From Current Literature

FINE STRUCTURE OF THE CONTINUOUS SPECTRUM OF X-RAYS

In order to obtain a more accurate value of the ratio \(\frac{h}{e}\), many authors have used the isochromat method.

It is known that the short-wavelength limit of the continuous X-ray spectrum is determined by the relation \(\lambda_{\min}=\frac{hc}{eV}\) (\(V\) is the voltage). Thus, by measuring \(V\) and the corresponding value \(\lambda_{\min}\) with high accuracy, one can determine the ratio \(\frac{h}{e}\).

For these measurements X-ray spectrographs are used. In particular, Du Mond\(^1\) used a double-crystal spectrometer. The measurement of the intensity was made by an ionization chamber, whose slit was set to a quite definite wavelength coinciding with the Mo \(K\alpha_1\) line. By gradually increasing the voltage on the tube and simultaneously measuring the current in the ionization chamber, one obtains a curve of the dependence of the spectral intensity on the voltage at constant \(\lambda\). This curve is called an isochromat. It begins at the voltage \(V_1=\frac{hc}{e\lambda}\), and, according to theoretical concepts\(^2\), in the first approximation has the form of a straight line.

However, on more careful examination, one can observe near the short-wavelength limit of the isochromat, within 10–20 V, a series of maxima and minima. These irregularities are especially clearly expressed in isochromats for which \(V_1=3\text{–}5\ \mathrm{kV}^3\). However, even at voltages of 10–20 kV\(^1\), fluctuations in the slope of the isochromat are observed at individual points.

An interpretation of both facts—the presence of irregularities and their dependence on voltage—was given by Du Mond\(^4\).

The continuous spectrum of X-rays from a thick anticathode may be represented, in simplified form\(^1\), as a superposition of separate spectra from all the thin layers into which the entire anticathode can be divided. The spectral intensity from a thin layer is constant over its entire extent (as a function of frequency). In passing through each layer, the electrons lose energy approximately in proportion to its thickness. These losses cause a shift of the short-wavelength limit of the spectrum toward larger \(\lambda\), and when summed over all layers we obtain the usual distribution for the continuous spectrum.

The structure of the spectrum from a thin anticathode near the short-wavelength limit can be explained by taking into account the allowed and forbidden bands (zones) of electron energy in the crystal lattice. In the process of braking, the incident electron gives rise to an X-ray quantum. The energy

of the electron after the radiative transition must belong to one of the permitted bands of crystal energy. If this condition is not fulfilled, then the corresponding transition will have a very low probability, which will show up as a sharp decrease in the intensity of the continuous spectrum. With a thick anticathode the entire fine structure almost disappears. This occurs, first, owing to superposition and to the above-mentioned displacement of the individual components of the spectrum, and, secondly, owing to averaging over all directions of the electrons that have made a radiative transition (chaotic orientation of the crystallites in the anticathode). Distortions of the fine structure due to the latter effect should be even more considerable than is actually observed.

Fig. 1.

However, the isochromats obtained with a thick anticathode, but at low voltages, are entirely similar to those that would be obtained with a thin anticathode. DuMond\(^4\) explains this fact by taking into account 1) the loss of energy by electrons when they penetrate into the anticathode and 2) the absorption of continuous radiation in the material of the anticathode. Let us consider Fig. 1 (anticathodes of Cu and W).

Here \(\tau\)—for the curves \(A\) and \(B\) (Cu and W)—is the depth of penetration of the electron into the anticathode at various initial energies, with a loss of energy of 10 eV occurring over this thickness (linear absorption). For the remaining curves \(\tau\) is the thickness, measured along the direction of incidence of the electrons, on passing through which the hardest component of the continuous spectrum is weakened by a factor of \(1/e\). From the curves it can be seen that, at voltages \(3\)–\(5\ \mathrm{kv}\), the absorption of radiation limits the effective layer of the anticathode to such a thickness that the loss of electron energy in this layer does not exceed \(6\)–\(12\ \mathrm{eV}\) (for W). At \(V_1 = 10\ \mathrm{kv}\) and at small atomic numbers the losses exceed \(30\ \mathrm{eV}\), which causes smoothing of the fine structure, since different parts of the effective layer of the anticathode are bombarded by electrons with very different energies.

Fig. 2.

DuMond further proposes a theoretical method that makes it possible, to a considerable degree, to correct the experimental isochromat by eliminating

FINE STRUCTURE OF THE CONTINUOUS SPECTRUM

...of the influence of both the first and the second effects. If \(f(v)\) is the isochromat of interest to us for an ideally thin anticathode, then the experimental isochromat \(F(v)\) is obtained by multiplying \(f\) by \(g\) and integrating over the entire spectral region. Thus \(g\) must describe the absorption effect. The author chooses \(g\) in the form of an exponential. To determine it, only one parameter is sufficient—the energy loss \(\Delta V\) (in volts) at the point where \(g(v)\) decreases by a factor of \(1/e\). It is clear that \(\Delta V\) corresponds to the energy loss by an electron that has penetrated into the anticathode to such a depth that the intensity of the emerging radiation is reduced, owing to absorption, by a factor of \(1/e\). It is significant that for each anticathode material and voltage, \(\Delta V\) is a constant quantity, decreasing with increasing absorption of radiation and increasing with increasing energy loss by the electron. The author calculated \(\Delta V\) for anticathodes of Cu and W for the voltages used in [3]. It varied from 6 V to 25 V. Thus we have:

Fig. 3.

\[ F(v)=\frac{1}{\Delta V}\int_{0}^{\infty} f(v-u)\,e^{-u/\Delta V}\,du, \tag{1} \]

where \((\Delta V)^{-1}\) is a normalizing factor. The variable \(u\) represents the energy loss by the electron. Integrating (1) by parts and solving for \(f(v)\), we obtain:

\[ f(v)=F(v)+\Delta V\,\frac{dF}{dV}. \tag{2} \]

With the aid of (2), the author corrected the isochromats for Cu and W anticathodes. These curves are shown in Fig. 2 for Cu (dash-dot line). On them one can clearly see the places where the intensity falls almost to zero, which corresponds to the energy of electrons (after the radiative transition has occurred) falling into the forbidden energy band of the crystal.

Fig. 3 shows the course of the potential, as well as of the total electron energy, in the cathode, anticathode, and between them. The forbidden energy bands are shaded. The conduction levels are blackened. From comparison of Figs. 2 and 3 it is seen that the first peak in the corrected isochromat corresponds to such an electron transition when its final state is located somewhere in the allowed band directly at the surface Fermi level \((e\Phi)\). For a Cu anticathode, the first dip is separated from the short-wavelength boundary by 10 eV. This result is in complete agreement with the results of recently performed experiments by Davisson and Germer (unpublished) on the bombardment of a Cu crystal by slow electrons.

Thus we obtain a new, very effective method for investigating the energy spectrum of electrons in a crystal located above the Fermi surface.

It is of interest to compare the above calculations with an investigation of the fine structure on the short-wavelength side of the absorption edges of the same elements. As is known, the theoretical explanation of the latter has been given by Kronig and Kostarev. It is still difficult to choose between these theories.^5

R. Barinskii

References

1) Du Mond and Bollman, Phys. Rev., 51, 400 (1937).
2) G. Wentzel, Zeits. f. Physik, 27, 257 (1924).
3) Ohlin, Arkiv f. Mat. Astr. o. Fysik, 33A, No. 23 (1946).
4) Du Mond, Phys. Rev., 72, No. 4 (1947).
5) Kostarev, ZhETF, 11, 60 (1940); ZhETF, 9, 267 (1939); ZhETF, 16, 738 (1946).