ELECTRICAL DISCHARGES IN GASES AT LOW PRESSURES*

Irving Langmuir, Schenectady, U.S.A.

Synopsis of the contents of the article

An analysis of the behavior of electrons and positive ions situated in relatively large quantities in a gas at low pressure shows that many fundamental properties of an electrical discharge can be explained by knowing their motion.

We investigate the potential distribution produced by a current formed by charge carriers of one sign. When the current is formed by positive ions, two cases are considered: the case of emission of ions by the electrode surface and the case of uniform formation of ions between the electrodes. In the latter case, when many ions are formed, the potential has a maximum in which slow electrons accumulate. As a result there appears a region, called the plasma, where the field is almost absent. The plasma possesses properties sharply different from those of regions with strong fields, called sheaths, which cover the electrodes and the walls of the tube.

The motion of electrons in the plasma is considered as thermal motion corresponding to the temperature \(T_e\). Owing to this thermal motion, electrons can overcome small retarding fields, the magnitude of which is determined from Boltzmann’s equation. This motion makes the electrons move from the center of the plasma to its boundary until the difference in the electron densities between the center and the boundary produces a potential difference of the order of a volt, under the influence of which the positive ions will begin to move toward the walls.

The motion of positive ions in the plasma may be roughly regarded as corresponding to the temperature

\[ T_p = \frac{T_e}{2}. \]

In a discharge in mercury vapor the current density of positive ions \(I_p\) was found to be equal to \(1/411\) of the electron current density \(I_e\), which is in full agreement with theory.

We investigate the properties of sheaths and give a method for calculating the potential drop in the sheath covering the glass walls of a discharge tube. By means of the sheath of positive ions and the sheath of electrons covering, depending on the potential, probes located in the discharge, one can measure \(T_e\), \(T_p\), \(I_e\), and the electron density \(n_e\).

Thereafter the theory of the positive column is set forth, according to which the properties of the column are determined by the following equations: 1) the equation of plasma equilibrium, 2) the equation of ion current, 3) the equation of ion formation, 4) the conductivity equation, 5) the energy-balance equation.

Some experimentally observed properties of the positive column are explained on the basis of the assumption that the ratio of the directed current to the random current has a definite value under given conditions and that this ratio is constant along the discharge.

* A report read at the International Electrotechnical Congress in Paris in 1932. Translation by S. Gvozdover.

When a potential difference is applied to two electrodes in a gas and current flows through the gas, the distribution of potential in the space between the electrodes assumes very diverse forms. Some of them are in sharp contrast to the distribution of potential in a metallic conductor. For example, there are many types of discharges in which almost the entire potential drop is concentrated near the cathode, while the rest of the space has, practically, the potential of the anode. Moreover, usually in the space between the electrodes the potential has maxima and minima; and it often happens that one of the maxima has a potential higher than the potential of the anode, or that a minimum has a potential lower than the potential of the cathode.

These outwardly anomalous phenomena, as investigations of recent years have shown, give a clear idea of the fundamental electrical properties of gases, which are the subject of the present article.

THE CASE OF CURRENT FORMED BY CHARGE CARRIERS OF ONE SIGN

The gradient of the potential at any point of space is determined by Poisson’s equation:

\[ \Delta V=\frac{\partial^2 V}{\partial x^2}+\frac{\partial^2 V}{\partial y^2}+\frac{\partial^2 V}{\partial z^2}=4\pi e(n_e-n_p), \tag{1} \]

where \(e\) is the charge of the electron, \(n_e\) is the density of electrons, and \(n_p\) is the density of ions formed by single ionization. The number of negative ions and of multiply ionized ions is usually so small that their influence may be neglected.

If the current is carried only by charge carriers of one sign, then Poisson’s equation can be solved for cases when, as a consequence of symmetry, it is possible to eliminate one of the three independent variables.

A simple example of a discharge formed by charge carriers of one sign is a purely electronic discharge in a high vacuum.

