Pulsed Breakdown of Solid Dielectrics

A. A. VOROB'EV, G. A. VOROB'EV

Introduction

In designing insulation, it is necessary to know the electric strength of insulating materials. In addition, in physical studies of dielectrics it is often important to know the dependence of their electric strength on various factors. In this connection, especially in the second case, pulsed tests of dielectrics for breakdown have advantages over breakdown tests at direct and alternating voltages.

When a solid dielectric is subjected to a voltage pulse of sufficiently short duration, it is possible to avoid the influence of space charges and heat release on the magnitude of the electric strength and to obtain its true value.

Hippel and Alger [1] determined the temperature dependence of the electric strength of potassium bromide single crystals for different durations of voltage application. The temperature dependence they obtained at direct voltage and on pulses with a duration of not less than \(10^{-3}\) sec has a maximum, which the authors explain by the influence of space charges. At the shortest exposure, of the order of \(10^{-6}\) sec, a monotonic increase of electric strength with increasing temperature was obtained. The authors consider that in this case space charges had no time to form, and the true temperature dependence was obtained.

A. M. Trubitsyn [2], taking into account the possibility of the formation of space charges, the influence of heat release, and also the influence of discharge delay, came to the conclusion that, in order to obtain the true value of the electric strength of solid dielectrics with a thickness on the order of tenths of a millimeter, their breakdown must be produced on the front of a pulse with a duration up to breakdown on the order of \(10^{-6}\) sec. A. M. Trubitsyn emphasized that, in order to avoid the influence of cumulative effects (heat release, space charges, incomplete breakdowns), breakdown should be produced on the front of a single pulse.

In individual cases it is necessary to determine the electric strength of dielectrics and semiconductors possessing appreciable electrical conductivity. In this case the tests can be carried out only at pulsed voltage.

Only at pulsed voltage can one obtain characteristics important for the theory of breakdown and for lightning-protection practice, such as the discharge delay time and the discharge propagation velocity.

1. Dependence of the electric strength of solid dielectrics on the duration of action

Crystals of alkali-halide salts

Hippel and Alger [1] investigated the dependence of the electric strength of potassium bromide single crystals on exposure. Figure 1 presents the dependence obtained, which the authors explain by the influence on the magnitude of the electric strength of space charges: electronic and ionic. Each space charge separately, in the authors’ opinion, creates a nonuniformity of the field in the dielectric and reduces its electric strength. The simultaneous presence in the dielectric of both types of charges leads to compensation of their influence, and under these conditions the true value of the electric strength is obtained at an exposure of more than \(10^{-4}\) sec (Fig. 1). The increase in electric strength at short times is explained by the lag in the formation of the electronic space charge.

Fig. 1. Dependence of the electric strength of KBr crystals on the rate of voltage application (after Hippel and Alger).

G. A. Vorob’ev [3] obtained time dependences of the electric strength of single crystals of rock salt, and of potassium chloride, bromide, and iodide, presented in Fig. 2. The thickness of the specimens was \(0.15\) mm. Over the entire range of exposures there is observed a proportionality of the electric strength to the energy of the crystal lattice, which had previously been established by A. A. Vorob’ev [4]. A certain increase in the electric strength of the indicated single crystals at an exposure of \(10^{-5}\) sec and longer is explained by the author as the influence of space charge; the increase in electric strength at an exposure of \(1 \cdot 10^{-7}\) sec and less is explained by the lag of the discharge. The author believes that at exposures of \(4 \cdot 10^{-7}\)–\(6 \cdot 10^{-6}\) sec, when the lag of the discharge no longer has an effect and there is not yet any space charge, for specimens of alkali-halide single crystals with thickness of the order of tenths of a millimeter the true value of the electric strength is obtained. This is in agreement with the assumption of A. M. Trubitsyn [2].

Fig. 2. Dependence of the electric strength of single crystals of rock salt, and of potassium chloride, bromide, and iodide, on the duration of voltage action (after G. A. Vorob’ev). Breakdown probability \(=90\%\).

Inuishi and Suita [5] carried out breakdown of single crystals of potassium chloride and sulfur in weakly nonuniform¹ and strongly nonuniform fields, with pulses close to rectangular. The authors took as the discharge delay time the time that elapsed from the moment when the voltage reached its maximum to the moment of voltage drop as a result of breakdown. This time decreased with increasing pulse amplitude.

