X-RAY OPTICS.

D. N. Nasledov, Kiev.

Refraction and reflection of X-rays.

Compton, in his address delivered in Bad-Kissingen in September of last year, emphasized that the phenomena of refraction and reflection should be regarded as a single problem. This follows from the fact that we can observe reflection only when a light ray falls on the boundary surface of two media with different refractive powers. Thus, by observing one phenomenon, we thereby also prove the existence of the other.

Beginning with Roentgen and up to very recent times, many unsuccessful attempts were made to detect the refraction of X-rays1.

Chance helped here to come closer to solving the problem. Thanks to Zsigmondy’s greatly improved method of X-ray spectrography, Stenström2, studying the spectra of X-rays of large wavelength with the aid of crystals of sugar and gypsum, discovered certain deviations from Bragg’s law. Later Ylmar investigated this question comprehensively and proposed a formula replacing Bragg’s formula with greater accuracy. These deviations from Bragg’s law proved possible to explain by the refraction of the ray in the crystal. In doing so it was only necessary to assume that the refractive index of X-rays is less than unity.

Compton drew from this the natural conclusion that X-rays, in passing from air into some body, may undergo total reflection. By proving the existence of total reflection, we thereby prove the refraction of X-rays. Moreover, by determining the limiting angle of total reflection, we obtain the possibility of calculating the refractive index of X-rays.

D. N. NASLEDOV

In 1923, Compton1 published the results of his investigations. Fig. 1 shows the arrangement of his experiment. A beam of X-rays, defined by the slits \(S_1\) and \(S_2\), falls upon the polished surface of silver or glass at an angle of several minutes and is reflected from it. The rays then fall on a crystal and subsequently enter an ionization chamber. In this way Compton proved that the refractive index of the materials investigated is indeed less than unity.

Fig. 1.

of the materials is indeed less than unity. Table 1 gives Compton’s measurement results for several substances.

Table 1.

Here \(\alpha\) is the limiting angle and \(\delta = 1 - \mu\), where \(\mu\) is the refractive index. Quite recently analogous investigations were carried out by Doan.2 Fig. 2 reproduces his photographs. From consideration of these photographs it is evident that the accuracy of measurement of the limiting angle can be very high. In these photographs the line \(P\) corresponds to the direct beam, and the line \(C\) to the critical angle of the totally reflected beam.

It should be noted that still earlier Doan, Linnik, and Lashkarev3 studied the refraction of X-rays. By means of an extremely simple method they succeeded in observing the phenomenon of total reflection. The limiting angle \(\alpha\) for the above-mentioned physicists could be ob-

measured with an accuracy of up to 1%. Fig. 3 shows a photograph for quartz. As can be seen, quartz gives a very sharp boundary.

Fig. 2.

Fig. 3.

Table 2 gives the results of the measurements of Linnik and Lashkarev.

Table 2.

Here \(\alpha\) and \(\delta\) have the same meaning as in Compton’s table of data.

An extremely curious circumstance is that Linnik and Lashkarev obtained ordinary reflection of X-rays from a silvered mirror even at an angle of incidence of about \(2^\circ\)!

Thus, it is now possible to speak with certainty about the refraction and reflection of X-rays. If this is so, then it would be of the utmost interest to have the possibility of obtaining an X-ray spectrum with the aid of an ordinary glass prism. And so, at the end of 1924, Siegbahn, Larsson, and Waller1 published an investigation in which they succeeded in obtaining an X-ray spectrum with the aid of a glass prism with a large refracting angle. As was to be expected, the rays were deflected by the prism toward the side opposite its base. This, as is easy to see, is a consequence of the fact that for X-rays \(\mu < 1\). In this way an X-ray spectrum with sufficiently sharp lines was obtained on the photographic plate; in addition, the reflected ray is also visible on it.

The data from measurements of the quantity \(\delta\) for the lines of the \(K\)-series of Fe, Cu, and Mo are found in Table 3.

Table 3.

It is also necessary to mention that measurements of the refractive index for X-rays agree very well with the consequences of the dispersion theory of Drude and Lorentz. According to Compton’s remark, the agreement between theory and experiment reaches 1%, which must undoubtedly be regarded as the fullest confirmation of those assumptions that underlie the theory. On the other hand, if the dispersion equation of the electron theory is regarded as correct, then on the basis of the experimental data one can calculate the number of electrons in the atom of the refracting medium. The calculations show that “the difference between the number of electrons in the atom and the atomic number of the element is less than 0.5%”!

Diffraction of X-rays.