Let us consider\(^{1,2}\) the transport of electrons from an incandescent plane cathode, which is at zero potential, to a plane parallel anode at potential \(V_a\).

If the initial velocities with which the electrons leave the cathode are neglected, then at a point with potential \(V\)

\[ \frac{1}{2}m_e v^2=Ve \tag{2} \]

and

\[ n_e e=\frac{I_e}{v}, \tag{3} \]

where \(I_e\) is the electron-current density, \(e\) and \(m_e\) are the charge and mass of the electron, and \(v\) is the velocity of the electron.

These equations will, obviously, also be applicable to a current carried by positive ions, if one replaces

\[ V,\ e,\ n_e,\ m_e \text{ and } I_e \quad \text{by} \quad -V,\ -e,\ n_p,\ m_p \text{ and } I_p . \]

In a discharge formed by charge carriers of one sign, let us call one electrode the source and the other the collector. The solution of Poisson’s equation for plane parallel electrodes has the form:

\[ I=\frac{\left(\frac{2e}{m}\right)^{\frac12} V^{\frac32}}{9\pi x^2} \tag{4} \]

or, substituting \(\dfrac{e}{m}=5.279\cdot 10^{17}\) \(CGSE\) and expressing \(V\) in volts, we obtain:

\[ I=5.462\cdot 10^{-8}M^{-\frac12}V^{\frac32}x^{-2}\ \text{ampere}\cdot\text{cm}^{-2}, \tag{5} \]

where \(I\) is the density of the electron or ion current, \(M\) is the molecular weight of the charge carriers (for oxygen \(M=16\)), and \(x\) is the distance from the source to the collector. For an electron current \(M=5.479\cdot 10^{-4}\), therefore,

\[ I=2.334\cdot 10^{-6}V^{\frac32}x^{-2}\ \text{ampere}\cdot\text{cm}^{-2}. \tag{6} \]

Let us consider the case of a current between plane electrodes and see in what way the potential in the space between the electrodes depends on the current passing between them. In Fig. 1 the distribution of the potential is shown when electrons move from a plane source in the plane \(x=0\) to a plane collector in the plane \(x=a\), having potential \(V_a\).

Substituting \(V=V_a\) and \(x=a\) into equation (6), we obtain the greatest current that can flow from one electrode to the other. When the current reaches this limiting value, the space charge between the electrodes prevents any further increase of the current.

Since the magnitude of the current is determined by the number of electrons leaving the source, it often happens that in reality the current attains a smaller value than the current given by equation (6) (the saturation current). In this case the space charge may be insignificant, and the [[unclear: word beginning “ras…”]]

the potential distribution will be represented by a straight line 1 (Fig. 1). When \(e n_e\) is sufficiently large, the curve of the potential distribution, determined by the equation \(\dfrac{d^2 V}{d x^2}=4\pi e n_e\), has positive curvature.

One of the curves of this type is shown in Fig. 1 (curve 2). With increasing \(n_e\), the curvature of these curves also increases.

With increasing \(n_e\), the limiting current given by equation (6) increases until it reaches the value at which the curvature of the corresponding curve will be such that \(\dfrac{dV}{dx}=0\) at the point \(x=0\), or, in other words, until the electric field at the surface of the source becomes zero.

Fig. 1. Potential distributions between plane parallel electrodes for a current of charges of one sign passing between them

Curve 2 gives the potential distribution for this particular case.

If the electrons leave the cathode with initial velocities, then at the point \(x=x_M\) a potential minimum may appear, as shown in curve 3.

Under such conditions, only those electrons can reach the collector which leave the cathode with a normal component of velocity sufficient to break through from \(x=0\) to \(x=x_M\) through the retarding field.