K. K. Sonchik [6] obtained an analogous dependence for single crystals of rock salt, potassium chloride, bromide, and iodide during breakdown in uniform and nonuniform fields with pulses close to rectangular.

Kawamura, Okura, and Kikushi [7] carried out breakdown of several solid dielectrics with a series of rectangular pulses, each of duration \(3\cdot 10^{-3}\) sec. Samples of potassium chloride single crystals \(10^{-3}\) cm thick were punctured by the very first pulse; therefore the authors consider that the delay time in the indicated samples is no more than \(3\cdot 10^{-8}\) sec.

Plesner [8] investigated the dependence of the electric strength of polycrystalline films of potassium bromide, sodium fluoride, calcium fluoride, quartz, and polystyrene on thickness and on the duration of voltage application. For films of potassium bromide and of sodium and calcium fluoride with thickness \(10^{-2}\div 10^{-5}\) cm, a monotonic increase of electric strength was obtained as the exposure decreased from 10 to \(10^{-5}\) sec. In the author’s opinion, this increase is not due to discharge delay. Referring to Fröhlich’s work, the author assumed that electron motion takes place in a time no greater than \(10^{-8}\) sec. Plesner supposes that the dependence he obtained is caused by thermal effects and polarization surface phenomena.

A. F. Walter and L. D. Inge [9] obtained volt-second characteristics of rock-salt plates 2.7 mm thick during breakdown in an electric point—plane field. The dependences they obtained are presented in Fig. 3. The rise in breakdown voltage with decreasing exposure, in the authors’ opinion, is due to discharge delay.

G. A. Vorob’ev [3] obtained volt-second characteristics of several solid dielectrics, broken down in a nonuniform field at a dielectric thickness of 0.7 mm. In these experiments attention was paid to eliminating the influence of edge discharges. If a dielectric plate is clamped between point—plane electrodes and a voltage is applied to them, then at a certain voltage value discharges appear near the point along the surface of the dielectric plate. These discharges can reduce the breakdown voltage. The influence of surface discharges on the breakdown voltage was shown in the work of A. A. Vorob’ev and

Fig. 3. Volt-second characteristics of rock salt during breakdown in a nonuniform field. Thickness 2.7 mm (after A. F. Walter and L. D. Inge).

N. A. Prikhod’ko [10], where, for one and the same solid dielectric, different polarity effects and different temperature dependences of the breakdown voltage are obtained when the specimen is placed, during breakdown, in different liquid media. The specimens were taken in the form of plates, on which sharp and plane electrodes were placed.

G. A. Vorob’ev made conical recesses in the specimens, which eliminated the influence of surface discharges. The dependences obtained by him are shown in Figs. 4 and 5. The author explains a certain increase in the breakdown voltage of rock salt at exposures of \(10^{-7}\) sec and longer in the same way as

Fig. 4. Volt-second characteristics of rock salt during breakdown in a nonuniform field (according to G. A. Vorob’ev).
1—sharp electrode (−); 2—sharp electrode (+).

in the case of a uniform field, by the influence of the space charge; the increase in the breakdown voltage of single crystals of rock salt, potassium chloride, bromide, and iodide at exposures of \(1 \div 2 \cdot 10^{-7}\) sec—by the lag of the discharge.

It may be said that in crystals of alkali-halide salts, for specimen thicknesses on the order of tenths of a millimeter and greater, in the case of breakdown both in uniform and in nonuniform fields, an increase in breakdown voltage with decreasing exposure has been found. Most investigators believe that this is due to the lag of the discharge.

Fig. 5. Volt-time characteristics during breakdown of single crystals of potassium chloride, bromide, and iodide in a nonuniform field (according to G. A. Vorob’ev).

Crystals Containing Water Molecules

Kawamura, Okura, and Kikuchi [7], in breakdown of mica specimens \(10^{-4}\) cm thick by a series of pulses, found that the number of pulses applied before breakdown is smaller the greater the pulse amplitude. The product of the number of applied pulses “\(n\)” and the pulse duration “\(T\),” i.e., the quantity \(nT\), was taken by the authors as the discharge lag time. This quantity at a 10-percent overvoltage is \(10^{-4}\) sec.

