THE PHYSICAL SOCIETY OF LONDON. REPORT ON RADIATION AND THE QUANTUM-THEORY. BY J. H. JEANS. SECOND EDITION. Price, 7s. 6d. net, post free 7s. 10d. Bound in cloth, 10s. 6d., post free 10S. 10d. LONDON: FLEETWAY PRESS, LTD., 3-9 DANE STREET, HIGH HOLBORN, W.C.I. 1924. LONDON: THE FLEETWAY PRESS, LTD., Dane Street, Holborn, W.C.I. 530.12 J34r 1924 PREFACE TO THE SECOND EDITION. When the first edition of this report was written in 1914, the Quantum-Theory was but little understood in this country, and in so far as it was understood, was an object of suspicion to most physicists. These considerations determined the form assumed by the original report; it had to be an apologia as well as an exposition. In preparing a second edition I have retained the arrangement and the main bulk of the old report. Such omissions as have been made have been prompted by the feeling that the Quantum-Theory need no longer be considered on the defensive ; � the additions, which are much more numerous than the omissions, represent merely the filling in of the old framework which has been necessitated by the rapid growth of the theory. No attempt has been made to expand the report into a complete treatise� nothing less than ten times the space available would have sufficed for this. In particular, I have been content to treat questions of atomic structure and spectral lines only in the very broadest way ; the recent English translation of Sommerfeld's "Atombau und Spektrallinien" would have made any detailed treatment of these questions in the present report superfluous. I am indebted to Dr. D. Owen for help in reading the proofs and for the preparation of the index which accompanies the report. J. H. JEANS. 552471 CONTENTS. Chapter Page I.�Introductory: On the Need for a Quantum-Theory............ 1 IL�The Radiation Problem, according to the Classical Mechanics....................................................................... 6 III.�The Development of the Quantum-Theory......................... 19 IV.�The Spectra of the Elements and the Emission of Radiation 32 V.�The Photo-electric Effect and the Absorption of Radiation 45 VI.�The Specific Heat of Solids............................................... 51 VII.�The Dynamics of the Quantum-Theory................................ 61 VIII.�The Physical Basis for the Quantum-Theory..................... 78 Index.......................................................................................... 85 CHAPTER I. ON THE NEED FOR A QUANTUM-THEORY. Introductory. 1. The quantum-theory, which is now found to be applicable 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., and again in Chapter VII.), the photoelectric effect (Chapter V.), and the specific heats of solids (Chapter VI.). Chapter VII. will report the progress made towards unravelling the general system of quantum dynamics, after which a final chapter (Chapter VIII.) will be 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 mass as resisting explanation in terms of these laws, and the conviction that something outside the Newtonian laws is needed for their explanation has now become firmly established. 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 throughout 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 demonstrating 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-established phenomenon of Nature, then the logical necessity of rejecting the impossibility becomes unanswerable. The phenomenon which provides the simplest crucial test as to the universal validity of the Newtonian mechanics is found to be the following : When radiant B 2 Radiation and the Quantum-Theory, energy is in temperature-equilibrium with matter, the total radiant energy per unit volume is finite, and not infinite. We shall see almost at once that this phenomenon 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-equilibrium with matter, or that energy dissociated from matter is a figment of the imagination. These possibilities must, of course, be discussed. It may be argued that the mathematical proof is open to question : the proof was originally given by Poincare, and the essential validity of his mathematical reasoning has never been seriously challenged, but this cannot cany absolute conviction. 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 arguments. 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 Newtonian laws have undoubtedly 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, 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 necessarily 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 the next chapter the mathematical analysis is given, but both in the present and the succeeding chapters an effort has been made to arrange the report so that the mathematical parts may be omitted, without losing the thread of the argument, by those readers whose interest centres mainly in physical ideas rather than in abstract reasoning. 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 in temperature-equilibrium with matter is finite, and not infinite. To understand the meaning of this phenomenon, and at the same time 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 iron at 0�C, and let us suppose that there is a state of equilibrium inside this enclosure. The iron is continually radiating energy from its surface into the space inside the enclosure, and is also absorbing energy from this space. 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 absorbing paint, then in point The Need for a Quantum-Theory. 3 of fact each square centimetre of surface radiates 3 X105 ergs of energy per second into the surrounding space, and also absorbs 3xl05 ergs per second of radiation falling on to it from this space. The energy in those parts of the enclosure which are not occupied by matter is of density 4 X 10~5 ergs per cubic centimetre; the heat energy in the iron is of the order of 8xl09 ergs per cubic centimetre, or 2xlOu times the energy density of the unoccupied space. The heat energy of the iron, as we knowT, resides in the oscillations of its atoms, each atom moving with an average velocity of about 30,000 cm. per second, and we have seen that after thermo-dynamical equilibrium has been attained, practically all the energy inside the enclosure will reside in these atomic oscillations. 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 in which is floated a system of corks (to represent 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 practically 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 equilibrium, 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 simplify the 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 will be -|NRT, and if there are n spheres, the energy of their Brownian movements will be f^RT, where R is the gas constant. The total energy in the steady state will be (|N+fw)RT, and T will be determined from the condition that this quantity must be equal to the total energy of the original oscillations of the spheres. Since N, the number of molecules, will be enormously large compared with n, the number b2 4 Radiation and the Quantum-Theory. f spheres, it is clear that practically all the energy will be contributed by the term NRT. By the time the steady state is reached the energy is almost entirely trans-erred 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 application 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 (cf. Chapter VI. below). If the molecular 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 4:7zKn-*dX ...........(1) 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 X and X+dX 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 between adjacent molecules. But a good approximation can be obtained by supposing that the formula holds accurately from A=oo to a limiting wave-length Xm, and that there are no waves at all of wave-length shorter than Xm. The total energy is then 4ttRTA-^=� y4>.......(2) and Xm can be found by equating this to the total energy of the molecules. Of the energy under 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 2Xm, while seven-eighths is of wave-length intermediate between this and Xm. In a chamber rilled with ordinary air Xm will be of the order of 10~7 cm. Only one-millionth part of the total energy will be of wave-length greater than 10 ~5 cm. Thus we may say that the energy tends to run almost entirely into the shortest wave-lengths which are possible in the medium. 6. This special result is onfy one case of a quite general law. In all known media there is a tendency for the energy of any systems moving in the medium to be transferred to the medium and ultimately, when a steady state has been reached, to be found 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 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. 7. It may perhaps be objected that the cases we have considered are not analogous to the radiation phenomenon, in that our corks and shot have been immersed in a material medium, whereas our radiating mass of iron is immersed * Phil. Mag. 17, p. 229, Feb. (1909). The Need for a Quantum-Theory. 5 perhaps in non-material ether and perhaps in nothing at all. Thus, it may be argued, the difference in the cases we have been discussing may perhaps not be one of obedience or disobedience to the classical mechanics, but of the existence or nonexistence of a surrounding medium. Our arguments may not prove the breakdown of the classical mechanics, but merely the non-existence of the ether. It is greatly to be wished that the question of the existence or non-existence of the ether could be settled by such simple considerations ; unfortunately it cannot. The theory of relativity in effect requires that it shall be impossible to decide as to whether an ether exists or not, either by these or by any other purely mechanical considerations ; the equations of radiation and absorption of energy are precisely the same whether the energy is radiated into, and absorbed from, an ether or empty space. The analysis we shall now give will show that the existence or non-existence of an ether is wholly irrelevant to the question, so that if our analogies break down, it is not on the question of the reality of the ether. Leaving analogies behind, we proceed to discuss the real physical problem at issue. CHAPTER II. THE RADIATION PROBLEM, ACCORDING TO THE CLASSICAL MECHANICS. 8. Any formula which is to represent the partition of radiant energy in a steady state of dynamical equilibrium with matter must be derived by expressing that the amount of energy radiated by the matter is equal to the amount absorbed. If T)dl is the law of partition of energy according to wave-length, there must be an equation of the form where E stands for the rate of emission of radiation by the matter and A for the rate of absorption ; and the steady state is determined by the condition ^�=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 the famous Paper* in which the conceptions of the Quantum-Theory were first introduced, supposed that the emission and absorption were accomplished 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 ER, AR. 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, AE. A further agency is the photo-electric effect; when an 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 EP, AP. Treating all other agencies in the same way, we shall have an equation of the form �\ ^=(EB-AB)+(EB-AE)+(Ep-AP)+.......(4) The absorptions Au, AE, 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 us then put AR=BR9, &c. In general, the emissions E&, 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=FB)+GjEl9, &c., in which any or all of the G's may vanish. We then have ^==(Fa+GB9�Bfi9)+(FE+GB9�BE9)+(Fp+GP9�BP9)+ . � and the equation giving the steady state, obtained by equating the right-hand member of this equation to zero, is FR + FE + FP + .......(5) (Bn�GR) + (BE� GE)+(BP� GP)+ ..." It is to be expected that the ratios of the different terms, both in the numerator and * Annalen. d. Physik, 4, p. 553 (1901). The Radiation Problem. 7 denominator, will vary from substance to substance. For instance, FE and BB � GE will be small for insulators or bad conductors in which there are few electrons, Fr, and B^�GPt will be small for substances which possess few resonators of the frequency under consideration, and FP and BP�GP will be small for substances hi 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 FR, FE, Fp, &c, may be, it is necessary, as a matter of algebra, to have giving at once ER=AB, EE=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 fpom its true value, while the fraction corresponding to this agency in equation (6) might have a value widely different from 9. The numerator and denominator wpuld 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 final steady state energy should continuously flow in a closed cycle through the different mechanisms in turn, a possibility which may be excluded from serious consideration. We 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, and let its motion be assumed, for the present, to obey the classical laws. Let its frequency be fifon, and let its kinetic and potential energies, T and V/be supposed given by T=4|89a. V=4a92, where 9 is: the co-ordinate' expressing the state of the resonator arid & &ii& ft are' constants such that p2=a/fi. 51 The equation 'of motion, expressing the change in 9, is Pi+a9 = Q>K .........: .z (7), where $M ...........(8) * Cf. Poincare, Journ. de Phys., 11, p. 34 (1912). 8 Radiation and the Quantum-Theory, The solution of equation (7) can be written down in the usual way.* Suppose that at time �=0, the values of 9 and 9 are the same as if the resonator were performing an oscillation specified by 9=A cos (pt�s). The impulse (JPfit, acting through the small interval it at time t', produces a superimposed oscillation such that the value of 9 initially is - Q>t>it, and, therefore, such that the subsequent value of 9 is - Jt'=o and the total absorption in time t, as given by expression (8), is j <$}yit=\\ ( OVOV cos p(t'-l")it'it" + f A cos (pt-e) <$>it. Jo f>J oJ 0 J 0 On summing over a great number of oscillations, the second term on the right may be neglected, and the equation written in the form f Oyit^^f j �t>�rcosp(t'-t'yt'it\.....(9) J 0 *pJ 0J 0 the divisor 2 appearing because the integration is over a square in the t't" plane instead of over the half-square bounded by the line t"~t\ The value of t>$inpt'it'...........(11) Jo and in terms of these coefficients equation (9) may clearly be expressed in the form ( 0>9^1 (F/+G/) .......(12) But from a theorem given by Lord Rayleigh,f f W=- f (F/+G/)#, Jo ^ 0 so that the mean value of where c is a constant. Assume the density of radiant energy, resolved into its ,00 constituent frequencies, to be / ~Rlj>)dp per unit volume; then the mean value Jo of X2 will be � R{p)dp, and c2 times this will be ), 71 and the mean rate of absorption of radiant energy by the resonator isr^/(^>), or If the radiating mechanism is electrical, we may take the emission in time t to be given by either of the equivalent forms I C-acp)^. pJo On integrating (10)' and (11) by parts K=-J f 'GV cos pfdf ; G�=\ f 0V sin �^V PU PU so that and the emission in time t is Cr* pJ 0 p. Thus the average rate of emission is C�-- C. where a bar over any quantity denotes that its average value is to be taken over the interval of time from 0 to t. io Radiation and the Quantum-Theory. On equating emission and absorption, the condition for a steady state is found to be 3C so that, in the steady state, the partition of radiant energy must be 3C ~2tcV ^)=^^^ *WP=^more rigorous proof and discussion will be found in a Paper in Phil. Mag.-, p. 14> Jan. (1914). The Radiation Problem. II (ii.) Oscillations parallel to the axes of x and y, each of a harmonic nature and of frequency /cw/2tz, where w=w0+V, this latter being, of couise, obtained by modifying the frequency of the incident light in accordance with the Doppler principle, the relativity modification being disregarded since the velocity of the electron will be small in comparison with that of light. According to the classical dynamics the electron will absorb light of frequency kV/2tc. The emitted light will be of different frequencies for different directions of emission; the frequency corresponding to any direction will be obtained by modifying the frequency kwj2tc 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 0=0. In any direction 6, 9 the component of velocity of the electron will be � w0 cos Q+v0 sin 6 cos (?