X-RAY ABSORPTION IN SOLID SOLUTIONS OF ALKALI-HALIDE SALTS

A. D. Shchelokov

(Presented by Professor Doctor A. A. Vorob’ev)

The expressions obtained in [1] for diatomic compounds, for the mass absorption coefficient and for the absorption constant \((C)\) of a substance, are extended in the present work to four-atom systems.

The measurements were carried out on the apparatus and according to the procedure described in [1]. The X-rays were obtained at a tube voltage of 102 kV, with a filter of 0.3 mm Cu + 0.6 mm Al, and had \(\Delta = 0.45\) mm Al, \(\lambda_{\mathrm{eff}} = 0.22\) Å; the beam inhomogeneity index was \(h = 1.38\).

The specimens were prepared from single crystals of solid solutions KCl—KBr, KCl—RbCl, and NaCl—NaBr, grown from specified mixtures of chemically pure salts by the Kyropoulos method. The permissible error in the concentration of the components was \(\sim 10\%\).

From the absorption curves obtained, presented in Fig. 1 (KCl—KBr), 2—(KCl—RbCl), and 3—(NaCl—NaBr), the effective linear absorption coefficients \(\mu_{AB}\) were determined for thicknesses of 0–1 mm; from these, graphs of the dependence of \(\mu_{AB}\) on composition were constructed (Fig. 4).

The position of the experimental points relative to the additivity straight lines (Fig. 4) shows the presence of a slight positive nonadditivity. For example, for all three systems, for a 50-percent composition the deviation of the experimental points in the positive direction from the additive straight lines reaches 6%.

A small excess of the experimental value of \(\mu_{AB}\) over the additive value with approach to the 50-percent composition may be explained by a relative densification of the packing of the atoms forming the solid solution; this should lead to negative nonadditivity of the “lattice constant—composition” curve, which is confirmed by the data of A. N. Kislina [2] for KCl—KBr.

The considerable dependence of \(\mu_{AB}\) on composition (Figs. 1–3 and 4) can be used to determine the concentration of components in a solid solution (the absorption method). By this method, for example, we were able to detect a considerable difference in the concentration of components along the height of a KCl—KBr single crystal grown by the Kyropoulos method.

A certain scatter of the points in Fig. 4 relative to the additivity straight lines is determined to a considerable degree by the inaccuracy of the data on the percentage content of the components. The error in the measurement of \(\mu_{AB}\) is no more than 3%.

Fig. 1. Absorption curves of X-rays ($\lambda_{\mathrm{eff}} = 0.22\,\text{\AA}$) in crystals of the solid solution KCl—KBr of different composition

Fig. 2. Absorption curves of X-rays ($\lambda_{\mathrm{eff}} = 0.22\,\text{\AA}$) in crystals of the solid solution KCl—RbCl of different composition

Fig. 3. Absorption curves of X-rays ($\lambda_{\mathrm{eff}} = 0.22\,\text{\AA}$) in crystals of the solid solution NaCl—NaBr

Fig. 4. Dependence of the absorption coefficient $\mu_{AB}$ of X-rays ($\lambda_{\mathrm{eff}} = 0.22\,\text{\AA}$) in solid solutions of alkali-halide salts on composition; 1, 2, and 3 — additivity straight lines

The calculation of \((\mu/\rho)_{AB}\) for two-component solid solutions of alkali-halide salts on the basis of the law of additivity and Glocker’s formula [3] for \(\mu/\rho\) of monoatomic substances gives the expression:

\[ (\mu/\rho)_{AB}=1.95\cdot 10^{-2}\lambda^{2.8}\cdot C_{AB}, \qquad (\lambda \leq \lambda_K) \tag{1} \]

where

\[ C_{AB}= \begin{cases} p_A C_A + p_B C_B, & \text{for composition by weight,} \tag{2a}\\[6pt] \dfrac{\alpha_A M_A C_A+\alpha_B M_B C_B}{M_{AB}}, & \text{for molar composition} \tag{2б} \end{cases} \]

represents the absorption constant [1] of the solid solution; \(C_A\) and \(C_B\) are the absorption constants [1] of components \(A\) and \(B\); \(p_A\) and \(p_B\) are the weight fractions of components \(A\) and \(B\), \(\alpha_A\) and \(\alpha_B\) are the molar fractions of the components; \(M_A\) and \(M_B\) are their molecular weights;

\[ M_{AB}=\alpha_A M_A+\alpha_B M_B \]

is the “additive molecular weight” of the solid solution.

