CALCULATION OF THE SURFACE ENERGY OF CRYSTALS OF ALKALI-HALIDE COMPOUNDS

A. A. MEDVINSKII

Introduction

The consideration of an infinite crystal proves to be an inapplicable idealization in the study of many physicochemical phenomena in real crystals. At the same time, if a crystal has not very small dimensions, so that the influence of edges and vertices may be neglected, then it is sufficient to take into account the change of properties only on the faces and in the regions immediately adjacent to them, as compared with the bulk of the crystal. In this connection, it is important to take account of the energetic nonequivalence of faces having different crystallographic orientations. The energetic nonequivalence of faceting planes of different crystallographic orientation is considered on the basis of the concept of specific surface energy; the determination of this physical quantity may be approached in the following way. Let us select, in an infinite crystal, a certain crystallographic plane \((l, m, n)\), where \(l, m, n\) are Miller indices, and imagine that the crystal is divided along this plane, with the subsequent separation of the resulting semi-infinite crystals to such a distance that their interaction would be excluded. In order to carry out the indicated division, it is necessary to expend a certain amount of work, which determines the difference between the energy of the two semi-infinite crystals and the energy of the infinite crystal. The difference between the energy of the two semi-infinite crystals and the energy of the infinite crystal, referred to a unit area of the surface formed, is called the specific surface energy of the face \((lmn)\); its magnitude is determined by the crystallographic orientation of the plane under consideration. The change in the energy of the crystal in the course of the division considered consists of the change in the self-energy and in the interaction energy, mainly for the ions forming the crystal in the surface layer immediately adjacent to the faceting plane.

The specific surface energy is an important characteristic of a bounded crystal [1], so the problem of determining its numerical values for different faces of different crystals is topical. The difficulties of experimental determination of surface energy [2] increase interest in theoretical calculations, but

the latter encounter their own considerable difficulties, analyzed below.

In what follows we shall consider only ionic crystals of alkali-halide compounds, since the ions act as the crystal-forming units. The equilibrium conditions for surface ions differ from the corresponding conditions for internal ions: surface ions are in a sharply asymmetric field, having neighbors only on one side (or, generally speaking, an unequal number of neighbors above and below). The surface ions must arrange themselves so that the forces acting on them as a result of the unequal environment are compensated—the resulting displacements of the ionic networks (lattice sites) propagate to some depth into the crystal. A sharply asymmetric field causes internal changes of the surface ions—their deformation. We shall call both of these effects, i.e., changes in the equilibrium positions of the nuclei and also in the density of the electronic distribution of the ions in the surface region, as compared with the internal regions of the crystal, surface distortions. The specific surface energy of the face \((lmn)\), \(\sigma_{(lmn)}\), can be represented in the form

\[ \sigma_{(lmn)}=\sigma^{(1)}_{(lmn)}+\sigma^{(2)}_{(lmn)}, \tag{1.0} \]

where \(\sigma^{(1)}_{(lmn)}\) is the specific surface energy determined without taking into account distortions in the surface region of the crystal; \(\sigma^{(2)}_{(lmn)}\) is that part of the specific surface energy which is due to surface distortions. In accordance with the principle of minimum energy, the appearance of distortions in the surface region must be accompanied by a lowering of the total energy of the crystal, so that

\[ \sigma_{(lmn)}<\sigma^{(1)}_{(lmn)}. \tag{1.1} \]

Naturally, the calculation of the surface energy without allowance for surface distortions should be regarded only as a first approximation, which may prove to be rather crude.

We note that everywhere below the crystal is assumed to be studied at absolute zero temperature, so that the contribution of ion vibrations to the surface energy, including the contribution of zero-point vibrations, is not taken into account. An ideal crystal lattice is considered.

Calculation of \(\sigma^{(1)}_{(lmn)}\) for a definite approximation of the binding forces in the crystal

Extensive calculations of \(\sigma^{(1)}_{(lmn)}\) for neutral faces of alkali-halide crystals were carried out by Zegeren and Benson [3, 4], and also by Shuttleworth [5], using the Born–Mayer approximation for the binding forces in the crystal. In this case the interionic forces were assumed to be central and additive, and the interaction energy of ions with charges \(Z_i\) and \(Z_k\), located at a distance \(R_{ik}\) from one another, was specified by the expression

\[ \varepsilon_{ik}(R_{ik})= \frac{Z_i Z_k}{R_{ik}}e^2 -\frac{c_{ik}}{R_{ik}^{6}} -\frac{d_{ik}}{R_{ik}^{8}} +b_i b_k f_{ik} e^{-R_{ik}/\rho}; \tag{2.0} \]

the coefficients \(c_{ik}\) and \(d_{ik}\) in the two-term approximation of the van der Waals forces were taken from [6], and the constants of the repulsive forces from [7].

It is known [8, 9] that, if surface distortions are neglected, the specific surface energy is expressed in the form

\[ \sigma^{(1)}_{(lmn)}=-\frac{U_{12}}{S}, \tag{2.1} \]

where \(S\) is the surface area, in the cleavage plane, per pair of ions; \(U_{12}\) is the interaction energy between two parts of an infinite crystal separated by an imaginary interface whose orientation coincides with the plane under consideration; in this case

\[ U_{12}=\frac{1}{2} \sum_{\substack{l_1,l_2=-\infty\\(l_1+l_2+l_3)\ \mathrm{even}}}^{\infty} \sum_{l_3=1}^{\infty} l_3\,[\varepsilon_{++}(R_l)+\varepsilon_{--}(R_l)] + \sum_{\substack{l_1,l_2=-\infty\\(l_1+l_2+l_3)\ \mathrm{odd}}}^{\infty} \sum_{l_3=1}^{\infty} l_3\,\varepsilon_{+-}(R_l); \tag{2.2} \]

\[ S=2\cdot|\vec a_1\times \vec a_2|;\ \ldots \tag{2.3} \]

\[ \vec R_l=l_1\vec a_1+l_2\vec a_2+l_3\vec a_3. \tag{2.4} \]