-\-wQ sin 6 sin 9, so that the frequency of the radiation emitted in this direction will be pl%7i, where p is given by p�icwV J (V�u0 cos 6~v0 sin 6 cos 9� wQ sin 0 sin 9) .... . (16) Let the distribution of intensity of radiation in different directions be supposed given by the formula 1(0, 9) sin Odddy ; let the mean value of u02+v02+w02 averaged over a great length of time be c2, and let the proportion of the whole time in which the velocity components lie within a small range duQ, dv0, dw0, be A/f---------��-----jdu0dv0dWQ. �> ' < Then the average total radiation emitted by the electron per unit time will be ffff J^�*+V�+W�*)l{0, 9) sin 6 dOd�duQdvQdw0, where the integration is over all values of u0, v0, wQ) 0and 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 u0, v0, w0, 6 and 9 to u0, w0) 0, 9 and p by the help of equation (16) and then integrating with respect to uQ) wQ, 6 and 9. The final result is necessarily of the form* /��&*)*� �:.......m where p0 stands for kV, 2tc times the frequency of the incident light, and O 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 6f 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 R(ft0)dp0, this energy being distributed at random as regards * For details, see the Paper already referred to. 12 Radiation and the Quantum-Theory. direction. Suppose that a unit amount of energy of frequency p0 is, as the result of interaction with the free electron, replaced by an amount of energy ip of the original frequency p0, and a spectrum fF(j)Q, p)dp of scattered energy. From formula (17), F(p0, p) must be of the form and from the conservation of energy we must have 'p. ,2yp 1-V=/ F(Po>P)dp=f nf'c7/.......(18* J o J o To lro Clearly on integration this last is a function of c2 only, so that tp is independent After unit time the law of partition of the whole amount of radiation will be y>fR(P0)dp0+jdpjy(p0)F(p0, p)dp, or, arranged according to frequency py j[vR(P)+fR{p0W(p�,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 fR(p)dp, so that wre must have or WR(P) + fR(pQ)F(p0, p)dpQ=R(fi) Jo R(p)(l-v)=fR(Po)�{~o> c*)^. The solution of this integral equation will give the form assumed by R(p) in the steady state. If we put Pq=xP> ^e e(lua^on becomes so that R(xp)jR{p) must be independent of p. It follows that R(p) must be of the form R(p)=Bpn, where B and n are constants, so that the partition of energy in the ether must be given by ' R(p)dp=Bpndp. The values of the constants B and n can at once be found by comparison with formula (15), which gives the limiting form of R(fi) when p=0. Clearly we must fttti2 have n=2 and B=-^=^, so that the general value of R(j>) must be given by R{p)dp=^dp..........m The Radiation Problem. 13 Thus the result obtained from the classical dynamics is that if K(p) has this form there can be permanent equilibrium between the free electrons and radiation, but if R(ft) has any other form, there will be a continued adjustment and exchange of ^energy until R(/>) has assumed this form. 11. According to the kinetic theory of matter, the mean value of mu2 for an electron, which has been denoted by mu2, ought to be equal to RT, where R is the gas constant and T the absolute temperature of the matter. Richardson and others have determined the value of mu2 by direct experiment, and find that it is in point of fact exactly equal to RT. If we substitute this value for mu2, formula (20) becomes R(j>)dp^sp\ip.........(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 W2=RT, so that formula (14), expressing the condition for equilibrium between the resonators and radiant energy, according to classical dynamical theory, becomes R(/>)#=~W.........(22) which is identical with (21). Radiation from Electron Orbits. 12. In � 10 we considered 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 radiation at each encounter with the atoms of the matter. The problem was first considered by Lorentz,* 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, f who confirmed Lorentz's result by a different method, and showed how it could be extended 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 RT RWP=~^P2dp ........(23) which, again, is identical both with (21) and (22). 13. The above formula, giving the partition of radiant energy according to frequency, is easily transformed into one giving the partition according to wave-. * " On the Emission and Absorption by Metals of Rays of Heat of Great Wave-length," Koninklijke Akad. van Wetenschappen, Amsterdam, April 24 (1903). t Phil. Mag., p. 773, Jnne (1909), and p. 209, July (1909). 14 Radiation and the Quantum-Theory. length. Let the energy within a range dX of wave-length be v(X,T)dX ; then, since 27rV A==-----, the above formula becomes P 9(A, T)rfA=87rRT^W........(24) On comparison with formula (1) it is seen that the final partition of radiarnt energy predicted by the classical dynamics is exactly analogous to the partition of the energy of sound waves which occurs in a steady state in a gas. The sole difference lies in a multiplying factor 2, which, as we shall at once see, is readily explained by the difference between sound-waves and light-waves. 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 pjliz and the corresponding wave-length X. When the wave-length is sufficiently small, the number of free vibrations for which X lies within even a small range of values will be very great. Let us, in general, suppose that the number lying within a range dX i$f(X)dX. Then clearly, f(X) 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=CvX~*dX, where C is a constant. In other words, the number of vibrations per unit volume of the medium of wave-length between X and X+dX is C?r*dX. The determination of the constant C for any medium is not difficult.* It is found that for the vibrations of sound in a gas C=4tu, for the light vibrations in free ether, if an ether exists, C=8rc, and for the elastic solid vibrations C=127u. If we like to assume that an ether does not exist, we may fall back on the general principle to which appeal has already been made, namely, that all observable phenomena must be precisely the same as if an ether did exist, and we shall find that C==87r for the radiant energy in empty space. The reason why the three values of C (to, 8tt, and 12tz) are in the ratio 1 : 2 : 3 is obvious. In a gas there is only the one set of normal vibrations; in the ether there are only transverse vibrations, but there are two independent 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, iizKTX^dX, we note that the number of vibrations within the range dX of wave-length is k-xX-HX, and therefore the average energy of each vibration is RT. In the radiation formula SnRTX^dX ..........(25) the number of vibrations is 8-nX~*dX, so that the average energy of each vibration is again RT. Thus the classical mechanics requires that, in the steady state of equilibrium, 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. * Phil. Mag., p. 91, July (1905); H. A. Lorentz, The Theory of Electrons, � 73 ; also a purely mathematical Paper by H. Weyl, Math. Annalen, 71, p. 441. The Radiation Problem. 15 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 light vibration set up by resonators, or by free electrons radiating energy as they experience accelerations in an electric field, 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, omitting all complications which have no bearing 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 condition be determined by n Lagrangian co-ordinates and the n corresponding momenta. Let these 2n quantities be denoted by 0lt 02 . . . 02n. Then the condition of the system at any instant may be represented graphically by a point in an imaginary space of 2n dimensions, this point having for its Cartesian co-ordinates dlt d2 . . . 02n. As the system follows out its natural motion this point will trace out a curve in the 2n dimensional space. We may suppose that the whole space is filled with representative points tracing out their appropriate curves, and in this way we can imagine that we have a graphical representation which enables us to study simultaneously all 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 obliterated�the system invariably possesses some definite property P. This might be from one of two reasons : Either that the representative points tend in their motion to cluster in those regions of the generalised space in which the property P holds, or that the property P is common to the whole of the generalised space. 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 insignificant exceptions, these exceptions being 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 generalised space, except for insignificant exceptions. These somewhat abstract considerations may be illustrated by a concrete example of a non-dynamical kind. Suppose that we have an army of a million men of average height 5 ft. 8 in. This army can be divided into two wings each of 500,000 men in approximately 10301,027 different wTays. There will, perhaps, be a million ways of arranging the * Boltzmann, Vorlesungen liber Gastheorie, Vol. 2., pp. 66, 67 ; or Jeans, Dynamical Theory of Gases (3rd Edition), p 73. 16 Radiation and the Quantum-Theory. men so that the average heights in the two wings may differ by as much as 2 in. ; there may be (say) 10100 ways in which the averages may differ by TJF in. ; there will be, perhaps, 101000 ways in which the average will differ ToV^n �f an inch. But for the majority of arrangements the average heights will be almost exactly equal. With the figures we have taken the chance will be 10300'000 to 1 that the average height in the two wings will be the same to within i^V^li of an 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 10300'000 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 insignificant 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 limits 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 JRT, then the value of the group of p terms will be i/>RT, while that of the. group of q terms will be fgRT. This is, in effect, the theorem of equipartition of energy.* 17. It will at once be seen that this result embraces all the results which have been obtained in this chapter. First let the system under consideration consist of ether together with matter of all kinds. The number of vibrations of the ether of wave-length between X and X+dX will be SizvX'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 will be IQizvX-^dX, and therefore their energy will be equal to this number multiplied by JRT, or to SnRTvX'^dX. Dividing by v to reduce to energy per unit volume of ether, we obtain for the radiant energy of wavelength between X and X-\-dX ZtzKIX-HX, which is identical with formula (24). Again, appealing to the principle that it cannot possibly make any difference to observed phenomena whether an ether exists or not, we find that the same law must give the partition of radiation in empty space in an etherless universe. This formula was given by Lord Rayleighand the present author in 1900 as being the formula which ought, on the Newtonian mechanics, to govern the partition of energy in the spectrum. It cannot be the true law, for the total energy obtained by integrating from X=0 to X=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, in 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 in the form of radiation into the adjoining space, just as in the analogies of * For references to the various proofs which have been given see Jeans, Dynamical Theory of Gases, Ch. V.,orI^a Theorie du Rayonnement et les Quanta (Gauthier-Villars, Paris, 1912), p. 71. Detailed proofs of the theorem will also be found in both these places. The Radiation Problem. 17 ��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. 18. Before proceeding to discuss the development of the quantum-theory, we may draw attention to various attempts which were made, before the quantum-theory had become as well established as it now is, to escape from the necessity, imposed by the classical mechanics, that all available energy should be degraded into radiant energy of very short wave-length. The classical theory requires that this degradation should occur before a steady state of thermodynamical equilibrium is attained ; the obvious loophole of escape was the supposition that the observed partition of energy represented states which were not true states of thermodynamical equilibrium at all. If it is supposed that the emitted radiation of the highest frequencies is in some way drained off or allowed to escape, so that the density of radiation always remains very small, then it is possible to obtain a definite law, which, in certain cases, may show some of the characteristics of the observed radiation law. 19. There is, however, no longer any reason why different assumed mechanisms of radiation should all lead to the same law of partition of radiant energy, and it is, in point of fact, found that they do not. The case which has received most attention is that of the radiation emitted by accelerated free electrons describing orbits under atomic forces. It is assumed at the outset that the electron describes its orbit in a space which is almost devoid of radiant energy, and this, as is now obvious, amounts to assuming that there is no thermodynamical equilibrium between the radiant energ}' and the matter. If the acceleration of the electron is produced by collisions with atoms, it is found that instead of obtaining the Rayleigh formula SizRTX^dX...........(26) we obtain* a formula of the type 8TTRTA'4/ (j)dX.........(27) in which c is a quantity such that ��; 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/ (-) must approximate to unity, while, if the collisions are all of equal duration, it can be shown that, when X is very small, f\ - ) will tend to zero in the same way as e X Making the assumption that the motion of each electron is made up of a series of exactly similar collisions, separated by rectilinear free-paths, Sir J. J. Thomsonf arrived at the formula 8KRTX-*e~xdX..........(28) which is, of course, a special case of (27). * Iya Theorie du Rayonnement et les Quanta, p. 69 ; and J. H. Jeans, Phil. Mag., July and August (1909). | Phil. Mag., 14, p. 225 (1907). C i8 Radiation and the Quantum-Theory. The following considerations at once are suggested : � I. To reconcile formula (27) with Wien's law (cf. �20 below), c must vary as 1/T, so that the duration of a collision must be exactly proportional to 1/T. This would be the case only if the atoms were centres of force repelling according to the law of the inverse cube,* 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. II. 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 collision would have to be of the order of 10~14 second. Remembering that the velocity of the electron is of the order of 107, it is clear that this time of collision is too large to reconcile with what is known about molecular or atomic dimensions, f 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 Richardson and Brown J and of others 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 integrating for all possible velocities, it is found that the limiting form for / (� -c \/~c when X is very small, is not proportional to e a but to e~ I, and this cannot be reconciled with observation^ * J. J. Thomson, I.e. ante. f Phil. Mag., 20, p. 651 (1910). % Phil. Mag., 16, p. 353 (1908). � Phil. Mag., 20, p. 650 (1910). CHAPTER III. THE DEVELOPMENT OF THE QUANTUM-THEORY. 20. The last chapter showed that, consistently with the assumption of thermo-dynamical equilibrium, the Newtonian system of equations could lead to only one formula for the partition of radiant energy, namely, 8^RT^_ HI, 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 perfect agreement with observation, and how Poincare has shown that the observed radiation formula can be derived from only one set of physical assumptions, namely, those of the quantum-theory. The radiation which is in thermodynamical equilibrium 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 T4; according to Wien's displacement law, its energy, distributed according to wavelength, is of the form �{XI)1-H1 ..........