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THEORY AND MECHANISM OF GAS BREAKDOWN
A. S. Zingerman, Leningrad
INTRODUCTION
Under ordinary conditions a gas does not conduct electric current. In order for a gas to become conducting, free carriers of electricity—electrons and ions—must appear in it. This may occur as a result of irradiation of the gas or of the negative electrode by a radioactive substance, by ultraviolet, X-ray, and other rays. The more intense the irradiation, the greater the conductivity of the gas. When a certain definite voltage is applied to a gas gap, depending on the size of the gap, its geometry, the kind of gas, its state, pressure, and temperature, the gas becomes conducting even after the irradiation is stopped. The transition of a gas from a nonconducting state to a state in which it conducts electric current takes place extremely rapidly and has the character of an explosion. This process, called breakdown, is the subject of the present review.
After breakdown, the discharge does not cease and, depending on a number of conditions, assumes various forms: a glow discharge or an arc. Depending on the character of the voltage, the discharge may either cease or exist for a long time. In nonuniform fields with a very sharp change of gradient, a discharge, once it has arisen, may not lead to complete breakdown. Such a retarded discharge may exist for a long time and is usually called corona.
The final stages of the discharge—the glow discharge and the arc—have been studied for a long time; there is an extensive literature on this question. At the same time, serious investigations of the first stage of the discharge were undertaken only in the last few years.
This state of affairs is quite natural, since the glow discharge or the arc are stable phenomena, existing for a long time, which considerably facilitates their study.
In contrast to the final stage of the discharge, breakdown is an unstable phenomenon, which moreover occurs extremely rapidly. Its study requires considerable skill on the part of the experimenter himself and a high level of experimental technique.
At the present time experimental technique has reached such a level that the study of such phenomena no longer presents special difficulties. In the last few years many exper-
imental and theoretical works devoted to this question. These works provide material so valuable and so extensive that, on the one hand, there has arisen a need to systematize it in order to clarify the paths of further research, while, on the other hand, it is possible to construct a fairly clear picture of the physical processes occurring during gas breakdown.
I. TOWNSEND’S LAW
The first theory of discharge in a gas was developed by Townsend1. Townsend assumed that a discharge in a gas occurs as a consequence of the appearance in it of carriers of electricity—electrons and ions. Electrons and positive ions are the product of ionization of neutral atoms and molecules of the gas when they collide with free electrons. If \(N_x\) free electrons enter a gas layer \(dx\), then they additionally form
\[ dN_x = \alpha N_x\,dx \]
pairs of ions; i.e., the number of additionally formed ion pairs in the gas layer is directly proportional to the thickness of the layer \(dx\) and to the number \(N_x\) of free electrons entering the layer. The proportionality coefficient \(\alpha\), evidently, must have the following physical meaning: \(\alpha\) is the number of ion pairs formed by one electron over a path length of \(1\ \mathrm{cm}\). In order for an electron to be able to ionize, it is necessary that it either possess sufficient energy upon entering the layer \(dx\), or be able to acquire this energy while passing through the layer \(dx\). In the latter case, an electric field accelerating the electron must be applied to the layer \(dx\), perpendicular to the planes bounding the layer \(dx\).
Thus, one should distinguish two ionization coefficients[^2] by electrons: \(\alpha_W\)—the coefficient of ionization by an electron having some initial energy \(W\), but not undergoing acceleration in the process of ionization, and \(\alpha\)—the coefficient of ionization by an electron having no initial energy, but accelerated in the process of ionization. In the first case, the electron will lose its energy as ionization proceeds. Therefore \(\alpha_W\) refers, strictly speaking, to the first portions of the electron’s path. \(\alpha_W\) depends on the energy of the electron, on the kind, state, and pressure of the gas. In the second case, the electron, being in the field accelerating it, will increase its energy and will begin to ionize when the energy it has accumulated is sufficient for this. In the act of ionization the electron loses all or part of its energy and, after ionization, begins again to be accelerated by the field, as does the secondary electron it has produced (and the positive ion), which in turn begins to ionize. Evidently, this case occurs in a gas discharge when some potential difference is applied to the gas gap.
Thus, passing through a layer \(dx\), the avalanche of electrons will increase by the amount \(dN_x\). If the electrons pass through a distance \(x\) in the gas from the place
of its occurrence, then at the end of the interval \(x\) the electron avalanche will be equal to
\[ N_x = N_0 e^{\int_0^x \alpha\, dx}, \tag{1} \]
where \(N_0\) is the number of electrons that arose at the beginning of the interval.
This Townsend law was verified experimentally by Townsend himself\(^1\) and by many investigators\({}^{2-11}\) after him. This law was tested in the following manner. To two plane electrodes, the distance between which was equal to \(L\), a certain potential difference was applied. The surface of the negative electrode was illuminated with ultraviolet light, and as a result of the photoelectric effect electrons were torn from it. Then the distance between the electrodes and, simultaneously, the potential difference between them were varied so that the gradient remained constant, and the current flowing in the electrode circuit was measured. The measured values in the coordinate plane \((\lg \frac{i}{i_0},\, L)\) must lie strictly on a straight line. \(i_0\) is the value of the current at some initial distance between the electrodes \(L_0\).
Fig. 1. Dependence of \(\lg\) of the current in the discharge gap on the distance between plane electrodes according to Posin’s experiments
\(L\)—distance between the electrodes
Fig. 2. Dependence of \(\dfrac{\alpha}{p}\) on \(\dfrac{E}{p}\)
I—Posin (pure nitrogen, \(22^\circ\)), II—Zairis (nitrogen), III—Malsch (air), IV—Sanders (dry air)
In Fig. 1 are shown data borrowed from Malsch\(^5\).
| Curve | \(E\) | \(\alpha\) | Curve | \(E\) | \(\alpha\) |
|---|---|---|---|---|---|
| 1 | \(20\ \mathrm{kV/cm}\) | 0.23 | 6 | \(30\ \mathrm{kV/cm}\) | 7.7 |
| 2 | 22 " | 0.735 | 7 | 32 " | 12.0 |
| 3 | 24 " | 1.54 | 8 | 34 " | 17.4 |
| 4 | 26 " | 2.44 | 9 | 36 " | 22.4 |
| 5 | 28 " | 4.83 |
As is seen from the figure, the points fit very well on straight lines whose slope, determining the value of \(a\), depends on the gradient and the gas pressure.
Calculating \(a\) from such measurements, one usually constructs the dependence
\[ \frac{a}{p}=f\left(\frac{E}{p}\right), \]
where \(p\) is the gas pressure in torr, and \(E\) is the field gradient. In Fig. 2 there are also given curves borrowed from the works of Posin \(^{8}\), Sanders \(^{2}\), Masch \(^{5}\), and Ayris \(^{11a}\). Each of these curves cannot be expressed analytically by a single equation, but its separate portions are well expressed by different equations.
The measurement results of different authors disagree somewhat with one another. Thus, Posin \(^{8}\), Paavola \(^{4}\), and Sanders \(^{2}\) give the following dependences of \(\frac{a}{p}\) on \(\frac{E}{p}\) (Table 1):
Table 1
| \(\frac{E}{p}\) range, \(V/\text{cm torr}\) | \(\frac{a}{p}=f\left(\frac{E}{p}\right)\) | Kind and state of gas |
|---|---|---|
| \(20 \leq \frac{E}{p} \leq 38\) | Posin \(\displaystyle \frac{a}{p}=(5.76\pm1.56)\cdot10^{-7}\cdot e^{(0.245\pm0.03)\frac{E}{p}}\) |
Pure nitrogen \(\theta=22^\circ\) Uniform field |
| \(44 \leq \frac{E}{p} \leq 176\) | \(\displaystyle \frac{a}{p}=(1.166\pm0.022)\cdot10^{-4}\cdot\left(\frac{E}{p}-32.1\pm1.4\right)^2\) | Uniform field |
| \(200 \leq \frac{E}{p} \leq 1000\) | \(\displaystyle \frac{a}{p}=-3.65+\left(0.21\cdot\frac{E}{p}\right)^{1/2}\) | Uniform field |
| \(30 \leq \frac{E}{p} \leq 45\) | Paavola \(\displaystyle \frac{a}{p}=1.56\cdot10^{-4}\left(\frac{E}{p}-30.1\right)^2\) |
Air, atmospheric pressure |
| \(20 \leq \frac{E}{p} \leq 36\) | Sanders \(\displaystyle \frac{a}{p}=(2.76\pm0.26)\cdot10^{-8}\cdot e^{(0.35\pm0.002)\frac{E}{p}}\) |
Dry air |
Iol’dauer \(^{9}\) showed that Sanders’ data in the range
\[ 36.5 \leq \frac{E}{p} \leq 120 \]
are well satisfied by the following expression:
\[ \frac{a}{p}=1.35\cdot10^{-4}\left(\frac{E}{p}-28.8\right)^2. \]
It can be shown that Masch’s \(^{5}\) data for air in the range
\[ 32 \leq \frac{E}{p} \leq 160 \]
are well satisfied by the following expression:
\[ \frac{a}{p}=1.26\cdot10^{-4}\left(\frac{E}{p}-28\right)^2. \]
Rotovsky3 showed that Masch’s data in the range \(40 \leq \dfrac{E}{p} \leq 400\) also satisfy the following expression:
\[ \frac{\alpha}{p}=10.5 \cdot e^{-\frac{267}{E/p}} . \]
Townsend’s data in the range \(\dfrac{E}{p} > 300\) satisfy the expression:
\[ \frac{\alpha}{p}=14.6 \cdot e^{-\frac{365}{E/p}} . \]
At smaller values of \(\dfrac{E}{p}\) this expression is not satisfied.
Especially careful experiments were carried out by Bowles2. He used pure platinum electrodes and took special measures against the penetration of mercury into the space containing the gas under investigation. For pure nitrogen he found values for \(\dfrac{\alpha}{p}\) lower than those of all other authors who had worked before him. At values of \(\dfrac{E}{p} > 300\), the values for \(\dfrac{\alpha}{p}\) obtained were 17% lower than those obtained by Townsend. When the gas was contaminated with mercury vapor and the electrode surfaces were coated with it, the values for \(\dfrac{\alpha}{p}\) coincided with those obtained by Townsend.
Apparently, the different values for \(\dfrac{\alpha}{p}\) must be explained by different contaminations of the gas and by its state.
The verification of Townsend’s law indicated above is indirect. In 1937, Raether4 carried out a direct verification of Townsend’s law. Fig. 3 shows photographs obtained by Raether4 in a Wilson chamber. A rectangular voltage wave was applied to two plane electrodes located in the chamber. Each spot (in the shape of a falling drop), with its point turned toward the negative electrode, represents an avalanche of positive ions left after
Fig. 3. Photographs of ion avalanches formed by impact ionization by electrons, obtained by Raether in a Wilson chamber
ionization from one initial electron. As is known, at each ion a droplet condenses at the moment of expansion. On the original photograph one can distinguish the individual droplets and count their number in the avalanche. This number agrees very well with the number obtained from Townsend’s law \(e^{\alpha l}\), where \(l\) is the length of the avalanche measured on the photograph. The shape of the avalanche in the form of a falling drop, as calculation shows, must be explained by the diffusion of the electrons.
Thus, avalanche-like ionization by electrons, for the corresponding gradients and distances between the electrodes, finds its complete confirmation.
2. TOWNSEND’S THEORY OF BREAKDOWN
Experiments show that, for large distances between the electrodes, the points in the coordinate system \(\left(\lg \dfrac{i}{i_0}, L\right)\) cease to lie on a straight line. The straight lines begin to curve upward, as is seen in Fig. 4. Each gradient has its own critical distance, beginning with which the avalanche grows faster than according to the exponential law. The increased growth of the avalanche can be explained either by the fact that a new ionizing agent comes to the aid of the electrons, or by the fact that, beginning with a certain distance, the electrons begin to ionize more strongly.
Townsend adopted the first cause to explain the observed phenomena and assumed that additional ionization is produced by positive ions1. Denoting by \(\beta\) the number of pairs of ions that one positive ion produces over \(1\ \mathrm{cm}\) of path in the direction of the field, Townsend found that the current flowing between plane electrodes, when a certain potential difference is applied to them and when the distance between them is \(L\), will be equal to
\[ i = i_0 \frac{(\alpha-\beta)e^{(\alpha-\beta)L}}{\alpha-\beta e^{(\alpha-\beta)L}} . \tag{2} \]
\(i_0\) is the current due to electrons produced as a result of ionization by an external source, i.e. the initial electrons.
Fig. 4. Dependence of \(\lg \dfrac{i}{i_0}\) on the distance between the electrodes according to Sanders’s experiments.
