DETERMINATION OF THE ABSOLUTE VALUES OF X-RAY WAVELENGTHS

M. I. Korsunskii, Tomsk

The wavelengths of X-rays, determined with the aid of crystals, are certain conventional numbers whose character is determined by the extent of our knowledge of the absolute dimensions of crystal lattices. The basis for determinations of this kind is Bragg’s equation, which represents the condition for the diffraction of monochromatic X-rays of wavelength \(\lambda\) by a crystal, considered by us as a system of parallel planes spaced from one another by a distance \(d\), and passing through the centers of the atoms or ions of the elements forming the given crystal. Diffraction by such a crystal occurs as if the ray incident upon the crystal were “reflected” from the above-mentioned system of planes, the phenomenon of “reflection” itself occurring when the angle formed by the incident ray with the “reflecting” plane of the crystal satisfies Bragg’s condition

Fig. 1

\[ 2d \sin \vartheta = n \lambda . \tag{1} \]

Here \(n\) is an integer characterizing the order of reflection.

Equation (1) makes it possible for us to solve two extremely important problems: 1) to determine the wavelengths of X-rays from experimentally found values of the angle \(\vartheta\) and known values of \(d\); 2) to determine the values of the constants \(d\) from known wavelengths. The quantities \(d\) and \(\lambda\) are interrelated; therefore, in order to solve the problems posed, it is necessary to have a starting point—the value either of the constant \(d\) or of \(\lambda\), determined independently of equation (1).

Such a determination was first carried out by Moseley for the crystal NaCl, whose structure had been determined by Bragg by means of a diffraction method.

comparisons of the intensities of “reflections” from the planes (100), (110), and (111) for different orders.

Figure 1 shows a part of a rock-salt crystal. The Na and Cl ions occupy the vertices of cubes, alternating with one another. The distance \(a\) between planes parallel to the faces of the cube will be equal to the distance between neighboring Na and Cl ions. It is not difficult to see that the volume occupied by one NaCl molecule is equal to \(2a^3\). Multiplying this volume by the density of the crystal \(\rho\), we obtain the weight of an NaCl molecule, which at the same time will be equal to the gram-molecule of NaCl divided by Loschmidt’s number \(L\),

\[ 2a^3\rho=\frac{M_{\mathrm{NaCl}}}{L} \]

or

\[ a=\sqrt[3]{\frac{M_{\mathrm{NaCl}}}{2\rho L}}. \tag{2} \]

The numerical values of the quantities entering formula (2) are as follows:

\[ \begin{aligned} M_{\mathrm{NaCl}}&=58.44,\\ \rho&=2.164,\\ L&=6.06\cdot 10^{23}. \end{aligned} \]

Of all these numbers, Loschmidt’s number is determined with the least accuracy. However, the error in determining the number \(L\) does not exceed one percent, and therefore the error in determining the constant \(a\) is no more than \(1/3\%\).

The numerical value of \(a\), calculated by Moseley from formula (2), is equal to 2.814. It is quite obvious that the value of the lattice constant of rock salt calculated by Moseley in no way corresponds to the present state of the art in the measurement of wavelengths. The methods used in spectrometry for determining glancing angles \(\delta\) make it possible to measure them with an accuracy of up to \(1''\). The latest work of Siegbahn’s school has brought the accuracy of determining angles to \(0.1''\). The increased technique of wavelength measurement therefore required a more accurate value for the constant \(a\) than could be obtained by calculation from formula (2). It was therefore necessary to resort to conventional values of the constant \(a\). Siegbahn’s school assigned to the lattice constant of rock salt the value

\[ a_{\mathrm{NaCl}}=2.81400, \]

and more recently

\[ a_{\mathrm{NaCl}}=2.814000. \]

Using this accepted value for \(a_{\mathrm{NaCl}}\), one can determine the wavelengths of X-rays with an accuracy up to the seventh significant figure.