E. A. Konorova [11] carried out breakdown of mica specimens \(5\cdot 10^{-4}\) cm thick on the front of a single pulse. She found no change in the electric strength of mica when the exposure was varied from \(10^{-6}\) to \(5\cdot 10^{-8}\) sec. E. A. Konorova explains the difference between her results and the results of the Japanese investigators [7] by the possibly different influence of space charge in the two cases. At small thicknesses \((10^{-5}\div 10^{-4}\) cm) the influence of space charge may be large.

M. A. Melnikov [26], investigating the volt-second characteristics of mica 25 microns thick, found a delay of the discharge only at an exposure of \(5\cdot 10^{-9}\) sec.

K. M. Kevroleva [12] investigated the volt-second characteristics of certain crystals containing water of crystallization. The author explains the increase in electric strength at exposures of \(10^{-7}\) sec and less by a delay of the discharge.

A. V. Astafurov [13] investigated the volt-second characteristics of river ice at distances between the electrodes of \(0.5\div 2.0\) cm. The corresponding data are presented in Fig. 6. The increase in breakdown voltage at exposures less than \(10^{-5}\) sec is explained by the author as due to a delay of the discharge.

Fig. 6. Volt-second characteristics of ice (...) and distilled water (000) during breakdown in a uniform field (after A. V. Astafurov).

Thus, in single crystals containing water of crystallization, with thicknesses of the order of tenths of a mm and about 1 cm, an increase in electric strength at small exposures has been found, which is explained by a delay of the discharge.

Amorphous dielectrics

A. F. Walter and L. D. Inge [14] investigated the volt-second characteristics of glass in a uniform field. The corresponding data are given in Fig. 7. No increase in electric strength at small exposures was found.

Keller [15] investigated the dependence of the electric strength of Thuringian glass on exposure and on temperature. The corresponding data are given in Figs. 8 and 9. An increase in electric strength

Fig. 7. Dependence of the breakdown voltage on hold time for glass of various thicknesses in a uniform field (after A. F. Walter and L. D. Inge).

Fig. 8. Dependence of the electric strength of Thuringian glass on the time of voltage application (after Keller).

Fig. 9. Dependence of the electric strength of Thuringian glass on temperature (after Keller).

the glass as the voltage exposure decreases should mainly be explained by the influence of thermal effects, as the author himself also indicates.

A similar dependence on voltage exposure was obtained by Farmer [16] for Pyrex glass, which is also explained by the influence of thermal effects.

M. A. Melnikov [26] investigated the volt-second characteristics of organic glass, polystyrene, and Teflon with a specimen thickness of 30–40 microns. A discharge delay was found only at exposures of \(5 \cdot 10^{-9}\) sec.

One should not look for a contradiction between the data of Keller and Farmer, on the one hand, and the data of A. F. Walter and L. D. Inge, on the other. Apparently, in the experiments of A. F. Walter and L. D. Inge the glass had low electrical conductivity, and breakdown under pulsed voltage was electrical in character.

Porcelain

Up to now we have considered homogeneous dielectrics. Porcelain is a complex solid dielectric: in addition to the solid structure, it contains pores. Both the solid structure and the pores may determine the character of the volt-time dependence.

A. A. Vorob’ev and A. F. Walter [17] give the following experimental data on the breakdown of porcelain: at a wavelength of 3500 μsec the impulse coefficient was 1.02; at a wavelength of 4 μsec, 1.33.

A. F. Walter and L. D. Inge [9] found that the electric strength of porcelain (thickness 0.8 mm) does not depend on the voltage exposure. They obtained the following values of the breakdown voltages:

at alternating voltage — 28 kV
for an impulse duration of 0.9 μsec — 27 kV
" " " 0.95 " — 28.5 kV
" " " 0.8 " — 29.4 kV

The difference in the data of the cited works [17, 9] can possibly be explained by different porosity of the porcelain.

Fibrous Materials

Fibrous materials (bakelized paper, pressboard, cardboard, etc.) are also complex solid dielectrics.

