Refraction of X-rays.

E. V. Shpolsky.

Attempts to detect the refraction of X-rays had already been made from the time of the discovery of these rays. Numerous variations of the experimental conditions were tried. Thus, Röntgen used prisms made of ebonite, glass, aluminum, and water; Perrin1 carried out experiments with prisms of wax and paraffin; Chapman2 filled a prism with vapors of \(C_2H_5Br\), and the radiation he studied included also frequencies corresponding to the characteristic radiation of bromine. In all cases the result was negative, although Chapman, with his apparatus, could have detected refraction if the refractive index had lain within the limits

\[ 1 \pm 0.003. \]

Subsequently similar attempts were made by Barkla3, Webster and Clark4, Ledoux-Lebard and Dauvillier5, these investigators working in those wavelength regions where anomalous dispersion might be expected, and hence a rapid change of the refractive index. However, these attempts also remained fruitless.

Thus the refractive index for X-rays must be very close to unity, which, incidentally, is in full agreement with theoretical predictions. Nevertheless, it does differ from unity, and this circumstance became evident as soon as X-ray spectroscopy attained modern striking accuracy. The point is that calculation of the wavelength is made by Bragg’s formula \(n\lambda = 2d\sin\theta\), in whose derivation the refractive index \((\mu)\) is taken to be equal to unity. Stenström6 first showed that, for wavelengths greater than \(2.5\,\text{\AA}\), deviations from this simple formula are already observed in spectra of the second order if a calcite crystal is used. Ewald then showed that these deviations can be fully explained if one assumes that \(\mu \ne 1\), i.e. that X-rays are refracted to an appreciable extent. Such a possibility had already been foreseen in 1914 by Darwin, who, among other things, showed that if \(\lambda_1\) and \(\lambda_2\) are the wavelengths observed in the \(n_1\) and \(n_2\) orders, then

\[ \delta = 1 - \mu = \frac{\lambda_1 - \lambda_2}{\lambda_2} = \frac{n_2^2}{n_2^2 - n_1^2}\sin^2\theta_1 . \tag{1} \]

If the refractive index is known, then the correct wavelength can be found from the formula

\[ n\lambda = 2d\sin\theta\left(1 - \frac{\delta}{\sin^2\theta}\right). \]

Duane and Patterson7 found for the tungsten \(L_\alpha\) line an apparent change in wavelength in the first and second orders of the spectrum, obtained with a calcite crystal, \(\lambda_1 - \lambda_2 = 0.00015\,\text{\AA}\). Using equation (1), we find, in this case, that

the refractive index must be less than unity by the amount \(\delta = 8 \cdot 10^{-6}\). Measurements by the same investigators gave, for wavelengths \(1.279\) and \(1.096\) Å respectively, \(\delta = 10 \cdot 10^{-6}\) and \(3 \cdot 10^{-6}\).

Thus these experiments show that the refractive index for X-rays must be less than unity. But in that case, by making a beam of rays fall on a mirror at a sufficiently large angle, one can obtain total reflection. The limiting angle is found from the formula: \(\cos \theta = \mu\), or—which is the same thing—

\[ \sin \theta = \sqrt{2\delta}, \]

where \(\theta\) is the angle of glancing incidence, and \(\delta\), as before, is \(1-\mu\). The value of \(\delta\) for glass (crown glass) can be calculated from Lorentz’s dispersion formula (see below); it is equal to \(5.2 \cdot 10^{-6}\). In that case \(\theta = 11'\), a value quite accessible to measurement. And indeed, Compton1 in 1922 was the first to observe such total reflection of X-rays from glass, silver, and a surface coated with lacquer. Compton’s results are compared in the following table:

