Gases in the Plasma State. I[^1]

R. Rompe, M. Steenbeck

Submitted 1941 | SovietRxiv: ru-194101.45579 | Translated from Russian

Full Text

Gases in the Plasma State. I1

R. Rompe, Berlin, and M. Steenbeck, Berlin-Siemensstadt

Introduction

I. Experimental data: Langmuir probes, quasineutrality, isothermal and nonisothermal plasmas
II. Energy exchange among plasma components under Coulomb interaction; microfield; reaction length
III. Electrical conductivity of plasma in constant fields
IV. Reduction of the work of ionization in plasma; work function; cohesion forces
V. Plasma oscillations
    a. Electrostatic oscillations of electrons
    b. Allowance for electron pressure
    c. Ion oscillations
    d. Origin of plasma oscillations
VI. Dielectric and magnetic properties of plasma
    a. Dielectric properties
    b. Magnetic properties
VII. Allowance for the individual properties of atoms
VIII. Isothermal plasma
    a. Calculation of component concentrations
    b. Specific heat of plasma
    c. Elementary processes in an isothermal plasma
    d. Coefficient of thermal conductivity of plasma
    e. Influence of the plasma state on atomic properties
IX. Nonisothermal plasma
    a. The concept of “temperatures” in a nonisothermal plasma
    b. Elementary processes in a nonisothermal plasma
    c. Energy balance in a nonisothermal plasma

Bibliography

Introduction

A gas the majority of whose particles are electrically charged, i.e., a highly ionized gas, differs in many respects from an ordinary gas. Thus, for example, in a number of phenomena such a gas is found to resemble electrolytes and solid conducting bodies such as semiconductors, as well as metals. A highly ionized gas also possesses properties characteristic only

GASES IN THE PLASMA STATE

to it alone. These properties are consequences either of strong electric fields between charged particles, or of the presence of a large number of excited particles, which in turn is connected with a high degree of ionization. Langmuir, to whom we owe a large part of our knowledge of such highly ionized gases, gave them the special name “plasma.” The introduction of such a term is intended to indicate that plasma, in many respects, may be regarded as a special state of matter, which we encounter, for example, in the stars, as well as in the Earth’s ionosphere, in gas discharges, in gases heated to very high temperatures, in the flame of explosions, and so forth.

Faraday^1) in 1816–1819 searched for a fourth state of aggregation, extrapolating the transformations of matter under heating, i.e., the transition of a body from the solid to the liquid, from the liquid to the gaseous state. Crookes^67 in 1879 thought that the “fourth state of aggregation, or radiant matter,” had been found by him in discharge tubes. One may thus say that Faraday and Crookes came close to the concept of plasma.

Approximately fifteen years of painstaking research work has still not, up to the present day, clarified all the details of the general picture of plasma. However, this work is already sufficiently extensive and profound to permit the writing of the first complete review.^2) The present article is intended to be such a review.

We regard the plasma of a gas as given, and do not go into the details of its origin. This makes it possible to avoid describing the whole complex of phenomena that occurs when current passes through gases. Nevertheless, to orient the reader, something about the formation of plasma will be said in the first section.

The greater part of our knowledge of plasma has been obtained from investigations of a gas discharge occurring with a fairly large current strength. Here typical plasmas arise in regions sufficiently far from the electrodes and the walls. Valuable information on plasma has also been obtained by studies of the passage of electromagnetic waves through the ionosphere, by spectroscopic study of stars and flames.

In general, by the specific properties of plasma we shall understand only those of its properties whose cause lies in the prolonged interaction or in the joint action of the numerous components of the plasma. Conversely, plasma properties that represent the sum of the properties inherent in an individual particle will not be taken into account; otherwise this article would turn into a textbook of atomic physics. The material presented is naturally divided into two parts: Sections I–VI cover facts that are a consequence of the presence in plasma of a very large number of positive and negative charges, as well as of their Coulomb interaction;

^1) Quoted from Crookes.^67

^2) See the bibliography. Reviews, however, not so detailed as the present one, devoted to the question of plasma, are given in the bibliography under numbers 466, 467, 468, 120, 497, 71, 510, 62, 63, 527.

In this, the nature of the gas plays no essential role. In the last three sections, on the contrary, those properties of plasma are discussed which depend on the individual features of excited and ionized atoms.

With rare exceptions, this article considers plasmas that are homogeneous in space and stationary in time; from this range of questions, phenomena have been selected for presentation which are based on problems sufficiently clear from the point of view of experiment and theory. However, even after such a double restriction, the material remains so extensive that the possibility is not excluded that the authors have overlooked individual important investigations.

In order to make possible a survey of numerical values of quantities characterizing the plasma properties under discussion, at the end of some sections data are given for typical discharge plasmas; at the same time the formulas used in calculating the figures presented in the tables are indicated. For comparison, the same quantities for known substances—not plasmas—are placed in the tables. In view of the limited space, numerical examples are not calculated in the text.

1. EXPERIMENTAL DATA: LANGMUIR PROBES, QUASI-NEUTRALITY, ISOTHERMAL AND NONISOTHERMAL PLASMAS

A highly ionized gas contains freely moving charges and therefore has a large—usually non-ohmic—conductivity. In many cases, as, for example, in arcs, the conductivity of such a gas approaches, in order of magnitude, metallic conductivity and in any case may take values much greater than electrolytic conductivity. In an undisturbed plasma, i.e. in a plasma situated sufficiently far from electrodes and walls, large potential differences do not arise. The motion of the plasma charges equalizes the potentials of its various parts in just the same way as occurs when electrostatic equilibrium is established in metals. This means, among other things, that in an undisturbed plasma there do not arise large volume electrostatic charges. Thus it is impossible for the number of positive charges in individual regions of the plasma to differ sharply from the number of negative ones. On the contrary, the mean values of the concentrations (formed in a not very small volume) of charges of both signs at one and the same place must be approximately equal. We shall therefore often denote in what follows the number of positive charges per unit volume $N_+$, and the number of negative charges per unit volume $N_-$, by the common symbol $N$:

\[ N_+ \simeq N_- \simeq N \tag{1,1} \]

\[ |N_+ - N_-| \ll N. \tag{1,1a} \]

Here, of course, $N$ may be a function of space and time. The property of a plasma of not forming any significant effective volume charges, expressed by equations (1,1), we

we shall call, following Schottky, quasineutrality. In this same sense, and for the same reasons as plasma, concentrated liquid electrolytes are quasineutral; likewise a metal with its electron gas, whose space charge is compensated by the charge of the ionic lattice, is a kind of quasineutral plasma.

We can define plasma as a quasineutral state, assuming that only small, in percentage terms, differences between the concentrations of charges of the two signs are possible.

At a minimum ion concentration of $10^9 — 10^{10}\ \mathrm{cm}^{-3}$, in laboratory experiments one may, according to this definition, regard a gas as “plasma.” In large ionized regions the condition of quasineutrality is necessarily fulfilled already at much smaller concentrations. Since, according to Poisson’s equation, the fields created by space charges increase linearly with the geometrical expansion of the region of volume charges, in very large ionized regions only very small effective space charges can exist. Therefore, in the earth’s ionosphere, extending over a thickness of many kilometers, for which $N \simeq 10^6\ \mathrm{cm}^{-3}$, the conditions of quasineutrality are fulfilled even more strictly than is given by inequality (1,1a), i.e., the differences in the concentrations $|N_+ - N_-|$ are many orders of magnitude smaller than $10^6\ \mathrm{cm}^{-3}$. Under laboratory conditions a concentration of $10^6\ \mathrm{cm}^{-3}$ can be realized in a very small volume, say in an ionization chamber, for charge of one sign; in this case one can no longer speak of plasma.

In order to maintain the degree of ionization of a plasma at one and the same level, it is necessary to make up for losses of charge carriers. These latter arise from recombination of charges of opposite signs, and also due to diffusion of carriers into regions poor in charge. Losses by diffusion grow linearly with concentration; losses by recombination grow at least quadratically with concentration. In order to attain a high degree of ionization, it is necessary above all to keep recombination of carriers at a low level. This is possible at low pressures, since the recombination coefficient here is proportional to the pressure and is therefore small. The coefficient of recombination of electrons with positive ions, according to the available far from complete data, is considerably smaller than the coefficient of recombination of negative ions with positive ions $^{465, 323, 344}$ $^{1)}$. Therefore typical plasmas with a high degree of ionization arise primarily in pure noble gases or metal vapors at low pressures ($10^{-3}$ to $1$ torr), where the probability of formation of a negative ion (attachment $^{344, 17, 18, 19, 20, 480, 431, 118, 325, 34, 327, 491, 492, 216}$ of an electron to a neutral atom) is least of all $^{2)}$. In non-

$^{1)}$ In the latest works $^{335, 360, 361, 362, 266, 167, 337, 326, 263, 152}$ there are numerical data concerning recombination.

$^{2)}$ Monatomic gases! The energy of rotation and vibration of the molecules is equal to zero.

In discharge gases there exist typical plasmas, for example in arc discharges, in which the gas is heated to such a high temperature that negative ions can again be neglected. This can be done because negative ions, possessing a small binding energy, immediately dissociate at the high temperature of the arc into an electron and a neutral atom^438. Of the negative charges of a plasma it is customary to take into account only free electrons, disregarding negative ions altogether. In the greater part of our exposition this assumption is adopted by us; it is necessary, however, to note that a number of the works cited indicate the presence in the plasma of a large number of negative ions, which under certain circumstances may even exceed the number of free electrons (impurities, such as HgH⁻?). One may dispute the fact of the existence of negative ions in such a quantity, since the majority of the properties of a plasma (conductivity, work function, cohesive forces, plasma oscillations, dielectric and magnetic properties, optical behavior, etc.) are practically determined by the content of free electrons; therefore, if the negative ions are removed from the plasma and (in order to preserve the conditions of quasineutrality) the corresponding number of positive ions, then the properties of the plasma will not change qualitatively and even quantitatively. Only if the number of negative ions exceeds by an order of magnitude the number of free electrons (it is possible that such a situation occurs in the lower ionized layers of the ionosphere) can negative ions become determining for the properties of the plasma.

Plasma resembles both a metal and an electrolyte. It resembles a metal in that in both cases a larger or smaller number of mobile electrons fills the space in which the electrostatic actions of these electrons are destroyed by the action of positive ions of the same concentration. True, the plasma ions do not form a regular lattice, as do the ions of a metal; however, owing to their greater mass (in comparison with the mass of the electron), with respect to rapid processes the plasma ions may be regarded as immobile. The “lattice” of positive ions is a statistically disordered system. In addition, the concentrations of the charge carriers are so small, and the mean kinetic energy so large, that degeneracy of the electron gas does not occur. It should be noted that phenomena connected with degeneracy of the electron gas have never played a role in a discharge and are unlikely ever to be significant for it, since the concentrations required for the occurrence of these phenomena cannot be attained even in a discharge at high pressures^1). For these reasons, the kinematic interactions of the plasma particles are treated, with one exception (compare the end of Section IV), classically (compare Sections II and IV). Plasma is thus similar to the old classical model of the metallic state (see Section III). The above-mentioned sim—

^1) Hg 1000 atm, 10 A and more would give a plasma temperature of the order of magnitudes at which degeneracy plays a role.

...of plasma with an electrolyte is based on the fact that carriers of charges of both signs, present in approximately equal concentrations, move in a medium with friction, namely in the always-present neutral gas. At the same time, the electrostatic interaction of the charge carriers is so great that phenomena occur which are known from the Debye–Hückel theory of strong electrolytes.¹

The electrostatic coupling between positive and negative charges manifests itself in plasma in a distinctive way, for example in diffusion processes. If the concentration $N$ in a plasma (for which, in accordance with (1.1), $N_+ \simeq N_- \simeq N$) depends on the point in space, and, owing to the quasineutrality condition, in the same way for positive and negative particles, then equalization of the concentrations of the charge carriers occurs because particles from regions of high concentration diffuse into regions of low concentration.

The diffusion coefficient for negative and positive particles is, generally speaking, not the same; this difference is especially large if the negative particles are electrons. We shall adopt the latter assumption. With the same drop in concentration for positive and negative particles, the particles with the larger diffusion coefficient (in our case, electrons) diffuse at first more rapidly. Owing to the electrons running ahead, small percentage differences arise in the concentrations of ions and electrons: in places of high concentration the remaining positive ions predominate, while in places of low concentration the electrons that have run ahead predominate. Thus volume charges arise, by virtue of which a field is created in such a direction that it hinders the motion of the too rapidly diffusing electrons and accelerates the slowly diffusing ions. So long as a still more rapid motion of electrons than of ions is possible (despite the existence of the retarding field), the space charges, and together with them the retarding field, will increase. The stationary state will set in when the retarding field reaches a magnitude that equalizes the diffusion of electrons and ions from regions of high to regions of low concentration. As Schottky showed⁴⁷³,⁴⁷⁴,⁴⁷⁵, this equalization of charge carriers spontaneously

¹ Compare Section VIII, e. In this connection it is interesting to note that from the well-known electrolytic conductivity, for example the conductivity of weak aqueous solutions in common salt, there exists a continuous transition to the conductivity of the plasma of a gas discharge. If the proper amount of salt solution is heated above the critical temperature (the solution is placed in a vessel that withstands the pressure), so that the liquid continuously passes from the liquid into the gaseous state, then in the gaseous phase a part of the salt remains in the dissolved and dissociated state. Thus the electrolytic conductivity of the liquid phase continuously passes into gaseous conductivity¹⁹⁷, ¹⁹⁸, ¹³¹, ²⁶¹, ²⁶². If strong currents are passed through this “gaseous electrolyte,” an increase in the number of ions begins by impact ionization, which ultimately leads to an independent gas discharge. In this transition the sharp boundary between electrolytic conductivity and conductivity in the plasma of a gas discharge is erased.

formed fields proceeds according to the ordinary laws of diffusion, with a velocity the same for positive and negative carriers. In other words, there exists a common diffusion coefficient $D_a$, called ambipolar, which is formed from the diffusion coefficients for carriers of both signs, $D_+$ and $D_-$, as an average with respect to the carrier mobilities $b_+$ and $b_-$$^{1}$:

\[ D_a=\frac{D_-b_+ + D_+b_-}{b_+ + b_-}. \tag{1,2} \]

Some consequences will be considered below.

