that 
 the energy tends to run almost entirely into the shortest wave- 
 lengths which are possible in the medium. 
 
 6. It is now possible to explain (a statement of the proof 
 TDeing reserved for the next chapter) why it is that the Newtonian 
 mechanics are inadequate to account for the facts of radiation. 
 For whatever is regarded as certain or uncertain about the 
 ether, it must be granted as quite certain that it approaches 
 more closely to a continuous medium than to a gas. If it has 
 any grained structure at all the distance between adjacent 
 ^ains must be enormously less than the 10~'cms. assumed 
 for the corresponding distance in the gas. But even if this 
 distance were as great as 10"' cms., only one-milhonth of the 
 total radiant energy ought to be of wave-length as great as 
 10"^ cms. — entirely contrary to what is observed. And if, as 
 seems most probable, the ether is a perfectly grainless structure, 
 then the quantity l,„ wiU be zero ; the integral corresponding to 
 integral (2) must be evaluated from /=0 to A= oo , and the total 
 energy will be infinite. This is why the crucial fact we have 
 mentioned above is believed to decide against the Nt■^^'tonian 
 mechanics. 
 
 To put the matter shortly : in all baown media there is a 
 tendency for the energy of any systems moving in the medium 
 to be transferred to the medium and ultimately to be found, 
 when a steady state has been reached, in the shortest vibrations 
 of which the medium is capable. This tendency can be shown 
 (Chapter II.) to be a direct consequence of the Newtonian laws. 
 
THE NEED FOE A QUANTUM-THEORY. 7 
 
 This tendency is not observed in the crucial phenomenon of 
 radiation; the inference is that the radiation phenomenon is 
 determined by laws other than the Newtonian laws. 
 
 A brief sketch of the line of proof used to establish the 
 quantum-theory may appropriately be given here before the 
 proof itself is given. The first step is not essential to the 
 establishment of the quantum-theory, but is helpful as con- 
 firming the theory. It consists in examining the final state of 
 a medium in which the motion is governed by the Newtonian 
 laws. The tendency of the energy to rmi into the vibrations of 
 shortest wave-length is found to admit of rigorous proof. The 
 second step is the really essential one. It consists in working 
 back from the observed final partition of energy in the ether to 
 the laws by which this partition of energy must be produced. 
 It can be shown that, if the final partition of energy is that 
 given by the well-known law of Planck, then the motion of the 
 medium must be governed by laws which involve the quantum- 
 theory. Further, if Planck's law, instead of expressing the 
 exact truth, expresses only an approximation to the truth, then 
 the fundamental laws must be generally similar to those of the 
 quantum-theory; at least they must be based on discontinuities, 
 and not on the ideas of continuity involved in the classical 
 mechanics. 
 
CHAPTER IL 
 
 GENERAL DISCUSSION OF THE RADIATION PROB- 
 LEM, ACCORDING TO THE CLASSICAL MECHANICS. 
 
 7. The radiation which is in thermodynamical equihbrium 
 with matter at temperature T is spoken of as " full radiation " 
 or " black-body radiation " at temperature T. According to 
 Stefan's law its total intensity is proportional to T* ; according 
 to Wien's displacement law, its energy, distributed according 
 to wave-length, is of the form 
 
 F(AT)2-S(?A, (3) 
 
 where F is a function, undetermined so far as Wien's law goes, 
 of the product 2T. From Wien's law, Stefan's law can be 
 deduced at once on integrating from 2=0 to A= oo. 
 
 Both Stefan's law and Wien's law can be deduced from 
 general thermodjmamical considerations, but sueh considera- 
 tions do not sufE-ce to determine the actual form of the function 
 F in Wien's law. Both of these laws can be tested experi- 
 mentally, and they are found to be in agreement with observa- 
 tion. The form of the function F can be determined experi- 
 mentally, and the form which has been found to agree most 
 closely with experiment is that suggested by Planck in 1901 in 
 a paper* which has since provided the basis for the quantum- 
 theory. Planck's form, which is derived from considerations 
 similar to those on which the quantum-theory now rests,")" is. 
 
 where v is the frequency corresponding to wave-length X, and 
 h is the constant, now known as Planck's constant, of which 
 the value is approximately 
 
 6-6x 10-2' (ergg ^ sec). 
 Substituting this form for F(AT) into expression (3), the 
 
 * " Annalen der Physik," 4 (1901), p. 553. 
 t Cf. below, Chapter VII., §§ 62, 63. 
 
DISCUSSION OF THE RADIATION PROBLEM. & 
 
 partition of full black-body radiation at temperature T is 
 found to be 
 
 Sn^TX-m-^ (4) 
 
 where x stands for hv/RT. This formula is beheved to agree 
 extremely well with experiment.* 
 
 8. Any radiation formula corresponding to a steady state 
 must be derived by expressing that the amount of energy 
 gained by the ether is equal to the amount absorbed. If 
 tp(A,T)cZA is the law of partition of energy according to wave- 
 length, there must be an equation of the form 
 
 |P=E-A, 
 dt 
 
 where B stands for the rate of emission and A for the rate of 
 absorption ; and the steady state is determined by the con- 
 dition -Z=0, or E=A. 
 
 In general, it may be expected that several agencies will be 
 at work in the processes of emission and absorption. Planck, 
 in his theory already referred to, supposed that the emission 
 and absorption were accomphshed by " resonators " of 
 perfectly definite periods. These cannot of course account for 
 the whole of the emission and absorption ; let the parts due to 
 them be denoted by Ee, Ah- An agency which must contribute 
 something to the emission and absorption is to be found in 
 the motion of free electrons in the matter ; let us. denote their 
 contributions by Ee, A^. A further agency is probably the 
 photo-electric efiect ; when ah electron is discharged there is 
 an absorption of radiation ; when one recombines there will be 
 an emission of radiation. Let the contributions of the photo- 
 electric effect be denoted by Bp, Ap. Treating any other 
 agencies in the same way, we shall have an equation of the 
 form 
 
 |'=(ER-AB)+(EB-A„)+(Ep-Ap)-f .... 
 
 The absorptions Ajj, Ajj, Ap . . . may each be expected to 
 contain 9 as a factor ; if we double the radiant energy of any 
 wave-length we expect twice as much to be absorbed. Let 
 
 * For a discussion see " La Theorie du Rayonnement et les Quanta " 
 Gauthier-ViUars, 1912). 
 
10 KADIATION AND THE QUANTUM-THEORY. 
 
 US then put Ak=Br9, &c. In general, the emissions En, Ee, ■ • • 
 will not depend on 9, but let us make the matter as general as 
 possible by assuming that they each contain a term in 9. Let 
 us then put Eb=Fr+Ge9, &c., in which any or all of the G's 
 may vanish. We then have 
 
 ?2=(Fk+Gr9-Br9)+(Fe+Gb<P-Bb9)+(Fp+Gp9-Bp9)+- 
 
 ■and the equation giving the steady state is 
 
 Fr + Fb + Fp +.^ _ _ ^5^ 
 
 ^ (BR-GR)+(BE-GE)+(Bp-Gp)-t- . . . 
 
 It is to be expected that the ratios of the different terms, 
 both in the numerator and denominator, will vary from 
 substance to substance. For instance, Fe and Be— Ge will be 
 small for insulators or bad conductors in which there are few 
 -electrons, Fr and Br— Gr will be small for substances which 
 possess few resonators of the frequency under consideration, and 
 Fp and Bp— Gp will be small for substances in which the photo- 
 electric effect is slight at this frequency. On the other hand, 
 observation shows that 9 has precisely the same value for all 
 substances. Now in order that 9 may be the same for all 
 substances, no matter what the relative values of Fe, Fe and Fp 
 may be, it is necessary, as a matter of algebra, to have 
 
 Fii Fe Fp ,„. 
 
 — . =9, . . (6) 
 
 Br— Gr Be— Ge Bp— Gp 
 
 giving at once Er=Ab, Eb=Ae and Ep=Ap. 
 
 In other words, each agency in the mechanism of emission 
 -and absorption must, considered by itself, be capable of setting 
 up the full black-body radiation, and must not set up any 
 radiation different from this ; if this were not so, the black- 
 body radiation would vary from substance to substance. 
 
 Algebraically, it is of course conceivable that one agency 
 might have so slight an influence that 9, as given by equation 
 (5) might have a value indistinguishable from its true value, 
 while the fraction corresponding to this agency in equation (6) 
 might have a value widely difEerent from 9. The numerator 
 and denominator would simply have both to be very small. 
 But physically this would be inconsistent with the second law 
 of thermodynamics* ; and, in fact, it would require that in the 
 
 • Cf. Poincare," Jourii. de Phys.," 11 (1912), p. 34 
 
DISCUSSION OF THE KADIATION PROBLEM. 11 
 
 inal steady state energy should continuously flow in a closed 
 xjycle through the different mechanisms in turn, a possibiUty 
 which may be excluded from serious consideration. 
 
 We may now proceed to consider the radiation from these 
 different agencies in turn, beginning with that from 
 " resonators." 
 
 Radiation from Resonators. 
 
 9. Let a resonator be considered as a dynamical system, 
 obeying the classical laws. Let its frequency be p/27i, and 
 let its kinetic and potential energies T and V be supposed 
 given by 
 
 where <p is the co-ordinate expressing the state of the resonator 
 and a and fi are constants such that f'^=ajfi. 
 
 The equation of motion, expressing the change in (p, is 
 
 ^;^+acp=a), . (7) 
 
 where is the generalised force acting to increase 9. On 
 .multiplying by 9, we obtain 
 
 so that the energy gained by the resonator in the time from 
 '0 to t is 
 
 / ^ts^dt. 
 Jo 
 
 m 
 
 The solution of equation (7) can be written down in the 
 usual way.* Suppose that at time t=0, the values of tp and 
 9 are the same as if the resonator were preforming an os- 
 cillation specified by (f=A cos (pt—s). The impulse Ofdt, 
 acting through the small interval dt at time t', produces a 
 superimposed oscillation such that the value of 9 initially is 
 
 - (Di'di, and, therefore, such that the subsequent value of 9 is 
 
 -Of cos p{t—t')dt. Hence the total value of 9 after time t h 
 
 1 /■' -* 
 <?=-n Oj' cos p{t—t')dt+A cos (vt—B), 
 Ph'=a 
 
 * Lord Eayleigh, " Theory of Sound," §G6. 
 
12 RADIATION AND THE QUANTUM-THEORY. 
 
 and the total absorption in time t, as given by expression 
 (8), is 
 
 A -l ^( d- ft 
 
 ^^dt=- / O,'0t'cosp{t'— t")dt'dt"+ 1 A cos (pt—e)Odt. 
 
 Jo pJoJo " 
 
 On summing over a great number of oscillations, the second 
 term on the right may be neglected, and the equation written 
 in the form 
 
 f(!)<pdt=^( f Of Oi' cos p(t'-t")dt'dt", . . (9) 
 
 the divisor 2 appearing because the integration is over a square 
 
 in the ft" plane, instead of over the half-square bounded by the 
 
 line t"=t'. 
 
 The value of O from time to i can be expressed as a Fourier 
 
 series in the form 
 
 1 /■" 
 0)=- / (Fj, cos ft+Gp sin pt)dp, 
 
 where the coefficients F^, Gp are given by 
 
 Fj,= [ Of cos pt'dt', ■ . (10) 
 
 Gp=( Ofsiapt'dt', (11) 
 
 Jo 
 and in terms of these coefficients equation (9) may clearly be 
 expressed in the form 
 
 jy^dt=^^{¥p^+Gl^) (12) 
 
 But from a theorem given by Lord Rayleigh,* 
 
 fo'dt=-r(-E/+G\)dp, 
 
 so that the mean value of O^ from to t, say O^, is given by 
 
 o^=rf{p)dp, 
 
 where /(p)= — (F/-t-Gj,^), and the mean rate of absorption is» 
 nt 
 
 from equation (12), 
 
 ,'{0cp..=^-i(F/+G.^)=g/(p). 
 
 * ■■ Phil. Mag.," [5], 27, p. 466. 
 
DISCUSSION OF THE RADIATION PROBLEM. 13 
 
 If the force <I) arises from an electric intensity X in any 
 direction in the ether, we may assume (l)=cX, where c is a 
 constant. Assume the density of radiant energy in the ether, 
 
 resolved into its constituent frequencies, to be I 'R(p)d'p per unit 
 
 J 
 
 volume, then the mean value of X^ will be 4 I R(P)'^P. andc^ 
 
 o / 
 
 times this will be O^, which is equal to | f{p)dp. Thus, 
 
 J 
 
 and the mean rate of absorption of radiant energy by the 
 
 -resonator is go/lp). or -g^R(p). 
 
 If the radiating mechanism is electrical, we may take the 
 emission in time t to be given by either of the equivalent forms 
 
 Jo J a 
 
 Tvhere C is a constant. (The left-hand member will give the 
 expression of Larmor, the right-hand that of Lorentz.) On 
 substituting for 9 from equation (7), the emission in time t is 
 
 On integrating (10) and (11) by parts 
 
 Y.=--[ Of cos pt'dt' ; G„=-[ (bfsui pt'dt', 
 
 pU PJo 
 
 so that 
 
 / <p6*=^(GjF,-F^Gp)=0, 
 
 .and the emission in time t is 
 
 Thus the average rate of emission is 
 
 .where a bar over any quantity denotes that its average value 
 IS to be taken over the interval of time from to t. 
 
14 RADIATION AND THE QUANTUM-THEORY. 
 
 On equating emission and absorption, the condition for a. 
 steady state is found to be 
 
 so that, in the steady state, the partition of radiant energy in. 
 the ether must be 
 
 R{p)dp=^^^P^Ydp (13) 
 
 As soon as the nature of the resonator is definitely known^ 
 
 2 
 C and c will be known. For a Hertzian oscillator* C/c^=-r^,,. 
 
 where V is the velocity of light, so that 
 
 ^iP)<ip=%V'dp (14). 
 
 If the resonator consists of a single electron capable for 
 
 e.2 2 
 
 oscillating with frequency p, 0=^^^, c=e,so that G/c^=-^^ as 
 
 before, and formula (14) again gives the partition of energy in. 
 the ether. 
 
 Radiation from Free Electrons. 
 
 10. A free electron may to some extent be treated as a 
 resonator of zero frequency in the sense that it is the limit of 
 an oscillating electron when the forces acting on it are made to 
 vanish. We can suppose the 9 of the preceding analysis to be 
 a co-ordinate x measured in any direction ; the kinetic energy 
 
 g2 
 
 T of this motion is |mx2=|mM^, so that ;S=m. Again C=f i^-^, 
 
 c=e, so that formula (14) is true, but now only for the limit 
 in which p=0. The formula becomes 
 
 2Y3t 
 
 ^ip)dp=z^3P^dp (15) 
 
 This only gives the limiting form of R(p) when p=0. We 
 can find the general form in the following way. 
 
 * Planck gives a slightly different proof for the Hertzian oscillator, see 
 " Acht Vorlesungen iiber Theoretische Physik," Lecture V., or " Ann. d. 
 Physik," 4 (1901), p. 556. 
 
DISCUSSION OF THE RADIATION PROBLEM. 15 
 
 A single ray of light propagated parallel to the axis of x 
 will be specified by equations of the form 
 
 X=0, Y=A cos ^(x+VO, Z=0, 
 a=0, ^=0, 7=-AcosK(a;+V«). 
 
 The motion of an electron moving in the field of force 
 arising irom this wave of fight may be regarded as compoimded 
 of—* 
 
 (i.) A uniform velocity of translation of components u^, «o, 
 ifo — the undisturbed velocity of the electron. 
 
 (ii.) Oscillations parallel to the axes of x and y, each of a 
 purely harmonic nature and of frequency /cw, where w=Mo+V, 
 this latter being, of course, obtained by modifying the fre- 
 quency of the incident light in accordance with the Doppler 
 principle. 
 
 According to the classical dynamics the electron will absorb 
 hght of frequency /cV. The emitted fight wiU be of different 
 frequencies for different directions of emission ; the frequency 
 corresponding to any direction will be obtained by modifying 
 the frequency icw of the oscillation in accordance with Doppler's 
 principle. 
 
 Let the usual polar co-ordinates r, 6, 9 be taken, the electron 
 being the origin and the axis of x being 6=0. In any direc- 
 tion Q, 9 the component of velocity of the electron will be 
 
 Mo COS 9-|-«o sin cos 9+Wo sin B sin 9, 
 
 so that the frequency p of the radiation emitted in this direc- 
 tion will be given by 
 
 j9=«wV/(V— Mo cos 9 — Vq sin cos 9— Wq sin sin 9). . (16) 
 Let the distribution of intensity of radiation in different 
 
 directions be supposed given by the formula 1(6, 9) sin ddOd<^ ; 
 let the mean value of Mo^+Vq'+Wo^ averaged over a great 
 length of time be c^, and let the proportion of the whole time 
 in which the velocity components fie within a smaU range 
 
 ('IJ I ?J I 7/J \ 
 -^ \ ~ jduQctvodwo. 
 
 A more rigorous proof and discusaion will be found in a Paper in " PLil 
 
16 RADIATION AND THE QUANTUM-THEORY. 
 
 Then the average total radiation emitted by the electron 
 per unit time will be 
 
 ///// A/( "°'^^°I'^"°' )l(g, 9) sin e ddd<fdUodvodwo, 
 
 where the integration is over all values of %, Vq, w^, 6 and 9, 
 and the frequency of any particular element is given by 
 equation (16). 
 
 This radiation can be analysed into its different frequencies 
 by changing the variables from Uq, Vq, Wq, d and 9 to Uq, Wq, d, 9 
 and p by the help of equation (16) and then integrating with 
 respect to Uq, Wq, d and 9. The final result is found to be of the 
 form* 
 
 where po stands for /cV, 2n times the frequency of the incident 
 light, and is a function of which the particular form does not 
 matter for the present purpose. 
 
 Clearly if the electron is acted on by a number of waves 
 traversing the ether simultaneously the resulting emission 
 of radiation will be the sum of a number of integrals, each of 
 the type of (17). 
 
 Now, suppose that the electron is in a region of space in 
 which the law of partition of radiant energy is E(y)o)(^po. this 
 «nergy being distributed at random as regards direction. 
 Suppose that a unit amount of energy of frequency po is, as 
 the result of interaction with the free electron, replaced by an 
 amount of energy y) of the original frequency po, and a spectrum 
 /F(po. 'P)^f of scattered energy. From formula (17), F(po, p) 
 must be of the form 
 
 and from the conservation of energy we must have 
 
 l-y,=\\iV„f)d'p=fm,A^. . . (18) 
 ■'0 .' \po / Vo 
 
 Clearly on integration this last is a function of c^ only, so that 
 y) is independent of pa- 
 
 * For details, ae€ the Paper already referred to. 
 
DISCUSSION OP THE RADIATION PROBLEM. 17 
 
 - After unit time the law of partition of the whole amount of 
 radiation will be 
 
 vj '^iPo)dpo+j dpj R(po)F(po. P)dp, 
 
 or, arranged according to frequency p, 
 
 J\y^{p)+J^ R(Po)F(po, P)dp^'\dp. . . (19) 
 
 The condition that the system shall have reached a final 
 steady state is that the partition of radiant energy shall be 
 unaltered by the interaction with the electron. The final 
 partition of energy (19) must accordingly be identical with the 
 initial partition of energy fJi{p)dp, and so we must have 
 
 y>'R{p)+ f ^{poWiPo, p)dp„='R{p) 
 ■J 
 
 or E(?,)(l-v,)=/\(po) o(^^, c^)^». 
 
 The solution of this integral equation will give the form 
 assumed by R(p) in the steady state. If we put po=%P> the 
 equation becomes 
 
 4o ~&{p)'^Vy:')x ^' 
 
 so that R(;^p)/R(p) must be independent of p. It follows that 
 R(p) must be of the form R(p)=Bp", where B and n are con- 
 stants, so that the partition of energy in the ether must be 
 given by 
 
 'R{p)dp='EpHp. 
 
 The values of the constants B and n can at once be found by 
 comparison with formula (15), which gives the hmiting form 
 
 of E(«) when p=0. Clearly we must have »i=2 and B=-^;^-. 
 Thus the general value of R(p) must be given by 
 
 mw 
 
 2 
 
 ^ip)dp=~^3P'dp (20) 
 
 Thus the result obtained from the classical dynamics is that 
 i R(p) has this form there can be permanent equilibrium 
 between the free electrons and radiation, but if R(p) has anv 
 
 C 
 
18 RADIATION AND THE QUANTUM-THEOKY. 
 
 other form, there will be a continued adjustment and exchange 
 of energy until R(p) has assumed this form. "^ 
 
 11. According to the kinetic theory of matter, the mean 
 value of mu^ for an electron, which has been denoted by mu^, 
 ought to be equal to RT, where R is the gas constant and T 
 the absolute temperature of the matter. Richardson and 
 Brown* have determined the value of mu^ by direct experi- 
 ment, and find that it is in point of fact exactly equal to RT. 
 If we substitute this value for mu^, formula (20) becomes 
 
 RT 
 ^ip)dp=^,p'dp (21) 
 
 Again, the kinetic theory of matter tells us that the mean 
 value of the kinetic energy of the resonator considered in § 9 
 ought, if the classical dynamics is true, to be given by 
 
 /V=RT, 
 
 so that formula (14), expressing the condition for equiUbrium 
 between the resonators and radiant energy, becomes 
 
 RT 
 ^(P)dp=-^'dp, (22) 
 
 which is identical with (21). 
 
 Radiation from Electron Orbits. 
 
