The Continuous X-Ray Spectrum

Abstract

Under the assumption that the target of an x-ray tube is a plane mirror which completely reflects the electron waves, the characteristics of the continuous x-ray spectrum are deduced from the principles of wave-mechanics. It is also assumed that the velocity of the electrons is so small that its square may be neglected in comparison to $c^2$. The conclusions are: (1) The component of momentum parallel to the plane of the target is conserved, the momentum of the emitted radiation being taken to be $h\kappa$, ($\kappa =\mathrm{w}\mathrm{a}\mathrm{v}\mathrm{e}-\mathrm{n}\mathrm{u}\mathrm{m}\mathrm{b}\mathrm{e}\mathrm{r}$). The radiation due to a single transition is practically unidirectional, the direction of the beam (Nadelstrahl) being determined by the conservation laws of energy and momentum. The angular opening of the beam agrees with the uncertainty principle, using the linear dimensions of the target as $\Delta x$ in the formula $\Delta x\Delta p_x=\frac{h}{2\pi }$. (2) No light of $\kappa >{\kappa }_m$ is emitted (short wave-length limit). ${\kappa }_m$ is determined primarily by the energy of motion normal to the target, and not by the total energy. This result is due to the simplified model, and not to general principles. (3) The total intensity is proportional to ${{\kappa }_m}^2$ (empirically verified). The intensity in the wave-number interval $d\kappa$ is proportional to ${\left[{\kappa }_m\right({\kappa }_m-\kappa \left)\right]}^{\frac{1}{2}}d\kappa$. (4) The intensity has a maximum in the azimuth of the velocity of the incident electron, the normal to the target being the polar axis. The plane of the target is one of mirror symmetry for the intensity distribution—a result which is again inseparable from the model. (5) For very small velocities, the radiation is linearly polarized parallel to the normal of the target (spectroscopic sense of the phrase). For higher velocities, the polarization cannot be so simply described, but in any direction whose azimuth is that of the incident velocity, the light should be plane polarized, the electric vector lying in the plane determined by the direction of observation and the normal to the target. These results also have no very general validity. The conclusions seem in satisfactory agreement with experiment, the schematic character of the model being taken into account.