(29) where F is a function, undetermined so far as Wien's law goes, of the product AT. From Wien's law, Stefan's law can be deduced at once on integrating from 1=0 to X= oo. Both Stefan's law and Wien's law can be deduced from general thermodynamical considerations, but such considerations do not suffice to determine the actual form of the function F in Wien's law. Both of these laws can be tested experimentally, and they are found to be in agreement with observation. The form of the function F can be determined experimentally, 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, derived from considerations similar to those on which the quantum-theory now rests, is where v is the frequency corresponding to wave-length 1, and h is the constant, now known as Planck's constant, of which the value is approximately j* ^=6-55xl0-27 (ergxsec.) ....... (30} Substituting this form for F(2T) into expression (29), the partition of full black-body radiation at temperature T is found to be toiRTlr*dX-^--, .........(31) where % stands for hv/RT. This formula is found to agree with experiment to * Annalen der Physik, 4, p. 553 (1901). f Cf. below, � 62. C2 20 Radiation and the Quantum-Theory. within the errors of observation. We shall nowT see how it can be derived by making use of a system of dynamics which differs widely from the classical system. 21. In the ordinary theory of gases, the " probability " that a system shall have its co-ordinates (pv ft2, . . . ) and momenta (qv q2, . . . ) within a range dpxdp2 . . . dqxdq2 ... is found to be of the form Ae~2WdpLdpx2 . . . dq�dq2 . . ., where E is the energy of the system in this configuration, A is a constant and k is given by 2&RT�1. Hence, if e is any amount of energy, the probabilities of the system having energies 0, s, 2s, . . . , will stand in the ratios 1 : e~2k* : The total energy of all these vibrations must be sNe"2he+2eNe'AJCG+Zs'Ne-Q7ce+ . . ., of which the value, on summing and using relation (33), is found to be Ms (34) If the particular vibrations are those of wave-length between X and X~{-dA in a unit volume of ether, the value of M must be taken to be (cf. � 14) STt^^dl, and formula (34) assumes the form 8^-^^rT...........<35) We can pass to the problem contemplated in � 15, namely, that of determining the radiation-formula predicted by the classical mechanics, by passing to the limit e=0 ; for the vibrations are supposed capable of having energies 0, s, 2s, 3e, . . . 00, which, when e=0, means simply that the energy can have any value. Now, when 1 ' e 1 s is small, the limiting value of -^�: is �or RT, so that formula (35) reduces to C �1 Art �7iKn~Hl...........(36) This agrees with formula (25), as of course it should, and so does not agree with observation. Development of the Quantum-Theory. 21 The general formula (35), can, however, be put in the form e ftnrPT2-4,7 2v ^T ........(37) and this will agree with Planck's formula (31), and so also with observation, if we define e by e=hv,............(38) where h is Planck's constant, and v is the number of vibrations per second. The assigning of a finite value of h is in opposition to the classical mechanics, since it does not permit the energy to vary continuously. If the energy changes at all when h has a finite value, it can do so only by taking jumps of finite amount. The foregoing method of arriving at Planck's radiation formula was first given by the present writer.* A very similar method has recently been proposed by Darwin and Fowler, f Such methods probably provide the shortest way of illustrating the connection between Planck's fundamental equation (38) and his radiation formula (4). There are obviously, however, grave objections to supposing that the energies of the vibrations in the ether must be multiples of s. For this involves that these energies can only alter by sudden jumps, 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. 22. The original method of PlanckJ was based on somewhat different physical ideas. Suppose that the M vibrations considered in � 21 are the vibrations of M resonators, each of frequency v. Their total energy will be given by expression s (34), so that the average energy of each vibrator will be -^----, and the average kinetic energy will be one-half of this, or iRTx^rzr...........(39) � where #=-�. The conditions for equilibrium between resonators and ether have Kl been considered in � 9 ; we note that the average kinetic energy of a resonator must % now, from formula (39), be supposed to be ��- times what it was supposed to be 6 X in � 9, and the result is that the radiation formula is equal to that previously found x multiplied by x-. In other words, the radiation formula now obtained is that of Planck. This is the way in which Planck originally derived his formula, but this method also is open to serious objections. For, in considering the partition of energy between the various resonators, it is assumed that the energy can only vary by jumps of * Phil. Mag., 20, p. 953 (1910). f C. G. Darwin and R. H. Fowler, Phil. Mag., 44, p. 450 (1922). { Annalen der Physik, 4, p. 556 (1901). 22 Radiation and the Quantum-Theory. 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. We may notice that the method assumes 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, may still 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 (36) of the classical mechanics. The result would be a compromise between the two laws. Worse than this, it would be a compromise which would depend on the relative numbers of Planck resonators and of free electrons (cf. � 8), and as the ratio of these would vary from one substance to another, there would be no definite law of radiation�the same for all substances�such as is demanded by observation. 23. Still another method of obtaining Planck's radiation formula has been suggested by Einstein.* Again it is supposed that a great number of molecules (or atoms or resonators) are engaged in the emission and absorption of radiation, but in place of assuming that the energy of these molecules can be only multiples of a fixed quantum e, it is assumed that the molecules can exist only in certain definite states, which we shall number as 1, 2, 3, . . . , in which their energies have specified values elt s2, es..... It is supposed that the number of molecules in states 1, 2, 3, ... is Nx, N2, N3 . . . , and that there is thermodynamical equilibrium with a field of radiation in which the energy per unit volume is of the type required by Wien's displacement law (cf. equation (29)), namely F(XT)X~*dX As a result of the action of the radiation of wave-length intermediate between I and X-\-dl, let it be supposed that a certain number of molecules pass from state m 1o state n, and a certain number from state n to state m. The number passing per unit time from state m to state n may reasonably be assumed to be proportional to Nm, the number of molecules in state m. We may also assume it to be proportional to the density of the radiation in question, and so may take it to be aNmF(2T)2-5^2 By similar reasoning, the number which pass per unit time from state n to state m may be assumed to be pNnF(AT)?-5dl Einstein supposes that in addition to these transitions, which are produced by the incidence of radiation, there occur a further number of spontaneous transitions from the state of higher energy m to the state of lower energy n. Such transitions are of course strictly analogous to those which occur in the degradation of radioactive substances. Einstein supposes such spontaneous transitions from state m to state n to occur at a rate per unit time. * Phys. Zeitschrift, 18, p. 122 (1917). Development of the Quantum-Theory. 23 The condition for a steady state requires that the total number of molecules passing from state m to state n shall be precisely equal to that passing in the reverse direction. With the assumptions we have made the condition becomes F(AT);r5iA[i3N,~aNJ=7Nm..........(40) and this may be regarded as an equation to determine F(AT), and hence also the law of distribution of radiant energy. In precise analogy to our formula (32), Einstein assumes the numbers of molecules in states 1, 2, 3, . . . . , corresponding to energies sv e2, �3, . . ? . to be Nm=^-e�/M xn=pne-*nW, . . . etc.,.....(41) The factors pm, pn, . . . may now be different, since it is no longer essential, as it was before, to assume that the ranges of the differentials must be of equal extent in each of the configurations ni, n, ... of the molecules. Substituting these values of Nm, Nn in equation (40), and dividing throughout by e~�m/RT, we obtain F(XT)^dX[fipn^^n)/B.T^apm]==ypM........(42) Obviously the right-hand member of this equation is independent of the temperature. When T=oo , the factor F(AT) on the left maybe supposed to become infinite, so that the factor in square brackets on the left must vanish when T is infinite, and this requires that fipn~apm. The equation accordingly assumes the form F {n)X-HX=y~------W-T.........(43) On the left-hand T enters only through the product IT, so that the same must be true on the right-hand. Thus em�en must be proportional to 1/?., or, what is the same thing, to the frequency v. Let us take the relation to be em�en==hv..........(44) where h is a constant (which will ultimately be identified with Planck's constant). The factor yja on the right-hand does not involve the temperature at all. Einstein determines its value from the assumption that at infinite temperature the radiation must have the value assigned to it by the classical mechanics, so that when T�go , the right-hand member of equation (43) must reduce to SjtRTX'^dL Thus we readily find as the general form of the radiation law F(AT)^W=&7rRTA-^^.........(45) where % is, as previously, equal to hv/RT; but h is now defined through equation (44). We may notice here that Einstein's deduction of the law of radiation does not require that the energy of the molecules should occur in complete quanta. It requires that the molecules can exist only in certain definite states, and that when a molecule jumps from one state to another the accompanying loss or gain of energy 24 Radiation and the Quantum-Theory. is always one complete quantum of the radiant energy emitted or absorbed. We may sum up Einstein's hypothesis as follows :� The interchange of energy between matter and radiation is always by single complete quanta of radiant energy. For comparison, we may remark that the hypothesis underlying our first deduction of the radiation law (�21) may be summed up in the words� Radiant energy, regarded as vibrations of an ethereal medium, can exist only in complete quanta, while Planck's original deduction of his law (� 22) was based on the supposition that The material energy of vibrating resonators can exist only in complete quanta. 24. 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 for the partition of radiant energy in temperature-equilibrium with matter, while the assumption that these changes occur by jumps of amount s=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.* In 1921 R. H. Fowlerf pointed out a want of rigour on the purely mathematical side in Poincare's proof, and strengthened the argument by supplying the missing steps. The result obtained by Poincare and confirmed by Fowler is, in brief, that no system can possibly lead to Planck's law, except one in which the assumption of the quantum-theory is satisfied: " L'hypothese des quanta est la seule qui conduise a la loi de Planck."J Unfortunately, Poincare's Paper is of such an abstruse mathematical 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 Poincare, is based on similar ideas, and leads to the same conclusions. Consider the system discussed in � 16, and let the values of the quantities dv 02, � � . be represented in a generalised space, just as before ; 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 classical mechanics. It followed from the theorem of Liouville that there was no concentration of the swarms of points; they moved indifferently through the whole space, and, as equipartition was a property of the whole space, it followed that equipartition ultimately ensued for all the 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 * Journal de Phys., January (1912). f Proc. Roy. Soc, 99 A, p. 462 (1921). X Poincare; I.e., p. 37. � Phil. Mag., 40, p. 943 (1910). Development of the Quantum-Theory. 25 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 discussion 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 moving points 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 densit}^ 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. 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 (25) 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 equipartion formula is to suppose that the density of the swarms of points must be zero throughout the whole, except for infinitesimally small regions, of the generalised space. There must be isolated small regions Rv R2,. . . in the generalised space occupied bj^ dense swarms of points ; in all other regions the density of points must be zero or infinitesimal. And, in order to satisfy the hydrodynamical equation of continuity in the generalised space, the motion of the points must consist of sudden jumps from one of the regions Rr, R2, ... to another. It appears in this way that, as soon as we seek to avoid the equipartition formula (25), we are compelled to assume motion involving discontinuities of some kind. 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 " All the states of the system which are represented in any one of these regions [called Rx, 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 physical 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 simplified illustration, that the state of the system depended only on three co-ordinates, x, yy z, so that we could represent it by a point in ordinary space. The swarm of points representing the different possible states will 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.* * The argument in more detail will be found in Poincare's Paper or the Paper already referred to, Phil. Mag., 40, p. 943 (1910). f Dernieres Pensees, p. 185. (I have translated freely.) * 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, &c, 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 calculations given in � 29 of the present report.�J. H. J. 26 Radiation and the Quantum-Theory. " 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 probable, 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 appliances we have been led to believe 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 liquid suggests at first that its structure is continuous ; observation shows that the liquid is incompressible, so that the volume of any portion of its matter remains constant. Certain reasons then lead us to suppose that this liquid is composed of molecules, very small and very numerous, but discrete. We cannot now suppose the distribution of these molecules arranged in any way we fancy; the 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/' 25. All this has followed merely from the hypothesis, or rather the fact, that the energy in the spectrum does not obey the equipartition formula of the classical mechanics. 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 Ex, E2. . . , this division being made in such a way that E2 depends only on one certain group of the co-ordinates 0V d2, . . , E2 depends only on a certain other group of co-ordinates and so on, no one co-ordinate entering into more than one group. Let WFl^Ex denote the proportion of the whole number of points for which K� has a value between Ex and Ei+dE-t; let WE^E2 denote the proportion for which E2 has a value between E2 and E2+dE2, and so on. Then* the proportion for * For details, see the Paper already referred to, rhil. Mag., December (1910) Development of the Quantum-Theory. 27 which Ev E2, . . . all lie within these specified ranges may be denoted byW^Ex^E2 . . . and will be given by WdE,dE2.. =(WBldE1)(WBjiE2)(...)