 12. In § 10 we considered only the transfer of energy 
 between electrons and radiation on the supposition that the 
 electrons were absolutely free, and acted on only by the forces 
 arising from the radiation. A somewhat different problem 
 arises in the more natural case in which the free electrons are 
 threading their way through the interstices of matter, and are 
 in consequence experiencing accelerations and emitting radia- 
 tion at each encounter with the atoms of the matter. The 
 problem was first considered by Lorentz,f who found a formula 
 giving the partition of radiant energy in the steady state for 
 waves of great wave-length. Later the question was again 
 attacked by the present writer, t who confirmed Lorentz's 
 
 * " Phil. Mag.," XVI., p. 353 ; also XVII., p. 890, and SVIII, p. 681. 
 
 f " On the Emission and Absorption by Metals of Rays of Heat of Great 
 Wave-length," " KoninkUjlco Akad. van Wetenschappon, "Amsterdam, 
 April 24, 1903. 
 
 t " The Motion of Elootrona in Solids," " Phil. Mag.," June, 1909, p. 773 
 and July, 1909, p. 209. 
 
DISCUSSION OF THE RADIATION PROBLEM. 19 
 
 result by a different method, and showed how it could be ex- 
 tended to waves of all wave-lengths. The analysis of these 
 Papers is too long to be reproduced here, even in abstract, but 
 the final result obtained is that the partition of energy of all 
 wave-lengths must, in the final state of equilibrium, be given by 
 
 T?T 
 mp)dv=^^^'dp. (23) 
 
 which, again, is identical both with (21) and (22). 
 
 13. The above formula gives the partition of radiant energy 
 according to frequency, but it is easily transformed into one 
 giving the partition according to wave-length. Let the energy 
 within a range dX of wave-length be tf(k,T}dX, then, since 
 
 , 2jrV , , 
 
 /.= , the above formula becomes 
 
 P 
 
 <f{X,i:)dX=8jiRTX-*dX (24) 
 
 On comparison with formula (1) it is seen that the final par- 
 tition of energy in the ether predicted by the classical dynamics 
 is exactly analogoiis to that known to occur in a steady state 
 in a gas. The sole difference lies in a multiplying factor 2, 
 which is readily explained by the difference between sound- 
 waves and hght-waves (c/. § 14, below). 
 
 14. The physical interpretation of these formulae is easily 
 found. Consider any finite volume, v, of any homogeneous 
 and continuous medium. The medium enclosed within this 
 volume will be capable of executing a certain number of free 
 vibrations, and associated with each free vibration will be its 
 frequency p/2n and the corresponding wave-length L When 
 the wave-length is sufficiently small, the number of free vibra- 
 tions for which A hes within even a small range of values will 
 be very great. Let us, in general, suppose that the number 
 lying within a range dX is f{X)dX. Then, clearly, /(A) will be 
 proportional to v, and hence, from a consideration of physical 
 dimensions, since f{X)dX is to be a pure number, it must be of the 
 form 
 
 f{X)dX=GvX'HX, 
 ■where C is a constant. In other words, the number of vibra- 
 tions per unit volume of the medium of wave-length between 
 X and X-\-dX is CX'^dX. The determination of the constant C 
 for any medium is not difficult.* It is found that for the vibra- 
 
 * " Phil. Mag.," July, 1905, p. 91 ; H. A. Lorentz, " The Theory ol 
 Electrons," § 73 ; also a purely mathematical Paper by H. Weyl, " Math. 
 Annalen," LSXI., p. 441. 
 
 C2 
 
20 EADIATION AND THE QUANTUM-THEORY. 
 
 tions of sound in a gas C=in, for the light vibrations in free 
 ether C=87r, and for the elastic solid vibrations C=12;r. The 
 reason why these numbers are in the ratio 1 : 2 : 3 is obvious. 
 In a gas there is only the one set of normal vibrations, in ether 
 there are only transverse vibrations, but there are two indepen- 
 dent transverse vibrations, corresponding to two planes of 
 polarisation, for each normal vibration in the gas, and in the 
 elastic solid there are both normal and transverse vibrations. 
 
 15. In the sound formula (1) given in § 5, namely, 
 
 ve note that the number of vibrations within the range dX oi 
 vsrave-length is 4:nX~*dl, and therefore the average energy of 
 zack vibration is RT. In the radiation formula 
 
 SnUTk'HX (25) 
 
 the number of vibrations is SnXr^X, so that the average energy 
 of each vibration is again RT. Thus the classical mechanics 
 require that, in the steady state of equihbrium, the average 
 energy of each vibration shall be RT, whether in the ether or in 
 a gas. A physical discussion of this is given later. 
 
 The Theorem of Equipartition of Energy. 
 
 16. It has now been seen how the general result can be 
 obtained that in the steady state the average energy of each 
 vibration is RT, whether the vibration is a sound vibration in a, 
 gas, or a Ught vibration set up by resonators, or by free electrons 
 in ether, or by free electrons colliding with matter. This result 
 is only part of a much more general result which can be obtained 
 from the well-known theorem of equipartition of energy, a 
 brief account of the proof of which may now be given, omit- 
 ting all comphcations which have no bearings on the problems 
 we are now concerned with. 
 
 Consider any dynamical system which will be supposed to 
 move in accordance with the Newtonian laws, and let its con- 
 dition be determined by n Lagrangian co-ordinates and the n- 
 corresponding momenta. Let these 2w quantities be denoted 
 
 by e„ e,... 0,„. 
 
 Then the condition of the system at any instant may be repre- 
 sented graphically by a point in an imaginary space of 2n 
 dimensions, this point having for its Cartesian co-ordmates 
 6i> ^2 • • • ^2n. As the system follows out its natural motion 
 this point wiU trace out a curve in the 2w dimensional space. 
 
DISCUSSION OF THE RADIATION PROBLEM. 21 
 
 We may now suppose that the whole space is filled with repre- 
 sentative points tracing out their appropriate curves, and in 
 this way we can imagine that we have a graphical representa- 
 tion which enables us to study simultaneously aU motions 
 possible for our dynamical system. 
 
 According to the well-known theorem of Liouville,* the 
 density of any group of points in this space does not change 
 as the points follow out the paths descriptive of the natural 
 motion of the system, provided this motion is in accordance 
 with the Newtonian mechanics. For instance, if the moving 
 representative points are initially sprinkled with uniform 
 density throughout the space they will remain of uniform 
 density for ever. 
 
 Suppose it is found that after the steady state has been 
 reached — i.e., after the motion has gone on for so long that the 
 influence of the initial conditions has been obhterated— the 
 system invariably possesses some definite property P. This 
 might be from one of two reasons : Either that the represen- 
 tative points tend in their motion to cluster in those regions 
 of the generahsed space in which the property P holds, or that 
 the property P is common to the whole of the generahsed space. 
 But Liouville 's theorem shows that the first of these cannot be 
 the true reason. The property P must be common, then, to the 
 whole of the generalised space. Strictly speaking, no property 
 of any physical interest can be found which is common to 
 absolutely every point in the generalised space, but a number 
 of properties can be found which are true with very insig- 
 nificant exceptions, these exceptions beiag such as would 
 escape notice in experimenting. Thus we may say that the 
 properties to be looked for in the steady state of a system are 
 those which are common to the whole of the generahsed space. 
 
 These somewhat abstract considerations may be illustrated 
 by a perfectly non-dynamical concrete example. 
 
 Suppose that we have an army of a milhon men of average 
 height 5 ft. 8 in. This army can be divided into two wings 
 each of 500,000 men in approximately lO^wo^? different ways. 
 There will, perhaps, be a milhon ways of arranging the men so 
 that the average heights in the two wings may differ by as 
 much as 2 in. ; there may be (say) lOi"" ways in which the 
 averages may differ by -p^ in. ; there will be, perhaps, lOi"*"* 
 ways in which the average will differ by , q'^ p th of an inch. But 
 
 * Boltzmann, " Vorlesungen uber Gastheorie," VoL II., pp. 66, 67 ; or 
 Jeans, " Dynamical Theory of Gases," p. 62. 
 
22 RADIATION AND THE QUANTUM-THEORY. 
 
 for the majority of arrangements the average heights will be 
 almost exactly equal. With the figures we have taken the 
 chance will be 10^ ""■'""' to 1 that the average height in the two 
 wings will be the same to within to'o6 ^^ o^ ^^ inch. It is not true 
 to say that for all arrangements of the men the average heights 
 in the two wings will be only imperceptibly different, and yet 
 the odds are 10^ ""'""' to 1 against this statement being untrue 
 if the men are arranged at random. 
 
 So, in the generalised space under consideration, there will 
 be no property absolutely true of the whole space, but a number 
 of properties can be found which are true for all points in the 
 space except for a number which constitute a perfectly insig- 
 nificant fraction of the whole. In particular, the following is 
 easily shown to be such a property. 
 
 Let two groups of terms be taken out of the expression for 
 the energy, each consisting of a very large number (p, q) of 
 squared terms. Then the property which is true within the 
 hmits explained is that the average value of each term in the 
 group of p terms is equal to the average value of the q terms. 
 If, as in the kinetic theory of gases, we take the average value 
 of terms in either group to be |RT, then the value of the group 
 of p terms will be JpRT, while that of the group of q terms will 
 be I^RT. This is, in effect, the theorem of equipartition of 
 energy.* 
 
 17. It will be seen at once that this result embraces all the 
 results which have been obtained in this chapter. For let the 
 system under consideration consist of ether together with 
 matter of aU kinds. The number of Adbrations of the ether of 
 wave-length between A and ?,-\-d^ will be SnvX'HX, where v is 
 the volume of ether considered, and each vibration will give 
 rise to two squared terms in the energy, one kinetic and one 
 potential. The total number of squared terms representing 
 the energy of all these vibrations wiU be l&nvX'HX, and there- 
 fore their energy will be equal to this number multiphed by 
 JRT, or to 87iRT«A"*<iA. Dividing by v to reduce to energy 
 per unit volume of ether, we obtain for the radiant energy of 
 wave-length between A and X-\-dX 
 
 StlRTX-HX, 
 which is identical with formula (24). 
 
 * For references to the various proofs which have been given see Jeans, 
 ' Dynamical Theory of Gases," Ch. V., or " La Throne du Rayonnement et 
 !e3 Quanta " (Gauthier-Villars, Paris, 1912), p. 71. Detailed proofs of tb* 
 theorem will abo be found in both these places. 
 
DISCUSSION OF THE EADIATION PROBLEM. 23 
 
 This formula was given by Lord Rayleigh and the present 
 author in 1900 as being the formula which ought, on the New- 
 tonian mechanics, to govern the partition of energy in the 
 spectrum. It cannot be the true law, for the total energy 
 obtained by iategrating from A=0 to A= oo would be infinite 
 for any finite value of T. And if the total energy were finite 
 the only possible value for T would be T=0. ^ 
 
 This, m fact, is the prediction of the classical mechanics as 
 to the final steady state. We are led to expect that all the 
 energy of the matter will be dissipated away into radiation in 
 the ether, just as in the analogies of §§ 4 and 5, where it was 
 seen that a continuous medium had the capacity of extracting 
 all the kinetic energy from a system immersed in it. It is to 
 escape from this necessary consequence of the classical mechanics 
 that the quantum-theory has been brought into being. 
 
 Attempts to Reconcile Radiation Phenomena with the Classical 
 Mechanics. 
 
 18. At this stage, before an account is given of the quantum- 
 theory, it is appropriate that some reference should be made 
 to the various attempts at reconcihng the observed radiation 
 formula with the Newtonian mechanics. A report such as the 
 present ought to aim at explaining the various opinions held 
 on controversial matters, in order that the reader may be 
 amply supphed with materials for forming his own judgment. 
 The recent discussion at the Birmingham meeting of the 
 British Association made it abimdantly clear that the quantum- 
 theory is far from being regarded as inevitable yet by many 
 of the English school of physicists. The following accounts of 
 the remarks made by Sir J. Larmor and Prof. Love, in addi- 
 tion to being of great intrinsic interest, will sufficiently indicate 
 what variety of views is possible.* 
 
 " Sir J. Larmor derived the impression from the trend of the 
 discussion that it would turn out that in the new low-tem- 
 perature determinations there was nothing in direct conflict 
 with the classical dynamical principles. The essential argu- 
 ment for equipartition among vibrational types of energy is, 
 briefly, that these types enter similarly into the total energy, 
 and thus, other things being iadifEerent, there is no reason that 
 can be assigned to the contrary. They enter similarly merely 
 
 • "British Association Report" (1913), Birmingham, p. 385. 
 
'ii BADIATION AND THE QUANTUM-THEORY. 
 
 because tlie energy is a sum of squares of their ' momentoids.' 
 But other things may not be indifierent ; for example, in the 
 kinetics of a rotating atmosphere the distribution of energy 
 must be modified so as to maintain constancy of the angular 
 momentum as well as of the energy, giving as the result equi- 
 partition relative to the rotation instead of absolutely. More- 
 over, in an isolated region of ether there is no way open for 
 any interchange of energy at all between one type of vibration 
 and another ; here also other things are not indifferent. The 
 exchange must be effected through the mediation of material 
 molecules. It is true that a single electron, moving erratically 
 between complete reflections from the ideal impervious walls 
 of the chamber, would suffice ; but the structure of an electron, 
 including the mechanisms by which it exchanges energy with 
 the ether, is totally unknown. Such a fundamental fact as the 
 pressure of radiation is involved in that structure ; we can only 
 estabhsh it theoretically as pressure on systems of electrons ; 
 it must be transmitted by the ether in some way, but we do not 
 know how, except by speculating, for it is a second-order 
 phenomenon not involved iu the Maxwellian linear scheme 
 of equations. In the very intense kinetic phenomena in the 
 mechanism of the electrons or molecules, by which they serve 
 to transfer energy from one type of ethereal vibration to other 
 types, the energy must be expressible as a sum of squares of 
 definite momentoids if the transfer is to lead ultimately to 
 eqxiipartition. This restriction in its form is unlikely ; the 
 transfer may even be of a discontinuous character, involving 
 release of electrons into freedom. The Planck formula for 
 the constitution of natural free radiation may be obtained by 
 statistical reasoning, strictly on the lines of Boltzmann's 
 entropy theory for gases, in which atoms or vibrators are not 
 considered at all, but the pressure of radiation is introduced 
 instead, as I have tried to show (" Proc. Roy. Soc," 1906) ; the 
 only imphcation is that the process of interchange of ethereal 
 energy between different vibrational types, by the mediation 
 of matter, though unknown, must be such as to provide a 
 pressure of radiation. If, then, there is no reason to press 
 equipartition as regards free natural radiation, the atomic 
 vibrations, which are set up by its agency and must be in 
 equihbrium with it, are also absolved therefrom. 
 
 " The new knowledge relating to specific heats at very low 
 temperatures has already suggested most interesting specula- 
 tions and tentative adjustments, and will certainly lead to 
 
DISCUSSION OF THE RADIATION PKOBLEM. 25 
 
 definite expansion of our theoretical schemes ; but it can be 
 held that there is nothing in it that is destructive to the 
 principles of physics which have led to so rich a harvest of 
 discovery and synthesis in the past." 
 
 " Prof. A. E. H. Love : I am unable to accept the view that, 
 in order to account for the facts about radiation, existing 
 theories of dynamics and electro-dynamics need to be supple- 
 mented by the theory of quanta. Part of the evidence in 
 iavour of this view has been derived from an apphcation of the 
 principle of equipartition of energy to a system consisting of 
 •ether and matter in an enclosure bounded by perfectly reflecting 
 walls. Such a system has an infinite number of degrees of 
 ireedom, and the principle of equipartition cannot be apphed 
 without modification to any such system. The ethereal kinetic 
 energy would be expressed by an infinite series of the form 
 
 in which there is one term answering to each degree of freedom, 
 and the order of the terms is that of increase of the correspond- 
 ing frequencies. In order that an infinite series may represent 
 •ethereal kinetic energy, or anything else, it is necessary that 
 it should be convergent. In order that it may be convergent 
 it is necessary that all the terms which are far advanced in the 
 order of the series should be very small compared with some 
 •of those which precede them. It is, therefore, impossible for 
 a\\ the terms to be equal as required by the principle of equi- 
 partition. If, however, the principle of equipartition is hmited 
 hj the principle of convergence, it yields some information as 
 to the distribution of energy in the spectrum of a black body. 
 In so far as it is legitimate to regard the infinite series, by way 
 •of approximation, as the sum of a large but finite number of 
 terms, the principle of equipartition should be apphcable, also 
 .as an approximation, and it yields Lord Rayleigh's experi- 
 mentally verified formula for the emissivity answering to long 
 waves. The principle of convergence shows, on the other hand, 
 that for short waves the curve obtained by plotting emissivity 
 .against wave-length should fall towards the origin, as it is 
 known to do. But this principle yields no information as to 
 "the position in the spectrum of the longest waves for which 
 lord Rayleigh's formula fails to give a valid approximation. 
 
 " The arguments on which the theory of quanta was founded 
 ■oannot be -regarded as satisfactory. Indeed, the most con- 
 vincing evidence in favour of the theory would seem to be the 
 
26 RADIATION AND THE QUANTUM-THEORY. 
 
 agreement with experiment of M. Planck's formula, according 
 to wliich the emissivity of a black body is given as a function 
 of the wave-length X and the absolute temperature T, by an 
 expression of the form 
 
 ^;i-5(gB/Xl_l)-l^ 
 
 where A and B are properly determined constants. It may, 
 therefore, be pertinent to remark that from a mathematical 
 point of view there must be infinitely many formulae which 
 would agree equally well with the experiments. In illustration 
 of this statement, it may be mentioned that a formula proposed 
 recently by A. Korn, according to which the emissivity of a 
 black body would be given by an expression of the form 
 
 where C and D are properly determined constants, when tested 
 arithmetically over a wide range, yields results showing just 
 about as good an agreement with the facts as Planck's. It 
 may be, however, that there is no simple formula, like those of 
 Planck and Korn, which is applicable to all wave-lengths. 
 However this may be, there seems to be no sufiicient reason for 
 regarding Planck's formula as expressing a law of Nature. 
 
 " In further illustration of the contention that the resources 
 of the ordinary theories are not exhausted, it may be pointed out 
 that it is possible to extend to some additional cases the cal- 
 culation, first carried out by H. A. Lorentz in the case of long 
 waves, of the emissivity of a thin metal plate. He supposed 
 the radiation to be generated in collisions between free electrons 
 and atoms, and calculated the emissivity for waves of periods 
 long compared with the times occupied in describing free paths. 
 For this calculation he required to evaluate approximately 
 a certain integral. Such an evaluation can be effected alsa 
 in the case of waves which have their periods comparable witk 
 the times occupied in describing free paths, and, for a number 
 of laws of variation of the acceleration during a collision, in, 
 the case of waves which have their periods comparable with 
 the times occupied by collisions. As the wave-length dimi- 
 nishes the emissivity of the thin plate at first increases, then 
 reaches a maximum, and finally diminishes to zero ; as the 
 temperature rises the wave-length answering to the maximum 
 emissivity diminishes, as would be expected. But there is no 
 simple analytical formula which represents the emissivity of 
 the thin plate over the whole range of wave-lengths." 
 
DISCUSSION OF THE RADIATION PROBLEM. 27 
 
 To the remarks of Prof. Love the following reply waa made 
 by Prof. Lorentz : — 
 
 " Prof. Lorentz : Li reply to Prof. Love's interesting remarks, 
 I should hke to say that it is precisely one of the objects of the 
 physical theory of radiation to explain why the energy of the 
 black radiation can be represented by a convergent series. 
 The old theories lead to a different result, but this in itself 
 would not be a physical contradiction or impossibihty ; it 
 would simply mean that all energy will in the end be trans- 
 ferred to the ether and that it will continually take the form of 
 shorter waves, a really final state never being reached. Our 
 aim must be to account for a true state of equihbrium, in which 
 there is a finite ratio between the parts of the energy that are 
 found in a ponderable body enclosed in an envelope and in the 
 surrounding ether. We shall also have to assign a physical 
 meaning to the universal constant h occurring in the formula 
 by which Planck calculates this ratio." 
 
 19. The following additional comments may also be made 
 with reference to Prof. Love's suggestions. As regards the 
 convergence of Prof. Love's infinite series, we may suppose 
 those terms which represent vibrations in the ether to be 
 arranged in order of iescendmg wave-length. It will then be 
 seen that the equations of the classical dynamics demand that 
 the energy should, so to spe^k, be transferred along the terms 
 of the series from left to right, and no steady state can be 
 attained imtil the terms are all equal. Now the time required, 
 on the classical dynamics, for the energy to reach the more 
 distant terms admits of approximate calculation, and it can be 
 shown that before we have got very far along the series this 
 time becomes a matter of milHons of years.* Let us then cut 
 off the tail of Prof. Love's infinite series, and retain, say, only 
 
 the first 10^" terms ; this gives us a finite series of terms 
 which, until after the lapse of milhons of years, is perfectly 
 capable of representing physical conditions, and for which 
 purely mathematical questions of convergence have no meaning. 
 The suggestion that a satisfactory radiation formula might 
 be obtained by the old mechanics from an analysis of electron 
 orbits is one to which a good deal of attention had already been 
 given. As has been seen, the only formula which can possibly be 
 obtained is Lord Rayleigh's law (§§ 12, 17), unless it is assumed 
 that the observed law is not a true final state of thermodyna- 
 
 • C/. " The Dynamical Theory of Gases," Chapter IS. 
 