\(L\) — distance between the electrodes.
The new Townsend theory, which takes into account additional ionization by positive ions, is a significant step forward in the study of the mechanism of gas breakdown. The point is that ionization by electrons alone cannot in general produce the phenomenon of discharge in gases. Indeed, suppose that a group of initial electrons, arising on the surface of the cathode under the action of an external source, moves toward the anode and ionizes together with the secondary electrons newly formed by them. This process is called the primary process. The avalanche of electrons, having reached the anode, is absorbed by it. If, after the first group of initial electrons, no subsequent groups of initial electrons arise on the surface of the cathode, then the process in the gas will end there and no breakdown will occur. If, however, after the first group, subsequent groups continuously arise on the surface of the cathode under the action of the external source and move, like the first and following it, toward the anode, ionizing along the way, a discharge will occur in the gas; this is called a non-self-sustained discharge.
If the first group of initial electrons, produced under the action of an external source, creates an avalanche and, behind the avalanche, new electrons are formed by the ionization products of the initial electrons, which can obviously in turn serve as subsequent initial electrons, a discharge will begin in the gas even if the action of the external source ceases. Such a discharge is called a self-sustained discharge.
If, for each initial electron, one or more electrons are formed behind the avalanche by the products of its ionization, the discharge will develop and lead to breakdown. If, however, fewer than one electron is formed by the products of its ionization for each initial electron, the discharge will die out. The actions of the ionization products produced by the primary electrons are called secondary processes.
Since positive ions move in the direction opposite to that of the electrons, their ionization will take place behind the electron avalanche that has passed and of which they are products. Thus a theory based on ionization by electrons and positive ions describes a mechanism that can explain the transition from a non-self-sustained to a self-sustained discharge. Mathematically, the condition for the transition from a non-self-sustained discharge to a self-sustained one is expressed by the fact that the denominator of the right-hand side of expression (2) becomes equal to zero. Then the current between the electrodes becomes arbitrarily large and does not depend on the magnitude of the initial current $i_0$ caused by the action of the external source.
We see that the original theory or, more precisely, Townsend’s law could not explain a self-sustained discharge in gases either physically or mathematically. The chief value of Townsend’s new theory consisted precisely in the fact that it eliminated this fundamental defect.
In addition, the second theory was able to explain a growth of current stronger than that given by the exponential law. The fact that the devia-
tion from the exponential law begins only at large distances shows that ionization by positive ions must be less effective than ionization by electrons. Indeed, the values for \(\beta\) are obtained experimentally as roughly 1,000 times smaller than for \(a\) under the same conditions in the interval \(\frac{E}{p}\) from 100 to 160. The values of \(\beta\) are obtained experimentally in the following way. From the rectilinear part of the curves \(\lg \frac{i}{i_0}=f(L)\) (see Fig. 4) one obtains the values for \(a\). Then, neglecting \(\beta\) in comparison with \(a\) in expression (2), we have
\[ i=i_0\,\frac{ae^{aL}}{a-\beta e^{aL}}, \]
whence \(\beta\) is determined.
The values for \(\beta\) obtained in the first experiments in this way also proved to depend on the gradient, the kind, state, and pressure of the gas. Therefore the values for \(\beta\) are usually given, as for \(a\), by the curve
\[ \frac{\beta}{p}=\varphi\left(\frac{E}{p}\right). \]
These values for \(\frac{\beta}{p}\) are given in Fig. 5.
Fig. 5. Dependence of \(\frac{\beta}{p}\) on \(\frac{E}{p}\)
A—Sanders (dry air), B—Ayris (nitrogen)
The dependence found from the first experiments
\[ \frac{\beta}{p}=\varphi\left(\frac{E}{p}\right) \]
turned out to be a monotonically increasing function\(^{3,11a}\). Such a character of the dependence for \(\frac{\beta}{p}\) and the numerical values for \(\beta\) agreed well with Townsend’s ideas about the physical meaning of \(\beta\) as the ionization coefficient for positive ions.
If one assigns certain values of \(L\) and \(p\), it is possible to determine the value of \(E\) at which the discharge passes into the self-sustained phase, i.e., satisfies the relation
\[ 1=\frac{\beta}{a}e^{(a-\beta)L} = \frac{ f\left(\frac{E}{p}\right) }{ \varphi\left(\frac{E}{p}\right) } e^{ \left[ f\left(\frac{E}{p}\right)-\varphi\left(\frac{E}{p}\right) \right]L }. \tag{3} \]
For the same values of \(L\) and \(p\), one can find experimentally the magnitude of the gradient \(E\) at which breakdown occurs. It turns out that these two values agree very well\(^{11a}\). Table 2 gives data borrowed from Ayris\(^{11a}\). This is the third substantial proof of the correctness of Townsend’s theory.
Alongside the first experiments confirming the correctness of Townsend’s second theory, facts emerged which could not be explained by the theory or did not agree with it.
Table 2
\(L\)—distance between plane electrodes, \(E\)—field gradient, \(p\)—gas pressure (nitrogen), \(U\)—breakdown voltage obtained experimentally, \(EL\)—voltage determined from expression (3)
| \(\dfrac{E}{p}\), V/cm torr | \(\dfrac{\alpha}{p}\) | \(\dfrac{\beta}{p}\) | \(Lp\), cm torr | \(EL\), V | \(U\), V |
|---|---|---|---|---|---|
| 125 | 0.85 | 0.0013 | 7.65 | 956 | 965 |
| 150 | 1.32 | 0.0048 | 4.27 | 641 | 648 |
| 175 | 1.84 | 0.0084 | 2.95 | 516 | 518 |
| 200 | 2.5 | 0.013 | 2.12 | 424 | 425 |
| 300 | 4.1 | 0.058 | 1.06 | 318 | 318 |
| 400 | 5.43 | 0.103 | 0.75 | 300 | 298 |
| 500 | 6.29 | 0.157 | 0.60 | 300 | 302 |
| 750 | 7.91 | 0.301 | 0.43 | 323 | 322 |
| 1,000 | 9.02 | 0.422 | 0.36 | 357 | 350 |
Experiment shows that breakdown occurs some time after the breakdown voltage has been applied to the gap \(^{14-33}\); this time is called the delay time. Although the phenomenon of delay can be explained on the basis of Townsend’s theory, the values obtained in this way are excessively large and differ from the experimental data by two or three orders of magnitude and more \(^{15,18-23,31-33a,56,84}\).
Then a number of authors \(^{34-42}\) observed that the breakdown voltages decrease with increasing initial current \(i_0\), caused by an external source of ionization. This fact not only fails to agree with Townsend’s theory, but is in direct contradiction to it. According to Townsend’s theory, the breakdown voltage determined from expression (3) should not depend on the initial current.
The first volt-second oscillograms, giving the variation of the voltage across the breakdown gap as a function of time, taken by Tamm \(^{18}\), Rogovskii \(^{19}\), and Rogovskii and Tamm \(^{43}\) (Figs. 6a and 6b), showed that during the discharge the voltage drops sharply and at the same time the current increases strongly. This phenomenon likewise cannot be explained in any way by Townsend’s theory.
Particular doubt is raised by the role of positive ions as ionizing agents.
Quantum mechanics treats the process of ionization in the following way. When an ionizing particle collides with an atom, the latter may remain in its normal energy state (elastic collision) or may pass into an excited state (inelastic collision). If, upon excitation, the atom passes into a state corresponding to a continuous spectrum, an electron flies out of the atom and the atom becomes ionized.
Bethe^44, using the approximate Born method, determined the probability of ionization in the collision of an ionizing particle with an atom. This probability depends only on the charge and velocity of the ionizing particle and does not depend on its mass. Thus, one and the same ionization probability in one and the same gas will occur both for an electron and for any positive ion with the same charges and with energies proportional to their masses. Thus, a proton begins to ionize at an energy of approximately 300 eV, and a molecular hydrogen ion \(H_2^+\) at an energy of about 600 eV. The minimum energy of nitrogen ions is still higher.
The range of applicability of the Born method is limited by the fact that the velocity of the ionizing particle must be large in comparison with the orbital
Fig. 6a and 6b. Volt-second oscillograms of a discharge in air between plane electrodes at atmospheric pressure, taken by Rogowski
velocity of the electron in the atom. For slow ions it gives exaggerated values. The method of perturbed wave functions of Mott and Massey does not have this limitation. The minimum energy which a proton must possess in order to ionize hydrogen atoms, as determined by this method, is approximately 400 eV^45. The probability of ionization of hydrogen atoms by protons according to the method of perturbed wave functions is one hundred times less than the probability determined by the Born method, and is of the order of \(10^{-17}\).
Let us now turn to the experimental data. Goldmann^46 could not detect any ionization of hydrogen by protons with energies of 4,000 eV and below. Wolf’s experiments^47 showed that hydrogen \(H_2\) and nitrogen \(N_2\) are not ionized by \(H_2^+\) and \(N_2^+\) ions with energies of 1,000 eV.
In an apparatus that makes it possible to detect one act of ionization for every thousand primary positive ions, no indications were obtained^48 of the ionization of hydrogen by sodium or potassium ions up to velocities of about 1,000 eV^1).
^1) In noble gases, ionization by potassium ions could be detected at small velocities of the order of 100 eV.
In the ordinary conditions of discharge in gases, at atmospheric and even much lower pressure, ions cannot acquire the large amount of energy that is necessary for ionization.
In 1939 Townsend[^49] again defended the possibility of explaining secondary processes by ionization by positive ions. His arguments are based on the following.
Townsend takes the ionization potential for air to be 15 eV. Since the masses of the positive ion and the atom are equal, the energy of the positive ion must be 30 eV. Positive ions possess a Maxwellian energy distribution and, consequently, the mean energy may be smaller. Townsend takes it to be 20 eV. Then, from the theory of viscosity, Townsend estimates the mean free path of positive ions at a temperature of 20 eV as \(7 \cdot 10^{-5}\) cm. Further, Townsend assumes a discharge voltage of 33 kV in a uniform field and a distance between the electrodes of 1 cm. Over the length of one path the positive ion will acquire 2.3 eV, and the number of paths will be 14,300, assuming that all paths coincide with the direction of the field. Consequently, the number of paths on which the positive ion will acquire an energy equal to or greater than 30 eV will be
\[ Ce^{-\frac{V_0}{E\lambda}} = 14300e^{-\frac{30}{33000 \cdot 7 \cdot 10^{-5}}}. \]
This number is \(6 \cdot 10^5\) times greater than the value of \(\beta\) determined for these conditions from expression (3). However, according to Thomson, an energy of 30 eV is insufficient for the ionization of air by nitrogen ions. According to his theory, ionization occurs when the ionizing particle, colliding with an electron of the atom, transfers to it an energy equal to or greater than the binding energy of the electron with the atom, i.e. the ionization energy[^5]. The electron is torn away from the atom and thereby the atom becomes ionized. In the most favorable case, with a central impact and when the kinetic energy of the secondary electron is zero, the minimum energy of the ionizing particle must be equal to \(\frac{M}{4m}V\), where \(M\) and \(m\) are the masses of the ionizing particle and the electron, and \(V\) is the ionization potential. Hence the minimum energy of nitrogen ions for the ionization of air is about 100,000 eV.
In 1940 Varney[^51] published a short note in response to Townsend’s article. His reply amounts to the following. New data (the author does not indicate by whom they were obtained, apparently by himself) show that for the ionization of air the energy of the positive ion in the most favorable case must be at least 3 times greater than the energy of the electron. The probability of acquiring this energy is determined by the mean free path of the ion, which at an ion temperature of 20 eV is not much greater than the mean free path of the molecule.
Taking these results into account, the probability of ionization by positive ions, as determined by wave mechanics, proves to be extremely small. Even if Townsend’s method is used, the new data cited above give, in the most favorable case, a number of ionizations per 1 cm of path \(10^{10}\) times smaller than the value of \(\beta\)1.
3. THE ROLE OF SPACE CHARGE
In order to explain the increase of the current, stronger than that given by an exponential law, with increasing distance between the electrodes, Rogowski\(^{53}\) and, independently of him, Loeb\(^{54}\) adopted a viewpoint different from Townsend’s. They assumed that the reason for this is an increase in the coefficient of ionization by electrons as a result of a change, in the course of the discharge, in the geometry of the field in the discharge space. While the electrons move toward the anode, the positive ions will move toward the cathode. Gradually a positive space charge is formed at the cathode, which greatly increases the gradient at the cathode. The subsequent electron avalanches will ionize much more strongly. Rogowski assumed that the time of formation of the space charge, when the distance between the electrodes is \(1\ \text{cm}\), is approximately equal to the time required for a positive ion to pass from the anode to the cathode. Having determined, for a uniform field with a gradient of \(30\ \text{kV}/\text{cm}\), the mobility of positive ions in air at atmospheric pressure, Rogowski estimated this time as \(10^{-4}—10^{-5}\ \text{sec}\).
