Dielectric Losses in Crystals at High Frequencies
Abstract Generated abstract
This paper reports high-frequency measurements of dielectric permittivity and dielectric-loss angle in ionic crystals, mixed alkali-halide crystals, and inorganic crystals containing polar molecules. Using an improved method for measuring very small loss angles, the authors relate losses in pure alkali-halide crystals to lattice energy, activation energy, temperature, and chemical structure, and interpret them as arising from motion of weakly bound ions. Mixed crystals show substantially larger losses, with composition-dependent maxima attributed to lattice distortions and enhanced ionic motion. For crystalline hydrates such as gypsum, copper sulfate, and talc, temperature and frequency maxima of the loss angle indicate relaxation processes associated with orientational polarization of polar groups.
Full Text
DIELECTRIC LOSSES IN CRYSTALS AT HIGH FREQUENCY
K. A. VODOP’YANOV and G. I. GALIBINA
Introduction
The study of the dielectric properties of solid dielectrics at high frequency is of great scientific and practical interest, especially for the rapidly developing electrical and radio-engineering industry.
In order consciously to understand the requirements imposed on electrical insulating materials, it is first of all necessary to study those physical phenomena that occur in dielectrics situated in an electric field.
To clarify the mechanism of dielectric losses, it is necessary to establish the connection between the electrical and physicochemical properties of dielectrics, i.e., to find the dependence of the principal parameters of a dielectric—the dielectric constant and the dielectric-loss angle—on their physicochemical characteristics (chemical composition, structure) and external conditions (temperature and frequency).
At the present time, theoretical and experimental investigations aimed at elucidating the mechanism of dielectric losses are in the process of being resolved. The difficulty of this problem is due to two circumstances: the poorly developed molecular-kinetic theory of the processes determining the electrical properties of solids, and the absence of refined methods for measuring the dielectric constant and the dielectric-loss angle, especially in the region of high and ultrahigh frequencies.
The present work gives experimental data characterizing dielectric losses in simple and complex ionic crystals, both not containing and containing polar molecules.
Until recently no reliable values had been obtained for the loss angles in dielectric crystals. Most investigators believed that in crystals, both those containing polar molecules and those not containing them, the dielectric losses are of an ohmic character. There existed the erroneous opinion that, because of the small value of the dielectric-loss angle in crystals, it is impossible to establish a connection between their dielectric properties and their physicochemical characteristics.
We have experimentally proved the unfoundedness of these assertions. Using the improved method developed in [1] for measuring small loss angles (less than \(10^{-4}\) radian with an accuracy of up to 30%), we are able, in the high-frequency region, to measure the dielectric constant and the dielectric-loss angle for a whole series of crystals.
We have also shown that there is a substantial difference in the character of dielectric losses in crystals that do not contain and that do contain polar molecules.
Dielectric Losses in Ionic Alkali-Halide Crystals
On the basis of the existing relation between dielectric permittivity, conductivity, and electric strength and their physical properties for alkali-halide crystals [2], we continued investigations to establish the relation between the dielectric-loss angle in certain alkali-halide crystals and their physicochemical characteristics.
Fig. 1. Relation between the crystal-lattice energy
and the dielectric-loss angle of crystals.
\[ f = 10^6\ \text{Hz} \]
As can be seen from Fig. 1, the dielectric-loss angle in crystals of alkali-halide salts decreases as the lattice energy increases. A similar dependence can also be represented for the activation energy, heat of formation, heat of sublimation, and melting temperature. The dielectric-loss angle increases with increasing crystal-lattice constant.
Fig. 2. Dependence of the dielectric-loss angle
of crystals on temperature.
\[ f = 10^6\ \text{Hz} \]
On the basis of the dependences obtained, one may conclude that the dielectric losses of alkali-halide crystals are determined by the chemical composition and the strength of the chemical compound; i.e., the dielectric-loss angle is a characteristic of the base substance and its structure, while impurities present in small amounts, which are found in
in real crystalline dielectrics apparently play a secondary role.
The temperature dependence of the dielectric-loss angle in crystals of different strength must be different. The smaller the activation energy and the lattice energy, the lower the temperatures at which the temperature behavior of the loss angle should be observed.
To verify these expectations, the temperature dependences of the loss angle at a frequency of \(10^6\) cps were measured for alkali-halide crystals NaCl, KCl, KBr, KJ, and CsBr, which have different crystal-lattice energies.
From Fig. 2 it is seen that, as the temperature is raised to \(175^\circ\)C, the loss angle changes little in all crystals. With a further increase in temperature, a noticeable increase in the loss angle begins. This temperature point is the lower, the smaller the energy of the crystal lattice, and conversely.