An exhaustive study of this case was given by us earlier.\(^3\) If the initial velocities of the electrons have a Maxwellian distribution with temperature \(T\), then instead of equation (6) we obtain:

\[ I=5.462\cdot 10^{-8}M^{-\frac12}V^{\frac32}x^{-2}\left[1+0.0247\left(\frac{T}{V}\right)^{\frac12}\right]\ \text{ampere}\cdot\text{cm}^{-2}. \tag{7} \]

Here \(V\) is expressed in volts, and \(x\) and \(V\) are measured from the point \(M\) in Fig. 1, taken as the origin.

Corresponding space-charge equations were also obtained for the greatest current carried by identically charged particles between electrodes in the form of coaxial cylinders and concentric spheres.\(^4\)

Current of Positive Ions

All the equations obtained are applicable to the current carried by positive ions which are emitted by a plane source.

A more general case of the current of positive ions is, however, the case when ions are formed throughout the entire space between the electrodes.

Let us consider, for simplicity, the case when ions are formed uniformly in the space between a plane electrode \(A\) and an electrode \(C\) parallel to it, which is at a distance \(a\) from it and has potential \(V_c\) relative to \(A\).

Let us suppose that \(S\) ions are formed per unit volume in one second, that they are formed without initial velocities, and that no electrons appear when they are formed.

It was shown,\(^5\) that when the volume charge and the corresponding curvature of the potential-distribution curve are such that at the point \(x=0\), \(dV/dx=0\), then

\[ I_c=\frac{\left(\frac{2e}{m}\right)^{\frac{1}{2}}\left(-V_c\right)^{\frac{3}{2}}}{\pi^2 a^2}, \tag{8} \]

where \(I_c\) is the current per unit surface reaching \(C\).

Comparing equation (8) with equation (5), which gives the greatest current \(I\) that can flow when all the ions arrive from \(A\), we find that

\[ I_c=\left(\frac{9}{\pi}\right)I=2.865\,I. \tag{9} \]

Under conditions satisfying equation (8), the potential distribution is represented by curve 1 in Fig. 2, which is a parabola. The relations expressed by these equations hold only when ions are formed in a definite amount \(S_1\), determined by the relation

\[ I_c=S_1ea. \tag{10} \]

If \(S>S_1\), then a maximum of potential appears between \(A\) and \(C\), and equation (8) may be applied separately to the two branches of the curve on both sides of the maximum. The potential-distribution curve will still be a parabola, but its vertex will no longer be at \(A\).

Curves 2, 3, 5, and 10 in Fig. 2 were obtained for values of \(S\) that are 2, 3, 5, and 10 times greater than \(S_1\). We therefore arrive at the conclusion that when ions are formed uniformly throughout the whole volume between two parallel planes, and no appreciable accumulation of electrons occurs, the potential-distribution curve is a parabola, i.e. the potential is determined by the equation \(V=Br^2\), where \(r\) is measured from the plane at which the potential has a maximum.

The ion density \(n_p\) is then the same throughout the entire space and is determined by the equation:

\[ n = 0.0112 M^{\frac{1}{3}} S^{\frac{2}{3}}\ \text{ions}\cdot \text{cm}^{-3}, \tag{11} \]

where \(S\) is expressed in \(\text{ions}\cdot \text{cm}^{-3}\cdot \text{sec}^{-1}\).

If the region of ionization, instead of being bounded by parallel planes, is bounded by a cylindrical or spherical surface, then the parabolic distribution is preserved, i.e. \(V = Br^2\), where \(r\) is the distance from the axis of the cylinder or from the center of the sphere. In these cases, similarly, the ion density is the same throughout the whole space, so that equation (11) is again applicable, with the only difference that the value of the coefficient is equal to \(0.0105\) for the cylindrical case and \(0.0102\) for the spherical case.

INFLUENCE OF ELECTRONS FORMED DURING IONIZATION

If positive ions are formed as a result of gas ionization, then simultaneously with them, and in equal number, electrons appear.

If \(S < S_1\), then the potential between the electrodes has no maximum, and all electrons, as they are formed, move toward electrode \(A\). Thus the electrode current to electrode \(A\) is equal to and opposite to the current of ions moving toward electrode \(C\).