Dittert [18] investigated the volt-time characteristics of bakelized paper of thickness 0.1 mm, 0.3 mm, and 0.6 mm, and of ebonite of thickness 0.5 mm and 1.0 mm. He found that the breakdown voltage of the indicated dielectrics does not depend on the voltage exposure within the range from \(10^{-1}\)–1 sec to \(10^{-8}\) sec.

Yost [18] and Strigel [19] investigated the volt-time characteristics of a large number of solid dielectrics: glass, porcelain, mica, pressboard, cardboard, micanite, ebonite, and others. These experiments had serious methodological shortcomings, of which two principal ones may be pointed out.

1. The dielectric specimens were taken in the form of plates, onto which electrodes were applied: in the case of a homogeneous field, flat electrodes (in Strigel’s work) or rounded electrodes (in Yost’s work); in the case of a nonuniform field, a point and a plane. In this case, edge discharges appear at the edges of the electrodes in the surrounding medium, which affect the breakdown of the solid dielectric. The breakdown voltage in this case characterizes not the solid dielectric, but the combination of the solid dielectric and the surrounding medium.

1. In carrying out the experiments, no electronic oscillograph was used, and in measuring the impulse breakdown voltage and especially the duration of the voltage action, significant errors could have occurred.

Therefore the data of Yost [18] and Strigel [19] may be far from the truth and should be checked using a more refined experimental technique.

2. Discharge time

At the present time it is believed that the electrical breakdown of solid dielectrics is an electronic process. In the general case, the discharge time may consist of two principal components: the statistical time lag and the time of formation of the discharge.

There is no established opinion regarding the existence of a statistical time in solid dielectrics. Whitehead [20] points out that in solid dielectrics there is always a certain number of free electrons as a result of absorption of radiation, thermal vibrations, and imperfections of the crystal, and that the statistical time lag in solid dielectrics should be absent.

Seitz [21] developed a theory of the statistical time lag in solid dielectrics. Seitz considers that, for the destruction of a solid dielectric, some minimum number of electrons in the avalanche is required. Seitz takes into account that electrons in a dielectric have different velocities. Electrons with high velocity have a greater probability of being accelerated to the ionization energy. Therefore, at any value of the electric-field strength there is some probability that, as a result of ionization processes, a number of electrons sufficient to destroy the dielectric will be obtained. The time between the appearances of two destructive avalanches was called by Seitz the statistical time lag.

The statistical time lag decreases with increasing field strength and, in the limit, is determined by the probability of the appearance of free electrons in the dielectric.

Crowe [22] investigated the delay of discharge in certain highly purified liquid hydrocarbons having an electric strength of the order of \(10^6\) V/cm. He considers that in these liquids there is no statistical time lag, since under strong prebreakdown fields there must occur cold emission of electrons from the cathode, which is the supplier of electrons for the formation of the discharge.

It is indicated [23] that, for the onset of cold emission of electrons from the cathode, a field strength of 300 kV/cm is required.

A. F. Walter and L. D. Inge [24] and G. A. Vorob’ev [3] investigated the delay of discharge in uncolored rock-salt crystals and in colored crystals, broken down in darkness and under strong illumination. In all three cases the authors obtained identical values of the breakdown voltage with discharge delays both in a uniform field (G. A. Vorob’ev) and in nonuniform fields (A. F. Walter and L. D. Inge). In colored crystals, under strong illumination, a large number of free electrons appears.

If in uncolored rock-salt crystals there were a statistical time lag, then in colored crystals it should, at least, be smaller. However, the experiments do not show this. Therefore it should be concluded that in rock salt, at the shortest exposure of \(2 \div 3 \cdot 10^{-8}\) s [3], the statistical time lag is practically absent. It is possible that at exposures considerably shorter

$2 \div 3 \cdot 10^{-8}$ sec, the statistical delay time will become comparable with the exposure and will affect the magnitude of the breakdown voltage.

Apparently, in other homogeneous solid dielectrics as well, having an electric strength of $10^6$ V/cm or greater at exposures of $2 \div 3 \cdot 10^{-8}$ sec, the statistical delay time is practically absent. Probably the source of electrons in the indicated solid dielectrics is cold emission from the cathode and electrostatic ionization in the dielectric.