28 RADIATION AND THE QUANTUM-THEORY. 
 
 mical equilibrium. If, however, it is supposed that the emitted 
 radiation of the highest frequencies is in some way drained ofi 
 or allowed to escape, so that the density of radiation in the 
 ether always remains very small, then it is possible to obtain 
 a definite law, which, as Love remarks, shows some of the 
 characteristics of the observed radiation law. Instead of 
 obtaining the Eayleigh formula 
 
 STiRTtH?, ........ (26; 
 
 we obtain* a formula of the type 
 
 8:?iRTA 
 
 ri(j)dl, ....... (27) 
 
 in which c is a quantity such that s-^is comparable with the 
 
 time of a collision between the electron and an atom. From 
 what has already been said, it is clear that for very long waves 
 
 n T J must approximate to unity, while, if the coUi^ons are all of 
 equal duration, it can be shown that, when A is very small, /( ^ ) 
 
 c 
 
 will tend to zero in the same way as e K Making the assump- 
 tion that the motion of each electron is made up of a series of 
 exactly similar colUsions, separated by rectilinear free-paths. 
 Sir J. J. Thomsont has arrived at the formula 
 
 STiRTX-^e-idX (28) 
 
 which is, of course, a special case of (27). 
 
 The following considerations at once are suggested : — 
 
 I. To reconcile formula (27) with Wien's law (3), c must vary 
 
 us pp, so that the duration of a colhsion must be exactly pro- 
 portional to p=. This would be the case if the atoms were 
 
 centres of force repelhng according to the law of the inverse 
 cube, I but this is not a condition which can be easily reconciled 
 with what is known about the structure of atoms and the 
 motion of electrons. 
 
 * " La Th'orie du Eayonnement et les Quanta," p. 69 ; and J. H. Jeana 
 " Phil. Mag.," July and August, 1909. 
 t " Phil. Mag." 14 (1907), p. 225. 
 } J. J. Thomson, I.e. ante. 
 
DISCUSSION OF THE RADIATION PROBLEM. 2& 
 
 n. It is, of course, possible to find the value of c which would 
 be required to make formula (27) agree with Planck's formula. 
 At ordinary temperatures, the time of coUision would have to- 
 be of the order of 10"^* seconds. Eemembering that the 
 velocity of the electron is of the order of 10', it is clear that this 
 time of collision is too large to reconcile with what is known 
 about molecular or atomic dimensions.* 
 
 III. For formula (27) to agree with the observed black-body 
 radiation, the value of c, and the time of a collision, would have 
 to be exactly the same for all substances, a condition which 
 cannot be reconciled with the known diversity of structure of 
 different substances. 
 
 IV. The experiments of Eichardson and Brownt have shown 
 that the velocities of the electrons in a solid are distributed 
 according to Maxwell's law, so that the values of c must be 
 different for different electrons and for different collisions. On 
 intergating for aU possible velocities, it is found that the 
 
 limiting form for f(-j), when A is very small, is not proportional 
 
 to e ^ but ti 
 observation.^ 
 
 -i -V- 
 
 to e '^ but to e \ and this cannot be reconciled with_ 
 
 * "Phil. Mag." 20, (1910), p. 651. 
 t " Phil. Mag." 16 (1908), p. 353. 
 i • Phil.Mag."20(1910), p. 650. 
 
CHAPTEE 111. 
 
 THE DEVELOPMENT OF THE QUANTUM-THEORY. 
 
 20. The last chapter showed that, consistently with the 
 assumption of thermodynamical equilibrium, the Newtonian 
 system of equations could lead to only one formula for the 
 partition of radiant energy, namely, 8nRTX~*d}., a formula 
 which does not agree with experiment. In the present chapter 
 it will be explained how Planck, starting from conceptions 
 entirely different from those of the Newtonian mechanics, has 
 arrived ' at a radiation formula which is in agreement with 
 observation, and how Poincar^ has shown that the observed 
 radiation formula can be derived from only one set of physical 
 assumptions, namely, those of the quantum-theory. 
 
 In the ordinary theory of gases, the " probability " that a 
 system shall have its co-ordinates (pj^, pg' • ■ • ) ^^^ momenta 
 (ji, ^2; • • • ) within a range dpjdp2 ■ . . dq^dq^, ... is found to 
 be of the form 
 
 Ae~^''^dpjdp2 . . . dq-fdq^ . . ., 
 
 where E is the energy of the system in this configuration, A ia 
 a constant and h is given by 2^RT=1. Hence, if e is any 
 amount of energy, the probabilities of the system having 
 snergies 0, e, 2e, . . . , will stand in the ratios 
 
 Strictly speaking, the probabilities we are discussing are not 
 those of the system having energies 0, e, 2e, . . . but of its 
 co-ordinates lying within equal infinitesimal ranges of values 
 dp-jdp^ . . . dqi dq^ ■ ■ . surrounding these energies, but this 
 complication is immaterial for our present purpose. 
 
 Suppose that we are considering a very great number, M, oi 
 vibrations, and suppose that of these N have zero energy. Then 
 the number which may be expected to have energy e will be 
 Ne"^*', the number which may be expected to have energy 2t 
 will be Ne"**', and so on. If we suppose, purely as a con- 
 jectural hypothesis at present, that all of the M vibrations have 
 
UEVELOrMENT OF THE QUANTUM-THEORY. 61 
 
 their energies cqiial to one or other of the values 0, e, 2e, . . . 
 then we must have 
 
 M=N{l+e-2'"'+e-"'+e-«'-+ . . ■) = , ^-2^. - • (29) 
 
 The total energy of all these vibrations must be 
 
 Ne-2*' . e+Ne-"' . 2e+Ne-«'^' . 3e+. . ., 
 of which the value, on summing and using relation (29), is found 
 
 ""^ .^ -...(30, 
 
 If the particular vibrations are those of wave-length between 
 A and X-\-dk in a imit volume of ether, the value of M must be 
 taken to be (c/. § 14) 8nX~*d}., and formula (30) assumes the form 
 
 8:zX-m^ (31) 
 
 We can pass to the problem contemplated in § 15 by 
 
 passing to tli3 limit e=0 ; for the vibrations are supposed 
 
 capable of having energies 0, e, 2e, 3e, . . . oo, which, when e=0, 
 
 means simply that the energy can have any value. Now, when e 
 
 e . 1 
 IS small, the Umiting value of -^^ — r- is ^^ or RT, so that formula 
 
 (31) reduces to 
 
 8:rRTA-*rfA (32* 
 
 This agrees with formula (25), as of course it should, and so 
 does not agree with observation. Formula (31), in which e is 
 not taken equal to zero, may be put in the form 
 
 e 
 Sn'Ria-^dU f^ , (33) 
 
 and so will agree with observation (c/. § 7) if 
 
 e=hv, .' . . (34) 
 
 where h is now Planck's constant and v is the number of 
 vibrations per second. 
 
 21. This is probably the shortest way of illustrating the 
 connection between Planck's fundamental equation (34) and 
 his radiation formula (4). There are obviously, however, 
 
32 RADIATION AND THE QUANTUM-THEORY. 
 
 grave objections to supposing that the energies of the vibra- 
 tions in the ether must be multiples of e. For this involves 
 that these energies can only alter by sudden jumps of amounts 
 which are themselves multiples of e, and the occurrence of 
 these jumps would seem to be inconsistent with the supposition 
 that the energy is spread throughout the whole of the ether. 
 
 The original, method of Planck* was based on somewhat 
 different physical ideas. Suppose that the M vibrations con- 
 sidered in §20 are the vibrations of M resonators, each of 
 frequency v. Their total energy will be given by expression 
 
 (30), so that the average energy of each vibrator will be -^^ — -, 
 
 and the average kinetic energy will be one-half of this, or 
 
 RTx---"-, (35) 
 
 ^ e^-l' 
 
 where x^z^^. The conditions for equilibrium between re- 
 
 senators and ether have been considered in § 9 ; we note that 
 the average kinetic energy of a resonator must now, from 
 
 formula (35), be supposed to be - — - times what it was supposed 
 
 to be in § 9, and the result is that the radiation formula is equal 
 
 to that previously foimd multiphed by - — r. In other words, 
 
 the radiation formula now obtained is that of Planck. 
 
 22. It is in this way that Planck obtains his formula, but 
 this method also is open to objections. For, in considering the 
 partition of energy between the various resonators, it is as- 
 sumed that the energy can only vary by jumps of amount s, 
 while, in considering the partition of energy between resonators 
 and ether, it has to be assumed, as in § 9, that the energy of the 
 resonators can vary continuously. Moreover, it has to be 
 assumed also that the energy of the various vibrations in the 
 ether can vary continuously, and, this being so, the condition 
 for equilibrium between ether and, for instance, free electrons, 
 would be expected to be that obtained in § 10. In other words, 
 we should expect that the resonators would, so to speak, strive 
 to set up Planck's partition of radiant energy, while the free 
 electrons and other mechanisms would strive to establish the 
 equi-partition formula (32) of Lord Rayleigh. The result 
 
 * " Annaleu iler Physik," i (1901), p. 556. 
 
DEVELOPMENT OF THE QUANTUM-THEORY. 33 
 
 would be a compromise between the two laws, as a result of 
 which the final law would differ for different substances (c/. § 8), 
 while the physical processes by which it was determined would 
 violate the second law of thermodynamics. 
 
 The present chapter is intended to deal only with the 
 mathematical development of the quantum-theory, so that the 
 physical difi&culties just mentioned may be reserved for later 
 discussion (Chapter VII. below). 
 
 23. The discussion of another mathematical problem finds 
 a suitable place in the present chapter. The assumption that 
 changes in the energy of resonators or vibrations follow the 
 Newtonian laws leads, as we have seen, to the formula of 
 Lord Rayleigh, while the assumption that these changes occur 
 by jumps of amount B=hv leads to Planck's law. The converse 
 problem demands consideration, at least in the special case of 
 Planck's law, which is known to agree with observation. The 
 problem is the following : Given that the final partition of 
 energy is that given by Planck's law, what laws of motion must 
 be postulated for the system in order to obtain this law ? 
 
 This problem has been solved with great completeness by 
 Poincare.* The result obtained is, in brief, that no system can 
 possibly lead to Planck's law, except one in which the assump- 
 tion of the quantum-theory is satisfied : " L'hypothese des 
 quanta est la seule qui conduise a la loi de Planck, "f Un- 
 fortunately Poincar^'s Paper is of such an abstruse mathe- 
 matical nature that it is impossible to do any sort of justice to it 
 in an abstract ; the reader who wishes to understand it must 
 turn to the original Paper. 
 
 The following investigation of the same question,^ although 
 less complete than that of Poincard, is based on similar ideas, 
 and leads to the same conclusions. 
 
 Consider the system discussed in § 16, and let the values of 
 the quantities 6^, 0^, . . . be represented in a generahsed 
 space, just as before ; and, again, let all possible states and 
 changes of the system be represented by swarms of moving 
 points. 
 
 In the former investigation, the systems represented by the 
 moving points were supposed to obey the laws of the old 
 mechanics. It followed that there was no concentration of the 
 swarms of points ; they moved indifierently through the whole 
 
 * " Journal de Phys.," January, 1912. 
 
 tic, p. 37. 
 
 j " Phil. Mag.," December, 1910, p. 943. 
 
M RADIATION AND THE QUANTUM-THEORY. 
 
 space, and, as equipartition was a property of the whole space, 
 it followed that equipartition ultimately ensued for all th>i 
 systems. 
 
 We are now searching for a final result different from 
 equipartition, so that we must no longer suppose that the 
 swarms of points move without any concentration taking 
 place, as they would do if the Newtonian laws held. Whatever 
 laws or systems of equations govern the motion of the system, 
 we must suppose some definite law of causation to hold — this 
 is a necessary condition for the problem being capable of dis- 
 cussion at all. A system of particles endowed with free-will 
 would not be a fit subject for mathematical treatment. It 
 must be supposed, then, that the movingpoints in the generalised 
 space follow definite tracks through the space, and, this being so, 
 it is possible so to arrange the initial swarm of particles that the 
 density at any point of the space remains always the same,* and 
 does not vary with the progress of the motion. It is now only 
 necessary to consider this permanent arrangement of density. 
 
 Now, if the density of these swarms of points differed only 
 by finite amounts, we should still be led to equipartition of 
 energy and the radiation formula (32) of the last chapter. 
 For, always excepting infinitesimal fractions, equipartition 
 holds for all of the generalised space, and so, if the density were 
 finite everywhere, would hold for all the representative points. 
 The only way of avoiding the equipartition formula is to suppose 
 that the density of the swarms of points must be zero through- 
 out the whole, except for infinitesimally small regions, of the 
 generalised space. There must be isolated small regions 
 El, Rg, ... in the generahsed space occupied by dense swarms 
 of points ; in all other regions the density of points must be 
 zero or infinitesimal. And, in order to satisfy the hydro- 
 dynamical equation of continuity in the generalised space, the 
 motion of the points must consist of sudden jumps from one 
 of the regions Rj, Rj, ... to another. It appears in this way 
 that, as soon as we seek to avoid the equipartition formula (32), 
 we are compelled to assume motion involving discontinuities of 
 some kind. 
 
 24. This result is so entirely identical with that obtained by 
 Poincare that we may, without break of thought, pass to his 
 discussion of it.f 
 
 * The argument in more detail will be found in Poinoar^'s Paper or the 
 Paper already referred to, " Phil. Mag.," December, 1910. 
 t " Demieres Pens6es," p. 185. (I have translated freely.) 
 
DEVELOPMENT OF THE QUANTUM-THEORY. 35 
 
 " All the states of the system which are represented in any 
 one of these regions [called R^, R2 above] are indistinguishable 
 from one another ; they constitute one single definite state, so 
 that we are led to the following proposition, one which is more 
 definite than that of M. Planck, but not, I think, contrary to 
 his ideas : — 
 
 " A fhysical system is only susceptible of a finite number of 
 distinct states ; it- jumps from one of these states to another without 
 passing through a continuous series of intermediate states. 
 
 " Suppose, to take a simpHfied illustration, that the state of 
 the system depended only on three co-ordinates, x, y, 2, so that 
 we could represent it by a point in ordinary space. The 
 swarm of points representing the difEerent possible states wiU 
 no longer fill the whole of this space or a region of this space in 
 the ordinary sense ; there will be a great number of isolated 
 points scattered through space. These points, it is true, are 
 very close together, which gives us the illusion of continuity.* 
 
 " All these states must be regarded as equally probable. 
 Indeed, if we admit determinism, to each of these states must 
 necessarily succeed another state, which is just equally prob- 
 able, since it is certain that the first is followed by the second. 
 Thus one can see step by step that if we start from any initial 
 state, all the states to which we shall arrive in time must be 
 equally probable ; all others must not be regarded as possible 
 states at all. 
 
 " But our representative isolated points must not be scattered 
 in space in any fashion we please ; they must be distributed in 
 such a way that in observing them with our coarse-grained 
 apphances (avec nos sens grossiers) we have been led to beheve 
 in the ordinary dynamical laws, in particular in the laws of 
 Hamilton. A comparison which approaches nearer to the 
 reality than might be thought will make this clearer. The 
 appearance of a Uquid suggests at first that its structure is 
 continuous ; observation shows that the hquid is incompres- 
 sible, so that the volume of any portion of its matter remains 
 constant. Certain reasons then lead us to suppose that this 
 Hquid is composed of molecules, very small and very numerous, 
 
 * That is to say, when we look at them on the scale of . the ordinary 
 mass- phenomena of nature. But in considering phenomena concerned with 
 atoms, electrons, &g., we assume, so to speak, a mental microscope, and on 
 looking at the points through a mental microscope of this power, they are seen 
 to be widely separated. How wide, judged on this scale, is the gap that 
 separates one point from its nearest neighbour will be clear from the calcula- 
 tions given in § 29 of the present report. — J. H. J. 
 
 D2 
 
36 RADIATION AND THE QUANTUM- THEORY. 
 
 but discrete. We cannot now suppose the distribution of theso 
 molecules arranged in any way we fancy ; tbe incompressibility 
 of the liquid compels us to suppose that two equal small 
 volumes contain the same number of molecules. In his 
 distribution of possible states, M. Planck finds himself limited 
 in a similar way. 
 
 " One might, it is true, imagine hybrid hypotheses. Let us 
 return to the system which we supposed specified by three co- 
 ordinates, so that its state could be represented by a point in 
 space. The regions occupied by the representative points 
 might be neither a continuous region in space nor a collection 
 of isolated spots : it might be composed of a great number of 
 small distinct surfaces, or of small distinct curves. It might 
 be, for instance, that one of the material points of a system 
 might describe only certain trajectories, and might describe 
 them in a continuous manner except when it jumps from one 
 trajectory to another under the influence of its neighbours* ; 
 this might be the case with the resonators considered above. 
 Or, again, the state of the ponderable matter might vary in a 
 discontinuous way, with only a finite number of possible states, 
 while the state of the ether varied in a continuous manner. 
 Nothing in all this would be incompatible with the ideas of M. 
 Planck. 
 
 " But the reader will doubtless prefer the first solution, a 
 solution which is free from all these bastard hypotheses. Only 
 he must note the consequences which it involves. What has 
 been said must apply to any isolated system whatever, and 
 even to the universe as a whole. Thus the universe would 
 jump suddenly from one state to another, but in the interval 
 between it would stand at rest. The different instants during 
 which it stayed in the same state could not be distinguished 
 from one another, so that we should be led to the discontinuous 
 variation of time, to the atom of time." 
 
 25. All this has followed merely from the hypothesis, or the 
 fact, that the energy in the spectrum does not obey the 
 equipartition formula of Lord Rayleigh. Let us next consider 
 what particular form of discontinuities must be postulated in 
 order to arrive at Planck's law. 
 
 Let the total energy E of the system be supposed made 
 up of separate parts E^, Ej, . . . , this division being made in 
 such a way that Ej depends only on one certain group of the 
 
 * Compare Bohr's theory of the structure of the hydrogen atom ; § 37 o£ 
 the present report. 
 
DEVELOPMENT OE THE QUANTUM- THEORY. 37 
 
 co-ordinates 6^^, 6^, . . ., Ej depends only on a certain other 
 group of co-ordinates and so on, no one co-ordinate entering 
 into more than one group. 
 
 Let WeiCZE^ denote the proportion of the whole number ot 
 points for which E^ has a value betv/een E^ and Ej+rfEj ; 
 let Wg^ffEo denote the proportion for which Ea has a value 
 between Eg and Ea+c^Ej, and so on. Then* the proportion 
 for which Ej, Ej, ... all lie within these specified ranges may 
 be denoted by 'WdE^dEi^ ■ ■ ■ and will be given by 
 
 Wd^^d^, . . .=(WE,rfEi)(WE,(iE,)(. ..)... 
 in which Wbi will be a function of E^ only, Wb^ of Ej only, 
 and so on. 
 
 The most Ukely values for Ej, Ej, ... to have will, of course, 
 be those which make W a maximum, or 6W=0. We have 
 
 Whatever system of laws governs the motion we may, on 
 experimental grounds, suppose the conservation of energy to be 
 satisfied so that the total energy E retains the same value 
 throughout. Now 
 
 E=Ei+E,+ . . ., 
 so that 
 
 aE=aEi4-aE2+...=0, .... (37) 
 
 and the condition that ^W, as given by equation (36), shall 
 vanish when ^Ej^, ^Ej, . . . are subject to the Umitations ex- 
 pressed by equation (37) is 
 
 A ?Wbj_J_9Wb,_ 
 
 Wb, SE,-Wb, 9E,- ^•^^' 
 
 If one part of the system, say that of energy E^, is supposed 
 
 to be a gas thermometer, the value of the corresponding 
 
 1 3Wb, 
 differential ^57- -5=^"' can be readily calculated and is found to 
 I Wb, oEj ■^ 
 
 be z^. Thus, the most hkely set of values for E^, Ej, , , . are 
 those given by 
 
 8S 9S 1 
 
 «^ aE;=9E;="-=f (^^ 
 
 * For details see the Paper already referred to " PhQ Mag.," December, 
 1910. 
 
38 RADIATION AND THE QUANTUM-THEORY. 
 
 where S=R log W. It is now possible to identify S with the 
 entropy* ; the most hkely values for E^, Eg, . . . are simply 
 those which make the entropy a maximum. It is easily shown 
 that, for any system consisting of an enormously great number 
 of parts, the regions in the representative space for which W is 
 actually a maximum are enormously greater than all the others 
 together. Thus, 6W=0 or (5S=0 expresses a true steady state. 
 Since S=R log W, it is the state simply in which the entropy is 
 a maximum. The equations which express this are equations 
 (40), which are now seen to be merely the expression of the 
 second law of thermodynamics. 
 
 26. To examine what conditions will lead to Planck's law 
 for the steady state, we have simply to identify the condition 
 expressed by equation (40) or (39) with that expressed in 
 Planck's formula. 
 
 According to Planck's formula (30), the energy E^ of M 
 vibrations of frequency v vibrations per second is given by 
 
 r, _ Me 
 
 where e^hv ; whence 
 
 IE, f, , Mel 
 
 and, from relations (39), 
 
 ^=Eg-|-(l0gWBj. 
 
 Thus 8|;(logWBj=Jlog(l+|j), 
 
 giving on integration 
 
 log W. =(m+^ ) log (M+^)-|^ log ^^+ 
 
 Write P for E^/e and use Stirling approximation for fac- 
 torials, and we obtain 
 
 (M+P)! . 
 where C is a constant. But — pj — is the number of ways in 
 
 which P articles can be put into M pigeon-holes, or, for our 
 ♦ Boltzmann, " Vorlesungen iiber Gastheorie," I., § 6. 
 
 cons. 
 
DEVELOPMENT OF THE QUANTUM-THEORY. 39 
 
 ■present purpose, is the number of ways in wMch P imits of 
 energy can be distributed between M vibrations capable of 
 holding energy. Thus Planck's formula is obtained by sup- 
 posing that the total energy Ej is divided into P units (each 
 
 E,\ . . 
 
 of amount e or hv, since P= — J and that these are distri- 
 buted among the M vibrations. Moreover, it is readily seen 
 that this way of arriving at Planck's formula must be unique. 
 