... in which WFl will be a function of Ex only, WB of E2 only, and so on. The most likely values for E1( E2 . . . to have will, of course, be those which make W a maximum, or eE or again ��. If this quantity, the negative energy of the orbit, is denoted by W, it is readily found that eE V2 W^ 2#=�- ; a>z=z--------- . . . # ..... i (55) These equations have been obtained by the Newtonian mechanics. According to the Newtonian mechanics the loss of energy W ought to go on increasing as energy was dissipated 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, the 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. To avoid all these difficulties, Bohr introduces 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 s ~hv. Bohr's assumption, introduced at present only with reference to the atom with one electron, is contained in the equation W=JtA 2^2^E' co=-^".......<57> Instead of r varying continuously, as it would in the older mechanics, it is now restricted to integral values. How complete a difference there is between the two Spectra and Emission of Radiation. 37 systems will be realised on considering the variations of angular momentum demanded in the two cases. According to the classical mechanics, the electron experiences e2 a retarding force, F, equal to |^ (2na&) from its interaction with the ether, and the rate of loss of its angular momentum xh/2n is given by h dx 2jt dt while, according to the new hypotheses of Bohr, dv/dt has no meaning at all! The effect of restricting x to integral values is that W, co and a are limited to certain definite values. Thus a cannot gradually shrink, but the electron is limited to a circular orbit of radius equal to one of the values of a given by __ r2A2 kn?nieYJ where t2�1, 4, 9, 16, 25, . . . . Since a cannot undergo slight changes, there can be no oscillations in the plane of the orbit, and the circumstance, pointed out originally 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 x=1. On putting in numerical values, the diameter of the orbit is found to be given by 2a=1-06 x 10~8 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, 25, . . . 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 x=xx 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 experience a loss of energy of amount <5W given by W.W^W.-^^i,-^........(58) Bohr supposes that the whole of this energy, suddenly set free from the atom, passes away into space in the form of absolutely monochromatic radiation. He further supposes that the amount of this radiation is exactly one quantum. Thus (5W, given by equation (58), must be the same as e in Planck's equation (38), and therefore equal to hv. It is also, of course, equal to the quantity sm�sn in Einstein's equation (44), and so again equal to hv.* The equation SW=hv, according to Bohr, determines the frequency of the monochromatic light emitted. Putting the right-hand of equation (58) equal to hv, the frequency is seen to be given by <�=*5-8>- This is commonly known as the Paschen series. Its first two lines (w=4f 5) were observed by Pascheni in 1908, and the next three (w=6, 7, 8) were observed bv F. S. Brackett� in 1922. The value r2=4 gives the series "=N(^4> of which the first two lines, far in the infra-red, have also been observed by Brackett.|| * The Electron, p.210. f Nature, Vol! 93, p. 314, May 7 (1914). J Ann. d. Physik, 27, p. 565 (1908). � Nature, Vol. 109, p. 209, Feb. 16 (1922). I] L.c. ante. Spectra and Emission of Radiation. 39 The series r2=5, 6, 7 ... . are so far in the infra-red as not yet to have been observed. . ; This completes the list of lines obtainable from formula (59), and it will be noticed that certain lines, formerly ascribed to hydrogen, have not been accounted for. Bohr suggested, and the suggestion subsequently received full experimental confirmation, that these missing lines are readily explained as belonging to helium. 39. Helium Spectrum.�Vox helium the nuclear charge E is 2e, and in the neutral helium atom there are two electrons describing orbits. A helium atom with one positive charge will, 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=2# in equation (59). If we keep for N the value assigned to it in equation (61), this spectrum can be put in the form '-Ks'-.TO-Kab-THi).........<62> The series t2=1 lies in the remote ultra-violet and has so far eluded observation. Two lines of the series r2=2, also in the ultra-violet, have recently been discovered by Lyman.* The series r2=3 may be regarded as falling into two parts according as xx is even or odd, being equivalent to the two series "=N(�-4). ("=2,3,4.. .)......... (63) ' � '=N(r-^Pip). (-=2-3,4...)........ (64) These two series were observed by Fowler t in 1912 in a mixture of hydrogen and helium. They were at first attributed to hydrogen, but after the appearance of Bohr's theory it was suspected that they might belong to the spectrum of ionised helium. All doubt has now been removed by their having been observed, first by E. J. Evans,J and subsequently by Paschen,� in tubes from which all traces, of hydrogen had been removed, and from which no traces of known hydrogen lines could be obtained. The series r2=4: may again be regarded as falling into two parts, according as^ ti is odd or even, namely� .': h 4 (n+lf The first of these is obviously Balmer's series, so that the lines of this series can be * Nature, Vol. 104, p. 314 (1919). f Monthly Notices R.A.S., 73, p. 62, Dec. (1912), andPhil. Trans., 214, p. 254, Ap. (1914). % Phil. Mag., 29, p. 284 (1915). � Ann. d. Physik, 50, p. 901 (1916). "=Nl-,-7^LlW.......� � � � (66) 40 Radiation and the Quantum-Theory. emitted by helium as well as by hydrogen. The second is the series observed by Pickering in the star f-Puppis in 1896.* This series was attributed by Pickering to hydrogen simply from the analogy, now seen to be inadequate, that the series (65) was emitted by hydrogen. In the �-Puppis spectrum, in point of fact, the series (65) is observed to be more intense than the series (66), presumably indicating that the source of light is a mixture of hydrogen and helium. 40. One point of interest in connection both with stellar and laboratory spectra may be noticed here. We have already calculated the diameter of the hydrogen atom (� 36) to be 1-06 x 10~8 cm. in the state r=l, but have further seen that an atom in the general state defined by the integer r will have a diameter equal to t2 times this�e.g., for t=30, the diameter will be approximately 10~5 cm. i; In stellar spectra lines are observed which, on Bohr's theory, require the existence of atoms in states specified by large values of r, and so which must be of great diameter. For instance, Pickering observed no fewer than 33 lines in the f-Puppis series (.66), while Dyson f and Evershedj; were able to identify 29 lines of the Balmer series (65) in the spectrum of the solar chromosphere. The twenty-ninth line in the Balmer series must, on Bohr's theory, be emitted by an atom jumping from the state rx=^31 to r2=2, and in the state ^=31 its diameter must be nearly 1,000 times that of the normal hydrogen atom, or about 1-02 xlO-5 cm. There is no difficulty in conceding that atoms having these huge diameters may exist either in the stars or in the solar chromosphere ; there is no doubt that the pressures are in both cases very low, and it is quite conceivable that atoms of the inflated size required by Bohr's theory may exist without any occurrence of overcrowding. The question of laboratory spectra is rather more difficult. In 1919 Merton and Nicholson� observed the Balmer series down to its twelfth member (r1==14) in hydrogen at a pressure of 41 mm. Now, hydrogen atoms in the state r=14 have a diameter of 2-08 x 10"6 cm., whereas the mean distance apart of atoms at 41 mm. pressure is only l-0xl0~6 cm. Superficially, at least, there would seem to be the difficulty that the packing of the atoms must have been so close as to preclude the possibility of atoms expanding to the state r�14. Fortunately, it seems probable that the difficulty is only superficial. The observed line was so excessively feeble that its explanation only needed an insignificant fraction of the total number of atoms present to be in the state r=14, and, moreover, on Bohr's theory, hydrogen atoms in this state, although of diameter 2-08 xl0~6 cm. in two out of the three directions in space, are of only infinitesimal diameter in the third direction, so that the volume occupied by a hydrogen atom, even in state t�14, is incomparably less than that of a cube of edge 10~6 cm., which is the average space available for an atom. Further light was thrown on the matter by some experiments made by R. W. Wood.|| In these the Balmer series was observed down to its twentieth line, but it was noticed that as the pressure was gradually raised, the highest members of the series disappeared in succession, exactly as required by Bohr's theory. * Astrophysical Journal, 4, p. 369 (1896), and 5, p. 92 (1897). t Proc. Roy. Soc, 68 A., p. 33 (1901). % Phil. Trans., 197 A., p. 381 (1901). � Proc. Roy. Soc, 96 A., p. 112 (1919). || Proc. Roy. Soc, 97 A., p. 455 (1920). Spectra and Emission of Radiation. 41 41. Our discussion of Bohr's theory has so far been limited by the assumption, made in � 35, that the mass M of the nucleus is very great compared with the mass m of the negative electron. Bohr has shown* that if the ratio m/M is not neglected, the value of N, the Rydberg constant, will no longer be given by equation (60), but by XT 2tcVE2 wM N=�^�^p-�..........(67) Thus, if MH denote the mass of the hydrogen nucleus, so that that of the helium nucleus is 4MH, the ratio of the two N's for helium and hydrogen will no longer be precisely 4 to 1 ; its exact value will be 4 # ___1_ ^-n+im * MH+ra" From the observed ratio of the two N's, taken from the best measurements of the hydrogen and helium lines, Fowlerf obtains for the ratio MH/m the value 1,836*' with a probable error of �12. Subsequently Paschen,{ allowing for the relativity correction and other refinements, obtained the value 1,843-7. These values are in extraordinarily good agreement with the values of M/m observed in other and more direct ways. The best direct value for their ratio is probably that of Miliikan, namely, Mjj/m=1,845. ' The Continuous Spectrum of Hydrogen. 42. Starting from the head of the Balmer series, and extending indefinitely towards the ultra-violet, hydrogen shows a faint continuous spectrum, which has been observed both in the laboratory,� and in solar protuberances and planetary nebulae. || This spectrum receives a simple explanation in terms of Bohr's theory, and also provides information of value. The lines of the Balmer's series are produced by electrons falling into the orbit t=2 from exterior orbits t=3, 4, .. . oo , the fall from the orbit t�oo providing the head of the series. Obviously then a continuous spectrum beyond this head can be produced by electrons falling into the orbit r�2 from orbits possessing more energy than the orbit t=oo �i.e., by free electrons moving with a total energy which is positive. Thus, on Bohr's theory, the continuous spectrum is emitted as a result of the capture of free electrons by a positively charged hydrogen nucleus ; that is to say, by the ordinary process of re-combination. The circumstance that the spectrum is, so far as is known, absolutely continuous shows that the free electrons must be moving before re-combination, with all possible amounts of kinetic energy. It suggests (in opposition to a theory which has been developed by Epstein^) that free electrons move with any kinetic energy they please, and that the quantum-theory imposes no restrictions on their motion of the kind which we have seen that it imposes on bound electrons. * Phil. Mag., 27, p. 509 (1914). | Royal Society, Bakerian Lecture (1914). % Ann. d. Phys., 50, p. 901 (1916). <> Stark, Ann. d. Phys., 52, p. 255 (1917). || Bvershed, Phil. Trans. R.S., 197 A., p. 399 (1901), and W. H. Wright, Lick Obs. Bulletin, No. 291 (1917). If Ann. d. Phys., 50, p. 85 (1916). 42 Radiation and the Quantum-Theory. 43. This completes our discussion of the hydrogen and helium spectra. One other case in which Bohr's 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 tx=6 gives, in addition to lines coinciding with those of the Balmer series, a pair of series .v=sN(i-rar-)' � � � �......m some of the lines of which have been identified by Nicholson* in the spectra of the Wolf-Rayet stars. 44. In the cases we have discussed, the agreement between the predictions of Bohr's theory and spectroscopic observation is so complete that little doubt can be felt that the theory is founded on a very firm basis of truth. Nevertheless it will be noticed that the cases we have considered have all been those of systems consisting of two constituents only, a positive nucleus and a negative electron revolving around it. Attempts have naturally been made, both by Bohr and others, to extend the theory to more complex systems. In particular, reference may be made to attempts by Bohr himselff to give a general explanation of the series spectra of the elements. It must, however, be admitted that the results of these efforts, interesting and suggestive as they are, do not carry conviction of the same inevitable kind as those of the simpler problems we have already discussed. This implies no failure in Bohr's theory ; it arises merely out of an essential difference in the two problems. In the case of the systems with two constituents so far considered, all the elements of the problem are known and the physical constants involved are known to a high degree of accuracy, so that Bohr's theory predicts a spectral formula in which every numerical quantity is fully known; there are no unknown or " adjustable" constants. Comparison with observatipn is accordingly simple and, when the theory is found to survive the test, extraordinarily convincing. � The case of the atom of, say, sodium is very different. No doubt the atom of sodium consists of a central nucleus of charge lie, with eleven negative electrons revolving round it; but the way in which the orbits of these electrons are arranged is still unknown to us. As a consequence, the theory can explain the sodium spectrum only in quite general terms, and the prediction, in terms of known physical constants, of the frequency of the spectral lines, which forms so arresting a feature of Bohr's discussion of the hydrogen spectrum, is perforce absent. For this reason we shall not discuss the general question of the spectra of the elements in the present report. We shall, however, devote the remainder of the present chapter to the special problem of the X-ray spectra of the elements, since, * Monthly Notices R.A.S., p. 382, March (1913). f See especially " TJeber die Serienspektra der Elemente," Zeitschrift fur Physik, 2, p. 423 (1920) ; or English translation in The Theory of Spectra and Atomic Constitution, by N. Bohr, Camb. Univ. Press (1923). �Spectra and Emission of Radiation. 43 ;as will immediately be seen, these provide a very clear and convincing test of the validity of Bohr's theory of atomic structure. The X-ray Spectra of the Elements. 45. When an element is suitably excited its atoms emit X-ray radiation, and the spectrum of this radiation consists of lines of definite frequency, analogous to the ordinary optical line-spectra. In 1913 Moseley showed that the frequencies in the X-ray spectrum of any element satisfy a formula of the general type v=N(tt_ff)2( 1 i_)........, . . . <69) in which N is the ordinary Rydberg constant for hydrogen, of which we have found the value to be given by equation (61), rlt z2 are integers, n is a constant for any clement, being in point of fact equal to the atomic number of the element, and a is a quantity, always positive and small in comparison with n, which varies from one element to another, and also depends on r2 and rv If it were not for the limitation that or is always positive and always small in comparison with n, the statement we have just made would be valueless, for if a were entirely at our disposal, and free to depend, to any extent, on nlt r1 and r2, we could, of course, fit any observed series �of lines into formula (69). Thus, the essence of the matter is the ascertained small-�ness of a, and this immediately suggests an explanation of the X-ray spectrum. If a were equal to zero, Moseley's formula (69) for the frequencies of the X-ray spectrum would become '-""Kf..-^)..........'