For the further discussion of the general properties of a plasma, let us assume, for greater clarity, that the plasma is produced by a gas discharge in the apparatus shown in Fig. 1. $K$ is the cathode, heated by the battery $H$, and $A$ is the anode. Both electrodes are situated in a fairly wide discharge space of approximately spherical form; thus the walls are as far as possible from the region between the electrodes. The discharge space is filled with Hg vapor or with a noble gas at a pressure from $10^{-2}$ to $10^{-1}$ torr; with the aid of the battery $U$ one can maintain a discharge which, at a current of 1 A, requires a striking voltage between $K$ and $A$ of the order of 5–15 V. In this case, in the discharge space there is formed a plasma which fills the whole vessel with an ionization density continuously decreasing toward the walls. By placing small electrodes, the so-called probes, $S_1,\ldots,S_3$, inside the plasma and measuring with an electrometer the voltage between them, one can show that within the plasma there are in fact no appreciable potential differences. In the apparatus described, the potential differences between probes reach only a few tenths of a volt. A more detailed investigation shows that the entire potential drop between cathode and anode (5–15 V) occurs almost completely in the immediate vicinity of the electrodes, chiefly near the cathode. If, however, one disregards these “regions of disturbance,” then the plasma itself at all its points has practically the same potential. The potential measured with the aid of an insu-

Fig. 1. Low-pressure discharge tube with three probes.

$^{1}$ Derivation: equate the flux densities of particles of both signs,

\[ \mathbf n_+ = \mathbf n_- = \mathbf n. \]

Each flux is the sum of the diffusion current and the current produced by the field:

\[ \mathbf n_+=+b_+N_+\mathbf E-D_+\operatorname{grad} N_+ =+b_+N\mathbf E-D_+\operatorname{grad} N=\mathbf n, \]

\[ \mathbf n_-=-b_-N_-\mathbf E-D_-\operatorname{grad} N_- =-b_-N\mathbf E-D_-\operatorname{grad} N=\mathbf n. \]

Eliminating $\mathbf E$ from these equations, we find

\[ \mathbf n=-\frac{D_-b_+ + D_+b_-}{b_+ + b_-}\operatorname{grad} N, \]

whence (1,2) follows.

of an isolated probe cannot be equated with the true potential of the plasma in the region surrounding the probe. A “contact potential difference” is formed between the plasma and the probe in the sense that the probe is charged to several volts negative with respect to the plasma. When an isolated probe is immersed in a plasma, it is surrounded by electrons and positive ions; since no current whatsoever flows out of the isolated probe, the probe must assume such a potential that, on the average, equal numbers of positive electrons and ions approach it. We know that the concentrations of positive ions and electrons in a plasma are approximately equal; owing to the difference in the masses of ions and electrons, the velocities of the electrons, of course, greatly exceed the velocities of the ions. Therefore, electrons will enter a probe that has assumed the plasma potential much more often than ions. Since the probe, being isolated, cannot carry away this excess of electrons, it will begin to acquire a negative charge. This negative charge will increase until the probe begins to repel most of the electrons arriving at it from the plasma. The probe will be reached only by the fastest electrons of the plasma. The resulting decrease in the influx of electrons continues until a state is established in which the influxes of ions and electrons are equal; the potential of this state is the one assumed by isolated probes. Below we give the exact value of the contact potential of a probe in a plasma.

Langmuir and his collaborators ²⁹², ²⁹³, ²⁹⁴, ²⁹⁹, ²⁹⁵, ³⁰⁰, ³⁰¹, ³⁰², ³⁰⁴, ³⁰⁶, ³⁰⁷, ²⁹⁸, ⁵²³ indicated a way by following which one can not only establish the true value of the plasma potential, but also find most of the quantities characterizing the plasma. Although this method has been set forth more than once in a number of monographs¹²³, we must briefly dwell on it here in order to introduce a number of concepts that will be discussed further on.

In Langmuir’s method it is not the probe potential that is measured; an alternating voltage $U_s$ is applied between the probe and the cathode, and the current $i_s$ flowing through the probe is measured. The probe characteristic $i_s=f(U_s)$ makes it possible to draw far-reaching conclusions about the structure of the plasma. If the probe is charged strongly negatively with respect to the plasma potential $U_p$ existing in the region where the probe is located (“space potential”), then it repels from itself all electrons and negative ions (the latter, however, we shall not consider); only positive ions arrive at the probe. The current of positive ions has the direction plasma—probe (Fig. 3, left-hand part). Let the potential of the probe become more positive, while still remaining negative with respect to the space potential; then the fastest electrons can already reach the probe. Thus an electron influx into the probe arises which, being superposed on the existing ion current, reduces the total current (Fig. 3, middle part). When the voltage reaches such a value that the ion and electron currents are equal, the total current ceases; this value of the voltage $U_0$ is assumed by an isolated probe.

The quantity $U_0$, as was shown above, is still negative with respect to the space potential $U_p$. With a further change of the probe potential toward positive values up to the value $U_p$, the probe continues to repel electrons. Nevertheless, the influx of electrons to the probe at voltage values $U_0 < U_s < U_p$ is already greater than the ion current. Therefore the total current, starting from $U_0$, has the direction probe—plasma. Only when $U_s$ becomes greater than the space potential $U_p$ does the probe begin to attract electrons. The laws of this attraction are different from the laws of repulsion; therefore the course of the probe characteristic changes rather sharply at $U_s = U_p$. As Fig. 3 shows, the curve $i_s(U)$ undergoes an inflection. The point of inflection of the curve establishes the value of the space potential $U_p$1). This also determines the contact potential difference $\Delta U = U_0 - U_p$, which was discussed above.

Fig. 2. Apparatus for recording a probe characteristic

Fig. 3. Example of a probe characteristic

From the probe characteristic in the region of electron repulsion one can obtain, by differentiation2), the distribution of electron velocities in the plasma; the principle of this method coincides with the well-known method of analyzing photoelectron velocities by means of a retarding field. These investigations lead to an astonishing result. It turns out that plasma electrons often satisfy Maxwell’s velocity distribution with surprising accuracy. One may say that Maxwell’s velocity distribution for plasma electrons has been confirmed experimentally to a greater degree than is the case for the molecules of an ordinary neutral gas.

1) Langmuir proposed yet another precise experiment, based on the regularities of the attraction of electrons by the probe (for $U_s > U_p$). A description of this experiment would lead us too far into the physics of the discharge123.

2) For the apparatus by means of which this differentiation is carried out automatically, see the papers of Sloan and Gregor482 and Van-Gorkum173; for an improvement of the method for obtaining the probe characteristic, see Geddes211. For a study of the influence on the measurements of rapidly varying voltage on the probe, see Van-Berckel31; for measurement of rapidly varying discharges, see Koch248, 249.

The Maxwellian distribution of electron velocities is the first indication of the intense interaction of the electrons of the entire plasma; for the Maxwell distribution is the equilibrium distribution in a gas that is in complete thermal equilibrium.

The temperature corresponding to this distribution, the so-called “electron temperature” $T_-$, has, in the apparatus described above, a value greater by an order of magnitude than the temperature of the neutral gas in which the discharge takes place. At low pressures $T_-$ can reach $80\,000^\circ$ (the mean kinetic energy of one electron $\simeq 10\ \mathrm{eV}$) while the temperature of the neutral gas is of the order of room temperature. In other respects, the established electron temperature depends strongly on the discharge conditions1. Here there occurs the remarkable unique case in which two completely mixed gases—the neutral gas and the electron gas—simultaneously and in one and the same place possess entirely different temperatures. Such a plasma we shall, for brevity, call nonisothermal, in contrast to isothermal2 plasmas, in which all components of the plasma (Plasmapartner) are at one and the same temperature.

After the velocity of the electrons has been found by probe measurements, the electron concentration $N$ can be determined from the values of the current flowing through the probe. The probe neither attracts nor repels electrons in the case when the probe voltage $U_S$ is equal to the plasma potential $U_p$. Under this condition the electrons impinge upon the probe with thermal velocities. The number of particles $n$ striking a unit surface of the probe per unit time can be found from the formula of the kinetic theory of gases

\[ n=\frac{Nw}{4}, \]

where $w$ is the mean thermal velocity, equal to $\sqrt{\frac{8kT_-}{m}}$. The value of $T_-$ is known; $n$ is found by measuring the current density flowing through the probe at $U_S=U_p$, as the quotient $\frac{j}{e}$. From the formula given one can thus calculate the electron concentration $N$.

A probe (a wire of diameter $d$) charged strongly negatively relative to the plasma repels from itself electrons and negative ions. Around the probe there is formed a layer of space charge, produced by positive ions. The action of the negative charges of the probe on distant regions is thereby reduced to nothing by this layer of positive charges. The ion layer must be such

of thickness (external diameter $a$), so that it contains an amount of positive electricity equal to the negative charge of the probe. Then outside the layer there is an undisturbed (to a first approximation) plasma, on which the probe exerts no electrostatic action1. Therefore, strictly speaking, the assertion that a negative probe extracts positive ions from the plasma is incorrect. The ions reach the outer boundary of the space-charge region with the velocities inherent to the given plasma; here these ions enter the sphere of action of the negative charges of the probe and are attracted to the latter. The current of positive ions in the direction of the negative probe is therefore a diffusion current of density

\[ j_+ = e\frac{N w_+}{4}, \tag{1,3} \]

where $w_+$ is the mean velocity of the disordered ions in the plasma. The total ionic current flowing to the probe depends on the voltage at the probe $U_S$ only insofar as this voltage changes the volume of the space-charge layer. The thickness $s=\frac{1}{2}(a-d)$ of the space-charge layer can be calculated, at least for $s \ll d$, from the Langmuir–Child law

\[ j_+ = \frac{1}{9\pi}\sqrt{\frac{2e}{M}\frac{(U_p-U_S)^{3/2}}{s^2}}. \tag{1,4} \]

In the case of sufficiently high plasma densities $N$ [when, according to (1,3), $j_+$ is large and, consequently, $s$ is small], the volume of the space-charge layer is only slightly greater than the volume of the probe. If so, then the inflow of ions to the probe is practically independent of the voltage at the probe as long as the probe is negative with respect to the surrounding plasma. Therefore the left-hand part of the probe characteristic, where the probe is sufficiently negative to completely deflect the inflow of electrons to itself, is approximately horizontal. The electron current, arising for less negative probes, can be quite clearly separated from the extrapolated course of the ion current shown by the dotted line in Fig. 3. As a result, it becomes possible to decompose the total current into ionic and electronic parts.

All these considerations are valid, generally speaking, only at low pressure of the neutral gas—the mean free path

very large compared with the thickness of the layer $s$. Only in this case is it possible to use equation (1.4), which is strictly applicable only to high vacuum. Collisions within the layer not only violate the law of free flight, which underlies the derivation of (1.4), but also lead to ionization, owing to which they affect the established equilibrium of the space charge$^1$.

In the case where there are no negative ions in the plasma, but only positive ions and electrons, probe measurements give a very complete description of the plasma. We have already determined above the electron velocity (and correspondingly $T_{-}$) and the electron concentration $N$. On the basis of the quasineutrality condition, the concentration of positive ions is also known to us—it is likewise equal to $N$. By measuring the ion current $j_{+}$, one can, with the aid of equation (1.3), find the mean velocity of the disordered ions $w_{+}$. From the formula $w_{+}=\sqrt{\dfrac{8kT_{+}}{\pi M}}$ one obtains the “ion temperature”; however, the ion velocity distribution need not coincide with the Maxwellian distribution. Such a calculation leads in many cases$^2$ to impossibly large values of the mean kinetic energy of the ions, which (see Section II) can in no way be taken as greater than the mean “thermal” energy of the electrons.

In order to explain the magnitude of the observed positive-ion currents, it is necessary to assume that the concentration of positive ions in these cases is much greater than the concentration of electrons. The quasineutrality condition compels one to suppose that in the plasma, even in the case of metallic vapors and noble gases, there is a large number of negative ions. Indeed, analyses of plasma particles, mainly by the canal-ray method, indicate the presence of negative ions. Direct determination of the number of negative ions $^{334,500,374}$ (and, consequently, of the number of additional positive ions) in a plasma by a method analogous to the probe method is as yet impossible. This, however, is immaterial for most plasma properties, which are determined primarily by the content of free electrons in the plasma.

The contact potential $\Delta U$ between the plasma and an isolated probe is determined by the equality of the electron current to the ion current. The electron current is smaller than the influx of electrons to a probe situated at the plasma potential,

\[ \frac{Nw}{4}=\frac{N}{4}\sqrt{\frac{8kT}{\pi m}}, \]

by a factor equal to the fac—

$^1$ Probe measurements have repeatedly been carried out at high pressures, where these assumptions are not fulfilled. As examples we may cite the works of Nottingham $^{377}$, Zibold $^{476}$, Mazon $^{343}$, Franckstein and Arndt $^{157}$, Meyer $^{576}$, in which an arc burning in a free atmosphere was investigated; Nyumena $^{373}$, Sommermeyer $^{490}$, and Bonnies $^{40}$, who investigated, partly experimentally and partly theoretically, discharge phenomena at low pressures. See also the works of Granovsky, Klarfeld, and Fabrikant $^{177,178}$.

