Electrical Properties of Fused Quartz

B. I. Vorozhtsov

(Presented by Corresponding Member of the Academy of Sciences of the USSR V. D. Kuznetsov)

Modern electrical and radio engineering imposes very stringent requirements on electrical insulating materials with respect to electrical, thermal, and mechanical properties. Of the enormous variety of materials currently in use, only a few fully meet the totality of these requirements. This applies especially to such severe operating conditions as high temperature, elevated voltage, high frequency, a chemically active medium, etc.

Among the dielectrics capable of withstanding the entire set of these conditions, fused quartz should be placed first. Insulators for electric gas-cleaning installations that operate in a chemically active medium at a temperature of 500–600°C and an operating voltage of 70 kV; high-voltage arc-resistant insulators in arresters, switches, etc.; cathode insulators of metallic mercury rectifiers; components of powerful generator tubes; various insulation for high-frequency apparatus, etc.—this is far from a complete list of products made of fused quartz, showing that it must possess remarkable properties.

However, the existing monographs [1, 2, 3] and journal articles [4, 5, 6] describing the physicochemical properties of fused quartz reflect its electrical characteristics to an extremely insufficient extent. Meanwhile, the broad and ever-increasing use of fused quartz in the electrical and radio industries makes this necessary.

The present article gives a review of work on the electrical properties of fused quartz¹. Along with a description of the properties of fused quartz, the article gives data relating to crystalline quartz. This makes it possible to understand more deeply the phenomena occurring in fused quartz when an electric current passes through it.

1. Electrical Conductivity

Fused quartz at room temperature has a very low volume electrical conductivity: $\sigma = 10^{-18}\ \Omega^{-1}\text{cm}^{-1}$.

Figure 1 shows the dependence of the logarithm of the volume electrical conductivity of fused and crystalline quartz on the reciprocal tem-

literature according to the data of Sosman [3, p. 145], N. P. Bogoroditskii and V. N. Malyshev [7].

It is clear from the figure that fused quartz (curve 1), in its insulating properties, surpasses crystalline quartz (curves 2 and 3), which has a reputation as an excellent insulator.

Fig. 1. Dependence of the electrical conductivity of fused and crystalline quartz on temperature.

1. Fused quartz.
2. Crystalline quartz, electric field directed perpendicular to the optical axis.
3. Crystalline quartz, electric field directed along the optical axis.
   \(o\)—Sosman’s data, \(x\)—Bogoroditskii and Malyshev’s data.

In addition, the figure reveals the dependence of the electrical conductivity of crystalline quartz on the crystallographic direction. The electrical conductivity along the optical axis (curve 3) is hundreds of times greater than that in the transverse direction (curve 2). This latter fact has long attracted the attention of investigators [8, 9, 10], who believe that the increased conductivity of crystalline quartz along the optical axis is due to peculiar channels formed by silicon–oxygen tetrahedra arranged in a spiral.

The difference in the magnitude of the conductivity of crystalline and fused quartz is probably due to the fact that, upon melting crystalline quartz, some of the impurity ions, fixed mainly in its “channels” by weak polarization forces, become incorporated into breaks in the silicon–oxygen framework, where they are bound by strong valence bonds. This leads to an increase in the activation energy of the current carriers, which is reflected in a decrease in the magnitude of the electrical conductivity and an increase in the slope of the curve

\[ \ln \sigma = f\left(\frac{1}{T}\right) \]

(Fig. 1).

The break in the curve

\[ \ln \sigma = f\left(\frac{1}{T}\right) \]

for fused quartz (curve 1) is associated with the polymorphic transformations of quartz. Indeed, it follows from a number of studies [2], [9], [11], [12] that the electrical conductivity of crystalline quartz undergoes a sharp change at the transformation point of the \(\alpha \leftrightarrows \beta\) modifications of quartz. Thus, according to Rakhmi [9], in a specimen cut perpendicular to the optical axis of the crystal (Fig. 2a), at the transformation point (\(T = 573^\circ\)C) there occurs either a sharp decrease in conductivity, followed by a slight subsequent increase with temperature, or a halt. In a specimen cut parallel to the optical axis (Fig. 2b), in the vicinity of the transformation point something like a threshold is observed; thereafter the conductivity increases in accordance with the preceding law. Fused quartz has no threshold (Fig. 2c), but in the vicin-

...near the transformation point decreases the slope of the curve \(\ln \sigma = f\!\left(\frac{1}{T}\right)\).