��... (70) E, while it is assumed that the chance of two quanta falling on the atom simultaneously, and so bringing energy 2hv to the task, is so small as to be negligible. The " threshold-frequency," v0, below which light cannot liberate electrons at all, now receives a simple explanation ; it is the frequency for which the quantum of energy is exactly equal to E and so is given by hvQ=E. If light of higher frequency than v0 falls on a metal, each quantum carries with it energy hv which is greater than E. It is impossible that an atom should absorb part only of the energy of a quantum, since this would leave a fractional part of a quantum of radiant energy in the ether. We accordingly suppose that an atom absorbs a whole quantum of energy hv, of which energy E goes to dislodging an electron from the atom, and energy hv�E or h(v�v0) is left over to be absorbed by the atom in other ways. If the whole of this residual energy goes in endowing the liberated electron with kinetic energy, then the velocity of the liberated electron will clearly be given hy equation This was Einstein's original interpretation of his equation (71), but the hypothesis of " light-quanta " on which it was based, namely that radiant energy can exist only in indivisible quanta, has not received general acceptance among physicists (cf. � 85 below). It may, however, be remarked that precisely the same equation could have been deduced from another hypothesis which we have already had under consideration, namely, the hypothesis that matter emits and absorbs radiant energy only in complete quanta of the radiant energy. On this view, again, the absorption of energy must be hv, of which an amount hvQ is used in detaching the electron from the atom, and the remainder h(v�v0) is then available for endowing it with kinetic energy. On either view hv0 must be the energy required to detach an electron, and so * Phys. Review, 4, p. 73 (1914). fPhys. Review, 6, p. 55 (1915). J Ann. d. Phys., 20, p. 132 (1906). See also 17, p. 132 (19C5) and 22, p. 180 (1907). Absorption of Radiation:., 4? must be ecjual to eV, where V is the ionisation potential of the substance in question. A comparison between the ionisation potential calculated in this way and that deduced more directly from observation provides yet another test for the theoretical explanation of equation (71). So far as the results go, they are favourable to the theory, but there is extreme difficulty in determining ionisation potentials directly with any high degree of accuracy. 48. It is worthy of notice* that this 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 hv0 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 Wx to be the lost energy in the normal state, corresponding to t=1 in the last chapter, and W^ to be the lost energy in the state r= oo, in which the electron is free at infinity, then W1=hv0 and W^ =0. Thus equation (71) may be written in the form i Radiation and the Quantum-Theory. hv0 is the energy required to detach an electron from an atom of the substance in question. When the photo-electric effect first comes into play v is exactly equal to vQ, and for all the applications to visible light v0 is comparable with v. But when we pass to X-radiation v has a value equal to thousands oi times its value for visible light, so that v0 is now insignificant in comparison with v, and the equation assumes the approximate form �mv2=hv. This has the simple interpretation that the energy of the ejected, electron is equal to that of a quantum of the incident X-radiation, and this in turn is equal, as we know, to the energy of the electrons which excited the X-radiation in the first instance. There is an extension of this phenomenon of a kind which was not possible in the case of visible light. We have seen that to pull off an electron from an atom requires an expenditure of energy which is about equal to the quantum of energy for visible radiation. But after one electron has been pulled off others remain, and we can imagine the process of an atom being stripped of its electrons one by one until only the nucleus is left. As we pass further and further into the atom the energy required to detach an electron continually increases, that required to detach an electron from one of the inner shells being known to be of the c rder of magnitude of a quantum of X-radiation. Indeed, the limits of the X-ray spectrum of an element will, as we have seen (� 45) be of frequencies such that the corresponding quanta represent the -energies necessary to detach electrons from the inner shells�the so-called K and L rings. Thus if X-ray radiation is used to detach electrons from these, inner shells, the velocity of the detached electron ought, according to the quantum theory, to be given by Einstein's equation (71) where v is the frequency of the incident radiation and v0 is the frequency of the limit of the X-ray spectrum corresponding to the detachment of the electron in question. De Broglie* by photographing the magnetic spectrum of the ejected electrons has been able to calculate their velocities of ejection. The spectrum is found to be a " line-spectrum," so that each electron is shot off with one or other of a definite series of velocities, and these velocities are found to be given precisely by equation (71) where vQ is the frequency of one or other of the limits of the X-ray spectrum. Ellisf has applied a similar method to the electrons ejected by the y-rays of radium B from the heavy elements Ur, Pb, Pt, W and Ba, and has obtained a similar result, except that in this case the frequencies of the exciting y-radiations were unknown. Since, however, the frequencies v0 are known, it is possible to use equation (71) for the determination of v. In this way Ellis calculated the frequencies of the y radiation emitted by radium B, radium C and thorium D.J A discussion of the values obtained shows that it is very probable that the y-rays of radioactive bodies are emitted from the nucleus. The constituents of the nucleus seem to be limited to a definite discrete series of configurations, the ejection of a y-ray being caused by the step down from a configuration of higher energy to one of lower energy, so that even in the nucleus of�an atom the principles of quantum-dynamics appear to hold sway. * Comptes Rendus Nos. 5, 9, 12, Vol. 172 (1921), or " I,es Rayons X," or Report to the 1921 Solvay Congress of Physics, published in " Atonies et Electrons." f Proc. R. S., 99A, p. 261 (1921). % Proc. R. S., 101 A, p. 1 (1922). CHAPTER VI. THE SPECIFIC HEAT OF SOLIDS. 51. Still a further application of the fundamental principles of the quantum-theory is required for a satisfactory explanation of the specific heats of solids. 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. This product is called the atomic heat, and is approximately equal to 5*95. A simple explanation of this law is provided by the usual kinetic theory of matter. Suppose that 1 gramme of the substance in question contains N atoms, each of mass m, so that Nw=l. Let each atom have s degrees of freedom�that is to say, if all the other atoms are held at rest, let it be possible for each atom to have 5 entirely independent motions of its own. Then the expression for the energy of each atom will be that of the energy of s vibrations, and so it follows from the theorem of equi-partition of energy (� 16) that the average energy of each atom will be sRT, ...........(75) where T is the absolute temperature, and the total energy E of the substance per 3E. . 1 dE gramme will be NsRT. The specific heat will be ^ in energy units, or - ^ in heat units, where J is the mechanical equivalent of heat. Calling this specific heat c, we have 1 3E 1 3 ^T wx NsR ^Jaf^pT1 NsRTH-y-.......(76) Let a be the atomic weight of the element, that of oxygen being taken to be 16. Then the mass of an atom m of the element is am0/16, where m0 is the mass of an atom of oxygen. For the oxygen atom, R/m0=520 x 104, so that for the element now under consideration R 16 �^~x 520x104, m a and since 'Nm=l, this is also the value of RN. The value of J is known to be 4-184 x.107, so that equation (76) becomes c=l-98-. a The constancy of ca, which is postulated by the law of Dulong and Petit, is seen now to depend simply on the constancy of s. Moreover, 5 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 ac=5-95. Each atom must have a value of sat 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 E2 5^ Radiation and the Quantum-Theory. value of the product ac is thus accounted for by supposing that the atom has no further independent motions ; from the point of view of specific heats the atoms are to be thought of as rigid points. 52. 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 experimental determinations of specific heats of solids were concerned with the specific heats at constant pressure, but recently Nernst and Lindeniann* 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 � 51 refers. The constancy of ac ought only to be true if c represents the specific heat at constant volume. In 1911 a series of 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, and all being corrected so as to refer to specific heats at constant volume, t It was found 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 accompanying figure will'show the general nature of the results obtained. The curves give the atomic heats of four elements�lead, silver, aluminium, and carbon in the form of a diamond�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 limiting atomic heat 5-95 as. the temperature increases. A still more striking discovery is that all the curves are exactly similar, except for differences in the temperature scale. Thus, if we take the curve for silver and compress it horizontally 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 coincide with the silver curve, and compressed in the ratio 1 : 4T it would coincide with the lead curve. 53. The simple investigation of � 51 would apply equally at all temperatures if the fundamental principles on which it is based were sound. The principles on which the investigation 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 applicable 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 suggestive- * Zeitschrift fiir Electrochem., p. 818 (1911). f See especially Papers by Nernst, Annalen der Physik, 36, p. 395 (1911), and a report presented to the first Solvay Congress at Brussels, 1911, "I^a Theorie du Rayonnement et les Quanta," p. 254. Also Papers in the Sitzungsberichte d. Preuss. Akad. (1911), and in the Zeitschrift fiir Blectrochemie (1911) and (1912). J For the theory and details of this correction, see Nernst and I^indemann, Zeitschrift f iir-Blectrochem., p. 817 (1911). Specific Heats of Solids, 53 ness, were attempted by Einstein* and by Nernst and Lindemann.* Both of these had, however, 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 finality 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. 54. In the analysis of � 51 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 20� 40� 60� 80s 100� 120� 140� Fig. 2. 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 solid 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. But 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 vibrations of wave-length between 1 and X+dl has been * The best account of these will be found in the reports of the first Solvay Congress at Brussels (1911), "La Theorie du Rayonnement 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 (� 59). | Zur Theorie der Spezifischen Warm en, Annalen der Physik, 39, p. 789 (1912). 54 Radiation and the Quantum-Theory, seen (� 14) to be 12n?r*dL Of these vibrations two-thirds are waves of distortion and one-third are waves of compression. Let the former be supposed propagated with velocity Vx and the latter with velocity V2, then the frequencies of the former vibrations are V-J2 and of the latter V2/A. It readily follows that the total number of vibrations with frequencies between v and v-j-dv is 47r(2Vr3+V2->2^ . ......(77) There must be a limit 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 (77) is true from v=0 up to a value vmt which is the maximum of all the frequencies, and is determined by the condition 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 [vm An 3N=47r(2Vr3+V2-3) v^dv=�(2V1-s+V,^)vm3,......(78) J 0 6 and formula (77) may be replaced by and falling off in intensity towards the low-frequency end. The value of vm can be calculated from the elastic constants of the solid by formula (78), and is found to be generally of the order of 1012. It will be noticed that this formula is derived theoretically by laying great stress on just that part of Debye's theory which is obviously most imperfect, namely, the supposition that the vibrations of the solid fall off absolutely suddenly at a certain frequency vm. It is, therefore, hardly to be expected that the values of vm 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 (cf., table below). 58. Lindemannf has given the following formula, to some extent empirical, which is found to give the values of vm with surprising accuracy for the substances examined. If Ts denotes the melting point of the substance in degrees absolute, m the atomic weight and V the atomic volume, Lindemann's formula^ is :� v^S-OSxlOi'xx/-^,........(85) This agrees rather better with the observed values of vm than do the values calculated directly from the elastic constants, as the table below shows� :� Element. vm calculated vm observed vm calculated (Iyindemann). (spec, heats). (elastic constants). I Al ...... 8-3 X1012 8-3 X1012 8-0 X 1012 Cu ...... 7-5 X1012 6-6 X1012 6-6 X1012 Zn ...... 4-8 X 10ia 4-8 X1012 Ag ...... 4-8 X 1012 4-5 X1012 4-3 XlO12 Pb ...... 2-0 X 1012 1-9 X1012 1-44 XlO12 Diamond 35-0 X1012 40-0 X1012 1 * See also a Paper by H. Thirring, Phys. Zeitsch., 14, p. 867 (1913), and also � 60 below. ff Physikalische Zeitschrift, 11, p. 609 (1910). { In Iyindemann's original Paper the multiplying constant, which is purely empirical, was taken to be 2-06 X 1012 and vm had a slightly different meaning. � In this table the first two columns are taken from the report by Nernst in " Vortrage uber die Kinetische Theorie der Materie," p. 77 (1914), and are calculated by giving Lindemann's formula the form (69). Einstein takes the multiplying factor in Iyindemann's formula to be 2-12 X 1012, and a table of values calculated in this way is given by him in "La Theorie du Rayonnement et les Quanta," p. 415. 58 Radiation and the Quantum-Theory. 59. Mention must now be made of the earlier theory by which 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 (80), he arrived at the expression E=3NRTxn=^r.........(86) ex�l for the total energy of the solid, as compared with the value 3NRT given by the classical mechanics (� 51). On differentiation 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, beyond question, be regarded as superseded by that of Debye, but is of importance 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 of 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 of the atoms relatively to one another. As a matter of geometry, there must be three degrees of freedom 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 hv2 mass is now given by Einstein's formula (86) in which %=��, where v2, for any substance, represents the observed frequency of the infra-red absorption band. Nernst supposes the energy of motion of the molecules as wholes to be given by Debye's formula (82), 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 diffeientiation 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 Deb}^e. If we sum these terms directly as derived from formulae (82) and (86), we shall obtain twice the atomic heat, for we are now supposing that there are N molecules per unit volume. The specimen table following, selected from several given by Nernst J in the report already referred to, will indicate the degree of closeness of agreement with observation. The Einstein term in the second column is calculated by assigning to v the value actually observed for it by Rubens ( � =166); the Debye term in the third column is calculated by assigning to vm the value given for it by Lindemann's formula * Nernst, report in " Vortrage ifber die Kinetische Theorie der Materie." f Sitzungsber. d. Preuss. Akad, Berlin, June 5 (1913). t L. c. p. 81. Specific Heats of Solids. 