$^2$ See, for example, Kömnik $^{252}$; an excess of positive ions had already earlier been indicated by Langmuir and Mott-Smith $^{305}$, Compton, Turner, and Mac-Karli $^{64}$. On the need to take negative ions into account, see p. 194.

Boltzmann factor $e^{-\frac{e\Delta U}{kT_-}}$. The ion current retains the value

\[ \frac{Nv_+}{4}=\frac{N}{4}\sqrt{\frac{8kT_+}{\pi M}}. \]

From the equation

\[ \frac{N}{4}\sqrt{\frac{8kT_-}{\pi m}}\, e^{-\frac{e\Delta U}{kT_-}} = \frac{N}{4}\sqrt{\frac{8kT_+}{\pi M}} \]

one finds the contact potential

\[ \Delta U=\frac{kT_-}{2e}\ln\frac{T_-M}{T_+m}. \]

Langmuir probes have been used in a very large number of works as an auxiliary means for studying discharges. A mere enumeration of all these works would force us beyond the scope of the present article. We shall therefore point only to certain works which either extend this method$^{421,493,101,206}$, or criticize it and indicate the limits of its applicability$^{115,58,82,81,76,449,471,260}$, or, finally, works in which the same results were obtained by means of probes in other ways$^{174,364,234,378}$$^{1}$.

In gas discharges at high pressure, in contrast to the case just considered of a nonisothermal discharge plasma at low pressure, the temperatures of the electrons and of the other plasma components differ little from one another (for the basis of this circumstance see Section II). Thus, for example, the plasmas of a high-pressure mercury lamp ($p\simeq 25$ atm) and of an ordinary arc burning in atmospheric air are to a considerable degree isothermal, i.e., between the highest electron temperature $T_-$ occurring in the plasma and the lowest temperature of the neutral gas $T_0$ there is only an insignificant percentage difference$^{2}$. The temperature of the neutral gas has been determined by many methods$^{3}$. Measurements show that the temperature of the neutral gas depends sharply on the external cooling conditions$^{4}$ and, for an arc in air, lies in the interval $5\,000$—$18\,000^\circ$. Determination of the electron temperature

$^{1}$ The predecessor of the modern probe method is Brockman$^{57}$.

$^{2}$ Thus, for example, Mannkopf$^{341}$ considers that in an arc burning in air the difference $T_- - T_0$ is less than $20^\circ$ when the arc temperature is of the order of $6\,000$—$8\,000^\circ$. Mazon (private communication) considers, on the basis of new measurements made by him at the Utrecht Institute, that this difference is considerably larger.

$^{3}$ For a review of works on optical determinations of temperature by comparing the intensities of lines and bands, see the literature, for example, 209, 309, 310, 502, 130, 272, 245, 125—127, 408, 29. For a review of other experiments see, for example, 395, 388, 390, 387, 393, 385, 389, 392, 394, 224, 546, 321, 322. In addition, one may mention the works of Richter$^{423}$ and of Engel and Steenbeck$^{126,128}$, as well as $^{130}$ (where a detailed review of older works is given).

$^{4}$ The influence of cooling lies in a direction that at first glance seems paradoxical. With weak cooling, the arc has relatively low temperatures (a broad arc); with intensive cooling, the arc forms a hot (thin) current filament. See on this in Kirchstein and Koppelmann$^{242}$.

is carried out, depending on the circumstances, with the aid of probes (see on similar work footnote 1 on p. 201) or also spectroscopically; this will be discussed in more detail in sections VII and VIII. A very important indication of the far-reaching isothermality of such a high-pressure arc is given by the fact that its electrical behavior both in the stationary state$^{396,126,223}$ and under dynamic changes$^{128,507,479a}$ is well explained by the assumption of equilibrium with respect to the degree of ionization (and, consequently, also of the conductivity of the gas) with the existing gas temperature (“thermal ionization” of the gas, see section VIII, a).

In the formulas of Eggert and Saha for isothermal ionization, calculated from the condition of equilibrium of chemical dissociation (which proceeds according to the reaction neutral atom + work of ionization = ion + electron), the pressures of the electron gas and the ion gas are completely equivalent to the pressures of the other plasma components; the sum of the partial pressures of particles of all kinds is equal to the total pressure under which the gas is found. It should be noted that the same can be justified also for a nonisothermal plasma$^{577}$. Let us return to the calculation carried out in deriving (1, 2). There is a fall of concentration in the direction in which the positive and negative particles diffuse; the flux densities of the particles are equal to each other and are equal to $n=-D_a \operatorname{grad} N$, where $D_a$ is the ambipolar (ambipolar) diffusion coefficient. The particles move through the neutral gas with friction and therefore act on a unit volume of gas with the force $n(\rho_+ + \rho_-)$. Here $\rho_+$ and $\rho_-$ are friction coefficients determining the magnitude of the friction force acting on a particle moving with unit velocity in the neutral gas. Thus, in the stationary case in a neutral gas, owing to the fall of the neutral pressure $p_0$, there must arise a hydrodynamic force equal in absolute value and opposite in sign to the friction force

\[ n(\rho_+ + \rho_-)=-D_a(\rho_+ + \rho_-)\operatorname{grad} N, \]

carried by the ions. From the equation

\[ \operatorname{grad} p_0=-D_a(\rho_+ + \rho_-) \]

we find by integration

\[ p_0 + D_a(\rho_+ + \rho_-)N=\text{const}. \]

The friction coefficient $\rho=\dfrac{e}{b}$; for each kind of particle Townsend’s relation$^{369}$ holds,

\[ \frac{D}{b}=\frac{kT}{e}. \]

Taking these relations into account and substituting for $D_a$ its value, according to formula (1, 2), we find

\[ p_0 + NkT_- + NkT_+=\text{const}. \]

Table 1

Characteristic data for frequently encountered typical plasmas

No.	Gas	Occurs in the following cases:	Pressure	Discharge current in cm	Length in cm	Electron concentration in cm$^{-3}$	Electron temperature in °K	Ion temperature in °K	Neutral-gas temperature in °K
I	Mercury vapor	Discharge at low pressures. Heated cathode	$5\cdot 10^{-3}$ torr	$\simeq 0.1$	Sphere $\varnothing\,15$	$1\cdot 10^{10}$	30 000	1 000 ?	300
II	Air	Heaviside layer	$\lesssim 1\cdot 10^{-2}$ torr $^{1)}$	—	$>10^{6}$	$\lesssim 10^{6\,1)}$	$\simeq 250$	$\simeq 250$	$\simeq 250$
III	Mercury vapor	Mercury rectifier	$5\cdot 10^{-2}$ torr	50—100	Tube $\varnothing\,6$	$1\cdot 10^{13}$	15 000	1 500 ?	425
IV	Neon	Advertising neon tube	1—5 torr $^{1)}$	$\simeq 1$	Tube $\varnothing\,2$	$5\cdot 10^{12}$	25 000	1 500 ?	400
V	Air	Arc in air	1 atm	$\simeq 10$	Columns $\varnothing\,1$	$1\cdot 10^{14}$	$\simeq 6\,500$	$\simeq 6\,500$	$\simeq 6\,500$
VI	Mercury vapor	Normal mercury lamp	1 atm	$\simeq 5$	Columns $\varnothing\,0.5$	$1\cdot 10^{15}$	$\simeq 6\,500$	$\simeq 6\,500$	$\simeq 6\,500$
VII	Mercury vapor	High-pressure mercury lamp	10 atm	$\simeq 5$	Columns $\varnothing\,0.2$	$1\cdot 10^{16}$	$\simeq 7\,500$	$\simeq 7\,500$	$\simeq 7\,500$
VIII	Mercury vapor	Mercury lamp of the highest pressure	100 atm	$\simeq 2$	Columns $\varnothing\,0.2$ Columns $\varnothing\,0.2$	$1\cdot 10^{17}$ $1\cdot 10^{17}$	$\simeq 7\,500$ $\simeq 8\,000$	$\simeq 7\,500$ $\simeq 8\,000$	$\simeq 7\,500$ $\simeq 8\,000$

$^{1)}$ In the following tables we shall take the pressure for plasma II to be equal to $1\cdot 10^{-2}$ torr and for plasma IV—1 torr.

Since $NkT_-$ and $NkT_+$ are the partial pressures of the electron and ion gases, the last equality represents the law of constancy of the sum of partial pressures in a stationary process and for a nonisothermal plasma. Especially significant may be the pressure of the electron gas in a low-pressure plasma, owing to the high electron temperatures occurring there.

Table 2

Survey of the numerical values of certain quantities characterizing a plasma, the measurement and calculation of which were discussed in Section I

No. of plasma according to Table 1	Partial pressure $^{1)}$, in torr	Contact voltage [eq. (1.5)], in V	Current density of ordered electrons $\left(e\dfrac{Nw_-}{4}\right)^{3)}$, A/cm$^2$	Current density of ordered ions $\left(e\dfrac{Nw_+}{4}\right)$, [eqs. (1.3)], A/cm$^2$	Thickness of the space-charge layer. Probe at 20 V negative relative to the plasma [eq. (1.4)], in cm	Mean free path of gas molecules (and ions)
I	$3.1\cdot10^{-5}$	$-20.9$	$4.3\cdot10^{-2}$	$1.3\cdot10^{-5}$	0.16	0.363
II	$2.6\cdot10^{-11}$	$-0.12$	$4\cdot10^{-7}$	$2\cdot10^{-9}$	$21.2^{4)}$	0.423
III	$1.6\cdot10^{-2}\,{}^{2)}$	$-9.7$	30	$1.6\cdot10^{-2}$	$4.7\cdot10^{-3}$	$5.15\cdot10^{-2}$
IV	$1.3\cdot10^{-2}$	$-14.4$	19	$2.3\cdot10^{-2}$	$6.9\cdot10^{-3}$	$1.4\cdot10^{-3}$
V	$6.7\cdot10^{-2}$	$-3.05$	$2\cdot10^{2}$	0.85	$1\cdot10^{-3}\,{}^{4)}$	$1.5\cdot10^{-4}$
VI	$6.7\cdot10^{-1}$	$-3.58$	$2\cdot10^{3}$	3.3	$3\cdot10^{-4}\,{}^{4)}$	$5.2\cdot10^{-5}$
VII	7.5	$-4.13$	$2.1\cdot10^{4}$	34	$1\cdot10^{-4}\,{}^{4)}$	$6\cdot10^{-6}$
VIII	82	$-4.37$	$2.2\cdot10^{5}$	360	$3\cdot10^{-5}\,{}^{4)}$	$6\cdot10^{-7}$

$^{1)}$ Equal to $NkT_-$; to convert to torr, multiply by $\dfrac{1}{981\cdot1.36}$.

$^{2)}$ Of the order of the total pressure.

$^{3)}$ See text, p. 199.

$^{4)}$ Considerably greater than the mean free path. This contradicts the assumptions of the derivation. Correct measurements with a probe are in principle impossible here.

It is considered that the magnitude of the electron pressure explains (Tonks $^{517}$) the large mechanical forces in the form of reaction pressures between the cathode and the plasma $^{503,432}$. These forces are especially significant because, at the cathode, not only is the electron energy $\approx kT_-$ large (acceleration in the region of the cathode fall!), but also the electron density.

II. EXCHANGE OF ENERGY OF THE PLASMA COMPONENTS IN COULOMB INTERACTION; MICROFIELD; RELAXATION LENGTH

The facts set forth in the preceding section—namely, the existence of a definite electron temperature, which in a nonisothermal plasma may differ by an order of magnitude from the temperature of the other plasma components—are quite easy to explain qualitatively (see, for example, M. Steenbeck $^{496}$, V. Utterhoeven $^{530}$, K. Sommermeyer $^{487}$). Despite the fact that the external electric field, which

is produced by the voltage applied to the electrodes, acts with equal force both on electrons and on ions, the energy imparted by the field per unit time to an electron is much greater (on the average by a factor

\[ \sqrt{\frac{m_+}{m}} \]

than this same quantity for an ion; this is understandable, since the lighter electrons follow these forces more rapidly than the ions. In the stationary case, electrons and ions must transfer the power imparted to them to the plasma components that possess less energy—these are, first of all, the neutral gas molecules. On the basis of the law of momentum, an electron in an elastic collision with an atom transfers to the latter only an insignificant part of its energy; in the most favorable case this part is equal to

\[ \frac{4m}{M_0}, \]

i.e. $2 \cdot 10^{-3}$ for $H_1$ and $1 \cdot 10^{-5}$ for Hg. Therefore, on the average the kinetic energy of the electrons must be so many times greater than the energy of the neutral gas that this part alone would be sufficient to transfer the power obtained by the electrons from the field. Inelastic, exciting or ionizing collisions do not affect what has been stated; these collisions occur so rarely that all together they take from the electrons an energy of the same order as the loss of energy in an elastic impact1. On the contrary, ions, possessing a large mass, can lose all their energy in a single impact. Therefore the mean kinetic energy of ions exceeds the energy of the neutral molecule only by the amount of the energy gained by the ion in its last free path2.

Thus, the mean kinetic energy of the electrons can be much greater than the energy of the ions and neutral atoms. From the simple picture considered it also follows that the energy of the electrons is the more

less it exceeds the energy of neutral atoms, the more collisions per unit time the electrons undergo with neutral molecules of the gas, in other words, the greater the pressure of the gas in which the discharge occurs. Thus, the plasma of an arc burning in the open atmosphere is almost isothermal1 (by convention $5\,000—20\,000^\circ$), whereas in a discharge in mercury vapor at a pressure of several thousandths of a torr electron temperatures of the order of $60\,000—80\,000^\circ$ are possible, while the gas temperature remains almost room temperature. Despite complete mixing of the gas, in this case there is very poor “thermal contact” between the electron gas, on the one hand, and the ionic and neutral gas, on the other.

