A Quantum Theory of the Scattering of X-rays by Light Elements

A quantum theory of the scattering of X-rays and p-rays by light elements. —The hypothesis is suggested that when an X-ray quantum is scattered it spends all of its energy and momentum upon some particular electron. This electron in turn scatters the ray in some definite direction. The change in momentum of the X-ray quantum due to the change in its direction of propagation results in a recoil of the scattering electron. The energy in the scattered quantum is thus less than the energy in the primary quantum by the kinetic energy of recoil of the scattering electron. The correspondingincrease in the wave-length of the scattered beam is Xg —Xp = (2h/mc) sin'-,'9 = o.o484 sin'-', 8, where h is the Planck constant, m is the mass of the scattering electron, c is the velocity of light, and 0 is the angle between the incident and the scattered ray. Hence the increase is independent of the wave-length. The distribution of the scattered radiation is found, by an indirect and not quite rigid method, to be concentrated in the forward direction according to a definite law (Eq. 27). The total energy removed from the primary beam comes out less than that given by the classical Thomson theory in the ratio z/(x + 2n), where a = h/mcXp = 0.0242/P p. Of this energy a fraction (r + n)/(I + 2n) reappears as scattered radiation, while the remainder is truly absorbed and transformed into kinetic energy of recoil of the scattering electrons. Hence, if op is the scattering absorption coePcient according to the classical theory, the coefficient according to this theory is 0 = o.p/(I + 2n) = 0. + cr„where 0. , is the true scattering coefficient [(I + a)0/(j: + 2a)'], and 0, is the coefficient of absorption due to scattering [no-/(z + 2n)'j. Unpublished experimental results are given which show that for graphite and the Mo—K radiation the scattered radiation is longer than the primary, the observed difference ()~f2 —Xp =,022) being close to the computed value . 02'. In the case of scattered p-rays, the wave-length has been found to vary with 9 in agreement with the theory, increasing from . 022 A (primary) to . o68 A (9 = 135'). Also the velocity of secondary P-rays excited in light elements by y-rays agrees with the suggestion that they are recoil electrons. As for the predicted variation of absorption with X, Hewlett's results for carbon for wave-lengths below o.5 A are in excellent agreement with this theory; also the predicted concentration in the forward direction is shown to be in agreement with the experimental results, AR THUR H. CO3IIPTOX both for X-rays and y-rays. This remarkable of;reemenf betmeeN experimenf, amd theory indicates clearly that scattering is a quantum phenomenon and can be explained without introducing any new hypothesis as to the size of the electron or any new constants; also that a radiation quantum carries with it momentum as well as energy. The restriction to light elements is due to the assumption that the constraining forces acting on the scattering electrons are negligible, which is probably legitimate only for the lighter elements, Spectrum of K-rays from Mo scattered by graphite, as compared with the spectrum of the primary rays, is given in Fig. g, showing the change of wavelength. Radiation from a moving isotropic radiator. —It is found that in a direction 0 with the velocity, I0/I' = (J —p)'/(I —p cos 8)' = (lg/v')4. For the total radiation from a black body in motion to an observer at rest, I/I' = (T/T')' = (s /v ')', where the primed quantities refer to the body at rest. J. J. Thomson's classical theory of the scattering of X-rays, though supported by the early experiments of Barkla and others, has been found incapable of explaining many of the more recent experiments. This theory, based upon the usual electrodynamics, leads to the result that the energy scattered by an electron traversed by an X-ray beam of unit intensity is the same whatever may be the wave-length of the incident rays. Moreover, when the X-rays traverse a thin layer of matter, the intensity of the scattered radiation on the two sides of the layer should be the same. Experiments on the scattering of X-rays by light elements have shown that these predictions are correct when X-rays of moderate hardness are employed; but when very hard X-rays or y-rays are employed, the scattered energy is found to be decidedly less than Thomson's theoretical value, and to be strongly concentrated on the emergent side of the scattering plate. Several years ago the writer suggested that this reduced scattering of the very short wave-length X-rays might be the result of interference between the rays scattered by different parts of the electron, if the electron s diameter is comparable with the wave-length of the radiation. By assuming the proper radius for the electron, this hypothesis supplied a quantitative explanation of the scattering for any particular wavelength. But recent experiments have shown that the size of the electron which must thus be assumed increases with the wave-length of the X-rays employed, ' and the conception of an electron whose size varies with the wave-length of the incident rays is difficult to defend. Recently an even more serious difficulty with the classical theory of X-ray scattering has appeared. It has long been known that secondary p-rays are softer than the primary rays which excite them, and recent experiments have shown that this is also true of X-rays. By a spectroscopic examination of the secondary X-rays from graphite, I have, indeed, ~ A. H. Compton, Bull. Nat. Research Council, No. 2o, p. xo (Oct. , 1922). SCA'l-, TEEING OF X-AA. YS BY LIGHT ELEMENTS been able to show that only a small part, if any, of the secondary Xradiation is of the same wave-length as the primary. While the energy of the secondary X-radiation is so nearly equal to that calculated from Thomson s classical theory that it is difficult to attribute it to anything other than true scattering, ' these results show that if there is any scattering comparable in magnitude with that predicted by Thomson, it is of a greater wave-length than the primary X-rays. Such a change in wave-length is directly counter to Thomson's theory of scattering, for this demands that the scattering electrons, radiating as they do because of their forced vibrations when traversed by a primary X-ray, shall give rise to radiation of exactly the same frequency as that of the radiation falling upon them. Nor does any modification of the theory such as the hypothesis of a large electron suggest a way out of the difficulty. This failure makes it appear improbable that a satisfactory explanation of the scattering of X-rays can be reached on the basis of the classical electrodynamics. THE QUANTUM HYPQTHEsIs or SGATTERING According to the classical theory, each X-ray affects every electron in the matter traversed, and the scattering observed is that due to the combined effects of all the electrons. From the point of view of the quantum theory, we may suppose that any particular quantum of X-rays is not scattered by all the electrons in the radiator, but spends all of its energy upon some particular electron. This electron will in turn scatter the ray in some definite direction, at an angle with the incident beam. This bending of the path of the quantum of radiation results in a change in its momentum. As a consequence, the scattering electron will recoil with a momentum equal to the change in momentum of the X-ray. The energy in the scattered ray will be equal to that in the incident ray minus the kinetic energy of the recoil of the scattering electron; and since the scattered ray must be a complete quantum, the frequency will be reduced in the same ratio as is the energy. Thus on the quantum theory we should expect the wave-length of the scattered X-rays to be greater than that of the incident rays. The effect of the momentum of the X-ray quantum is to set the In previous papers (Phil. Mag. 4x, &49, I92r; Phys. Rev. x8, 96, I92I) I have defended the view that the softening of the secondary X-radiation was due to a considerable admixture of a form of fluorescent radiation. Gray (Phil. Mag. 26, 6I I, I9I3; Frank. Inst. Journ. , Nov. , j:920, p, 643) and Florance (Phil. Mag. ry, 225, 19r4) have considered that the evidence favored true scattering, and that the softening is in some way an accompaniment of the scattering process. The considerations brought forward in the present paper indicate that the latter view' is the correct one. ' A. H. Compton, loc. cit., p. I6. A R TH UA H. CO MPT0N scattering electron in motion at an angle of less than 90 with the primary beam. But it is well known that the energy radiated by a moving body is greater in the direction of its motion. We should therefore expect, as is experimentally observed, that the intensity of the scattered radiation should be greater in the general direction of the primary X-rays than in the reverse direction. The change in wave-length due to scattering. —Imagine, as in Fig. IA,