PHYSICS

Academician of the Academy of Sciences of the BSSR N. N. SIROTA, N. M. OLEKHNOVICH

X-RAY DETERMINATION OF THE DIAMAGNETIC SUSCEPTIBILITY OF CERTAIN IONIC AND SEMICONDUCTOR COMPOUNDS

The experimentally determined lattice magnetic susceptibility (\chi) of ionic and semiconductor compounds and other many-atom systems can be represented as the sum of two terms—the diamagnetic component (Langevin term) (\chi_d) and the paramagnetic component (Van Vleck term) (\chi_p):

[
\chi=\chi_d+\chi_p
=-\frac{Ne^2}{6mc^2}\sum_i \overline{r_i^2}
+\frac{2}{3}N\sum_{j\ne i}\frac{|M(j,i)|^2}{E_j-E_i},
\tag{1}
]

where (M(j,i)) is the off-diagonal matrix element of the magnetic moment; (E_j-E_i) is the width of the forbidden band (gap); (\sum_i \overline{r_i^2}) is the sum of the mean squares of the radii of the electron orbits. The first term is determined by the distribution of electron density in the lattice of the compounds; the second term, by the deviation of the electron-density distribution from sphericity.

As Ya. G. Dorfman emphasizes ((^1)), it is of considerable interest to find independent ways of separating the experimentally determined magnetic susceptibility (\chi) into the components (\chi_d) and (\chi_p). It is known ((^2)) that the diamagnetic component of the lattice susceptibility (the Langevin term) can be determined from data on the distribution of electron density, determined experimentally by the x-ray method.

The aim of the present investigation was to determine the diamagnetic component of the magnetic susceptibility of the arsenides and antimonides of aluminum, gallium, and indium from data on the distribution of electron densities in the lattice of these compounds obtained in our laboratory ((^{3-8})), and of the compounds NaCl, KCl, CaF(_2), and Cu(_2)O, using literature data ([( \mathrm{NaCl}, \mathrm{KCl})(^9), (\mathrm{CaF}_2)(^{10}), (\mathrm{Cu}_2\mathrm{O})(^{11})]).

The magnitude of the diamagnetic component was determined, in accordance with the method described earlier ((^2)), using for each kind of ion the relation

[
\chi=-\frac{4\pi Ne^3}{6mc^2}\int_0^\infty(\rho_1'+\rho_2')r^4\,dr
=-23.64\cdot10^{-6}\frac{A_1}{a_1^{5/2}}
-35.55\cdot10^{10}\sum_i \rho_{2i}' r_1^4\cdot \Delta r,
\tag{2}
]

where (\rho_1'=A_1 e^{-\alpha_1 r^2}); (\rho_2') was determined from the series (\sum_h\sum_k\sum_l F_2\exp(-2\pi i\bar r \bar H)); (F_2) was determined from the difference between the experimentally determined structure amplitude and the structure amplitude calculated for a given distribution of electron density (\rho_1=\sum_i A_i e^{-\alpha_i(r-r_{0i})^2}). The quantities (A) and (\alpha) were determined experimentally from the logarithm of the (f)-curves in the region of the linear portion at comparatively large values of (\sum_i h_i^2).

Since the temperature-independent diamagnetic component of the lattice susceptibility is being considered, the calculation was carried out by extrapolating the electron-density distribution to zero temperature for the ion at rest, also eliminating zero-point vibrations.

The quantities (A_{01}) and (\alpha_{01}), characterizing the distribution (\rho'_1) in the case of an ion at rest, are related to (A_1) and (\alpha_1), determined at temperature (T), by the relations

[
\alpha_{01}=\frac{\alpha_1}{1-2u_T^2\alpha_1};
\tag{3}
]

[
A_{01}=A_1\frac{\alpha_{01}^{3/2}}{\alpha_1^{3/2}}.
\tag{4}
]

The extrapolation of (\rho'_2) to absolute zero was also carried out with allowance for the temperature factor. However, as calculations and direct experiments have shown, the effect of temperature on the electron-density distribution (\rho_2) and, correspondingly, on (\chi_2) is negligibly small.

Table 1

Diamagnetic component of the magnetic susceptibility of NaCl, KCl, CaF(_2), and Cu(_2)O, determined from X-ray data

Table 1 gives the results of calculating the diamagnetic susceptibility of the indicated ionic compounds.

It should be noted that, in determining the electron density, the error is of the order of (0.05) el/Å(^3). This leads to an error in the determinations of the diamagnetic susceptibility of about 5%, i.e., of the same order as the accuracy of direct experimental measurements.

On the basis of analysis of the results presented in Table 1, it may be concluded that the diamagnetic susceptibility calculated from X-ray measurements for ionic compounds agrees well with the experimental values, which is due to the smallness of the paramagnetic component in the case of ionic compounds. This, in turn, makes it possible to regard the method for determining the diamagnetic component of the magnetic susceptibility from X-ray data as a sufficiently reliable independent method. When there are significant differences between the experimental values and those calculated from X-ray data, it may be assumed that the difference is determined mainly by the magnitude of the paramagnetic component (the Van Vleck term).

Table 2 gives the results of determining the diamagnetic susceptibility of compounds (A^{\mathrm{III}}B^{\mathrm{V}}) with the sphalerite structure—arsenides and antimonides of aluminum, gallium, and indium ((\Delta E) is the forbidden-band width; the quantity (M_{01}) was calculated from the values of (\Delta\chi) and (\Delta E)).

It follows from Table 2 that the difference (\Delta\chi) between the experimental values and the values determined from X-ray data, corresponding to the paramagnetic component of the magnetic susceptibility,

significantly exceed the possible experimental errors. Thus, determination of the diamagnetic component from X-ray data makes it possible to separate the lattice magnetic susceptibility of the simplest compounds into $\chi_d$ and $\chi_p=\Delta\chi$.

Table 2

Diamagnetic component of the magnetic susceptibility of arsenides and antimonides of aluminum, gallium, and indium, determined from X-ray data
(experimental and theoretical values)

The regularities in the variation of $\chi_d$ and $\chi_p$ as functions of the positions of the components in the periodic system of D. I. Mendeleev are noteworthy. In particular, it should be noted that the value of $\Delta\chi$ for gallium arsenide and antimonide is considerably larger than for the arsenides and antimonides of indium. The absolute values of $M_{01}$ amount to tenths or hundredths of a Bohr magneton. Estimates of $\chi_p$ from the magnitude of the polarizability are apparently less reliable than those determined from X-ray data. The X-ray data obtained make it possible to solve the inverse problem and determine the magnitude of the polarizability of the compounds.

Department of Solid-State and Semiconductor Physics
Academy of Sciences of the BSSR

Received
27 VII 1962

CITED LITERATURE

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