Using approximation (2.0), for \(\sigma^{(1)}_{(lmn)}\) we obtain

\[ \sigma^{(1)}_{(lmn)}=S^{-1}\cdot\left\{ \frac{e^2}{a_0}\cdot B+\frac{1}{a_0^6}\cdot \left[ B_6'\cdot c_{+-}+ \right.\right. \]

\[ \left. +B_6^{*}\cdot\left(\frac{c_{++}+c_{--}}{2}\right) \right] +\frac{1}{a_0^8}\cdot \left[ B_8'\cdot d_{+-}+B_8''\cdot\left(\frac{d_{++}+d_{--}}{2}\right) \right] \left.-R\right\}, \tag{2.5} \]

where \(a_0\) is the distance between nearest neighbors in the crystal; \(B\), \(B_6'\), \(B_6^{*}\), \(B_8'\), and \(B_8''\) are expressed in terms of two basic sums over the lattice sites \(S_{2\nu}\) and \(T_{2\nu}\) by the relations

\[ B=-T_1;\qquad B_6'=\frac{1}{2}(S_6-T_6);\qquad B_6^{*}=\frac{1}{2}(S_6+T_6); \]

\[ B_8'=\frac{1}{2}(S_8-T_8);\qquad B_8''=\frac{1}{2}(S_8+T_8); \tag{2.6} \]

\[ S_{2\nu}=a_0^{2\nu}\cdot \sum_{l_1,l_2=-\infty}^{\infty} \sum_{l_3=1}^{\infty} l_3\cdot R_l^{-2\nu}; \qquad T_{2\nu}=a_0^{2\nu}\cdot \sum_{l_1,l_2=-\infty}^{\infty} \sum_{l_3=1}^{\infty} (-1)^{l_1+l_2+l_3}\cdot l_3\cdot R_l^{-2\nu}. \tag{2.7} \]

In [3, 4] a “generalized crystal of the NaCl type” is considered—a structure that can be obtained from the NaCl structure by deforming the angles between the principal axes together with various changes of the lattice parameter in the directions of the three axes; and a general expression is obtained from which, as special cases, follow the calculation formulas for \(\sigma^{(1)}_{(100)}\) and \(\sigma^{(1)}_{(110)}\) for the NaCl-type structure and \(\sigma^{(1)}_{(110)}\) for the CsCl-type structure. In this case the sums (2.7) are transformed into rapidly convergent series on the basis of the method presented in [10]; numerical values of the sums \(S_6, S_8, T_1, T_6, T_8\) are tabulated in [3]. In calculating the \(R\)-term, caused by repulsive forces, the interaction of ions with only the two layers of nearest neighbors is taken into account. Zegen and Benson [3, 4] carried out calculations for the (100) and (110) faces of the NaCl structure and the (110) face of the CsCl structure, considering compounds of Li, Na, K, Rb, Cs with F, Cl, Br, I; in obtaining numerical

values, the theoretical values of the lattice constant were used, corresponding to the minimum of the binding energy for the given approximation of the interionic interaction. The values of the surface energy tabulated in [3, 4] essentially coincide with Shuttelworth’s data [5], who carried out the calculation of \(\sigma^{(1)}_{(100)}\) and \(\sigma^{(1)}_{(110)}\) for the halide compounds of Na and K and used another method for computing sums over lattice sites. In the tables given in [3—5], the contributions to \(\sigma^{(1)}_{(lmn)}\) are presented separately: from electrostatic Coulomb forces (1); from dipole–dipole (2) and dipole–quadrupole (3) van der Waals interactions; and from repulsive forces (4). By way of illustration we give some results.

Table 1

If, in calculating the binding energy, the electrostatic term is predominant, while the terms associated with the repulsive forces and the van der Waals forces amount to approximately \(1/10\) and \(1/30\) of the electrostatic term, respectively, then in the case of \(\sigma^{(1)}_{(lmn)}\) the non-Coulomb contribution is very substantial. In particular, in approximation (2.0) the contribution of the van der Waals forces is very substantial. A calculation was made in order to check whether the large magnitude of the non-Coulomb contribution is a consequence of inconsistency in the choice of parameters in the approximate formula. In doing so, using the equilibrium equations, the parameters for the repulsive forces were expressed through the experimental values of the elastic moduli and the values of the van der Waals constants \(c_{ik}\) and \(d_{ik}\). It was found that variation of the values of the van der Waals constants leads to large changes in the values of the surface energy (the case \(\sigma^{(1)}_{(100)}\) was considered), while the total non-Coulomb contribution to the surface energy is nevertheless significant.

The substantial nature of the non-Coulomb contribution to \(\sigma^{(1)}_{(lmn)}\) and the corresponding strong sensitivity of its numerical values to the approximation of the interaction in the crystal create an unpleasant situation when the approximate method is used. Indeed, the question of binding forces in the modern theory of the solid state is treated only approximately; the approximate formulas used are imperfect both as regards their theoretical justification and as regards the agreement of the results obtained with their aid with experimental data [11]; complication of the approximate expressions is accompanied by an increase in computational work, but does not lead to a reliable improvement of the numerical values obtained [11]. In view of this, it is hardly possible to give preference to the data

[3–5] in comparison with calculations based on the approximation of the binding forces by the simple two-term Born formula

\[ \varepsilon_{ik}(R_{ik})=\frac{Z_i Z_k}{R_{ik}}\cdot e^2+\frac{b_{ik}}{R_{ik}^{n_{ik}}}, \tag{2.8} \]

as was done, for example, in [9] and [12]. At the same time, the strong sensitivity of \(\sigma^{(1)}_{(lmn)}\) to the approximation of the non-Coulomb interaction determines the discrepancy between the numerical values of this quantity obtained by different authors using different interaction laws. Thus, in contrast to the value \(188\ \text{erg}/\text{cm}^2\) obtained for \(\sigma^{(1)}_{(100)}\) of the NaCl crystal in [3–5], in [9] the value \(150.2\ \text{erg}/\text{cm}^2\) is indicated, while in [12] \(96\ \text{erg}/\text{cm}^2\) is given as the numerical value of this quantity. In [9] the exponents \(n_{ik}\) were taken to be the same (\(n=10\)) for all interactions, whereas in [12] \(n=9\) in the case of interaction of argon-like ions, \(n=11\) in the case of interaction of neon-like ions, and \(n=10\) for the interaction of ions of one type with ions of the other type. The insurmountable difficulties that arise in approximating the non-Coulomb interaction in an ionic crystal diminish the value of quantitative results of calculations when the approximation method is used.

Consideration of surface distortions and \(\sigma^{(2)}_{(lmn)}\) on the basis of the approximation method

It is very important to determine surface distortions and their contribution to the surface energy. Distortions in the surface region to a considerable extent determine the special physicochemical properties of the surface as compared with the bulk of the crystal.