Somewhat later these assumptions concerning the velocity of motion of positive ions in the discharge were confirmed by the experiments of Lawrence and Dennington\(^{21}\), who measured the velocities of motion of positive ions in a discharge between plates by means of a Kerr shutter and a spectrograph. The authors measured the time from the beginning of the discharge to the appearance of lines associated with the material of the electrodes, at a given distance from the electrode. An alternating 60-cycle voltage of \(25\ \text{kV}\) was applied to the electrodes, with a distance between the electrodes of \(6\ \text{mm}\). The velocities of the positive ions of zinc, cadmium, and magnesium proved to be from \(10^{5}\) to \(2\cdot 10^{5}\ \text{cm}/\text{sec}\).
Thus, the delay time of the discharge should be \(10^{-4}—10^{-5}\ \text{sec}\). However, volt-second oscillograms taken by Rogowski\(^{20}\), Tamm\(^{18}\), and Rogowski, Flegler, and Tamm\(^{19}\), for such a case with static voltage, showed that the delay time of the discharge is \(10^{-7}\), or several units times \(10^{-7}\ \text{sec}\); the discharge itself takes place in a time less than \(10^{-8}\ \text{sec}\) (approximately \(3\cdot 10^{-9}\ \text{sec}\)). With an overvoltage of \(30\%\), the delay time decreases to \(10^{-8}\ \text{sec}\). In Fig. 6 two such oscillograms are shown. In such a short interval of time the positive ions practically cannot have time to move from their place.
These data of Rogowski on the small delay time and on the extremely rapid development of the discharge agree with the results of works by other authors\(^{15,\,21—23,\,31—33,\,33a,\,56,\,84}\).
Tippel and Frank\(^{55}\) believe that, for the formation of a space charge, there is no need at all for the assumption of the motion of positive ions, which, as is evident, was not confirmed by experiment. The formation of a space charge and the small delay times can readily be explained in the following way. An electron, torn from the surface of the cathode by an external source, moving toward the anode in a uniform field, forms an avalanche of positive ions equal to \(e^{\alpha L}-1\), where \(L\) is the distance between the electrodes. Since the electrons
move a thousand times faster than positive ions, it must practically be assumed that, during the motion of the electrons, the positive space charge remains in place.
The avalanche of electrons, having reached the anode, will be absorbed by it. The remaining positive space charge will evidently also be distributed exponentially as the distance from the cathode increases. Thus, almost the entire positive charge will be concentrated near the anode. The potential difference between the cathode and this space charge, according to the calculations of Hippel and Frank, will be
\[ U_1=\frac{4\pi \varepsilon}{\alpha_1 \Delta f_1}\left(e^{\alpha_1 L}-1\right), \]
where \(\varepsilon\) is the charge of the electron, \(\Delta f_1\) is the cross-sectional area of the avalanche, which is determined from the following considerations. If an external source tears from the cathode a stream of electrons that give a current \(I\ \mathrm{A/cm^2}\), then, in order to tear one electron from the electrode surface in the time \(\tau_1\), the channel cross-section must be
\[ \Delta f_1=\frac{\varepsilon}{I\tau_1}. \]
The first avalanche somewhat changes the distribution of potential in the discharge gap. The new potential distribution is shown in Fig. 7, where curve \(A\) is the initial potential distribution, curve \(B\) is the true potential distribution after passage of the first avalanche, and curve \(C\) is the calculated potential distribution after passage of the first avalanche. Replacing the true curve \(B\) by the calculated curve \(C\) and taking a new \(\tau_2\) (ten times smaller than \(\tau_1\)), one can carry out the same calculation with the second avalanche.
Fig. 7. Potential distribution in the discharge gap in the process of discharge development according to the theory of Hippel and Frank
For the case of a uniform field with a gradient of \(31.7\ \mathrm{kV/cm}\), distance between electrodes \(1\ \mathrm{cm}\), \(\tau_1=10^{-7}\ \mathrm{s}\), \(I=1.5\cdot10^{-15}\ \mathrm{A/cm^2}\), Hippel and Frank obtain \(\Delta f_1=1\ \mathrm{mm^2}\) and \(U_1=1.06\ \mathrm{kV}\). The second electron, torn from the cathode surface in an interval of \(10^{-8}\ \mathrm{s}\), gives a channel cross-section \(\Delta f_2=10\ \mathrm{mm^2}\) and \(U_2=2.5\ \mathrm{kV}\); the third, in the same time \((10^{-8}\ \mathrm{s})\), gives \(U_3=37\ \mathrm{kV}\). Thus, three successive initial electrons are more than sufficient to form a powerful space charge, which will fill the whole gap along its length and will reach almost to the cathode itself. The delay time is a statistical time—the time necessary for the appearance of the first electron and the formation of the first avalanche; it is \(10^{-7}\ \mathrm{s}\). The time necessary for the motion of the electrons from cathode to anode is \(10^{-8}\ \mathrm{s}\). The time for the formation of the discharge and completion of breakdown will be of the same order. As can be seen, the calculations carried out...
obtained by Gippel and Frank give the same values for the delay and formation of the discharge as were obtained by Rogowski and his students from volt-second oscillograms.
Gippel and Frank’s assumption about the mechanism of formation of the space charge is confirmed by the experiments of Gamos[^23]. With the aid of a Kerr shutter, Gamos photographed the glow of the discharge between two small spheres at a distance of 5 mm. Already \(2 \cdot 10^{-9}\) sec after the voltage is applied, a glow appears at the anode with strongly broadened lines owing to the Stark effect, which, in the author’s opinion, is caused by the presence of a large space charge of positive ions. As the discharge develops, this glow spreads from the anode to the cathode.
Thus, Gippel and Frank’s idea of the rapid formation of a space charge sufficient for the discharge is based on two assumptions: on the rapid motion of electrons, which leave behind them a positive space charge, and on the sufficiency of a relatively small number of successive initial electrons.
Gippel and Frank’s theory explains well the rapid formation of space charge, but it cannot explain all the phenomena of discharge in gases. Even in explaining the simple case considered by the authors, there are assumptions that hardly agree with the facts. A channel cross-section of \(10\ \mathrm{mm}^2\) is too large. According to the experiments of White[^56], Dunnington[^22], Buss[^57], Tepler[^58], Slepyan[^59], and the calculations of Ollendorff[^60], the channel cross-section is no more than several tenths of a square millimeter. If the channel cross-section \(\Delta f\) is reduced, the delay and formation time of the discharge will increase. The electron velocity \(10^8\ \mathrm{cm/sec}\) adopted by Gippel and Frank is also too large. Theoretically, the electron velocity under these conditions is determined by a value of the order of \(10^7\ \mathrm{cm/sec}\), which is confirmed by Rezer’s experiments in a Wilson chamber4. Moreover, in long sparks the electrons are not absorbed by the anode, and they cannot be neglected in calculating the field of the space charges. In such a simple form, Gippel and Frank’s calculation is not applicable to them1.
A more rigorous theory of the discharge, based on the action of the space charge alone, was developed by Verne, White, Loeb, and Pozin[^62], and by Rogowski and Vollrath[^63]. That a space charge takes place in the discharge space follows at least from the following. Breakdown between points occurs at voltages that give average gradients too small, considerably smaller than those at which any ionization whatever can be noticeable. Breakdown in such a field can be explained only by postulating the movement of large gradients, existing at the points, toward the middle of the gap. The latter can be caused only by the formation of a space charge.
In addition to the experiments of Gamos cited above, the existence of space charge in the discharge is confirmed by analogous experiments of Lawrence and Dunnington[^21]. Lawrence and Dunnington recorded
spectrum of the discharge between two plates with the distance between them equal to 6 mm. From the broadening of the spectral lines (45 Å), caused by the presence of an ionic field, the authors determined the degree of ionization of the air molecules to be 33%.
Calculations by Verney, White, Loeb, and Posin \(^{64}\), taking into account the influence of space charge, showed that the presence of a space charge formed solely by primary electron ionization in the gap between two flat electrodes can indeed cause a curvature of the straight line \(\lg \dfrac{i}{i_0}\) as a function of \(L\), and eventual spark breakdown. This curvature depends on the distance between the electrodes, the pressure, the voltage, and the intensity of irradiation. The probability of such phenomena as a result of primary electron ionization alone is determined by the form of \(f\left(\dfrac{E}{p}\right)\), which gives the value \(\dfrac{\alpha}{p}\). If \(f\left(\dfrac{E}{p}\right)\) has a power-law form with an exponent greater than unity, then distortion of the field caused by the space charge will occur. The larger the exponent, the greater the distortion of the field and the breakdown for this reason. Further calculations by Verney \(^{64}\) for nitrogen at atmospheric pressure and a uniform field with gradients of 30 kV/cm showed that, in the presence of primary-electron ionization alone and with a density of the initial photoelectric current from an external source \(i_0 = 5 \cdot 10^{-12}\ \mathrm{A}/\mathrm{cm}^2\), breakdown will occur when the distance between the flat electrodes is 1 cm. When the density of the initial current is reduced to \(i_0 = 5 \cdot 10^{-15}\ \mathrm{A}/\mathrm{cm}^2\) and \(i_0 = 5 \cdot 10^{-17}\ \mathrm{A}/\mathrm{cm}^2\), the distance between the electrodes must be increased respectively to 1.7 and 2 cm. In reality, under the above conditions, breakdown always takes place at a distance between the flat electrodes of 1 cm, and no changes in the breakdown voltage were observed when the value of the initial current from the external source ranged from \(10^{-17}\) to \(10^{-12}\ \mathrm{A}/\mathrm{cm}^2\) \(^{35,41,65}\). Increasing the density of the initial current from the external source above \(10^{-12}\ \mathrm{A}/\mathrm{cm}^2\) leads to a decrease in the breakdown voltage. According to the theory of Schade \(^{37}\), Rogowski, Fuchs, and Vollrath \(^{12,34,66,67}\), which takes into account not only the formation of space charge but also secondary processes, the decrease in the breakdown voltage must be proportional to the square root of the density of the initial current from the external source,
\[ \frac{\Delta U}{U}\bigg|_0 = k\sqrt{i_0}. \]
Here
\[ \Delta U = U - U_a, \]
where \(U\) is the breakdown voltage without irradiation, and \(U_a\) is the breakdown voltage under irradiation.
The experiments of Rogowski and Vollrath \(^{35}\), Schade \(^{37}\), and the especially careful experiments of Fuchs and Bongards \(^{41}\) and Brinkmann \(^{68,69,39}\) confirmed this theory.
Thus, the theoretical and experimental works of the authors mentioned above make it possible to conclude that, owing to the action of secondary processes, breakdown in reality occurs long before the space charge formed solely
primary electron ionization reaches such values at which breakdown would have to occur.
Rogowski and Vollrath\(^ {63}\), in their theoretical work devoted specifically to the question of whether breakdown occurs as a result of the action of primary electron ionization alone or under the action of secondary processes, arrive at the same conclusion. The authors concluded that, for a given mean gradient \(E_m\), breakdown can occur under the action of the space charge formed by primary electron ionization alone only when the density of the initial current caused by an external source is not less than some definite value \(i_{0m}\). Only when the initial current reaches this critical value do the processes in the discharge become unstable and lead to breakdown.
If one takes from experiment the value of the mean gradient at which breakdown occurs and calculates the critical values of the initial current from an external source \(i_{0m}\), then for distances between plane electrodes \(L\) the following data are obtained:
| \(L\), cm | 0.01 | 0.1 | 1 | 10 |
|---|---|---|---|---|
| \(i_{0m}\), A/cm\(^2\) | \(8.1\cdot10^{-6}\) | \(3.6\cdot10^{-8}\) | \(4.8\cdot10^{-12}\) | \(1.9\cdot10^{-23}\) |
For small distances, values that are much too large are obtained. In reality, breakdown occurs at values of the initial current from an external source even of the order of \(10^{-16}\)—\(10^{-17}\) A/cm\(^2\). This means that, at small distances, besides space charge, other processes also contribute to the formation of breakdown. For large distances the question still remains uncertain.
Fig. 8.