Naturally, researchers in whose experiments a discharge delay was found attempted to estimate the statistical delay time and the discharge formation time. In doing so, two different approaches to this question were found.

Strigel [19] carried out a statistical treatment of experimentally obtained results for the breakdown of solid dielectrics. He plotted graphs in which the time of discharge delay was laid off along the abscissa axis, and along the ordinate axis the probability of this delay time or a greater one. Figure 10 presents graphs of the statistical treatment for nitrocellulose, presspahn, and mica.

Fig. 10. Graphs for determining the discharge formation time and the statistical delay time from breakdown data for nitrocellulose, presspahn, and mica (according to Strigel).
1—nitrocellulose, thickness 0.05 mm, impulse voltage 18.3 kV; 2—presspahn, thickness 0.3 mm, impulse voltage 20 kV; 3—mica, thickness 0.3 mm, impulse voltage 20 kV.

As can be seen, the experimental points fit well near straight lines. The smallest delay time was taken by the author as the discharge formation time, and the delay time at a probability of 37% minus the discharge formation time was taken as the mean statistical delay time.

Such a determination of the discharge formation time and the mean statistical delay time should be regarded as unjustified for two main reasons.

1. No physical justification is given for the existence of a statistical delay time in solid dielectrics. As was indicated above, the statistical delay time is practically absent up to exposures of $10^{-8}$ sec.

2. From the above method of treating the results it follows that the discharge formation time is the smallest delay time and therefore is a constant quantity, with which one cannot agree.

The discharge develops each time under conditions different from the preceding case.

An analogous method for processing results on the breakdown of solid dielectrics was adopted by Inushi and Suida [5]. Therefore the data they give for \(\sigma\) and \(\tau\) should be regarded as unfounded.

G. A. Vorob’ev [3] assumed that the statistical lag time in solid dielectrics is practically absent, and that the discharge lag time is in fact the discharge-formation time. In the experiments of G. A. Vorob’ev, breakdown was produced on the front of the pulse. The determination of the discharge-formation time in solid dielectrics was carried out from breakdown oscillograms, according to the scheme shown in Fig. 11, where \(U_0\) is the breakdown voltage at such an exposure that the influence of the discharge lag on the breakdown voltage is no longer manifested and the influence of space charges is not yet present. This condition was satisfied by an exposure of \(4 \cdot 10^{-7} \div 6 \cdot 10^{-6}\) sec. In the experiments of G. A. Vorob’ev [3] it was established that, at exposures where the influence of the discharge is manifested, the scatter in the values of the electric strength of individual specimens is considerably greater than at larger exposures. In the latter case the scatter in the values of the electric strength of crystals is due to structural differences among individual specimens. At smaller exposures, to this scatter there is added scatter due to statistical fluctuations in the magnitude of the discharge-formation time. Therefore it may be considered that the greatest value of the electric strength at small exposures corresponds to the greatest value of the discharge-formation time, and the smallest value of the electric strength to the smallest value of the discharge-formation time.

Fig. 11. Plot for determining the discharge-formation time \(t_{\phi}\) and the voltage-fall time during breakdown \((t_{\text{sp}})\).

K. M. Kevrol’eva [12], A. V. Astafurov [13], and V. D. Kuchin [25] also used this method for determining the discharge-formation time.

K. K. Sonchik [6], breaking down crystals with voltage pulses close to rectangular, determined the mean discharge-formation time from a plot of the type of Fig. 10. He took \(t_{\phi \text{av}} = \tau + \sigma\) (Fig. 11).

3. Velocity of discharge propagation

Inushi and Suida [5] indicate that, with an increase in the amplitude of the pulse, the statistical lag time decreases, while the discharge-formation time changes little. They defined the probability of the appearance of free electrons \(n_i\) in the dielectric as a quantity reciprocal to the minimum statistical lag time, and the electron velocity \(v\) (this quantity should more correctly be called the average velo-

…the discharge-propagation rate), as the ratio of the dielectric thickness to the discharge-formation time, and the electron mobility \(\kappa\). (The results are presented in the table.)

Table

The authors consider the electron velocity \((v)\) to be small and assume that breakdown in the crystals they studied occurs not by a single avalanche, but by several avalanches.