 27. This result is the same as is arrived at in Poincare's 
 Paper, referred to above,* but Poincare's method is one which 
 makes it possible to proceed one stage further. Planck's law, 
 if absolutely true, must, as has been seen, require the discon- 
 tinuities of the theory of quanta. But, as Poincare remarks, f 
 a law found experimentally is never more than an approxima- 
 tion. Could we then imagine laws such that their differences 
 from Planck's law would be within the errors of observation, 
 but which would at the same time lead to a continuous system 
 of dynamical laws ? - 
 
 This question has already been answered in § 23, and in 
 Poincare's Paper a second decided negative is again given. It 
 is shown that no small, or even finite, departures from Planck's 
 law will dispose of the necessity for discontinuity. Poincare 
 shows definitely and conclusively that the mere fact that the 
 total radiation at a finite temperature is finite (the crucial fact 
 referred to in § 2) requires that the ultimate motion should be 
 in some way discontinuous — % 
 
 " Quelle que soit la loi du rayonnement, si I'on suppose que 
 le rayonnement total est fini, on serait conduit a une fonction 
 presentant des discontinuites analogues a celles que donne 
 I'hypothfese des quanta." 
 
 It seems then to be abundantly proved that the transfer of 
 energy must in some way take place by jumps or jerks of 
 amount e=hv, but mathematical analysis gives no indication 
 as to the physical nature of these processes. The physical 
 problem as to when, where and how the jumps occur can be 
 solved with much less certainty than the mathematical problem 
 of which the solution has predicted the occurrence of the 
 energy jumps with a high degree of certainty. The considera- 
 tion of the physical problem is deferred to the last chapter ; 
 
 * " Journal de Physique," January, 1912. 
 t L.c, p. 27. 
 } L.c , p. 29. 
 
40 
 
 RADIATION AND THE QUANTUM-THEORY. 
 
 we now proceed to consider some purely numerical conse- 
 quences of the mathematical solution which has been obtained. 
 28. The formula given by the Newtonian mechanics for 
 the partition of black-body radiation was 
 
 87iRTX-Hl, (41) 
 
 while the formula given by the quantum-theory is the same 
 formula multiplied by 
 
 X 1 
 
 in which x=.^=;=:^^. From the above expansion it is clear 
 RT Jtli 
 
 that the energy given by Planck's formula is always less than 
 
 that given by formula (41), and the divergence between the two 
 
 formulae increases as x increases — i.e., as we pass to higher 
 
 frequencies or lower temperatures. The following table gives 
 
 values of x/{e'~l) for different values of x : — 
 
 X. 
 
 x/(e^-l). 
 
 X. 
 
 x/{e^-l). 
 
 0-0 
 
 1-000 
 
 2-0 
 
 0-313 
 
 0-2 
 
 0-903 
 
 2-4 
 
 0-239 
 
 0-4 
 
 0-813 
 
 2-8 
 
 0-181 
 
 0-6 
 
 0-730 
 
 3-2 
 
 0-136 
 
 0-8 
 
 0-653 
 
 3-6 
 
 0-101 
 
 10 
 
 0-582 
 
 4-0 
 
 0-0764 
 
 1-2 
 
 0-517 
 
 4-5 
 
 00505 
 
 1-4 
 
 0-458 
 
 5-0 
 
 0-0339 
 
 1-6 
 
 0-405 
 
 6-0 
 
 00149 
 
 1-8 
 
 0-357 
 
 7-0 
 
 0-0064 
 
 A graphical representation of Planck's formula and of 
 formula (41), both arranged according to frequency, is given in 
 Fig. 1, the thin line representing formula (41) and the thick 
 Une the formula of Planck. The characteristic difference 
 between the two formulae is, of course, that formula (41) in- 
 creases indefinitely as the frequency increases, whereas Planck's 
 formula attains a maximum, and then very rapidly falls off 
 again to zero. 
 
 29. It is foimd that Planck's formida represents the ex- 
 perimentally observed distribution of energy if 
 h—6Sx 10"" ergs X seconds. 
 
 The value of the quantum of energy for any wave-length can 
 be found at once. For the D-lines, v=5x 1(F*, so that 
 
 e=Ai;=3.3x 10-12 ergs. 
 
DEVELOPMENT OF THE QUANTUM-THEORY. 
 
 41 
 
 This may be compared with the energies met with in the 
 theory of gases. At 0°C. ET=3-6xlO-" ergs, so that the 
 energy of an atom of mercury vapour at 0°C. is 5-4x lO"** ergs, 
 and the quantum of yellow light is about that of sixty atoms of 
 mercury, or other monatomic gas, at 0°C. 
 
 Again, the energy per atom of a hot solid is 3RT, or 4 X lO'i'T. 
 The radiation emitted by a solid at temperature T has its 
 
 maximum ordinate A^ax. given by Aa,ai.T=0-29 cm. X deg. The 
 energy in a quantum of hght of this wave-length is given by 
 
 fi=- 
 
 -=7x 10-i«T, 
 
 so that the quantum of Kght of any colour contains as much 
 energy as nearly two atoms of the solid at the temperature at 
 which hght of this colour is emitted. 
 
 Thus it appears that the quantum is by no means a small 
 amount of energy when compared with atomic energies. 
 
42 RADIATION AND THE QUANTUM-THEORY. 
 
 30. Of the M vibrations, whether of material resonators or 
 
 of light, considered in § 20, N were supposed to possess no 
 
 energy at all. Thus, out of the whole number, only a fraction 
 
 M— N 
 
 — ^r^ possess any energy, and the value of this fraction, by 
 
 equation (29), is e~^'" or e''^^^. If e/RT is large, only a small 
 fraction of the vibrations will possess any energy ; the majority 
 will be perfectly at rest. 
 
 For instance, at 0°C., RT=3-6x 10"" ergs, and the quantum 
 for yellow light is 3-3xl0"^2gj,gg ggthat e/RT=about90, and 
 e"''^^=about 10"'^. If we suppose that each atom in a mass, 
 of sodium at deg. has associated with it a vibration of fre- 
 quency equal to that of the D lines, then only one out of every 
 10^^ of these vibrations will have any energy. Only one atom 
 in 5x 10^" tons of sodium will have any D-hne energy to give 
 out when in its steady state at 0°C. 
 
 Corresponding to the wave-length Amax., the value of the 
 quantum is given by e=4-965 RT, so that even as regards light 
 of wave-length A„aj at any temperature only one in e*'^^°, or 
 about 1 in 140, of the vibrations has any energy. Thus, in a. 
 red-hot mass of iron, less than 1 per cent, of the red vibrations 
 have any energy : the remaining 99 per cent, are perfectly- 
 dead. 
 
CHAPTEE IV. 
 
 THE LINE SPECTRA OF THE ELEMENTS. 
 
 3L In the last two chapters it has been seen that mathe- 
 matical analysis leads to the following quite definite results: 
 (i.) The final steady state of any system in which the laws of 
 Newtonian dynamics are obeyed must be one in which the 
 partition of energy in the ether is given by the formula 
 
 SjiRTtkU (42) 
 
 (ii.) To obtain a final steady state in which the partition of 
 energy is given by Planck's law, and so agrees with observation, 
 the motion of the system must be governed by the laws of the 
 quantum-theory. 
 
 The laws of the quantum-theory have so far been obtained 
 only in the mathematical form e—hv, with the restriction that 
 exchanges of energy equal to fractions of the quantum s cannot 
 occur. These laws do not amount to a complete system of 
 dynamical laws ; indeed, it could hardly be expected that the 
 complete system of laws governing the ultimate processes of 
 Nature could be obtained from a study of the one phenomenon 
 of black-body radiation. The laws, in so far as we have been 
 able to obtain them, give no information as to motion which 
 is not of a vibratory nature ; they give no information as to 
 any vibrations except perfectly isochronous ones ; they 
 require that exchanges of energy should take place by multiples 
 of the quantum e, but give no information as to when or why 
 these exchanges of energy may be expected to take place. 
 Finally, they give no indication at all as to the seat of these 
 quanta of energy, whether they are to be looked for in the 
 ether or in matter. The phenomenon of black-body radiation 
 has yielded what information there is to be extracted from it ; 
 to complete our knowledge we must survey other phenomena 
 of Nature. Incidentally, the discovery that other phenomena 
 require laws similar to those of the quantum-theory will help 
 to confirm the results obtained from the study of black-body 
 radiation. 
 
44 RADIATION AND THE QUANTUM-THEORY. 
 
 So far as concerns the problem of black-body radiation, 
 the formula derived from the Newtonian laws was found te 
 give a good approximation to the truth so long as x is small, 
 where 
 
 --^ (43) 
 
 Thus, perhaps, what is to be hoped from the quantum-theory 
 is not so much that it shall annihilate our belief in the New- 
 tonian laws as that it shall extend and amplify these laws so 
 ■as to cover large values of a; in the radiation problem, and cor- 
 respondingly in other problems. (Compare, however, Poia- 
 ■care's remarks given in § 24 of the last chapter.) 
 
 The physical meaning of the smallness of x, as given by 
 ■equation (43), must first be considered. The numerator of 
 X is Jiv, the quantum of energy ; the denominator is RT, which 
 is the mean energy of a vibration of great wave-length, and 
 would be the mean energy of all vibrations if the Newtonian 
 laws held. When this quantity RT contains many quanta, x is 
 small ; when it contains few quanta x is large. The New- 
 tonian laws begin to fail as soon as the average energy RT does 
 not contain a great number of quanta. 
 
 Many similar cases of apparent failures of laws in physics 
 at once suggest themselves. The laws of gases begin to fail 
 when the gas does not contain a great number of molecules, 
 the laws of electricity fail when the charges or currents do not 
 contain a great number of electrons, and so on. The laws 
 "which appear to fail are apphcable to large-scale phenomena 
 only ; they take no account of the atomic nature of some 
 entity, and so fail when appHed to small-scale phenomena. 
 Boyle's law is, so to speak, too coarse-grained to apply to single 
 molecules, Ohm's law is too coarse-grained to apply to a few 
 electrons. In the same way the Newtonian laws are too 
 ■coarse-grained to apply to one or a few quanta. 
 
 The Newtonian laws apply, as an approximation, to the 
 radiation problem when x is small — i.e., when T is very large 
 ■or when v is very small. When T is very large, energy is very 
 abundant ; there are a great many quanta involved, and the 
 -atomicity of the quanta is of no account. When v is very 
 small, the quanta Qi,v) are very small, and again the atomieity 
 is of no account. 
 
 The atomicity involved is not necessarily an atomicity of 
 energy. RT is average energy, Ijv is the time of a vibration, so 
 
LINE SPECTRA. 45 
 
 that UT/v is a physical quantity of dimensions (energy) 
 X (time). A quantity of these physical dimensions is spoken of 
 as action. Thus, the Newtonian laws hold when the action 
 RT/i/ is great compared with the action h, of which the amount 
 is known experimentally to be ^=6-6x10"" ergs X seconds. 
 An alternative way of looking at the matter would be to sup- 
 pose that " action-" is atomic, that there is a universal atom of 
 action of amount h, that the Newtonian laws take no account 
 of this atomicity of action, and so are only valid approximations 
 when the amount of action involved is a very great multiple 
 of h. This way of regarding the situation is not hmited to- 
 problems dealing with isochronous vibrations. 
 
 In searching for other phenomena in which the Newtonian 
 laws break down it is natural to examine closely those in which 
 the temperature is very low, or the frequency of a vibration 
 very high, or the " action " of a process very small. In point 
 of fact, each of these alternatives is found to lead to one phe- 
 nomenon of importance in connection with the quantum- 
 theory, these being respectively the phenomena of specific 
 heats at low temperatures, of the line spectra of the elements, 
 and of the photo-electric effect. Purely for convenience of 
 arrangement, we may take the line-spectra first. 
 
 The Problem of the Line Spectra of the Elements. 
 
 32. So long as physicists thought in terms of the New- 
 tonian mechanics almost every characteristic of the Hne- 
 spectrum was a source of difficulty. The evidence both of 
 emission and absorption suggested that the spectrum gave 
 indication of either oscillations about a state of steady motion 
 or vibrations about a state of rest of the dynamical system 
 forming the atom. The discovery of the Zeeman effect and the 
 explanation of it given by Lorentz in terms of the electron 
 theory seemed to confirm these conjectures, and supphed the 
 further information that the moving charges which were the 
 origin of the emitted hght were the moving electrons. 
 
 The first serious difficulty in this view was one which pre- 
 sented itself to Maxwell and Lord Kelvin. Each spectral line 
 represented a separate vibration of the atom, and each vibra- 
 tion ought, according to the theorem of equipartition of energy, 
 to have energy equal to RT. Thus a gas with n lines in its. 
 spectrum ought to have energy wRT per atom from its spectral 
 vibrations alone. But the total energy of mercury is only 
 
46 RADIATION AND THE QUANTUM-THEORY. 
 
 fRT per atom, which is fully accouiited for by the kinetie 
 energy of motion of the atoms ; the total energy of hydrogen 
 is only -|RT per molecule and so on. 
 
 A further difficulty arose in connection with the number of 
 moving electrons per atom. For, if the atom were a system of 
 vibrating or oscillating electrons, there ought to be a number 
 of spectral fines not greater than three (at most) times the 
 number of electrons. But various fines of evidence have for 
 some time combined in indicating that the number of optically- 
 active electrons in an atom was at most of the order of magni- 
 tude of the atomic weight. It was clearly impossible to account 
 for the compUcated spectra of hydrogen and hefium in this 
 way. 
 
 Finally, it was foimd that the frequencies of the spectral 
 fines fall into the wefi-known spectral series, which could not 
 be interpreted on the older mechanics ; indeed the mere cir- 
 cumstance that the atom gave sharply defined lines at aU was 
 sufficiently difficult of interpretation. The foUowing quota- 
 tions from a Paper by Lord Rayleigh* wiU indicate the nature 
 of some of the difficultiec- encountered : — 
 
 " In the calculation of frequencies for a cloud of electrons 
 the undisturbed condition is one of equifibrium, and the fre- 
 quencies of radiatior. are those of vibration about this con- 
 dition of equifibrium. Almost every theory of this kind is 
 open to the objection that I put forward some years ago (' Phil. 
 Mag.,' 1897, 44, p. 362; ' Scientific Papers,' IV., p. 345), viz., 
 that p^ and not f is given in the first instance. It is difficult 
 to explain on this basis the simple expressions found for p, and 
 the constant differences manifested in the formulae of Rydberg 
 and of Kayser and Runge 
 
 " In recent years theories of atomic structure have found 
 favour, in which the electrons are regarded as describing orbits, 
 probably with great rapidity. If the electrons are sufficiently 
 mmierous there may be an approach to steady motion. In 
 case of disturbance osciUations about this steady motion may 
 ensue, and these osciUations are regarded as the origin of 
 luminous waves of the same frequency. But, in view of the 
 discrete character of electrons, such a motion can never be 
 fuUy steady, and the system must tend to radiate even when 
 undisturbed (c/. Larmor, ' Matter and .^ther ').... 
 
 " An apparently formidable difficulty .... stands in the 
 
 * " PhU. Mag.," January, 1906. 
 
LINE SPECTRA. 47 
 
 way of all theories of this character. How can the atom have 
 the definiteness which the spectroscope demands ? It would 
 seem that variations must exist in (say) hydrogen atoms which 
 would be fatal to the sharpness of the observed radiation ; and, 
 indeed, the gradual change of an atom is directly contemplated 
 in view of the phenomenon of radioactivity." 
 
 In brief, as the atom radiates, its total energy changes, so 
 that the orbits of its electrons change, and consequently the 
 frequencies of the oscillations about these orbits change. Each 
 atom in a gas would thus have its own spectrum, so that the 
 spectroscope ought, on this view, to show continuous radiation 
 and not sharp hues. 
 
 33. Since Lord Rayleigh wrote the Paper (1906) from which 
 the foregoing quotations have been taken, our knowledge of 
 atomic structure has progressed from speculation to something 
 approaching fairly near to certainty. There seems to be httle 
 room for doubt now that the atom is built after what may be 
 called the Rutherford* model, according to which the atom 
 consists of a central nucleus of dimensions that may be re- 
 garded as infinitesimal, and of electrons revolving in planetary 
 manner around this nucleus. 
 
 The argument for this type of atom may, perhaps, con- 
 veniently start from a consideration of the helium atom. The 
 «-ray particle has been shown to be a hehum atom with a 
 positive charge equal to exactly twice the electronic charge — 
 in other words, the a-particle is a hehum atom from which two 
 electrons have been removed, or, again, the normal hehum atom 
 consists of an a-ray particle and two electrons. The extra- 
 ordinary penetrating power of the a-particle shows that it 
 must be something extremely small and compact — of size, 
 perhaps, something less than 10~i^ cms. and of compactness 
 3uch that nothing seems to afiect its structure. This is the 
 nucleus of the helium atom ; ordinary methods of gas theory 
 indicate that the size of the complete atom is of the order of 
 10"* cms., which gives information as to the size of the orbits 
 described by the two electrons around the nucleus. The nucleus 
 possesses practically all the mass of the atom, and so may be 
 regarded as a massive fixed point around which the compara- 
 tively weightless electrons move. 
 
 There is strong evidence that this structure is not a pecu- 
 liarity of the hehum atom, except as regards the special number 
 
 * " PhU. Mag.," 21 (1911), p. 669, and 27 (1914), p. 488. 
 
48 RADIATION AND THE QUANTUM-THEORY. 
 
 of electrons (two) involved. According to the experiments of 
 Geiger and Marsden* and the calculations of Rutherfordf and 
 DarwinJ, the scattering of both a and /3-particles by every 
 kind of matter examined was exactly what was to be expected 
 if matter of all kinds had the structure suggested by Ruther- 
 ford. A great deal of work has naturally been expended on 
 the determination of the numbers of electrons in the different 
 elements. It now seems highly probable§ that the number of 
 positive unit charges contained in the nucleus is equal to what 
 is called the atomic number, which, in turn, is the number 
 identifying the position of the element when all elements are 
 arranged in order beginning with the hghtest. Calling their 
 number N, we have for hydrogen N=l, for helium N=2 (as 
 has already been seen), and so on. 
 
 Thus, according to this view, which is supported by a much 
 greater volume of evidence than can be referred to here, the 
 hydrogen atom consists of a negative electron revolving in an 
 orbit round a positive nucleus of equal charge, while the helium 
 atom consists of two electrons revolving in orbits round a 
 nucleus with a positive charge equal in amomit to twice that 
 of an electron. 
 
 The inabihty of the Newtonian mechanics to explain hne 
 spectra becomes evident if we fix our attention on the hydrogen 
 atom, which consists of two point charges only, one positive 
 and one negative, and yet gives the highly complicated spec- 
 trum associated with hydrogen. We are at once led to inquire 
 why these two charges continue rotating round one another, 
 instead of falling into one another, as the energy of their motion 
 is dissipated by radiation. And how can such a simple system 
 give a spectrum containing so many lines ? The number of 
 degrees of the system is only six, three of which represent its 
 ability to move in space. How, then, can so many vibrations 
 be possible ? And, again, why does not the energy of these 
 vibrations appear in the measure of the specific heats ? To 
 all these questions the Newtonian mechanics can give no 
 answer. The answer given by the quantum-theory is, per- 
 haps, only partial, but, as we shall see, it is so convincing that 
 little doubt can be felt of its fundamental accuracy. 
 
 * " Proe." Roy. Soo., A. 82, p. 495 (1909), and later Papers. 
 
 t " Phil. Mag.," 21 (1911), p. 669. 
 
 t •' Phil. Mag.," 25 (1913), p. 201, and 23 (1912), p. 901. 
 
 § H. G. J. Moseley, " Phil. Mag." 26 (1913), p. 1024, and 27 (1914), p. 703. 
 
LINE SPECTRA. 49 
 
 Nicholson's Theories of the Line-Spectrum. 
 
 34. The first serious progress made towards the interpreta- 
 tion of the hne-spectrum is du6 to J. W. Nicholson.* Nicholson 
 studied in particular the spectrum which was to be expected 
 from atoms built on the Rutherford model, and having four 
 and five positive unit charges respectively in the nucleus. The 
 former of these elements he called nebulium, and the latter 
 protofluorine. These names were allotted in the behef, then 
 held by Nicholson, that they were primitive forms of matter, 
 more fundamental than hydrogen, the former being observable 
 in the nebular, and the latter in the coronal, spectrum. The 
 electrons were in each case supposed to move in steady motion 
 in a circular ring about the nucleus, and the possible oscilla- 
 tions about this state of steady motion were first calculated by 
 the Newtonian mechanics. The possible oscillations fall, as 
 can easily be seen, into two quite distinct classes — oscillations 
 normal to the plane of the ring, in which each electron keeps 
 its distance from the nucleus unaltered, and oscillations in the 
 plane of the ring. There is, of course, nothing in the assumed 
 structure of the atom to fix the radius of the ring of electrons : 
 this depends on the energy of the atom, and so ought con- 
 tinually to change as the atom loses energy by radiation. But 
 for an assumed radius a the frequencies of the oscillations can 
 be calculated, and vice versa, and certain ratios of frequencies 
 of oscillations will not depend on a. 
 
 Nicholson calculates the frequencies of the oscillations 
 normal to the plane and certain of their ratios which do not 
 involve a. It is found to be possible to pick out pairs of lines 
 in the observed spectra of nebulae or the corona, as the case 
 may be, having exactly the calculated ratio of frequencies ; 
 the number of such ratios for which the observed and calcu- 
 lated values are in almost exact agreement is believed by 
 Nicholson to put any explanation in terms of the theory of 
 coincidences out of court entirely. From any such pair of lines, 
 when it has been found, it is, of course, possible to calculate a, 
 the radius of the ring of electrons. The radius of the ring is not 
 however, found to have always the same value when deter- 
 mined from different pairs of lines. 
 