If the number of electrons per unit volume is not small in comparison with the number of molecules, then, as is easy to see, the plasma electrons exchange energy among themselves in such a way that a Maxwellian velocity distribution is established. In a “collision” of an electron with another electron, one of them may transfer to the other all its kinetic energy; if the frequency of electron–electron collisions is to some degree comparable with the frequency of electron collisions with neutral atoms (at least the former is a hundredth or a thousandth part of the latter), then the energy exchange according to the electron–electron scheme is considerably greater than the transfer of energy by an electron to a neutral atom or ion. The mutual exchange of energies of the gas particles, overcoming all kinds of obstacles, is sufficient for a Maxwellian velocity distribution to be established for these particles. Under these conditions it makes sense to speak of the “temperature” of the electrons, which, naturally, has large values in accordance with the large kinetic energy of the electrons. It is easy to see that a more or less definite “temperature” can be ascribed to the ionic gas only when it is isothermal with the neutral gas2.

This picture, so simple in qualitative terms, becomes very obscure if one attempts to trace quantitatively the establishment of the electron temperature. In Maxwell’s classical derivation, even a small number of collisions is already sufficient for a particle that has had an unusually small or unusually large velocity to acquire the value of the mean velocity. Any perturbation of the Max—

of the Maxwellian velocity distribution is thereby destroyed within a short interval of time, sufficient for all particles to undergo several collisions. Carrying out an analogous consideration for the electron gas of a plasma runs up against the impossibility of defining, for infinitely extended Coulomb force fields, a concept analogous to an “impact.” This is achieved without difficulty for neutral particles, because their fields decrease very rapidly at high powers. In the case of Coulomb fields, divergent integrals enter the formulae of the kinetic theory of gases; moreover, the integrals diverge for large distances between particles. This means that “impacts” at a considerable distance play an essential role. In interaction at large distances, the velocities of the particles change only slightly both in absolute magnitude and in direction. Thus, what are essential are those collisions in which the velocity vector changes only slightly (L. Landau $^{285}$). One might try, for a rough estimate, to define an “impact” as the approach of electrons to a small segment $p$, in which a considerable part (say one half) of the energy $kT_{-}$ is transferred (where $T_{-}$ is the electron temperature), and to disregard all other interactions of the particles. Then

\[ \frac{e^{2}}{p} \simeq \frac{1}{2} kT_{-} \]

and the cross section of the impact is

\[ Q = p^{2}\pi \simeq \frac{4\pi e^{4}}{(kT_{-})^{2}}; \]

if the number of electrons per unit volume is $N$, then the mean free path would be determined, under these assumptions, as

\[ \lambda \simeq \frac{1}{NQ} = \frac{(kT_{-})^{2}}{4\pi Ne^{4}}. \]

Further, analogously to the kinetic theory of gases, one could introduce a “relaxation length” $s$, i.e. the length of the segment which an electron must traverse in order to enter into a Maxwellian distribution; this length is equal to several (approximately four) mean free paths, i.e.

\[ s_{1} \simeq \frac{(kT_{-})^{2}}{\pi Ne^{4}} = \gamma_{1}\frac{(kT_{-})^{2}}{Ne^{4}}. \tag{2,1} \]

There is no doubt that the segment $s_{1}$ computed in this way is sufficient for the establishment of a Maxwellian distribution; moreover, there is no doubt that a substantially shorter segment is sufficient for this. In fact, the establishment of equilibrium proceeds much faster than according to (2,1). Thus, for example, Langmuir and Mott-Smith $^{303}$ found for an Hg plasma with $N = 4 \cdot 10^{9}\ \mathrm{cm}^{-3}$ and $T = 36\,000^\circ$, that already at a distance of $3\ \mathrm{cm}$ from the disturbing electrode, which drew off from the plasma all the fast electrons, there exists a complete Maxwellian distribution, although with a lower temperature (a decrease in energy due to the fact that all ...

fast “electrons”). Calculations by equation (2,1), however, give for these conditions a relaxation length of several tens of thousands of centimeters, which exceeds the value given by experiment by at least a thousand times.

Energy exchange does not occur between two electrons situated in especially close proximity; on the contrary, each electron is acted upon simultaneously by the aggregate of all the others; at least one must speak of the action, on the given electron, of a large number of its neighbors. We must investigate the change in the energy of a single charge by the electric field of the plasma; therefore this field must first be considered in greater detail. It must be admitted that our knowledge is still far from sufficient to explain such an intense exchange of energies as was established by the experiment described above.

As was noted above, at a sufficiently large electron concentration $N$, the field inside the plasma is determined to a far greater extent by the charges of the plasma than by the electrodes. The charges of ions and electrons have equal significance for the instantaneous picture of the field (approximately equal effective cross sections of ions and electrons for momentum exchange). Conversely, for energy exchange the ions play a secondary role, since they possess large masses (large effective cross section for electrons, small for ions). With respect to the motion of the electrons the ions may be regarded as at rest; therefore the presence of ions is the cause of scattering of electrons in various directions; in other words, it increases the circuitous paths of the electrons, but does not take from them a noticeable amount of energy. The mean distance between charges is a quantity of order $N^{-\frac{1}{3}}$; consequently, the field is a quantity of order $4eN^{\frac{2}{3}}$.

For a discharge in mercury vapor $N = 10^{12}\ \mathrm{cm}^{-3}$, and the field is approximately $60\ \mathrm{V/cm}$, i.e. 100 times greater than the electrode field maintaining the discharge. It is therefore possible in what follows not to consider the field of the electrodes (for instantaneous accelerations)¹.

The field between the individual particles of the plasma depends, of course, on the instantaneous arrangement of the charge carriers and therefore changes in magnitude and direction. The value of the field must be specified within a very small volume and a very small time interval. Therefore the field between the plasma particles is naturally called a “microfield.” A two-dimensional analogue of the picture of equipotential surfaces of a plasma is a very restless surface of water, separate regions of which rise and fall relative to one another (see, however, Section V on traveling waves). A sphere (at whose center the electron is located) of such a size that, on the average, one electron falls within it, we shall call a cell of the microfield or, more briefly, a microcell. Within such a cell the oscillations

¹ The energy acquired by the electrons is determined by the field of the electrodes, which creates acceleration throughout the entire region of the discharge. The “microfield,” owing to its oscillations in magnitude and direction, on the average does not transmit energy to the electron gas.

of the electric field are small in percentage terms. Let us assume that the plasma charges are distributed throughout the volume completely at random (we shall correct this assumption later); then the presence of an electron at a given place in the volume does not depend on the instantaneous positions of the remaining electrons.

The mean value of the intensity of the electric microfield is, of course, zero, since the quantity $-E$ occurs just as often as the quantity $+E$. The mean value of the modulus of the electric vector $\overline{|E|}$ can be specified. As was mentioned above, the dimension of this quantity corresponds to the dimension of the product $eN^{2/3}$.

Gabor $^{162}$ calculated the value of $\overline{|E|}$, which to a considerable degree does not depend on the exact configuration of the charges:

\[ \overline{|E|}=12.2eN^{\frac{2}{3}}. \tag{2,2} \]

Taking into account the field of the ions, Langmuir $^{296}$ gives for $\overline{|E|}$ the value $13.7eN^{2/3}$.

Engel and Steenbeck consider that $\overline{|E|}$ is greater than $12\left(\frac{\pi}{3}\right)^{2/3} eN^{2/3}$ and is equal to approximately $20eN^{2/3}$. For a given point of the plasma this field varies with time in magnitude and direction with a frequency that corresponds approximately to the mean flight time of an electron along the mean path between two plasma charges.

An effective value of the electric-field intensity $\sqrt{\overline{E^2}}$ can also be specified.

If, at the observation point where one wishes to determine the field, there is a point electron, then the quantity $\sqrt{\overline{E^2}}$ tends to infinity. To calculate this quantity, one mentally cuts out from the plasma a sphere of radius $a$ around the observation point and assumes a uniform distribution of probability density outside this sphere. Then, according to Holtsmark, and especially simply according to Ornstein $^{384}$, the following value is obtained for the effective field intensity in a specified direction $x$:

\[ \sqrt{\overline{E_x^2}}=\sqrt{\frac{4\pi e^2N}{3a}}. \tag{2,3} \]

The effective value of the field in an arbitrary direction is the geometric sum of the three components, i.e. it is $\sqrt{3}$ times larger than (2,3). The quantity $a$ (the minimal distance), whose introduction was necessary for mathematical reasons, also has a clear physical meaning. If, for example, the exciting action of the field on an atom is being investigated, then $\pi a^2$ denotes the effective cross section of the atom; in the case of the action of the field on an electron at the zero point, the quantity $a=\frac{e^2}{kT}$, i.e. it is equal to the minimal distance at which some electron possessing-

...which is the most probable velocity of the Maxwellian distribution, can approach the electron at the point of observation1.

The arithmetic mean of the potential is also equal to zero (or to a constant). It is possible, however, to determine the fluctuations of the arithmetic mean value of the potential inside a sphere of radius $a$. For the two-dimensional analogy of the plasma at the surface of water this means fluctuations of the height of a piece of wood of radius $a$ floating in the water. Fluctuations of the mean value of the potential are also analogous to changes in the potential of the plasma in a conducting sphere of radius $a$ suspended in it. In order that there be no perturbations in the plasma, it must be assumed that the sphere is in thermal equilibrium with the plasma, i.e., either it emits (for example, thermally) into the plasma as many electrons as arrive at it from the plasma, or it is transparent to the electrons of the plasma. On the average such a sphere contains $\frac{4}{3}a^3\pi N$ electrons; the effective deviation due to statistical fluctuations is equal to $\sqrt{\frac{4}{3}\pi a^3N}$, and the effective charge is equal to $e\sqrt{\frac{4}{3}\pi a^3N}$. Consequently, the effective potential

\[ \sqrt{\overline{U^2}}=\frac{1}{a}e\sqrt{\frac{4}{3}\pi a^3N} =\sqrt{\frac{4\pi e^2N}{3}}\cdot a. \tag{2,4} \]

The rate of the potential fluctuations $\sqrt{\overline{U^2}}$ is readily calculated from the statistical fluctuations of the influx and outflow of charges into the sphere. To within a numerical factor of order unity, the mean time interval between two returns of the value $\sqrt{\overline{U^2}}$ to zero is equal to the time necessary for an electron of average velocity to traverse a path segment equal to the radius of the sphere; this result is quite understandable from the standpoint of general physical considerations. Attempts have been made, with the aid of such a model, to calculate the exchange of energy between an electron moving through the plasma and the plasma. Such attempts are certainly unjustified; equation (2,4) leads to excessively large potential fluctuations, which even become arbitrarily large as the size of the sphere increases without bound!

In reality, some of the electrons which should have penetrated into the uncharged sphere from the surrounding space will be repelled by the sphere, charged negatively, and their penetration into the sphere will be impeded. Therefore the probability of finding an electron at a given point of the plasma depends on the positions of neighboring electrons and cannot be regarded as constant in space if the electron temperature is finite. For example, the probability of finding

the electron in the neighborhood of a positive ion is greater than the spatially averaged probability. Therefore the mean density of negative charges at these points is greater than is required by the condition of quasineutrality. Around each positive charge there forms a cloud of negative space charge, excessive relative to the mean statistical value, and conversely. These clouds of opposite sign screen the action of the central charge. A calculation based on Poisson’s equation and the Boltzmann principle corresponds completely to the Debye—Hückel theory of strong electrolytes1. The theory shows that the potential of the central charge varies with distance not as in a vacuum—proportionally to $\frac{1}{r}$, but much more rapidly, namely as

\[ \frac{1}{r}\cdot e^{-\frac{r}{D}}, \]

where the quantity

\[ D=\sqrt{\frac{kT}{4\pi e^2 N}} \tag{2,5} \]

is called the Debye radius ($T$ is the electron temperature). It is often assumed that the screening of the central charge, owing to the polarization of the plasma that occurs, consists in the following: inside a sphere of Debye radius there is an undistorted central field, while outside the sphere this field does not act. In order that the averaging underlying the calculation be legitimate, it is necessary to require that the number of plasma charges inside the sphere of Debye radius be large compared with unity, i.e.

\[ N\cdot \frac{4}{3}D^3\pi = \frac{(kT)^{\frac{3}{2}}}{6\pi^{\frac{1}{2}} e^3 N^{\frac{1}{2}}} \gg 1. \tag{2,6} \]

Then, on average, the number of charges unlike the central one will exceed by unity the number of like charges. In practice, equation (2,6) is satisfied for all gas-discharge plasmas encountered—

GASES IN THE PLASMA STATE

of series. Abstracting from numerical factors, one may represent (2.6) in the form

\[ kT \gg e^2 N^{\frac{1}{3}}; \]

thus, the thermal energy of the electron must be large in comparison with the potential energy between two neighboring particles of the plasma (the distance $\simeq N^{-\frac{1}{3}}$!).

