PHYSICOCHEMICAL PROPERTIES OF IONIC CRYSTALS AND MOLECULAR CONCENTRATION

P. A. SAVINTSEV

(Presented by the scientific seminar on dielectrics)

Many physical properties of ionic crystals depend on the interaction of the particles forming the crystal lattice. Therefore it proves possible to compare a large number of mechanical, thermal, electrical, and optical properties of crystals with the energy of the crystal lattice \(U\) [1].

According to A. F. Kapustinskii,

\[ U = 256 \frac{z \eta_1 \eta_2}{r}, \tag{1} \]

i.e., the energy of the crystal lattice is determined by the number of ions \(z\) in the molecule of the compound, the valence of the ions, and the distance between the ions \(r\). Let us note that, expressing \(U\) through \(r\), we use a quantity characterizing the linear dimensions of the crystal. For judging processes occurring in crystals, it is often necessary to use quantities characterizing the volume properties of the crystal (density, molecular volume, etc.). In this case it is desirable to transform equation (1), replacing \(r\) by another constant, for example, by the volume of the molecule \(v_1 = \frac{4}{3}\pi r^3\). Then the expression for the energy of \(N\) interacting particles will have the form:

\[ U = 256 \sqrt[3]{\frac{4\pi}{3}} \frac{z \eta_1 \eta_2}{\sqrt[3]{v_1}}. \]

It is expedient to pass from the volume of the molecule \(v_1\) to the volume of the gram-molecule \(v\). In this case

\[ U = 256 \sqrt[3]{\frac{4\pi N}{3}} \frac{z \eta_1 \eta_2}{\sqrt[3]{v}}. \]

Let us replace the gram-molecular volume \(v\) by the density of the crystal \(D\) and the molecular weight \(M\). Then

\[ U = 256 \sqrt[3]{\frac{4\pi N}{3}} z \eta_1 \eta_2 \sqrt[3]{\frac{D}{M}}. \]

Let us denote: \(256 \sqrt[3]{\frac{4\pi N}{3}} = C\). We obtain:

\[ U = C z \eta_1 \eta_2 \sqrt[3]{\frac{D}{M}}. \]

Thus, the energy of the crystal lattice is proportional to the cube root of the ratio \(\dfrac{D}{M}\), which is equal to the number of gram-molecules per unit volume and may be called the molecular concentration. In the case of ionic crystals, \(\dfrac{D}{M}\) is equal to the ionic concentration.

Let us consider the relation of \(\dfrac{D}{M}\) to other quantities characterizing the properties of a crystal.

A number of authors [2, 3] compare the constants of crystals with the characteristic temperature \(\Theta = \dfrac{h\nu}{\kappa}\), where \(h\) is Planck’s constant, \(\kappa\) is Boltzmann’s constant, and \(\nu\) is the frequency of vibration. Let us find the relation of \(\Theta\) to the ratio \(\dfrac{D}{M}\).

For calculating \(\Theta\) of an ionic crystal, the formula [2] is proposed:

\[ \Theta=\frac{3\cdot 10^3}{r}\sqrt{\frac{z\eta_1\eta_2}{A}}, \]
where \(A\) is the mean atomic weight.

In accordance with the foregoing, let us replace \(r\) by the expression
\[ \sqrt[3]{\frac{3M}{4\pi ND}}. \]

We obtain:
\[ \Theta=3\cdot 10^3\sqrt[3]{\frac{4\pi N}{3}}\sqrt{\frac{z\eta_1\eta_2}{A}}\sqrt[3]{\frac{D}{M}}. \]

Let us denote
\[ 3\cdot 10^3\sqrt[3]{\frac{4\pi N}{3}}=C_1. \]

Then
\[ \Theta=\frac{z\eta_1\eta_2}{A}C_1\sqrt[3]{\frac{D}{M}}. \]

Consequently, the characteristic temperature is proportional to the cube root of the molecular concentration.