 Nicholson also calculates the frequencies of the oscillations 
 m the plane. Many of these oscillations are found to be unstable. 
 
 * Nicholson's Papers will be found in the Monthly Notices of the Royal 
 Astr. Soo. from 1912 to 1914. 
 
50 RADIATION AND THE QUAXTU.M-THEOEY. 
 
 Thus the ring of electrons, according to the Newtonian 
 mechanics, could not continue for any length of time in exis- 
 tence as a ring ; the shghtest jar which it might receive would 
 cause it either to explode or collapse. Consequently the 
 oscillations normal to the plane of the rmg can only have a real 
 existence if it is supposed that in some way the oscillations in 
 the plane of the ring are either prevented from taking place 
 altogether or are governed by some system of dynamics 
 different from the Newtonian system. The sharpness of the 
 observed spectral lines demands that the radius of the ring of 
 electrons should remain invariable, or should have only a 
 definite number of possible values, jumping from one to the 
 other instantaneously, perhaps, as the energy is lost by radia- 
 tion. Whatever view is taken it is obvious that something 
 quite difierent from the Newtonian mechanics will be required 
 to explain the motion. Nicholson believes he can find quite 
 distinct evidence of the existence of Planck's quantum from a 
 study of the radii of the different rings, but his work is hardly 
 complete enough yet to admit of abstraction as a finished 
 theory. We may take leave of it with the remark that, in 
 addition to whatever it may achieve when more fuUy developed, 
 it has probably already succeeded m paving the way for the 
 ultimate explanation of the phenomenon of the line spectrum. 
 
 Bohr's Theories of ilie Line Spectrum. 
 
 •35. In July, 1913, there appeared the first of a series of very 
 remarkable and intensely interesting Papers by Dr. Bohr, of 
 Copenhagen,* in which he explams how the Rutherford con- 
 ception of the atom together with a system of mechanics based 
 on the quantum-theory can be made to yield an explanation of 
 the series observed in the line spectra of hydrogen, and of other 
 phenomena. The theory advanced by Dr. Bohr explains so 
 many facts, and its numerical agreement with observation is 
 so complete, that httle doubt will be felt that it is at least based 
 on a very substantial substratum of truth. 
 
 Bohr considers first the simplest case of a Rutherford atom, 
 in which a single electron of mass m and charge — e is supposed to 
 describe an orbit of radius a round a nucleus of charge E and 
 of mass M so large that the nucleus may be supposed to remain 
 at rest, at any rate for preUminary approximate calculations. 
 
 * " Phil. Mag.," 26 (1913), p. 1, and 476. and later Papers, also in the 
 " Phil. Mag." 
 
LINE SPECTRA. 51 
 
 Bohr takes co to be the frequency of revolution of the electron 
 in its orbit, so that 2na> is the angular velocity with which this 
 orbit is described. Then, according to the Newtonian mecha- 
 nics, the condition that a circular orbit of radius a may be 
 possible is 
 
 eE 
 
 —={2nco)^'ma (44) 
 
 The kinetic energy of the electron is Jm(27rcoa)^, or | — . The 
 
 work required to move the electron from its orbit to a position 
 
 of rest at infinity is .^T^\'m{2jcmaf, or again | — . Denoting 
 
 this quantity, the negative energy of the orbit, by W, it is readily 
 found that 3 
 
 eE V2~ W^^ 
 
 2a=--; co=— — -= (45) 
 
 W n eiLi\/m 
 
 These equations have been obtained by the Newtonian 
 mechanics. Now, according to the Newtonian mechanics the 
 loss of energy W ought to go on increasing as energy was dissi- 
 pated by radiation, so that a would continually decrease and 
 CO increase. The orbit would get smaller and more rapid until 
 the electron fell into the nucleus, and as this process was going 
 on any spectrum emitted by the system would naturally 
 undergo continual change. These considerations illustrate 
 in a forcible way the inability of the old mechanics to account 
 for the line spectrum. 
 
 36. Bohr introduces, to avoid all these difficulties, an 
 assumption which is not inconsistent with the quantum- 
 theory and is closely related to it. It has been noticed already 
 that the quantum-theory is not in itself a complete system of 
 dynamics, and so must not be expected to give a full account 
 of phenomena by itself. The complete system of dynamics, 
 of which it is a part, has not yet been found, and Bohr's 
 assumption only amounts to adding tentatively one other brick 
 to the foundation already laid by Planck's equation e=hv. 
 
 Bohr's assumption, introduced at present only with reference 
 to the atom with one electron, is contained in the equation 
 
 W=tA| (46) 
 
 in which h is Planck's constant and z is an integer. Various 
 
 £2 
 
52 RADIATIOX AND THE QUANTUM-THEORY. 
 
 physical interpretations can be given to this equation. Per- 
 haps the simplest is the following : The angular momentum 
 of the electron in its orbit is 2nmcoa^ or W/nw. Bohr's 
 .assumption, expressed in equation (46), makes this equal to 
 -i:h/2n, i.e., to an integral multiple of the universal constant h/27i, 
 ■and so is equivalent to assuming that angular momentum is 
 atomic, and that h/27t is the unit. An alternative physical 
 interpretation is suggested by Bohr, and will be found in his 
 Paper (p. 4, last para.). 
 
 The effect of Bohr's assumption is, of course, to prevent the 
 continuous variations of W, a and co, which would be demanded 
 by the Newtonian mechanics. The values of W, a and co are 
 found, from equations (45) and (46), to be 
 
 w=-^5p-> ^-=2^,^;^' '"=^%^- (^') 
 
 Instead of t varying continuously, as it would in the older 
 mechanics, it is now restricted to integral values. How com- 
 plete a difference there is between the two systems will be 
 reahsed on considering the variations of angular momentum 
 demanded in the two cases. According to the classical 
 mechanics, the electron experiences a retarding force, F, equal 
 
 to ^^{2rcaa>) from its interaction with the ether, and the rate 
 
 .of loss of its angular momentum rh/27i is given by 
 
 dt\2nj 
 
 while, according to the new hypotheses of Bohr, dr/dt has no 
 meaning at all ! The. effect of restricting r to integral values 
 is, of course, to limit W, co and a to certain definite values. 
 For instance, a cannot gradually shrink, but the electron is 
 limited to one of the values of a given by 
 
 -where t2=1, 4, 9, 16, 25, ... . 
 
 Since a cannot undergo shght changes, there can be no 
 oscillations in the plane of the orbit, and the circumstance 
 discovered by Nicholson, that such oscillations would, on the 
 old mechanics, be unstable, is no longer an objection. The 
 normal hydrogen atom will be that in which the loss of energy 
 .has been greatest, and so is given by t=1. On putting in 
 
LINE SPECTRA. 53 
 
 numerical values, the diameter of the orbit is fouad to be given 
 by 2a=l-lxlO"^ cm., which is certainly of the right order of 
 magnitude. But it is an essential part of Bohr's theory that 
 there can be hydrogen atoms of sizes 4, 9, 16, 2.5, . . . times this. 
 
 37. The radius of the electron orbit of a particular atom is 
 not supposed to be fixed for all time, and we have to consider 
 the possibility of a sudden shrinkage from, say, the orbit given 
 by T=Ti to the orbit given by t=T2. No suggestion is made 
 by Bohr's theory as to what determines when this shrinkage 
 shall take place, or how it happens, but the formulae already 
 obtained show that when it does happen the system must ex-- 
 perience a loss of energy of amount dW given by 
 
 dW=W^-W,=^-^^^^(J-^~^^. . . (48) 
 
 h'- 
 
 Bohr supposes that the whole of this energy, suddenly set 
 free from the atom, passes away into space in the form of abso- 
 lutely monochromatic radiation. He further supposes that 
 the amount of this radiation is exactly one quantum. Thus dW, 
 given by equation (48), must be the same as e in Planck's 
 equation, and therefore equal to hv. This condition, accord- 
 ing to Bohr, determines the frequency of the monochromatic 
 light emitted. Putting the right-hand of equation (48) equal 
 to hv, the frequency is seen tc be given by 
 
 "-(^-^■)- 
 
 (49) 
 
 where N= — — (M 
 
 h^ 
 
 According to Bohr's theory, the different values of Tj and r^ 
 which can be inserted in this equation must give the fre- 
 quencies of the difierent spectral lines of the substance. The 
 lines can be sorted into series corresponding to different values 
 of Tj. For instance, there will be a series given by T2=1, and 
 the lines of this series will be given by Ti=2, 3, 4, ... ; there 
 will be a series T2=2, with hues given by Ti=3, 4, 5, . . . ; and 
 so on. 
 
 38. Hydrogen Spectrum. To obtain the hydrogen spectrum, 
 we simply take E=e, and so 
 
 ^=-hr (-^^i 
 

 54 RADIATION AND THE QUANTUM- THEORY. 
 
 The series T2=2 gives the series of lines 
 
 where n has the values 3, 4, 5, ... , and this is exactly the well- 
 known Balmer's series. The numerical agreement is almost 
 perfect, for on substituting the best known values for m, e and 
 h, the calculated value of N is 3-26 XlO^', whereas the ob- 
 served value is 3-290X lO^^, the error thus being well within the 
 probable errors in the assumed values of e and h. 
 
 The value T2=1 in equation (49) would give the series 
 
 i'=n(i-~), (n=2, 3, 4, . . .), 
 
 all the lines of which would be in the extreme ultra-violet. 
 None of the lines of this series had been observed when Bohr 
 j)ubhshed his Paper, but the series has since been discovered by 
 JLyman.* 
 
 The value T2=3 gives the series 
 
 which is exactly the series observed by Paschenf in the infra- 
 red. The series T2=4, 5, 6, . . . are too far in the infra-red to be 
 observed. ; 
 
 This completes the Hst of lines obtainable from formula (J 9), 
 and it will be noticed that certain Hues, usually ascribed to 
 hydrogen, have not been accounted for. Bohr shows that 
 many of these missing lines are readily explained as belonging 
 to helium. 
 
 39. Helium Spectrum. — For hehum the nuclear charge E 
 is 2e, and in the neutral hehum atom there are two electrons 
 describing orbits. A hehum atom with one positive charge 
 wiU, however, have only one electron describing an orbit 
 around a charge 2e, so that the spectrum of such an atom 
 ought to be obtained on putting E=2e in equation (49). If 
 we keep for N the value assigned to it in equation (51), this 
 spectrum can be put in the form 
 
 ■■2 
 
 '■'^"'(©■"(i 
 
 The series T2=1 and T2=2 he in the extreme ultra-violet, and 
 
 * Details have not yet been published. 
 
 t " Annalen der Physik," 27 (1908), p. 565. 
 
LINE SPECTRA. 55 
 
 have not been observed. The series T2=3 may be regarded as 
 falling into two parts according as Tj is odd or even, namely 
 
 '^-Kl-w^'^-il-dw)' 
 
 ^nd these are two of the series recently observed by Fowler * 
 in a mixture of hydrogen and hehum. These series were at 
 first attributed by Fowler to hydrogen, although from the first 
 it was apparent that they could not be obtained without an 
 admixture of hehum. In view of Bohr's explanation it now 
 ■seems reasonable to attribute them to positively charged helium 
 atoms, the hydrogen presumably being necessary to effect the 
 ionisation of the helium. 
 
 The series T2=4 may again be regarded as falling into two 
 parts, according as z^ is odd or even, namely — 
 
 -"(j-s) '''> 
 
 •=«( rdw) ' '^*' 
 
 The first of these is obviously Bahner's series, so that the 
 lines of this series can be emitted by helium as' well as by 
 hydrogen. The second is the series observed by Pickering in 
 the star f-Puppis in 1896.1 This series was attributed by 
 Pickering to hydrogen simply from the analogy, now seen to be 
 inadequate, that the series (53) was emitted by hydrogen.. In 
 the f-Puppis spectrum, in point of fact, the series. (53) is ob- 
 served to be more intense than the series (54), presumably 
 indicating that the source of hght is a mixture of hydrogen and 
 hehum. 
 
 40. One point of interest arises in connection with these 
 stellar spectra. In the laboratory it is not possible to observe 
 the Balmer series beyond the line w=12, while in the stellar 
 spectra this series appears as far as n=33. The inference, on 
 Bohr's theory, is that in the vacuum-tubes available in the 
 laboratory there are no hydrogen atoms of diameter greater 
 than that given by Tj=12, whereas in the stars there are atoms 
 of diameter up to that given by ri=33. For t=12 the calcu- 
 lated diameter is 1-6x10"* cms., which is equal to the mean 
 distance between the molecules in a gas at a pressure of about 
 
 * Monthly Notices Royal Ast. See, 73, December, 1912. 
 
 t " Astrophysical Journal," i (1896), p. 369, and o (1897), p. 92. 
 
8b RADIATION AND THE QUANTUM-THEORY. 
 
 7 mm. of mercury ; for t=33, the corresponding diameter is 
 l-2x 10~^ cms., and the pressure about 0-02 mm. of mercury. 
 Thus, Bohr's theory leads us to suppose that in the stars there 
 may be hydrogen atoms of diameter about one thousand times 
 that of the normal atom. 
 
 41. The brief calculations given above have proceeded on 
 the supposition that the mass of the nucleus M is very great 
 compared with that of the electron m. If this assumption is no 
 longer retained, the value of N, the Rydberg's constant for the 
 various series, instead of being given by equation (50), is found 
 to be given by* 
 
 2jrVE2_mM_ 
 ^ ¥~ (M+m) '""^'^ 
 
 Thus, if M refers to the hydrogen nucleus, the mass of the 
 helium nucleus will be 4M, and instead of the ratio of the two 
 N's being, as was given by the simple approximate calculations, 
 4 to 1, it will be 
 
 4 . 1 
 M-f|w ■ M+m 
 
 From the observed ratio of the two N's, taken from the best 
 measurements of the hydrogen and hehum lines. Fowler f 
 obtains for the ratio M/m the value 1,836, with a probable 
 error of not more than 12. It need hardly be said in what 
 goo.d agreement this value is with other determinations of M/m. 
 
 42. The foregoing remarks refer only to Bohr's interpreta- 
 tion of the hydrogen and hehum spectra. One other case in 
 which his theory seems to be successful is that of the single 
 electron rotating round a nucleus of charge 3e. This atom, on 
 Bohr's view, is the lithium atom with a double positive charge, 
 and it ought to show spectral lines of frequencies 
 
 '-<^'} 
 
 The special value 1^=6 gives, in addition to lines coinciding 
 with those of the Bahner series, a pair of series 
 
 some of the lines of which have been identified by Nicholson { 
 in the spectra of the Wolf-Rayet stars. 
 
 * Bohr, " PhU. Mag.," March, 1914, p. 509. 
 
 t Royal Society, Bakerian Lecture, 1914. 
 
 + Monthly Notices of the Royal Ast. Soc , March, 1913, p. 382. 
 
LINE SPECTRA. 57 
 
 43. This brief summary of Dr. Bohr's work has only 
 attempted to touch on a few of the very interesting points 
 covered by his original Papers. Enough has, however, per- 
 haps been said to indicate how rich a field is opened by the 
 application of the quantum-theory to problems of atomic 
 structure. 
 
 It will be seen, on examination of Bohr's work, that its most 
 marked successes occur in the case of a single electron rotating 
 round a nucleus ; indeed, in more complex cases, the success 
 of BohY's theory is at least open to doubt. It almost would 
 appear as though the motion of the simple system consisting 
 of a nucleus and one electron was governed by the laws 
 assumed by Bohr, while the mechanics of the more complex 
 system consisting of more than two parts had not been 
 fathomed yet.* 
 
 One general reflection may perhaps be made about the 
 general problem of the line spectrum. One of the most 
 noticeable features of the general system of line spectra is the 
 way in which the well-kno'wn Rydberg constant enters simi- 
 larly into the spectra of all substances. The natural inter- 
 pretation of this would be to suppose that this constant de- 
 pended only on quantities common to all kinds of matter, or, 
 more mathematically, to suppose that the constant was a 
 function only of e, m and V, the velocity of light. But it is at 
 once found that no constant of the same physical dimensions 
 as Rydberg's constant can be constructed out of the quan- 
 tities e, m and V. The quantum-theory, however, puts the 
 additional quantity h at our disposal. As we have seen, it is 
 now possible to construct the Rydberg constant as a f miction 
 of e, m, V and h. The constant so constructed has exactly the 
 right numerical value, and even its slight variations can per- 
 haps (c/. § 41) be accounted for by the different values of 
 m/M for the different elements. The point which is worth 
 considering is briefly this : The existence of Rydberg's con- 
 stant indicated that there must be some universal constant 
 common to all matter in addition to the previously known 
 constants e, m and V, and the constant h of the quantum-theory 
 has been found to supply the omission exactly. 
 
 * Su a Paper by J. W. Nicholson, Monthly Notices of the Royal Ast 
 Soc, March, 1914, p. 425. 
 
CHAPTER V. 
 
 THE PHOTO-ELECTRIC EFFECT. 
 
 44. No single feature of the quantum-theory causes more 
 difficulty in its acceptance than the postulate that radiant 
 energy must, so to speak, be in some way tied up in bundles -of 
 amount hv. Whether this postulate is absolutely essential to 
 the quantum-theory will, in the present report, be discussed 
 in a later chapter, after all the evidence has been reviewed. 
 The view that radiant energy exists in the form of indivisible 
 " hght-quanta " has been put forward and defended by 
 Einstein,* and has been the centre of much discussion. The 
 photo-electric phenomenon, which will be discussed in the 
 present chapter, is of special interest, as throwing light on this 
 important question, as well as giving evidence as to the validity 
 of the general theory of quanta. 
 
 The general features of the phenomenon are well-known. t 
 For some time it has been known that the incidence of high- 
 frequency hght on to the surface of a negatively charged con- 
 ductor tended to precipitate a discharge, while Hertz showed 
 that the incidence of the hght on an uncharged conductor 
 resulted in its acquiring a positive charge. These phenomena 
 have been shown quite conclusively to depend on the emission 
 of electrons from the surface of the metal, the electrons being 
 set free in some way by the incidence of the light. 
 
 In any particular experiment, the velocities with which 
 individual electrons leave the metal have all values from zero 
 up to a certain maximum velocity v, which depends on the 
 conditions of the particular experiment. No electron is found 
 to leave the metal with a velocity greater than this maxi- 
 
 * " Ann. der Physik," 17 (1905), p. 132, 20 (1906), p. 199, 22 (1907), 180. 
 See also various letters in the " Physikalische Zeitschrift," and §§ 64, 65 of 
 the present report. 
 
 t A very good summary will be found in Campbell's " Modern Electrical 
 Theory" (2nd edition, 1913), while a mere complete account is given by 
 Hughes, "Pho'o-Ele^t icity " (Camh. Univ. Press, 1914). I have drawn 
 freely from both of these s uices in writing the present chapter. See also 
 Richardson and Compton, "Phil. Mag.," 24 (1912), p. 575, and Hughes, 
 " Phil. Trans. Eoy. Soc," 212, A (1912), p. 205. 
 
THE PHOTO-ELECTRIC EFFECT. 53 
 
 mum V. It seems probable that in any one experiment all the 
 electrons are initially shot off with the same velocity v, but 
 that those which come from a small distance below the sur- 
 face lose part of their velocity in fighting their way out to the 
 surface. 
 
 Leaving out of account such disturbing influences as films of 
 impurities on the metallic surface, it appears to be a general 
 law that the maximum velocity v depends only on the nature 
 of the metal and Bn the frequency of the incident light. It 
 does not depend on the intensity of the light, and within the 
 range of temperature within which experiments are possible 
 it does not depend on the temperature of the metal. The 
 Tion-dependence or intensity was first established by Lenard,* 
 -and has subsequently been confirmed in an interesting way 
 by Pohl and Pringsheimf and byMilhkan.f who found that the 
 maximum velocity was the same whether using either very 
 intense spark-hght, or arc light of the same wave length. 
 And, as regards dependence on temperature, LadenburgJ 
 lias examined the photoelectric effect for three metals (Au, Pt 
 and Ir) up to 800°C., and found it independent of the tem- 
 perature, while Lienhop§ has worked down to — 180°C. with 
 the same result. 
 
 For a given metal this maximum velocity increases regu- 
 larly as the frequency of the light is increased, but there is a 
 ■certain frequency below which no emission takes place at all. 
 There is almost general agreement now that if v is the maximum 
 velocity for light of frequency v falling on a given metal, the 
 kinetic energy |mD^ is given by 
 
 ^mv^=l-i'—Wa, (57) 
 
 where k, Wq are constants for the metal. The frequency vq, 
 below which no emission takes place, is, of course, given by 
 vg^w^/k. As an instance may be taken its value for 
 sodium, for which j^o=5-15x 10", which is the frequency for 
 green light. Light more red than this can fall on sodium for 
 acenturies without producing any photo-electric effect at all ; 
 the incidence of hght more violet than this will immediately 
 result in an emission of electrons witli a maximum kinetic 
 'energy k{v—vo). 
 
 *" Ann. d. Phys.," 8 (1902), p. 149. 
 t " Verh. d. Deutsch. Phys. Gesell," 15 (1912), p. 974. 
 + " Verh. d. Deutsch. Phys. Gesell," 9 (1907), p. 165. 
 I " Ann. d. Pliys.," 21 (1906), p. 281. 
 
60 
 
 EADIATION AND THE QUANTUM-THEOEY. 
 
 45. As a typical set of good experimental determinations- 
 may be taken those of Hughes.* The quantity m'o, being of the- 
 ph}-sical dimensions of energy, may be put equal to eVo, where 
 Vq is the P.D. through which an electron would have to fall to 
 acquire kinetic energy Wq. The values obtaiaed by Hughes for 
 k and Vq are given in the following table (the last column but 
 one is different from that actually printed in Hughes' Paper, 
 because he uses kin a different sense) : — 
 
 Element. 
 