Thus Larsson (1) found the following values for \(K_\alpha\) of iron and molybdenum:

\[ K_{\alpha_1}\ \mathrm{Fe} = 1.932058\ \text{\AA} \]

\[ K_{\alpha_1}\ \mathrm{Mo} = 0.707830\ \text{\AA}. \]

It is quite obvious that these values have the same conditional character as the value for \(d_{\mathrm{NaCl}}\). The difference of these conventional numbers from the true values of the wavelengths cannot be more than \(1/3\%\).

The discovery by Compton (2) in 1923 of the phenomenon of total internal reflection of X-rays gave an impetus to finding another method, which makes it possible—at least in principle—to determine the value of \(d\) more accurately than can be done by using formula (2). In such a case formula (2) would become a means for a more precise determination of the number \(L\), by means of which, in turn, the value of certain fundamental constants, such as, for example, the charge of the electron \(e\) and Planck’s constant \(h\), can be checked.

Fig. 2

Fig. 3

The new method of determining \(d\) is based on the phenomenon of diffraction of X-rays from a linear grating discovered by Compton and Doan (3). Fig. 2 schematically represents a section of an ordinary glass grating (\(AB\), \(A'B'\), etc.—parts of the glass plate not touched by the cutter). This grating will work as a diffraction grating either when the intensity of the rays scattered by the “smooth” parts of the grating \(AB\), \(A'B'\), etc., differs appreciably from the intensity of the rays scattered by the matte parts of the grating, or when rays, strongly absorbed by the material of the grating, fall upon it at an acute angle. In Fig. 3 it is seen that the parts of the grating \(BCK\), \(B'C'K'\), etc., will not be illuminated by the rays at all, and therefore will not scatter at all. The first condition is always fulfilled for visible rays, for which the coefficient of scattering from a smooth surface differs sharply from the coefficient of scattering from a matte surface. Is this condition fulfilled for X-rays? No; the coefficient of scattering of X-rays is the same for all points of the grating, and even in the case of oblique incidence of the rays the grating shown in Fig. 2 will not serve as a diffraction grating, since the coefficient of absorption of X-rays by glass is not so large that along the path \(AL\) there would occur a noticeable weakening of them.^1 That is why a grating which is diffractive for visible rays cannot be such for X-rays, and only Compton’s discovery made it possible to make

an ordinary glass diffraction grating also for X-ray beams.

Compton, basing himself on the work of Siegbahn and his students, who had established that the refractive index of X-rays is less than unity, showed that when X-rays pass from a less dense medium into a denser medium, the phenomenon of total internal reflection can be observed. What is essential here is that no special smoothness of the separating surface is required, since total internal reflection of X-rays is observed not only from a glass plate polished to \( \frac{1}{4} \) of the wavelength of visible light, but also from an ordinary unpolished plate. Linnik and Lashkarev (4) observed total internal reflection from a polished metal surface.

Fig. 4

The limiting angle of the ray with the plane, at which total internal reflection still occurs, is determined by the condition

\[ \beta = \sqrt{2\delta} \]

where \(\delta = 1-\mu\) denotes the magnitude of the deviation of the refractive index \(\mu\) of X-rays from unity. In the case where the frequency of the X-rays is sufficiently far from the edge of the absorption band of the medium, the angle \(\beta\) is approximately determined from the condition

\[ \beta = \sqrt{2.7 \cdot \rho \cdot \lambda \cdot 10^{-3}} \]

where \(\rho\) is the density of the medium, and \(\lambda\) the wavelength, expressed in Ångströms.

The limiting angle of total internal reflection of X-rays is very small; thus, for \(K\alpha\mathrm{Cu}\), on passing from air into glass, \(\beta\) has a value of approximately \(15''\).

If we direct a beam of X-rays in such a way that the angle it makes with the plane of the grating is less than the limiting angle, then the smooth parts of the grating, owing to total internal reflection, will scatter differently from the rough (matte) parts of the grating, and consequently the grating will become a diffraction apparatus.