59 (85), 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 reality 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 remarkable, the more so in view of the fact that equally good agreement is obtained in the case of other substances for which v2 (the frequency of the rest-strahlen absorption band) is known�namely, NaCl, AgCl, KBr and PbCl2. Values of 2cp for KC1. Einstein Debye Correction T. term term term Calculated Observed in 2cv. in 2cv. 2{cp�cv). 2cp. 2cp. 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 1 550 5-87 5-93 0-90 12-70 13-18 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 solids, 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 Born and Karmanf to space lattices in which two or more different masses are supposed to alternate. In this way it is found possible to calculate the relation between the elasticity and specific heat for regular crystals. The calculation is carried out for NaCl, KC1, CaF2 and FeS2.. The author finds a mean deviation of 2-3 per cent., with a maximum of 4 per cent, from observed values through the range of temperature considered by him, a range much less than that covered by Nernst's tables. 60. 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. Nernst and Lindemann have found no general difference between the specific heats of good and bad conductors ; RichterJ has examined the specific heats of a series of Bi-Sn and * Phys. Zeitschr., 15, p. 180, and also p. 127 (1914). f See above, �56. % Ann. d. Physik, 39, p. 1590 (1912). 6o Radiation and the Quantum-Theory. Bi-Pb alloys with special reference to this question, and concludes that, in these alloys 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 certainly 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 the atoms inside the solid are so large that the corresponding vibrations are of very high frequency, and so normally possess very little energy. As far as pure theory goes, there is no question that to the terms in the specific heat contemplated by Nernst's theory (� 59) there ought to be added an additional term, of a form exactly similar to the Einstein term, but having x=^f where v3 is the frequency (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 in the specific heats, beyond that accounted for by the theories we have considered, as the fusion-point is approached, f when presumably the intensity of the forces 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 in normal cases these deviations from the simple theory are too small to be detected experimentally, and it may be stated as a general rule (not applicable close to the fusion-point) that the motion of translation of the atoms suffices to account for the observed specific heats. 61. Debye's result (�� 54, 55) may be summarised in the statement that the whole heat-energy of an element resides in the energy of its elastic-solid vibrations, the atoms being treated as the particles of the solid, 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 solid, and to the energy of the elastic vibrations of the solid must be added the energy of the internal vibrations of the molecules, each vibration again having exactly the energy allotted to it by the quantum-theory. Both these theories, as well as the theories of Born 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 having the same energy as a light vibration of identical frequency. This recalls the result obtained in � 9, that the condition for equilibrium between a material vibration and light vibrations in the ether is the equality 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 must be true in the quantum-mechanics also. * See Griineisen, Molekulartheorie der Festen Korper, report presented to the second Solvay Congress of Physics (Brussels, 1913), also A. B. Oxley, Proc. Camb. Phil. Soc, 17, p. 450 (1914). f Oxley, I.e. CHAPTER VII. THE DYNAMICS OF THE QUANTUM-THEORY. 62. We have now briefly discussed the four phenomena which give the most direct evidence of the failure of the classical dynamics and of the need for a new system of dynamics which we may term quantum-dynamics. The four phenomena that we have had under consideration have been� (i.) Black body radiation. (ii.) The spectra of the elements (iii.) The photo-electric effect, (iv.) The specific heats of solids. That the phenomena agree in showing the inadequacy of the Newtonian mechanics will need no further argument; that they agree in pointing to the same new system of quantum-dynamics will become clear on comparing the values of h which are deduced from the four phenomena in question. These values are as follows :�- (i.) From black-body radiation*............^=6*55 xlO"27 ^=6-547 xlO-27 (ii) (From Rydberg's constantf | From K-ray spectra J A=6-555xl0-27 A^6-561 x 10-27 /*=6-585xJ0-27 A=6-59 xlO~27 (iii.) Photo-electric effect (sodium) � (lithium) || ...... (iv.) Specific heats at low temperatures^ Thus the same constant h appears as being intimately connected with all the four phenomena, which we have found the old mechanics unable to explain. What is h, and what is the new dynamics wThich appears to centre round its existence ? We found that the phenomenon of black-body radiation could be explained if we assumed that radiant energy existed only in complete quanta e, defined by e~hv, where v is the frequency of the radiation ; oi if we assumed (with Planck) that the energy of matter consisted wholly of energy of isochronous vibrators, and that the energy of each vibrator occurred only in complete quanta of energy �, again defined by e=hv, where v is now the frequency of the vibrator ; or if we assumed (with Einstein) that atoms of matter could exist only in a certain series of definite states and that the transfer of energy between matter and the surrounding field of radiation, produced by the sudden jump of an atom from one state to another, occurred only by single complete quanta of the radiant energy, these quanta again being defined by the equation e~hv. Turning next to the line spectra of the simplest elements, we found that these * W. W. Coblentz, Scientific Papers, Bureau of Standards, No. 360 (1920). f R. A. Millikan, Phil. Mag., 34, p. 15 (1917). X Blake and Duane, Phys. Review, 10, p. 624 (1917). � R. A. Millikan, Phys. Review, 4, p. 73 (1914). || R. A. Millikan, Phys. Review, 6, p. 55 (1915). ^f From experiments of Keesom and Kamerlingh-Onnes (cf. � 56 of the present Report). 62 Radiation and the Quantum-Theory. admitted of no such variety of explanation. Bohr's theory has been found to lead to results in such complete agreement with observation that it can hardly be doubted any longer that it contains the true explanation of the line spectrum. Now the requirements of this theory are perfectly precise :� (i.) Atoms consisting only of a nucleus and one electron can exist only in certain definite states, and these are specified by the equation (cf. equation (56)) W=Mv, where v is the frequency of the electron in its orbit, W is the negative energy of the atom, and x is necessarily integral; and (ii.) The transfer of energy from matter to radiation is by single complet e quanta of the radiated energy, and so is given by the equation AW=Av. We have seen that the photo-electric effects also can be explained very simply and convincingly in terms of Bohr's theory ; the only additional hypothesis required is the converse of hypothesis (ii.) above, and may be stated in the form (iii.) The transfer of energy from radiation to matter is by single complete quanta of the radiant energy, and so is given by the equation -AW=Av. Thus it appears that the three hypotheses we have numbered as (i.), (ii.), and (iii.) are adequate to explain three out of the four phenomena we have had under consideration. To explain the fourth phenomenon, the specific heats of solids at low temperatures, we have found it necessary to introduce a fourth hypothesis� (iv.) The energy of the wave-vibrations in an elastic solid occurs only in definite quanta, so that if E is the energy of any vibration E � xhv, where x is an integer. 63. These four hypotheses may naturally be taken as foundation stones in the attempt to build up a system of quantum-dynamics. We have seen that they are adequate to explain the four principal phenomena we have had under discussion ; we can almost say that they are necessary, since the explanations they give are so convincing that it is difficult to imagine that any other set of hypotheses could meet with equal success. The four hypotheses fall naturally into two pairs. Numbers (ii.) and (iii.) deal with the interchange of energy between radiation and matter, while numbers (i.) and (iv.) are not concerned with radiation at all, but deal with the partition of energy in matter. Hypotheses (ii.) and (iii.) can be combined in the single hypothesis that all interchange of energy between matter and radiation is by single complete quanta of the radiation; or in mathematical symbols, AW=�Av...........(87) where v is the frequency of the radiation, emitted or absorbed. The two pairs of hypotheses are closely related by the so-called " combination Dynamics of the Quantum-Theory. 63 principle " of Ritz,* according to which all the series lines in the spectrum of an element have frequencies which can be expressed as differences of other frequencies. To take, for instance, the spectrum of hydrogen ; if we write N ".=�-,.............(88) where N is the Rydberg constant, and xs is any integer from 1 to 00 ; then the whole spectrum of hydrogen, as we have seen in Chapter IV., is obtained by putting v=Vm�Vn ...........(89) where m, n have all possible integral values from 1 to 00. In general, the spectrum of any element shows frequencies which can be expressed as simple differences of more fundamental frequencies in the manner of equation (89). The truth and exactness of the combination principle are undisputed. The quantum hypothesis expressed in equation (87) requires us to suppose that radiation is emitted and absorbed in finite chunks, and the principle of conservation of energy now requires that either emission or absorption shall be accompanied by finite jumps in the energy of the material system which emits or absorbs energy. If we put Wm=hvm> Wn=hvn, etc., so that Wm, Wn, .... have the physical dimensions of energy, the combination principle expressed in equation (89) may be put in the form and the quantum hypothesis (87) now assumes the form AW=WW-WB. In words, this equation shows that all possible jumps of energy in the radiation-emitting or radiation-absorbing mechanism may be expressed as simple differences of certain more fundamental energies Wx, W2, � . � Ww, Ww. ... It would be difficult to interpret this in any more natural way than by supposing that the mechanism can exist only in certain definite states corresponding to energies Wv W2, .... The quantum-hypotheses which have been labelled (i.) and (iv.) confirm us in this interpretation, amounting indeed merely to special cases of the general result to which the combination principle has led us. Hypothesis (i.) requires that the possible states of an atom consisting of a nucleus and only one electron shall be specified by W=irhv............(90) while hypothesis (iv.) requires that the possible states of vibrations of an elastic solid shall be specified by E=x*v ............(91) 64. It is natural to inquire at this stage whether the special hypotheses expressed by equations (90) and (91) can be regarded as special cases of a single more general * Phys. Zeitschr., 9, p. 521 (1908), or Astrophys. Journ., 28, p. 237 (1908). For a simple discussion see N. Campbell, "Series Spectra " (Camb. Univ. Press, 1921),p. 11. See also Fowler, " Report on Series in Iyine Spectra," p. 23. 64 Radiation and the Quantum-Theory. hypothesis applicable to all dynamical systems. Each of the two equations refers to a motion in which only one co-ordinate varies, and if T, V denote the kinetic and potential energies of the motion we have in the special case to which equation (90) refers (c/. �36), T=W, so that the equation becomes 2T=rhv............(92) In the special case to which equation (91) refers, namely, an ordinary wave-motion of small amplitude, we have T=V=iE, where a bar over a symbol denotes an average value throughout a complete period, so that equation (91) can be written in the form 2T=rhv............(93) In equation (92) T is constant throughout the motion, and so is the same thing as T. Thus the two equations (90) and (91) which expressed our original hypotheses (i.) and (iv.) are covered by the single equation (93). Allowed and Disallowed Motions. 65. The classical mechanics permits of an infinite series of motions for every dynamical series, and these motions merge continuously into one another�for instance, an electron can describe a circular orbit of radius r about a positive nucleus, and all values of r are possible. It is already clear that the quantum-mechanics allows only a certain number of these motions to be performed ; it takes the infinite series of motions permitted by the classical mechanics and divides them sharply into " allowed " and " disallowed " motions, the former being, in all cases so far considered, only infinitesimal in number in comparison with the latter. To return to the illustration of the circular electron orbit, the " allowed " motions are those in which r has the series of values (cf. equation (57)) 4:7i2meE while all other values of r, although permitted by the classical mechanics, are " disallowed " by the quantum mechanics. In this way we arrive at the conception of the special equations of the quantum-dynamics being additional restrictions which have to be superposed on to the equations of the classical mechanics, and it is already obvious that it is through these additional equations that the quantum-constant h enters into the scheme of nature. Some function of the co-ordinates and velocity, which may have any value we please under the classical mechanics, is restricted by the quantum-equations to having values which are integral multiples of h. It is not likely that the functions which are restricted to integral values of h will be entirely different for different problems ; we must rather expect that some general function or functions will ultimately be discovered which will cover all possible dynamical problems. We have, for instance, seen that the relation ?I=tA...........i (94) v , covers the two problems of the Bohr atom with one electron and the vibrations Dynamics of the Quantum-Theory. 65 of an elastic solid. Obviously, however, this cannot be the most general form of qtiantuim-eqiration, for v has no meaning, and T has not been defined, except in reference to systems which repeat their configuration after definite isochronous intervals, each equal to 1/v. Adiabatic Invariants. 66, The number of functions which can be placed on the left hand of an equation such as (94) is strictly limited, for, as we shall now see, these functions must satisfy a very definite condition. Let F be such a quantity, and let a dynamical system be describing a motion which conforms to the equation F=rh.........., (95) where x is an integer. Let one of the quantities which specify the strength of an external field of force acting on the system be slowly and gradually changed�for example, let us slowly and gradually create an electrostatic field in the neighbourhood of the system, or create, increase or diminish a magnetic field, or slowly and1 gradually introduce or iremove a mechanical constraint. Let a be the quantity which' gradf&ally changes; then as is a universal " adiabatic invariant," while the angular momentum is not, indicates that the most significant expression is that provided by equation (94). 68. The " adiabatic invariant " 2T/i> has reference only to systems for which the motion repeats itself after definite equal intervals 1/v, for the symbol v has no meaning except with reference to such systems. If we write a for 1/v, the period of the motion, we may put ?I=2fa=fP2T<&..........