PHYSICOCHEMICAL PROPERTIES OF SOLID SOLUTIONS AND MOLECULAR CONCENTRATION

P. A. SAVINTSEV

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

Many physical properties of ionic dielectrics depend on the crystalline structure of the substance, on the interaction and arrangement of the particles forming the crystal lattice. It has been shown [1] that the electrical strength of crystals depends on the energy of the crystal lattice and on the liberation energy of the most weakly bound electrons. For the electrical breakdown of solid dielectrics, two energy processes are of great importance: the liberation of electrons and the destruction of the lattice. The liberation energy of electrons is of the order of magnitude of the lattice energy. The energy of the electric field is transferred to the lattice and is expended on separating it into ions in the breakdown channel. Therefore it is reasonable to seek a connection between the electrical strength of dielectrics and quantities characterizing the crystal lattice.

Dielectric losses are connected with the motion of ions in dielectrics, with their ionic structure [1]. Dielectric losses increase as the binding energy of ions in the lattice decreases. Ceramic materials with small dielectric losses can be obtained by using compositions with a higher binding energy between particles.

N. P. Bogoroditskii [2] indicates that in ceramics the ionic relaxation losses caused by displacements of ions under the influence of an electric field deserve the greatest attention. The losses depend on the structure of the material, on the character of the packing of ions in the lattice. This gives grounds for seeking a connection between dielectric losses and quantities characterizing the crystal lattice.

B. M. Rovinskii [3] points to the dependence of the mechanical properties of crystals on their atomic structure.

N. N. Sirota [4], discussing the question of the connection between the interaction energy of components and the elastic constants of solid solutions and chemical compounds, comes to the conclusion that with an increase in the interaction energy the elastic modulus of solid solutions and chemical compounds will deviate from the additive value in the direction of increase, and, conversely, with a decrease in the interaction energy between the components one should expect the value of Young’s modulus to deviate from the additive value in the direction of decrease.

The change in the electrical resistance and the temperature coefficient of resistance of crystals when impurities are introduced into them

is explained [4] by a change in the interaction between the particles forming the crystal lattice.

With an increase in the energy of interaction between atoms, the heat of sublimation and the characteristic vibration frequency increase. Grüneisen [4] established a relation between the characteristic temperature and the coefficient of linear expansion, and the change of the latter with temperature. When the specific volume of a crystal changes, its internal energy changes [4].

Lindemann [5] found a relation between the melting temperature \(T_s\), the characteristic temperature \(\Theta\), the molecular volume \(v\), and the atomic weight of the crystal \(A\) in the form:

\[ \Theta = \frac{137}{v^3}\sqrt{\frac{T_s}{A}}. \]

A quantity proportional to the energy of the crystal lattice [6] and characterizing interatomic interactions is the molecular concentration

\[ \alpha=\frac{D}{M}10^3, \]

where \(D\) is the density of the crystal, and \(M\) is the molecular weight of the substance; \(\alpha\) is measured in \(\mathrm{cm}^{-3}\,\mathrm{mol}\).

The relation of the molecular concentration to certain electrical and mechanical properties of crystals has been shown [6]. Let us determine whether there is a relation between \(\alpha\) and the properties of solid solutions. In this case the molecular weight \(M=P_1M_1+P_2M_2\), where \(P_1\) and \(P_2\) are the molecular percentages of the alloy components, and \(M_1\) and \(M_2\) are the molecular weights of the substances entering into the composition of the solid solution.

A comparison of the electric strength \(E_{\mathrm{pr}}\) of some alkali-halide crystals containing impurities with the value of \(\alpha\) is shown in Table 1, from which it follows that \(E_{\mathrm{pr}}\) increases as the molecular concentration \(\alpha\) increases. The values of the electric strength are taken from works [7, 8, 9].