Surface distortions were first considered by Madelung [13], who studied the displacements of the constituents of a crystal in the surface region without taking into account deformations of the constituents. The structure of NaCl and the plane (100) were considered as the surface; the interaction law was not specified, and only the requirement of centrality of the acting forces was imposed. Madelung considered displacements perpendicular to the surface plane of the crystal; the displacements were assumed to be so small that their quadratic terms could be neglected. The result of the study: at the surface, ions of one kind are displaced vertically toward the surface in the direction opposite to ions of the other kind—a double electric layer arises. Madelung’s attempts to detect experimentally the presence of the double layer were unsuccessful, which Born [8] explains by the smallness of the effect. Thus, first, Madelung puts forward the proposition of the smallness of surface displacements (essentially without any basis); second, he arrives at the conclusion that the displacements of ions of different signs have opposite character. There is a substantial flaw in Madelung’s solution: he restricts himself to considering the interaction of an ion only with the two nearest-neighbor layers, which is inadmissible, since long-range Coulomb forces are a component part of the interaction.

Surface distortions only in the form of displacements of surface ions were considered by Glauberman [14], who used approximation (2.8) for the binding forces in the crystal. For the repulsive forces, the interaction of an ion only with the first layer of nearest neighbors was taken into account; for the Coulomb interaction, the action on

the given ion relative to all the others. Displacements in a chain normal to the (100) surface of an NaCl crystal were investigated. When an ion is displaced, an equilibrium is established between the change in the repulsive forces and the electrostatic force \(E_z\) \((z > 0)\) acting on the surface ion; from the condition of force equilibrium the magnitude of the displacement was determined. It was found that the displacement of the first surface plane is \(+0.008 \cdot a_0\), while that of the second plane is \(-0.000097 \cdot a_0\) (displacements directed into the crystal are taken as positive). In the approximation used by Glauberman, different displacements of ions of different signs belonging to the same surface plane, in principle, cannot be obtained. The calculated displacements were used to determine \(\sigma^{(2)}_{(100)}\); the value \(-20.4\ \mathrm{erg}/\mathrm{cm}^2\) was obtained, which amounts to 13.3% of the value of \(\sigma^{(1)}_{(100)}\).

As early as 1928, Lennard-Jones and Dent [15] considered displacements of surface ions with allowance for their deformation. It turned out that the polarization-field effect is decisive with respect to the magnitude of the surface displacements. In [15] they investigated (in the approximation of independent effects) the change in the interplanar distances in the surface region and the change in the distances between ions in the cleavage plane. By considering an isolated (100) plane of the NaCl structure, a limiting value was found for the latter effect: a contraction of 6% in the case of NaCl (ion polarization was not considered); the actual contraction must, of course, be substantially smaller owing to the compensating influence of the lower layers. In calculating the change in the distance between the surface plane and the plane below it, changes in the distances between other planes were neglected. The interaction potential was also approximated by Born’s two-term formula (2.8), with the constants \(b_{ik}\) and \(n_{ik}\) of the repulsive forces taken to be different for the interaction of plus—plus, plus—minus, and minus—minus ions, and the summation was carried out over the entire crystal. The average displacement of a cell lying in the cleavage plane and containing a positive and a negative ion was determined. Ion polarization was investigated in the dipole approximation; however, in calculating the electrostatic force acting on the surface ions and due to point charges and dipoles, an error was made, as was first pointed out in [16]: the force was overestimated by a factor of two. The magnitude of the displacements obtained by Lennard-Jones and Dent must therefore be reduced by approximately a factor of two. Halide compounds of Na and K were considered. For NaCl, without allowance for polarization of the surface ions, a contraction of the interplanar distance by 0.1% was obtained; with allowance for polarization this contraction was 5%, and for the compounds considered it varies within the range from 5 to 8%. The results of [15] were used by Dent [17] to determine \(\sigma^{(2)}_{(100)}\), and for the indicated compounds \(\sigma^{(2)}_{(100)}\) amounts to from 10 to 26% of \(\sigma^{(1)}_{(100)}\); in particular, for NaCl, \(\sigma^{(2)}_{(100)}\) amounts to 17% of \(\sigma^{(1)}_{(100)}\) (Dent uses for NaCl the value \(\sigma^{(1)}_{(100)} = 93\ \mathrm{erg}/\mathrm{cm}^2\)).

Among earlier investigations one may note the work of Bimüller [18], who considered surface distortions in the form of polarization of the ions of the surface plane in the absence of their displacements; the bonding forces were approximated by expression (2.8). In accordance with the calculations, \(\sigma^{(2)}_{(lmn)}\) is expressed as a certain, \(\kappa\)-th part of the electrostatic contribution to \(\sigma^{(1)}_{(lmn)}\),

where

\[ k=\gamma\cdot\frac{\alpha_{+}+\alpha_{-}}{a_0^3}; \tag{3.1} \]

\(\alpha_{+}\) and \(\alpha_{-}\) are the polarizability coefficients of the cation and anion, taken from [19]; the numerical values of \(\gamma\) for the NaCl structure are 0.671 in the case of the (100) plane and 1.386 in the case of the (110) plane; for the (110) plane of the CsCl structure \(\gamma=0.755\). For NaCl \(\sigma_{(100)}=\sigma^{(1)}_{(100)}+\sigma^{(2)}_{(100)}=86.7\ \mathrm{erg/cm^2}\).

The possibility of not only normal but also tangential displacements of surface ions, allowed for in [15], was investigated more fully in [20]. Approximation (2.8) was used; polarization was considered in the dipole approximation, but the parameters of the repulsive forces and the polarizabilities were taken to be the same for ions of different signs. It was found that, for almost half of the alkali halides, the normal coordination structure in the surface plane proves to be unstable: ions of opposite signs group into pairs and a lattice of molecular complexes is formed; the grouping of ions and the instability of the coordination structure are favored by the large size of the anion and the small size of the cation. The distorted surface plane is displaced in the direction toward the crystal. The results of [20] were analyzed by Patterson [21], who showed that the crude simplifications used in [20] generally have the consequence of decreasing the stability of the structure in the surface plane; however, Patterson also does not deny the possibility of tangential distortions.