Gabor believes that equation (2.2) should, to a good approximation, hold for a plasma with a finite electron temperature. An equation analogous to (2.3) for the effective field strength in a plasma, taking into account the interaction of the plasma electrons, has not yet, so far as we know, been formulated. On the other hand, it is easy to formulate an equation replacing (2.4) for the potential averaged over a sphere of radius $a$. A sphere of radius $a$, suspended in the plasma and at potential $U$, has the energy $\frac{1}{2} aU^2$. Since the sphere has one degree of freedom, on the other hand it has the energy $\frac{1}{2} kT$. Hence it follows directly that

\[ \sqrt{\overline{U^2}}=\sqrt{\frac{kT}{a}}. \tag{2.7} \]

Regarding the calculation of the speed of oscillations, the remarks made in the discussion of equation (2.4) are valid. For large radii $a$, equation (2.7) gives smaller voltage oscillations than (2.4); on the contrary, for small spheres (2.7) gives larger quantities. However, charge oscillations cannot exceed the value given by equation (2.4) (which does not take back action into account). Consequently, for small radii (2.4) gives more correct figures. The roots of the incorrectness in the thermodynamic derivation of (2.7) lie in the small number of particles inside the sphere: the second law of thermodynamics is violated. The transition region from the applicability of equation (2.4) to the applicability of (2.7) has, to within an inessential numerical factor, the magnitude of the Debye radius $D$ [formula (2.5)].

With the aid of the information considered about the field of the plasma, one can estimate more accurately the values of the relaxation lengths. Between two neighboring regions, each of which has the size of the Debye radius, in the first approximation there is no interaction (see above). The potential oscillations of these regions occur independently of one another; therefore the relative effective change in potential is $\sqrt{2}$ times larger than the value given by equation (2.7) for $a=D$. Thus, an electron which has flown through two Debye regions will change, in the field of the plasma, its energy by an amount equal to

\[ \sqrt{2}\cdot e\sqrt{\frac{kT}{D}}. \]

On the path $s=n\cdot 2D$, i.e. after traversing $n$ Debye regions, it will change its energy by

\[ \sqrt{n}\cdot e\sqrt{\frac{kT}{D}}; \]

if the total change of energy is equal to $kT_{-}$, then the quantities $s$ may be regarded as relaxation lengths. From the equality

\[ kT_{-}=\sqrt{\frac{s}{2D}}\cdot e\,\sqrt{\frac{kT_{-}}{D}} \]

we find, with the aid of (2,5),

\[ s_2=\frac{(kT_{-})^2}{2\pi e^4 N}=\gamma_2\cdot\frac{(kT_{-})^2}{e^4 N}. \tag{2,8} \]

This equation differs from (2,1) only by the factor $\frac{1}{2}$; the functional dependence on the quantities characterizing the plasma has remained the same. Consequently, this mechanism also does not consider the exchange of energies so accurately as to yield agreement with the experimental data. Considerably better results are obtained by a calculation in which the Coulomb interaction of electrons located farther from one another than was assumed in the derivation of (2,1) is taken into account. In this case the exchange of energies of the given electron with one electron of the plasma is considered as an isolated collision; the other electrons do not interact with the given ones if their distance is greater than the Debye radius $D$. Drayvesteyn $^{97}$ finds

\[ s_3 \simeq \frac{9\sqrt{6}}{2\pi\ln\left(2.7\,\dfrac{kT_{-}}{e^2 N^{1/3}}\right)}\cdot\frac{(kT_{-})^2}{e^4N} =\gamma_3\cdot\frac{(kT_{-})^2}{e^4N}. \tag{2,9} \]

The expression standing under the logarithm is, to within a numerical factor, equal to the ratio of the mean thermal energy of the electrons to the mean potential energy of two neighboring plasma particles. In discussing equation (2,6), we have already indicated that this ratio is much greater than unity.

Landau $^{285}$, under approximately similar assumptions, arrives at the formula

\[ s_4\simeq \frac{1}{3\ln\left(\dfrac{kT_{-}}{e^2 N^{1/3}}\right)}\cdot\frac{(kT_{-})^2}{e^4N} =\gamma_4\frac{(kT_{-})^2}{e^4N}. \tag{2,10} \]

Davydov’s $^{75}$ investigation gives

\[ s_5=\frac{2}{3\pi\ln\left(0.43\,\dfrac{kT_{-}}{e^2N^{1/3}}\right)}\cdot\frac{(kT_{-})^2}{e^4N} =\gamma_5\frac{(kT_{-})^2}{e^4N}. \tag{2,11} \]

Thomas $^{508}$ and Gvosdover $^{196}$ $^{1}$ give approximately the same formulas. Starting from formula (2,2) for the intensity of the electric microfield and taking the intensities of the fields in neighboring microcells to be independent, Engel and Steenbeck $^{122}$ propose the following formula [taking into account an error in the calculation: an error by a factor of 9, for instead of $\left(\frac{3}{2}kT_{-}\right)^2$ one must have $\left(\frac{1}{2}kT_{-}\right)^2$]

\[ s_6\simeq \frac{1}{20}\cdot\frac{(kT_{-})^2}{e^4N} =\gamma_6\frac{(kT_{-})^2}{e^4N}. \tag{2,12} \]

$^{1}$ See also S. Pekar $^{401}$.

In an entirely different way, namely by taking into account the plasma field arising during plasma oscillations, Langmuir $^{297}$ outlined (on this, see Section IV):

\[ s_7 \approx \frac{4}{\pi^2 \ln \left(0.29\,\frac{kT}{e^2 N^{1/3}}\right)} \cdot \frac{(kT_-)^2}{e^4 N} = \gamma_7 \frac{(kT_-)^2}{e^4 N}. \tag{2,13} \]

It is extremely interesting that models differing from one another lead to one and the same functional dependence on $T$ and $N$; moreover, the last four formulas give, for plasmas encountered in practice, almost identical numbers. It would seem that all calculations leading to the values $s_1,\ldots,s_7$, being based on different principles, confirm one another; nevertheless, the relaxation lengths calculated from

\[ \frac{1}{s}=\sum \frac{1}{s_i} \]

are approximately 30 times larger than the measured values $^{1}$.

One might think that the interaction between electrons is underestimated by these calculations and that the action of distant particles should be integrated not over a sphere of Debye radius, but over the volume of the vessel, in other words, over all points of the plasma. It turns out, however, that this changes the results of the preceding calculations only by a factor of two and at the same time contradicts general ideas about plasma. If the integration is carried out with the limits extended to infinity, then in principle the interaction becomes (logarithmically) infinitely large and, consequently, the segments $s$ become infinitely small $^{2}$. Therefore one cannot proceed in this way in explaining the observed small relaxation lengths. There are a number of other processes that can affect the exchange of energy in an electron gas; among them are second-kind collisions $^{3}$ with excited neutral particles. As an explanation, there is also put forward the energy exchange of two electrons in the field of a neutral atom in the form of a triple collision, mentioned by Scherzer $^{458}$. It may also be that more accurate experimental investigations will lead to larger relaxation lengths.

The appearance of the function

\[ \frac{(kT_-)^2}{e^4 N} \]

in the calculation of $s$ from individual collisions into some effective cross sections is not difficult to explain. If, as for individual collisions, $s$ must be proportional to $\frac{1}{N}$, then from $kT$, $N$, and $e$, by dimensional considerations, a length can be formed only in the form written above. This does not mean, however, that any

$^{1}$ If one takes into account that the actual plasma oscillations may be much more intense than corresponds to equilibrium at $T_-$ (see below, Section V, d), then (2,13) may give much smaller values of $s_7$. It may be that this path will succeed in avoiding the difficulties.

$^{2}$ The establishment of an upper limit of integration is necessary here for the same reasons and on analogous grounds as in the case (see below, Section VIII, a) of the cutoff of the term scheme in calculating the concentration of excited plasma particles.

$^{3}$ On the influence of metastable states on the electron temperature, see, for example, Spivak and Reichrudel $^{494}$.

Table 3

Microfield

No. of plasma according to Table 1	Debye radius for electrons [eq. (2,5)], cm	Debye radius for ions [eq. (2,5)], cm	Number of charges in the Debye sphere [eq. (2,6)]	Ratio of the mean interaction potential to the mean kinetic energy of electrons, $\dfrac{e^2 N^{1/3}}{kT}$	Linear mean value of the microfield strength [eq. (2,2)], V/cm	Effective value of the microfield strength [eq. (2,3)$^2$], V/cm	Mean frequency of exchange of places, $w \sim N'^{1/3}$, sec$^{-1}$	Relaxation length $s_1$, cm [eq. (2,1)]	Relaxation length $s_3$, cm [eq. (2,11)]	Relaxation length $s_8$, cm [eq. (2,14)]	Relaxation length $s_9$, cm [eq. (2,15)]
I	$1.2\cdot10^{-2}$	$1.2\cdot10^{-2}$	$7.2\cdot10^{4}$	$1.2\cdot10^{-4}$	8.2	215	$2.3\cdot10^{11}$	$1.0\cdot10^{4}$	$8.7\cdot10^{2}$	$3.7\cdot10^{1}$	5.4
II	$1.1\cdot10^{-1}$	$7.8\cdot10^{-2}$	$5.5\cdot10^{3}$	$6.7\cdot10^{-4}$	0.018	0.2	$1\cdot10^{9}$	$7.2\cdot10^{3}$	$7.4\cdot10^{2}$	$9.4\cdot10^{1}$	$4.9\cdot10^{1}$
III	$2.7\cdot10^{-4}$	$2.7\cdot10^{-4}$	810	$2.4\cdot10^{-3}$	820	4 860	$1.6\cdot10^{12}$	2.6	$3.3\cdot10^{-1}$	$9.0\cdot10^{-2}$	$1.2\cdot10^{-1}$
IV	$4.9\cdot10^{-4}$	$4.9\cdot10^{-4}$	$2.4\cdot10^{3}$	$1.1\cdot10^{-3}$	510	4 380	$1.7\cdot10^{12}$	14	1.6	$2.8\cdot10^{-1}$	$2.2\cdot10^{-1}$
V	$5.6\cdot10^{-5}$	$4.0\cdot10^{-5}$	73	$1.2\cdot10^{-2}$	3 900	10 000	$2.3\cdot10^{12}$	$4.8\cdot10^{-2}$	$9.0\cdot10^{-2}$	$5.5\cdot10^{-3}$	$2.5\cdot10^{-2}$
VI	$1.8\cdot10^{-5}$	$1.3\cdot10^{-5}$	23	$2.6\cdot10^{-2}$	17 500	33 600	$5.0\cdot10^{12}$	$4.8\cdot10^{-3}$	$1.1\cdot10^{-3}$	$9.2\cdot10^{-4}$	$7.9\cdot10^{-3}$
VII	$6.0\cdot10^{-6}$	$4.2\cdot10^{-6}$	9	$4.8\cdot10^{-2}$	82 000	106 000	$1.2\cdot10^{13}$	$6.5\cdot10^{-4}$	$2.0\cdot10^{-4}$	$2.1\cdot10^{-4}$	$2.7\cdot10^{-3}$
VIII	$2.0\cdot10^{-6}$	$1.4\cdot10^{-6}$	$3\ ^3)$	$9.7\cdot10^{-2}$	390 000	350 000	$2.6\cdot10^{13}$	$7.4\cdot10^{-5}$	$3.3\cdot10^{-5}$	$4.0\cdot10^{-5}$	$8.8\cdot10^{-4}$

$^1$ Take into account the footnote on p. 212.

$^2$ Here $a=\dfrac{e^2}{kT}$; see the text. The effective value of the field strength at the location of the electron under consideration is determined; the electron’s own field is not taken into account.

$^3$ Inequality (2,6) is already invalid; here lies the boundary of applicability of the considerations set forth in Section II.

another mechanism of energy exchange, in which more than two electrons interact, cannot lead to a different functional dependence. We do not think, as Landau does, that a formula with a different functional dependence is a priori based on incorrect premises. This applies, for example, to two formulas for relaxation lengths proposed by Gabor $^{163,164,165}$. The derivation of these formulas, especially of the second of them, requires an unusually large amount of space; the physical assumptions on which the calculations are based are also rather difficult to review. Since we have no possibility of successfully discussing the derivation of these equations, we shall confine ourselves to writing down these two equations:

\[ s_8 = 0.43 \frac{(kT_-)^{\frac{5}{4}}}{e^{\frac{5}{2}} N^{\frac{3}{4}}} \tag{2,14} \]

and

\[ s_9 = 125 \frac{(kT_-)^{\frac{1}{2}}}{e N^{\frac{1}{2}}}. \tag{2,15} \]

We have indicated the numerical values of the constants, since in these cases they have dimensions. In the literature there is neither a critique of Gabor’s theory nor any investigations of the dependence of the relaxation length on $T_-$ and $N$. Despite the great difficulty of such measurements, it would undoubtedly have been possible to discover so large a difference between the powers of $T_-$ and $N$ in (2,15) and in other formulas. For the Langmuir—Mott—Smith example cited above, equation (2,14) gives for the relaxation length the value $64\ \text{cm}$, and equation (2,15) even $2.8\ \text{cm}$. In addition, one must take into account the curvature of the path as the electron traverses the relaxation length; for this Gabor introduces a factor of order $3$—$5$. In this case Gabor’s formulas give numbers that agree considerably better with experiment; it is to be regretted that the obscurity of the derivation prevents one from assessing its rigor. In his second paper Gabor takes into account the electrostatic opposing field created around the electron by an asymmetric distribution of the Debye—Hückel space charge. The plasma electrons do not have time to move aside from the path of the given flying electron; consequently, at these points a layer of positive charges is not formed immediately; on the other hand, behind the given electron the plasma electrons do not have time to return to their places (from which they have been driven off by the given flying electron). Such a charge distribution should retard the flight of the electron; thus, the kinetic energy of the given flying electron is transferred to the aggregate of all the other plasma electrons$^{1}$.

$^{1}$ The authors consider it possible that from the normalization of the equation drawn up for this case (unpublished) it follows that

\[ s \simeq \frac{(kT_-)^2}{e^4 N}. \]

Thus, Landau’s objection would seem to have a real basis for this case.