Further, Rogowski and Vollrath\(^ {63}\), proceeding from two assumptions, determined the magnitude of the lowering of the breakdown voltage when the discharge gap is irradiated by a source producing ionization in it. They found that, if only the action of the space charge formed by primary electron ionization is taken into account, the lowering of the voltage should be approximately proportional to the increment of the density of the initial current caused by the external source:
\[ \left.\frac{\Delta U}{U}\right|_{0}\sim \frac{\Delta i_{0}}{i_{0}} . \]
If the combined action of space charge and secondary processes is taken into account, the lowering of the breakdown voltage should be proportional to the square root of the density of the initial current caused by the external source:
\[ -\frac{\Delta U}{U}\sim \sqrt{i_{0}} . \]
The experiments performed, as has already been mentioned above, convincingly showed that the second assumption is correct. Figure 8 gives a graph borrowed from the work of Fucks and Bontars\(^ {41}\).
Along the ordinate axis is plotted the decrease of the voltage in percent; along the abscissa, the square root of the density of the initial current produced by the external source. Breakdown was produced between spheres in air at atmospheric pressure. The three curves correspond to three different distances between the spheres or to different initial voltages. The gap was irradiated by a quartz-mercury arc. The initial photoelectric current \(i_0\) was measured with a sensitive galvanometer with amber insulation. From the graph it is seen that even at large distances the second supposition of Rogowski and Wolff, concerning the joint action of the positive space charge with the secondary processes, was brilliantly confirmed. From all that has been said above it follows that the formation of a positive space charge can lead to a discharge only in the presence of a continuous primary current produced by an external source. Such a discharge will not be self-sustained. In reality, in the presence of a continuous primary current produced by an external source, the positive space charge formed by it very strongly promotes the development of the discharge, but breakdown actually occurs through the joint action of the space charge and the secondary processes.
4. SECONDARY PROCESSES
It was indicated above that ionization by positive ions, as a secondary process in discharges, under ordinary conditions falls away\(^{1}\). Even before the works on the influence of the space charge on the development of the discharge, Townsend proposed another mechanism of secondary processes. Positive ions located near the cathode in the region of large gradients, under the action of the strong field, move toward the cathode. Having reached the surface of the cathode, they strike it and tear electrons out of it. These electrons, formed behind the already passed electron avalanche by the products of ionization of this avalanche, may, if sufficiently numerous, cause a self-sustained discharge. The current flowing between plane electrodes in this case will be equal to
\[ i = i_0 \frac{e^{\alpha L}}{1-\gamma\left(e^{\alpha L}-1\right)}, \tag{4} \]
where \(\gamma\) is the number of electrons torn from the surface of the cathode per one pair of ions formed in the discharge. Since the mechanism of secondary processes under consideration was proposed by Townsend before Rogowski’s work with the oscillograph, Townsend believed that all positive ions arrive at the cathode. Therefore, by \(\gamma\) he understood the number of electrons torn from the cathode by one positive ion. This is how the coefficient \(\gamma\) is usually interpreted. However, since not all ions manage during breakdown to reach the cathode, one must either introduce a certain coefficient taking into account that fraction of the ions which does manage to reach
\(^{1}\) Under unusual, exceptional conditions—at very low pressures and very high voltages—ionization by positive ions may have a predominant significance in the discharge\(^{70–72}\).
to the cathode, or assign \(\gamma\) to one pair of ions formed in the discharge space.
The condition for the transition of a non-self-sustained discharge into a self-sustained one will be
\[ 1=\gamma(e^{\alpha L}-1). \tag{5} \]
In addition to the above, a number of authors have proposed other mechanisms of secondary processes: photoionization of the gas, the ejection of electrons from the cathode surface by the photoelectric effect, and ionization by metastable atoms through collisions of the second kind.
When an electron collides with atoms, besides ionization, excitation of the atom also occurs. Since excitation involving the transfer of electrons of the outer shells of the atom requires less energy than ionization, such processes are more probable. Returning to the normal state, the atom emits radiation. Recombination of ions is also accompanied by radiation; the radiation propagates in all directions. Part of it is absorbed by the gas itself, and part of it reaches the cathode surface and ejects electrons from it.
In addition to the photoelectric effect at the cathode surface, the radiation may also produce photoionization of the gas itself. Photoionization of the gas is considerably less probable than the photoelectric effect at the cathode surface, since the ionization potential of any gas is considerably higher than the work function. Atoms of the same gas can be ionized only by radiation emitted during recombination or in the case when the atom has been excited as a result of an electron transition from a low energy level. The latter circumstance is very unlikely. In mixed gases, when the ionization potential of one gas is lower than that of the other, even small admixtures of the first gas may be sufficient for photoionization of the gas to serve as a mechanism of secondary processes.
Ionization by metastable atoms through collisions of the second kind is also important in mixed gases, when the excitation energy of a metastable level of one gas is higher than the ionization potential of the second. Since the lifetime of metastable levels is long—from \(10^{-4}\) to \(10^{-2}\) sec—and since metastable atoms, owing to diffusion, are capable of moving against the field, their role in the development of the discharge may be very considerable. Here too, small admixtures of another gas may very strongly promote the development of the discharge.
If the photoelectric effect at the cathode is the mechanism of the secondary processes, then the current flowing between plane electrodes will be
\[ i=i_0\frac{e^{\alpha L}}{1-\frac{\eta\sigma\vartheta}{\alpha-\nu}\left[e^{(\alpha-\nu)L}-1\right]}, \tag{6} \]
where \(\vartheta\) is the number of atoms excited by one electron over \(1\ \mathrm{cm}\) of path in the direction of the field, \(\nu\) is the coefficient of absorption of radiation in the gas, \(\sigma\) is the fraction of all the radiation that reaches the cathode surface, and \(\eta\) is the electron yield in the photoelectric effect, i.e. the number of electrons ejected by one photon.
The condition for the transition of a non-self-sustained discharge into a self-sustained one will have the following form:
\[ 1=\frac{\eta\sigma^{\beta}}{\alpha-\nu}\left[e^{(\alpha-\nu)L}-1\right]. \tag{7} \]
Loeb\({}^{64}\) showed that there is no essential difference between equations (2), (4), and (6). In practice these equations reduce to one, and consequently expressions (3), (5), and (7) also reduce to one form. Indeed, if one sets
\[ \gamma'=\frac{\beta}{\alpha-\beta}, \]
\[ \gamma''=\frac{\eta\sigma^{\beta}}{\alpha-\gamma}, \]
then equations (2) and (6) become
\[ i=i_0\frac{e^{\alpha L}}{1-\gamma'\left[e^{\frac{\alpha}{1-\gamma'}L}-1\right]}, \tag{2a} \]
\[ i=i_0\frac{e^{\alpha L}}{1+\gamma''\left[e^{(\alpha-\gamma)L}-1\right]}. \tag{6a} \]
If, further, \(\gamma'\) is neglected in comparison with 1, and \(\nu\) in comparison with \(\alpha\), then all these equations will formally be identical and similar to equation (4), while expressions (3) and (7) will be similar to (5). Expression (5) can be interpreted physically as follows. If a free initial electron is formed at the cathode surface, then, on reaching the anode, it forms on its path \((e^{\alpha L}-1)\) ion pairs. If for each ion pair, by some mechanism of secondary processes, \(\gamma\) electrons are torn from the cathode surface, then for each initial electron the secondary processes will form, at the cathode surface,
\[ \mu=\gamma\left(e^{\alpha L}-1\right) \tag{8} \]
electrons. If \(\mu<1\), then the number of initial electrons will gradually decrease and the discharge will die out. If \(\mu>1\), then the initial number of electrons will gradually increase and the discharge will develop. At \(\mu=1\) and \(1=\gamma(e^{\alpha L}-1)\), an equilibrium state sets in, and at the same time a state in which the non-self-sustained discharge passes into a self-sustained one.
If the secondary processes occur in the bulk of the gas, and not at the cathode surface, then formally mathematically the process can be described as if they occurred at the cathode surface\({}^{67}\).
Electrons formed in the gas by photoionization or by ionization by positive ions are equivalent to some smaller number of electrons formed at the cathode surface. Indeed, if in a unit volume of gas in 1 sec, for each ion pair, secondary processes form \(N_0\) electrons, then, moving toward the anode, all the electrons formed in the gas will produce an avalanche of electrons equal to
\[ N=N_0\int_0^L e^{\alpha(L-x)}\,dx=\frac{N_0}{\alpha}\left(1-e^{-\alpha L}\right)e^{\alpha L}. \]
If the same number of electrons were formed at the surface of the cathode, then, upon reaching the anode, they would produce an avalanche of electrons \(N_0Le^{aL}\). In order for the avalanches to be the same, a smaller number of initial electrons \(\chi N_0L\) must be formed at the cathode, where \(\chi < 1\). One electron formed in the gas is equivalent to \(\chi=\dfrac{1}{aL}(1-e^{-aL})\) electrons formed at the surface of the cathode.
If the value \(\chi\) is included in the coefficient \(\gamma\), then expression (5) remains valid in all cases, independently of the mechanism of the secondary processes. Hence it is clear that the mechanism of the secondary processes must be concealed in the value and character of the coefficient \(\gamma\).
Fig. 9. Dependence of \(\gamma\) on \(\dfrac{E}{p}\)
○ — Bowls (pure nitrogen); △ — Bowls (nitrogen contaminated with mercury vapor); ● — Posin (nitrogen)
From experiments in which the values of \(a\) and \(\beta\) were found, the values of \(\gamma\) can be determined, if these experiments are carried out under conditions in which the formation of space charge is negligibly small.
Posin’s experiments\(^8\) are irreproachable in this respect. In Fig. 9 is shown the curve obtained by Posin for the dependence of \(\gamma\) on \(\dfrac{E}{p}\) for nitrogen. With increasing \(\dfrac{E}{p}\), the value of \(\gamma\) increases monotonically. Such a character of the curve for \(\gamma\) as a function of \(\dfrac{E}{p}\) is to be expected if the mechanism of the secondary processes consists in the extraction of electrons from the surface of the cathode by bombardment with positive ions. The higher the gradient, the greater the velocity of the positive ions, and the more electrons they must extract from the surface of the cathode. Experiments carried out by Bowls\(^ {10}\) after Posin did not, however, confirm the character of the \(\gamma\) curve. As was indicated above, Bowls took special measures so that mercury vapor would not penetrate into the discharge gap with pure nitrogen and would not settle on the surface of the cathode. The curve obtained in this case, as is seen in Fig. 9, for \(\gamma=\dfrac{\beta}{a}\) has an entirely different form. The curve lies considerably higher and has a very sharp and high maximum. When mercury vapor was admitted into the chamber, the values obtained for \(\gamma\) were the same as those of Posin.
The same result was obtained by Hale\(^ {11}\). By the same method as Bowls, Hale found values of \(\gamma\) for pure hydrogen. Hale determined the values of \(\gamma\) from expression (4). As is seen from Fig. 10, the curve for \(\gamma\) as a function of \(\dfrac{E}{p}\) in pure hydrogen has a maximum.
Both authors believe that such a character of the curves can be explained by the photoelectric effect at the surface of the cathode. The decrease in the value
The contamination of a gas by mercury vapor is due to the absorption of radiation by mercury vapor.
Schäfer’s experiments\(^ {73}\) for several gases with electrodes made of different materials showed that \(\gamma\) either does not depend at all, or depends only very weakly, on \(\dfrac{E}{p}\). The same author investigated the discharge in argon, hydrogen, and air occurring in a cylindrical condenser. The diameters of the cylinders were 1 and 2 cm. Upon reversal of polarity the breakdown voltage changed hardly at all (for air, \(11\,100\) as against \(11\,000\) V at \(p=500\) torr; for hydrogen, \(5\,930\) as against \(5\,950\) V at \(p=480\) torr); consequently, the coefficient \(\gamma\) did not change either, whereas the gradient at the cathode changed very appreciably.
Fig. 10. Dependence of \(\gamma\) on \(\dfrac{E}{p}\) for pure hydrogen according to Keil’s experiments
All these experiments indicate that the mechanism of electron extraction which was usually ascribed to ion bombardment unquestionably does not take place. The mechanism of extracting electrons from the metal surface was usually explained as follows. An ion, striking the surface of the metal, transfers to it a large amount of thermal energy. Because of the short duration of the process, the heat does not have time to propagate over any appreciable distance. Therefore the place where the ion strikes is heated to a very high temperature. From this place electrons begin to evaporate owing to thermionic emission. The greater the ion velocity, the higher the temperature of the emitting region and the more electrons are emitted. As we see, the experiments of Bowles, Keil, and Schäfer did not confirm the existence of such a mechanism. However, this still does not prove that the extraction of electrons from the cathode surface takes place without the participation of positive ions.