According to (1), (2a), and (2б), the mass absorption coefficient of a solid solution, as also for pure components [1], is a linear function of the absorption constant \(C_{AB}\) of the solid solution, which in turn, for different compositions, is formed additively from the absorption constants of the components \(C_A\) and \(C_B\) when the composition is expressed by weight (2a), or, as the calculation has shown, according to the law of positive nonadditivity when the composition is molar (2б).

Thus \((\mu/\rho)_{AB}\), when the composition is expressed by weight (\(p_A\) and \(p_B\)), varies linearly with composition, whereas for molar composition the function \((\mu/\rho)_{AB}=f(\alpha_B)\) is represented by a curve with a small positive nonadditivity.

The linear absorption coefficient \(\mu_{AB}\) of the solid solution, according to (1), will be:

\[ \mu_{AB}=1.95\cdot 10^{-2}\lambda^{2.8}C_{AB}\cdot \rho_{AB}, \tag{3} \]

where \(\rho_{AB}\) is the density of the system, which also depends on composition.

Relying on unpublished measurements by L. G. Olert (TPI) of density as a function of composition for several solid solutions of alkali-halide salts, and also on the dependence of the lattice constant of binary metallic alloys on composition [4], one may assume that the function \(\rho_{AB}=f(\text{composition})\), depending on the system, may have positive, negative, or mixed nonadditivity.

Consequently, in the character of the function \(\mu_{AB}=f(\text{composition})\) for two-component solid solutions of alkali-halide salts, depending on the components, as well as on the composition and the way it is expressed, one may expect different signs of deviation from the law of additivity, whereas for \((\mu/\rho)_{AB}=f(\text{composition})\) there may be either additivity (weight percent) or positive nonadditivity (molar percent).

In Fig. 5, the experimental curve 2 for \(\mu_{AB}=f(\text{composition})\) is compared with curve 1, calculated from (3) on the assumption that the lattice constant obeys Vegard’s law of additivity. Both have a positive deviation from the additive straight line 3. A certain discrepancy between curves 1 and 2 is explained by the error in measuring the X-ray wavelength \(\lambda\), which amounts to \(\sim 0.006\ \text{Å}\) (\(\sim 3\%\)).

The density \(\rho_{AB}\) of the solid solution can be represented by the formula:

\[ \rho_{AB}=\frac{M_{AB}}{2Na_{AB}^{3}}\,n_{0}, \tag{4} \]

where \(M_{AB}=\alpha_A M_A+\alpha_B M_B\) is the additive molecular weight of the solid solution,
\(N\) is Avogadro’s number,
\(a_{AB}\) is the smallest period of the crystal lattice of the solid solution (the distance between the nearest crystallographic planes),
\(a_{AB}^{3}\) is the volume of the small [4] cell,
\(n_0\) is the number of particles (independent of sort) in the small cell.

Fig. 5.

Using (4) and (2b), we represent (3) in the form

\[ \mu_{AB}=1.95\cdot 10^{-2}\lambda^{2.8}\frac{1}{2N} \left(\alpha_A M_A C_A+\right. \]

\[ \left.+\alpha_B M_B C_B\right)\cdot\frac{n_0}{a_{AB}^{3}}. \tag{5} \]

The magnitude and sign of the nonadditivity for \(\mu_{AB}=f\) (composition) depend only on the “structural” factor \(n_0/a_{AB}^{3}\). The number \(n_0\), characterizing the type of solution, may be \(n_0=1\) (substitutional solution), \(n_0>1\) (interstitial solution), and \(n_0<1\) (subtraction solution), if components \(A\) and \(B\) have \(n_0=1\) (for cesium halides, except CsCl, \(n_0=2\)). As a function of composition, \(n_0\) may change from 1 to 1, passing through a maximum (interstitial solution), or through a minimum (subtraction solution), or remain unchanged, \(n_0=1\) (substitutional solution).

The lattice constant \(a_{AB}\), as a function of composition, as already mentioned, may, for different constituent atoms, have different signs and magnitudes of nonadditivity [4].

By combining the absorption method (\(\mu_{AB}\)) and X-ray structural analysis (\(a_{AB}\)), one can find the very important curve \(n_0=f\) (composition), characterizing the typical nature of the solution for different compositions.

LITERATURE

1. Shchelokov A. D. Absorption of X-rays in crystals of alkali-halide salts (present collection).

2. Kislina A. N. X-ray investigation of single crystals of solid solutions of alkali-halide salts (present collection).

3. Korsunskii M. I. Physics of X-rays, ONTI, Moscow–Leningrad, 1936.

4. Umanskii Ya. S., Finkelshtein B. N., et al. Physical Metallurgy, Moscow, 1955.

Errata