If we denote by \(a_0\) the angle made with the plane of the grating (Fig. 4), and by \(b\) the distance between neighboring strokes of the grating, then the direction of the maximum intensity of the rays scattered

\(^1\) For soft X-rays the absorption coefficient assumes so large a value that the grating begins to work as a diffraction grating, following the second principle.

by the grating, which in what follows we shall call the “diffraction maximum,” is determined from the condition

\[ b(\cos \alpha_0-\cos \alpha)=n\lambda . \tag{3} \]

The value \(n=0\) corresponds to the ray reflected from the grating. The diffraction maxima corresponding to a positive value of \(n\) (angle \(\alpha>\alpha_0\)) are called positive, while those corresponding to a negative value of \(n\) are called negative.

Equation (3) may be rewritten in the form

\[ 2b\sin\frac{\alpha+\alpha_0}{2}\sin\frac{\alpha-\alpha_0}{2}=n\lambda, \]

and since the angles \(\alpha\) and \(\alpha_0\) are small, then

\[ b(\alpha+\alpha_0)\cdot\frac{\alpha-\alpha_0}{2}=n\lambda. \]

or

\[ b\cdot\frac{\delta_n}{2}\cdot(2\alpha_0+\delta_n)=n\lambda \tag{4} \]

where \(\delta_n\) is the angle between the diffraction maximum of order \(n\) and the reflected ray.

From formula (4), using the experimentally found values of \(\delta_n\) and \(\alpha_0\), one can calculate the wavelength \(\lambda\) of the X-rays, since the value of \(b\) can be determined by independent experiments on the diffraction of visible rays.

Formula (4), according to which calculations are usually made in the spectroscopy of visible rays, is valid only in the case of a parallel beam of incident rays. For X-rays this condition cannot be fulfilled, since a collecting lens for them, owing to the small deviation of the refractive index from unity (\(\delta \cong 10^{-6}\)), is impossible. The beam of incident X-rays is always divergent. Porter (5) took into account the correction that must be made to formula (4), which determines the positions of the diffraction maximum, in the case of a divergent beam. Namely,

\[ b\cdot\frac{\delta_n}{2}(2\alpha_0+\delta_n)\left[1+\frac{3}{20}\cdot\frac{x^2}{l^2}\right]=n\lambda, \tag{5} \]

where \(2x\) is the size of the diffraction grating, \(l\) is the distance from the light source to the grating, reckoned along the mean ray, and from the grating to the photographic plate (Porter considers both distances equal).

The following Table 1 gives the numerical value of the correction factor

Table I

As is seen from the table, the correction for the divergence of the beam is not very significant and, with increasing distance, rapidly decreases.

Further, Tibo (6) showed that the divergence of the positive maxima is smaller than the divergence of the incident beam, i.e. the grating to a certain degree focuses the rays incident upon it. Differentiating (3), we obtain

\[ b(\sin\alpha_0\cdot d\alpha_0-\sin\alpha\, d\alpha)=0 \]

or

\[ \frac{d\alpha}{d\alpha_0} = \frac{\sin\alpha_0}{\sin\alpha} = \frac{\alpha_0}{\alpha} = \frac{\alpha_0}{\alpha_0+\delta} <1. \]

The focusing of the rays will be the more perfect the larger \(\delta_n\) is in comparison with \(\alpha_0\). For large \(\delta_n\) the quantity \(\dfrac{d\alpha}{d\alpha_0}\) will be close to zero, and the rays emerging from the grating may be regarded as parallel even in the case of a considerable divergence of the incident rays.

Fig. 5.