A similar field, opposing the instantaneous direction of the velocity, can also be determined by considering only the Coulomb interaction between electrons; this is how, for example, Dreivesten proceeds. The moving electron under consideration encounters in front of it plasma electrons in greater number and, on average, with greater velocity than it leaves behind it. Therefore braking will occur here as well. When the electron velocity is zero, the opposing field is, naturally, equal to zero. As the velocity increases this field appears and grows, but for very fast electrons it again begins to decrease. This happens because the short time during which the given electron encounters the plasma electrons prevents the transfer of a significant momentum. The maximum braking occurs approximately at the mean thermal velocity. The braking field of the asymmetric Debye charge cloud obeys the same rules; one may suppose that these fields are identical. The field opposing an electron with energy $\frac{mv^2}{2} \gg \frac{3kT_{-}}{2}$ has, according to Dreivesten $^{1}$, the value

\[ G=\frac{6\pi Ne^4}{mv^2}\cdot \ln \frac{mv^2}{2\sqrt{\pi Ne^2}} . \tag{2,16} \]

A quantitative confirmation of this equation by the experiments of Langmuir and his collaborators $^{294,295}$ was not achieved. The experiments consisted in gradually changing the velocity of an electron beam passing through the plasma. It turned out that at the beam intensities necessary for making the observations, a significant interaction of the beam electrons begins, which in this experiment is a disturbance. The qualitative relations are illustrated quite correctly by equation (2,16).

A number of problems connected with the most complex processes affecting the establishment of the Maxwellian distribution have been investigated; however, unfortunately, still more work is needed both in theory and in experiment in order to achieve a quantitative understanding of these phenomena.

III. ELECTRICAL CONDUCTIVITY OF PLASMA IN CONSTANT FIELDS

If a plasma is in an external electric field, then its positive and negative charges acquire accelerations parallel and antiparallel to the direction of the field; a directed current is superposed on the disordered thermal motion of the charges. The density of the total electric current is the sum of the electron and ion currents; however, the velocity of the electron flow considerably exceeds—

$^{1}$ Fast electrons cannot form around themselves a real Debye layer; therefore, in order to obtain convergent integrals in the calculation by the method of Coulomb fields, one must introduce into consideration a certain auxiliary layer, which, of course, has an artificial character. The magnitude of this layer enters only into the logarithmic term (2,16), and therefore in quantitative terms this indeterminacy plays no role.

the ion velocity enters, owing to the much smaller mass of the electrons. If the concentration of electrons is equal to or at least comparable with the concentration of ions, which is always the case owing to the condition of quasineutrality (provided only that the number of negative ions does not exceed the number of electrons in order of magnitude), then the density of the total current is practically equal to the density of the electron current:

\[ j \simeq j_{-}=Nev_{-}. \tag{3,1} \]

Thus, in order to determine the conductivity of the plasma

\[ \sigma=\frac{j}{E}, \]

it is necessary, if $N$ is known, to know only the velocity of motion of the electrons $v_{-}$ as a function of the electric-field strength, in other words, the mobility

\[ b_{-}=\frac{v_{-}}{E}: \]

\[ \sigma=eN_{-}b_{-}. \tag{3,2} \]

In this section we shall consider the motion of charges in a constant field, and also in a slowly varying field. For the latter field we shall compute the instantaneous velocity from the instantaneous field strength and from the mobility for constant fields. The motion of plasma charges at high frequencies will be considered in Section VI, a—Dielectric properties of plasma.

In order to establish the characteristic features of the conductivity of plasma, let us first dwell on the known considerations (E. Riecke$^{424}$, P. Drude$^{92}$, P. Langevin$^{288}$) concerning the motion of charges in a neutral gas. Between two collisions with molecules of the neutral gas an electron moves under the action of the electric field alone, with acceleration

\[ \frac{eE}{m}. \]

If the field $E$ is sufficiently small$^{1}$, then the increment of kinetic energy acquired by the particle during the time $\tau$ between two collisions is small in comparison with the already existing thermal energy

\[ \frac{3}{2}kT_{-}; \]

therefore the time of free flight $\tau$ may be expressed in terms of the mean free path $\lambda$ and the thermal velocity $w$, namely

\[ \tau=\frac{\lambda}{w}. \]

The path traveled in the direction of the field during the time $\tau$ is equal to

\[ z=\frac{1}{2}\frac{eE}{m}\tau^{2}; \]

therefore the mean drift velocity is

\[ v_{-}=\frac{e\lambda}{2mw}E. \]

Proper averaging over statistically distributed free-path lengths gives

\[ v_{-}=\frac{e\lambda}{mw}E=b_{-}\cdot E, \]

where $\lambda$ is the “mean” free path.

$^{1}$ A precise definition will be given below (see footnote 3 on p. 220).

The definition in the case under consideration of the “mean” free path depends on what interaction of particles flying past one another (up to what angle of deflection) is to be regarded as a collision. There is no such arbitrariness only in the case when the colliding particles are regarded as elastic spheres. The simplified Drude–Langevin formula, whose derivation scheme has just been presented, is based on the assumption that after a collision the directed velocity (which depends on the direction of the preceding free path) is on the average equal to zero1. This is by no means true in the general case, but only in the collision of a light solid elastic sphere with one of equal weight. Therefore, in the calculation of the mobility there appears an undetermined factor $a$ of order unity, unless in each individual case a very detailed calculation is made, in which account is taken of the probability distribution of deflection through a given angle as a function of the velocity at collision (the Ramsauer effect). For this, however, in most cases there are not enough experimental data. Another reason for the necessity of introducing an undetermined factor $a$ is that the velocity distribution of the electrons will be Maxwellian only in the case when the electrons are in intense energy interaction. For this a considerable concentration of charge carriers is necessary (see the preceding section). If the electron concentration is small, then in the field there arises another velocity distribution, depending on the kind of gas in which the electrons move; if, for such a velocity distribution, one substitutes in the mobility formula the arithmetic mean value instead of $\overline{w}$, an error of order unity is thereby made. For these reasons one obtains

\[ b = a \frac{e\lambda}{m\overline{w}}, \tag{3,3} \]

where for the most part $a \simeq 0.75$2.

At very low field strength the thermal velocity corresponds to the temperature of the gas; then $b$ does not depend on the field strength and $v \sim E$. If the field strength is large, then the following approximate relations hold: $T_- \sim E$, consequently, $\overline{w} \sim \sqrt{E}$, therefore $b \sim \frac{1}{\sqrt{E}}$ and $v \sim \sqrt{E}$3. The temperature of the electrons may be

is determined, for example, by measurements with probes; therefore, if only $\lambda$$^{1}$ is known, one can find from equation (3.3) the mobility up to an undetermined factor $a$, and from equation (3.2) the conductivity $\sigma$; the quantity $N$ can also be measured by probes.

If the concentration of plasma charges is small, and the density of neutral molecules is sufficiently large, then in practice the electron flight is limited by collision with a neutral molecule. Then $\lambda$ is calculated from the formula

\[ \frac{1}{\lambda}=N_{0}Q, \tag{3.4} \]

where $N_{0}$ is the concentration of neutral molecules (per cubic centimeter), and $Q$ is the effective cross section (in square centimeters) of the molecule with respect to electrons. The value of $Q$ may be taken from the measurements of Ramsauer and his collaborators$^{250}$; if the quantity $Q$ depends sharply on the electron velocities, then it must be averaged in accordance with the distribution of these velocities$^{349}$. At higher concentrations of electrons and ions, i.e., in a truly typical plasma, “collisions” between charges play, of course, a very important role, and in the case of a high degree of ionization—a decisive one. A clear understanding of this role encounters great difficulties, the causes of which are ultimately identical with the causes of the difficulties in calculating relaxation lengths (see the preceding section).

Already in calculating collisions between electrons and neutral molecules it was necessary to introduce an undetermined factor $a$

of the random motion, and that, in this way, it is the free-flight time that is specified, and not its length. The velocity of the electron after impact has the character of a “thermal” velocity if the electron loses an entirely negligible part of its velocity, but in direction the subsequent distribution in space is completely random. Therefore formula (3.3) is also applicable in the case when, in an impact, the electron loses on average a negligible part of its kinetic energy; the latter condition is automatically satisfied if the number of elastic collisions experienced by the electron is much greater than the number of exciting or ionizing collisions, i.e., practically always$^{213}$. It should be noted, however, that the behavior of ions is quite different; ions lose all their energy in an elastic collision, and therefore begin a new flight with zero velocity. Here, in strong fields, the flight time is determined by the length of the free path, according to the equality $\lambda=\dfrac{eE}{2m}\tau^{2}$. The mean velocity $v=\dfrac{\lambda}{\tau}$ will then also be proportional to $\sqrt{E}$, but for quite different reasons—not as a consequence of equation (3.3) for $T\sim E$ (V. Rogovskii$^{434}$). In the latter case one cannot speak of thermal motion in the strict sense of the word.

$^{1}$ On the contrary, one can calculate $\lambda$ by measuring the electron temperature, the electron concentration, and the conductivity; see, for example, Engel and Steenbeck$^{129}$, where the literature on this question is also cited. See also Killian$^{240}$. This method of determining $\lambda$ is particularly interesting because in this way anomalies in the density of the neutral gas that affect the discharge can be investigated. The anomalously large values of $\lambda$ found by Engel and Steenbeck undoubtedly indicate such distortions, for example the suction effect (Saugeffekt) of the discharge [not found in Töpler’s work$^{520}$] or a decrease in the pressure of the neutral gas as a consequence of high partial electron pressure (see above, Section I and$^{577}$).

[equation (3.3)], which also took into account the rather rapid decrease of the interaction forces with distance (in the case of a collision between elastic spheres the forces act instantaneously). Still less applicable for describing the process of “collision” of two plasma charges is the model of an elastic collision of two spheres with prescribed effective cross sections. It is true that we are in a favorable position in the sense that we know exactly the law of interaction of the particles—this is Coulomb’s law; moreover, unlike the collision of an electron with a neutral atom, the interaction does not depend on the nature of the ions. Consequently, in the present case there is no need to introduce any factor analogous to the indefinite factor $a$; the calculation can be carried out completely with known accuracy. This does not mean, of course, that it will be entirely free of arbitrary assumptions.

In a crude approximation one may use the following model. The state of an electron which has passed, under the action of the microfield of the plasma, through a segment $s$ (the relaxation length which we calculated above) does not depend on its previous velocity, either in magnitude or in direction. It is as if the electrons, over a path equal to $s$, undergo one collision which destroys the existing velocity. The effective cross section of these model collisions is equal to $\frac{1}{s}$; together with the effective cross section $N_0 Q$ of the neutral molecule, it gives the “effective” mean free path—the result of the combined action of microfields and neutral particles:

\[ \frac{1}{\lambda_{\mathrm{eff}}}=N_0Q+\frac{1}{s}=\frac{1}{\lambda}+\frac{1}{s}. \tag{3,5} \]

Since, according to Section II, the quantity $s$ is equal to

\[ \gamma\cdot \frac{(kT_{-})^2}{e^4N}, \]

where $\gamma$ is a numerical factor having different values depending on the method of calculation, it follows from (3,5) that the following formula for the mobility is obtained:

\[ b=a\,\frac{e}{mv}\cdot \frac{1}{\displaystyle \frac{1}{\lambda}+\frac{1}{\gamma}\frac{e^4}{(kT_{-})^2}N}. \tag{3,6} \]

This method of calculation $\left(\frac{1}{\lambda_{\mathrm{eff}}}=\frac{1}{\lambda}+\frac{1}{s}\right)$ was proposed by Engel and Steenbeck$^{123}$. A more accurate analysis shows that, despite actual agreement with the correct formula, the assumptions underlying the derivation are fundamentally incorrect. In the preceding section $s$ was calculated from the interaction of the plasma electrons; the action of the ions was not taken into account, because, owing to their large mass, the ions change the direction of motion, i.e. the momentum, but not the kinetic energy of the electrons, and consequently are immaterial for calculating the relaxation length, which is determined by the exchange of energy. On the contrary, precisely when calculating mo-

of the mobility of electrons, the decisive factor is the change in momentum, the deflection of the electrons, and not their energy losses. In an arbitrary interaction between electrons their momentum remains unchanged, $\sum m v = m \sum v$, and consequently also the corresponding part of the current, $\sum e v = e \sum v$ (Drude$^{97}$). Collisions between electrons in general have no influence on the process of transport of electricity, and consequently on the electrical conductivity and on the mobility. On the contrary, scattering by positive ions impedes the transport of charges, since in this case the flying charge is deflected from the direction of its motion, just as occurs in a collision with a neutral atom$^{1}$.

For scattering there also exists an effective mean free path, of the order of magnitude of the relaxation length; the reason for this circumstance lies in the equality of the concentrations of ions and electrons, and also in the identity (to within the sign) of the laws of interaction of an electron with an electron and of an electron with an ion.

Kirschtein and Kopelman$^{242}$ calculated the scattering action of an individual positive ion; as the radius of the effective sphere they took the radius of such a cross section within which the mean velocity after the collision has no component in the direction of the initial velocity$^{2}$. It is not difficult to verify that the calculation (its details were not published by Kirschtein and Kopelman) leads to equation (3.6); in this case $\gamma \simeq 80$.

In the preceding section it was shown that the transfer of energy as a result of a small number of intense individual impacts is very small, and that the decisive factor for the transfer of energy is the action of numerous collisions with weak interaction; analogously, in the case under consideration, the indicated ways of calculating the transfer of momentum, which determines the mobility, are incomplete. Allowance for the small scattering action of more distant ions was made by Gvozdover$^{194,195}$$^{3}$.