A number of experiments carried out by Güntherschulze, Behr, and Winter\(^ {74-76}\) led the authors to the conclusion that the extraction of electrons from the cathode surface in their experiments occurs not by impact, but by the capture of electrons by positive ions. This conclusion of the authors, however, is not especially convincing, since the results they obtained can also be explained by the photoelectric effect at the cathode.
Thus, the experiments already cited above give a number of indirect proofs that the photoelectric effect at the cathode is the mechanism of secondary processes in the discharge. However, there are also direct experiments confirming this. First of all it must be shown that the radiation of the discharge has photoionizing properties.
Although the photoionizing action of spark radiation has been known since the time of Hertz, we shall nevertheless cite two studies on the properties of this radiation carried out during the last two years.
One of these works was Brinkman’s^68 study of the photoelectric effect on the surface of a metal; the second was Rezer’s^77 study of the photoionization of a gas produced by spark radiation. Brinkman^68 investigated the radiation of a spark between two spheres made of different materials, in air at atmospheric pressure and at pressures considerably below and above atmospheric pressure. The spark illuminated a flat electrode, having the shape of a Rogowski surface, to which a string electrometer on amber insulation was connected. The second electrode was a grid through which the radiation from the spark passed. Brinkman found, in agreement with all preceding works, that spark radiation is indeed capable of producing a photoelectric effect on the surface of the electrode; moreover, the number of electrons torn from \(1\ \mathrm{cm}^2\) of surface at a distance from the spark to the electrode of \(10\ \mathrm{cm}\) can reach \(2\cdot10^9\). The radiation is not monochromatic; the most short-wave part is very strongly absorbed by air. The spectrum of the radiation depends on the material of the spark electrodes. The yield of photoelectrons depends on the material of the irradiated electrode and on the distance to the spark.
Figs. 11 and 12. Photographs obtained by Rezer in a Wilson chamber. The arrow indicates the position of the slit through which the chamber was irradiated by the spark.
When the spark is moved away from the electrode, the intensity of the photocurrent falls either inversely proportionally to the square of the distance, or somewhat more strongly, depending on the radiation spectrum and the material of the electrodes.
Rezer^77 investigated in a Wilson chamber the ability of spark radiation to ionize a gas. Figs. 11 and 12 show the experimental arrangements and photographs obtained by Rezer in the Wilson chamber. The slit in the Wilson chamber, through which the chamber was irradiated, was positioned perpendicular to the spark and was covered with a celluloid film \(7.5\ \mu\) thick. As can be seen from Figs. 11 and 12, all tracks begin on a straight line running from the slit. This proves that all the tracks were produced by initial electrons torn from the atoms of the gas as a result of photoionization. From the decrease in the number of tracks per unit length, Rezer determined the absorption of the radiation.
tion for the given gas. For air the absorption coefficient is equal to \(1.8—2.2\ \text{cm}^{-1}\), and for oxygen \(5\ \text{cm}^{-1}\), reduced to atmospheric pressure. From a comparison of these absorption coefficients the author comes to the conclusion that in air both oxygen and nitrogen are ionized. From the ionization energy of the gas the author estimates the wavelength in a discharge in air as \(1\,000\ \text{Å}\), and from the absorption coefficient and the solid angle, the total number of quanta from the whole spark at atmospheric pressure as \(10^7—10^8\). From the duration of the rectangular voltage wave on the chamber and from the length of the tracks, knowing the rate of their formation from his previous works, the author determines the time of emission and establishes that it occurs when the voltage on the spark falls before its transition into an arc. Since all the tracks are of approximately equal length, Rezer concludes that the radiation occurs over a sufficiently short interval of time.
Fig. 13. Diagram of Costa’s apparatus
\(Q\)—light from a mercury-quartz arc, \(P\)—to the pump, \(G\)—mirror galvanometer, \(E\)—electrometer, \(a\)—amber
White\(^{56}\), on the basis of his own experiments and Dennington’s\(^{22}\) experiments, found that the spark already begins to emit noticeably \(2\cdot10^{-9}\) sec after the beginning of the sharp drop of voltage across the discharge gap.
From these works one may draw the following indisputable conclusion. The radiation emitted by the spark already in the very initial period of its formation is capable of producing a very intense photocurrent from the surface of the electrode and of quite effectively producing photoionization of the very same gas in which the discharge occurs. It would be quite incomprehensible if the radiation of the discharge exerted a strong action on the development of a discharge in an identical discharge gap located next to it, and did not exert the same action on the development of its own discharge. Thus, there is no reason to doubt that the photoionizing action of its own radiation promotes the development of the discharge. The question is to what extent the photoionizing action of the discharge radiation is the principal mechanism of the secondary processes.
Costa\(^{78}\) gives an experimental answer to the question posed above. Costa’s apparatus is shown in Fig. 13. The electrode is illuminated by a quartz-mercury arc. In the absence of electron ionization in the gap, the initial current will be \(i_0\). When a certain higher voltage is applied between the electrodes, the current between them will be
\[ i=i_0\frac{e^{\alpha L}}{1-\gamma(e^{\alpha L}-1)}=(i_0+i_0')e^{\alpha L}, \]
where \(i'_0\) is the additional photocurrent caused by the photoeffect on the surface of the electrode due to its own radiation. Measuring \(i\) and \(i_0\) with a mirror galvanometer and determining in the usual way the value of \(a\) for the given conditions, one can determine \(i'_0\), provided that no space charge is formed in the space between the electrodes. The latter circumstance was checked experimentally by Kosta. Hence
\[ i'_0=\frac{i}{e^{aL}}-i_0. \]
A small positive potential was applied to the grid \(S_2\), which impeded the penetration of charges owing to diffusion from one space into the other.
The radiation from the discharge between the electrodes \(K_1A_1\) passes through the grids, enters the space between the electrodes \(K_2A_2\) and reaches the electrode \(K_2\), where it produces a photocurrent \(i''_0\). This current will be amplified as a result of electron ionization in the gap between the electrodes \(K_2A_2\):
\[ i'=i''_0 e^{aL}. \]
Measuring \(i'\) with a string electrometer, Kosta could determine \(i''_0\). The author then determines the absorption of the radiation in the gas and in the grids, and the smaller solid angle under which the electrode \(K_2\) is seen from the space \(K_1A_1\); from these data one can calculate \(i'''_0\)—the photocurrent between the electrodes \(K_2A_2\), if both gaps were under identical conditions.
The ratio
\[ \frac{i'_0}{i'''_0} \]
for \(L\) equal to from 1 to 2 cm, voltages from 150 to 400 V/cm, and pressure \(p=0.8\) torr, varied from 1.16 to 2; for \(p=7.1\) torr—from 1.43 to 2.73. Since the accuracy of the measurements was, for \(i'_0\), from 10 to 65%, and for \(i'''_0\)—8%, Kosta considers that the two currents are identical. From this Kosta concludes that in the discharge, in secondary processes, the photoeffect plays the principal role. Kosta further notes that already in the very initial stage of a non-self-sustained discharge the current \(i''_0\) is proportional to the current \(i\), and, consequently, radiation with photoionizing properties is already present at that time. The coefficients \(a\), \(\gamma\), and \(\nu\) obtained by Kosta were in complete agreement with the data of Posin, Rezer, and other authors.
Above was cited the work of Schaefer \(^{73}\), in which it was found that the coefficient \(\gamma\) does not depend on \(\dfrac{E}{p}\), in particular also for air (experiments on the study of discharge in a cylindrical condenser). On the other hand, in the same work Schaefer, studying discharge in air in a uniform field, found that Paschen’s law is obeyed very accurately. Loeb \(^{64}\) showed that Paschen’s law follows from expression (5), if one assumes that \(\gamma=\varphi\!\left(\dfrac{E}{p}\right)\). Consequently, the validity of Paschen’s law for air simultaneously denotes the dependence of \(\gamma\) on
\[ \frac{E}{p}. \]
Schade \(^{79}\) showed that these two contradictory facts are in reality consistent with one another, if one takes into account the dependence of the absorp-
tion of radiation on the gas pressure. The dependence of the coefficient \(\gamma\) on \(\frac{E}{p}\) also does not contradict its physical interpretation as a coefficient taking into account processes caused by the photoelectric effect.
For the case when the secondary processes are due to the photoelectric effect,
\[ \gamma=\frac{\eta\sigma\vartheta}{\alpha-\nu}. \]
\(\vartheta\)—the number of atoms excited by one electron over \(1\ \mathrm{cm}\) of path in the direction of the field—will also depend on the gradient and the gas pressure. \(\frac{\vartheta}{p}\), like \(\frac{\alpha}{p}\), is a function of \(\frac{E}{p}\), i.e. \(\frac{\vartheta}{p}=\psi\!\left(\frac{E}{p}\right)\). The absorption of radiation depends on the pressure and is directly proportional to it. Therefore one may put \(\nu=\nu_0 p\). Then
\[ \gamma= \frac{\eta\sigma\,\dfrac{\vartheta}{p}} {\dfrac{\alpha}{p}-\nu_0} = \frac{\eta\sigma\psi\!\left(\dfrac{E}{p}\right)} {f\!\left(\dfrac{E}{p}\right)-\nu_0} = \varphi\!\left(\eta,\sigma,\nu_0,\frac{E}{p}\right), \tag{9} \]
where \(\eta\), \(\sigma\), and \(\nu_0\) are parameters. Hence it is clear that (depending on the values of the parameters and of the functions \(\psi\) and \(f\)) \(\gamma\) may be any function of \(\frac{E}{p}\)—increasing, decreasing, or remaining practically constant.
Expression (5) will have the following form:
\[ 1=\varphi\!\left(\eta,\sigma,\nu_0,\frac{E}{p}\right) \left(e^{\,f\left(\frac{E}{p}\right)pL}-1\right) = \varphi\!\left(\eta,\sigma,\nu_0,\frac{U}{pL}\right) \left(e^{\,f\left(\frac{U}{pL}\right)pL}-1\right) \]
or
\[ \Phi\!\left(\eta,\sigma,\nu_0,\frac{U}{pL},\,pL\right)=1. \tag{10} \]
The last equation expresses Paschen’s law.
In deriving equation (6) it was assumed that \(\sigma\) is a constant independent of \(x\) (the distance between the cathode and the element under consideration), and in the integration \(\sigma\) was taken outside the integral sign.
Properly speaking, \(\sigma\) denotes the solid angle under which, from the point where the radiating atom \((A)\) is located, the part of the cathode surface \(S\) cut out by the discharge channel \(K\) is visible (Fig. 14).
At small distances between the electrodes and low pressures, when the discharge covers the entire cathode surface, \(\sigma\) may practically be regarded as constant. In a discharge at atmospheric pressure or somewhat lower, the discharge channel is very small and the length of the channel is considerably greater than the diameter of its cross section. Then \(\sigma=\frac{K}{x^2}\), where \(K\)—the cross section of the channel—may also depend on \(x\). The equation becomes more complicated. Whether Paschen’s law remains valid in this case will be shown by an additional check.
Fig. 14
5. THE THEORY OF HIPPEL AND ROGOWSKI
Replacing equation (2) by equation (6) does not eliminate all the shortcomings possessed by the discharge theory that takes into account only the primary and secondary processes discussed above. From the standpoint of this theory, the phenomena of discharge lag, the dependence of the breakdown voltage on the strength of the initial current caused by an external source, and the voltage drop during breakdown with a simultaneous increase of the current still remain unexplained.
The chief merit of this theory—that expression (7) gives the correct value of the breakdown voltage, agreeing with experimental data—is, as Leb\(^80\) justly observes, purely accidental.