G. A. Vorob’ev [3] also determined the mean propagation velocity of the discharge \(v_{\mathrm{av}}\). The mean discharge-propagation velocities calculated by him proved to be an order of magnitude greater than in the experiments of Inushi and Suita [5] for potassium chloride, i.e., of the order of \(10^6\ \mathrm{V/cm}\).

For a nonuniform field, according to the data of G. A. Vorob’ev [3], \(v_{\mathrm{av}}\) is also of the order of \(10^6\ \mathrm{V/cm}\), but in the case of positive polarity of the point \(v_{\mathrm{av}}\) is several times greater than in the case of negative polarity. The same regularity is characteristic of air. This analogy shows that in solid dielectrics the principal role in breakdown is apparently played by impact ionization.

According to the measurements of K. M. Kevroleva [12], A. V. Astafurov [13], K. K. Sonchik [6], and V. D. Kuchin [25], \(v_{\mathrm{av}}\) is also of the order of \(10^6\ \mathrm{V/cm}\). However, according to the data of A. V. Astafurov [13], \(v_{\mathrm{av}}\) is several times greater than for other authors [3], [12], [6], [25]. In his experiments the thickness of the specimens was \(0.5 \div 1.5\ \mathrm{cm}\). In air, the discharge-propagation velocity also increases with increasing interelectrode distance, which is connected with the streamer mechanism of the discharge. It is possible that in solid dielectrics, at thicknesses of the order of centimeters, a streamer mechanism of discharge exists.

4. Polarity effect

By the polarity effect in breakdown of dielectrics is meant the influence of the polarity of the electrode with the greater curvature on the magnitude of the breakdown voltage. The polarity effect is connected with the breakdown mechanism, as is evident from the example of air.

Knowledge of the polarity effect for solid dielectrics may help in elucidating the mechanism of discharge in them. If one assumes that breakdown of solid dielectrics is a mechanical rupture under the action of electrostatic forces, then no polarity effect should be expected.

If one assumes that breakdown of solid dielectrics is due to ionization processes, then a polarity effect should be expected. In this connection A. A. Vorob’ev and N. A. Prikhod’ko [10] put forward two assumptions.

1. If the breakdown process is caused by cold emission from the cathode, then a decrease in the breakdown voltage should occur with negative polarity of the point.

2. If impact ionization plays the determining role in the breakdown process, then there should be a lowering of the breakdown voltage with positive polarity of the point, as in air, owing to the influence of the positive space charge formed near the point.

If electrostatic ionization played the main role in the breakdown process of solid dielectrics, then a lowering of the breakdown voltage should also be expected with positive polarity of the point. A. A. Vorob’ev and N. A. Prikhod’ko [10] took dielectrics in the form of plates on which electrodes—a point and a plane—were placed. Only in the case of ice was the point, under pressure, driven into the ice. For the dielectrics, different polarity effects were obtained depending on the type of voltage (direct or pulsed) and on the surrounding medium. However, from the data of [10] the following regularity can be established: for crystalline dielectrics (sulfur, mica, calcite) placed in water or glycerin, under pulsed voltage one and the same polarity effect was obtained—a lowering of the breakdown voltage with positive polarity of the point. Water and glycerin have a large value of the dielectric permittivity; therefore it is possible that in these media, under pulsed voltage, surface discharges were absent and the true polarity effect was obtained. The same polarity effect was obtained in the breakdown of ice. In the breakdown of amorphous dielectrics (glass, paraffin, ebonite) no polarity effect was observed.

A lowering of the breakdown voltage of rock salt with positive polarity of the point was obtained in the experiments of A. F. Val’ter and L. D. Inge [9] (Fig. 3).

In the experiments of G. A. Vorob’ev [3], in which the influence of discharges in the surrounding medium was eliminated, for all the dielectrics investigated a lowering of the breakdown voltage was likewise obtained with positive polarity of the point.

An analogous polarity effect was obtained by Inuishi and Suwita [5] in the breakdown of potassium chloride plates in transformer oil in a nonuniform field, and the reverse effect—in the breakdown of sulfur-crystal plates.

