ON THE RELATIONSHIP BETWEEN THE DIELECTRIC STRENGTH OF SOLID DIELECTRICS AND ENTROPY

A. F. GORODETSKII

Since, in the electrical breakdown of solid dielectrics, the bonds between the particles that constitute them are destroyed, it is of interest to compare the magnitude of dielectric strength with quantities related to the energy characteristics of the lattice.

The most detailed investigation of dielectrics from this point of view was carried out by A. A. Vorob'ev [1], A. A. Vorob'ev and E. K. Zavadovskaya [2], and E. K. Zavadovskaya [3].

The authors point to the existence of a dependence between the dielectric strength of crystals of alkali-halide salts and the lattice energy, heat of formation, lattice constant, etc.

On the basis of such energetic considerations, E. K. Zavadovskaya [4] examined a large group of dielectrics possessing high dielectric strength. The cited work gives an empirical formula that makes it possible to express the dielectric strength of crystals through quantities characterizing the packing density and the valence bond of the particles. The values of dielectric strength calculated by this formula proved to be in good agreement with the experimental data for alkali-halide salts.

This latter circumstance made it possible to draw certain conclusions about the high values of dielectric strength of a number of oxides, sulfides, and fluorides.

The fruitfulness of investigations in this direction led to the idea of the advisability of comparing dielectric strength with various thermodynamic characteristics and, in particular, with entropy.

Table 1 and Fig. 1 give data on the dielectric strength and standard entropies [5] of several salts of the alkali-halide series.

Table 1

As is evident from the table and from the graph, there is a quite distinct dependence between the quantities being compared.

Thus, it may be asserted that an increase in the standard entropies for substances with the same structure characterizes a corresponding decrease in their dielectric strength.

Consequently, for substances with the same structure, the values of the standard entropies can be used to judge the relationships among the dielectric strengths of these substances.

Fig. 1. Standard entropy

Thus, for some oxides having a lattice of the NaCl type, cited in work [4], the values of the standard entropies are given in Table 2.

Table 2

Proceeding from our considerations, one should expect the dielectric strength to change in the direction of decrease in the sequence: MgO, CaO, FeO, BaO.

According to the calculations of E. K. Zavadovskaya, the corresponding dielectric strengths have the values: FeO—25.0; MgO—24.0; CaO—21.7; BaO—19.5.

Unfortunately, there are no experimental data for these dielectrics, with the exception of MgO, for which the value \(E_{\mathrm{br}} = 4.4 \cdot 10^2\ \mathrm{kV/cm}\) is given; the latter result differs sharply from that calculated in [4].

Let us attempt to give an interpretation of the results obtained.

As is known from statistical physics, the entropy of a system can be represented by the expression:

\[ S = 3Nk \ln \frac{T}{\theta_c} + 4Nk \quad \text{for } T \gg \theta_c, \tag{1} \]

where

\[ \theta_c=\frac{h\nu_{\max}}{k}=\frac{h}{k}\left(\frac{9N}{4\pi V}\right)^{1/3} \left(\frac{c_l^3 c_t^3}{c_l^3+c_t^3}\right)^{1/3} \tag{2} \]

is the characteristic temperature of the crystal, and the quantities \(h\), \(N\), \(k\) are universal constants having their usual meaning; \(\dfrac{N}{V}\) is the density of the crystal; \(c_l\), \(c_t\) are the velocities of sound in the crystal.

Proceeding from the values of the quantities entering into (1) and (2), one may put forward the following hypothesis.

Electrons, having accumulated energy in the electric field, transfer their energy to the lattice sites by collisions with them. The energy received by the ions is dissipated in the form of elastic waves.

If the mean energy accumulated by the electrons is denoted by:

\[ w=\frac{ne^2E^2}{m}\tau(w), \tag{3} \]

where \(n\) is the number of electrons per unit volume, \(e\) is the charge of the electron, \(m\) is the mass of the electron, \(\tau(w)\) is the time of free motion of the electron, and \(E\) is the electric-field strength [1], and if the energy carried by a sound wave in one second through \(\text{cm}^2\) is expressed, as usual, as

\[ w'=\frac{1}{2}\rho c \xi_0^2\omega^2, \tag{4} \]

where \(\rho\) is the density, \(c\) is the velocity of sound, \(\xi_0\) is the oscillatory velocity (amplitude value), and \(\omega\) is the frequency of oscillations, then it seems possible to us to assume that breakdown of a solid dielectric can take place only under the condition that

\[ w-w'>0. \tag{5} \]

It follows from these considerations that, in those substances in which the velocity of sound is higher, one may expect a greater dielectric strength.

Table 3

The data of Table 3, in which the values calculated by us for the velocity of propagation of sound in several alkali-halide crystals are given, show that our conclusion has a basis.

In conclusion I express my gratitude to Professor Doctor A. A. Vorob’ev for discussion of the questions touched upon in the article.

References Cited

1. Vorob’ev A. A. Izvestiya TPI, vol. 63, 3, 1944.
2. Vorob’ev A. A. and Zavadovskaya E. K. DAN SSSR, 81, 375, 1951.
3. Zavadovskaya E. K. Izvestiya TPI, vol. 73, 15, 1952.
4. Zavadovskaya E. K. DAN SSSR, vol. 82, 709, 1952.
5. Handbook of Chemistry, vol. 1, Goskhimizdat, 1901.