In 1946 Verwey [22] considered a number of alkali-halide compounds for which the polarizability coefficients of the cation and anion differ greatly from one another, and proposed the following model of distortions of the surface plane: only anions are polarized; displacements of ions of different signs are different, but occur only in the direction normal to the plane. The (100) plane of the NaCl structure was investigated. The non-Coulomb interaction was reduced to repulsive forces:

\[ \varepsilon_{ik}(R_{ik})=b\cdot e^{-\frac{R_{ik}-(r_i+r_k)}{\rho}}; \tag{3.2} \]

here \(r_i\) and \(r_k\) are the radii of the \(i\)-th and \(k\)-th ions, and \(b\) and \(\rho\) are constants; for distances smaller than the sum of the ionic radii, (3.2) was replaced by

\[ \varepsilon_{ik}(R_{ik})=\frac{B}{R_{ik}^{12}}. \]

Interaction by repulsive forces was taken into account within two layers of nearest neighbors. The calculation showed that in all cases the positive ions are displaced in the direction toward the crystal; the displacements of the negative polarized ions are always smaller than the displacements of the cations and, in most compounds, occur in the opposite direction. Thus, an electric double layer is formed at the surface, with a negative ion on the outside. As in other works, in [22] the method of force equilibrium was used to determine numerical values of the surface distortions, as being more sensitive than the method of the energy minimum. The calculated values of the distortions were used to estimate \(\sigma^{(2)}_{(100)}\). Calculations were carried out for NaBr: the displacement of the cation at the surface is \(z_{+}=0.070\cdot a_0\), the displacement of the anion is \(z_{-}=-0.021\cdot a_0\); \(\sigma^{(1)}_{(100)}=122\ \mathrm{erg/cm^2}\) (in [9] \(\sigma^{(1)}_{(100)}=118.7\ \mathrm{erg/cm^2}\) is given); \(\sigma^{(2)}_{(100)}=-84\ \mathrm{erg/cm^2}\), so that \(\sigma_{(100)}=\)

\(= \sigma^{(1)}_{(100)} + \sigma^{(2)}_{(100)} = 38\) erg/cm\(^2\), i.e., the Verwey model leads to a very strong decrease in the surface energy.

The complexity of calculating the surface distortions and \(\sigma^{(2)}_{(lmn)}\) increases rapidly with an increase in the number of variables used to determine the equilibrium configuration; therefore, as is evident from the foregoing, different authors, in order to simplify the calculations, propose different models of surface distortions. The problem of modeling distortions in the surface region was investigated in the interesting works [23, 24]. In [23] the approximation (3.2) was used for the repulsive forces, replaced by \(A + B/R_{ik}^{12}\) for \(R_{ik} < r_i + r_k\). For NaCl, 7 different surface configurations were considered; the results are presented in Table 2 (\(\mu\) is the dipole moment measured in Debye units; \(\sigma^{(2)}_{(100)}\) is given in erg/cm\(^2\)).

Table 2

The Verwey model (7) leads to a considerably larger change in the surface energy than do the other models, i.e., the determining factors with respect to the magnitude of \(\sigma^{(2)}_{(100)}\) are the polarization of the anion and the difference in the displacements of the surface cations and anions. The magnitude of \(\sigma^{(2)}_{(100)}\) increases as the difference between the cation and the anion increases, so that in the case of LiF and LiJ the calculation of \(\sigma^{(2)}_{(100)}\) according to the Verwey model gives negative values of \(\sigma_{(100)}\). Although negative values of the total surface energy cannot be rejected a priori, values less than zero, as well as very small positive values, seem implausible.

Work [24] should be regarded as a continuation of the study [23]. In [24] a comparison of different models is carried out using the approximation (2.0) for the binding forces, i.e., with the van der Waals interaction taken into account; in this case the values of \(\sigma^{(2)}_{(100)}\) can be compared with the values of \(\sigma^{(1)}_{(100)}\) from [3–5]. The results of the calculation of eight different models for the (100) face of the NaCl crystal are presented in Table 3, borrowed from [24]. For comparison, [24] gives the numerical values of the parameters determining the surface configuration, and \(\sigma^{(2)}_{(100)}\) for NaCl, calculated within the framework of model 8 (\(z_+ \ne z_- \ne 0;\ \mu_+ \ne \mu_- \ne 0\)), using approximation (3.2) for the repulsive forces and neglecting the van der Waals interaction: \(z_+ = 0.1194 \cdot a_0;\ z_- = 0.0295 \cdot a_0;\ \mu_+ = 0.0711;\ \mu_- = 1.1294;\ \sigma^{(2)}_{(100)} = -102.7\) erg/cm\(^2\). Thus, the calculation with inclusion of the van der Waals interaction also leads to the conclusion that the determining factors for the magnitude of \(\sigma^{(2)}_{(100)}\) are the polarization of the anion and different displacements of cations and anions at the surface. The most

Table 3

The greatest stability is possessed by the surface configuration described by the eighth model; this is quite natural, since in this model the largest number of variable parameters is allowed, in comparison with the others. However, the numerical values of $\sigma^{(2)}_{(100)}$ for the eighth model and the Verwey model are close both when the van der Waals interaction is taken into account and when this interaction is neglected. Taking the van der Waals interaction into account leads to appreciably smaller numerical values of the surface distortions, and, in the case of the Verwey model and the eighth model, gives values for $\sigma^{(2)}_{(100)}$ that are one and a half times smaller. The calculation of the surface distortions described by model 8, and of $\sigma^{(2)}_{(100)}$, was carried out in [24] for all alkali halides, with the exception of LiF, using the approximation (2.0) for the binding forces, and also for all compounds except LiF and LiCl without taking the van der Waals interaction into account. In these three cases the approximations used for the binding forces do not lead to a stable surface configuration corresponding to the minimum energy of a semi-infinite crystal. Apparently, one should not take too seriously the numerical values calculated for the lithium halides either. As is seen from the table given in [24], the numerical differences in the surface distortions and in $\sigma^{(2)}_{(100)}$ (when calculated on the basis of approximation (2.0) or (3.2)) are very large for the lithium halides and large for the sodium halides; in the case of compounds with heavier cations (K, Rb, Cs), the differences in the magnitude of the distortions and in $\sigma^{(2)}_{(100)}$ are small. In all cases the surface cations are displaced into the crystal; the surface anions undergo much smaller displacements into the crystal, and in the case of NaI and KI the displacements of the anions occur outward.