This is also shown in Fig. 1¹. However, recent measurements by V. P. Pryanishnikov and N. A. Konovalova [2] show that for fused quartz as well, on the curve \(\ln \sigma = f\!\left(\frac{1}{T}\right)\), in the vicinity of 573°C, sometimes instead of a simple break there is observed a sharp decrease in conductivity, similar to the decrease in the electrical conductivity of crystalline quartz obtained by Rahimi (Fig. 3 and Fig. 2a).

Fig. 2. Temperature dependence of the electrical conductivity of quartz (after Rahimi [9]).

a) Plate cut perpendicular to the optical axis.

b) Plate cut parallel to the optical axis.

c) Plate of fused quartz.

Interpreting the change in electrical conductivity at a temperature of 573°C as the transformation of the \(\alpha \rightleftarrows \beta\) modifications of quartz, one may expect the same effects at the transformation points of other modifications of silica. However

results indicating a change in the character of the temperature dependence of the electrical conductivity of fused quartz in the region of the polymorphic transformations of tridymite (117, 163°C) and cristobalite (200–275°C). Probably in this case the stability (according to R. L. Müller [13]) of the Si—O bond is manifested, excluding the possibility of structural transformations.

Fig. 3. Dependence of the electrical conductivity of fused quartz on temperature near the point of polymorphic transformation (according to Konovalov and Pryanishnikov [2]).
1. For transparent fused quartz.
2. For opaque fused quartz.

Surface Electrical Conductivity of Fused Quartz

The experiments of Curtis [1] showed that the surface conductivity of a fused-quartz specimen preliminarily cleaned with a chromic mixture is small (\(10^{16}\ \text{ohm}^{-1}\text{cm}^{-1}\)) up to 100% relative humidity of the air. At the same time, the surface electrical conductivity of an uncleaned specimen increases with changes in the humidity of the surrounding air, increasing by several orders of magnitude (from \(10^{-16}\ \text{ohm}^{-1}\text{cm}^{-1}\) to \(10^{-8}\ \text{ohm}^{-1}\text{cm}^{-1}\)) when the relative humidity of the air is raised to 90%. Later experiments by N. Chirkov [14] testify to the same thing: the surface electrical conductivity of fused quartz is due to the presence on the surface of the specimen of ion-forming substances and solvent in the form of an adsorbed polymolecular layer.

The presence of capillaries, as Gnanan [15] showed, only slightly lowers the electrical conductivity of fused quartz, which indicates the small amount of impurity contained in the capillaries.

Mechanism of Electrical Conductivity

The question of the mechanism of the electrical conductivity of quartz has long interested researchers. The works of Russian scientists A. F. Ioffe [16] and A. A. Shaposhnikov [17] led to the conclusion that the electrical conductivity of quartz is caused mainly by impurity ions. The possibility of ion motion in quartz was demonstrated by the experiments of Warburg and later of Vogel and Gibson [18], who passed sodium and lithium ions through quartz crystals in quantities fully corresponding to Faraday’s law.

The strong dependence of the electrical conductivity of quartz on temperature, even at low temperatures, also indicates its ionic character.

Let us turn again to Fig. 2. It is clear from the figure that in the temperature range 200–300°C the temperature dependence of the electrical conductivity undergoes a break. Rakhim [9] explains this break by the fact that below the break point the conductivity is mixed, while above it is ionic. P. E. Sarzhevskii [19], studying the nature of the additional conductivity of quartz in strong fields, also obtained a similar break in the curve \(\ln \sigma = f\left(\frac{1}{T}\right)\).

in the same temperature interval. Studying galvanomagnetic effects, the detection of which is direct evidence of electronic (hole) conductivity of the dielectric, he found that these effects are observed only at temperatures below 300°C. One may also list a number of other works likewise indicating that up to 200–300°C the conductivity of quartz is mixed, and above this it is ionic (with cations moving [16], [10]), which completely obscures the electronic component. It should be noted here that R. L. Müller, in his theory of the electrical conductivity of solid iono-atomovalent substances [13], considers the break in the curve

\[ \ln \sigma = f\left(\frac{1}{T}\right) \]

at 200–300°C to be caused not by a change in the current carriers, but by the result of a decrease in activation energy due to a change in entropy during the transition from a stable state to a labile one (the critical temperature for Si–O lies in the range 150–300°C). Thus, according to R. L. Müller, the conductivity mechanism of quartz should be the same (mixed electrical conductivity) over the entire temperature interval; only at high temperature does the influence of entropy begin, which leads to a change in the slope of the curve

\[ \ln \sigma = f\left(\frac{1}{T}\right). \]