It may be concluded that in those cases when the influence of the surrounding medium on the breakdown voltage was eliminated, one and the same polarity effect was obtained: a lowering of the breakdown voltage with positive polarity of the point. This indicates that, in the breakdown of solid dielectrics, the main role is played by impact or electrostatic ionization in the volume of the dielectric.

LITERATURE

1. Hippel A. and A. Alger R. S., Breakdown of ionic crystals by electron avalanches. Phys. Rev., 76, 1, 127, 1949.

2. Trubitsyn A. M. Investigation of the pulsed electric strength of single crystals of certain alkali-halide salts and their solid solutions with partial replacement by gallium. Dissertation, Tomsk, 1954.

3. Vorob’ev G. A. Investigation of electrical breakdown of solid dielectrics at various durations of voltage application. Dissertation, Tomsk, 1956.

4. Vorob’ev A. A. On the electrical breakdown of insulating crystals. ZhTF, 10, 1183, 1940.

5. Inuishi Y., Suwita. T. On electrical breakdown under direct and pulsed voltage and the time lag in solids. Z. Inst. El. Eng. Japan, 74, 2, 150, 1954.

1. Sonchik K. K. On the delay time of discharge in ionic crystals. Report at the All-Union Conference on the Physicochemical Properties of Dielectrics, Dnepropetrovsk, 1956.

2. Kawamura H., Ohkura H., Kikushi T. Statistical delay of dielectric breakdown in mica, glass, and KCl. J. Phys. Soc. Japan, 9, 541, 1954.

3. Plessner K. W. Electrical strength of dielectric films, Proc. Phys. Soc., 60, 339, 242, 1948.

4. Val’ter A. F. and Inge L. D. Breakdown of dielectrics under impulse voltages. ZhTF, III, 6, 840, 1933.

5. Vorob’ev A. A. and Prikhod’ko N. A. Electrical breakdown of dielectrics in a nonuniform field. Trudy Sib. FTI, 4, 3, 112, 1936.

6. Konorova E. A. On the question of the statistical delay of breakdown in solid dielectrics. Izv. Tomsk. Politekhn. In-ta, 91, 73, 1956.

7. Kevroleva K. M. Dependence of the electrical strength of crystallohydrates on the time of action of the voltage and on the crystallographic direction, (present collection).

8. Astafurov A. V. Volt-time characteristics of river ice in large thicknesses. Izv. Tomskogo Politekhn. In-ta, vol. 94, in press.

9. Val’ter A. F. and Inge L. D. Breakdown of glass in a nonuniform electric field. ZhTF, 6, 6, 33, 1929.

10. Keller K. J. Breakdown and electrical conductivity of glass, Physica, XVII, 5, 511, 1951.

11. Vermer J. Impulse electrical strength of Pyrex glass. Physica, XX, 6, 313, 1954.

12. Vorob’ev A. A. and Val’ter A. F. Breakdown of porcelain under impulse voltages. Proceedings of the 3rd All-Union Conference on Electrical-Insulating Materials, 3, 173, 1931.

13. Binder L. Traveling waves in electrical networks. Translated from German. Moscow—Leningrad, 1935.

14. Strigel R. Electrical impulse strength. Berlin, 1939.

15. Whitehead S. Electrical breakdown of solid bodies. Oxford, Clarendon Press, London, 1951.

16. Seitz F. On the theory of multiplication of electrons in crystals. Phys. Rev., 76, 9, 1376, 1949.

17. Crowe R. W. Inertia in the formation of electrical breakdown in liquid hydrocarbons. J. Appl. Phys., 27, 2, 156, 1956.

18. Edited by Prof. L. I. Sirotinskii. High-Voltage Engineering, Part I, p. 143, 1951.

19. Val’ter A. F. and Inge L. D. Influence of the concentration of conduction electrons on the electrical strength of crystals. DAN SSSR, 3, 2, 65, 1934.

20. Kuchin V. D. Dependence of the electrical strength of ionic crystals on temperature. DAN SSSR, 114, 301, 1957.

21. Mel’nikov M. A. Izvestiya MVO SSSR, “Elektromekhanika,” in press.

ERRATA

1. The electrodes were a hemisphere opposite a plane, but the radius of curvature of the hemisphere was close to the thickness of the sample. In such a case the field, even at the thinnest point, will not be uniform; the field will be weakly nonuniform. ↩