A Quantum Theory of the Scattering of X-rays by Light Elements (1923)

A quantum theory of the scattering of X-rays and $\ensuremath{\gamma}$-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 corresponding increase in the wave-length of the scattered beam is ${\ensuremath{\lambda}}_{\ensuremath{\theta}}\ensuremath{-}{\ensuremath{\lambda}}_{0}=(\frac{2h}{\mathrm{mc}}) {sin}^{2} \frac{1}{2}\ensuremath{\theta}=0.0484 {sin}^{2} \frac{1}{2}\ensuremath{\theta}$, where $h$ is the Planck constant, $m$ is the mass of the scattering electron, $c$ is the velocity of light, and $\ensuremath{\theta}$ 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 $\frac{1}{(1+2\ensuremath{\alpha})}$, where $\ensuremath{\alpha}=\frac{h}{\mathrm{mc}{\ensuremath{\lambda}}_{0}}=\frac{0.0242}{{\ensuremath{\lambda}}_{0}}$. Of this energy a fraction $\frac{(1+\ensuremath{\alpha})}{(1+2\ensuremath{\alpha})}$ reappears as scattered radiation, while the remainder is truly absorbed and transformed into kinetic energy of recoil of the scattering electrons. Hence, if ${\ensuremath{\sigma}}_{0}$ is the scattering absorption coefficient according to the classical theory, the coefficient according to this theory is $\ensuremath{\sigma}=\frac{{\ensuremath{\sigma}}_{0}}{(1+2\ensuremath{\alpha})}={\ensuremath{\sigma}}_{s}+{\ensuremath{\sigma}}_{a}$, where ${\ensuremath{\sigma}}_{s}$ is the true scattering coefficient [$\frac{(1+\ensuremath{\alpha})\ensuremath{\sigma}}{{(1+2\ensuremath{\alpha})}^{2}}$], and ${\ensuremath{\sigma}}_{a}$ is the coefficient of absorption due to scattering [$\frac{\ensuremath{\alpha}\ensuremath{\sigma}}{{(1+2\ensuremath{\alpha})}^{2}}$]. Unpublished experimental results are given which show that for graphite and the Mo-K radiation the scattered radiation is longer than the primary, the observed difference (${\ensuremath{\lambda}}_{\frac{\ensuremath{\pi}}{2}}\ensuremath{-}{\ensuremath{\lambda}}_{0}=.022$) being close to the computed value.024. In the case of scattered $\ensuremath{\gamma}$-rays, the wave-length has been found to vary with $\ensuremath{\theta}$ in agreement with the theory, increasing from.022 A (primary) to.068 A ($\ensuremath{\theta}={135}^{\ensuremath{\circ}}$). Also the velocity of secondary $\ensuremath{\beta}$-rays excited in light elements by $\ensuremath{\gamma}$-rays agrees with the suggestion that they are recoil electrons. As for the predicted variation of absorption with $\ensuremath{\lambda}$, Hewlett's results for carbon for wave-lengths below 0.5 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, both for X-rays and $\ensuremath{\gamma}$-rays. This remarkable agreement between experiment and 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. 4, showing the change of wave-length.Radiation from a moving isotropic radiator.---It is found that in a direction $\ensuremath{\theta}$ with the velocity, $\frac{{I}_{\ensuremath{\theta}}}{{I}^{\ensuremath{'}}}=\frac{{(1\ensuremath{-}\ensuremath{\beta})}^{2}}{{(1\ensuremath{-}\ensuremath{\beta} cos \ensuremath{\theta})}^{4}}={(\frac{{\ensuremath{\nu}}_{\ensuremath{\theta}}}{{\ensuremath{\nu}}^{\ensuremath{'}}})}^{4}$. For the total radiation from a black body in motion to an observer at rest, $\frac{I}{{I}^{\ensuremath{'}}}={(\frac{T}{{T}^{\ensuremath{'}}})}^{4}={(\frac{{\ensuremath{\nu}}_{m}}{{{\ensuremath{\nu}}_{m}}^{\ensuremath{'}}})}^{4}$, where the primed quantities refer to the body at rest.