The surface distortions and $\sigma^{(2)}_{(lmn)}$ are sensitive to the approximation of the binding forces in the crystal, so that the difficulty noted in § 2 remains. However, a new difficulty arises: the question remains open of the applicability of an approximation, valid in the interior regions, to the description of interionic interaction in the surface layer, since deformation of the surface ions may lead to a substantial change in the character of the interaction. As we have seen, all authors retain not only the form of the approximation of the binding forces, but even the numerical values of the interaction parameters, at best merely adding terms that describe the interaction caused by the fact that a surface ion possesses not only

with a definite charge (which is assumed to be equal to the charge of the corresponding ion in the bulk of the crystal), but also with a definite dipole moment:

\[ \begin{aligned} U_{ik}(R_{ik})=\varepsilon_{ik}(R_{ik}) &-\frac{e_i\cdot(\vec R_{ik}\cdot \vec p_k)}{R_{ik}^{3}} +\frac{e_k\cdot(\vec R_{ik}\cdot \vec p_i)}{R_{ik}^{3}}+ \\[4pt] &+\frac{(\vec p_i \vec p_k)}{R_{ik}^{3}} -3\frac{(\vec R_{ik}\cdot \vec p_i)\cdot(\vec R_{ik}\cdot \vec p_k)}{R_{ik}^{5}}; \end{aligned} \tag{3.3} \]

\(\varepsilon_{ik}(R_{ik})\) is the approximation in the bulk of the crystal; \(U_{ik}(R_{ik})\) is the approximation in the surface region; \(\vec p_i, \vec p_k\) are the dipole moments, and \(e_i, e_k\) are the charges of the \(i\)-th and \(k\)-th ions. However, this is a crude simplification of the essence of the problem: first, ionic deformation can be reduced to dipole polarization only in the case where the field within the ions may be regarded as homogeneous—it is obvious that this condition is not satisfied for the surface region of the crystal; second, the non-Coulomb interaction (i.e., the “repulsive forces” and the “van der Waals forces” as separate parts of this interaction) must be modified with allowance for the state of polarization of the ions; the importance of such a modification is indicated by a number of authors, in particular [25]. The very large value of \(\varepsilon^{(2)}_{(100)}\) in the case of LiF and LiJ apparently testifies to the inapplicability, for these compounds, of the classical (approximation) method in the simple form used in [22, 23]. In light of the foregoing, it should be clear that the quantitative results of works on the consideration of surface distortions and their contribution to the surface energy using the method of approximation of the binding forces are unreliable. These works are important only as preliminary studies indicating the possibility of significant distortions in the surface region of the crystal and the determining role of these distortions in assessing the energetic nonequivalence of the crystal-forming units at different points of the surface among themselves and in comparison with ions inside the crystal.

Quantum-mechanical consideration of the specific surface energy

The need for a definite approximation of the interaction between the crystal-forming units disappears in a quantum-mechanical treatment of the crystal, since in such a treatment the character of the binding forces is determined by the electronic structure of the crystal-forming units, i.e., by the character of the states (wave functions) of the electrons of the outer shells of the crystal-forming units. In a consistent quantum-mechanical study there is no need to use experimental data to find the interaction constants, so that a purely theoretical determination of the physical characteristics of the crystal becomes possible. The relevance of a non-approximation treatment increases in connection with the difficulties, noted above, of the approximation method in calculating the surface energy. However, the quantum-mechanical study of a finite crystal leads to a problem considerably more complicated than the already complex problem of an infinite crystal, and the latter admits solution only after crude simplifications [26]. The application to a finite crystal of even very simplified quantum-mechanical models leads to the necessity of

the performance of enormous computational work; it is therefore not surprising that, as far as we know, the attempt at a quantum-mechanical calculation of the surface energy of ionic crystals is exhausted by three papers [27—29].

A simplified quantum-mechanical investigation of alkali halides was carried out in [27]: the binding energy, elastic constants, and \(\sigma^{(1)}_{(100)}\) were calculated. In this work the smaller ion of the compound was treated as a point charge, while the states of the valence electrons of the larger ion were described by Slater functions:

\[ \varphi_{n,l,m}=g\cdot r^{\,n^*-1}\cdot l^{-\frac{z-s}{n^*}r}\cdot \Phi_{l,m}(\Theta,\varphi). \tag{4.0} \]

Here \(g\) is a normalizing factor, \(n^*\) is the effective principal quantum number, \(z\) is the charge of the ion nucleus, and \(s\) is the screening constant. The interaction was treated in the pair approximation; an interaction parameter \(\delta=(z-s)/n^*\) was introduced, whose numerical value was found from the experimental value of the lattice constant. The non-Coulomb interaction in the crystal, due to the non-point nature of the large ions, was then described by the product of an exponential by a polynomial in \(\delta\) and the lattice constant. The calculated values of \(\sigma^{(1)}_{(100)}\) proved to exceed the data of [9] by almost a factor of one and a half; for example, for NaCl \(\sigma^{(1)}_{(100)}=210\ \text{erg}/\text{cm}^2\).

A quantum-mechanical calculation of the surface energy without taking account of distortions in the surface region, for the (100) face of the NaCl structure, was carried out by Zegeren and Benson [28]. They used the model and numerical method of Löwdin [30]: a one-electron approximation was employed, and Hartree—Fock functions for free ions were used as one-electron wave functions for the electrons of the outer shells of the ions forming the crystal; relativistic effects and thermal vibrations of the atomic cores were not taken into account; mutual deformation of the ions was neglected, which corresponds to considering energy quantities in the first approximation of perturbation theory. No experimental data were used, with the exception of the crystal structure and the numerical values of the fundamental atomic constants—\(e\), \(h\), etc. From the interaction energies \(E_{ik}\) of the \(i\)-th and \(k\)-th ions in the crystal, obtained with allowance for the electronic structure of the ions, it is easy to separate the part corresponding to the interaction of these ions treated as point charges, while the remaining energy of non-Coulomb interaction may conventionally be divided into certain parts: a correction to the Madelung energy, due to the purely electrostatic interaction of charges distributed in space; the exchange energy; and the so-called \(S\)-energy, whose magnitude depends on the electron states not only of the interacting ions but also of all the other ions of the crystal. In calculating the non-Coulomb interaction, the overlap of the wave functions was taken into account only for ions that are nearest neighbors. The values of \(\sigma^{(1)}_{(100)}\) were determined for 5 alkali halides—NaCl, ZiCl, KCl, NaF, ZiF—and were found to be equal to 187, 177, 184, 219, 145 \(\text{erg}/\text{cm}^2\), respectively. In the case of LiF the theoretical model is clearly unsatisfactory, since it leads to a lattice constant very different from the experimental value (2.4 Å instead of the experimental value 2.0 Å). Consequently, the numerical value of \(\sigma^{(1)}_{(100)}\) for LiF is especially unreliable.