The outer boundary of the effective sphere, whose introduction is necessary in all problems where Coulomb fields figure, is determined in principle in the same way as was done in the pre—

$^{1}$ An analogous state of affairs obtains in a metal, where the resistance is likewise determined not by the intense interaction of electrons, but by the scattering of electrons by ions (more precisely, by inhomogeneities in the structure of the ions).

$^{2}$ In a central collision, and in collisions close to it, the electron is reflected in the direction from which it came; thus the electron retains the component of the initial velocity, but with a negative sign. An electron passing at a large distance from the ion is deflected only slightly; in this case the component in the direction of the initial velocity is preserved with a positive sign. In averaging over the whole area of the effective sphere Kirschtein and Kopelman’s negative velocity components in a central collision are annihilated by the positive components of passages at a large distance. This is undoubtedly the most natural model of the collision of elastic spheres for our case—the action of individual scattering centers in Coulomb fields.

$^{3}$ See also L. A. Sena$^{471,472}$.

in the preceding section in calculating the relaxation length. Gvozdover arrives at an equation of type (3,6) with

\[ \gamma=\frac{1}{\frac{\pi}{2}\ln\left(\frac{3}{2}\frac{kT_-}{e^2N^{1/3}}\right)}; \tag{3,7} \]

as in Section II, under the logarithm sign stands the ratio of the thermal energy to the mean interaction energy between two neighboring plasma charges.

According to Gvozdover, the mobility of electrons scattered by ions is constant in small fields, i.e. $v_-$ is proportional to $E$; in strong

Table 4

Electrical Conductivity of the Plasma

Plasma No. (from Table 1)	Electrical conductivity without allowance for the ionic field [Eqs. (3,2) and (3,3)], $\Omega^{-1}\,\mathrm{cm}^{-1}$	Decrease in electrical conductivity when scattering by ions is taken into account, in $v/v_0$	Electrical conductivity according to experimental data
I	$1\cdot10^{-2}$	0.1	By order of magnitude corresponds to the calculations
II	$3\cdot10^{-5}$	4	Not measured
III	2	30	By order of magnitude corresponds to the calculations
IV	3	17	The same
V	$2\cdot10^{-1}$	7	” ”
VI	$3\cdot10^{-1}$	8	” ”
VII	$3\cdot10^{-1}$	6	” ”
VIII	$4\cdot10^{-1}$	4	” ”
Storage-battery acid	—	—	$\simeq 0.5$
Graphite	—	—	$\simeq 150$

fields $v_-$ is again $\sim \sqrt{E}$, but on grounds not entirely analogous to those which hold for scattering by neutral molecules.

These formulas have not yet been confirmed by experiment to a sufficient degree. Elenbaas$^{114}$ finds very good agreement with experiment, though by a rather indirect method. Böckner and Moller$^{36}$ find an influence of ions (of approximately the expected magnitude, though still somewhat smaller) on the mobility of electrons in a cesium-vapor plasma; here the mobility is determined from measured values of the field strength by calculation from $N$ and $T_-$. A number of other works (B. Kirshtein and F. Koppelman$^{242}$, T. D. Killian$^{240}$, A. Engel and M. Steenbeck$^{129}$), which, it is true, were not carried out directly for the purpose of testing the theory

...goes over, does not reveal this influence, or reveals it to a much smaller degree than the theory requires. Here, too, much work remains to be done in order to bring complete clarity.

IV. DECREASE OF THE WORK OF IONIZATION IN A PLASMA; WORK FUNCTION; COHESIVE FORCES

We have already spoken in Section II of the picture according to which, on the statistical average, around each charge of the plasma there is formed an accumulation of charges of the opposite sign1; in this case the cloud of charges screens the field of the central charge from acting on distant regions. This screening, which is manifested in the fact that the decrease of the potential does not occur according to the law $\frac{1}{r}$ ($r$ is the distance from the central charge), but more rapidly, according to the law $\frac{1}{r}e^{-\frac{r}{D}}$, may be represented as an “absorption” of the lines of force emanating from the charge in a medium with absorption coefficient $\frac{1}{D}$, where $D$ is determined by equation (2,5). In a rough approximation the length of the lines of force is taken to be equal to $D$: the lines of force end on charges of the opposite sign. In this connection we wish to draw attention to a number of further circumstances.

What work must be done, on the average, in order to ionize a neutral, unexcited atom? Let us imagine an electron slowly receding from the atom. When the electron is at a distance $r$ from the remaining positive atomic residue, the force that we shall have to overcome will in vacuum be equal to $\frac{e^2}{r^2}$. In a plasma this force will be diminished by the cloud of charges that has formed. Thus, in order to liberate an electron from an atom in a plasma, we must expend less work; on the average the work of ionization is reduced. Assuming, in a rough approximation, that beyond the distance $D$ the field of the central ion is almost completely screened, one may suppose that the work of removing an electron from the atom to infinity is reduced in comparison with

compared with the case of vacuum by an amount approximately equal to $\dfrac{e^2}{D}$, i.e., by the amount of work which in vacuum we would have had to perform in order to carry an electron from the point $r = D$ to infinity. We cannot even carry the charge of the plasma to an infinitely large distance from a charge of opposite sign, since unlike charges are on average at distances $\simeq D$ from one another. The sharp fluctuations of statistical origin of the potential of the microfield in the immediate vicinity of the atom are the reason why the work of ionization in individual acts of ionization may differ from the mean work of ionization by a quantity of order $kT$ (in the case of a non-isothermal plasma, $kT_-^{1)}$). For more details on the influence of these factors on the state of ionization and excitation of a plasma, see Section VIII, e.

If, from a large but finite volume of plasma, we remove to infinity first some electron and then some positive ion, then, by bringing these charges together at infinity to a single point, we perform the full work of ionization, although in order to separate them in the plasma the full work of ionization is not required. Thus the law of conservation of energy requires that, when charges are removed from the plasma, “work of emission” be performed.²) In this case the sum of the emission works of the electron and ion must be equal to the difference of the ionization voltages in vacuum and in the plasma.

If, from a quasi-neutral plasma of very large but finite volume, all charges are removed to infinity one after another (alternately ion and electron), a certain work will be expended on this. The same work will be expended on the infinite expansion of the plasma, i.e., on such a change in which the charges of the plasma will be at infinitely large distances from one another and, consequently, will not interact with one another. The work expended on such an expansion goes to overcoming the forces of cohesion—

¹) Ionization by an ordinary individual electron impact is, in this connection, interpreted as follows: when the “striking” electron of the plasma microfield approaches very closely in the immediate vicinity of the atom considered, it experiences for a sufficiently long time such a sharp change of potential that one of the electrons of the atom “independently,” i.e., without overcoming the “work of ionization,” leaves the atomic residue. In the work of ionization the totality of the electron gas is used, from which, through interactions of the kind described in Section II, the “striking” electron draws its energy; this latter is sufficient to create, in the neighborhood of the atom considered, a sufficient amplitude of the microfield potential. For an individual impact this formulation seems artificial; however, if one takes into account the fact that, in a sufficiently strongly ionized plasma, around each atom there are many electrons perturbing the potential of the microfield, then the reasonableness of this formulation becomes obvious (see also VIII, e).

²) The work of emission of an electron from a metal is represented in a similar way. The electrons in the field of the periodic lattice of positive metal ions are on average at a distance from an ion smaller than the mean geometrical one. Similarly, a conduction electron in a semiconductor is bound, owing to the polarization of the surrounding ions, to its layer of spatial charge against the background of the layer. Debye–Hückel. Therefore here, too, work of emission is necessary to remove an electron.

of motion. These cohesive forces correspond completely to the tension which, in strong electrolytes, reduces the osmotic pressure owing to the electrostatic interaction between ions.

The cyclic process from which this tension and the closely connected work functions and ionization work can be calculated was carried out by Debye and Hückel^77,78 for strong electrolytes^1); we shall therefore not repeat these calculations here. The difference from an electrolyte consists, as was indicated in Section II, only in the fact that in a plasma the layer of space charges is always created by an excess or deficiency of electrons (and only in an isothermal plasma do the ions take part in creating the charge cloud around ions), whereas in an electrolyte—always isothermal—the carriers of electricity of both signs participate equally in the formation of volume charges. The quantity $D_+$ has the value $\sqrt{\dfrac{kT}{8\pi e^2 N}}$ (the same as in electrolytes) only for an isothermal plasma; otherwise its value is $\sqrt{2}$ times larger and is given by equation (2,5). The calculation gives, for a quasineutral plasma, the work function $\dfrac{1}{3}\dfrac{e^2}{D}$ both for the electron and for the ion.

In the case of an excess charge of one sign, to this one must add the electrostatic work for the excess charge, taking the sign into account. The decrease of the ionization work, $-e\Delta V^*$, can then be represented as the sum of the work functions $\dfrac{1}{3}\dfrac{e^2}{D_+}+\dfrac{1}{3}\dfrac{e^2}{D_-}$. For a nonisothermal plasma

\[ D_+=D_-=D=\sqrt{\frac{kT_-}{4\pi e^2 N}} \]

and

\[ e\Delta V^*=-\frac{2}{3}\frac{e^2}{D} =-\frac{4}{3}\sqrt{\pi e^3}\sqrt{\frac{N}{kT_-}}; \tag{4,1} \]

for an isothermal plasma we obtain a value larger by the factor $\left(\dfrac{1+\sqrt{2}}{2}\right)$, i.e. by $20\%$^2). The calculation of the reduction of ionization voltages gives quantities of the order $10^{-5}$ to $10^{-3}\ \mathrm{V}$; they may therefore always^3) be neglected, and its significance is only a matter of principle.

^1) See also G. R. Fowler^156 and A. Kohn and G. Young^60.

^2) See the footnote on p. 212.

^3) It is possible that this work function plays a role in the emission of electrons from a cathode, where the strong field of the adjacent positive ion cloud tears out autoelectrons (“Feldbogen”). If so, then the work function cathode–plasma would have to be less than the work function cathode–vacuum, and consequently electron emission could occur at smaller fields. The potential barrier (Schottky) would be less steep than in the case of the metal–vacuum boundary and therefore could be more easily overcome. The barrier would be destroyed and the transition from the metal to the plasma would be continuous if the cathode-focus temperature reached the critical boiling temperature of the cathode metal. The considerations indicated should for the time being be regarded only as uncertain hypotheses.

Table 5

Decrease of ionization work; work function; tension due to cohesion forces (Kohäsionszug)

Plasma no. (according to Table 1)	Decrease of the mean ionization, eV	Electron work function, eV	Work function for ions, eV	Tension due to cohesion forces. Polarization part according to (4,2), dyn/cm²	Tension due to cohesion forces. Exchange forces according to (4,3), dyn/cm²
I	$8\cdot 10^{-6}$	$4\cdot 10^{-6}$	$4\cdot 10^{-6}$	$3\cdot 10^{-8}$	$1\cdot 10^{-14}$
II	$1\cdot 10^{-6}$	$4\cdot 10^{-7}$	$6\cdot 10^{-7}$	$3\cdot 10^{-13}$	$1\cdot 10^{-20}$
III	$4\cdot 10^{-4}$	$2\cdot 10^{-4}$	$2\cdot 10^{-4}$	$1\cdot 10^{-3}$	$2\cdot 10^{-8}$
IV	$2\cdot 10^{-4}$	$1\cdot 10^{-4}$	$1\cdot 10^{-4}$	$4\cdot 10^{-4}$	$3\cdot 10^{-9}$
V	$3\cdot 10^{-3}$	$1\cdot 10^{-3}$	$1\cdot 10^{-3}$	$7\cdot 10^{-2}$	$5\cdot 10^{-6}$
VI	$7\cdot 10^{-3}$	$3\cdot 10^{-3}$	$4\cdot 10^{-3}$	$2$	$5\cdot 10^{-4}$
VII	$3\cdot 10^{-2}$	$1\cdot 10^{-2}$	$1\cdot 10^{-2}$	$60$	$4\cdot 10^{-2}$
VIII	$6\cdot 10^{-2}$	$2\cdot 10^{-2}$	$3\cdot 10^{-3}$	$2\cdot 10^{3}$	$4$

The tension produced by the cohesion forces $Z$, in the case of a nonisothermal plasma, satisfies the formula:

\[ Z=\frac{kT}{24\pi D^3}=\frac{\sqrt{\pi}}{3}\,e_3\,\frac{N^{\frac{3}{2}}}{(kT)^{\frac{1}{2}}}; \tag{4,2} \]

the values of $Z$ for an isothermal plasma are $200/_0$ times larger1. The tension for an isothermal luminous arc (several amperes, $N\simeq 10^{14}\ \mathrm{cm}^{-3}$, $T\simeq 5\,000^\circ\mathrm{K}$) has, according to (4,2), the order of magnitude of tenths of $\mathrm{dyn}/\mathrm{cm}^2$. Forces of such a small order are in general completely masked by various disturbances (air currents, electrodynamic forces, etc.).

Steenbeck[^499] attempted to exclude the influence of these disturbances by investigating an arc in a space where gravity does not act (a freely falling box). He found in the arc an elastic tension whose order of magnitude could be measured by studying the deflection of the arc in a transverse magnetic field. It cannot, however, be regarded as established that this tension is produced wholly or partly by cohesion forces according to (4,2).