The results obtained make it possible to pass from comparing the physical properties of crystals with the lattice energy or with the characteristic temperature to comparing the properties with a simpler physical quantity—the concentration of ions and molecules in them. P. P. Lazarev [4] pointed to the dependence of the mechanical properties of elements on atomic concentration. He calculated the force \(F\) required in order to bring an atom out of its equilibrium position and transfer it to a neighboring position, \(F=\pi N F(R)\), where \(N\) is the concentration of atoms,

\[ F(R)=\int_0^R \varphi(r)r^2\,dr, \]

with \(\varphi(r)\) being the force of attraction of neighboring atoms, depending on the distance \(r\) and equal to zero for \(r>R\); \(R\) is the radius of the sphere of molecular action.

P. P. Lazarev assumes that \(F(R)\) does not depend on the nature of the body. Then \(F=\kappa N\), where \(\kappa = N F(R)\) is a constant. The author considered that \(F\) is equal to the hardness of the body, and therefore the hardness of the body must be proportional to the atomic concentration of the elements. Processing ex-

experimental data, P. P. Lazarev constructed curves of hardness—atomic weight and atomic concentration—atomic weight. It turned out that the course of these curves is approximately the same. This gave the author the right to regard his conclusions as correct.

It must be noted that P. P. Lazarev gave an overly simplified definition of hardness. One cannot agree with P. P. Lazarev’s assertion that \(F(R)\) and \(\kappa\) do not depend on the nature of the body. The author did not compare the properties of chemical compounds and alloys with particle concentration.

We believe that comparison of the properties of crystals with particle concentration can be carried out only for similar ionic compounds of the type \(AB\), \(AC\), \(AD\), etc.

Let us compare the mechanical properties of crystals with molecular concentration.

V. D. Kuznetsov and E. V. Lavrent’eva [5] determined the hardness of crystals by the method of damped oscillations. The results obtained by them are presented in Fig. 1, on which we have written the values of \(\alpha\) to the right of the curves. In the damped-oscillation method, the amplitude of some oscillation is chosen as a measure of hardness. Using Fig. 1, let us take the amplitude of the tenth oscillation as such a characteristic and compare its values with the value of \(\alpha\). The values thus obtained are entered in Table 1.

Fig. 1. Curves of damped oscillations.

Table 1

Relation between the hardness of crystals, determined by the method of damped oscillations, and the molecular concentration \(\alpha\)

From Table 1 and Fig. 1 it is evident that the hardness of crystals \(H\) increases with increasing \(\alpha\). Moreover, the dependence of \(H\) on \(\alpha\) for alkali-halo-

ionic crystals and compounds of the second group of chemical elements will be different.

K. V. Savitskii [5] determined the hardness of alkali-halide crystals by the drilling method. The measure of hardness here is taken to be the diameter of the pit for a definite drill geometry and a constant pressing load. The larger the pit diameter \(d\), the lower the hardness of the crystals. The results of the experiments are given in Table 2.

Table 2

Comparison of hardness by drilling with molecular concentration

From Table 2 it is evident that the hardness of alkali-halide crystals increases with increasing molecular concentration.

In Fig. 2 we show the dependence of the scratch hardness \(H\) on \(\alpha\). The latter proves to be linearly related to the molecular concentration.

Fig. 2. Dependence of scratch hardness on \(\alpha\)

Goldschmidt [6] attempted to establish a relation between the hardness of crystals and their structure.

We used the results of his experiments in order to find the relation between Mohs hardness and molecular concentration. The dependences thus obtained are given in Figs. 3 and 4, from which it is evident that Mohs hardness is regularly related to molecular concentration. It can also be seen that the straight lines corresponding to compounds of the elements of Group 2 of D. I. Mendeleev’s table lie above the straight line constructed for the alkali-halide salts. In the case of divalent compounds, a sharper dependence of hardness on molecular concentration is observed than in the case of alka-

…alkali-halide salts. Thus, in finding the relation of hardness to molecular concentration, the influence of valence must be taken into account.