In experimental respects, work with a diffraction grating presents no difficulties. In particular, the setting of the diffraction grating relative to the incident beam is carried out very simply, since from the whole incident complex of rays the grating automatically selects only those rays which form with the plane of the grating angles smaller than the limiting angle of total internal reflection. The measurement of the angles \(\delta_n\) and \(\alpha_0\) is performed in many ways. The most accurate of them, as Birden (7) indicates, consists in measuring the distance between the lines corresponding to diffraction maxima obtained on two mutually parallel photographic plates placed at different distances from the grating. Let \(ss'\) be the ray incident on the grating (Fig. 5), \(OA\) the ray deflected by the angle of total internal reflection, \(OB\) the ray corresponding to the maximum of order \(n\), \(P\) and \(P'\) two positions of the photographic plate, and \(C\) and \(C'\) the traces of the incident ray on the photographic plates. If \(D\) is the distance between them, then

\[ 2\alpha_0+\delta_n = \frac{B'C'(A'C'-AC)}{A'C'\cdot D}; \qquad \delta_n = \frac{A'B'}{A'C'}\cdot \frac{A'C'-AC}{D} \]

and

\[ n\lambda = \frac{b}{2D}\cdot \frac{A'B'\cdot B'C'(A'C'-AC)^2}{A'C'^2}. \tag{6} \]

DETERMINATION OF THE WAVELENGTHS OF X-RAYS

In recent years a considerable number of works have appeared devoted to the measurement of X-ray wavelengths by means of a diffraction grating. Some of these works set as their task the determination of the wavelengths of soft X-rays.^1 Others set as their task the determination of the absolute value of some wavelength, so that from the wavelength value found one could determine the constant \(a\) entering into formula (1), and at the same time verify the value of the number \(L\) and of the physical quantities connected with it. The following Table II gives an idea of how far the determination of the wavelengths of soft X-rays has advanced. The wavelength values are taken from the works of Soderman (8) and Thibaud (9).

Table II

Works by Beclin (9), Valdound (10), and Bearden (7) are devoted to the determination of the absolute value of wavelengths. Beclin measured the absolute value of the wavelength of aluminum \(K\alpha\). The value he obtained is 0.13% greater than the value obtained for \(K\alpha\)Al from measurements with a crystal. The value \(L\), calculated by formula (2) from Beclin’s data, is equal to

\[ L = 6.036 \times 10^{23} \]

and for \(e\) we obtain the value^1

\[ e = 4.792 \times 10^{-10}. \]

^1 The advantage of the diffraction grating, as a spectral apparatus in comparison with a crystal, is the possibility of measuring any wavelengths from hard X-rays up to the visible.

completely coinciding with the value calculated by Edington (11) from theoretical considerations.

Valund made measurements with the wavelengths \(K\alpha\mathrm{Cu}\), \(\mathrm{CuFe}\), and \(K\alpha\mathrm{Mo}\). The results of his work are as follows:

\[ \begin{array}{r|r} K\alpha\mathrm{Fe}=1.937(6)\pm0.002 & 1.93230\\ K\alpha\mathrm{Cu}=1.5373\pm0.0008 & 1.53730\\ K\alpha\mathrm{Mo}=0.708(3)\pm0.001 & 0.70759 \end{array} \]

At the side are indicated the values of the wavelengths obtained by Ziegban from measurements with a crystal.

Valund computes from his data the value

\[ L=(6.061\pm0.009)\times10^{23} \qquad e=(4.774\pm0.007)\times10^{-10}. \]

Excellent agreement was obtained with Millikan’s measurement for the quantity \(e\). However, the results obtained by Valund cannot be regarded with complete confidence, since, of the three wavelengths measured by him, the greatest error, both absolute and relative, was obtained in measuring the wavelength \(K\alpha\mathrm{Fe}\), which has the greatest wavelength and which therefore, it would seem, should have been measured more accurately. Moreover, if one carries out the calculations for \(N\) and \(e\), using the author’s data for Fe \(K\alpha\) (not for Cu\(K\alpha\), as was done by Valund), then we obtain the following value

\[ e=4.802\times10^{-10}\ \text{and}\ L=6.024\times10^{23}. \]

Apparently the most accurate measurement of the absolute value of the length was made by Birden (1929), who gives the following value for \(K\alpha\) and \(K\beta\) of copper:

\[ \begin{aligned} \mathrm{Cu}K\alpha&=1.5422\pm0.0002\\ \mathrm{Cu}K\beta&=1.3926\pm0.0002. \end{aligned} \]

The value \(\mathrm{Cu}K\alpha\) differs from measurements with a crystal (taking the position of the “center of gravity” of the doublet as \(K\alpha\)) by \(0.23\%\).