A special model, developed in [31, 32] for calculating the binding energy of the molecule and crystal LiF, was applied in [29] to the consideration of surface distortions and of the total surface energy \(\sigma_{(100)}\). The ion \(\mathrm{Li}^{+}\) was treated as a point charge; the ion \(\mathrm{F}^{-}\) was represented in the form of an atomic core with charge \(+5\) and an electron shell containing 6 electrons. The six electrons of \(\mathrm{F}^{-}\) were placed pairwise in orbitals representing the \(2p_x\), \(2p_y\), \(2p_z\) orbitals of the free ion \(\mathrm{F}^{-}\), approximately orthogonalized with respect to the \(1s\) orbitals of the \(\mathrm{Li}^{+}\) ions that are nearest neighbors. Thus, the electronic structure of \(\mathrm{Li}^{+}\) was taken into account inconsistently and only indirectly—through the orthogonality relations. The \(2p\) orbitals of the free ion \(\mathrm{F}^{-}\), the \(1s\) orbital of the free ion \(\mathrm{Li}^{+}\), and also the \(3s\) orbital of the free ion \(\mathrm{F}^{-}\) and the \(2s\) orbital of the free ion \(\mathrm{Li}\) were taken as simple analytical approximations to orbitals calculated by the self-consistent-field method without allowance for exchange. Surface distortions were modeled as follows: first, a change in the distance between the surface plane and the next plane was allowed; second, the deformation of the electron distributions of the anions located in the surface plane was considered. The polarized electron shell of the surface \(\mathrm{F}^{-}\) ions loses spherical symmetry through hybridization of the \(2p\) states with the excited \(3s\) state of the free \(\mathrm{F}^{-}\) ion and the \(2s\) state of the \(\mathrm{Li}^{+}\) ion; in this case, at the surface the chemical bond acquires a partially covalent character, in contrast to the purely ionic bond in the bulk of the crystal. The electric field acting on the surface ions is described in its limiting classical form; polarization is considered in the dipole approximation. The calculated value \(\sigma_{(100)} = 557\ \mathrm{erg}/\mathrm{cm}^{2}\), i.e., considerably larger than the value \(\sigma^{(1)}_{(100)}\) for this salt obtained by the classical method (in [3] \(\sigma^{(1)}_{(100)} = 169\ \mathrm{erg}/\mathrm{cm}^{2}\) is given). Analyzing the experimental data, the authors regard the quantum-mechanical value as more acceptable than that obtained by means of the approximation method. The surface distortions consist in an increase of the interplanar distance by \(0.5\%\) and in the appearance of \(p_{-} = 0.079\).

Application of Statistical Theory to a Bounded Ionic Crystal

It seems expedient to attempt to apply statistical theory to the study of a bounded ionic crystal and to construct a model that would make it possible to analyze distortions in the surface region. Statistical theory, as a certain approximation to the quantum-mechanical theory of many-electron systems, makes it possible to consider a number of phenomena rather simply and has been successfully applied in the theory of atoms, molecules, and crystals [33]. Statistical treatment entails additional simplifications in comparison with the one-electron self-consistent-field method; however, the resulting simplification of the mathematical problem often makes it possible to avoid a number of further crude approximations associated with the practical application of the self-consistent-field method to a crystal, and substantially simplifies the calculations. Naturally, in a statistical treatment, when the individual features of the electronic states in the crystal are not taken into account, exact quantitative values of the physical quantities under consideration cannot be obtained. But, while pointing out the advisability of applying

In applying statistical theory to the study of a finite crystal, we have in mind that a theoretical treatment without the introduction of any empirical constants is of interest in itself and, in the present case, acquires special significance because of the difficulties of the classical approximation method; at the same time, a rigorous quantum-mechanical treatment of surface distortions and of the increase in the energy of a crystal caused by its boundary is impracticable. The quantum-mechanical works cited [27–29] are based on very crude simplifications of the problem. The rough approximations that must necessarily be introduced in a quantum-mechanical investigation lead to numerical values whose accuracy is difficult to analyze; the matter is complicated by the absence of reliable experimental data. Under conditions in which obtaining exact absolute quantitative values is impossible either experimentally or theoretically—and this is essentially the situation at present—qualitative investigations and approximate quantitative estimates of the absolute and relative values of the surface energies for various faces are of considerable interest. A statistical treatment, while retaining the advantages of a nonapproximate method, can serve the purposes of qualitative analysis and approximate quantitative estimates, at the same time making it possible to simplify substantially the necessary calculations. At the same time, by choosing an appropriate model and object of investigation, one can ensure that the accuracy of the statistical treatment will not be significantly inferior to that of an investigation based on the use of approximate electronic orbitals.

Below we describe a statistical model of an ionic crystal and its application to the calculation of the binding energy and \(\sigma^{(1)}_{(100)}\) of a crystal with the NaCl structure. The account, in this calculation, of the deformation of the electronic distribution caused by the transition from free ions to ions in the crystal is itself of considerable interest [33, pp. 296–297] and illustrates the possibilities of statistical theory.

The total energy of an infinite crystal with a known crystal structure at absolute zero temperature, within the framework of statistical theory, is given [33] by the following expression:

\[ E_n=\chi_k \int \rho(\vec r)^{5/3}\,dv +\frac{1}{2}e^2 \sum_{\substack{i,k\\ i\ne k}} \frac{Z_i Z_k}{r_{ik}} -e \int \left[ V_k(\vec r)+ \right. \]

\[ \left. +\frac{1}{2}V^e(\vec r)\right]\rho(\vec r)\,dv -\chi_a \int \rho(\vec r)^{4/3}\,dv; \tag{5.0} \]

Here \(\rho(\vec r)\) is the density of the electronic distribution in the crystal; \(r_{ik}\) is the distance between nuclei with charges \(Z_i\) and \(Z_k\), located at the \(i\)-th and \(k\)-th sites of the crystal lattice, respectively; \(\chi_k\) and \(\chi_a\) are constants; here and below summation is carried out over all sites of the crystal lattice, and integration over all infinite space.

\[ V_k(\vec r)=e\sum_i \frac{Z_i}{|\vec r-\vec r_i|}; \qquad V^e(\vec r)=-e\int \frac{\rho(\vec r')}{|\vec r-\vec r'|}\,dv' \tag{5.1} \]

are the electrostatic potentials created at the point \(\vec r\) by all nuclei and by the electronic distribution \(\rho(\vec r)\), respectively.

