NA TURE 



[December 23, 1922 



which have a natural frequency of vibration A„/27r, 

 the wave they scatter is given by 



E, = N 



mc- 



J- 



, cos pt. 



So if we identify N K e 2 /mc a with NA„«„ the expressions 

 are the same. But the only difference between the 

 phenomena of scattering and of the refractive index 

 lies in the matter of allowing for the mutual influence 

 of the atoms, an influence exerted by the waves 

 they send out and therefore the same on both theories. 

 So we may at once say that from ourresult will 

 follow the dispersion formula of Lorentz 



3j^ Li ) = ^4 I N^A a a n . 

 M a +2 " A„ 2 -p" 



From the linear way in which the chance of 

 excitation depends on the incident force, it follows 

 that the average effects of superposed waves is 

 additive ; in other words, the atoms act as Fourier 

 analysers, sort out the harmonic components of an 

 arbitrary incident wave and refract each component 

 in the proper degree. In all cases the characteristic 

 frequency with which the waves are really emitted 

 will entirely disappear by averaging. 



It will be necessary to consider the balance of 

 energy which is nearly but not quite exact, but the 

 present simple equations are not suited for this ; 

 they fail to give the balance even in the classical 

 case, and there it must occur. This question is 

 better treated in connexion with absorption. The 

 problem is complicated by the fact that the excited 

 wave may possibly have a phase differing slightly 

 (it may only be 'slightly) from that of a cosine. 

 I have assumed the form of the damping factor as 

 e -k„t on iy for convenience ; all that is necessary is 

 that the infinite end should be unimportant. An 

 alternative is to suppose that the wave is undamped 

 but that there is a chance \dt in every element of 

 time dt that it should stop. We have only discussed 

 waves polarised along the .r-axis and have supposed 

 that the excited waves have this axis as pole ; for 

 the general case the formulation must be somewhat 

 changed, but it would take too long to state and prove 

 the modification here. The essential points of the 

 theory are not altered, and it also appears that 

 there should be no particular difficulty in fitting 

 double refraction and rotatory dispersion into our 

 scheme. 



A theory of dispersion is not of course complete 

 without including selective absorption. If X„ is 

 retained in the integration of (2) the result is an 

 expression practically the same as that given in the 

 classical theory when a damping factor is included. 

 Observe that on the present theory, when the forced 

 period approaches the natural, there is no increase 

 either in the number of atoms excited or in the 

 strength of the waves they send out. The whole 

 change is due to the greater efficiency with which 

 they reinforce the primary beam. Our theory gi\ es 

 no explanation of the mechanism of conversion of 

 radiant energy into atomic heat, any more than does 

 the classical theory with its damping factor. The 

 conversion is probably better studied by the con- 

 sideration of other cases of absorption, such as 

 metallic reflection, and our method of argument, 

 applied to this last, should certainly give interesting 

 results. We shall have to find what emission of 

 spherical waves will diminish the aethereal energy 

 when superposed on the incident wave. Thus a 

 wave like that for dispersion would do for metallic 

 reflei tion, it the phase is suitably altered, or possibly 

 we may suppose that the wave is again in the form 



NO. 2773, VOL. I io] 



of a cosine, but that the chance of excitation is now 

 proportional to K x instead of to 3E„/3£. It seems 

 likely that a study of the optical constants of metals 

 would throw light on this question. Afterwards it 

 would be necessary to examine the balance of energy 

 between aether and matter, and this might help in 

 understanding the mechanism of the process. 



We may now review how these speculations will 

 modify the accepted theory. As we have made no 

 assumptions as to what goes on inside the atom, we 

 can take over the whole of the dynamics of stationary 

 states. We suppose that an atom is usually in its 

 lowest quantum state. The motions of the electrons 

 will sometimes lead to a favourable configuration, 

 and when this occurs in the presence of a changing 

 electric force, there is a chance that the atom may be 

 jerked into a condition in some way associated with 

 one of its higher quantised states. It at once starts 

 radiating with a frequency corresponding to the 

 return from that state to the lowest. Dispersion 

 throws no light on the amplitude of the wave, for 

 in the formula it always occurs multiplied by the 

 probability factor A„. It is rather tempting to sup- 

 pose that it actually goes into the higher quantised 

 state, and then gives a wave of such amplitude and 

 length that, but for the interference with the incident 

 light, it would emit energy hkJ2.1T. If this is so we 

 may perhaps extend our theory to cover pure emission; 

 for, though we have not postulated any precise 

 relationship between electric force and electrons, it 

 seems inevitable that there should be a rapidly 

 changing electric force near a moving electron, and 

 this force would have a chance of jerking the atom 

 into its higher state. On the other hand, difficulties 

 are raised in other directions. For the radiation 

 must be immediate and therefore the state would 

 not really be stationary at all, and the accepted theory 

 of specific heats requires that a molecule should be 

 able to remain in its higher states. In any case 

 there is a clear contradiction to the principle of 

 energy, but the phases of the outgoing waves are 

 so adjusted that for cases of pure scattering or 

 refraction, on the average, as much energy goes out 

 as comes in. 



There are many other points that will require 

 attention. In the first place the refractive index 

 is closely related to the dielectric constant. Now 

 though it is quite proper to treat the dielectric 

 constant as a limiting case of refraction, yet it can 

 be regarded electrostatically and it will be necessary 

 to see the physical meaning of this aspect. Again 

 it is possible to count the electrons in the atom by 

 X-ray reflection, and it follows that there must be 

 a relation between the e'jmc* of the classical theory 

 and our A„a„. In this connexion I owe to Prof. 

 P. S. Epstein the suggestion that the theory will 

 explain the defect observed in the scattering of 

 hard 7-rays below that predicted. Here the wave- 

 length of the incident light is much shorter than 

 the distances between the electrons and the incoherent 

 waves cannot recombine in the way they do under 

 the classical theory. Lastly, it will be necessary 

 to re-examine the deduction of the formula for black 

 radiation, for all present proofs are founded on 

 theorems following out of the conservation of 

 energy. 



In view of the great number of problems that are 

 suggested and the probability that it will take a 

 considerable time to deal with them, it appeared to 

 me that it might be of interest to publish this pre- 

 liminary account of a very incomplete theory. 



C. G. Darwin. 



Institute of California, 

 Pasadena, Cal. 