2. Dielectric losses and dielectric permittivity

Initial investigations, for example the work of N. P. Bogoroditskii and V. N. Malyshev [7], led to the conclusion that dielectric losses in fused quartz are purely ohmic. The authors based their conclusion on the following facts: dielectric losses measured at a technical frequency (the measurements were carried out from 20 to 400°C), and losses calculated from the conductivity, differ by only a factor of two. At high frequencies, owing to the low sensitivity of the apparatus, the authors were unable to measure \(\operatorname{tg}\delta\). Considering that the discrepancy between the calculated and measured values of the dielectric-loss angle is due to experimental inaccuracy, and noting the negligible value of \(\operatorname{tg}\delta\) at high frequencies, the authors conclude that the dielectric losses in fused quartz are of ohmic nature.

Subsequent experiments, carried out by many authors, showed that at low temperatures (from 0 to 400°C) and in the high-frequency range the dielectric-loss angle of fused quartz is very small and, within the limits of measurement error, does not depend on temperature or frequency [20], [21], [22], [23], [30], (Table 1)\(^1\).

At higher temperatures (from 400 to 1200°C), in the same frequency range (5–50 MHz), the dielectric-loss angle of fused quartz, as our investigations [28], [31] showed, begins to increase sharply, exhibiting regularities characteristic of the ohmic type of loss: the values of the dielectric-loss angle are inversely proportional to frequency and depend exponentially on temperature. In addition, the ohmic character of the high-temperature dielectric losses is indicated by the agreement of the measured values of the dielectric-loss angle with those calculated from the conductivity, and by the completely identical heating of the specimen when either a direct voltage or a high-frequency voltage equal to it is applied (Table 2).

\(^1\) In general, as N. P. Bogoroditskii and I. L. Fridberg [29] point out, there should be no absolute independence of the dielectric-loss angle of solid dielectrics from temperature and frequency.

Table 1

Values of the dielectric-loss angle of fused quartz at room temperature according to data of various authors

Table 2

Heating of a fused-quartz specimen upon application to it of a constant and an alternating voltage (500 volts for 30 sec) as a function of temperature

The study of dielectric losses in glasses at very low temperatures [32, 33] revealed the so-called deformation type of dielectric losses, which is due to the rocking motion of whole parts of the structural lattice, as a consequence of the presence in it of various kinds of defects. As V. A. Ioffe [33] showed, this loss mechanism leads to the appearance, in various silicate glasses, of a maximum of the dielectric-loss angle which, at a frequency of $10^6$ cps, lies in the region of 40–60°K. With increasing frequency the maximum shifts toward higher temperatures, which indicates the relaxation character of the dielectric losses. The low value of the activation energy (0.5–1 kcal/mole) explains the absence of a maximum in the region of higher temperatures.

Measurement of the dielectric-loss angle in crystalline quartz also leads to the detection of a low-temperature maximum of $\operatorname{tg}\delta$, or rather of several maxima [34], caused by different types of structural defects. In the same work there is a reference to the fact that fused quartz also possesses these effects: in it the presence of three low-temperature maxima was found, associated with such defects as the presence of iron ions, oxygen deficiency, etc.

Thus, in the temperature dependence of the dielectric-loss angle of fused quartz, three regions are clearly expressed: low-temperature maxima, a middle part that depends only weakly on temperature, and a sharp increase at high temperatures.

As regards the mechanism responsible for the weak dependence of the dielectric-loss angle at intermediate temperatures, no definite opinion has yet been formed. This effect may perhaps be due simply to the superposition of gradually decreasing deformation losses and increasing ohmic losses.