In accordance with the principle of minimum energy, the density of the electronic distribution \(\rho(\vec r)\) and the lattice constant \(\delta\) must be such as to give a minimum to expression (5.0). In finding the minimum for (5.0), the variation of \(\rho(\vec r)\) must be carried out subject to the additional condition

\[ \int \rho(\vec r)\,dv=N, \tag{5.2} \]

where \(N\) is the total number of electrons in the crystal.

The complexity of the variational problem that arises requires certain simplifications. The first basic simplification consists in representing \(\rho(\vec r)\) in the form

\[ \rho(\vec r)=\sum_k \rho_k\left(|\vec r-\vec r_k|\right). \tag{5.3} \]

The individual terms of the sum, \(\rho_k\), are assumed to be spherically symmetric with respect to the \(k\)-th lattice site and admit the obvious interpretation: \(\rho_k\) is the density of the electronic distribution of the \(k\)-th constituent of the crystal. In this case the electronic distributions of the constituents of the crystal differ, generally speaking, from the electronic distributions of the corresponding free atoms (ions). In the case of alkali halides, the simplification (5.3) has a good physical justification, since, as is known, the representation of these crystals as being formed by spherically symmetric metal (A) and halide (B) ions is widely used and, to a good approximation, is confirmed by the entire body of physicochemical properties of these compounds. At the same time, consideration of ions possessing filled electron shells is a particularly favorable case for the application of statistical theory, since in this case its shortcomings are manifested to the least degree.

In solving the variational problem for expression (5.0) with the additional condition (5.2), we shall not reduce it to the Thomas–Fermi–Dirac equation for the crystal, but shall use the direct Ritz method. In doing so, while giving up the exact solution of the variational problem, we obtain the possibility of somewhat improving the form of the electronic distribution as compared with the exact mathematical solution for the statistical expression (5.0). Success depends on a successful choice of functions approximating \(\rho(\vec r)\). Very good results were obtained with the Jensen approximation [34], which, when applied to free atoms and ions, gives a good approximation to the quantum-mechanical form of the electronic distribution. The next basic simplification consists in using the form of approximation proposed by Jensen for free atoms (ions) to approximate the electronic distribution of ions in the crystal. In accordance with the first approximation of Jensen’s approximation, we have

\[ \rho_a(y_a)=\frac{N_a}{4\pi\cdot A_a}\cdot\frac{e^{-y_a}}{y_a^3}\cdot(1+c_a\cdot y_a)^3; \]

\[ \rho_b(y_b)=\frac{N_b}{4\pi\cdot A_b}\cdot\frac{e^{-y_b}}{y_b^3}\cdot(1+c_b\cdot y_b)^3. \tag{5.4} \]

Here \(\rho_a\) and \(\rho_b\) are the electron distributions of the cation \(A\) and the anion \(B\) in the crystal; the dimensionless variables \(y_a\) and \(y_b\) are related to \(r_a\) and \(r_b\) as follows:

\[ r_a=\lambda_a\cdot y_a^2;\qquad r_b=\lambda_b\cdot y_b^2. \tag{5.5} \]

The constants \(A_a\) and \(A_b\) are found from the additional condition (5.2), which in the present case reduces to

\[ \int \rho_a(r_a)\,dv=N_a;\qquad \int \rho_b(r_b)\,dv=N_b, \tag{5.6} \]

where \(N_a\) and \(N_b\) specify the “number of electrons” in ions \(A\) and \(B\), respectively. If we consider a purely ionic bond between singly charged ions, then

\[ N_a=Z_a-1;\qquad N_b=Z_b+1. \tag{5.7} \]

However, instead of (5.7) we introduce

\[ N_a=Z_a-(1+n);\qquad N_b=Z_b+(1+n), \tag{5.8} \]

where \(n\) is a variational parameter whose numerical value is found from the minimum of (5.0); in this way it proves possible to take into account the nonionic component in the chemical bond realized in the crystal.

We write the expression for the total energy of the crystal in the form

\[ \begin{aligned} E_n=\sum_i \Bigg\{& \frac{1}{2}e^2 Z_i \sum_{\kappa\ne i}\frac{Z_\kappa}{r_{i\kappa}} - e^2 Z_i \sum_\kappa \int \frac{1}{r_i}\rho_\kappa(r_\kappa)\,dv \\ &-\frac{1}{2}e\sum_\kappa \int V_i^e(r_i)\rho_\kappa(r_\kappa)\,dv \Bigg\} +\varkappa_k\int\left[\left\{\sum_i\rho_i(r_i)\right\}^{5/3} \right. \\ &\left. -\sum_i \rho_i^{5/3}(r_i)\right]dv +\varkappa_k\sum_i\int \rho_i^{5/3}(r_i)\,dv -\varkappa_a\sum_i\int \rho_i^{4/3}(r_i)\,dv. \end{aligned} \tag{5.9} \]

Owing to the exponential decrease of the density of the electron distribution of an ion in the crystal with distance from the nucleus, the regions of essentially pairwise overlap are the most significant. In the pair approximation, \(I_k\)—the change in kinetic energy due to overlap of the electron distributions of the ions forming the crystal (the fourth term in (5.9))—is equal to

\[ I_k\simeq \frac{1}{2}\varkappa_k \sum_{\substack{i,\kappa\\ i\ne\kappa}} \int\left[ \{\rho_i(r_i)+\rho_\kappa(r_\kappa)\}^{5/3} -\rho_i^{5/3}(r_i)-\rho_\kappa^{5/3}(r_\kappa) \right]dv. \tag{5.10} \]

The method for calculating the integrals in (5.10) is constructed in such a way as to take into account essentially only the region of pair overlap. In the pair-interaction approximation we underestimate the value of \(I_k\); this underestimation is, in a definite way, compensated by the neglect, admitted by us in (5.9), of the change in exchange energy due to overlap of the electron distributions, i.e., by neglect of the quantity

\[ -\varkappa_a\int\left[ \left\{\sum_i\rho_i(r_i)\right\}^{4/3} -\sum_i\rho_i^{4/3}(r_i) \right]dv, \tag{5.11} \]

see in this connection [33, pp. 290–291].

