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Anomalous Dispersion in the X-ray Region (J. A. Prins, ZS. f. Phys. 47, 479, 1928). As is known, in recent years several investigators have succeeded in detecting refraction in the X-ray region1. It has turned out that the refractive index in this region is of the order of \(10^{-6}\) less than unity:
\[ n = 1 - \delta . \tag{1} \]
The quantity \(\delta\) in the first approximation can be easily calculated if one uses the Lorentz dispersion formula and makes certain simplifications admissible owing to the high frequency of X-rays. In this way one obtains1:
\[ \delta=\frac{e^2}{2\pi me^4}\lambda^2 N, \tag{2} \]
where \(N\) is the number of electrons in a cubic centimeter. In deriving this formula, among other things, it is assumed that all the electrons are free, i.e., that the forces binding them may be neglected in comparison with the quantity \(h\nu\) of the X-rays. Thus formula (2) determines the normal dispersion. But alongside this normal dispersion, for wavelengths close to the absorption edge of a given atom, one may also expect anomalous dispersion caused by more firmly bound electrons. The existence of anomalous dispersion has been established with complete certainty in the work under review by Prins (ZS. f. Phys. 47, 479, 1928, Naturwiss. 28, 555, 1928).
The usual method of determining the refractive index for X-rays consists in finding the limiting angle of total reflection. Since the refractive index for X-rays is always less than unity, these rays, on passing from vacuum into any medium, may undergo total reflection. The limiting angle of total reflection is calculated approximately from the formula:
\[ \varphi_m=\sqrt{2\delta}. \tag{3} \]
Thus, knowing \(\varphi_m\), one can find \(\delta\), and consequently also the refractive index \(n\).
To determine \(\varphi_m\) as a function of \(\lambda\), Prins used an ingenious method, reminiscent of the crossed-prism method, by means of which Kundt had shown with particular clarity the existence of anomalous dispersion in the optical region. The radiation of an X-ray tube with a tungsten anticathode, with the aid of a crystal capable of rotating about a horizontal axis, and of a horizontal slit, was spread out into a continuous spectrum. By means of two vertical slits, a narrow strip was selected from this spectrum and recorded on a plate in the form of a straight spectrum. In the path of the rays between the second vertical slit and the plate, a steel mirror was placed at a small angle to the beam of rays. In this way part of the beam was reflected from the mirror and gave a reflected spectrum. The mirror could rotate about a vertical axis. As a result, a whole series of reflected spectra was obtained, directly adjoining one another and forming one continuous band. It is quite clear that the outline
of the outer boundary of this band immediately gives us the required dependence of \(\varphi_m\) on \(\lambda\), just as the form of a spectrum obtained through crossed prisms gives the course of the refractive index as a function of \(\lambda\).
The principal component of the mirror was iron (the composition of the steel used was as follows: \(72\,\mathrm{Fe} + 19\,\mathrm{Cr} + 8\,\mathrm{Ni} + 1\,\mathrm{C}\)). The spectral region used in the experiments of Prince extended from 1675 XU to 1938 XU, i.e., it also included the \(K\)-absorption edge of Fe (1739 XU). In considering the reflected spectrum, two circumstances attract attention: 1) The brightness of the reflection changes abruptly at a certain definite place. The wavelength corresponding to the place of this jump is 1740 XU, i.e., it is equal to the wavelength of the \(K\)-absorption edge of iron. For wavelengths shorter than this edge the reflection is very weak; for longer wavelengths the reflection is complete. The cause of the jump is the onset of strong absorption on crossing the absorption edge toward shorter wavelengths; this circumstance alone can already serve as an indication of anomalous dispersion. 2) On the side of the longer wavelengths, as was said, the reflection is complete; the boundary of the reflected spectrum is sharply outlined, and its form characterizes the dependence of \(\varphi_m\) on \(\lambda\). Formulas (2) and (3) show that in the case of normal dispersion this dependence should be linear. Meanwhile the outer edge of the reflected spectrum near the place of the jump (the \(K\)-edge of iron) is a crooked line. This fact serves as qualitative proof of the existence of anomalous dispersion. Prince, however, also obtained quantitative proof. Namely, using formulas borrowed from the theory of anomalous dispersion in the X-ray region (H. Kallman und H. Mark, Ann. d. Phys. 82, 585, 1927), Prince constructed the curve of the dependence of \(\varphi_m\) on \(\lambda\). It turned out that the experimentally found points fit this curve quite satisfactorily.
E. Shpolsky.