At increased electric-field strengths, the dielectric losses in fused quartz begin to depend on it. In this case, the presence of gas inclusions in the specimen is obligatory. This dependence is shown in Fig. 4 [28]. The character of the temperature

Fig. 4. Dependence of the dielectric-loss angle of fused quartz containing gas inclusions on voltage (specimen thickness 3 mm).

dependence of the dielectric-loss angle also changes at a certain temperature determined by the number and size of the gas inclusions, by the frequency, and by the magnitude of the electric-field strength: the dielectric-loss angle of fused quartz begins to increase more sharply than in the case of conduction losses. These additional losses occur because of ionization of the gas inclusions and may thus serve as the cause of rapid heating of quartz insulators in high-frequency fields of increased intensity. For example, according to our observations, specimens containing gas inclusions could be heated to white heat within a few seconds (at \(E = 3\ \text{kV/cm}\)).

According to data from a number of authors [1], [9], [27], [35], the dielectric constant of fused quartz is \(3.5 \div 4.1\). The difference in values apparently lies in the difference between specimens. Especially large fluctuations in the value of the dielectric constant are observed for opaque quartz. As P. P. Kobeko, D. V. Korovnik, and others [27] indicate, these fluctuations are caused by absorption of moisture, which increases the value of \(\varepsilon\), and by the presence of air bubbles, which lowers it.

Of interest is the dependence of the dielectric constant of fused quartz on frequency and temperature. Eger’s measurements [1, p. 109] show that from 250 to \(3 \cdot 10^{7}\) Hz, \(\varepsilon\) is practically independent of frequency. The dielectric constant of fused quartz also depends only very weakly on temperature. Thus, P. P. Kobeko [27] and N. P. Bogoroditskii [36] found that up to \(400^\circ\text{C}\) \(\varepsilon\) of fused quartz varies-

changes only in the third decimal place. Upon reaching higher temperatures, the dielectric constant begins to increase. However, this increase is very small if compared with the increase of $\varepsilon$ for crystalline quartz (the plate is cut perpendicular to the optical axis) or glass. According to Rahimi [9] and M. S. Chmutin [37], the dielectric constant of crystalline quartz, beginning at $350^\circ$C, increases sharply, reaching values of 20–30 at $600^\circ$C (Figs. 5 and 6). This difference is evidently due to the greater binding energy of impurity ions in fused quartz, which leads to a weaker influence of the orienting action of the electric field on the thermal motion of the ions.

Fig. 5. Dependence of $\varepsilon$ of fused quartz on temperature (after Rahimi [9])
1—before electropurification, 2—after electropurification.

Fig. 6. Dependence of $\varepsilon$ of crystalline quartz on temperature
1—before electropurification, 2—after electropurification.

3. Influence of the History of a Fused-Quartz Specimen on Its Electrical Properties

In this section we shall consider only the influence on the electrical properties of fused quartz of preliminary thermal treatment and electropurification.

The fact of improvement in the electrical properties of a dielectric, in particular the decrease in electrical conductivity after a direct electric current has been passed through a specimen heated to a high temperature, has long been known [38], [16], [39], [9].

A thorough study of this phenomenon enabled A. F. Ioffe, as early as 1915 [16], to draw the following conclusions: 1) an irreversible decrease in conductivity after passing current through a specimen (“electrical purification”) should be observed at high temperatures, when the number of ions participating in the passage of current becomes comparable with the total amount of substance capable of dissociating; 2) at lower temperatures the decrease in electrical conductivity when current is passed through the specimen increasingly becomes the result of the growth of the reverse e.m.f. of polarization.

Rakhimi [9] confirmed this conclusion, showing (Fig. 2) that after electrical purification at 700°C the electrical conductivity of crystalline and fused quartz decreased by approximately one order of magnitude.

He also showed that electrical purification affects the value of the dielectric constant, which, as can be seen from Figs. 5 and 6, also decreases.

We have shown [28] that the dielectric loss angle of fused quartz, measured at elevated temperatures, does not remain inert with respect to electrical purification (Fig. 7)—it also decreases.

A number of investigators have found that preliminary heat treatment of quartz also improves its electrical characteristics. Curie [8], who investigated the dependence of the electrical conductivity of quartz on preliminary heating, showed that with an increase in the temperature and time of heating (from 500 to 1500°C) the conductivity becomes smaller and smaller. Rakhimi likewise notes a decrease in the electrical conductivity of quartz after holding it with electrodes at 700–800°C.

Fig. 7. Temperature dependence of the dielectric loss angle of fused quartz subjected to various preliminary treatments. ●—before treatment; ○—after electrical purification; ×—after subsequent heating at 500°C for 10 hours; ◌—after additional heat treatment at 1200°C for 3 hours.