Neither physically nor mathematically does equation (6), when equality (7) is satisfied, give a condition leading to a completely definite breakdown voltage. Mathematically, equation (6), when equality (7) is satisfied, gives only a certain indeterminacy. Physically, equality (7) must be interpreted only as a condition determining the equilibrium state of the discharge. It means that, for given values of \(E, L, p, \eta, \sigma\), and \(\nu_0\), for each initial electron one electron, referred to the cathode surface, is produced by secondary processes. Such a state of the discharge may lead to breakdown, but this is not necessary. The fulfillment of condition (7) is necessary, but not sufficient. Whether the discharge ends in breakdown or dies out depends also on a number of accidental circumstances. Therefore a voltage satisfying condition (7) may still not lead to breakdown. Practically, if the magnitude of the voltage is such that condition (7) is only just satisfied, i.e. that
\[ \mu=\frac{\eta\sigma\frac{\vartheta}{p}}{\frac{\alpha}{p}-\nu_0}\left(e^{\alpha L}-1\right) \]
only slightly exceeds unity, breakdown occurs very rarely. When the voltage is increased and, consequently, when \(\mu\) becomes greater than unity, the probability that breakdown will occur increases. The greater \(\mu\) is, the greater this probability. When \(\mu\) is sufficiently large, breakdown practically occurs in every case when voltage is applied. Thus, even upon applying a voltage greater than the minimum voltage that determines the equilibrium state of the discharge, there exists only a certain probability that breakdown will occur. Therefore the breakdown voltage is not a completely definite quantity and, under completely definite and constant conditions, has no unique value. In practice, a conventional value of the breakdown voltage may be established in the following way: the breakdown voltage is taken to be that voltage which, under given definite and constant conditions, leads to a statistically determined percentage of breakdowns in a definite, prescribed interval of time. The fact that the voltage determined from condition (7) practically coincides with the experimentally measured break-
... by the voltage, is explained by the form of the function for \(\mu\). In the range of gradients at which breakdown occurs in air at atmospheric pressure, the expression for \(\alpha\) has the following form: \(\alpha=Ae^{BE}\), and \(\mu=\gamma(e^{Ae^{BE}}-1)\). From this expression it is seen that, within these limits, \(\mu\) is very sensitive to the voltage. A small increase in voltage very strongly increases the value of \(\mu\) and raises the probability of breakdown. The illusion is created that there exists some definite breakdown voltage; in practice, the breakdown voltage is taken to be that voltage at which \(\mu\) is always greater than unity.
Thus, the physical meaning which, as was assumed, is contained in expression (7) is incorrect, and then equation (6), subject to condition (7), loses all physical and mathematical meaning.
In this respect the theory of Hippel \(^{81}\) and Rogowski \(^{82,12}\) is beyond reproach. Rogowski developed this theory in especially great detail. Both authors advanced their theory long before the question of the physical nature of the coefficient \(\gamma\) had been resolved. In their theory they assumed that the mechanism of the secondary processes consists in the tearing of electrons from the surface of the cathode by positive ions. Rogowski \(^{63,67,82}\) persistently defended this point of view for a long time. However, as new data appeared concerning the physical nature of \(\gamma\), he revised his theory, taking into account photoionization of the gas and the photoelectric effect at the cathode surface as the mechanism of the secondary processes \(^{12}\). It is precisely in this last form that it will be presented here.
One of the basic elements in Rogowski’s theory is the allowance for the formation of space charge. Space charge formation proceeds according to the ideas of Hippel and Franck \(^{55}\). At a certain voltage applied to the discharge gap, an accidental electron arising in the gap will begin to move toward the anode and to ionize along its path. If, at the given voltage, \(\mu<1\), the discharge will not develop but, on the contrary, will begin to die out. By increasing the voltage one can reach such a value that \(\mu=1\).
The space charge formed by the avalanche from the first initial electron will be located mainly near the anode. The field will be compressed, \(L\) will decrease, the gradients at the cathode will rise, and at the anode, on the contrary, will decrease. Simultaneously with the ion avalanche, the electrons leave behind an avalanche of excited atoms. The radiation of these atoms is scattered in all directions and is partially absorbed; only a fraction of it reaches the cathode surface and, by the photoelectric effect, tears one electron out of it. This new electron moves in a somewhat stronger field and therefore ionizes more strongly. With the compression of the field and the increase of the gradient, \(\alpha L\) increases, despite the decrease of \(L\). With increasing \(E\), \(\gamma\) either remains constant or increases, and in rare cases decreases slightly. This leads to an increase of \(\mu\). Now, for each initial electron, more than one electron is produced by secondary processes at the cathode. In the discharge gap the ionization is intensified because of the increase...
number of initial electrons and because of the increase in the ionizing ability of the electrons \(\alpha\). The space charge, which grows from the anode toward the cathode, increases. The field is compressed still more. This leads to a still greater increase in \(\mu\), and so on.
Thus, at the beginning of the discharge, when \(L\) is large and \(a\) is small, the state of the discharge in which \(\mu\) becomes equal to unity is an equilibrium state, but an unstable one. If the discharge reaches this state, it must then develop further by itself.
From expression (8) it is seen that for \(a=0\) and for \(L=0\), \(\mu=0\). \(\mu\) becomes equal to zero twice and, consequently, must have the value unity at least twice. As the gradient increases, \(\alpha\) first grows rapidly, and then the rate of growth of \(\alpha\) slows down. As the field is compressed, \(\mu\) will first increase, reach a maximum, and then begin to decrease, again reaching the value unity. The discharge will at first develop rapidly; then the rate of development of the discharge will slow down. At \(\mu=1\) an equilibrium state of the discharge again sets in. If in this case the field continues to be compressed, then, because \(\mu\) becomes less than unity, the discharge will begin to die out; the space charge will begin to dissipate. The field will be stretched until \(\mu\) returns back to unity. If, at \(\mu=1\), for some reason the field begins to stretch, \(\mu\) will become greater than unity, ionization will increase, and the field will again begin to be compressed until \(\mu\) again becomes equal to unity.
Fig. 15. Development of the discharge according to the theory of Ginpel and Rogowski
\(I\)—stable state, \(II\)—unstable state
Thus, the second equilibrium state of the discharge is stable. Rogowski showed that the second equilibrium state of the discharge must be identified with a glow discharge or an arc. The transition from the first to the second equilibrium state takes place extremely rapidly and has the character of a catastrophe. Rogowski identifies this transitional process with breakdown.
If, taking \(L\) as given, one plots the dependence of the voltage \(U\) on \(L\) for the equilibrium state \(\mu=1\), then the curve shown in Fig. 15 is obtained. The region inside the curve characterizes the state of the discharge for \(\mu>1\), outside the curve—for \(\mu<1\).
The motion of a point on the graph depicts the development of the discharge process. Suppose that for some gap length \(L\) a voltage \(U\) corresponding to an equilibrium state is applied to it. This is characterized by point \(A\) on the graph. The discharge begins to develop, and the point begins to move along the graph. If one assumes an energy source of infinitely large power, then as the discharge develops the voltage will not fall. The point will move along a straight line parallel to the abscissa axis and will reach position \(B\)—the second equilibrium, but stable, state. Since in practice the source has limited power, as the development of
of the discharge, the voltage will fall. The point will describe the curve \(AC\).
In the state of a glow discharge or an arc, which is characterized by the position \(C\), the discharge will exist as long as there is a corresponding voltage in the gap (which may be established by selecting suitable resistances).
If the voltage, owing to depletion of the source, continues to fall, the point will pass into the region \(\mu < 1\), and the discharge will begin to die out.
If the power of the source is small and the voltage, as the discharge develops, falls very sharply, the point will move along the curve \(AD\). The discharge will die out before it has time to pass into an arc; breakdown will not occur.
Thus, the right-hand branch of the curve corresponds to an unstable equilibrium state, and the left-hand branch of the curve to a stable equilibrium state. The right-hand branch characterizes the beginning of breakdown, the transition of the discharge from a non-self-sustained to a self-sustained one; the left-hand branch characterizes the end of breakdown and its transition into a glow discharge or an arc.
According to Paschen’s law, on the abscissa axis one should plot not \(L\), but the product \(pL\). From Paschen’s law or equation (10) it follows that every gas and electrodes have a certain minimum voltage \(U_m\), below which discharge cannot occur under any circumstances.
With the development of the discharge, ionization in the gap will increase, and along with it the current flowing between the electrodes. Thus, according to Rogowski’s theory, during breakdown the current must increase with a simultaneous sharp fall of the voltage.
Rogowski, proceeding from these ideas, and under certain assumptions, sometimes insufficiently justified, analytically obtained expressions for a number of regularities of discharge in a gas:
6. SHERPZER’S THEORY
Rogowski’s theory, although free from a number of shortcomings inherent in Townsend’s theory, is still not complete.
As positive charges accumulate at the anode, the positive space charge begins to exert a reverse action on the electrons of subsequent avalanches that pass through it. Under the action of this space charge the electrons begin to be retarded. The probability of the formation of negative ions increases as the velocity of the electrons decreases. The electrons settle, forming negative ions, which move very slowly toward the anode (in practice almost immobile). Thus there is formed the so-called plasma or, in Sherpzer’s apt name \(^{83}\), an “electron swamp.”
Rogowski did not take into account the reverse action of the positive charge and the formation of plasma. This circumstance was taken into account by Sherpzer \(^{83}\).
Sherpzer’s theory \(^{83}\) is a supplement to Rogowski’s theory. At first the discharge develops according to Rogowski, until such a space charge is formed that begins to exert a noticeable retarding action on the electrons. This occurs before stabilization
discharge according to Rogowski’s theory. Stabilization of the discharge according to Scherzer is caused by recombination of ions in the plasma. As positive and negative charges settle at the anode, their recombination begins.
The probability of recombination for each ion is proportional to the number of ions of the opposite sign in a unit volume. Therefore the recombination of ions (the number of recombining pairs of ions per unit volume) will be proportional to the product of the ion densities of both signs:
\[ \frac{dN}{dt}=a_p n_+ n_- \simeq a_p n^2. \]
Ionization (the number of ion pairs formed per unit volume) is proportional only to the first power of the electron density,
\[ \frac{dn}{dt}\sim n_e. \]
Thus, as ionization increases and ions accumulate, their recombination will increase. A moment will come when the number of ion pairs formed in the discharge will be equal to the number that have recombined. Under these conditions the discharge will pass into an equilibrium state, breakdown will end, and the current flowing between the electrodes will become constant.
Fig. 16. Stabilization of the discharge according to Scherzer’s theory
Stabilization of the discharge occurs as a result of the following causes. If for some reason, in the equilibrium state, recombination increases, the ion density decreases, the plasma is rarefied, and the region where ionization occurs increases. This causes an increase in ionization. Conversely, if in the equilibrium state ionization increases, recombination will increase to a greater degree and will bring the discharge again to the equilibrium state. This is well illustrated by Fig. 16. Here curve \(A\) represents the potential distribution in the steady equilibrium state; \(B\) is the potential distribution in the state when recombination decreases. The decrease in recombination causes a decrease in the plasma, an increase in the gradient at the anode, and, as a consequence of these causes, an increase in ionization. \(C\) is the potential distribution in the state when recombination increases. The increase in recombination causes growth of the plasma, a change in the sign of the gradient at the anode, a decrease in the region where ionization occurs, and, as a consequence of these causes, a decrease in ionization. \(D_A, D_B, C\) are the plasma regions in the corresponding indexed position.
The equilibrium condition (7) is supplemented by one more term, taking into account the processes in the plasma. Scherzer developed his theory mathematically, but since his theory concerns chiefly a glowing discharge, i.e. a steady equilibrium state, we shall not present it here and with this shall conclude its exposition.
7. MECHANISM OF BREAKDOWN
The theories set forth, although they do describe and explain many aspects of the discharge, nevertheless are not exhaustive. Thus, according to Rogowski’s theory, the time required for the formation of breakdown should be of the order of the time necessary for the propagation of the first avalanche from cathode to anode. Investigations carried out by Rezerov \(^{13}\) with a Wilson chamber showed that the velocity of avalanche propagation is identical with the velocity of electron motion, i.e. it is of the order of \(10^7\) cm/sec. Consequently, the time for the formation of breakdown in a gap of \(1\) cm should be of the order of \(10^{-7}\) sec. Experiments on the delay of breakdown \(^{14, 15, 18—24, 30—33, 84, 85}\) showed that at a pressure \(p > 300\) torr and gaps \(L > 3\) mm this time is about 5 times smaller. At overvoltages the electron velocities change only slightly, while the time required for the formation of breakdown decreases very strongly.
For large gaps breakdown does not develop owing to an avalanche proceeding from the cathode, but may begin at any of the electrodes, predominantly where the gradient is greater, independently of the sign, or even simultaneously at both electrodes \(^{86—83}\), or somewhere in the middle of the gap \(^{22, 56, 86}\).
If one makes use of the theory developed by Rogowski and Scherzer and supplements it with the considerations of other authors \(^{80, 86, 89}\), it is already possible at the present time to explain the mechanism of breakdown to a sufficient degree.