The change in the electric strength of solid solutions of KCl in KBr and KJ in KBr as a function of composition is shown in Fig. 1, from which it is seen that the KJ impurity lowers the electric strength of the KBr crystal more significantly than the KCl impurity. At the same time, it is necessary to note that \(\alpha\) of KCl alloys in KBr is higher than \(\alpha\) of KJ alloys in KBr, if the comparison is made at equal amounts of impurity. It follows from this that alloys with a larger value of \(\alpha\) possess greater electric strength.

Fig. 1

Table 2 compares the values of the tangent of the dielectric-loss angle [10, 11] of crystals at a frequency of \(10^6\) Hz and room temperature with the molecular concentration \(\alpha\). From Table 2 it is seen that the dielectric losses increase as \(\alpha\) decreases. The change in the tangent of the dielectric-loss angle of solid solutions of KCl in KBr

and KI in KBr as a function of composition is shown in Fig. 2, from which it follows that the dielectric losses of the KBr crystal are increased considerably more by a KI impurity than by a KCl impurity. We have already said that \(\alpha\) of KCl alloys in KBr is higher than \(\alpha\) of KI alloys in KBr at identical impurity percentages (KCl or KI). Thus, alloys with a larger value of \(\alpha\) have smaller dielectric losses.

Table 1

Comparison of the electric strength \(E_{\text{br}}\) of alkali-halide crystals containing impurities with molecular concentration

Comparison of the dielectric coefficient (at a frequency of \(10^6\) Hz) of solid solutions of MgO in ZrO\(_2\) and CaO in ZrO\(_2\) [10] with molecular concentration is shown in Table 3, from which it follows that in crystals with impurities the dielectric coefficient increases with increasing \(\alpha\).

Fig. 2

A comparison of the lattice constant \(a_0\) [8] of solid solutions with the value \(\alpha\) is given in Table 4, from which it is seen that \(a_0\) decreases with increasing \(\alpha\).

Table 2

Comparison of the dielectric-loss tangent of ionic crystals
at a frequency of \(10^6\) Hz and room temperature with the molecular concentration

Table 3

Comparison of the dielectric coefficient \(\varepsilon\) of solid solutions
with molecular concentration

Table 4

Comparison of the lattice constant of alloys
with molecular concentration

In some cases, for estimating the properties of alloys one may use the quantity \(\alpha_1 + \alpha_2\), where \(\alpha_1\) is the molecular concentration of the principal component and \(\alpha_2\) is the molecular concentration of the substance constituting the impurity. It is convenient to use the quantity \(\alpha_1 + \alpha_2\) when impurities are introduced successively into a given crystal in equal quanti-

cases; or when it is necessary to compare the course of curves indicating changes in the properties of crystals caused by different impurities.

In Table 5, \(\alpha_1+\alpha_2\) is compared with the quantity \(\Delta\rho/C\), characterizing the change in the electrical resistance of gold, silver, and copper caused by one atomic percent of impurity. In Table 5 the impurities are grouped according to the type of crystal lattice. In addition, cases are distinguished in which \(\alpha_1>\alpha_2\) and in which \(\alpha_1<\alpha_2\). The values of \(\Delta\rho/C\), expressed in

\[ \frac{\text{ohm}\cdot\text{cm}\cdot 10^{-6}}{\text{at}\%}, \]

are taken from [12, 13].

It is seen from Table 5 that, for all impurities except impurities with a simple cubic lattice, an increase in \(\Delta\rho/C\) is observed with increasing \(\alpha_1+\alpha_2\).

Table 6 shows a comparison of the coefficient of linear expansion in alloys of copper and gold with the quantity \(\alpha_1+\alpha_2\). The values of \(b\) are taken from [14, 15]. It is seen from Table 6 that, as \(\alpha_1+\alpha_2\) increases, the coefficient \(b\) decreases.