Our investigations show that the high-temperature dielectric losses in fused quartz also exhibit a similar decrease [28]. It should be noted, however, that according to our investigations the effect of heat treatment on the electrical characteristics of fused quartz begins to appear only from 1000–1100°C, and the process that decreases the dielectric losses is almost completely finished within 2–3 hours.

As can be seen, our data do not agree with the results of previous investigators and require a different approach to explaining this effect. If Curie explained the fact of the decrease in electrical conductivity after heating by dehydration of the specimens, the extent of which is greater the higher the temperature and the longer the duration of heat treatment, then our investigations contradict this for the following reasons: 1) the magni

the magnitude of the dielectric-loss angle reduced as a result of heat treatment does not depend on the temperature or duration of the heat treatment (this is true for \(T > 1000^\circ\mathrm{C}\) and \(t > 3\) hours); 2) repeated heat treatment does not cause any further change in the dielectric-loss angle; and 3) heat treatment in vacuum, which would facilitate degassing, gives the same decrease in the dielectric-loss angle as at atmospheric pressure.

We assumed that the effect of heat treatment is explained by the fixation of impurity ions through complex formation, and also by their thermionic emission from the specimen.

A study of the combined influence of heat treatment and electrical purification shows that heat treatment and electrical purification apparently act on the same current carriers, since, first, electrical purification after heat treatment does not noticeably change the magnitude of the dielectric-loss angle, and, second, heat treatment after electrical purification additionally lowers the magnitude of the dielectric-loss angle to the level determined by heat treatment alone.

Conclusion

Having considered the electrical characteristics of fused quartz and their variation with temperature, frequency, and electric-field strength, one may summarize as follows: fused quartz is one of the few electrical insulating materials possessing very high electrical properties \((\sigma = 10^{-18}\ \Omega^{-1}\mathrm{cm}^{-1},\ \tg \delta = 10^{-4})\), which are retained over a wide temperature range. At very high temperatures \((\sim 1000^\circ\mathrm{C})\) the electrical properties of fused quartz still remain quite good \((\sigma = 10^{-6}\ \Omega^{-1}\mathrm{cm}^{-1},\ \tg \delta = 10^{-2}\) at \(f = 25\ \mathrm{MHz})\).

It is undesirable for fused quartz to contain even a small number of gas bubbles, which, at elevated high-frequency electric-field strengths, cause the appearance of ionization losses that considerably increase the heating of the article.

References

1. Bagolev S. P. Quartz Glass. ONTI, 1934.

2. Pryanishnikov V. P. Quartz Glass. Moscow, 1956.

3. Shishakov N. A. Problems of the Structure of Silicate Glasses. Moscow, 1954.

4. Gareev B. M. Fused quartz, Elektrichestvo No. 6, 47, 1934.

5. Pulpan P. Fused quartz, Českosl. sklář a keramik, 3, No. 3, 1953.

6. Wright F. E. Fused quartz—a versatile industrial material, Mater. and Methods, 37, No. 2, 98, 1953.

7. Bogoroditskii N. and Malyshev V. Dielectric losses in crystals, ZhTF, 4, 7, 1306–1317, 1934.

8. Curie J. Study of the electrical conductivity of crystalline solids, Ann. d. chimie et d. phys., 17, 203, 1889.

9. Rahini N. G. Addition to the study of the electrical conductivity and dielectric constant of quartz and certain other crystals, Journ. Phys. Rad. No. 7, 291, 1938.

10. Chentsova L. G. and Vedeneeva N. E. The state of impurities introduced into the lattice of fused quartz, DAN SSSR, vol. 68, No. 2, 273, 1949.

11. Shimizu S. Preliminary report on the anomalous dependence of the electrical conductivity of quartz on temperature, Philos. Mag., 13, No. 87, 907, 1932.

12. Saegusa N. and Matsimoto T. On anomalous changes with temperature in the electrical conductivity of certain silicate minerals and of quartz cut in a special manner, Sci. Rep. Tôhoku Univ., 24, 538, 1935.

13. Müller R. L. Electrical conductivity of solid ion-atomic-valence substances, ZhTF, 25, issue 2, 236–245, 246–255, 1955.