Let us suppose that between the electrodes, at some place where the field is sufficiently effective, an electron arises accidentally. Under the action of the field the electron will begin to move toward the anode and ionize along its path. The secondary electrons will likewise move toward the anode and, as they advance, an avalanche of electrons and positive ions will grow. These charges will be arranged in the channel in the following manner. At the head of the channel the entire avalanche of electrons will be located. The positive ions, whose velocities are 1000 times smaller, will be arranged along the channel with an ever-decreasing density from head to tail. The cross-section of the channel will be determined by the diffusion of the electrons. Its mean radius in the given cross-section
\[ \rho_x = \sqrt{2D_x\tau_x}, \]
where \(D_x\) is the diffusion coefficient, will be proportional to the mean thermal velocity of the electrons and, therefore, will depend to some degree on the field strength. The diffusion coefficient \(D_x\), moreover, depends on the partial pressure, i.e. on the density of electrons. \(\tau_x\) is the time required for the passage of the avalanche of electrons to the given cross-section. The diffusion of ions is small in comparison with the diffusion of electrons, and after the avalanche of electrons has passed, the channel cross-section will expand only slightly. Thus, the channel cross-section must increase as it grows. Owing to the same diffusion, the electron avalanche should gradually occupy an ever larger volume \(^{1}\). Not
\(^{1}\) Calculations show that the electrostatic stretching of electrons is small in comparison with diffusion.
it will be a serious mistake to suppose that the electron avalanche occupies the volume of a sphere with mean radius $\bar{\rho}$ and is distributed with a density decreasing from the center to the periphery.
As the channel grows, the gradients in it will begin to change. At the tail the gradient will increase; then, as one approaches the head, it will decrease, and at the end of the head it will again increase. The distribution of the potential in the channel is shown in Fig. 18. Part of the electron avalanche falls into a retarding field. As a result, an ever larger fraction of the electrons, losing their velocity, will gradually begin to attach to the atoms and molecules of the gas, forming negative ions. The negative ions will be mixed with the positive ones and will form a plasma, which will be situated near the head of the channel.
Simultaneously with ionization, excitation of the atom is produced. The avalanche of excited atoms is located in the same place as the avalanche of positive ions. An atom remains in the excited state for approximately $10^{-8}$ sec.; consequently, soon after excitation the radiation of atoms begins. However, the radiation of excited atoms is important only when there is a mixture of two gases with different ionization potentials. With the growth of the ion avalanche, recombination radiation increases; it is especially intensified with the formation of plasma. Owing to the low mobility of the ions, the glow of recombination exists for a comparatively long time. The radiation will come chiefly from the head of the channel. It spreads equally in all directions. The short-wave part of the radiation, the most effective but also the most strongly absorbed, will ionize mainly near the head of the channel. This will lead to a uniform widening of the head of the channel in all directions. The more long-wave part of the radiation is absorbed less, can act at points comparatively remote from the head of the channel, but has a smaller probability of ionization. Cravath[^90], in agreement with Traynor[^91] and Rezer[^77], found for a discharge in air an absorption coefficient of the radiation of $2\ \mathrm{cm}^{-1}$ and $10\ \mathrm{cm}^{-1}$.
As a result of this photoionization a number of electrons may arise in the tail of the channel. The closer to the head of the channel, the more probable the appearance of an electron will be, but the less effective such an electron is, since the smaller the avalanche it can form. To maintain the processes in the discharge, i.e. in order that $\mu$ be equal to unity, it is necessary that an electron arise at a point at such a distance from the head that it could form approximately the same avalanche as the first. If, in the section from this point to the end of the tail of the channel, one electron arises, then $\mu=1$ and the probability that the discharge will continue will still be small. The more electrons arise in this section, the greater $\mu$ will be, and the greater the probability that the discharge will continue and intensify. Each new avalanche will be accompanied by the same processes as the first. The avalanches, following one after another in the same channel, will be superimposed upon one another. The plasma will grow back toward the tail; together with it there will expand
THEORY AND MECHANISM OF GAS BREAKDOWN
the cross-section of the channel. The channel will gradually lose its shape of a falling drop and will become more uniform.
The probability of an electron arising in all directions will be the same. But only electrons that have arisen in the tail of the channel and in front of the head in the direction of the greatest gradients will be of importance. Electrons that have arisen in front of the head of the first channel, but at such a distance from it where the gradients are sufficient for effective ionization, will begin, in the manner described, to form a second channel. The tail of the second channel will grow in the direction toward the head of the first channel and, having joined with it, will form one common channel.
Several second channels may form simultaneously, arranged in a chain along the discharge gap both in front of and behind the first channel. In this case, less time is required for the channel to grow over some length \(l\). If, along a length \(L\), \(m\) channels begin to form simultaneously at every centimeter, arranged along the gap \(L\) at equal distances, then the time required for the growth of the entire channel will be \(lm\) times less. Owing to the ever-increasing intense photoionization and to the increase of the gradients at the tail as the channel grows, the time for the advance of the subsequent avalanches will be small in comparison with the time required for the advance of the first avalanche. Therefore the time required for the growth of a channel of length \(L\) will be of the order
\[ \frac{L}{lm}\cdot 10^{-7}\ \text{sec.}, \]
whereas an electron would traverse the same distance in a time of approximately \(L\cdot 10^{-7}\ \text{sec.}\)
Fig. 17. Mechanism of breakdown development
The mechanism of breakdown development described above is illustrated by Figs. 17 and 18.
In Fig. 17. \(a\)—the channel is formed by one initial electron. At the head of the channel there is an avalanche of electrons and ions; in the tail, only ions. The channel grows with the velocity of motion of electrons, of the order \(10^{7}\ \text{cm/sec}\);
\(b\)—the head of the channel expands uniformly owing to ionization by short-wave radiation;
\(c\)—plasma has formed in the head of the first channel. The radiation has intensified, especially because of the increase in recombination. In front of and behind the first channel new channels have formed;
\(d\)—the new channels formed in the tail of the first channel have grown forward and have been superposed one on another, forming a continuous channel. Between the first channel and the channel formed in front of it, a number of channels have grown
small channels. The growth of the channel occurs at a rate of the order of \(10^8\) cm/sec;
\(e\)—all the channels have merged into one continuous channel. The whole channel is a plasma, radiating very intensely. At the end of the channel the positive ions predominate over the negative ones; in the head it is the opposite.
In Fig. 18 the distribution of the potential in the channel is shown.
Fig. 18. Potential distribution in the channel
1—potential before the beginning of the discharge, 2—potential in the presence of a positive space charge, 3—potential in the presence of a negative space charge, 4—resulting potential
For comparison with the discharge process described in detail above, we give photographs of individual stages of the discharge obtained in the Wilson chamber.
In Fig. 19 are shown photographs obtained by Raether in a Wilson chamber with air at a pressure of 273 torr and a temperature of 20°. A constant voltage was applied to plane electrodes; the field strength was 12 kV/cm; in \(\sim 1.5 \cdot 10^{-7}\) sec avalanches are obtained, marked in Fig. 19, \(b\) by arrows. The channels have the form of an elongated falling drop 1.9 cm long.
Secondary processes have not yet begun or are very weak. The exposure is increased by \(2 \cdot 10^{-8}\) sec. The appearance of the avalanches is shown in Fig. 19, \(a\). From the photograph it is seen that at first the head of the channel broadens (marked in photograph 19, \(a\) by an arrow), then the head broadens still more, and a new channel is formed in front of the first channel, which merges with the first. The length of the first channel remains almost unchanged. In darkness a weak diffuse blue glow of the channel head is visible. Secondary processes are already taking place in full measure. The exposure is increased by another \(2.7 \cdot 10^{-8}\) sec. (Fig. 19, \(b\)). The heads of the channels broaden still more, the cross-section of the channel becomes equalized along its length. Along its length the channel already almost bridges the distance between the electrodes. In darkness a weak glow around the channel heads is visible. Sometimes thin luminous threads from the head to the anode are noticeable. The exposure is increased by another \(2.7 \cdot 10^{-8}\) sec. (Fig. 19, \(c\)). The channel has broadened and bridges the gap between the cathode and the anode. In darkness thin continuous channels are visible.
Fig. 19. Photographs obtained by Raether in a Wilson chamber. Mechanism of discharge development
At one and the same voltage and exposure, a statistical scatter in the development of the discharge is observed. In Table 3 are given
the distribution of the discharge according to its degree of development for ten pulses.
Table 3
| Exposure (sec.) | \(1.55\cdot10^{-7}\) | \(1.83\cdot10^{-7}\) | \(2.10\cdot10^{-7}\) |
| Slight glow of the channel head | 10 | 5 | 1 |
| Continuous luminous filaments | 0 | 5 | 9 |
At an exposure of \(2.10\cdot10^{-7}\) sec. the continuous channels are always brighter than at an exposure of \(1.83\cdot10^{-7}\) sec. The same picture is observed at one and the same exposure when the voltage is increased. The photograph marked by an arrow in Fig. 19, \(b\), is obtained at \(11.4\ \mathrm{kV/cm}\); in Fig. 19, \(a\), it is given at \(12\ \mathrm{kV/cm}\), and in Fig. 19, \(c\)—at \(12.3\ \mathrm{kV/cm}\).
The velocity of propagation of the entire channel from cathode to anode (distance \(3.6\ \mathrm{cm}\)) is \(1.3\cdot10^{8}\ \mathrm{cm/sec}\), whereas the growth velocity of individual avalanches is only \(1.25\cdot10^{7}\ \mathrm{cm/sec}\), in full agreement with the experiments of White \(^{56}\).
If one calculates the quantity \(\alpha l\), where \(l\) is the avalanche length when the expansion of the head and its state become unstable, then independently of the avalanche length this value remains constant and approximately equal to 20.
Fig. 20. Photographs obtained by Rezer in a Wilson chamber. Mechanism of propagation of the channel from cathode to anode
In order to observe the details of the mechanism in the transition from a normal avalanche to a continuous channel, the photographs were also taken at a smaller expansion coefficient in the chamber; in this case weak traces (the beginning of the avalanche) do not appear, but the stronger ones (the head) appear more distinctly.
A similar photograph is shown in Fig. 20. The chamber was filled with air, the pressure—initial \(297\ \mathrm{torr}\), final—\(260\ \mathrm{torr}\), temperature \(20^\circ\). The photographs were obtained at constant exposure and variable voltage. Photograph 20, \(a\), was obtained at a voltage of \(11.8\ \mathrm{kV/cm}\). With increasing voltage, at first the channel grows from the side of the head, while the tail of the channel does not change (photographs 20, \(b\) and \(c\)), and at a voltage of \(12.2\ \mathrm{kV/cm}\) the channel bridges the entire gap (photographs 20, \(d\) and \(e\)). Thus, at first the head of the channel grows and, after it reaches the anode, the tail of the channel begins to grow. In addition, observations show that the growth of the head is less sensitive to voltage than the growth of the tail. Small changes in voltage can cause the tail either to grow rapidly or to be inhibited. Small changes in voltage do not have a substantial influence on the growth of the head. The growth velocity of the head is about \(7\cdot10^{7}\ \mathrm{cm/sec}\), whereas the tail grows considerably faster than \(1.2\cdot10^{8}\ \mathrm{cm/sec}\). At constant voltage and different exposures, the same picture is obtained.
In the photographs in Fig. 20 the first avalanche originated near the cathode; therefore the growth of the channel tail is not accessible to sufficiently detailed observation.
observation. In Fig. 21 are given photographs in which the first avalanche begins near the anode. Here one can see the separate stages of growth of the tail. A comparison of Figs. 20 and 21, as well as numerous visual observations, shows that the growth of the head always proceeds along the shortest path along the lines of force and gives a smooth channel, whereas the sprouting of the tail part of the channel often gives bends and branchings.
Fig. 21. Photographs obtained by Rezer in a Wilson chamber. Mechanism of the propagation of the channel from anode to cathode
For comparison with photographs 20 and 21 I give one more photograph (Fig. 22), made by Rezer in the same chamber, but in a nonuniform field. A small ball of diameter \(1\ \mathrm{mm}\) is mounted on one of the plane electrodes. The distance between the electrodes is \(3.6\ \mathrm{cm}\). In Fig. 21, \(a\) the ball is below. The polarity is negative, the voltage \(\sim 32\ \mathrm{kV}\); in Fig. 21, \(b\) the ball is above, the polarity is positive, the voltage \(\sim 37\ \mathrm{kV}\). The duration of the pulse is \(\sim 2.5\cdot 10^{-7}\ \mathrm{sec}\). The voltage wave is almost rectangular. The comparison shows complete identity of the processes at a positive and a negative point, with the corresponding ends of the channels.
The photographs presented and Rezer’s observations, as is evident, confirm very well the discharge mechanism described above.
The discharge begins where the gradients are sufficient for effective ionization by electrons. The higher the gradient, the greater will be the probability of the origination of a charge. Therefore the discharge always begins at the place of greatest gradient, at the point, independently of the sign of the potential. This is confirmed by the photographs of Rezer cited and by Allibone’s experiments\(^{86}\).