Table 5

Comparison of the change in electrical resistance upon introduction of one atomic percent of impurity \(\Delta\rho/C\) with the quantity \(\alpha_1+\alpha_2\)

Table 6

Comparison of the linear expansion coefficient \(b\) of copper and gold alloys with the quantity \(\alpha_1+\alpha_2\)

Table 7

Comparison of the Brinell hardness of copper and gold alloys with the quantity \(\alpha_1+\alpha_2\)

Table 7 shows a comparison of the Brinell hardness [16] of copper and gold alloys with the quantity \(\alpha_1+\alpha_2\). From Table 7 there follows an increase in Brinell hardness with increasing molecular concentration.

The change in the properties of crystals upon the introduction of impurities into them is explained by distortion of the crystal lattice. It is useful to compare the magnitude of the distortion with the molecular concentration of the alloys. From Table 8 it follows that the static distortions of the crystal lattice of solid solutions of nickel [17] decrease with increasing molecular concentration \(\alpha_1+\alpha_2\).

Table 8

Comparison of static distortions of the crystal lattice of nickel solid solutions with molecular concentration

Thus, the molecular concentration \(\alpha\) may be regarded as a quantity characterizing many electrical, thermal, mechanical, and other properties of alloys. With increasing molecular concentration, the electric strength of crystals increases, electrical losses decrease, the coefficient of linear expansion of crystals decreases, and the Brinell hardness increases.

References

1. Vorob'ev A. A. On the properties of ionic dielectrics. Izv. Tomsk Polytechnic Institute, 91, 173, 1956.

2. Bogoroditskii N. P. Dielectric losses in radioceramics. Reports of the Scientific-Technical Conference on Radioceramics. Leningrad Electrotechnical Institute named after V. I. Ulyanov (Lenin), 1955.

3. Rovinskii B. M. On the dependence of the mechanical properties of solids on atomic interaction in the lattice. Izv. AN SSSR OTN No. 9, 55, 1955.

4. Sirota N. N. On the physical properties of solid solutions and compounds in connection with the energy of interaction of the components. Izv. Sector of Physico-Chemical Analysis, 21, 90, 1952.

5. Lindemann F. A. On the calculation of the natural frequency of molecular vibrations. Phys. Z., 11, 609, 1910.

6. Savintsev P. A. On the question of the relationship between the electrical strength of crystals and their surface energy. Izvestiya TPI, 91, 199, 1956.

7. Trubitsyn A. M. Electrical strength of single crystals of solid solutions of alkali-halide salts with a substituted halide. Izv. Tomsk Polytechnic Institute, 91, 119, 1956.

8. Kislina A. N. Relationship of the electrical strength of single crystals of solid solutions of alkali-halide salts with certain physicochemical properties. Izv. Tomsk Polytechnic Institute, 91, 125, 1956.

9. Ivankina M. S. and Gorodetskii A. F. Electrical strength of solid solutions of a system. Izv. Tomsk Polytechnic Institute, 91, 159, 1956.

10. Avgustinik A. I. and Antselevich N. S. Electrical properties of solid solutions in the systems zirconium dioxide—magnesium oxide and zirconium dioxide—calcium oxide. ZhFKh, 27, No. 7, 973, 1953.

11. Volod'yanov K. A. and Galibina G. I. Dielectric losses in crystals at high frequency. Izv. Tomsk Polytechnic Institute, 91, 269, 1956.

12. Johanson C. H. and Linde J. O. Lattice structure and electrical conductivity of mixed crystals. Z. Metallkunde, 17, 213, 1925.

13. Collection of Physical Constants, edited by Ya. G. Dorfman and S. E. Frish. ONTI, 1937.

14. Technical Encyclopedia. Handbook of Physical, Chemical, and Technological Quantities, 2, 1929.

15. Encyclopedia of Metallophysics, edited by G. Mazing, 1, 1937. ONTI.

16. Kuznetsov V. D. Physics of the Solid State, Vol. 2, Tomsk, Krasnoe Znamya Publishing House, 1941.

17. Kurdyumov G. V. and Travina N. T. X-ray study of interatomic interaction in nickel-based solid solutions. DAN SSSR, 99, No. 1, 77, 1954.

ERRATA

Izv. TPI, vol. 95.