The mechanism of dielectric losses in ionic crystals appears to us as follows.
Under the influence of an external action (electrical, thermal, etc.), the number of weakly bound ions having different activation energies increases in the crystal.
A comparatively small number of ions possessing a higher activation energy, under the influence of the electric field in the crystal, will undergo translational motion, creating a through conduction current.
The greater part of the ions, having a lower activation energy, under the action of an alternating electric field will execute a reciprocating-translational displacement in a limited volume, causing dielectric losses of relaxation character.
The smaller the lattice energy, the greater will be the number of weakly bound ions responsible for dielectric losses.
Dielectric Losses in Mixed Crystals
Many investigators point out that the physical properties of mixed crystals differ from the properties of pure crystals even at a small concentration of the constituent component.
However, the structure of a mixed crystal has as yet been little studied.
In the theory of regular solutions it is assumed that, when a solid solution is formed, the type of unit cell of the initial components does not change; only the distance between the centers of the ions changes, owing to the fact that a lattice site is occupied by an ion with a different ionic radius.
Durham and Hodgkins [3] rejected the concepts of the theory of regular solutions (the preservation of the unit cell of the crystal in mixed crystals). They believe that the introduced ion is fixed at the normal site of the crystal lattice, but, owing to the fact that this ion is foreign and has a different radius, the proper ions are displaced by some distance from their positions in the undistorted lattice. This position corresponds to the minimum of the potential energy of the ions—the central impurity ion and the proper ions.
In either case it follows that the lattice constant in mixed crystals, with increasing concentration of the second component, changes monotonically up to the value of the lattice constant of the second component. The energy of the crystal lattice of mixed crystals changes analogously with change in composition.
Measurement of the crystal-lattice constant is carried out by X-ray methods. On X-ray photographs a completely sharp interference pattern is obtained; however, by X-ray methods we determine the average value of the lattice period of the solid solution, actually
the crystal lattice of a solid solution is distorted—the different elementary cells of the crystal have different dimensions.
Ions located at sites of distortion of the crystal lattice are the most weakly bound to their neighbors (the potential barriers are distorted). These ions can move within limited volumes, or can be torn away from their equilibrium positions and take part in the conduction process.
Fig. 3. Dependence of the loss angle on the composition of solid solutions at room temperature.
Fig. 4. Dependence of the loss angle on the composition of solid solutions at \(T = 20^\circ\).
If one takes into account that in mixed crystals consisting of components having different crystal-lattice constants there are distorted sites at which thermal dissociation of ions is facilitated, then, depending on external factors (temperature and frequency), one may expect a noticeable increase in the angle of dielectric losses.
We investigated the dielectric losses of a system of solid solutions of isomorphous crystals NaCl—NaBr, KCl—KBr, and KJ—KBr. The crystals were grown from the melt.
It is known that the melting temperature of solid solutions varies with the composition of the solid solution along a curve with a minimum.
The heat absorbed in the formation of a solid solution, as a function of the composition of the solid solution, varies along a curve with a maximum in the region of approximately equal concentration of the components [5, 6, 7, 8].
As is seen from Figs. 3, 4, 5, the angle of dielectric losses of mixed crystals, as a function of crystal composition, varies along a curve with a maximum at frequencies of \(10^7\) cps and lower. With decreasing frequency the maximum becomes more sharply expressed. A similar picture is observed with increasing temperature (Fig. 6).
The dependence of the loss angle on temperature for mixed crystals is presented in Figs. 7 and 8. It is seen that the onset of the increase in the loss angle with increasing temperature depends on the concentration of the second component.
At frequencies of \(10^7\) Hz and higher, a weaker dependence of the loss angle on the composition of the crystal is observed.
In absolute value, the maximum of the loss angle in the binary system KI—KBr is considerably higher than in the other systems studied (KCl—KBr and NaCl—NaBr). This was to be expected, since in the latter systems the minimum melting temperature is higher and the heat of formation is considerably lower than in the KI—KBr system.
Fig. 5. Dependence of the loss angle on the composition of solid solutions at \(T=20^\circ\).
Fig. 6. Dependence of the loss angle on the composition of solid solutions at different temperatures.
\[ f = 10^6\ \text{Hz} \]
If ions located at sites of distortion of the crystal lattice take part in conduction, increasing it in comparison with the conductivity of pure crystals (and the dependence of conductivity on composition for pressed specimens of mixed crystals is expressed by a curve with a maximum), then the presence of a maximum in the curve of the dependence of the loss angle on the composition of mixed crystals is understandable. With an increase in the concentration of the second component, the number of distortions in the crystal lattice increases, and the electrical conductivity increases. With such an explanation, the increase in the magnitude of the maximum with decreasing frequency is also understandable.