Taking into account (5.10) and symmetrizing the term describing the interaction of the electron clouds with the nuclei, the total energy of the crystal can be written in the form

\[ E_n=\sum_i U_i+\frac{1}{2}\sum_{\substack{i,\kappa\\ i\ne\kappa}}\Phi_{i,\kappa}, \tag{5.12} \]

where the total energy of the \(i\)-th ion in the crystal is

\[ \begin{aligned} U_i={}&\chi_\kappa\int \rho_i^{5/3}(r_i)\,dv-\chi_a\int \rho_i^{4/3}(r_i)\,dv \\ &{}-e^2 Z_i\int \frac{1}{r_i}\rho_i(r_i)\,dv -\frac{1}{2}e\int V_i^e(r_i)\rho_i(r_i)\,dv, \end{aligned} \tag{5.13} \]

and the interaction energy of the \(i\)-th and \(\kappa\)-th constituent ions is

\[ \begin{aligned} \Phi_{i,\kappa}={}&e^2\frac{Z_iZ_\kappa}{r_{i\kappa}} +\chi_\kappa\int \left[\{\rho_i(r_i)+\rho_\kappa(r_\kappa)\}^{5/3}\right.\\ &\left.{}-\rho_i^{5/3}(r_i)-\rho_\kappa^{5/3}(r_\kappa)\right]\,dv -e^2Z_i\int \frac{1}{r_i}\rho_\kappa(r_\kappa)\,dv\\ &{}-e^2Z_\kappa\int \frac{1}{r_\kappa}\rho_i(r_i)\,dv -e\int V_i^e(r_i)\rho_\kappa(r_\kappa)\,dv . \end{aligned} \tag{5.14} \]

When the overlap is taken into account only within two shells of nearest neighbors, for a structure of the NaCl type the total energy, calculated per ion pair, is

\[ U_n=U_a+U_b+6\cdot(U_{a,b}+U_{a,a}+U_{b,b}) -M\cdot\frac{(1+n)^2}{\delta}\cdot e^2, \tag{5.15} \]

where \(U_{a,b}\), \(U_{a,a}\), \(U_{b,b}\) are the interaction energies of ions \(A\) and \(B\) after subtracting the energy of the Coulomb interaction; \(M\) is the Madelung constant. The binding energy per ion pair is equal to

\[ \begin{aligned} U_{\text{bind}}={}&(U_a-U_{a,\text{free}})+(U_b-U_{b,\text{free}}) +6(U_{a,b}+U_{a,a}\\ &{}+U_{b,b})-M\cdot\frac{(1+n)^2}{\delta}\cdot e^2, \end{aligned} \tag{5.16} \]

where \(U_{a,\text{free}}\) and \(U_{b,\text{free}}\) are the energies of the free ions \(A\) and \(B\).

The system of equations arising upon minimization of the total energy, or, equivalently, the binding energy, with respect to the 6 variable parameters \(\lambda_a,\lambda_b,c_a,c_b,n,\delta\), was linearized.

Numerical calculations were carried out for RbBr, which was chosen in order to make possible comparison with the calculations of Jensen [34], and also because this compound represents the most favorable case for applying the model described, since it is formed by electron-rich ions of approximately “equal sizes.” The free ions Rb\(^+\) and Br\(^-\) were considered, and it was established that allowance for exchange interaction leads to a slight contraction, especially in the outer regions, of their electron distributions. Using the values of the parameters \(c\) and \(\lambda\) for the free ions and minimizing the binding energy with respect to \(\delta\), we obtained

\[ U_{\text{bind}}=-142.8\ \text{kcal/mol},\qquad \delta=3.74\ \text{\AA}. \]

Both values differ by approximately 9% from the corresponding experimental values, if for the latter one takes [35] \(U_{\text{bond}}=-157.6\ \text{kcal/mol}\); \(\delta=3.445\ \text{Å}\). When the deformation of the electronic distributions caused by the transition from free ions to ions in the crystal was taken into account, the values obtained were

\[ U_{\text{bond}}=-149\ \text{kcal/mol};\qquad \delta=3.61\ \text{Å}. \]

The deformation caused by the transition from the free state to the state in the crystal consists in a slight expansion of the positive ion and a somewhat greater compression of the negative one. For \((1+n)\) the value 0.96 was found. The good agreement of the calculated values of the binding energy and lattice constant with experimental data, as well as the reasonable difference of the effective charge of the ions in the crystal from unity, serve as a criterion for the applicability of the model described. As for the calculation of \(\sigma^{(1)}_{(100)}\), it can be shown that

\[ \sigma^{(1)}_{(100)}= \frac{1}{2\delta^2} \left\{ \frac{(1+n)^2}{\delta}\cdot e^2\cdot B - \left[U_{a,b}+2\cdot\left(U_{a,a}+U_{b,b}\right)\right] \right\} \tag{5.17} \]

where \(B\) is specified by relations (2.6)—(2.7).

\(\sigma^{(1)}_{(100)}\) was calculated in the zero approximation for the electronic distributions of ions in the crystal, when the electronic distributions of free ions were taken as such. In this case \(\sigma^{(1)}_{(100)}=83.7\ \text{erg/cm}^2\), with a Coulomb contribution equal to \(143.8\ \text{erg/cm}^2\). The considerably greater role of the non-Coulomb interaction in the calculation of \(\sigma^{(1)}_{(100)}\), as compared with its contribution to the binding energy, is directly evident from comparing (5.16), in the approximation under consideration, with (5.17). When the deformation of the electronic distributions of the ions caused by the transition from free ions to ions in the crystal is taken into account, for \(\sigma^{(1)}_{(100)}\) a value of \(112.5\ \text{erg/cm}^2\) is obtained (the Coulomb contribution is \(147.5\ \text{erg/cm}^2\)); RbBr is meant. Thus, an initial theoretical model has been defined, relative to which distortions in the surface region of the crystal must be considered. This work is currently being carried out.

Conclusion

Sufficiently reliable values of the specific surface energy of ionic crystals cannot be obtained by means of the classical (approximate) method; the results of studying surface distortions by means of this method are also unreliable. At the same time, a rigorously posed quantum-mechanical problem for a finite crystal proves insoluble, so that the problem of the theoretical determination of exact absolute values of the surface energy for alkali-halide compounds remains unsolved. Alongside searches for a solution to this problem, further approximate non-approximation studies of a finite ionic crystal are useful; it is precisely in this light that the expediency of applying statistical theory in the present study is considered.

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