Nevertheless, the electron current does not exert any noticeable influence on the space charge, since, combining equations (2) and (3), we find that

\[ en = I \left[\frac{m}{2Ve}\right]^{\frac{1}{2}}, \tag{12} \]

whence, for an electron current equal to the current of positive ions,

\[ \frac{n_e}{n_p} = \left(\frac{m_e}{m_p}\right)^{\frac{1}{2}}, \tag{13} \]

Consequently, the space charge formed by electrons is hundreds of times smaller than the space charge formed by positive ions.

However, the picture changes sharply if \(S > S_1\), because then there appears a tendency for a maximum to arise in the potential distribution, as illustrated in Fig. 2. Since at the maximum the potential is higher than the potentials of both electrodes, in this region (above \(ON\) in Fig. 2) there accumulate in large quantity the electrons formed during ionization and possessing small velocities,

The accumulation of electrons creates a negative space charge, as a result of which, in the region of the maximum, the potential falls until

Fig. 2. Potential distributions between plane electrodes, with uniform formation of ions between them.

it reaches the value at which electrons begin to reach the anode. The value of this potential is somewhat higher than the anode potential, since the electrons have initial velocities that allow them to move against weak retarding fields. Fig. 3 shows the potential distribution produced as a result only of the accumulation of electrons just described. Positive ions move in the accelerating field only from \(P\) to \(A\) and from \(Q\) to \(C\); electrons break through the retarding field only from \(P\) to \(A\). Practically speaking, in the region between \(Q\) and \(C\) there are no electrons, since the strong retarding field prevents their penetration into this region. Owing to the electrons accumulating in the space between \(P\) and \(Q\), the potential in this region is almost the same (assuming that in this region the disappearance of electrons occurs only at the expense of electrons moving toward \(A\)),

Fig. 3. Potential distribution caused by the accumulation of electrons in the region of maximum potential.

The reasons for such a leveling of the potential will become clear if we take into account that, owing to their high mobility, the electrons at once collect at the maximum of the potential and thus lower this potential, while, leaving the minimum, they raise the potential.

Since in the region between \(P\) and \(Q\) the field is very small, the number of electrons per unit volume in this region must be approximately equal to the number of positive ions. In other words, the electrons neutralize the space charge of the positive ions, and we have \(n_e = n_i\).

We now see that, when current is carried by positive ions and electrons, two different regions appear in the space between the electrodes. The regions of strong fields, formed by space charges that cover the electrodes, we shall call sheaths. \(PA\) represents the potential drop in the anode sheath, and \(QC\)—in the cathode sheath.

The region between the sheaths, where there is almost no field, we shall call the plasma. These regions must be studied separately, since they possess very different properties.

Many phenomena of the gas discharge, ordinarily visible to the eye, are caused by the different properties of these two regions. For example, when a current of only a few milliamperes passes through a discharge tube containing mercury vapor saturated at room temperature, at a potential difference of the order of 20 V, almost the entire tube is filled with the greenish-blue glow characteristic of mercury, which does not reach the walls.

A dark space separates the glow from the walls, as though the glow were being repelled by the glass. When the current increases, the dark space becomes thin, and the glow approaches the walls. At still larger currents the eye no longer distinguishes the dark space at all. The luminous region in this case is a typical plasma, and the dark space is a typical sheath of positive ions. The latter is a region containing positive ions and repelling electrons. Because of the absence of electrons, the brightness of the sheath is reduced. The glow of the plasma is therefore caused by the excitation of mercury atoms by electron impacts.

Conditions for the Formation of Plasma

The condition necessary for the formation of plasma in a discharge is that the number of ions formed be sufficient for the appearance of a potential maximum in the tube. From equation (9) we see that this will always occur if the number of ions formed per unit time throughout the whole space is several-

is \(K_0\) times greater than the number of ions, emitted by the anode, that form the ionic current.