Davis and his collaborators (Hardroth, Gatley, and others) took a somewhat different path. The principle of their ingenious “wedge method” is as follows. Let us imagine a crystal whose surface is ground at some angle \(\varphi\) to the cleavage planes. Let a beam of rays fall on this crystal, producing reflection at the glancing angle \(\theta\) between the ray and the reflecting molecular planes. It is easy to see (see Fig. 1) that the angle between the incident ray and the ground surface of the crystal \(AB\) will be \(\theta_1+\varphi\); the angle between the reflected ray and \(AB\) will be \(r=\theta-\varphi\). Since this latter angle will be small (\(<1^\circ\)), refraction must already become apparent here; therefore, on emerging from the crystal, the ray will travel not along \(SQ\), but along \(SQ'\), forming with \(AB\) an angle \(i>r\). Let us call \(i-r=\gamma\), i.e. \(i=r+\gamma\). Now suppose that the crystal is rotated about the axis \(xx'\) by \(180^\circ\). Obviously, this is equivalent to considering the incident ray as coming from the right rather than from the left. Consequently, refraction will now appear in the incident ray, and therefore when the crystal is rotated by exactly \(180^\circ\) we shall not obtain reflection.

Fig. 1.

REFRACTION OF X-RAYS

In order to obtain reflection, it is necessary to rotate the crystal through an additional angle \(\gamma\). In this way we can easily find the angle \(\gamma\) we need, which under known conditions reaches a considerable value, up to \(218.1''\). From this it is easy to determine also the refractive index of the crystal

\[ \mu=\frac{\cos i}{\cos r}=\frac{\cos(r+\gamma)}{\cos r}=1-\delta . \]

By these and analogous methods the refractive indices of several crystals for X-rays were determined. Thus, for molybdenum \(K_{\alpha}\) \(\left(\lambda=0.707717\,\text{\AA}\right)\) the following values of \(\delta\) were obtained:

\[ \begin{array}{lcl} & & \delta\cdot 10^{6}\\ \text{Calcite }(\mathrm{CaCO}_{3}) & \ldots & 2.03\pm0.9\\ \text{Pyrite }(\mathrm{FeS}_{2}) & \ldots & 3.35\pm0.2 . \end{array} \]

It should be noted that the values of the refractive index found experimentally agree very well with those calculated from the dispersion formula of Lorentz\(^1\)

\[ \mu^{2}-1=\sum_{m}\frac{N e^{2}}{\nu_{0}^{2}}\, \frac{1}{(\nu_{0}^{2}-\nu^{2})-\frac{1}{3}} . \]

If in this formula we pass from Lorentz’s rational units to the usual absolute units and make certain transformations and simplifications, using the fact that all \(\nu\) are very large and \(\mu\) is close to unity, then the formula takes the form

\[ \delta=\frac{e^{2}}{2\pi m} \left[ \frac{n_{1}}{\nu^{2}-\nu_{1}^{2}} + \frac{n_{2}}{\nu^{2}-\nu_{2}^{2}} +\cdots \right], \tag{2} \]

where \(e\) and \(m\) have their usual meanings, and \(n_{1}, n_{2}\ldots\) are the numbers of electrons per unit volume having the natural frequencies \(\nu_{1}, \nu_{2}\ldots\). We have already seen that Compton’s data agree well with this formula. The results obtained by the “wedge method” also satisfy it quite fully. For example:

The figures given in the third column were calculated on the assumption that the number of electrons in the \(K\)-shell is equal to two. Calculations made under other assumptions (0, 1, 3 electrons) give results less consistent with experiment. Thus we obtain the possibility of directly determining the number of electrons at the various energy levels.

In conclusion, attention should be given to two important works carried out in Siegbahn’s laboratory, but so far published only in the form of preliminary communications. By an appropriate choice of the experimental conditions (refracting angle, angle of incidence), Siegbahn, Larsson, and Waller succeeded in producing refraction

\(^1\) H. A. Lorentz, The Theory of Electrons, 2nd ed., p. 144.

of X-rays by a prism (crystalline and even simply glass). A very thin beam of X-rays fell on the prism at an angle close to the limiting one. Part of this beam passed above the edge of the prism and left on the plate the trace of the direct ray; part underwent total reflection at the front face of the prism, and the reflected ray was excellently recorded on the photographic plate; finally, the remaining rays, having passed through the prism, were refracted and gave a sharp line spectrum. Thus there was photographed the prismatic line spectrum of an anticathode consisting of iron and copper. Calculation of the value \(\delta = 1-\mu\) gave the following results:

Glass prism (density: 2.551).