N. S. Kurnakov [7] determined the flow pressure of some alkali-halide salts. We used N. S. Kurnakov’s data to establish the relation of the flow pressure to the value \(\alpha\).

Fig. 3. Dependence of Mohs hardness on \(\alpha\).

Fig. 4. Dependence of Mohs hardness on \(\alpha\).

The dependence found is presented in Fig. 5, from which it is evident that the flow pressure increases with increasing \(\alpha\).

Fig. 5. Dependence of flow pressure on \(\alpha\).

One of the important quantities characterizing the mechanical properties of a crystal is the coefficient of compressibility. The relationship between this quantity [8] and the molecular concentration is given in Table 3, from which it is seen that, for compounds of Li, Na, K, Rb, Cs, and Ag, the coefficient of compressibility decreases regularly with increasing molecular concentration.

Table 3

Relationship between the coefficient of compressibility of \(k\)-ionic crystals and molecular concentration

The most important elastic constant characterizing the interaction between ions is the shear modulus. The dependence of this quantity for alkali-halide crystals on molecular concentration is shown in Fig. 6, on the basis of which one may speak of the existence of a connection between the shear modulus and \(\alpha\).

V. D. Kuznetsov, in studying the mechanical properties of crystals, uses the methods of abrasive wear and mutual grinding of crystals [9]. From experiments on grinding alkali-halide crystals with abrasive paper, ratios of the friction force \(F\) to the ground-off mass \(M\) were found. These ratios \(\frac{F}{M}\) in Table 4 we compared with the molecular concentration \(\alpha\).

Table 4

From Table 4 it is seen that, for alkali-halide crystals, the ratio \(\frac{F}{M}\) increases with increasing molecular concentration.

In mutual grinding of alkali-halide polycrystals with the aid of abrasive powder of grade KZ-100, [9] obtained

Fig. 6. Dependence of the shear modulus on \(a\).

relations of the co-ground volumes \(\frac{v_2}{v_1}\), which in Table 5 are compared by us with the molecular concentration. A single crystal of NaCl was used as the reference crystal in these experiments. From Table 5 it is seen that the relative wear \(\frac{v_2}{v_1}\) decreases as the molecular concentration increases.

We carried out experiments [10] on mutual grinding of alkali-halide single crystals with the aid of abrasive powder KZ-100. The single crystals were grown from chemically pure salts by the Kyropoulos method. The results of these experiments are given in Table 6, in which a comparison is made of the ratio of the co-ground volumes \(\frac{v_2}{v_1}\) of alkali-halide single crystals with the molecular concentration \(a\). A single crystal of KCl was chosen as the reference in these experiments. From Table 6 it is seen that, with increasing molecular concentration, the abrasive wear of alkali-halide single crystals decreases.

Thus, with increasing molecular concentration, there increase the hardness determined by the methods of damped vibrations and drilling, Mohs hardness, abrasive wear resistance, pres—

flow. With increasing molecular concentration, the compressibility coefficient of ionic crystals decreases.

Table 5

Table 6

It is of interest to establish the connection of $\alpha$ with the thermal characteristics of ionic crystals. A comparison of $\alpha$ with the thermal constants of alkali-halide crystals is shown in Table 7.

In compiling this table we used the data of [7, 8, 11, 12].

From Table 7 it is seen that, with increasing molecular concentration, there is an increase in the melting temperature, the boiling temperature, the heat of fusion, and the heat of sublimation, and a decrease in the coefficient of linear expansion. It should also be emphasized that the coefficients of proportionality between the physical quantities under consideration and $\alpha$ change in passing from compounds of one halide to compounds of another halide.

In Figs. 7–10 we depict the dependence on $\alpha$ of the heat capacity, entropy, melting temperature, and heat of sublimation for oxides of the second group (BaO, SrO, CaO, MgO). From these figures it is seen that the heat of sublimation and the heat of fusion increase as $\alpha$ increases; heat capacity and entropy decrease as $\alpha$ increases.

