X-Ray Wave-Lengths by The Dispersion in Quartz

Abstract

It has been shown that the quantum theory of dispersion accounts satisfactorily for the measured refraction of x-rays. Since the x-ray wave-lengths as determined by crystals and by ruled gratings differ by 0.25 percent, the dispersion may be used to indicate which of these values is correct. Thus the quantum theory of dispersion may be written $\ensuremath{\lambda}={\ensuremath{\delta}}^{\frac{1}{2}}{\left[\frac{\ensuremath{\rho}}{W}\ifmmode\cdot\else\textperiodcentered\fi{}\frac{e}{m}\ifmmode\cdot\else\textperiodcentered\fi{}\frac{F}{2\ensuremath{\pi}}\ensuremath{\Sigma}\stackrel{s}{1}{N}_{s}\left\{1+\frac{log({x}^{2}\ensuremath{-}1)}{{x}^{2}}\ensuremath{-}\frac{2\ensuremath{\pi}q}{{x}^{3}}\right\}\right]}^{\ensuremath{-}\frac{1}{2}}$ where $\ensuremath{\delta}=1\ensuremath{-}\ensuremath{\mu}$, $\ensuremath{\rho}$ the density, $W$ the molecular weight, $\frac{e}{m}$ the ratio of the electronic charge to mass, $F$ the Faraday constant, ${N}_{s}$ the number of electrons per molecule of frequency ${\ensuremath{\nu}}_{s}$, $x=\frac{\ensuremath{\nu}}{{\ensuremath{\nu}}_{s}}$, $\ensuremath{\nu}$ the frequency of the incident radiation, $q=\frac{h}{{\ensuremath{\nu}}_{s}}$, $k$ the damping factor which can be obtained from the atomic absorption coefficient. Precise measurements of the refraction of the copper and molybdenum $K$ series in crystal quartz have been made. Both possible methods of using a prism were used. The 90\ifmmode^\circ\else\textdegree\fi{} edge of the prism was very carefully prepared. The results obtained are given in the following table. It is difficult to believe that such an agreement between the wave-lengths as determined by dispersion and by crystals is entirely fortuitous. The apparent precision obtained in the ruled grating measurements is so high that the above results must indicate a failure of the optical diffraction theory when applied to x-ray wave-lengths.