Fig. 22. Photographs obtained by Rezer in a Wilson chamber. Mechanism of propagation of the channel in a nonuniform field
\(a\)—the point (a ball of diameter \(1\ \mathrm{mm}\)) below;
\(b\)—the point above
The mechanism of the discharge at a positive and at a negative point is one and the same, with the difference that at the positive point it is chiefly the tail of the channel that grows, while at the negative point it is chiefly the head.
To decide the question of the preferential growth of the head or the tail of the channel, let us cite also the experiments of White\(^{56}\), Dunnington\(^{22}\), and Gamow\(^{23}\).
White\(^{56}\) observed the development of the discharge between balls with the aid of a Kerr cell. In White’s experiments, just as in the experiments
of the other two authors, the cathode was illuminated with ultraviolet light. In nitrogen, oxygen, a mixture of these gases, and in air at atmospheric pressure and small distances between the electrodes (up to 3.5 mm for nitrogen), the discharge proceeds as follows.
At the cathode a streamer1 appears in the form of a thin luminous thread, which grows in the direction of the anode. When it has reached approximately half the distance from the cathode to the anode, a similar streamer begins to grow from the anode toward it. The two streamers meet at approximately one quarter of the distance from the anode. At large distances two streamers appear simultaneously: at the cathode and in the middle of the gap. Both streamers grow simultaneously toward one another. The anodic end of the central streamer almost does not grow until it meets the cathode streamer. After the two streamers join, the process continues as in small gaps. The two streamers just mentioned are especially clearly visible in air. The velocity of cathode streamers is from \(0.75 \cdot 10^7\) to \(1.5 \cdot 10^7\) cm/sec in nitrogen. The velocity of growth of the central streamer toward the cathode is 3–4 times greater. In Fig. 23 are shown photographs obtained by White with a Kerr shutter. They show the development of the discharge in air at an electrode spacing of 6 mm. \(t_p\) is the travel time of the voltage wave from the discharge gap to the Kerr capacitor.
Fig. 23
The same discharge pattern between spheres 4 cm in diameter was observed by Dunnington[^22] in dry air at pressures from 20 cm of mercury to atmospheric. At atmospheric pressure Dunnington observed two streamers, a central one and a cathode one, at distances between the spheres beginning with 2.9 mm. Dunnington even establishes a dependence between the distance to the midpoint of the central streamer, the pressure, and the distance between the spheres, \(P^{1/3}L = f(d)\), where \(f(d)\) is a constant depending only on the distance to the midpoint of the central streamer. Dunnington also notes the fact that first the cathode and central streamers merge, and only then is the whole gap bridged. In the absence of a central streamer, almost exclusively the cathode streamer grows, which meets the anodic one near the anode (Figs. 24 and 25).
Gamos[^23] observed a discharge in air between spheres at a distance of 5 mm. The luminous channel, having the form of a reservoir avalanche, originated at the anode and grew with its tail from the anode toward the cathode. At the cathode a luminous spot was visible, which slightly increased and stretched toward the anode as the channel from the anode grew. The joining of the channel with the spot occurred near the cathode. In Fig. 26
shown are photographs obtained by Gamos with a Kerr shutter, which present the development of a discharge in air at atmospheric pressure and at a distance between spherical electrodes of 5 mm. Each—
Fig. 24. Dunnington photographs obtained with a Kerr shutter. Development of a discharge in air at atmospheric pressure and with a distance between electrodes of 5 mm
\(t_p\) — time of travel of the voltage wave from the discharge gap to the Kerr condenser; \(t\) — true time. In the first two photographs several discharges are shown; in the remaining ones, one discharge each
each subsequent stage was photographed after \(\sim 10^9\) sec. Each photograph was made with 10,000 discharges.
Stekolnikov and Belyakov \(^{88}\) studied the discharge between two rods to which an impulse voltage was supplied. According to the observations—
Fig. 25. Sketches made by Dunnington on the basis of visual observations with the aid of a Kerr shutter. Development of a discharge in dry air at atmospheric pressure
of the authors, from both electrodes there grew, toward each other, two streamers. The anode streamer had a greater velocity than the cathode streamer.
Fig. 26
After the gap is bridged, the channel, according to the observations of all authors, broadens very strongly, and its cross section becomes everywhere identical.
What Gamos observed is entirely consistent with the discharge mechanism described by us above. First of all, it must be noted that Gamos photographed the discharge through a Kerr condenser. Thus, with the shortest exposure, Gamos recorded in the photograph an early stage of the discharge, with comparatively intense light emission, corresponding to Fig. 17, \(c\) or \(d\).
according to our scheme, whereas Raether had the possibility of observing, and did in fact observe, considerably earlier stages of the discharge. The photographs of Gamos give discharges with a comparatively large density of space charge. Gamos notes that, according to the spectrum, which he recorded at the same time, this glow belongs to excited molecules situated in the strong field of a spatial positive charge. The photographs with the smallest exposure correspond to that stage when the first, and perhaps the very first, avalanches of electrons have passed from the cathode to the anode, creating at the anode a space charge with sufficient radiation. Although the avalanche begins at the cathode, because of the absence or very weak radiation the tail of the avalanche is not visible. Then the tail of the channel, by means of the mechanism drawn in the left-hand part of Fig. 17, grows from the anode toward the cathode. These photographs of Gamos correspond to Raether’s photographs in Fig. 21, when the avalanche is born at the anode and the channel grows from the anode toward the cathode. The velocity of propagation of the channel, according to Gamos’s measurements, is equal to \(0.5\cdot10^8\ \mathrm{cm/sec}\), somewhat lower than the velocity measured by Raether for the tail of the channel, from \(1.1\) to \(1.3\cdot10^8\ \mathrm{cm/sec}\). It is possible that the higher velocities in Raether are due to the overvoltage \((10—20\%)\) at which the experiments were carried out. The same mechanism operates in the propagation of the anode and central streamers of Wyatt and Dunnington until they join with the cathode streamers. The velocity of propagation of the central streamer toward the cathode, measured by Wyatt, from \(0.3\cdot10^8\) to \(0.6\cdot10^8\ \mathrm{cm/sec}\), coincides with the growth velocity of the channel in Gamos.
The propagation of the cathode streamers in Wyatt and Dunnington at first is nothing other than the growth of an ordinary avalanche. Their length, \(1—2\ \mathrm{mm}\), is small. The growth velocity, from \(1\) to \(1.5\cdot10^7\ \mathrm{cm/sec}\), measured by Wyatt, coincides with the electron velocity under these conditions and agrees with the growth velocity of Raether’s avalanches, \(1.25\cdot10^7\ \mathrm{cm/sec}\). The mechanism of growth of the cathode streamers corresponds to Fig. 17, \(a\) and \(b\).
As regards the mechanism of growth of the central streamers toward the anode, since their velocities have not been measured, its mechanism can be judged only from the reproductions given in the papers. Before the central streamer joins the cathode streamer, its growth toward the anode is small. It represents the growth of an ordinary avalanche with small velocities. Therefore the growth of the central streamer toward the anode, i.e. the growth of the head of the channel, is hardly noticeable in comparison with the growth of the tail (which is about 4 times greater). After the central streamer joins with the cathode streamer, the further growth of the common channel toward the anode apparently accelerates. The mechanism of further growth corresponds to Fig. 17, \(d\), and to Raether’s photographs (Fig. 20), when the avalanche is born near the cathode and grows first toward the anode.
Raether establishes the preferential growth of the head of the channel rather than of the tail (the channel first grows toward the anode, then toward the cathode). This agrees with the growth of cathode streamers at small distances in Wyatt and Dunnington (first, up to half the distance, a cathode streamer grows, and then an anode streamer begins to grow toward it). Gamos’s experiments do not contradict this, if one assumes that at a distance of \(5\ \mathrm{mm}\) Gamos could have had a central streamer which had already managed to grow as far as the anode. Here it should be noted that for every
a Gamow photograph yielded 10,000 discharges, and because of the statistics there could have been many cases of a central streamer that reached the anode. In the formation of a central streamer, White and Dunnington note, on the contrary, a preference for growth of the tail (first the central streamer connects with the cathode, and then grows toward the anode).
Stekol’nikov and Belyakov\(^{88}\) also note that the meeting place of the cathode and anode streamers is located closer to the cathode. However, on the basis of the material published by Stekol’nikov and Belyakov, it is impossible to say definitely whether this occurs because the anode streamer begins to grow earlier than the cathode one, or because of the greater growth rate of the anode streamer. The authors themselves prefer the latter reason.
It seems to me that the preferential growth of one or another end of the channel depends on the specific conditions. The preferential growth depends on the intensity of the radiation and the probability of the photoeffect, which are the same both for the tail and for the head of the channel, and on the magnitudes of the gradients. In whatever direction the gradients are greater under the given conditions (the magnitude and distribution of the space charge and the geometry of the initial field), in that direction the channel propagation will be more probable.
The mechanism of breakdown shows that breakdown consists of a number of separate processes which follow one another in time and space. The inception of each separate process at a definite place and at a definite time has a certain probability. Thus, breakdown and the time during which it occurs are statistical phenomena.
The phenomenon described above, which takes place both in the discharge at small distances between the electrodes and in long sparks, is a process of transition of the gas gap from a nonconducting state to a state with good conductivity. This process consists in the establishment of a conducting channel from the cathode to the anode, and ends at the moment when such a channel has been created.
In long sparks, because the time required for the growth of the channel, especially at low pressures, is comparatively large, the growth of the channel can be recorded on a photographic plate\(^{86–88, 92–93}\). By artificial means this growth can be slowed down still further. This is achieved by inserting a large resistance between the generator and the gap. The growth of the channel causes a voltage drop in the resistance and a decrease of it across the gap itself; moreover, the faster the channel grows, the lower the voltage across the gap becomes. This leads to automatic regulation of the channel-growth rate. The larger the resistance, the more slowly the channel grows. In order to record the very growth of the channel or the process of development of the discharge, the latter is photographed either on a rotating film or by means of a rotating mirror.
Figure 27 gives one of such photographs, taken from the work of Allibone and Meek\(^{87}\). The photograph, made on a rotating film with a quartz lens, represents a discharge between a rod and a grounded plane. The distance between the electrodes was 100 cm. A positive impulse was applied to the rod—
... voltage wave. Between the generator and the gap a resistance of \(1{,}000{,}000\ \Omega\) was inserted. The discharge begins at the rod. The letters \(a\), \(b\), and \(c\) indicate the different stages in the growth of the channel, while the line \(d\) gives the channel at the moment when it reaches the second plane electrode. Such a channel is usually called a leader. It is evident from the photograph that the channel grew in steps. The intervals between the steps are equal to 14, 22, and 41 \(\mu\mathrm{sec}\). Such a leader is called a stepped leader.
Fig. 27
\(a\), \(b\), \(c\), and \(d\)—steps of the leader;
\(e\)—main discharge
For the most part the leader has no steps. A stepped leader could be observed only when a large resistance was inserted between the generator and the discharge gap. This phenomenon can be explained by the voltage drop in the resistance during the growth of the channel and by its decrease in the discharge gap itself. The gradients at the end of the channel decrease, and the growth of the channel is retarded. The cessation of the channel growth causes the cessation of the current in the discharge circuit and the restoration of the voltage across the gap. The gradients at the end of the channel again increase, and the channel begins to grow farther. Such a stoppage in the growth of the channel may occur several times, until the channel closes the gap between the electrodes.
Fig. 28. Drawing by Allibone and Meek, made by them from a photograph of a discharge, and the voltage oscillogram synchronized with it
Fig. 28, made by Allibone and Meek\(^{87}\) from an original photograph of a discharge in air at a pressure of 10 cm and from a volt-second oscillogram synchronized with it, well confirms this explanation. It is evident from the figure that each fall of voltage corresponds to a step of the leader.
A stepped leader is observed both for a positive and for a negative voltage wave\(^{86–88,\ 92–93}\).
After the leader reaches the second electrode, i.e., when the breakdown of the gap is completed, a more powerful discharge begins to propagate from the second electrode in the reverse direction along the very same path; this is called the main discharge.
If two leaders simultaneously grow from both electrodes toward one another, the main discharge begins to propagate from the place where the two leaders meet^87 simultaneously toward both electrodes, in directions opposite to the growth of the leaders. The main discharge has the same tortuosities and branches as the leader.
The main discharge, which is a type of arc, unlike the leader has not been studied at all. Its nature and mechanism remain entirely unknown even at the present time. Moreover, there is still not a single hypothesis concerning the mechanism of the main discharge that deserves any attention. At the same time, the study of the main discharge is of not only purely scientific but also enormous practical interest.
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