In this connection one should point to the work of T. Nägelebauer[^372], in which the tension due to cohesion forces (Kohäsionszug) is calculated for a plasma by means of quantum-mechanical exchange forces between electrons; in this calculation is carried out for small electron concentrations,

so that Boltzmann statistics is applicable to the electron gas. Neigebauer gives for $Z$ the formula

\[ Z=\frac{e^2 h^2 N^2}{8\pi m k T}; \tag{4,3} \]

for large electron concentrations, $Z$ according to (4,3) exceeds the polarization tension (Polarisationszug) (4,2) (a more rapid increase with increasing $N$). True, for the example given above, the tension calculated from (4,3) is 3 orders of magnitude smaller than the tension calculated from (4,2). For very large concentrations ($\simeq 10^{19}\ \mathrm{cm}^{-3}$) and at temperatures of the order of several hundred degrees, the tensile stress due to cohesion may, according to calculations, be in equilibrium with the kinetic pressure of the gas of the order of $1\ \mathrm{atm}$, i.e., it is capable of holding strongly ionized masses of gas (ball lightning?).

(To be continued in the next issue)

In their review of the properties of a high-current arc, B. Kirshtéin and F. Kopelman took this tension into account in their calculations; however, these authors used the formula for $Z$ with an incorrect numerical factor. ↩↩↩↩↩↩↩↩↩↩
A survey of much older attempts to determine the coefficient $a$ exactly is given by K. H. Liboff. A more complete review of the same is given by L. B. Loeb. For recent works devoted to the question of $\lambda$ or velocity distributions, see under Nos. 1, 2, 26, 73–75, 87, 88, 95, 96, 212, 229, 311, 349, 367, 404, 407, 485, 519, 524, 525, 532. ↩↩↩↩
At large field strength (at which approximately $v \sim \sqrt{E}$) one may use formula (3,3) for the mobility, substituting, of course, $\overline{w}$ corresponding to the actual temperature of the electrons. This can be done if the assumption made in the derivation is fulfilled, namely that the free-flight time (see above, p. 219) is determined by the velocity ↩

Submission history

[v1] 1941

No. of plasma according to Table 1	Partial pressure \(^{1)}\), in torr	Contact voltage [eq. (1.5)], in V	Current density of ordered electrons \(\left(e\dfrac{Nw_-}{4}\right)^{3)}\), A/cm\(^2\)	Current density of ordered ions \(\left(e\dfrac{Nw_+}{4}\right)\), [eqs. (1.3)], A/cm\(^2\)	Thickness of the space-charge layer. Probe at 20 V negative relative to the plasma [eq. (1.4)], in cm	Mean free path of gas molecules (and ions)
I	\(3.1\cdot10^{-5}\)	\(-20.9\)	\(4.3\cdot10^{-2}\)	\(1.3\cdot10^{-5}\)	0.16	0.363
II	\(2.6\cdot10^{-11}\)	\(-0.12\)	\(4\cdot10^{-7}\)	\(2\cdot10^{-9}\)	\(21.2^{4)}\)	0.423
III	\(1.6\cdot10^{-2}\,{}^{2)}\)	\(-9.7\)	30	\(1.6\cdot10^{-2}\)	\(4.7\cdot10^{-3}\)	\(5.15\cdot10^{-2}\)
IV	\(1.3\cdot10^{-2}\)	\(-14.4\)	19	\(2.3\cdot10^{-2}\)	\(6.9\cdot10^{-3}\)	\(1.4\cdot10^{-3}\)
V	\(6.7\cdot10^{-2}\)	\(-3.05\)	\(2\cdot10^{2}\)	0.85	\(1\cdot10^{-3}\,{}^{4)}\)	\(1.5\cdot10^{-4}\)
VI	\(6.7\cdot10^{-1}\)	\(-3.58\)	\(2\cdot10^{3}\)	3.3	\(3\cdot10^{-4}\,{}^{4)}\)	\(5.2\cdot10^{-5}\)
VII	7.5	\(-4.13\)	\(2.1\cdot10^{4}\)	34	\(1\cdot10^{-4}\,{}^{4)}\)	\(6\cdot10^{-6}\)
VIII	82	\(-4.37\)	\(2.2\cdot10^{5}\)	360	\(3\cdot10^{-5}\,{}^{4)}\)	\(6\cdot10^{-7}\)

No. of plasma according to Table 1	Debye radius for electrons [eq. (2,5)], cm	Debye radius for ions [eq. (2,5)], cm	Number of charges in the Debye sphere [eq. (2,6)]	Ratio of the mean interaction potential to the mean kinetic energy of electrons, \(\dfrac{e^2 N^{1/3}}{kT}\)	Linear mean value of the microfield strength [eq. (2,2)], V/cm	Effective value of the microfield strength [eq. (2,3)\(^2\)], V/cm	Mean frequency of exchange of places, \(w \sim N'^{1/3}\), sec\(^{-1}\)	Relaxation length \(s_1\), cm [eq. (2,1)]	Relaxation length \(s_3\), cm [eq. (2,11)]	Relaxation length \(s_8\), cm [eq. (2,14)]	Relaxation length \(s_9\), cm [eq. (2,15)]
I	\(1.2\cdot10^{-2}\)	\(1.2\cdot10^{-2}\)	\(7.2\cdot10^{4}\)	\(1.2\cdot10^{-4}\)	8.2	215	\(2.3\cdot10^{11}\)	\(1.0\cdot10^{4}\)	\(8.7\cdot10^{2}\)	\(3.7\cdot10^{1}\)	5.4
II	\(1.1\cdot10^{-1}\)	\(7.8\cdot10^{-2}\)	\(5.5\cdot10^{3}\)	\(6.7\cdot10^{-4}\)	0.018	0.2	\(1\cdot10^{9}\)	\(7.2\cdot10^{3}\)	\(7.4\cdot10^{2}\)	\(9.4\cdot10^{1}\)	\(4.9\cdot10^{1}\)
III	\(2.7\cdot10^{-4}\)	\(2.7\cdot10^{-4}\)	810	\(2.4\cdot10^{-3}\)	820	4 860	\(1.6\cdot10^{12}\)	2.6	\(3.3\cdot10^{-1}\)	\(9.0\cdot10^{-2}\)	\(1.2\cdot10^{-1}\)
IV	\(4.9\cdot10^{-4}\)	\(4.9\cdot10^{-4}\)	\(2.4\cdot10^{3}\)	\(1.1\cdot10^{-3}\)	510	4 380	\(1.7\cdot10^{12}\)	14	1.6	\(2.8\cdot10^{-1}\)	\(2.2\cdot10^{-1}\)
V	\(5.6\cdot10^{-5}\)	\(4.0\cdot10^{-5}\)	73	\(1.2\cdot10^{-2}\)	3 900	10 000	\(2.3\cdot10^{12}\)	\(4.8\cdot10^{-2}\)	\(9.0\cdot10^{-2}\)	\(5.5\cdot10^{-3}\)	\(2.5\cdot10^{-2}\)
VI	\(1.8\cdot10^{-5}\)	\(1.3\cdot10^{-5}\)	23	\(2.6\cdot10^{-2}\)	17 500	33 600	\(5.0\cdot10^{12}\)	\(4.8\cdot10^{-3}\)	\(1.1\cdot10^{-3}\)	\(9.2\cdot10^{-4}\)	\(7.9\cdot10^{-3}\)
VII	\(6.0\cdot10^{-6}\)	\(4.2\cdot10^{-6}\)	9	\(4.8\cdot10^{-2}\)	82 000	106 000	\(1.2\cdot10^{13}\)	\(6.5\cdot10^{-4}\)	\(2.0\cdot10^{-4}\)	\(2.1\cdot10^{-4}\)	\(2.7\cdot10^{-3}\)
VIII	\(2.0\cdot10^{-6}\)	\(1.4\cdot10^{-6}\)	\(3\ ^3)\)	\(9.7\cdot10^{-2}\)	390 000	350 000	\(2.6\cdot10^{13}\)	\(7.4\cdot10^{-5}\)	\(3.3\cdot10^{-5}\)	\(4.0\cdot10^{-5}\)	\(8.8\cdot10^{-4}\)

Plasma no. (according to Table 1)	Decrease of the mean ionization, eV	Electron work function, eV	Work function for ions, eV	Tension due to cohesion forces. Polarization part according to (4,2), dyn/cm²	Tension due to cohesion forces. Exchange forces according to (4,3), dyn/cm²
I	\(8\cdot 10^{-6}\)	\(4\cdot 10^{-6}\)	\(4\cdot 10^{-6}\)	\(3\cdot 10^{-8}\)	\(1\cdot 10^{-14}\)
II	\(1\cdot 10^{-6}\)	\(4\cdot 10^{-7}\)	\(6\cdot 10^{-7}\)	\(3\cdot 10^{-13}\)	\(1\cdot 10^{-20}\)
III	\(4\cdot 10^{-4}\)	\(2\cdot 10^{-4}\)	\(2\cdot 10^{-4}\)	\(1\cdot 10^{-3}\)	\(2\cdot 10^{-8}\)
IV	\(2\cdot 10^{-4}\)	\(1\cdot 10^{-4}\)	\(1\cdot 10^{-4}\)	\(4\cdot 10^{-4}\)	\(3\cdot 10^{-9}\)
V	\(3\cdot 10^{-3}\)	\(1\cdot 10^{-3}\)	\(1\cdot 10^{-3}\)	\(7\cdot 10^{-2}\)	\(5\cdot 10^{-6}\)
VI	\(7\cdot 10^{-3}\)	\(3\cdot 10^{-3}\)	\(4\cdot 10^{-3}\)	\(2\)	\(5\cdot 10^{-4}\)
VII	\(3\cdot 10^{-2}\)	\(1\cdot 10^{-2}\)	\(1\cdot 10^{-2}\)	\(60\)	\(4\cdot 10^{-2}\)
VIII	\(6\cdot 10^{-2}\)	\(2\cdot 10^{-2}\)	\(3\cdot 10^{-3}\)	\(2\cdot 10^{3}\)	\(4\)

Full Text

Gases in the Plasma State. I1

Contents

Introduction

1. EXPERIMENTAL DATA: LANGMUIR PROBES, QUASI-NEUTRALITY, ISOTHERMAL AND NONISOTHERMAL PLASMAS

II. EXCHANGE OF ENERGY OF THE PLASMA COMPONENTS IN COULOMB INTERACTION; MICROFIELD; RELAXATION LENGTH

III. ELECTRICAL CONDUCTIVITY OF PLASMA IN CONSTANT FIELDS

IV. DECREASE OF THE WORK OF IONIZATION IN A PLASMA; WORK FUNCTION; COHESIVE FORCES

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Gases in the Plasma State. I[^1]

No.	Gas	Occurs in the following cases:	Pressure	Discharge current in cm	Length in cm	Electron concentration in cm\(^{-3}\)	Electron temperature in °K	Ion temperature in °K	Neutral-gas temperature in °K
I	Mercury vapor	Discharge at low pressures. Heated cathode	\(5\cdot 10^{-3}\) torr	\(\simeq 0.1\)	Sphere \(\varnothing\,15\)	\(1\cdot 10^{10}\)	30 000	1 000 ?	300
II	Air	Heaviside layer	\(\lesssim 1\cdot 10^{-2}\) torr \(^{1)}\)	—	\(>10^{6}\)	\(\lesssim 10^{6\,1)}\)	\(\simeq 250\)	\(\simeq 250\)	\(\simeq 250\)
III	Mercury vapor	Mercury rectifier	\(5\cdot 10^{-2}\) torr	50—100	Tube \(\varnothing\,6\)	\(1\cdot 10^{13}\)	15 000	1 500 ?	425
IV	Neon	Advertising neon tube	1—5 torr \(^{1)}\)	\(\simeq 1\)	Tube \(\varnothing\,2\)	\(5\cdot 10^{12}\)	25 000	1 500 ?	400
V	Air	Arc in air	1 atm	\(\simeq 10\)	Columns \(\varnothing\,1\)	\(1\cdot 10^{14}\)	\(\simeq 6\,500\)	\(\simeq 6\,500\)	\(\simeq 6\,500\)
VI	Mercury vapor	Normal mercury lamp	1 atm	\(\simeq 5\)	Columns \(\varnothing\,0.5\)	\(1\cdot 10^{15}\)	\(\simeq 6\,500\)	\(\simeq 6\,500\)	\(\simeq 6\,500\)
VII	Mercury vapor	High-pressure mercury lamp	10 atm	\(\simeq 5\)	Columns \(\varnothing\,0.2\)	\(1\cdot 10^{16}\)	\(\simeq 7\,500\)	\(\simeq 7\,500\)	\(\simeq 7\,500\)
VIII	Mercury vapor	Mercury lamp of the highest pressure	100 atm	\(\simeq 2\)	Columns \(\varnothing\,0.2\) Columns \(\varnothing\,0.2\)	\(1\cdot 10^{17}\) \(1\cdot 10^{17}\)	\(\simeq 7\,500\) \(\simeq 8\,000\)	\(\simeq 7\,500\) \(\simeq 8\,000\)	\(\simeq 7\,500\) \(\simeq 8\,000\)

Plasma No. (from Table 1)	Electrical conductivity without allowance for the ionic field [Eqs. (3,2) and (3,3)], \(\Omega^{-1}\,\mathrm{cm}^{-1}\)	Decrease in electrical conductivity when scattering by ions is taken into account, in \(v/v_0\)	Electrical conductivity according to experimental data
I	\(1\cdot10^{-2}\)	0.1	By order of magnitude corresponds to the calculations
II	\(3\cdot10^{-5}\)	4	Not measured
III	2	30	By order of magnitude corresponds to the calculations
IV	3	17	The same
V	\(2\cdot10^{-1}\)	7	” ”
VI	\(3\cdot10^{-1}\)	8	” ”
VII	\(3\cdot10^{-1}\)	6	” ”
VIII	\(4\cdot10^{-1}\)	4	” ”
Storage-battery acid	—	—	\(\simeq 0.5\)
Graphite	—	—	\(\simeq 150\)