Soon after Röntgen’s discovery and his unsuccessful experiments with a diffraction grating, Haga and Wind1 carried out a series of experiments with narrow slits. Their experiments were later repeated by Walter and Pohl2 with an even more delicate arrangement. Although the above-named authors did obtain some results, nevertheless their experiments were so unconvincing that it was necessary to verify and repeat all this many times. It is well known that this question was resolved by using natural diffraction gratings—crystals. With the help of crystals with large constants, when using vacuum spectrographs, Siegbahn and his pupils succeeded in extending the X-ray spectrum to 13.6 Å. On the other hand, Millikan investigated the spectrum of ultraviolet rays with the aid of diffraction gratings of very fine construction, working in vacuum, down to 144 Å. Thus there remained a very large gap between X-rays and ultraviolet rays.

It was possible to outline three ways of filling this gap: 1) the use of crystals with large constants, 2) the use of the photoelectric effect, and 3) the use of artificial diffraction gratings. It is quite natural that researchers at first paid greater attention to the first path. In this way they did indeed achieve something. Thus, for example, Thoraeus and Siegbahn¹ used crystals of palmitic acid, whose constant is equal to 35.5 Å. They measured the \(L\)-lines of Cr with a wavelength of about 21 Å. The same method was used by Dauvillier². Thus he obtained the carbon \(K\alpha\) line, for which \(\lambda = 45.3\) Å. These works already partially fill the gap in the general spectrum of radiation. Along the second path—by using Einstein’s law—many experimenters also went. Among them one should name Richardson, Hughes, Foote, Holweck, and Lukirsky. The work of the last of these is especially interesting³.

Lukirsky determined, among other things, the wavelength for the \(K\alpha C\) line. For it, according to Lukirsky, \(\lambda = 48.9\) Å. This number agrees rather well with the number determined by Dauvillier by an entirely different method. The photoelectric-effect method is, above all, an indirect method, and since we cannot measure with great accuracy either the velocity of the photoelectron or the work of extraction, an exact measurement of the wavelength by means of it is impossible. That is why this method was soon abandoned. Of course, the ideal solution of the problem should be considered to be the method of obtaining an X-ray spectrum by means of an artificial diffraction grating, for only it gives us the possibility of making an absolute measurement of the wavelength of a ray. Even the crystal method, in essence, is not capable of giving us this, since after all it is based on definite hypotheses concerning the structure of crystals. In this respect it would have been extremely important to verify these hypotheses by means of an absolute measurement of the wavelengths of X-rays. For this reason the attention of the best experimenters in recent years has been directed toward methods of obtaining X-ray spectra by means of artificial diffraction gratings. We shall now turn to a description of these works. But first we shall show that for rays of small wavelength the tangential method is far more preferable than the method of normal incidence.

For this purpose let us turn to Fig. 4. The fundamental equation of a diffraction grating, as is known, is written in the following form:

\[ n\lambda = d[\cos \Theta - \cos(\alpha + \Theta)]. \tag{1} \]

For a sufficiently small angle \(\Theta\) it may be rewritten:

\[ n\lambda = \frac{d}{2}(\alpha^2 + 2\alpha\Theta) \]

¹ Siegbahn u. Thoraeus, Arch. f. M. o. F., 19, 1, 1925; Thoraeus, Phil. Mag., 7, 312, 1926.
² Dauvillier, Comptes Rendus, 182, 1083, 1926.
³ Lukirsky, ZS. f. Physik, 22, 1924.

or, if we put:

$$ \alpha + 2\Theta = \Delta, $$

it is rewritten in the following form:

$$ n\lambda = \frac{d}{2}\,\alpha\Delta. \tag{2} $$

These formulas, of course, presuppose the conditions of tangential incidence of the ray.

Thibaud, in his work, notes the advantage of tangential incidence over normal incidence, consisting in the fact that in the former case the resolving power is greater than in the latter.

Fig. 4.

Indeed, suppose that we have two gratings with one and the same constant \(d\). Let rays of one and the same wavelength \(\lambda\) fall on them, but on the first tangentially, and on the second normally, at angles \(\alpha_1\) and \(\alpha_2\). Taking these angles to be sufficiently small, it is not difficult to obtain for them the following expressions:

$$ \alpha_1 = \sqrt{\frac{2\lambda}{d}} \qquad \text{and} \qquad \alpha_2 = \frac{\lambda}{d}. $$

The resolving powers will be equal, respectively, to:

$$ p_1 = \frac{\partial \alpha_1}{\partial \lambda} = \frac{1}{\sqrt{2\lambda d}} \qquad \text{and} \qquad p_2 = \frac{\partial \alpha_2}{\partial \lambda} = \frac{1}{d}. \tag{3} $$

From these formulas it is evident that for tangential incidence the resolving power depends on \(\lambda\) and, at very small wavelengths, can be very large. Thibaud\(^1\) indicates that for \(\lambda = 1860\,\text{Å}\) a grating of 20 lines under tangential incidence gives the same effect as a grating of 730 lines illuminated normally. For \(\lambda = 500\,\text{Å}\), a grating of 2400 lines may be replaced by a grating of 570 lines! It may also be shown that the method of tangential incidence gives us the possibility of obtaining very fine spectral lines.

