Relationship between the Magnitude of the Dielectric Strength of Crystals and the Components of the Energy of the Crystal Lattice

E. K. ZAVADOVSKAYA

The expression for the energy of an ionic crystal lattice is composed additively of the magnitude of the potential energy of the electrostatic forces of attraction, the potential energy of the repulsive forces, and the forces of molecular cohesion (van der Waals energy). In this case one obtains the part of the lattice energy, dependent on volume, at the temperature under consideration, and not at absolute zero. The difference between the lattice energy calculated in this way and the energy at absolute zero is small and does not exceed \(0.1\text{–}2\%\). To calculate the total vibrational energy of the lattice, one must also take into account the vibrational energy of the lattice at absolute zero. The vibrational energy at absolute zero is due to the fact that, according to the laws of quantum mechanics, each oscillator at the temperature of absolute zero possesses one half of a “vibrational” quantum, that is, an energy \(\dfrac{h\nu_0}{2}\). The vibrational energy of a lattice consisting of \(N\) atoms, at the temperature of absolute zero, is obtained in the form \(\dfrac{9}{4}Nh\nu_m\), where \(\nu_m\) is the frequency of the natural vibrations of the ions (determined, for example, by the residual-rays method).

Thus, the total lattice energy of an alkali-halide crystal per mole of substance at temperature \(T = 0\) can be expressed by the formula

\[ U_0 = -\Phi_0 = N\left[\frac{z^2 e^2 \alpha}{r_0} + \frac{c}{r^6} - B(r_0) - \frac{9}{4}h\nu_m\right], \tag{1} \]

where \(-\Phi_0\) is the lattice energy, a negative quantity; \(\dfrac{z^2 e^2 \alpha}{r_0}\) represents the part of the energy due to electrostatic attraction, where \(z\) is the smallest valence occurring in the lattice, \(r_0\) is the distance between nearest neighboring ions, and \(\alpha\) is the Madelung coefficient;

\[ \frac{c}{r^6} \]

is the van der Waals energy (the energy of the forces of molecular cohesion);

\(B(r_0)\) is the potential energy of the repulsive forces of the lattice.

The lattice energy of crystals of alkali-halide salts and its additive components are known. Figure 1 presents the relationship between the part of the lattice energy due to the electrostatic forces of attraction and the dielectric strength. Let us note that the electrostatic potential energy of the crystal lattice, due to the forces of attraction between oppositely charged ions, is a substantially negative quantity.

Table 1

Lattice energy of crystals of salts of the alkali-halide series and its additive components (according to Born and Göppert-Mayer), in \(\times 10^{12}\) erg/molecule.

Electrical strength and the electrostatic part of the lattice energy

Electrical strength and repulsion-force energy

Electrical strength, kV/cm

Electrostatic part of the lattice energy

\[ U_{\mathrm{e}}=-\frac{\alpha e^2}{r_0}\times 10^{12}\ \frac{\mathrm{erg}}{\mathrm{molecule}} \]

Fig. 1. Dependence between the part of the energy due to electrostatic forces of attraction and the electrical strength for crystals of alkali-halide salts.

Repulsion-force energy \(B(r_0)\) in \(10^{12}\) erg/molecule

Fig. 2. Dependence between the part of the energy due to electrostatic forces of repulsion and the electrical strength for crystals of the alkali-halide series.

According to the data shown in Fig. 1, the electrical strength of ionic crystals increases with increasing potential energy of the attractive forces in the lattice. This increase may be due both to an increa-

…increase of the numerator \(z^2 e^2 \alpha\), and also a decrease of the distance between the ions, that is, of the lattice constant.

The equation of the straight line in Fig. 1 has the form:

\[ E_{\mathrm{pr}} = 0.28\,U_{\mathrm{e}} - 2.51, \]

where \(E\) is in \(MV/cm\) and \(U_{\mathrm{e}}\) is in \(10^{12}\) erg/molecule.