From equations (9) and (5) we see that, at a voltage of the order of 100 V between electrodes situated at a distance of 5 cm from one another, plasma will begin to form when the current of positive ions exceeds, in a discharge in mercury vapor \((M = 200)\), \(4.5 \cdot 10^{-7}\) ampere \(\cdot\) cm\(^{-2}\), or \(3.2 \cdot 10^{-6}\) ampere \(\cdot\) cm\(^{-2}\) in a discharge in helium \((M = 4)\).

When the pressures are so high that the electron mean free path is small in comparison with the distance between the electrodes, the number of ions \(\beta\), formed by each electron emitted from the cathode (the total ionizing ability), does not depend on the pressure. For electrons with a velocity of 100 V in mercury vapor \(\beta = 2.7\), and for helium \(\beta = 2.9\). Thus, under the indicated conditions, the electron density necessary for the formation of plasma in mercury vapor is equal to \(1.7 \cdot 10^{-7}\) ampere \(\cdot\) cm\(^{-2}\), and for helium to \(1.1 \cdot 10^{-6}\) ampere \(\cdot\) cm\(^{-2}\). At lower pressures, when the electrons fly freely from the cathode to the anode, the ionizing ability is proportional to the pressure and to the distance between the electrodes. For electrons with a velocity of 100 V the ionizing ability of one electron in mercury vapor at \(28^\circ\)C is equal to \(0.0213\) cm\(^{-1}\), and in helium at a pressure of \(0.001\) mm it is equal to \(0.0016\) cm\(^{-1}\). Therefore, at an electron-current density of \(1\) mA \(\cdot\) cm\(^{-2}\), we can calculate the gas pressure at which plasma begins to form. This pressure for mercury vapor is equal to \(4 \cdot 10^{-6}\) mm, and for helium—\(4 \cdot 10^{-4}\) mm. These calculations serve as an illustration of the principal factors on which the phenomena characterizing a discharge in a gas depend, a discharge that is completely opposite to a purely electronic discharge in vacuum. In reality, the current necessary for the formation of a sharply bounded plasma is somewhat greater than that calculated above, owing to the disappearance of ions on the walls of the tube and to other secondary factors.

Motion of Electrons in Plasma

The fundamental property of plasma is that it represents a region in which electrons accumulate until their density becomes approximately equal to the density of the ions, so that \(n_e = n_p\).

Usually, in a plasma formed in a gas at low pressures of several millimeters or less, recombination of ions occurs to a negligible extent. The main reason for this is that the electrons move between the ions and only in rare cases collide with them; but even in the event of a collision the probability of recombination is very small.

In ionization, when a fast electron collides with a neutral gas molecule, the amount of motion of the electron is so small that it cannot impart appreciable kinetic energy to the ion, so that the ions formed possess the same kinetic energy as the neutral gas molecules. In ionization, however, a new electron is formed; therefore, instead of one primary electron, two appear. The energy of the primary electron, after subtracting the energy expended on ionization, is distributed between the two electrons. Therefore the electrons formed in ionization in the plasma usually possess considerable energies. A small number of the electrons possessing the greatest kinetic energy can, in this way, break through the anode layer, whereas electrons possessing less kinetic energy cannot escape and continue to move inside the plasma, being continuously reflected from the layers bounding it. As a result of the interaction between these electrons and thanks to the elastic collisions that can occur between electrons and gas molecules, a disordered motion arises among the electrons. The distribution of velocities approaches the ordinary Maxwellian distribution, which exists at a definite temperature among gas molecules. And indeed, the motion of the electrons inside the plasma may quite well be represented as thermal motion with temperature \(T_i\).