\(^1\) J. Thibaud, Journ. de Phys. et le Radium, VIII, 13, 1927; VIII, 447, 1927; Phys. Z. S., 29, 241, 1928.

We shall now turn to the description of the diffraction of X-rays by means of a grating with tangential incidence, which was obtained by Thibaud. Thibaud chose the angle of incidence \(\theta\) so that it would be smaller than the limiting angle of total internal reflection. In this way, of course, the maximum intensity of the diffracted beam of rays will be attained. From the fact that this angle, as was indicated above, is very small, the whole difficulty and delicacy of the work is already apparent. At first Thibaud made photographs in vacuum with a glass grating of 200 lines per 1 mm. The distance between the plate and the grating was equal to 445 and 1300 mm. The exposure ranged between 10 and 60 min. In Fig. 5 a photograph of the copper spectrum at a distance of 1300 mm is shown. In it we see the trace of the direct ray \(T\), then the trace of the ray that has undergone total reflection, and, finally, a system of beautiful spectral lines.

Fig. 5.

Fig. 6.

Thibaud tried to obtain the same spectra by means of a metallic grating with 570 lines per 1 mm, but even with a very long exposure he did not succeed in attaining any good results. He explains this by the fact that, for this kind of rays, a metallic mirror has a coefficient of reflection considerably smaller than that of glass.

Let us now see what the measurement of \(\lambda\) gave.

Thibaud obtained for the \(K\alpha Cu\) line:

\[ \lambda = 1.540 \ \text{\AA}. \]

The crystal, however, gives for the same line:

\[ \lambda = 1.538 \ \text{\AA}. \]

The agreement, as we see, is brilliant, which serves as proof of the correctness of all those principles that up to now have lain at the foundation of X-ray spectroscopy. Thus one can obtain the spectrum of X-rays by the most ordinary optical method, namely by the method of a glass diffraction grating. It would now be interesting to make use of this to fill the gap existing between ultraviolet and X-rays. This last question too was solved by Thibaud.

in his last paper, printed at the end of 1927. For this purpose he constructed a special metallic X-ray tube. The tube operated at voltages of 900—1,000 V and at currents from 10 to 50 mA. The grating was taken to be a permanent glass one on the basis of preceding investigations. The first grating used by him had 200 lines per 1 mm. The anticathode of the tube was carbon. The second grating, by Thibaud, had 1,180 lines per 1 mm. Fig. 6 shows the spectrum obtained with the aid of this grating. In the spectrum, in addition to the \(K\alpha C\) line, the \(K\alpha\)-line of oxygen is also noticeable. The constant of the second grating was measured with very great accuracy by means of the line \(\lambda = 5\,461\ \text{Å}\) of the mercury arc and proved to be equal to \(d = 8\,477\ \text{Å}\). Thus it was possible to measure wavelengths in absolute measure with sufficiently great accuracy. The results of the measurements are given in Table 4.

Table 4.

The absolute accuracy of the measurements, after all sources of error had already been taken into account, was approximately \(0.2\ \text{Å}\).

It is interesting to compare Thibaud’s measurement results with the results of earlier investigations. Thus, as we have already noted, for \(La\mathrm{Fe}\) Toreus obtained \(17.58\ \text{Å}\), Thibaud—\(17.7\ \text{Å}\). For \(K\alpha\mathrm{O}\) Dauvillier obtained \(24.8\ \text{Å}\), Thibaud—\(23.8\ \text{Å}\). Finally, for \(K\alpha C\) Dauvillier obtained \(45.5\ \text{Å}\), Lukirskii—\(48.9\ \text{Å}\), and Thibaud—\(44.9\ \text{Å}\). Of course, the greatest confidence should be placed in the numbers obtained by Thibaud.

It is also necessary to mention that no less successful results with an ordinary diffraction grating were obtained by a number of other authors as well. Among them the most notable are the works of Osgood, Hunt, and several others.

1. Haga u. Wind, Wied. Ann., 68, 884, 1899. ↩↩↩↩

2. Walter u. Pohl, Ann. d. Phys, 29, 331, 1909. ↩↩↩

3. W. Linnik u. W. Laschkarew, Z. f. Physik, 38, 659, 1926. ↩