(97) v J 0 since T is by definition equal to the value of T averaged through a time-interval g. This last integral is, of course, the " action " taken over a complete period. Thus while the Newtonian mechanics requires that the action shall be a minimum, the quantum-dynamics imposes the further restriction that the action, taken over a complete period in the special case now under consideration, shall be a multiple of h. 69. Consider for the moment the most general dynamical system specified by n generalised co-ordinates qlf q2, . . . qn and their rates of change qv #�,..* qn. Let E denote the total energy of the system, expressed as a function of qvq2, . . . qn, qlf q%> . . . qn> an(i let momenta plt p2, . . . pn be introduced, defined by Ps^^~> etc dls Since the kinetic energy T is a quadratic function of the velocities, qv q2, . . qn, we have 3T . 3E . OHs (Ms so that the action, taken over any time-interval 0 to t, is (%j;dt=Z [ pJ?sdt=Z f pbdqs .....(98) Jo Jo J o * Verslag Amsterdam Acad., 25, p. 412 (1916;. t On the Quantum Theory of I^ine Spectra, Kgl. Danske Selsk. Skrifter, Naturvidoog Mathemat., 8, 4, I., p. 10 (1918). % Phil. Mag., 33, p. 504 (1917). Dynamics of the Quantum-Theory. 67 70. In the special case in which the system has only one degree of freedom and a definite period cr, the generalised space of the type we introduced in �15 contains only two co-ordinates p and q and so reduces to a plane, while from equations (97):and(98) � 2T [* �= w �.........(") v - 0 Also by hypothesis the point in this plane which represents the motion of the system returns to its original position at successive intervals a, and so repeatedly describes a certain closed curve in this plane. The area of this closed curve is, of course, / pdq, so that equation (99) shows that the complete area of the curve is equal to 2T/v. The quantum-restriction (94) which has been found to give accurate results in the problems of the Bohr atom and of the vibrations of an elastic solid is now found to assume the form J pdq=xh..........(100) Thus the effect of the quantum-restrictions is to limit the representative point to describing certain definite closed curves in the p, q plane, these curves being defined by the condition that their areas are integral multiples of h. 71. The discovery of suitable generalisations of equation (100) constitutes one of the principal problems of quantum-dynamics. In the most general case to which this equation is applicable, namely, a system of two degrees of freedom �describing a recurrent motion with period a, it is highly probable, although not yet rigorously proved, that the equation expresses the true quantum-equation of restriction. For the left-hand member is an " adiabatic invariant " and is the only known " adiabatic invariant" which leads to the right result in the simple cases of the Bohr atom and the vibrations of an elastic solid. 72. In certain dynamical systems the total energy E can be put in the form E=Atfi,^+/^8.?�)+ � � � +fn(Pn,qn) .....(101) The motion of such systems, when calculated by the classical mechanics, is such that the changes in any one co-ordinate qs are not affected by the values of the remaining co-ordinates qv q2 . . . For certain of those co-ordinates the motion will be recurrent, having a definite period which is, of course, independent of the motions of the other co-ordinates.' The kinetic energy 2T of the system must, like E, fall into the sum of a number of separate contributions, one from each of the co-ordinates and its corresponding velocity. Now if qs is any one co-ordinate, of which the corresponding motion has a period P2 � � � ?i, ?� � � �)......(105) cannot be separated into a sum of terms in which each term depends only on one co-ordinate and its associated momentum. There will, however, in every special problem be a certain number of integrals of the classical equations of motion ; let us suppose that in a special problem under consideration these are $1>P2> ?2 � � .)=�1- 92(^1* 9v P* 92 � � .)=�2> etc-> ..... (106) where av a2, . . . are constants of integration which retain the same value throughout the motion of the system so long as this motion is in accordance with the classical dynamics. In general the momentum pS) being defined by Ps=^............(107) cannot be expressed as a function of qs only. In special problems it maybe possible to eliminate all co-ordinates and velocities other than ^>sand qsixom equations (106) and (107), and obtain, for certain co-ordinates at least, equations of the type Ps=fs(qs)...........(108) The equation fs(qs) =0 may have any number of real roots. Each time the value of qs reaches one of these roots ps vanishes, so that in general the direction of the momentum ps is reversed. If the equation fs{qs) =0 has no real roots, ps retains the same sign throughout the motion, as, for example, in the case of motion in a straight line under no forces. If the equation fs(qs) =0 has only one real roc-t Q, the motion will be one in which ps will retain the same sign until qs passes through the value qs=Q, after which it will remain continually of the opposite sign, as, for example, in the case of the motion of a particle thrown up in opposition to gravity. Except for these special cases, the equation fs(qs) =0 will have at least two real roots, and if Qx, Q2 are two adjacent roots there will be a possible motion in which qs will oscillate between the values Qv Q2. Such systems are called " conditionally periodic " systems. Each co-ordinate whose momentum can be expressed in the form (108) will oscillate between two fixed limits such as Qv 02, just as though the co-ordinate belonged to a system possessing only one degree of freedom. An instance is provided by a particle describing an elliptic orbit about a centre of force Dynamics of the Quantum-Theory. 69 attracting according to the law of the inverse square. If r is the radius vector, the corresponding momentum can be expressed as a function of r only, and the co-ordinate r oscillates between two values, rv r2, just as though the particle had only one degree of freedom. Sommerfeld, Epstein, Debye and others have assumed that in conditionally periodic systems the quantum-equations are of the type fpsdqs=Tsh,..........(109) in which the integral is taken over a complete liberation of the qs co-ordinate. There will, of course, be one such equation of restriction for each co-ordinate qs, whose momentum can be expressed in the form (109). This assumption was immediately justified by success. Subsequently further justification was provided when Burgers proved that the left-hand members of equations (108) are adiabatic invariants, an exception occurring only in the physically unimportant case in which the periods of the different oscillating co-ordinates are commensurable. The Bohr Atom with Elliptic Orbits. 74. The foregoing ideas derive their practical importance mainly from their application to the Bohr atom in the general case in which the electron is not limited to circular orbits. We have already seen that the supposition that the electrons are restricted to circular orbits will predict the ordinary " line "-spectra of atomic hydrogen and ionised helium; but each of the " lines " in the spectrum, for instance, Ha, Hp, &a, was predicted as a true mathematical line, having no breadth, and corresponding to a perfectly definite frequency. In actual fact, it is known that each of these "lines" shows a highly complicated "fine-structure" when viewed under sufficiently high dispersion�so far from being the single line predicted by Bohr's simple theory already described, it is seen to consist of a bunch of a great number of quite distinct lines. Sommerfeld has shown that this " fine-structure " can be accounted for as soon as we remove the restriction that the electron is confined to circular orbits.* The Fine Structure of Spectral Lines. ^ 75. Following Sommerfeld, let us first examine the form assumed by the qiian-tum-equations for an electron describing an elliptic orbit about a central nucleus. If for the present we neglect the dependence of the mass of the electron on its velocity, and neglect also the ratio of the mass of the electron to that of the nucleus, we may suppose the kinetic and potential energies to be given by e . , , , � T=^,V=------...........(110) Using polar co-ordinates 6, r, the value of T becomes *w(r202+f2), and the two momenta corresponding to the co-ordinates 6 and r are seen to be p^mr*�, p2=tnr......... (Ill * Ann. d. Phys., 51, pp. 1 and 125 (1916), or " Atombau und Spektrallinien " (Braunschweig, 1921), p. 263. (Third edition,, translated in Bnglish, 1923.) 70 Radiation and the Quantum-Theory. , Since the energy of the system does not involve 6, the first equation of motion of the classical mechanics reduces simply to p^constant, or w20=constant......... (112) In place of the second equation we may use the equation of energy, which may be written in the form eE \m(rH2-\-r2) = � +W=0 . where W is the constant negative energy of the orbit, or again, using the values of the momenta given in equation (111) ' * i(w+70-f+w=o........ ] [.-^J-.......<-� where W is the negative energy of the orbit, e is the eccentricity, and y is given by " ^~m ..." � . Sommerfeld now introduces the quantum-restrictions in the form already given in equations (114) and (115). Neglecting squares and higher powers of fineE/hC)2, a small quantity of the order of 5 X 10-\ we find that Wean be expanded in the form The first term in this expression for W represents of course the value of W when the relativity correction is neglected�i.e., the value which W would have if C were infinite�and this leads to the spectrum already considered in which each line is the superposition of a number of separate lines. When r2 is put equal to zero in equation (120) we obtain the values of W appropriate to circular orbits ; the difference between these values and those previously obtained arising from the relativity-correction which is required on account of the velocity of the electron in its circular orbit. In the simpler case previously discussed, the removal of the restriction r2�-0 led to no new values for W. In the present case in which W has its full value given by equation (120) we find that the values of W corresponding to different values of t2 are all distinct; they differ from the values of W appropriate to circular orbits by small quantities which are proportional to t2/Ti. Sommerfeld believes that the frequency differences resulting from these small differences of energy exactly explain the observed fine structure of spectral lines. The agreement between the predicted and observed frequencies in the fine-structure of lines cannot be claimed to be perfect. Some of the want of agreement must be attributed to the difficulties of measurement of the observed fine-structures of lines, but apart from this some lines which are predicted are not observed at all, suggesting that certain sets of values of rv r2) z'v t'2 are prohibited altogether. 78. As an illustration of the kind of agreement observed, let us consider the case of the Balmer series of hydrogen. The line n of this series is emitted when- Dynamics of the Quantum-Theory. 73 ever an electron falls from an orbit x\, x\ for which T,1+xf2=n to an orbit xvx2 for which xt+x2=2. The energy of the latter orbits, obtained by putting e=E and r1+x2='2> m equation (120), is w-^c >+(�)" g+#]......<-> while the energy of the former, say W, is similarly found to be Thus, if v0 is the frequency of the line corresponding to circular orbits (t'2=0 ; t2�0), the frequency of the lines corresponding to non-circular orbits iwill be given by �^ A3 LVaC/ x\ \4JiCs tJ V ; The value t2=2 is prohibited since this would make xx=0, and would require the electron to pass through the nucleus. Thus the possible values for r2 and xx are t2=1, Ti=1 and r2=0, x1=2f giving x^x-^�l and 0 respectively. Corresponding to any specified values of x\ and x\ there will be two values of v with a frequency difference *-^ ��-��� � and the complete line ought to consist of two similar structures separated by a frequency-difference 0-365. Since this is independent of n, it appears that the whole Balmer series may be supposed constituted of doublets of constant frequency difference 0-365, each constituent of these doublets showing its own fine structure. Measurement'of "so'small a "separation is naturally difficult, and the observational results are far from accordant. For the separation of the doublet of Ha(w=S), Mchelson has found (3^=0-32, while Fabry and Buisson find 0-307, Merton and Nicholson* find 0-313, and Merton alonef by a more refined method 0-344. For Hp, again, Mertonf has found 3v=0r372, and for Hy Michelson has found (5i>=0'42-^Quite recently G. M. Shrum has measured the doublet separations of Ha, H^, H7, H6 and He respectively as 0*33, 0'36, 0*37, 0*36 and 0*35, with a probable error of 0*01 for H^ and of 0*02 for the other lines. J More accurate measurements can be made on the spectrum of positively charged helium, and from these Paschen has determined the frequency-difference as 0-3645�0-0045. There seems to be little room for doubt that the frequency-difference predicted by Sommerfeld, namely, 0-365, is very near to the truth. We have given only one instance of the results of Sommerfeld's theory of fine-structure ; a large number of other instances, with a full discussion as to their agreement with observation, will be found in his " Atombau und Spektrallinien/' or its English translation " Atom Structure and Spectral Lines/' * Proc. Roy. Soc., 93, p. 27 (1917). t Proc. Roy. Soc, 97, p. 318 (1920). t British Assoc. (1923) Pres. Address to Section A. 74 Radiation and the Quantum-Theory. The Stark and Zeeman Effects. 79. When the atom is placed in an electric field the superposed lines predicted by the simple theory of � 75 are separated out to an extent which, in the case of a strong field, may be much greater than the permanent separation produced by the relativity correction. - This is the phenomenon known as the Stark effect. The amount of separation required by the quantum-dynamics, using the same equations of restriction as we have already employed, has been worked out by Epstein,* and has been found to be in excellent agreement with observation, j* A still wider separation can be produced by placing the atom in a powerful magnetic field, giving rise to the Zeeman effect. A theory of this in terms of quantum-dynamics has been given by Sommerfeld| and Debye,� and this is found to show satisfactory agreement, as regards its general features, with observation. This theory employs quantum restrictions of the type we have had under consideration, except that the co-ordinates are supposed to be measured with reference to a system of axes in slow rotation. Recently W. Wilson|| has shown that the necessity for rotating axes can be avoided by taking the quantum restrictions in a slightly more general form, and the theory has been further developed by Mosharrafa.lf Full details, with reference to both the Stark and the Zeeman effect, will be found in Sommerfeld's book. Bohr's Principle of Correspondence. 80. We found in � 64 that the problems which had so far been discussed appeared to be governed by general laws, which we then expressed by the equations 2T=rhv (t=1, 2, 3,.....) .....(124) &W=�hv .............(125) We have proceeded as far as is at present profitable with the generalisation of equation (124), and now turn to a consideration of equation (125). Bohr has shown that this provisional equation admits of an especially interesting interpretation in the special case in which AW is so small that continuous differential analysis may legitimately be used. Equation (124) has reference to a system describing a recurrent motion of period a=l/v, this motion being in accordance with the classical mechanics, so that if E is the total energy, expressed in the form (105), d^=Jl and dllJl ........ (126) at Ms dt dps * Ann. d. Physik, 50, p. 489 (1916). For an account in English, see Sommerfeld's book. just referred to, or J. H. Jeans, " Dynamical Theory of Gases" (3rd Edition), p. 427. t See also H. A. Kramers, Zeitschr. fur Physik, 3, p. 199 (1920). % Phys. Zeits., 17, p. 491 (1916). � Phys. Zeits., 17, p. 507 (1916). || Proc. Roy. Soc, 102, p. 478 (1923). If Proc. Roy. Soc, 102, p. 529 (1923). Dynamics of the Quantum-Theory. 