Let us consider the relation between $\alpha$ and quantities characterizing the electrical properties of bodies.

The relation of $\operatorname{tg}\delta$ at a frequency of $10^6$ cps to molecular concentration is presented in Fig. 11. The values of $\operatorname{tg}\delta$ were taken by us from [13]. From Fig. 11 a regular decrease of $\operatorname{tg}\delta$ with increasing molecular concentration is seen.

N. P. Bogoroditskii [14] gives, for some alkali-halide salts, values of the dielectric constant $\varepsilon$ and $\operatorname{tg}\delta$ at temperatures of 100, 200, and 300°C. On the basis of these data one can calculate the temperature coefficient of dielectric losses $\beta$ by the formula:

$$ \beta = \frac{\operatorname{tg}\delta_2 - \operatorname{tg}\delta_1}{t_2 - t_1}, $$

where $\operatorname{tg}\delta_1$ and $\operatorname{tg}\delta_2$ are the tangents of the dielectric-loss angle at temperatures $t_1$ and $t_2$, respectively.

Fig. 7. Dependence of heat capacity on \(x\)

Fig. 8. Dependence of entropy on \(x\)

Fig. 9. Dependence of the melting temperature on $\alpha$

Fig. 10. Dependence of the heat of sublimation on $\alpha$

Fig. 11. Relation of the tangent of the dielectric-loss angle at a frequency of $10^6$ cycles/s to the number $\alpha$

Table 7

Relation of $\alpha$ to thermal constants

Table 8

Comparison of the dielectric constant, dielectric losses, and temperature coefficient of dielectric losses with molecular concentration

From Table 8, in which the quantities $\varepsilon$, $\mathrm{tg}\,\delta$, and $\beta$ are compared with the molecular concentration $\alpha$, it is evident that, as $\alpha$ decreases, the dielectric constant, the dielectric losses, and the temperature coefficient of dielectric losses increase.

Figure 12 presents the dependence of the electronic component of the dielectric coefficient \(\tilde{\varepsilon}\) on the molecular concentration. From Fig. 12 one can see a regular decrease of \(\tilde{\varepsilon}\) with increasing \(\alpha\).

A comparison of the electric strength \(E_{pr}\) of alkali-halide crystals with molecular concentration is shown in Table 9 and in Fig. 13. The values of \(E_{pr}\) were taken by us from work [15]. From Fig. 13 and Table 9 it can be seen that the electric strength \(E_{pr}\) is related to the molecular concentration \(\alpha\) by the relation:

\[ E_{pr}=\kappa\alpha . \]

Using this relation, we determined the electric strength of LiCl, LiBr, CsCl, CsBr, and CsJ crystals, for which experimental values of \(E_{pr}\) are not yet known. These data are given in Table 9.

Table 9

Comparison of the electric strength of alkali-halide crystals with molecular concentration

It should be noted that the electric strength of the LiCl crystal was calculated earlier by E. K. Zavadovskaya [16], proceeding from the relation of \(E_{pr}\) to the energy of the crystal lattice. The value of \(E_{pr}\) for LiCl obtained by us coincided with that found by E. K. Zavadovskaya.

In addition, we found a dependence of the electric strength of certain liquid dielectrics on molecular concentration (Fig. 14). The values of \(E_{\mathrm{pr}}\) were taken from [17].

Fig. 12. Relation of the electronic component of the dielectric coefficient to the number \(z\)

Fig. 13. Dependence of the electric strength of alkali-halide crystals on molecular concentration

Fig. 14. Dependence of the electric strength of a homologous series of liquid dielectrics on molecular concentration

From Figs. 13 and 14 it is evident that the character of the dependence of electric strength on molecular concentration is different for alkali-halide crystals and liquid dielectrics.

Let us consider the relationship between the ionic conductivity of alkali-halide salts and the value \(a\). The ionic conductivity of solids increases with temperature and is determined by the Van’t Hoff equation: \(\sigma = Ae^{-B/kt}\).