 Atomic 
 weight. 
 
 Atomic 
 volume. 
 
 Valency. 
 
 k. 
 
 Vo (volts). 
 
 Ca 
 
 401 
 
 25-4 
 
 2 
 
 4-91 X 10-"-" 
 
 2-57 
 
 Jig 
 
 24-3 
 
 14-0 
 
 2 
 
 5-24 
 
 3-08 
 
 Cd 
 
 112-4 
 
 13-0 
 
 2 
 
 5-67 
 
 3-49 
 
 Zn 
 
 65-4 
 
 9-2 
 
 2 
 
 5-88 
 
 3-77 
 
 Pb 
 
 207-1 
 
 181 
 
 4 
 
 5-50 
 
 3-42 
 
 Bi 
 
 208-0 
 
 21-2 
 
 5 
 
 5-63 
 
 3-37 
 
 Sb 
 
 120-2 
 
 18-1 
 
 5 
 
 5-72 
 
 3-60 
 
 As 
 
 750 
 
 13-1 
 
 
 
 5-7 
 
 4-5 
 
 Se 
 
 79-2 
 
 17-6 
 
 6 
 
 • •■ 
 
 4-8 
 
 O2 
 
 160 
 
 12-6 
 
 6 
 
 ... 
 
 80 
 
 Hughes points out that in any set of elements of the same 
 valency there is a regular increase in k and Vq with decreasing 
 atomic volume, and that in passing from one valency to 
 another this relation is discontinuous, the discontinuity being 
 alwavs in the same direction. It wiU be noticed that Vo is 
 certainly of the same order of magnitude as the ionisation 
 potential, and may probably with fair certainty be identified 
 with it. For oxygen Hughes finds Vo=8-0 volts, while the 
 best direct determination of the ionisation potential gives 8-1 
 volts. f 
 
 46. If we assume this identification, Wg in equation (-57) 
 becomes identical with the amount of energy required to move 
 the electron out of its atom to rest at a point outside the 
 attraction of the atom. The amount of work needed to move 
 the electron away from its atom and endow it with a velocity 
 V is ^mv^-\-Wo, and this, by equation (-57), is equal to ki\ 
 
 Now the average value of k for the elements examined by 
 Hughes is about 5-5 XlO^^', while Planck's constant h is 
 6-6x10"". Thus the amount of energy kv absorbed by the 
 electron in setting itseK free from the atom is always nearly 
 equal to one quantum of energy, irrespective of the frequency 
 
 * " Phil. Trans.," 212, A (1912), p. 225. 
 f Hughes, I.e., p. 223. 
 
THE PHOTO-ELECTKIC EFFECT. 61 
 
 of the incident light. The whole phenomenon is very clearly 
 ^nd adequately explained on the quantum-theory, if we sup- 
 pose that the total energy absorbed by the atom and electron 
 together is exactly one quantum, and that about five-sixths of 
 this is absorbed by the electron, while the remaining one- 
 sixth is absorbed by the remainder of the atom. A general 
 explanation of the photo-electric phenomenon in terms of 
 the quantum-theory was first put forward by Einstein.* 
 
 This explanation of the photo-electric effect seems to de- 
 mand that energy of frequency v should exist in the ether in 
 indivisible bundles of amount hv, or, at least, that interchanges 
 of energy between matter and ether, at any rate so far as this 
 special type of interchange is concerned, should occur by 
 indivisible quanta of amount hv. On this view of the matter 
 it is obvious why light of frequency less than the critical 
 frequency v^^woik cannot produce any emission of electrons. 
 For one quantum of such energy is not adequate to remove the 
 electron from its orbit, and the chance of two quanta being inci- 
 dent on the atom simultaneously may be treated as negligible. 
 Again, it is obvious why the speed of emission must increase with 
 the frequency of the incident hght ; for the atom, if it absorbs 
 energy from the ether at all, cannot absorb less than the whole 
 of the quantum — the quantum being supposed indivisible — 
 and so any excess of energy in the quantum over that required 
 to remove the electron from its orbit must appear (mainly) as 
 kmetic energy of the electron (equation (57)). 
 
 47. It cannot be denied that very serious difficulties can 
 be formulated against Einstein's conception of the energy 
 quantum (c/. below, § 65). Without stopping to discuss these 
 difficulties now, we may inquire whether an alternative ex- 
 planation, more on the hues of the older mechanics, cannot Idc 
 given of the photo-electric phenomenon. 
 
 Those who attempt such an explanation have to fall back 
 on some kind of " trigger action " and resonance eSect.i" For 
 instance, the critical frequency for sodium is ^0=5-15x101* ; 
 Hght of frequency greater than this will liberate electrons, 
 while light of lower frequency, no matter how intense, will pro- 
 duce no effect at all. This has to be explained by supposing 
 that in a mass of sodium there are a number of atoms in which 
 
 * " Annalen der Physik," 17 (1905), p. 146. 
 
 t In particular, Lenard, " Ann. der Phyaik," 8 (1902), p. 149. See note 
 on next page. 
 
iiZ 1ADIATI0N AND THE QUANTUM-THEORY. 
 
 electrons are oscillating with frequencies, say, 5-2x10", 
 5-3x101*, 5-4x10", and so on, and that the amplitudes of 
 these vibrations are so great that each electron is just about to 
 jump out of its orbit and free itself from its atom. If this were 
 so, it can readily be imagined that a small amount of hght of 
 any frequency greater than 5-15x10^* would so increase the 
 amplitude of vibration of those atoms with which it was 
 approximately in resonance that they would soon escape 
 altogether. But, to account for the fact that hght of fre- 
 quency less than v,, can produce no efEect, it would have to be 
 supposed that there were no such oscillations of frequencies 
 51x 10", 5-Ox 10", and so on. 
 
 Even if all this could be accepted, it would only explain why 
 photo-electric action could take place, and not at aU why it 
 should be governed by an equation of the form of (57). But 
 the explanation is not in keeping with the teachings of the 
 sodium spectrum as to the frequencies of the oscillations in the 
 sodium atom, and even if it were, it would be difficult to 
 understand why it is that, if very faint hght of frequency 
 5-2 x 10^* can produce enough resonance for electrons to be set 
 free, very intense Hght of frequency 5-lX 10^* cannot produce 
 the same efEect. Moreover, the view that the sodium atoms have 
 large stores of energy of internal vibration is not one which 
 can be reconciled either with our loiowledge derived from 
 specific heats or spectroscopy. 
 
 48. If it is once granted that a " trigger action " of this tjrpe 
 is inadmissible* there is a very convincing argument against 
 any explanation in terms of the old undulatory theory of hght. 
 This is stated with great clearness by Campbell as follows t : — 
 
 " There is one remarkable feature connected with the 
 photo-electric efiect which shows that the older view is quite 
 inadequate, and that some reconstruction of our ideas is neces- 
 sary, while it is in complete agreement with Einstein's theory. 
 A photo-electric efiect can certainly be observed when the 
 energy falling per second on 1 sq. cm. of the substance is much 
 less than 1 erg (which is equivalent to a standard candle at a 
 distance of 2 metres), and the energy of each electron liberated 
 h}' it can certainly be greater than 10"^^ ergs (corresponding to 
 O-O volt). On the ordinary theory of light an electron cannot 
 
 " Lenard has recently agreed that the photo-eleotrio energy must come 
 from the incident light. See Ramsauer, " Physikalische Zeitsohrift," 12 
 (1911), p. 997. 
 
 t " Modern Electrical Theory " (2nd edition, 1912), p. 249. 
 
THE PHOTO-ELECTRIC EFFECT. 63 
 
 absorb more energy than falls on the molecule in which, it is 
 contained ; but the area covered by the section of a single 
 molecule is certainly less than 10"^^ cms.* It appears, then, that 
 no electron could acquire the energy with which it actually 
 emerges unless the light had acted for 10^ sec, or about a 
 quarter of an hour ; until this time has elapsed from the 
 moment when the light was turned on, there should be no 
 photo-electric effect. As a matter of fact, the effect appears 
 to start absolutely simultaneously with the action of the 
 light. 
 
 " But if the light energy is done up in bundles or quanta, 
 and an electron can take up a whole quantum instantaneously, 
 then there can be a photo-electric effect resulting in the emis- 
 sion of a single electron as soon as the total energy of the light 
 emitted is equal to one quantum. Now the source contemplated 
 is emitting light at the rate of 5,000 ergs a second, so that only 
 2xl0~i^ sees, need elapse before a quantum is emitted. This 
 time is, of course, wholly inappreciable." 
 
 These considerations show very clearly the inability of the 
 old undulatory theory to explain the photo-electric effect. 
 
 49. It is worthy of noticef that Einstein's theory, which 
 has been seen to give such a satisfactory explanation of the 
 photo-electric effect, falls into place very naturally as a logical 
 extension of Bohr's theory, which in the last chapter was 
 seen to account for the observed phenomena of line spectra. 
 The quantity Wg which has been used in this chapter is the 
 amount of energy required to set the electron free, without 
 velocity, from its orbit, and so is exactly equal to the quantity 
 denoted by W in the last chapter. If we suppose Wj to be the 
 lost energy in the normal state, corresponding to r=l in the 
 last chapter, and W^^ to be the lost energy in the state t=oo, 
 in which the electron is free at infinity, then Wi=Wo and 
 W^ =0. If, for simplicity in argument, we neglect the differ- 
 ence between h and h, i.e., if we assume that the whole 
 quantum of energy is absorbed by the electron ultimately set 
 free, then equation (57) becomes 
 
 \mv'^=liv—Wa=liv — (\Vi — W^ ), 
 
 or Az/=Wi-W„+|mt;2 (58) 
 
 * Sec Marx and Licliteueokcr, Ann. d. Phys., 41 (19rj), p. 121; alto 
 Matx, Ann. d. Phj-s., 41 (1913), p. 101. 
 t Bohr, " Phil. Mag.," 20 (1013), p. 17. 
 
64 RADIATION AND THE QUANTUM-THEORY. 
 
 Bohr's equation giving the line spectrum was (c/. equation 
 (48)) 
 
 hv=W, -W„ (59) 
 
 This equation referred to the emission of radiation accompany- 
 ing the shrinkage of the atom from the state t^ to the state Tj. 
 The physical process must, however, be reversible, so that the 
 incidence of a quantum of energy of frequency v given by 
 equation (59) on an atom in the state r^ ought to result in the 
 absorption of the radiation and the expansion of the atom to 
 the state r^. Thus a line of this frequency ought to appear 
 in the absorption spectrum of the gas, provided there are 
 atoms in the gas in the state tj ; if there are no atoms in this 
 state, there would be no absorption of light of this frequency. 
 For instance, in ordinary inert hydrogen there are presumably 
 no atoms in the state T2=2, so that the corresponding lines, 
 which constitute the Balmer series, do not appear as absorp- 
 tion lines. In general, when atoms occur in an inert gas, they 
 may all be expected to be in the final state in which t = 1, so 
 that the only lines of the emission spectrum which may be 
 looked for in the absorption spectrum will be the lines given by 
 T2=l) namely, lines of frequency given by 
 
 ^z;=Wi-W„...(w=2, 3, 4,...oo). . . . (60) 
 
 If radiation of one of these frequencies falls on a gas it will 
 be absorbed ; if radiation of a frequency intermediate between 
 these frequencies falls on a gas it will not be absorbed, for it 
 could only be absorbed by complete quanta, and one quantum 
 of energy would take the atom to a state intermediate between 
 two of the states given by integral values of n, which is im- 
 possible. But if radiation of a frequency higher than the 
 highest given by equation (60) (namely, that corresponding to 
 n=oo) falls on a gas it will be absorbed, for the absorption of 
 one quantum will carry the atom beyond the state w=Qo, i.e., 
 it will set one electron free altogether, and endow it with a 
 certain amount of kinetic energy in addition. This last 
 phenomenon is simply the photo-electric effect, interpreted 
 according to Einstein's theory ; it is now seen to be a neces- 
 sary logical extension of Bohr's theory of absorption. 
 
 The complete absorption spectrum ought, according to this 
 view, to consist of a series of lines given by equation (60) and a 
 continuous band stretching from n= oo (the head of this series 
 of lines, given by v=WJJi) up to i'=oo. In this spectrum the 
 
THE PHOTO-ELECTEIC EFFECT. 65 
 
 lines would represent the reversal of Bohr's emission efEect, 
 while the band would represent the photo-electric efiect. 
 
 R. W. Wood* experimenting on the absorption spectrum of 
 sodium vapour has observed a complete absorption spectrum 
 of exactly this type. Fifty lines were observed, agreeing 
 exactly in position with those of the principal sodium series, 
 and in addition a continuous absorption beginning at the head 
 of this series and extending to the extreme ultra-violet. 
 
 • Wood, " Physical Optics " (1911), p. 513 ; Bohr, I.e. p. 17. 
 
CHAPTER VI. 
 
 THE SPECIFIC HEAT OF SOLIDS. 
 
 50. According, to the well-known law of Dulong and Petit, 
 the product of the atomic weight and the specific heat of an 
 element has, at least at ordinary temperatures, a value which 
 is approximately the same for a great number of elements. 
 The product is called the atomic heat, and is approximately 
 equal to six. A simple explanation of this is provided by the 
 usual kinetic theory of matter. 
 
 Suppose that 1 gramme of the substance in question con- 
 tains N atoms, each of mass m, so that Nm=l. Let each atom 
 have s degrees of freedom — that is to say, if aU the other atoms 
 are held at rest, let it be possible for each atom to have s 
 entirely independent motions of its own. Then the expres- 
 sion for the energy of each atom will be that of the energy of s 
 vibrations, and so it follows from the theorem of equipartition 
 of energy (§ 16) that the average energy of each atom will be 
 
 sRT .- . (61) 
 
 where T is the absolute temperature, and the total energy E 
 of the substance per gramme will be NsRT. The specific heat 
 
 will be r=: in energy units, or t^ -- in heat units, where J is the 
 
 mechanical equivalent of heat. Calling this specific heat c, we 
 have 
 
 IDE 19 ,T,Tr.rrv NsR ,c„. 
 
 °=jaT=jyT(^^^^)=-j- (^'^ 
 
 Let a be the atomic weight of the element, then if m^ denotes 
 
 the mass of the hydrogen molecule, the mass m of the atom 
 
 now considered is m=^am^. For the hydrogen molecule the 
 
 value of R/wij is known to be l-lSx 10'', so that for the present 
 
 2 
 substance R/m=-x 4-13x10', and since Nm=l this is also 
 
 the value of RN. The value of J is 4-18x 10', so that equation 
 (C2) becomes 
 
 ^2s 4-13xl0' ^l-98s. 
 ""a 4-18x10'"' a 
 
SPECIFIC HEAT OF SOLIDS. 67 
 
 The constancy of ca, which is postulated by the law of 
 Dulong and Petit, is seen now to depend simply on the con- 
 stancy of s. Moreover, s must from its meaning be an integral 
 number, and to give to the product ac the value required by 
 Dulong and Petit's law we must obviously have s=3, giving 
 oc=5-95. 
 
 Each atom must have a value of s at least equal to 3, for it 
 has three independent motions in space — namely, those 
 parallel to the axes of x, y and z. The observed value of the 
 product ac is thus accounted for by supposing that the atom 
 has no further independent motions ; the atoms are to be 
 thought of as rigid points. 
 
 51. Just as in a gas, there is a distinction between specific 
 heats at constant volume and at constant pressure, the 
 difference between the two depending, of course, on the 
 amount of work required to compress the heated solid back to 
 the volume it occupied when unheated. The older experi- 
 mental determinations of specific heats of sohds were con- 
 cerned with the specific heats at constant pressure, but re- 
 cently Nemst and Lindemann * have shown how to correct 
 such values so as to deduce the specific heat at constant volume. 
 Now, it is to the specific heat at constant volume that the 
 theoretical evaluation of § 50 refers. The constancy of ac 
 ought only to be true if c represents the specific heat at con- 
 stant volume. 
 
 In 1911 a series of very fine determinations of specific heats 
 were undertaken at Berlin by Nernst and his collaborators 
 Lindemann, Koref and others, f these determinations covering 
 the whole range of temperature available in the laboratory. 
 All these determinations were corrected so as to refer to 
 specific heats at constant volume. J 
 
 In these experiments it was foimd that the product ac at 
 sufficiently high temperatures approximated very closely to 
 its theoretically predicted value 5-95, but that there was a 
 very remarkable falling off at low temperatures. The accom- 
 panying figure will show the general nature of the results 
 
 * " Zeitaohrat Jfiir Electrochem." (1911), p. 818. 
 
 t See especially Papers by Nernst, " Amalen der Physik," 36 (1911), 
 p. 395, and a report presented- to the first Solvay congress at Brussels, 
 1-911, " La Thtorie du Rayonnement et les Quanta," p. 254. Also Papers 
 in the " Sitzungsberichte d. Preuss. Akad.," 1911, and in the " Zeitsohiit fiir 
 Elefctrochemie," 1911 and 1912. 
 
 t For the theory and details of this correction, see Nernst and Lindemann, 
 "Zeitschrift fiir Elektroohem." (1911), p. 817. 
 
 P2 
 
68 
 
 RADIATION AND THE QUANTUM-THEORY. 
 
 obtained. The curves giving tlie atomic heats of four ele- 
 ments — ^lead, silver, aluminium and carbon in the form of a 
 diamond — are shown, the atomic heat being the ordinate and 
 the absolute temperature being the abscissa. It is found that 
 the curves of all the elements examined show zero atomic heat at 
 absolute zero of temperature, and tend asymptotically towards 
 the hmiting atomic heat 5-95 as the temperature increases. A 
 still more striking discovery is that all the curves are exactly 
 
 Diamond. 
 
 similar, except for differences in the temperature scale. Thus, 
 if, in Fig. 2, we take the curve for silver and compress it hori- 
 zontally in a certain ratio, which is about 1 : 2-3, it will be found 
 to coincide^ exactly with the lead curve. Similarly, the 
 aluminium curve, compressed in the ratio 1 : 1-8 would coin- 
 cide with the silver curve, and compressed in the ratio 1 : 4-1 
 it would coincide with the lead curve. 
 
 52. The simple investigation of § 50 ought to apply equally 
 at all temperatures if the fundamental principles on which it 
 is based were sound. The principles on which the investiga- 
 tion was based were essentially those of the classical mechanics, 
 so that the result of the experiments would seem to indicate 
 that the principles of the classical mechanics are apphcable 
 
SPECIFIC HEAT OF SOLIDS. 69 
 
 at high temperatures but not at low. This is exactly the 
 situation which it was seen in § 31, might be expected to arise 
 with regard to low-temperature phenomena. 'It is, therefore, 
 natural to anticipate that the explanation of low-temperature 
 specific heats is to be looked for in terms of the quantum-theory. 
 Explanations in terms of the quantum-theory, of great 
 interest and suggestiveness, have been attempted by Einstein* 
 and by Nernst and Lindemann.* These had, however, both 
 a somewhat artificial element in them, and their importance 
 lay rather in their paving the way for a final theory than in any 
 claim to finahty that they could profess. The explanation of 
 the phenomenon, which, both from its complete naturalness 
 and from its agreement with experiment, seems destined to be 
 final, was given by Debyef in 1912. 
 
 53. In the analysis of § 50 we supposed each gramme of the 
 matter under consideration to consist of N atoms, each having 
 three degrees of freedom as though they were rigid points. 
 There were 3N degrees of freedom in all, so that the total energy 
 was, according to the older mechanics, 3NRT, regardless of the 
 particular nature or properties of these degrees of freedom. 
 But according to the quantum-theory the average energy of a 
 degree of freedom is not entirely independent of its nature ; in 
 the case of a vibration the average energy depends on the 
 frequency of the vibration. 
 
 The atoms of the sohd do not, of course, possess independent 
 separate free vibrations. The oscillation of any one will set 
 its neighbours into oscillation, and the free vibrations of the 
 system are those of all the atoms simultaneously. Any motion, 
 no matter how complicated, of the atoms can be analysed into 
 trains of waves, just as was the case with the motion of the 
 molecules of a gas in § 5. The number of independent vibra- 
 tions of wave-length between 1 and l-^Al has been seen (§ 14) 
 to be 127tA'*(^A. Of these vibrations two-thirds are waves of 
 distortion and one-third are waves of compression. Let the 
 former be supposed propagated with velocity Vj and the latter 
 with velocity Vj, then the frequencies of the former vibrations 
 
 * The best account of these will be found in the reports of the first Solvay 
 Congress at Brussels (1911), " La Thcorie du Eayorniement et les Quanta" 
 (1912), pp. 254, 407, in which will be found valuable discussions of the general 
 problem. The theory of Einstein is explained later in the present report, 
 
 (§ 58). 
 
 ■f " Zur Theorie der Spezifischen Warmen, Annalen der Phyaik " 39 
 
 (912), p. 789. 
 