75 Following the classical mechanics, let us imagine that we pass to a slightly varied motion, of energy E+<5E and of period a+da. Let us write 2T rv ra I=� =2 [ Tdt= Zpsqsdt, v Jo Jo and suppose that in the varied motion I becomes I+<5I. We then find by the ordinary calculus of variations r* . . . (51= Z{qsdps+ps6qs)dt+{Spsqs)t=:(rdG J o = / Z(qsdps-psdqs)dt+(ZpsSqs+ZpsqsdG)t=(r J o Since Sg is given by the relation &7s+&A*=0, when t=a the last term is readily seen to vanish, while, by the use of equations (126) the first term gives 31= T Z (� dps+ � Sqs ) dt= ($Edt. . . (127) Jo sdPs d are true under the quantum-theory as they were under the old dynamics. But under the old dynamics X, Y, Z had a very precise meaning ; they were components of a force, and an electron of charge e placed in the field would experience a mechanical force Xe, Ye, Ze. Under the quantum-dynamics this cannot be the true meaning of X, Y, Z, for if it were, an electron placed in a field of radiation would be absorbing energy otherwise than by complete quanta. Indeed, we have already seen in �10 that the supposition that an electron in a field of radiation is acted upon by a force * Proc. Camb. Phil. Soc, 15, p. 114 (1909). t Proc. Roy. Soc, 104A, p. 115 (1923). Physical Basis for the Quantum-Theory. 81 Xe, Ye, Ze, leads to the equipartition law for the distribution of radiant energy, and this is in opposition to quantum-dynamics. 88. It seems possible that real progress towards a physical understanding of the quantum-theory may follow the adoption of the standpoint assumed by Einstein in his Paper " Zur Quantentheorie der Strahlung,"* which we have already explained in � 23 of the present report. On this view a dynamical system may jump from one state to another either spontaneously or as the result of the action of external agencies, and obviously quantum-dynamics must be intimately concerned with the probability of such jumps, We can avoid all contradiction with the conceptions of the quantum-theory, and at the same time retain equations (134), if we suppose that X, Y, Z, or some functions of these quantities, measure, in some way at present unknown, the probabilities of jumps in the velocity and perhaps also in the position of an electron which forms part of an atomic system. In the limiting case in which X, Y, Z change only very slowly with the time, the whole field of radiation may be analysed, by Fourier analysis, into trains of waves of very low frequency. For radiation of low frequency the quantum is very small, and the jumps required by quantum-theory are so slight that the motion of the material electrons approximates closely to continuous motion. When the jumps in the electron velocities are very slight, they must be very frequent if finite change is to be produced, and instead of considering the probability of a jump occurring in one second, it is more natural to discuss the " average number " of jumps per second. But the " average number " of jumps per second in the velocity of an electron is proportional to the increase per second in this velocity, and so again is proportional to the electron's acceleration. On the classical mechanics, X, Y, Z are proportional to the acceleration of the electron; on the quantum-mechanics we have seen that they are proportional to this acceleration in the special case of radiation of very long wave-length, but that they assume some other meaning in the general case of radiation for which the quantum is not of inappreciable amount. Thus the quantum-mechanics must include the classical electrodynamical theory as a special case appropriate to radiation of great wave-length ; this consideration explains why the true black-body radiation formula (Planck's formula) reduces to the equipartition formula when the wave-length of the radiation is v^ry great. 89. This train of thought goes some way, although indeed only a very short way, towards suggesting a picture of the physical processes at work in quantum phenomena. As an instance let us examine that phenomenon which is perhaps the most perplexing of all to one who thinks in terms of the old mechanics�the X-ray photo-electric effect. X-rays of frequency v are produced at a point P, and instantly ^-particles are shot off from atoms at distant points Q, Q', Q", . , . . Their velocity depends only on the frequency v of the X-rays, being given by Einstein's equation (71) �tnv2=h(v�v0) which, in view of the extreme smallness of v0 in comparison with v, assumes, as we saw in � 50, the simple form �mv2�hv ............(135) For any particular experiment it is easy to calculate the rate at which energy is radiated away from the source P, and hence to estimate the time necessary for an * Phys. Zeit., 18, p. 122 (1917). G 82 Radiation and the Quantum-Theory. amount hv or \mv2 to be absorbed by an atom at Q, For ordinary laboratory conditions the time comes out of the order of years,* whereas in practice the emission of /^-particles commences practically instantaneously. From the point of view of the old mechanics, this transfer of energy is entirely unintelligible. Even apart from the rate of transfer there is the further difficulty of imagining a mechanism by which the energy can be stored up in the atom until it is suddenly set free by the ejection of an electron as a /^-particle. It has been suggested, with entire fairness from the point of view of the old dynamics, that an analogous case would be if the accidental falling overboard of a deck-cargo of planks from a ship in the English Channel were to result in the shooting out of similar planks from the sides of ships scattered all over the Atlantic Ocean, the velocities of these latter planks each being exactly equal to that with which the original planks fell into the water. The interpretation of the quantum-theory which we are now considering promises to bring at least a little order out of all this chaos. We believe that, in accordance with Maxwell's equations, X-rays of a definite frequency v produced at P must spread out into space as spherical waves. According to the old mechanics, liny mechanical effect of these waves was proportional to the square of their amplitude, and so fell off inversely as the square of the distance. According to the quantum-theory, however, AW=/w, so that any mechanical effect depends only on the frequency v. Now, according to Maxwell's equations, the frequency v is the one and only quantity which a train of waves carries with it unchanged to all distances. Thus the circumstance that the energy of the ejected ^-particles is independent of their distance from the source of the X-rays now appears as a direct consequence of Maxwell's equations. According to the classical electrodynamics, the energy per unit area of a wave depended on the square of its amplitude, and so fell off as ljr2 as the distance from the source increased. Since the area of the wave-front increased as r2, the whole energy of the wave remained constant, and the conservation of energy was satisfied. According to our present view, we are not entitled to say that 1_ {X2+Y2+Z2+a2+P2+y2) oTt represents energy per unit volume, nor indeed are we entitled to think of energy as being localised in free space at all. A separate investigation is therefore needed to discover what becomes of the conservation of energy on the quantum-theory; For waves of very great wave-length, the quantum hv is infinitesimal, and we may assume the classical mechanics to hold in the limiting case of i>�0. Thus, if A is the amplitude at a distance r from the source, energy is transferred from the radiation to any material structures on which it falls at a rate which is proportional to A2. If we divide the transferred energy, quite artificially for the present, into chunks each of equal but quite arbitrary amount AW, then the classical mechanics requires that the number of chunks of energy transferred per unit time shall be proportional to * Soniinerfeld, Atomic Structure and Spectral i,mes, p. 44. Physical Basis for the Quantum-Theory. 83 To obtain the classical mechanics exactly we put AW=0. To obtain the quantum-mechanics we put AW=/w. Our artificial chunks of energy become quanta, and in place of formula (136) we have that the number of quanta of energy transferred per unit time is proportional to A2/hv. Or, since we do not contemplate the possibility of a single atom absorbing a whole succession of similar quanta, we slightly alter our statement and say that the probability of a quantum hv of energy being transferred in unit time is proportional to A2/hv. According to Maxwell's equations A2jhv varies as 1/r2, so that the probability of an atom absorbing a quantum of energy varies inversely as the square of its distance from the source of radiation. We no longer think of a spherical wave as carrying with it a store of energy in each element of its wave-front, to be yielded up to whatever matter it encounters; we rather think of it as carrying with it a potentiality of yielding up energy to such matter as it encounters in a suitable state to receive this energy. The amount of energy which it may yield up remains always equal to hv, but the chance of its yielding this amount up falls off as 1/r2. The total " expectation " of the energy to be yielded up by the whole wave front, of area inr2, is accordingly independent of r. The conservation of energy now reappears, but merely as a statistical law, which is not necessarily true in every individual case, but has the same probability of truth as, for instance, the second law of thermodynamics. We are obviously far removed from the conception of radiation as energy which can be localised in space, and which moves about as directed by the law of the Poynting Flux; but the abandonment of this conception need cause no regrets. 90. On this or any other view of the meaning of radiation, there remains the outstanding difficulty of imagining a structure of matter such that an atom can absorb radiation only when it is offered in specified doses hvv hv2> . . . , where vv v2 . . . are, so far as we know, perfectly definite frequencies. In this connection we may notice that a free electron cannot accept a quantum of radiation under any circumstances whatever. For if an electron moving freely with a velocity u accepted a quantum hv of energy from a field of radiation, its final velocity v would be given by \mv2=\mu2 -\-h v. For an observer moving with a velocity v. equal to the final velocity of the electron, this equation would reduce to so that the sum of two essentially positive quantities would have to be zero. Thus the simplest structure which can absorb a quantum of energy is a complete atom, and the same is obviously true with regard to emission. In an atom we must suppose the electrons bound to the nucleus not only by its electrostatic force, but also by the more rigid bonds of the quantum-theory which, in some way at present totally unknown, compel that certain quantities associated with the orbit shall exist in complete quanta only. The simplest conception we can form of the absorption or emission of a quantum of energy is that these latter bonds are momentarily dissolved, that the whole atom is for the time being put, so to speak, into the melting pot, and that a new system, subject to new quantum-bonds, is in due course formed out of the old constituents endowed with 84 Radiation and the Quantum-Thcoty. their new appropriate amount of energy. We are still very far indeeed from being able to fill in the details of this picture, although we know that in the limiting case of radiation of very great wave-length, for which the emitting or absorbing electron must be very distant from the nucleus and the quantum-bonds so feeble that they may be disregarded altogether, the whole process becomes identical with that predicted by the classical mechanics. 91. If the point of view from which we have just been considering the question is a legitimate one, then the classical electrodynamics, and, of course, also the Newtonian mechanics, fall into place very naturally in the general scheme of nature. But as soon as we pass beyond the minute range covered by the classical electrodynamics and the Newtonian mechanics, the picture of this general scheme presented to us by the quantum-dynamics is strangely unexpected; we find�or at least we at present think we find�that nature consists of a series of more or less abrupt jumps, although just in that one particular corner of the picture which we have so far been able to explore with some thoroughness these jumps are so minute and so close together as to produce the illusion of continuous motion. It is perhaps not altogether an idle question to inquire whether the surprise we all feel at the newrly-discovered discontinuity, or apparent discontinuity, of nature is philosophically justifiable, or whether it arises merely from the inertia of our minds. A man who unrolls what he had thought, judging from one corner of it only, to be a crayon-drawing, and finds that it is a line-engraving, in which the illusion of continuity is limited to the one corner he first saw, is, no doubt, surprised, but he has only himself to thank, for his surprise arose merely from his having, in an unscientific frame of mind, generalised too hastily and assumed that the unseen would be in essence a repetition of the seen. Is or is not his surprise a fair analogy to the surprise which the progress of the quantum-theory has excited in us ? The supposed discontinuity may itself prove to be unreal; the " jumps " may ultimately be found to consist of a succession of continuous steps. To borrow an analogy from Poincare : the first observer of a collision 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. If there is a real difficulty in imagining discontinuity to be the ultimate rule of nature, we can escape this difficulty by imagining a sort of sub-universe such that the apparently discontinuous processes of the quantum-theory result from a rapid succession of continuous changes in this sub-universe. The conception of there being a definite " probability " of a spontaneous " jump " of a system from one state to another, a conception which is an essential feature of the view we have just been considering, may be thought to point rather in this direction. On the other hand> to some minds it may appear as likely that the ultimate scheme of nature will prove to be discontinuous as that it will be continuous. The circumstance that discontinuity was an unexpected solution of the problem ought not to prejudice us against the possibility of its being the final solution. For the history of science teaches that each great advance towards ultimate reality has shown this reality to lie in an entirely unexpected direction. INDEX TO AUTHORS. Balmer, 38, 39, 40, 41, 47, 71, 73. Blake, 44, 61. Bohr, 35-44, 47, 48, 49, 62, 66, 67, 69, 74, 76, 77. Boltzmann, 15, 65. Born, 56, 59, 60. Brackett, 38. Bragg, W. H., 57. Broglie, de, 50. Brown, 18. Bucherer, 38. Buisson, 73. Burgers, 65, 69. Campbell, 63. Coblentz, 61. Darwin, 21. Davies, 49. Davis, 49. De Broglie, 50. Debye,53-58, 60, 69, 74. Duane, 44, 61. Dulong & Petit, 52. Dyson, 40. Ehrenfest, 65, 66. Einstein, from 22, continually quoted. Ellis, 50. Epstein, 41, 69, 74. Evans, 39. Evershed, 40, 41. Fabry, 73. Fourier, 77, 81. Fowler, A., 39, 41, 63. Fowler, R. H., 21, 24. Franck, 48. Gerlach, 48. Goucher, 49. Gruneisen, 60. Hamilton, 26. Hartmann, 48. Hertz, G., 48. Hertz, H., 45. Horton, 49. Hull, 44. Hunt, 44. Karman, 56, 59, 60. Keesom, 56, 61. Kelvin, 34. Koref, 52. Kramers, 74. Ladenburg, 45. Larmor, 9. Lenard, 45. Lienhop, 45. Ivindemann, 52, 53, 57, 58, 59, 60. Liouville, 15. Ivorentz, 9, 13, 14, 33, 79. Lyman, 38, 39. Maxwell, 4, 33, 80, 82, 83, &c. Merton, 40, 73. Michelson, 73. Millikan, 38, 41, 45, 46, 61. Moseley, 43. Mosharaffa, 74. Nernst, 52, 53, 57-60. Nicholson, 37, 40, 42, 66, 73. Onnes, Kamerlingh, 56, 61. Oxley, 60. Paschen, 38, 39, 41, 73. Pickering, 40. Planck, continually quoted, Pohl, 45. The numbers refer to the pages. 86 Index to Authors. Poineare, 7, 19, 24, 25, 28, 84. Poynting, 83. Pringsheim, 45. Rau, 49. Rayleigh, 8, 16, 17, 24. Richardson, 13, 18. Richter, 59. Ritz, 63. Rubens, 58. Rutherford, 34. Rydberg, 41, 61, 63, 76. Sommerfeld, 44, 48, 49, 69-74, 77, 82. Stark, 41, 74. Stefan, 19. Taylor, 80. Thirring, 57, 59, 60. Thomson, G. P., 80. Thomson, J. J., 17, 18, 79. Weyl, 14. Wien, 19. Wilson, W., 74. Wood, R. W., 40, 48. Wright, 41. Zeeman, 33, 74. smE3a-i