The frequency dependences of the loss angle of mixed crystals also speak in favor of such an explanation: a small dependence on frequency \((10^7—3\cdot10^6\ \text{Hz})\), and then an increase.
Thus, in mixed crystals we have a considerable increase in \(\operatorname{tg}\delta\) in comparison with pure alkali-halide crystals. \(\operatorname{tg}\delta\) depends noticeably on temperature and frequency.
Fig. 7. Dependence of the loss angle of mixed crystals on temperature. \(f = 10^6\) cycles.
1 — 100% KCl;
2 — 95 mol% KCl — 5 mol% KBr;
3 — 90 mol% KCl — 10 mol% KBr;
4 — 70 mol% KCl — 30 mol% KBr;
5 — 50 mol% KCl — 50 mol% KBr.
Fig. 8. Dependence of the loss angle of mixed crystals on temperature.
1 — 100% NaCl;
2 — 95% NaCl — 5% NaBr;
3 — 80% NaCl — 20% NaBr;
4 — 70% NaCl — 30% NaBr;
5 — 40% NaCl — 60% NaBr;
6 — 30% NaCl — 70% NaBr;
7 — 5% NaCl — 95% NaBr.
A general characteristic of ionic crystals is that the dielectric losses in them are ionic in nature; they are associated with the motion of ions and with the ionic structure of the dielectric.
Dielectric losses in inorganic crystals containing polar molecules
An essential factor determining the dielectric properties of polar dielectrics is thermal orientational polarization, caused by the orientation of polar molecules, or their radicals, in an electric field of the corresponding frequency and within a certain temperature interval.
Some of the most common solid dielectrics contain polar molecules in their composition. For the proper use of such dielectrics in electrical and radio-engineering installations, it is of interest to determine what their dielectric characteristics will be in an alternating electric field, especially at high frequency, as a function of temperature change. In addition, such characterization is very necessary for selecting the formulation and manufacturing technology of new electrical-insulating materials possessing high qualities.
We believe that, in order to characterize the dielectric polarization of ionic crystals with polar molecules, it is necessary to study the temperature–frequency dependence of the loss angle and of the dielectric permittivity. Many authors have investigated only the dielectric permittivity, which in some cases may vary only weakly as a function of frequency and temperature. A more sensitive characteristic of a dielectric is the dielectric loss angle, which, in the presence of relaxation processes, changes markedly as a function of external factors, provided only that conduction losses are not dominant. The latter, however, can easily be eliminated if the investigation is carried out at high frequency.
In our laboratory, systematic investigations were undertaken of the dielectric properties of crystalline polar dielectrics in high-frequency fields [9]. The temperature–frequency dependence of the dielectric loss angle and of the dielectric permittivity was studied in certain polar crystalline dielectrics containing, as polar molecules, water of crystallization: gypsum \((\mathrm{CaSO_4 \cdot 2H_2O})\), copper sulfate \((\mathrm{CuSO_4 \cdot 5H_2O})\), and talc of various kinds \((\mathrm{3MgO \cdot 4SiO_2 \cdot H_2O})\).
In posing this problem we had in mind the indication by G. I. Skanavi [10] that the orientational polarization of aqueous crystals is caused by the rotation of hydroxyl groups. These groups, possessing considerable polarity, are bound rather weakly in the molecule and, apparently, can rotate even when the water molecules are incorporated into the crystal lattice.
From the experimental material obtained [9, 11] it is evident that, in the high-frequency region, all the crystals investigated exhibit temperature and frequency maxima of \(\operatorname{tg}\delta\). Moreover, in the region of the maximum of \(\operatorname{tg}\delta\), a change in \(\varepsilon\) is observed.
The frequency–temperature dependences of \(\operatorname{tg}\delta\) for polar crystals are analogous to the dependences for polar liquid and amorphous substances. On this basis we believe that, in crystalline dielectrics containing water of crystallization, at certain temperatures and frequencies, orientation of polar molecules or their radicals is possible; i.e., a relaxation mechanism of dielectric losses is possible.
The presence of several temperature and frequency maxima corresponds to the structural formula of the crystalline hydrate, indicating the number
groups of OH dipoles, differently bound in the complex molecule. This circumstance is in agreement with the indications of D. I. Mendeleev concerning the release of water of crystallization in several stages.