The last column shows that \(\delta\), as was to be expected, is proportional to \(\lambda^2\) (see, for example, the dispersion formula).

Finally, in their last communication, Hjalmar and Zitban give data establishing anomalous dispersion in the region of X-rays. In order to determine more accurately the wavelengths of X-rays, the authors undertook a precise comparison of the lattice constants of calcite \((\mathrm{CaCO_3})\) and gypsum \((\mathrm{CaSO_4})\) for various wavelengths from 0.7 to \(5.2\,\mathring{\mathrm A}\). Plotting \(\lambda\) along the axis of abscissae, and

\[ \frac{d_1}{d_2}=\frac{\sin\theta_2}{\sin\theta_1} \]

along the axis of ordinates, one can find in the course of the curve two sharp jumps: one corresponds to the absorption edge of calcium, the other—to the absorption edge of sulfur. The explanation of this behavior lies in the following. It can be shown that in the Bragg formula, in exact calculations, when refraction is taken into account, \(d\) must be replaced by the following quantity:

\[ d=d_0\left[1-\frac{4d_0}{n^2}\cdot\frac{\delta}{\lambda^2}\right], \]

where \(d_0\) is the crude value obtained from the Bragg formula, \(n\) is the order of the spectrum, and \(\delta=1-\mu\) has the value determined by formula (2). Thus, for \(\nu\) equal to the natural frequencies \(K, L, M\) of the electrons, \(d\) must undergo sharp jumps.

The works cited undoubtedly open a new chapter in the optics of X-rays. In particular, knowledge of the refractive index makes it possible to calculate the wavelength in a new way, independent of knowledge of the structure of the crystal. On the other hand, the study of anomalous dispersion of X-rays will give a direct method for determining the number of electrons at various energy levels.

LITERATURE.

1) For the earlier works see Ledoux-Lebard et Dauvillier. La Physique des Rayons X. Paris, 1921, p.p. 109—114.

2) Stenström. Experimentelle Untersuchungen der Röntgenspektra. Diss. Lund. 1919.

1. Ewald, P. P. Abweichungen vom Braggschen Reflexionsgesetz der Röntgenstrahlen. — Phys. ZS. 21, p. 617 (1920).

2. Ewald, P. P. Zum Reflexionsgesetz der Röntgenstrahlen. — ZS. für Physik. 2, p. 332 (1920).

3. Duane and Patterson: On the X-Ray Spectrum of Tungsten. — Phys. Rev. 16, p. 526 (1920).

4. Compton, A. H. The Total Reflexion of X-Rays. — Phil. Mag., 45, p. 1121 (1923).

5. Nardroff, R. v. The Refraction of X-Rays in Iron Pyrites. — Phys. Rev. 24, p. 143 (1924).

6. Hatley, C. C. Index of Refraction of Calcite for X-Rays. — Phys. Rev. 24, p. 486 (1924).

7. Larsson, M. Siegbahn und J. Waller. Der experimentelle Nachweis der Brechung von Röntgenstrahlen. — Die Naturwissenschaften, 12, p. 1212 (1924).

8. Hjalmar, E. and Siegbahn, M. Anomalous Dispersion in the Field of X-Rays. — Nature, 115, p. 85 (1925).

9. Ewald, P. Über den Brechungsindex für Röntgenstrahlen und die Abweichungen vom Braggschen Reflexionsgesetz. ZS. für Phys., 30, 1 (1924).

1. A. H. Compton, Phil. Mag. 45, p. 1121 (1923). ↩↩

2. Chapman, Proc. Cambr. Phil. Soc., p. 574 (1912). ↩

3. Barkla, Phil. Mag., 31, p. 257 (1916). ↩

4. Webster and Clark, Phys. Rev., 8, p. 528 (1916). ↩

5. Cf. Ledoux-Lebard et Dauvillier, La Physique des Rayons X, p. 11, Paris, 1921. ↩

6. Stenström, Diss. Lund, 1919. ↩

7. Duane and Patterson, Phys. Rev. 26, p. 532 (1920). ↩