The same result was obtained by him also for the \(K\beta\) line. The wavelength \(K\beta\) measured by him is greater than that measured by Ziegban with the aid of a crystal by \(0.23\%\). From his data Birden determines

\[ L=6.022\times10^{23},\ e=4.804\times10^{-10}. \]

The experimental conditions under which Birden’s work was carried out give an idea of the care with which this work was done.

The widths of the slits \(s\) and \(s'\) were equal to \(0.01\ \mathrm{mm}\); their mutual distance was \(50\ \mathrm{mm}\). The distance between the two positions of the photographic plate was \(2\ \mathrm{m}\),

\(^{1}\) \(e\) is calculated by the formula \(e=\dfrac{MC}{10Lz}\), where \(M\) is the molecular weight of the substance (e.g., silver), \(z\) is its electrochemical equivalent, and \(c\) is the speed of light.

and this distance was measured by the author with an accuracy of up to 0.04 mm. The photographic plates were set parallel to one another by means of a Gauss telescope with an accuracy of up to 5″. The slits \(s\) and \(s'\) were set parallel to the plane of the grating with an accuracy of up to 10″. The grating itself was mounted on the table of a precision spectrograph of the firm Société Genévoise. The grating was centered on the axis of rotation and set parallel to it with the aid of a modified Michelson interferometer. During the exposure observations were made, and it was possible to notice a displacement of the grating by 0.1 of the wavelength of visible light, or a rotation of it by 1″. Bearden carried out the measurement with two different gratings made by Michelson under the supervision of Michelson. One grating had 50 divisions per millimeter, the other 600. The result obtained with both gratings was the same. After the publication of Backlin’s work, Bearden repeated the measurements using a grating similar to the grating used by Backlin. The result came out the same as with the other two gratings.

Bearden’s work, in addition to its thoroughness, also has a fundamental advantage over the works of Behnken and Backlin. Namely: Bearden measured the absolute value of the wavelength \(K\beta\), which is a single line, whereas the other authors made measurements with the \(K\alpha\) radiation, which is a doublet whose components differ in wavelength by 0.25% of its value.

The results of Bearden’s work (in view of their sharp disagreement with the data of other investigators) were, even before their publication, carefully checked by Compton, who carried out a series of independent measurements that gave the same results as Bearden’s measurements.

LITERATURE

1. Larsson, A. Z. Physik. 41, 507, 1927.
2. Compton, A. H. Phil. Mag. 45, 1121, 1922.
3. A. H. Compton and R. L. Doan, Proc. Nat. Acad. U. S. A. 11, 598, 1925.
4. Linnik und Laschkarew, Z. Physik. 38, 659, 1926.
5. Porter. Phil. Mag. 1064, 1928.
6. Thibaud. Phys. Zeitschr. 1928, No. 9, p. 241.
7. Bearden. Proc. Nat. Acad. Sci. U. S. A. 15, p. 528.
8. Sodermann. Z. Physik. 52, 795, 1929.
9. Bäklin. Diss. Upsala Univ. Arsskrift. 1928.
10. Wadlund. Phys. Rev. 32, 841, 1928.
11. Eddington. Proc. Royal Society. Jan. 1929.

Literature not mentioned in the text

Osgood. Phys. Rev. 30, 567, 1927.
Heint. Phys. Rev. 30, 227, 1927.
Weatherby. Phys. Rev. 32, 707, 1928.
Howe. Proc. Nat. Acad. Sci. 15, 251, 1929.
Kellström. Z. Physik 58, 511, 1929.