VU RADIATION AND THE QUANTUM-THEORY. 
 
 are V^/A and of the latter Vj/A. It readily follows that the 
 total number of vibrations with frequencies between v and 
 v-{-dv is 
 
 in{2Yi-^+YfyHv (63) 
 
 There must be a Umit to the values of v possible ; obviously 
 the formula breaks down when we come to frequencies such 
 that the wave-length is comparable with the distance between 
 adjoining atoms. Debye assumes, as an approximation, that 
 formula (63) is true from i'=0 up to a value Vm, which is the 
 maximum of all the frequencies, and is determined by the con- 
 dition that the total number of degrees of freedom obtained by 
 this assumption is equal to the known total number 3N. In 
 other words, we must have 
 
 3N 
 
 =i7t{2Y,-'+Y,-')fJvHv=^(2Y^-o+Y,-'K^ (64) 
 
 and formula (63) may be replaced by 
 
 QN'^ (65) 
 
 54. Debye proceeds by supposing that the energy of these 
 vibrations is equal to the amount assigned to them by the 
 quantum -theory as developed in § 18 of the present report. 
 That is to say, each vibration of frequency v is supposed to 
 have an average amount of energy 
 
 ETx^j, (66)- 
 
 where x stands as before for :5=,. Thus the total energy of the 
 
 solid, obtained by integrating over vibrations of all frequencies 
 will be 
 
 E=r9NRT-^,— ; (67) 
 
 Jo e^-1 vj 
 
 Incidentally, it may be noticed that, according to the classical 
 
 lechanics, we should have omitted t] 
 
 so the total energy would have been 
 
 / 9NRT— ^-; 
 
 leading back at once to the result obtained in § 50, namely, 
 
 mechanics, we should have omitted the factor -j — - in (66), and 
 
 6 '^"X 
 
 9NRT-^'-3NRT, 
 
SPECIFIC HEAT OF SOLIDS. 71 
 
 that the atomic heat ought to have the value 5-95 at all tem- 
 peratures. 
 
 On replacing x by hvj'&T, the value of E given by formula 
 (67) reduces to 
 
 The integral unfortunately cannot be evaluated in finite 
 terms. Details of its numerical treatment and computations 
 will be found in Debye's Paper already referred to. 
 
 55. A detailed statement of its agreement with experiment 
 will be found in the same Paper. For the purposes of the 
 present report the following instances will perhaps sufl&ce. 
 
 400 
 
 The various specific heat curves ought, as we have seen, to 
 be dependent only on a single parameter, and this may con- 
 veniently be taken to be a quantity 6 defined by 6=—^. It 
 is then easily seen that the specific heat at any temperature T 
 ought to be of the form j( - J, where / is a function of — , which 
 
 is the same for all substances. In Fig. 3, which is taken from 
 Debye's Paper,* the continuous curve represents the graph of 
 
 /f- J calculated from the integral (68), the scale of abscissae 
 • L.C. p. 812. 
 
72 RADIATION AND THE QUANTUM-THEOKY. 
 
 being arranged for the value 0=396°. The crosses represent 
 the actually observed values of the specific heat for aluminium. 
 This figure accordingly exhibits the agi-eement between 
 theory and experiment in the case of aluminium, provided that 
 G, in the theoretical formula, is treated as an adjustable 
 constant. The value of for any particular substance, how- 
 ever, depends on Vm, which is given by equation (64), and so 
 admits of calculation when the elastic constants of the sub- 
 stance are known. The following comparison is given by 
 Debye between observed and calculated values of 6 : — 
 
 Element. 9 observed. 9 calculated. 
 
 Al 396 399 
 
 Cu 309 329 
 
 Ag 215 212 
 
 Pb 95 72 
 
 In these comparisons, the value of " 6 observed " is ob- 
 tained by fitting a curve of the type predicted by theory to the 
 observations as closely as possible, as, for instance, has been 
 done for aluminium in Fig. 3 above. 
 
 This will give some idea of the agreement between Debye 's 
 theory and observation. The agreement could not be expected 
 to be perfect, for Debye's assumption of the existence of a 
 sharply defined maximum frequency v^ is obviously at best 
 only a somewhat rough approximation. Various attempts 
 have been made to improve Debye's theory in this respect ; in 
 particular, attention may be called to Papers by Bom and 
 K4rmdn,* in which the arrangement of atoms in the sohd is 
 treated as forming a " space-lattice " of the type with which 
 the work of Bragg on the theory of X-ray spectra has made us 
 famihar.f The results obtained in this way agree, perhaps, 
 somewhat better with experiment than those obtained for 
 the simpler theory, but it seems probable that nothing short 
 of a complete knowledge of the structure of matter will result 
 in any very substantial improvement on Debye's formula. 
 
 56. According to Debye's theory, the sohd itself must have 
 free vibrations of frequencies ranging from to the maximum 
 frequency v^,- The law of distribution of these frequencies is 
 given by formula (65), showing that the number within a given 
 small range di> is proportional to v^dv. Thus the spectrum of 
 these frequencies would be a band having its head at frequency 
 
 * " Physikalische Zeitechrift," U (1913), pp. 15 and 65. 
 t See also a Paper by H. Thirring, " Phys. Zeitsch.," 14 (1913), p. 867, 
 and also 5 -50 below. 
 
SPECIFIC HEAT OF SOLIDS. 
 
 73 
 
 Vm, and falling off in intensity towards the low-frequency end. 
 The value of v^ can be calculated from the elastic constants of 
 the solid by formula (64), and is found to be generally of the 
 order of lO^^. 
 
 It will be noticed that this formula is derived theoretic- 
 ally by laying great stress on just that part of Itebye's theory 
 which is obviously most imperfect, namely, the supposition 
 that the vibrations of the solid fall oS absolutely suddenly at a 
 certain frequency v^. It is, therefore, hardly to be expected 
 that the values of v^ calculated from it should agree exactly 
 with the values determined from observation of the specific 
 heat. Perhaps the most that can be expected is that the two 
 sets of values should be approximately the same. This they 
 may certainly be said to be, as will appear in the next section 
 (c/., table below). 
 
 57. Lindemann * has given the following formula, to some 
 extent empirical, which is found to give the values of v^ with 
 surprising accuracy for the substances examined. If T, 
 denotes the melting point of the substance in degrees absolute, 
 m the atomic weight and V the atomic volume, Lindemann 's 
 formulaf is : — 
 
 ,„=3.08X10-X^/A 
 
 (69) 
 
 This agrees rather better with the observed values of if^ 
 than do the values calculated directly from the elastic con- 
 stants, as the table below shows + : — 
 
 Element. 
 
 Vm calculated 
 (Lindemann). 
 
 Vra observed 
 (spec, heats). 
 
 Cm calculated, 
 (elastic constants). 
 
 Al 
 
 8-3x1012 
 7-5 X 1012 
 4-8x1012 
 4-8x1012 
 2-0x1012 
 35-0x1012 
 
 8-3 X 1012 
 6-6x1012 
 4-8x1012 
 4-5x1012 
 1-9x1012 
 400X1012 
 
 8-0x1012 
 6-6 X 1012 
 
 4-3x1012 
 1-44x1012 
 
 Cu 
 
 Zn 
 
 Ag 
 
 Pb 
 
 Diamond 
 
 * " Physikalisohe Zeitschrift," 11 (1910), p. 609. 
 
 -j- In Lindemann's original Paper the multiplying constant, which is 
 purely empirical, was taken to be 2-06 x IO12 and v^ had a sUghtly different 
 meaning. 
 
 J In this table the first two columns are taken from the report by Nernst 
 in " Vortrage uber die Kinetische Theorie der Materie," 1914, p. 77, and 
 are calculated by giving the Lindemann's formula the form (69). Einstein 
 takes the multiplying factor in Lindemann's formula to be 2-12x1012, and 
 a table of values calculated in this way is given by him in " La Theorie du 
 Rayonnement et les Quanta," p. 415. 
 
74 RADIATION AND THE QUANTUM-THEORY. 
 
 58. Mention must now be made of the theory by whicli 
 Einstein originally attempted to explain the specific heat 
 phenomenon. Instead of supposing the energy of the solid to 
 reside solely in the kinetic energy of the atoms, he supposed 
 that the atoms were absolutely at rest, and that each had 
 three degrees of internal vibration all of the same frequency v. 
 On assuming that the energy of these vibrations was governed 
 by Planck's formula (66), he arrived at the expression 
 
 E=3NRTx^, (70) 
 
 e — 1 
 
 for the total energy of the solid, as compared with the value 
 3NRT given by the classical mechanics (§ 50). On diSerentia- 
 tion with respect to the temperature, a formula is obtained 
 which reproduces the experimental curves of Fig. 2 in their 
 general features, but does not agree with them to within the 
 errors of experiment. 
 
 Einstein's explanation of specific heat must necessarily 
 be regarded as superseded by that of Debye, but is of impor- 
 tance on account of the use which Nernst* has made of it in 
 his explanation of the specific heats of compound substances. 
 The method will be sufficiently described by considering 
 a diatomic substance, such, for instance, as potassium 
 chloride. 
 
 Nernst supposes the heat energy of unit mass of potassium 
 chloride to be made up to two parts — the internal energy of the 
 molecules, and the motion of the molecules relative to one 
 another. The internal energy of the molecules is supposed to 
 arise from the vibrations ot the atoms relatively to one another. 
 As a matter of geometry, there must be three degrees of free- 
 dom in these internal vibrations. Nernst supposes them all to 
 have exactly the same frequency, and identifies this frequency 
 with the frequency of the sharply-defined absorption band 
 observed by Rubens in the infra-red by the method of rest- 
 strahlen.f The internal energy of the N molecules in unit mass 
 
 is now given by Einstein's formula (70) in whicli a;=— p, 
 
 H i 
 
 where v^, for any substance, represents the observea frequency 
 
 otthe infra-red absorption band. Nernst supposes the energy 
 
 of motion of the molecules as wholes to be given by Debye s 
 
 ♦ Nernst, report in " Vortrage uber die Kinetiache Theorie der Materie " 
 t " Sitzungsber. d. Preusa. Akad, Berlin," June 5, 1913. 
 
SPECIFIC HEAT OF SOLIDS. 
 
 75 
 
 formula (68), the molecules now playing the part assigned to the 
 atoms in Debye's formula. On these suppositions the total 
 energy of the N molecules can be written down, and by differen- 
 tiation the specific heat. The specific heat will be the sum of 
 two specific heat terms, one calculated from the formula of 
 Einstein, the other from that of Debye. If we sum these terms 
 directly as derived from formula (68) and (70), we shall obtain 
 twice the atomic heat, for we are now supposing that there are 
 N molecules per unit volume. 
 
 The following specimen table, selected from several given by 
 Nernst * in the report already referred to, will indicate the 
 degree of closeness of agreement with observation. 
 
 Valms of 2cp for KCl. 
 
 
 Einstein 
 
 Debye 
 
 Correction 
 
 Calculated 
 
 Observed 
 
 T. 
 
 term 
 
 term 
 
 term 
 
 2c. 
 
 2r 
 
 
 in 2c,.. 
 
 in2c„. 
 
 2(Cj,-c^) 
 
 £Aipt 
 
 ^^p. 
 
 22-8 
 
 0-046 
 
 1-04 
 
 
 1-086 
 
 1-16 
 
 26-9 
 
 0-13 
 
 1-48 
 
 • •■ 
 
 1-61 
 
 1-52 
 
 30-1 
 
 0-25 
 
 1-87 
 
 ■ •• 
 
 2-12 
 
 1-96 
 
 33-7 
 
 0-43 
 
 2-25 
 
 ... 
 
 2-68 
 
 2-50 
 
 48-3 
 
 1-43 
 
 3-52 
 
 ... 
 
 4-95 
 
 5-70 
 
 57-6 
 
 2-13 
 
 4-06 
 
 0-02 
 
 6-21 
 
 6-12 
 
 70-0 
 
 2-89 
 
 4-57 
 
 0-04 
 
 7-50 
 
 7-58 
 
 86-0 
 
 3-66 
 
 4-97 
 
 0-06 
 
 8-79 
 
 8-72 
 
 235 
 
 5-55 
 
 5-81 
 
 0-32 
 
 11-68 
 
 11-78 
 
 416 
 
 5-83 
 
 5-91 
 
 0-68 
 
 12-42 
 
 12-72 
 
 550 
 
 5-87 
 
 5-93 
 
 0-90 
 
 12-70 
 
 13-18 
 
 The Einstein-term in the second column is calculated by 
 assigning to v the value actually observed for it by Eubens 
 
 T— =166) ; the Debye term in the third column is calcu- 
 lated by assigning to v^. the value given for it by Lindemann's 
 formula (69), and the correction term is calculated from the 
 known physico-chemical constants of potassium-chloride. 
 Thus the calculated values in the fifth column are in reahty 
 derived from a formula in which every constant is evaluated 
 by experiment, and there are no adjustable constants at all. 
 Under these circumstances the agreement between the observed 
 and calculated values of 2cp must be regarded as very remark- 
 able, the more so in view of the fact that equally good agree- 
 ment is obtained in the case of other substances for which 
 
 * L.C. p. 81, 
 
76 RADIATION AND THE QUANTUM-THEOBY. 
 
 V2 (the frequency of the rest-strahlen absorption band) is 
 known — namely, NaCl, AgCl, KBr and PbCla- 
 
 Thus, it will be seen that the quantum-theory is able to 
 predict with striking success a relation between phenomena as 
 different in their nature as black-body radiation, the principal 
 infra-red absorption bands of schds, and their specific heats. 
 
 An attempt has also been made by Thirring* to account for 
 the specific heats of compounds by extending the ideas of Bom 
 and Karmanf to space lattices in which two or more different 
 masses are supposed to alternate. In this way it is found pos- 
 sible to calculate the relation between the elasticity and specific 
 heat for regular crystals. The calculation is carried out for 
 NaCl, KCl, CaFa and FeS^. The author fmds a mean devia- 
 tion of 2-3 per cent., with a maximum of 4 per cent, from ob- 
 served values through the range of temperature considered by 
 him, a range much less than that covered by Nemst's tables. 
 
 59. There is no doubt that there must, in theory at least, 
 be other contributions to the specific heat besides those we have 
 been considering ; what is remarkable is that the corresponding 
 deviations from the theoretical formulae we have had under 
 consideration seem to be hardly noticeable experimentally. 
 The presence of free electrons must add something to the 
 specific heat, but it appears as though the contribution were 
 too small to be detected experimentally. Nemst and Linde- 
 mann have found no general difference between the specific 
 heats of good and bad conductors ; Eichterl has examined the 
 specific heats of a series of Bi-Sn and Bi-Pb alloys with special 
 reference to this question, and concludes that, in these aUoj^ 
 at all events, the free electrons do not contribute sensibly to 
 the specific heat. 
 
 Again the rotations of the atoms or molecules must in theory 
 contribute something to the specific heats. The atoms cer- 
 tainly can rotate, for if a crystal is rotated in the hand its optical 
 axes will be found to have rotated with the body as a whole, 
 showing that each atom must have individually changed its 
 direction ; also if a magnet is turned, its magnetic field will 
 turn with it. The absence of a noticeable contribution to the 
 specific heats is accounted for, on the quantum-theory, by 
 supposing that the forces opposing rotational movements of 
 
 • " Phys. Zeitschr.," 15 (1914), p. 180, and also p. 127. 
 
 t See above, § 55. 
 
 t " Ann. d. Physik," 39 (1912), p. 1590. 
 
SPECIFIC HEAT OF SOLIDS. 77 
 
 the atoms inside the sohd are so large that the corresponding 
 vibrations are of very high frequency, and so normally possess 
 very httle energy. As far as pure theory goes, there is no 
 question that to the terms in the specific heat contemplated by 
 Nernst's theory (§ 58), there ought to be added an additional 
 term, of a form exactly similar to the Einstein term, but having 
 
 a;=^^^ . where v^ is the frequence- (or average frequency) of the 
 
 vibrations which depend on the rotation of the atoms.* 
 
 It is worthy of notice that sodium and mercury show an 
 increase, be3'ond that accounted for by the theories we have 
 considered, in the specific heats as the fusion-point is ap- 
 proached,! when I resumably the intensity of the iorces which 
 prevent the atom from rotating is relaxed, and Nernst and 
 Lindemann find that in general the same is true for the 
 substances they have examined. 
 
 But these deviations from the simple theory are, in normal 
 cases, too small to be detected experimentally, and it may be 
 stated as a general rule (not appUcable close to the fusion- 
 point) that the motion of translation of the atoms suffices to 
 account for the observed specific heats. 
 
 60. Debye's result (§§ 53, 54) may be summarised as 
 follows : 
 
 The whole heat-energy of an element resides in the energy 
 of its elastic-sohd vibrations, the atoms bemg treated as the 
 particles of the sohd, and each vibration having exactly the 
 energy allotted to it by the quantum-theory. Nernst has 
 shown that for the compounds examined by him the same is 
 true except that the molecules must be treated as the particles 
 of the sohd, and to the energy of the elastic vibrations of the 
 sohd must be added the energy of the internal vibrations of the 
 molecules, each vibration agam having exactly the energy 
 allotted to it by the quantum-theory. 
 
 Both these theories, as well as the theories of Bom and 
 Karman, and of Thirring, are covered by a general explanation 
 that the heat-energy of a solid is the energy of the vibrations 
 of the atoms of the solid, each vibration having exactly the 
 energy allotted to it by the quantum-theory, and therefore 
 
 * See Griineisen, " Molekulartheorie der Festen Korper," report pre 
 eented to the second Sol-vay Congress of Physios (Brussels, 1913) (not yet 
 published), also A. E. Oxley, " Proo." Camb. Phil. Soc, 17 (1914), p. 450. 
 
 t Oxley, i.e. 
 
78 RADIATION AND THE QUANTUM-THEORY. 
 
 having the same energy as a light vibration of identical fre- 
 quency. 
 
 This recalls the result obtained in § 9. that the condition for 
 equilibrium between a material vibration and light vibrations 
 in the ether is the equahty of mean energy of vibrations of the 
 same frequency. This result was, in § 9, obtained from the old 
 mechanics : the specific heat phenomenon suggests that it may 
 be true in the quantum-mechanics also. The physical bearing 
 of this will be discussed in the next chapter. 
 
CHAPTEE VII. 
 
 ON THE PHYSICAL BASIS FOR THE QUANTUM- 
 THEORY. 
 
 61. The foregoing chapters have given a summary of the 
 priacipal pieces of experimental evidence bearing on the 
 quantum-theory, and a statement of some ot the mathematical 
 analysis connected with it. It has by no means been found 
 possible to represent the quantum-theory completely by a set 
 of mathematical equations or of physical concepts. The 
 indications are that there is, underlyiag the most minute pro- 
 cesses of nature, a system of mechanical laws difierent from the 
 classical laws, expressible by equations in which probably the 
 quantum-constant h plays a prominent part. But these 
 general equations remain unknown, and at most all that has 
 been discovered is the main outhne of the nature of these 
 equations when appUed to isochronous vibrations. 
 
 Even if the complete set oi equations were known, it might 
 be no easy task to give a physical interpretation of them, or to 
 imagine the mechanism from which they originate. When the 
 equations themselves are still so incompletely known this task 
 becomes yet harder, and so Uttle progress has been made in 
 this direction that an attempt to report on it reduces mainly 
 to a chronicle of conflicting opinions. 
 
 62. The quantum-theory is beheved to have disclosed in 
 nature a certain atomicity of a kind unsuspected by the older 
 mechanics. In the form in which the quantum-theory was 
 originally put forward by Planck, this atomicity was in effect 
 an atomicity of energy, although this conception was not 
 exactly used by Planck. But the atomicity was dependent on 
 the frequency of the energy, and seemed to have httle or no 
 meaning except in reference to absolutely imonochromatic 
 vibrations. It is more natural to suppose that the real atomi- 
 city, if it exists, is that of the entity measured by h, or by some 
 fmiction of h and constants of nature. The constant Ji is 
 physically of the dimensions of energy multiplied by time, or 
 action. But an attempt to imagine a universe in which action 
 
80 RADIATION AND THE QUANTUM-THEOR"S . 
 
 is atomic leads the mind into a state of hopeless confusion. The 
 classical equations can be put in the form 5A=0, where A is 
 the action and is supposed to be capable of continuous varia- 
 tion. If A were not supposed capable of continuous variation, 
 but were limited to being a multiple of a unit of action, h, it 
 might be possible to construct the true system of equations, 
 including the quantum-theory as a special case. These 
 equations would, of course, reduce to an expression of the 
 principle of least action when the number of units h in the total 
 action was very great, and so would also include the Newtonian 
 mechanics as a special case. No progress towards the solution 
 of this general problem has been made as yet. Mention must, 
 however, be made of Papers by Sommerfeld,* in which a theory 
 
 of an " element of action," — or j, is applied to the explana- 
 
 tion of X-rays, 7-rays and the photo-electric effect. These 
 serve to suggest the type of general theory which may, perhaps, 
 be hoped for in the future, but if so, they at the same time 
 indicate how very far the physical interpretation must be from 
 anything at present imagined. 
 
 The quantity h ox—- is also of the same physical dimensions 
 
 as angular momentum, so that if we could imagine a state of 
 things in which angular momentum were atomic we might be 
 in the right state of mind for attempting a physical explanation 
 of the quantum-theory. It is to be noticed that Bohr's theory 
 implies that the angular momentum in an atom with only one 
 
 electron is always a multiple of -— , and the brilUant agreement 
 
 of this part of his theory with experiment may indicate that 
 in these cases the angular momentum of the single electron 
 certainly behaves as though it were atomic, but this does not 
 carry us any perceptible distance towards a physical explana- 
 tion of why this atomicity exists. 
 
 The quantity hN, or n~V, is of the physical dimensions of the 
 
 square of an electric charge. In point of fact, ^r— V, if not exactly 
 
 27!: 
 
 equal, is almost equal, to (4jie)^, i.e., to the Square of the 
 
 * Sommerfeld gives an admirable summary in his report to the first Solvey 
 Congress at Brussels (1911). " La Theorie de Rayoimement et les Quanta," 
 p. 313. 
 