We have shown that dielectric losses in polar crystals depend on the activation energy of polar molecules.
We calculated the activation energy from the formula
\[ U = k \frac{T_1 T_2}{T_1 - T_2} \left( \ln f_1 - \ln f_2 \right) \]
on the basis of experimental data on the shift of the frequency maxima of \(\operatorname{tg}\delta\) as a function of temperature.
It may be assumed that the number of maxima of \(\operatorname{tg}\delta = f(T,\omega)\) is determined by the structure of the crystal. The position of the frequency and temperature maxima of \(\operatorname{tg}\delta\) is governed by the magnitude of the activation energy of weakly bound dipoles. The magnitude of the maxima of \(\operatorname{tg}\delta\) is determined by the number of weakly bound dipoles participating in the process of dielectric relaxation.
We are inclined to believe that the regularities obtained by us are in qualitative agreement with Skanavi’s theory of dielectric losses, if in the latter one takes into account that polar molecules have different values of the relaxation time, whose absolute magnitude is determined by the activation energies of the molecules or their radicals.
It should be noted that, up to the present, there is no theoretical substantiation of the distribution function of relaxation times on the basis of the molecular characteristics of a dielectric. The investigations of foreign scientists in this field are essentially a mathematical treatment of experimental data.
CONCLUSION
The relationship we have found between the dielectric properties of a crystal and its physicochemical data should be made the basis both for the development of new dielectrics possessing high electrical properties and for elucidating the mechanism of dielectric losses in solid dielectrics.
LITERATURE
- K. A. Vodop’yanov and V. F. Ivlev. ZhTF, 8, 17, 1938.
- A. A. Vorob’ev and K. A. Vodop’yanov. DAN, USSR 94, No. 3, 1954.
- Durham and Hawkins. The J. of Chem. Phys. 19, 2, 1951.
- V. P. Radishchev. ZhOKh, vol. 5, issue 4, 1935.
- N. S. Kurnakov and S. F. Zhemchuzhnyi. Collected Selected Works of Kurnakov, vol. I.
- I. B. Vrzheshnevskii. Izv. S. P. (b) LI, vol. 15, issue 2, 1911.
- Grimm und Herzfeld. Z. f. Physik 16, 77, 1923.
- Fieneman. J. Chem. Phys. 18, 771, 1950.
- K. A. Vodop’yanov. ZhTF, 23, issue 1, 1954.
- G. I. Skanavi. Physics of Dielectrics, 1949.
Siberian Physico-Technical Institute.
Questions and Answers
A. A. Vorob’ev—are there data on measurements of the dielectric constant for mixed systems?
K. A. Vodop’yanov—there are data; the change of \(\varepsilon\) with composition is very slight, and the character of the changes is the same as for \(\operatorname{tg}\delta\).
A. A. Vorob’ev—was \(\operatorname{tg}\delta\) studied for aged specimens?
K. A. Vodop’yanov—measurements showed that for some mixed crystals \(\operatorname{tg}\delta\) decreases with aging.
S. S. Gutin—in what fields were the measurements carried out?
K. A. Vodop’yanov—the measurements were made in weak fields.
S. S. Gutin asks that the data obtained be compared with breakdown data.
K. A. Vodop’yanov says that, according to their data, a decrease of \(\operatorname{tg}\delta\) of the crystal is obtained with an increase in the lattice energy. Investigations carried out at TPI show an increase in the electric strength of the crystal with increasing lattice energy.
G. I. Skanavi—what was the situation with the influence of moisture and hygroscopicity?
K. A. Vodop’yanov—the specimens were heated in a vacuum furnace to \(200^\circ\), and then kept in a desiccator. During the measurements the influence of moisture is slight, as practice shows. The influence of moisture would above all change the character of the dependence of \(\operatorname{tg}\delta\) on \(t^\circ\); obviously there is no influence, since the forward and reverse runs of the curves coincided.
B. N. Matsonashvili—the report indicates that the losses are due to conductivity and to weakly bound ions. Are the losses relaxation losses, and do they give a maximum?
K. A. Vodop’yanov—a maximum is not manifested in these phenomena, but if the specimens are annealed, the maximum of \(\operatorname{tg}\delta\) is clearly manifested and shifts toward higher temperatures with increasing frequency.
B. N. Matsonashvili—was the influence of impurities and of the conditions of growing single crystals investigated?
K. A. Vodop’yanov—the influence of impurities was not investigated, but the impurities present in pure real crystals play a secondary role here.