The dependence of the constant \(B\) (activation energy) on molecular concentration is shown in Fig. 15. We took the values of \(B\) from work [18]. From Fig. 15 one can see a regular increase of \(B\) with increasing \(a\).

A comparison of the value \(a\) with the ionic conductivity of salts \(\sigma\) at the melting temperature is shown in Table 10, from which it is evident that at the melting temperature of salts an increase in electrical conductivity is observed with increasing \(a\).

In the case of solid ionic dielectrics, the conductivity decreases with increasing \(a\); \(a\) is a constant having the meaning of ionic concentration. Thus, an increase in the concentration of ions in the melt increases its electrical conductivity, whereas an increase in the concentration of ions in the lattice lowers the electrical conductivity. The latter finds its explanation on the basis of the relationships we have considered. An increase in \(a\) leads to an increase in mechanical, thermal, and electrical strength.

It is of interest to establish the relationship of \(a\) with the optical constants of the crystal.

Fig. 15. Dependence of the activation energy of the electrical-conduction process on \(a\)

Table 10

Comparison of the ionic conductivity of salts at the melting temperature \(\sigma\) with molecular concentration

In Fig. 16 we give the dependence of the refractive index \(n_D\) on the number \(a\). In this case a regular decrease in the refractive index is observed with increasing \(a\).

In alkali-halide crystals it is possible to observe color centers with a characteristic band of \(F\)-absorption in the visible part of the spec-

of the color. The properties of the \(F\)-band are connected with the structure of the crystal. E. K. Zavadovskaya [19] established a relation between the absorption maximum corresponding to the \(F\)-band and the lattice energy.

Fig. 16. Dependence of the refractive index \(n_D\) on molecular concentration.

In Fig. 17 we give the relation between the position of the maximum of the \(F\)-absorption band and molecular concentration. The dependence in this case is linear.

Thus, the optical constants are also connected with the value \(\alpha\). There is a relation between molecular concentration and certain properties of semiconductors. A. F. Ioffe [20] points out that the electrical properties of semiconductors are determined primarily by short-range order, i.e., by the chemical nature of the neighbors, the arrangement of the particles, and the magnitude of the interatomic distances. The presence of a strictly periodic lattice plays a secondary role. Molecular concentration depends on the arrangement of the particles, the interatomic distances, and the nature of the particles, and therefore can be used to characterize the properties of semiconductors.

N. P. Mokrovskii and A. R. Regel [21], measuring the electrical conductivity of silicon, found that upon melting its specific resistance decreases by a factor of 29. This is explained by an increase in the average coordination number in liquid silicon in comparison with solid silicon. The density of silicon in the liquid state is greater than in the solid state, which leads to a significant increase of \(\alpha\) upon melting.

They also [22] found a relation between changes in density and electronic conductivity upon melting of substances with structures of the diamond or zinc-blende type. It turned out that in substances (Ge, GaSb, InSb) for which the electrical conductivity increases sharply upon melting, there is a corresponding increase in density as well. In HgTe the change in electrical conductivity upon melting has a complex charac-

...ter, which corresponds to the complex dependence of the temperature behavior of density.

These experiments are convincing evidence of the possibility

Fig. 17. Relationship between the position of the maximum of the \(F\)-absorption band and the value of \(\alpha\).

of using molecular concentration to estimate the properties of substances.

With the aid of \(\alpha\) one can also estimate the Hall mobility of electrons \(\mu_H\) in crystals. Table 11 compares \(\mu_H\) with molecular concentration. The values of \(\mu_H\) are taken from [23]. It is seen from Table 11 that, for alkali-halide compounds, the mobility of electrons increases as the molecular concentration decreases.

Thus, concentration is connected with a large number of constants that characterize real crystals. With increasing molecular concentration, the electrical, thermal, and mechanical stability of ionic crystals increases.

Table 11

Comparison of the Hall mobility of electrons in crystals \(\mu_H\) with molecular concentration \(\alpha\)

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