PHYSICAL BASIS FOR THE QUANTUM-THEORY. 81 
 
 strength of a tube of force binding two electrons. This sug- 
 gests that the atomicity of h may be associated with the atom- 
 city of e. The atomicity of e will not lead to the quantum- 
 theory, otherwise the quantum-theory would have been fully 
 developed long ago, but there is, perhaps, a hope that the two 
 atomicities may be special aspects of some principle more 
 general than either of them. It must be remembered that the 
 atomicity of e has never received a physical explanation ; it 
 is in no way implied in the classical equations of Maxwell, and 
 no reason is known why an electron with a charge ^e should not 
 exist. Any attempt to refer back the atomicity of e to the 
 structure of the ether simply discloses the fact that the funda- 
 mental equations of the ether are not yet fully known ; it 
 imphes that if they were fully Imown they might be expected 
 to contain the quantity e, and this is perhaps the same thing 
 as saying that they would contain the quantity h. It may be 
 that if the equations of the ether were fully known they would 
 
 be seen to involve the quantum-theory. If r-V is the same 
 
 thing as {ineY, an attempt to give a physical explanation of 
 the quantum-theory might be based on the atomicity and 
 possible discrete existence of tubes of force of strength 47ie, 
 ideas with which we have been made familiar in the writings of 
 Sir J. J. Thomson.* These ideas seem to many physicists to 
 be at variance with experience, but they have to their credit 
 that they give a natural and simple explanation of the electro- 
 kinetic momentum in the ether,t such as I believe cannot be 
 given by any other series of physical conceptions. 
 
 63. It may, perhaps, be accepted that it is yet too early to 
 expect an explanation, in anything like fundamental terms, 
 of the meaning of the quantum-theory, and we may attempt 
 merely to understand some of the physical processes imphed 
 in the established results of this theory. Even as regards this 
 preliminary problem there is no general agreement. 
 
 Physically, the quantum-theory imdoubtedly centres round 
 the idea of spasmodic or instantaneous interchanges, losses or. 
 gains of energy of amount e^=hv. This idea involves the 
 conception of two receptacles of energy, but the quantum- 
 theory gives no indication as to what these receptacles are, or 
 when or why they lose or acquire energy. 
 
 * " Recent Researches in Electricity and Magnetism " (1892), Chapter I. 
 and later books. 
 
 •f " Elements of Electricity and Magnetism," 3rd Edition (1904), p 279. 
 
 G 
 
H'j! radiation and the quantum-theory. 
 
 In Planck's original theory,* it was supposed that all matter 
 contained large numbers of resonators of different frequencies ; 
 these were capable only of holding energy in quanta, and the 
 spasmodic interchanges of energy took place between different 
 resonators. The theory was open to somewhat obvious 
 objections which have been repeatedly urged against it. In 
 the first place an Laterchange of energy between resonators of 
 the same frequency does nothing to advance the progress 
 towards the final steady state, while interchanges of energy 
 between resonators of difierent frequencies are in general 
 impossible owing to the incommensurabihty of the quanta 
 corresponding to difierent frequencies. This difficulty might 
 be met by supposing that there is some other receptacle for 
 energy, in which the energy is not hmited to existence only in 
 complete quanta as, for instance, the ether or possibly free 
 electrons. This conception, however, is not part of Planck's 
 original theory, and is certainly open to many objections. 
 
 The specific heat formula of Debye imphes that in a sohd 
 there are no , receptacles of energy except the elastic-sohd 
 vibrations of the matter, and the ether. It would (c/. § 59) 
 be straining the evidence to conclude that there is no possi- 
 bihty of any further receptacles of energy in matter, but it is 
 significant that Planck's assumed resonators have given no 
 direct indication of their existence. Resonators of infinitely 
 varied frequencies, as assumed by Planck, would imply an 
 infinite variety of atoms or molecules in the sohd unless they 
 are of such a scale as to involve relations between different 
 atoms or molecules, in which case they merge into the elastic- 
 sohd vibrations of Debye. Resonators of a few definite fre- 
 quencies are not what is required by Planck's theory, and their 
 existence seems to be put out of question from their not pro- 
 claiming themselves spectroscopically. 
 
 Perhaps Planck's original theory seems best capable of 
 coming into line with the known facts of physics if his resona- 
 tors are definitely identified with the elastic vibrations of the 
 sohd. This identification removes one objection which has 
 frequently been brought against Planck's theory. The first 
 part of his proof consists in showing that in equilibrium the 
 mean energy of a resonator must be equal to the mean energy 
 of an ether-vibration of the same frequency. Planck bases 
 this proof on the classical mechanics ; in fact, his proof is in 
 
 ♦ " Annalen der Physik," 4 (1901), p. 553 
 
PHYSICAL BASIS FOE THE QUANTUM-THEORY. 83 
 
 essence the same as that given in § 9 of the present report. 
 Now, it is easy to point out that it is inconsistent to assume, as 
 Planck does in the first part of his theory, that the energy of 
 the resonators can change continuously according to the 
 classical laws, and afterwards to assume that the energy can 
 only change by complete quanta. If, however, the resonators 
 are interpreted simply as the elastic-vibrations of the sohd 
 then the equality of the mean energies in matter and ether may 
 be treated as an experimental fact ; the supposition that the 
 energies are equal is the basis of Debye's theory of specific 
 heat, and is fully confirmed by observation. The fact that 
 these energies are equal is perhaps (if the classical dynamics 
 are no longer valid) a fact which still demands explanation, but 
 there is no question that it is a true fact. But even if the 
 resonators are identified with the elastic- vibrations, we are no 
 nearer to discovering a mechanism which will make it possible 
 for the energies to change only by complete quanta — indeed, 
 the proposed identification makes any such hope seem in- 
 finitely remote. And it must also be noticed that the fre- 
 quencies of the " resonators " provided by Debye's elastic- 
 sohd vibrations are limited to the infra-red end of the spectrum, 
 whereas Planck's theory demands resonators of frequencies 
 ranging through the whole spectrum. 
 
 In the later forms assumed by Planck's theory it is supposed 
 that the resonators emit their radiation in complete quanta, 
 but that they absorb it in a continuous manner. Thus the 
 energy of a resonator at any instant may be anything : it is 
 not restricted to being an exact multiple of a quantum. This 
 idea is attractive physically, for it minimises the divergence 
 from the older electromagnetic theory, and is capable of recon- 
 cihation with the undulating theory of Ught, but it is question- 
 able whether it is really consistent with the quantum-theory 
 itself. If the energy of a resonator can vary continuously, we 
 must not deduce the radiation-formula from the supposition 
 that it must be an integral multiple of a quantum ; and 
 Poincare's analysis (§ 23) seems to indicate that Planck's 
 radiation formula can only be true if the energy of the radiation 
 cannot vary continuously. 
 
 It may be remarked that Planck's physical theories deal 
 only with one special mechanism of radiation, whereas the 
 considerations mentioned in § 8 of the present report indicate 
 that there must be many mechanisms, and that each one 
 independently must lead to Planck's formula. Thus the 
 
84 RADIATION AND THE QUANTUM-THEORY. 
 
 physical problem apparently cannot be solved by attributing 
 special properties to Planck's radiators ; what is ultimately 
 needed is a new system of mechanics which wiU endow all 
 radiators with these properties. 
 
 That this new mechanics must be of a very comprehensive 
 nature becomes evident on considering the result obtained in 
 § 10 of the present report. It was there seen that free electrons 
 and ether, interacting according to the classical djmamics, 
 would set up Lord Eayleigh's law of radiation. It is, therefore, 
 evident that the interaction between the ether and free elec- 
 trons cannot be governed by the classical mechanics : the new 
 mechanics must differ from the old even as regards the motion 
 of free electrons. 
 
 For this reason it seems useless to attempt to explain away 
 the conflict between the radiation -laws and the classical 
 mechanics by ingeniously devised special models of atoms, 
 or special detailed mechanisms of emission of radiation, which 
 might seem, while obeying the classical laws, to give something 
 approximating to Planck's law. Any such attempt would 
 first have to surmount the difi&culty that any system whatever, 
 if it obeys the classical laws, must also in its state of thermo- 
 dynamical equiUbrium obey th^ Jaw of equipartition of energy, 
 which is known in turn to lead to Eayleigh's radiation formula 
 (§ 17). And if this difficulty could be turned, as, for instance, 
 by postulating a final state which was not one of thermody- 
 namical equihbrium, the question of why all possible mechan- 
 isms of radiation lead to Planck's law, as they certainly appear 
 to do (§ 8) would remain untouched. And finally, if single 
 electrons do not obey the classical laws, there would seem to be 
 Uttle gain in proving, if it could be proved, that complicated 
 structures might possibly obey them. 
 
 64* The mechanism of the emission of radiation such as. 
 figures in the line-spectrum of a gas is very clearly pictured b}- 
 Bohr's theory, and the marked agreement of this theory with 
 observation certainly raises a strong presumption that his 
 premisses contain at least some truth. It is difficult to see 
 how his final equations could stand at all unless it is assumed 
 that the emission of radiation is in complete quanta, and is an 
 emission of monochromatic hght. Absolutely instantaneous 
 emission is not in any way required, but the emission must be 
 catastrophic in the sense that when once it has started nothing 
 can stop it until a whole quantum has been set free. 
 
 Again the whole evidence of the photo-electric effect is that, 
 
PHTSIOAL BASIS FOR THE QUANTUM-THEORY. 85 
 
 in this phenomenon at least, the absorption must be in com- 
 plete quanta, and moreover that the energy of each quantum 
 must be in such a concentrated form that it can all be absorbed 
 by one atom. 
 
 The whole of the phenomena associated with the quantum- 
 hypothesis could probably be given a consistent physical 
 explanation if we were free to imagine radiation to be atomic 
 in its structure, consisting of indivisible quanta of radiation, 
 each monochromatic and highly concentrated in space. The 
 quanta might, perhaps, almost be thought of as corpuscles of 
 radiation, travelling through space with the velocity of light, 
 emitted and absorbed as complete entities, and entirely in- 
 capable of any kind of sub-division. Ideas of this kind con- 
 stitute Einstein's theory (see above, §44) of " light-quanta." 
 
 Somewhat similar ideas had been formulated before the rise 
 of the quantum-theory, in particular by Sir J. J. Thomson,* 
 who regarded the Faraday tubes as having discrete existence 
 in the ether. This train of ideas is discussed by Whethamf as 
 follows : — 
 
 " Faraday's tubes, it is clear, give a very powerful and con- 
 venient method of studying the phenomena of the electro- 
 magnetic field, and indications are not wanting that they 
 represent something more than a useful mathematical fiction. 
 If the structure of the electric field be discontinuous in reality, 
 as our tube-picture oi it indicates ; if the electric and magnetic 
 effects of a charge of electricity are in reality exerted through- 
 out the surrounding space by means of discrete tubes of force- 
 vortex filaments in the ether, or whatever they may actually 
 be, an advancing wave of hght must be discontinuous also. 
 Could we look at such a wave from the front and magnify it 
 millions of times we should see not a uniform field of illu- 
 mination but a number of bright specks scattered over a dark 
 ground. Each tube of force would convey its own tremors, 
 and these would constitute light, but between them would lie 
 undisturbed seas of ether. 
 
 " Such an idea about the nature of a wave-front of light is 
 very unexpected and surprising. We are inclined at once to 
 relegate our tubes ot force to a museum of conceptional curi- 
 osities. But it is a remarkable thing that certain evidence in 
 favour of the discontinuous nature of a wave-front really does 
 exist." 
 
 * See, in particular, Camb. PMl. Soc. " Proc," 14 (1907), p. 417. 
 t " Experimental Electricity " (1905), p. 207. 
 
86 RADIATION AND THE QUANTUM-THEORY. 
 
 The evidence referred to arose from a study of the ionisation 
 of gases by X-rays. It was noticed that the ionisation was 
 such as would be expected if the X-rays did not spread out in 
 accordance with the undulatory theory of hght, but moved 
 with their energy concentrated as described.* It is now 
 beheved that the ionisation produced by the X-rays is produced 
 by the intermediary agency of fi-ia.js, and this somewhat 
 alters the problem. The question now becomes how the X- 
 rays, if they are not concentrated, can produce ;6-rays as they 
 do, and the problem really becomes similar to that of the photo- 
 electric effect, discussed in § 48 of the present report. In any 
 case the phenomena of the production of p-Tajs by X-rays and 
 the emission of electrons by ultra-violet light both suggest very 
 strongly the possibihty of radiation being propagated in the 
 form of concentrated " atoms " or quanta. 
 
 65. This view, however, has been unable to gain acceptance 
 owing to its obvious conflict with the well-established prin- 
 ciples of the undulatory theory of hght. For instance, Prof. 
 Lorentz says : — "f 
 
 " Now it must, I think, be taken for granted that the quanta 
 can have no individual and permanent existence in the ether, 
 that they cannot be regarded as accumulations of energy in 
 certain minute spaces flying about with the speed of light. 
 This would be in contradiction with many well-known pheno- 
 mena of interference and diffraction. It is clear that, if a beam 
 of hght consisted of separate quanta, which, of course, ought 
 to be considered as mutually independent and unconnected, 
 the bright and dark fringes to which it gives rise could never 
 be sharper than those that would be produced by a single 
 quantum. Hence, if by the use of a source of approximately 
 monochromatic light, we succeed in obtaining distinct inter- 
 ference bands with a difierence of phase of a great many, say 
 some miUions, of wave-lengths, we may conclude that each 
 quantum contains a regular succession of as many waves, and 
 that it extends therefore over a quite appreciable length in the 
 direction of propagation. Similarly, the superiority of a 
 telescope with wide aperture over a smaller instrument, in so 
 far as it consists in a greater sharpness of the image, can only 
 
 * See CampbeU, " Modem Electrical Theoiy," 1st Edition (1907), p. 226. 
 
 t Discussion on " Radiation " at the Birmingham Meeting of the British 
 Association (1913). See also a short article, also by Lorente, " Die Hypo- 
 these der Lichtquanten," " Phys. Zeitsch.," 11 (1910), p. 349. 
 
PHYSICAL BASIS FOE THE QUANTUM-THEORY. 87 
 
 be understood if eacli individual quantum can fill tte whole, 
 object-glass. 
 
 " These considerations show that a quantum ought at all 
 events to have a size that cannot be called very small. It may 
 be added that, according to Maxwell's equations of the electro- 
 magnetic field, an initial disturbance of equilibrium must 
 always be propagated over a continually increasing space." 
 
 Certainly every one of the objections stated in this short 
 paragraph seems to be insuperable. It may be added that the 
 question of the indivisible existence of quanta in the ether has 
 been tested directly, or as directly as is possible, by experiment. 
 
 If Hght occurred only in quanta, interference could only 
 occur at a point at which two or more quanta existed simul- 
 taneously. If the light were sufficiently feeble the simul- 
 taneous occurrence of two quanta at any point ought to be a 
 very rare occurrence, so that all phenomena, such as diffraction 
 patterns, which depend on interference, ought to disappear as 
 the quantity of light present is reduced. Taylor* has shown 
 that this is not the case ; he reduced the intensity of his hght 
 to such an extent that an exposure of 2,000 hours was necessary 
 to obtain a photograph, and yet obtained photographs of 
 diffraction patterns in which the alternation of light and dark 
 appeared with undiminished sharpness. In Taylor's experi- 
 ments the intensity of hght was 10-^^ ergs per cubic centimetre, 
 or about one light-quantum per 10,000 cubic cm., so that if the 
 quanta had been concentrated nothing of the nature of a 
 diffraction pattern could possibly have been observed. 
 
 Thus it appears that there is no hope of reconcilmg the 
 undulatory theory of light with the quantum-theory by regard- 
 ing the undulatory-theory as being, so to speak, only statistically 
 true when a great number of quanta are present. One theory 
 cannot be the hmit of the other in the sense in which the New- 
 tonian mechanics is the hmit of the quantum-mechanics, and 
 we are faced with the problem of combining two apparently 
 quite irreconcihable theories. 
 
 66. A question intimately involved in any attempt to find 
 a physical basis for the quantum-theory, is that of the degree 
 of reality or substantiality with which the ether must be sup- 
 posed endowed. The modern school of British physicists have 
 been inchned to regard the ether as a completely real substance 
 — indeed to some of them the ether is the primary real substance 
 
 * " Proe. Camb. Phil. Soc," XV. (1909), p. 114> 
 
«» RADIATION AND THE QUANTUM-THEORY. 
 
 of the universe. At the other extreme stands the relativity 
 school, who alnaost deny that the ether has any reality at all. 
 There are innumerable intermediate positions between these 
 two extremes. 
 
 For the physical interpretation of the quantum-theory the 
 precise question which is of importance is whether the ether 
 has so much of substantiality that it must be treated as part 
 of the dynamical system in connection with which it occurs. 
 To the relativity school, to speak of the " dynamics of the 
 ether " would be as meaningless as to speak of the " dyn:;mics 
 of space." ""■ 
 
 If the ether is part of the dynamical system, then the e,'^''' ^ 
 in the ether must be treated as part of the energy of the sySifem, 
 and the waves or vibrations which are possible in the ethe'f will 
 represent just so many additional degrees of freedom o'^the 
 whole system. The final state of this system can be tr?;:, jd 
 as a problem in statistical mechanics, and to arrive at Plaflfflc's 
 formula it woi'ld appear (cf. § 26) to be necessary to suppose 
 that the vibrations in the ether themselves gain or lose energy 
 by whole quanta. This is, of course, in accordance with the 
 suppositions of Bohr's theory, and with the evidence of the 
 photo-electric effect. It is worth noticing that this in itself 
 does not imply any departure from Maxwell's equations for free 
 ether, because these equations are not concerned with the inter- 
 change of energy between vibrations of different wave-lengths 
 in the ether. Such an interchange requires the presence of 
 matter in some form or other, and an interchange by quanta 
 would require only a modification of the classical equations for 
 the case in which ether and matter are present together. On 
 the other hand, the photo-electric effect seems to demand a 
 concentration in space of the energy of the quanta, and this 
 certainly does demand a modification of Maxwell's equations 
 even for free ether. For Maxwell's equations indicate that 
 the energy ought to spread out in all directions in space, and 
 under these equations no quantum could remain in a concen- 
 trated form. 
 
 67. If the ether is regarded as too devoid of substantiality, 
 to be treated as part of the dynamical system, it is necessary 
 to fall back on to the changes of energy in the mechanism by 
 which the radiation is produced. Almost all the Continental 
 physicists have regarded the quantum-theory from this point 
 of view, which has the disadvantage that it is necessary to 
 specify exactly the mechanism of radiation. Whatever this 
 
PHYSICAL BASIS FOR THE QUANTUM-THEORY. R9 
 
 mechanism may be, and however many types of mechanism 
 there may be, it is necessary that the energy of each should 
 jump by quanta. So far no success has attended any explana- 
 tion of why the energy of any type of mechanism should be 
 incapable of changing except by quanta : assuming the fact, 
 of course, the required results following, but the difficulty 
 lies in making the assumption of the fact seem physically 
 plausible. 
 
 68 Whatever role is assigned to the ether, the main 
 phy cal difficulty about the quantum-theory at present seems 
 is' ij the apparent impossibility of reconcihng it with the 
 est^ ;iished results of the undulatory theory of hght. The 
 expfrimental evidence — for instance, of the photo-electric 
 effe . and of interference — seems almost to indicate that both 
 tl ies are true simultaneously : that radiant energy both 
 rej' iins concentrated and indivisible, and at the same time 
 spreads and is divisible : the idea that these two phenomena 
 may be concerned with two totally different kinds of radiation 
 seems at present too fantastic for serious consideration. In 
 any case it may be asserted with confidence that until some 
 kind of reconciliation can be effected between the demands ox 
 the quantum-theory and those of the undulatory theory of 
 light, the physical interpretation of the quantum-theory is 
 likely to remain in a very unsatisfactory state. Probably the 
 hope of most physicists is that some sort of a compromise may 
 ultimately be effected, but at present any practical attempt at 
 a compromise appears to require the abandonment of some- 
 tiiing which is essential to one or other of the two theories. 
 And it must be remembered that any hope of a compromise is 
 to a very large extent excluded by Poincare's arguments, 
 already summarised in § 27 of the present report : the explana- 
 tion of the black-body spectrum demands the quantum-theory 
 and nothing but the quantum-theory, all the discontinuities 
 of the theory and their surprising physical consequences in- 
 cluded. The keynote oi the old mechanics was continuity, 
 natwa non facit saMus. The keynote of the new mechanics 
 is discontinuity ; in Poincare's words : — 
 
 " Un systeme physique n'est susceptible que d'un nombre 
 fini d'etats distincts ; il saute d'un de ces etats a I'autre sans 
 passer par une serie continue d'etats intermediares. " 
 
 The antithesis is obvious ; its resolution will not be easy. 
 Perhaps the present report cannot end better than by a free 
 
 H 
 
90 RADIATION AND THE QUANTUM-THEORY. 
 
 translation of Poincare's concluding remarks in his striking 
 article, " L'hypothese des Quanta ": — * 
 
 " We see now how this question stands. The old theories 
 which seemed until recently able to account for all known 
 phenomena have suddenly met with an unexpected check. 
 Some modification has been seen to be necessary. A hypo- 
 thesis has been suggested by M. Planck, but so strange a hypo- 
 thesis that every possible means was sought for escaping it. 
 The search has revealed no escape so far, although the new 
 theory bristles with difficulties, many of which are real and 
 not simple illusions caused by the inertia of our minds, which 
 resent change. 
 
 " It is impossible at present to predict the final issue. Will 
 some entirely different solution be found ? Or will the advo- 
 cates of the new theory succeed in removing the obstacles 
 which prevent us accepting it without reserve ? Is discon- 
 tinuity destined to reign over the physical universe, and will 
 its triumph be final ? Or will it finally be recognised that this 
 discontinuity is only apparent, and a disguise for a series of 
 continuous processes. The first observer of a colUsion thought 
 he was witnessing a discontinuous process, but we know to-day 
 that what he saw was the result of changes which, although 
 very rapid, were continuous. Any attempt at present to give 
 a judgment on these questions would be a waste of paper and 
 ink." 
 
 • " Dernieres penaees," H. Poincar^ (Flammarion. Paris. 1913) 
 
■Y^^ 
 
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