G. I. Skanavi says that for gypsum the maximum of \(\operatorname{tg}\delta\) is high, while \(\varepsilon\) changes almost not at all. Are the measurements for \(\varepsilon\) reliable?
K. A. Vodop’yanov—the changes of \(\varepsilon\) in the region of the maximum have been established accurately.
R. M. Kessenikh—you said that impurities do not play an essential role; however, in the graph the influence of impurities is manifested.
K. A. Vodop’yanov—notes that he was speaking of the secondary role of impurities, which occur in very small quantities in pure crystals.
Errata
| Page | Line | Printed | Should read |
|---|---|---|---|
| 5 | 15 from top | Zinnerman | Zingerman |
| 9 | 2 from bottom | by breakdown of voltage | by breakdown voltage |
| 13 | 4 from top | $\varepsilon_m \simeq 4\,\dfrac{h}{2\pi}$ | $\varepsilon \simeq 4\,\dfrac{h}{2\pi}\,\omega$ |
| 24 | caption under the figure | fig. | fig. 9. |
| 50 | 14 from top | $\displaystyle -\frac{1}{M_L^{ak}}=\left\{\frac{1}{M}\right.$ | $\displaystyle -\frac{1}{M_L^{ak}}=\left\{\frac{1}{M_-}\right.$ |
| 53 | 10 from top | $\displaystyle \frac{d f^{1/2}}{d\varepsilon}$ | $\displaystyle \frac{d f^{1/2}}{d\varepsilon}$ |
| 57 | 5 from top | $\displaystyle \left(1-\sqrt{1-\frac{1}{1.3\,\varepsilon}}\right)e^{\frac{q}{E}(\varepsilon-I)^2}$ | $\displaystyle \left(1-\sqrt{1-\frac{1}{1.3\,\varepsilon}}\right)e^{\frac{q}{E}(\varepsilon-I)^2}$ |
| 64 | Equation (87) | $\displaystyle l_2(\varepsilon_2,\varepsilon_1)=l_1 e^{-\frac{1}{l_0}}$ | $\displaystyle l_2(\varepsilon_2,\varepsilon_1)=l_1 e^{-\frac{l_1}{l_0}}$ |
| 65 | 18 from bottom | $I \div 2\varepsilon_p$ | $I+2\varepsilon_p$ |
| 85 | 7 from bottom | $\displaystyle 10^{14}\cdot \frac{\pi\,0.52\cdot10^{-2}}{4}\cdot 10-$ | $\displaystyle 10^{14}\cdot \frac{\pi\,0.52\cdot10^{-2}}{4}\,10^{-2}$ |
| 111 | 2 from top | producibility | conductivity |
| 111 | 8 and 10 from top | activation | activation |
| 120 | 5 from top | strength of chemical | strength from chemical |
| 135 | ref. 6 | p. … | p. 113 |
| 147 | 11 from bottom | $\varepsilon=1000$ | $\varepsilon\simeq1000$ |
| 148 | caption to Fig. 4 | $u_{\text{disch.}}\,u_{\text{break.}}$ | $u_{\text{disch.}}\;|\;u_{\text{break.}}$ |
| 204 | Fig. 1 | KbBr | RbBr |
| 253 | caption to Fig. 6 | $2-\ f=4\cdot10^3$ cycles; $3-\ f=10^2$ cycles | $2-\ f=4\cdot10^2$ cycles; $3-\ f=10^3$ cycles |
| 273 | 7 from top | KCJ | KCl |
| 280 | caption to Fig. 4 | $210^5$ | $2\cdot10^5$ |
| 294 | 1 from bottom | from 10 | from $10^{-4}$ at $f=2\cdot10^7$ to $6\cdot10$ |
| 282 | table | table 2 | |
| 324 | 3 from top | p. 269 | p. 251 |
| 336 | 16 from bottom | [3,4] | [3] |
| 337 | 10 from top | [5] | [4] |
| 337 | 4 from bottom | [6] | [5] |
| 337 | 1 from bottom | [7] | — |
| 338 | 2 from top | [8] | [6] |
| 338 | 4 and 5 from top | [9,10] | [7,8] |
| 339 | 8 from bottom | [11,12] | [9,10] |
| 340 | 14 from top | [13] | [11] |
| 347 | 23 from bottom | [14] | [12] |
| 347 | 3,4 from bottom | [15] | — |
| 348 | 17 from bottom | [16] | [13] |
| 359 | 4 from top | $\psi_\varphi[M]\,x=\mathrm{Const}$ | $\psi_\varphi[M]\,x=\mathrm{Const}$ |
| 451 | 1 from bottom | mechanical | technical |