1. Chirkov N. On the question of the surface electrical conductivity of solid dielectrics. ZhFKh, 21, 11, 1613, 1947.

2. Gnapau W. On the electrical conductivity of fused quartz. Zs. Physik. Bd. 66, H. 3–4, 436, 1930.

3. Ioffe A. F. Elastic and electrical properties of quartz. Izv. Petrogr. Polytech. Inst., Dept. of Technical and Materials Sciences, 24, 1, 27, 1915.

4. Shaposhnikov A. A. Effect of temperature on the electrical conductivity of crystalline quartz. ZhRFKhO, Physical Section, 46, 7, 265, 1914.

5. Vogel R. C., Gibson G. Migration of sodium ions through a quartz plate in an electric field. Journ. Chem. Phys. 18, No. 4, 490, 1950.

6. Sarzhevskii P. E. Dependence of the electrical conductivity of quartz on the electric field, temperature, and magnetic field. DAN SSSR, 82, No. 4, 571, 1952.

7. Khar’kov A. A. and Vodop’yanov K. A. Silicate glasses in ultrahigh-frequency fields. ZhTF, 8, 10, 910, 1938.

8. Aleksandrov B. I., Kobeko P. P. and Kuvshinskii E. V. Study of the amorphous state. ZhTF, 6, 1311, 1936.

9. Vodop’yanov K. A. On the question of dielectric losses in crystals at high frequency. Tr. SFTI, issue 24, 144, 1947.

10. Vodop’yanov K. A. and Krivtsov K. A. Investigation of the angle of dielectric losses in simple silicate and boron glasses as a function of the concentration of impurity ions and of the frequency of the electric field. Tr. SFTI, issue 24, 150, 1947.

11. Sapel’kov L. I. New electrical insulating materials. Elektrichestvo, No. 7, 3739, 1935.

12. Works C. N., Dakin T. N., Boggs F. Use of a resonator for measuring dielectric properties at ultrahigh frequencies. PIRE 33, No. 4, 245, 1945.

13. Race H. H., Leonard S. C. Measurement of dielectric losses in solids by a calorimetric method. Trans. Amer. Inst. El. Eng. vol. 55, p. 1347, 1936.

14. Kobeko P. P., Korovkin L. V., Kuvshinskii E. V. and Chelnokov I. Yu. Electrical properties of quartz manufactured by the Lomonosov Plant. IZST, No. 8–9, 51, 1936.

15. Vorozhtsov B. I. Certain electrical properties of fused quartz at high frequencies, elevated voltages, and high temperatures. Izv. Tomsk Polytech. Inst., vol. 91, 363, 1956.

16. Bogoroditskii N. P. and Fridberg I. D. On the nature of the temperature dependence of dielectric losses upon polarization of ionic compounds. ZhTF, 26, No. 9, 1884, 1956.

17. Khar’kov A. A. Dielectric losses in crystalline and fused quartz at high frequencies. SFTI, 1947.

18. Vorozhtsov B. I. Method for measuring dielectric losses at high temperatures and high frequencies. Tr. SFTI, issue 35, 50, 1956.

19. Dielectric losses in glass. Vestnik informatsii, No. 22, 13, 1952.

20. Ioffe V. A. Dielectric losses in silicate glasses. ZhTF, 24, 4, 611–621, 1954.

21. Volger J., Stevels J., Amerongen C. Dielectric losses of various quartz single crystals at very low temperatures. Philips Res. Rep. vol. 10, p. 260, 1955.

22. Jatkar S. K. K. and Yengar B. R. Y. Dielectric constant of quartz. Indian Journ. Phys. and Proc. Ind. Ass. for Cult. Science 4, vol. 23 32, 145, 1949.

23. Bogoroditskii N. P. and Malyshev V. N. Dielectric losses in glasses. ZhTF, 5, issue 4, 639, 1935.

24. Chmutin M. S. Dependence of the dielectric permittivity of a crystal on temperature. ZhTF, 26, issue 5, 640, 1954.

25. Shaposhnikov A. A. Electrical conductivity of crystalline quartz and deviation from Ohm’s law. ZhRFKhO, Physical Section, 42, issue 8, 1910.

26. Gokhberg B. M. Electrical conductivity of dielectrics. GTTI, 1933.

1. As our investigations have shown, in the dependence \(\ln \operatorname{tg}\delta = f\!\left(\frac{1}{T}\right)\), at 573°C a similar break in the curve is also observed. ↩↩