Thus, the greater the amount of energy expended in the formation of a given lattice and manifested in the part of the potential energy due to the forces of attraction \((-U_{\mathrm{e}})\), the greater is also its dielectric strength.

Figure 2 shows that, with an increase in the part of the potential energy of the lattice due to repulsive forces, the dielectric strength increases. This part of the lattice reaches 12% for RbI, and for LiF—20% of the total lattice energy. The simplest expression for the part of the energy due to repulsive forces is given in the form

\[ U_{\mathrm{ot}} = -\frac{N}{2}\,\frac{p}{r^n}, \]

where \(n\) is the number of particles in the crystal, \(p\) is a quantity depending both on the spatial arrangement of the particles of the crystal, on its structure, and on the structure of the ions. The increase in the energy due to repulsive forces is determined by the decrease in the distance between the ions \(r_0\). A decrease in the distance between the ions, that is, in the lattice constant, is accompanied by an increase in the dielectric strength of the dielectric. The greater the ability of a given structure to withstand the action of the repulsive force, the greater is its strength.

Dielectric strength and the van der Waals part of the energy of the crystal lattice

Fig. 3. Dependence between dielectric strength and the forces of molecular cohesion.

The data presented in Fig. 3 show that the dielectric strength of the dielectric decreases with increasing energy of the forces of molecular cohesion. Its magnitude is insignificant.

For LiF it amounts to 0.5%, and for RbI—2.8% \(U\).

This part of the lattice energy is due to the polarization of the ions. In the theory of the crystal lattice it is written in the form \(\dfrac{C}{r^6}\).

The universal forces of molecular cohesion are explained by quantum mechanics on the basis of the existence of motion at absolute zero temperature. The energy of the forces of molecular cohesion, in comparison with the internal energy of individual molecules, is always small.

The energy of the molecular binding forces, according to quantum mechanics, of two independent oscillators is expressed by the formula

\[ \Delta W_m = -\frac{C}{r^6} = -\frac{1}{2} h\nu_0\alpha^2, \]

where \(\dfrac{h\nu_0}{2}\) denotes the zero-point energy of an unbound oscillator, and \(\alpha\) is the polarizability of an individual ion with charge and the quasi-elastic coefficient \(a\) of the oscillator \((eE = ax,\) where \(x\) is the amplitude of oscillation of the oscillator). The minus sign in the expression for the energy \(\Delta W_m\) means that attraction takes place here.

Thus, the constant \(C\) can be determined from the natural frequencies and the polarizabilities, which are related by the relation

\[ \alpha\nu_0^2=\frac{e^2}{4\pi^2 m}. \]

From the preceding reasoning there follows the significant conclusion that the forces of molecular cohesion increase proportionally to the square of the polarizability of the ion \((\alpha^2)\).

With increasing polarizability of the particles forming the lattice, the electric strength of the crystal rapidly decreases. In the present case we also observe a decrease in the magnitude of the electric strength of the dielectric as a consequence of the increase in polarizability with increasing energy of the molecular binding forces in the lattice.

The decrease in the electric strength of the dielectric as a consequence of the increase in polarizability of its particles is so great that it outweighs the increase in electric strength with increasing zero-point energy described earlier. This is also evident from the following condition. The quantity \(h\nu_0=\dfrac{2}{3}\dfrac{e^2}{\alpha}\rho\), where \(\rho\) is the mean distance from the electron to the nucleus. Thus one obtains

\[ C=\frac{1}{3}\frac{e^2}{\alpha}\rho^2\alpha^2=\frac{1}{3}e^2\rho^2\alpha. \]

Between the principal components of the lattice energy, due to electrostatic forces of attraction and forces of repulsion, on the one hand, and electric strength, on the other hand, a satisfactory linear dependence is observed.

I express my gratitude to Prof. A. A. Vorob’ev for discussion of the questions raised in the article.