Experiments have shown that in most gas discharges, especially at low pressures, this Maxwellian distribution of electron velocities is established much more rapidly than could have been expected on the basis of any known phenomenon. \(^{73}\) This phenomenon is closely connected with the presence inside the plasma of oscillations of very high frequency, of the order

\[ \nu=e\left(\frac{n}{\pi m_{e}}\right)^{\frac{1}{2}}=8980\, n_{e}^{\frac{1}{2}}\ \mathrm{sec}^{-1}. \tag{14} \]

The temperatures of the electron distribution, obtained by the method described below, vary, beginning at \(5000^\circ K\), reaching in some cases \(80\,000^\circ K\).

The average energies of these electrons thus range approximately from 1 to 10 electron-volts. The current carried by electrons moving in this way in all directions, which in one second fly through a unit surface mentally drawn in the plasma, we shall call the density of the random current.

From the formulas of kinetic theory it will be found that the density of this current,

$$ I_e = e n_e \left(\frac{kT}{2\pi m}\right)^{\frac{1}{2}} = 2.48 \cdot 10^{-14} n_e T_e^{\frac{1}{2}} \ \text{ampere}\cdot\text{cm}^{-2}. \tag{15} $$

When the molecules of a gas, which is in thermal equilibrium, are subjected to the action of a steady force field, their densities, at places with different potential energies, become different. In such cases the distribution is determined by the Boltzmann equation.

In the case of the distribution of electrons, the Boltzmann equation takes the form:

$$ n = n_0 e^{\frac{Ve}{kT}}, \tag{16} $$

where \(n\) is the electron density at the place where the potential is equal to \(V\), and \(n_0\) is the density at the place where the potential is equal to zero. For calculations in this equation one may put \(\frac{e}{k} = 11.606\) degrees per volt. Thus we see that if our assumption concerning thermal equilibrium is valid, and if in different parts of the plasma there exists a difference in the electron densities, then there also exists the corresponding difference of potentials; and, conversely, a difference of potentials inevitably entails a difference in densities.

In the case of a discharge in a long tube, the region usually called the positive column is a typical plasma. Usually, when a considerable current passes, a small potential gradient is observed along the entire length of the cylindrical plasma. As a result, the electrons in the plasma receive acceleration from the field acting on them, but a steady state is soon reached in which the loss of energy, due to collisions of the electrons with the gas molecules, is exactly compensated by the energy imparted by the field.

In a long homogeneous positive column the temperature of the electron distribution is the same throughout the plasma; the electron density is also constant, despite the presence of a potential gradient. Consequently, the Boltzmann equation is not applicable for determining changes of density along the length of the discharge, but it can be used successfully for determining changes of density in the plane of the transverse cross-section.

Thus, generally speaking, we must distinguish two kinds of currents within the plasma: a random current with density \(I_e\), and a directed current of density \(I_d\), caused by the applied electric field.

Thus the amount of energy expended on

formation per unit volume of plasma, equal to the product \(I_a\) times the gradient of the potential.

Usually, in a discharge, the density of the random current is several times greater than the density of the directed current.

Motion of Positive Ions in Plasma

If the positive ions had approximately the same kinetic energies as the electrons, and likewise moved in all directions, then the density of their current \(I_p\) would be less than \(I_e\) in the ratio \(m_e^{\frac{1}{2}} : m_p^{\frac{1}{2}}\).

Thus, in the case of mercury vapor, \(I_p\) would be equal to \(\frac{1}{605} I_e\). Experimentally, by the methods described below, it was found that in a discharge in mercury vapor \(I_p\) usually was equal to \(\frac{1}{41} I_e\).

We have seen, however, that positive ions arise without possessing initial velocities. It is therefore highly improbable that the ions could acquire a Maxwellian distribution of velocities.

In any plasma that is in a steady state, i.e., when \(n_p\) does not increase with time, it is necessary that the ions, as they are formed, leave the regions of their formation. This requires the presence of electric fields that remove the ions from these regions; these fields can arise only from a slight excess of positive charge within the plasma. The fact that the electrons possess rather large energies causes them to tend to leave the plasma, leaving in the plasma an excess of positive charge. This positive charge is formed until it causes the ions to move out of the plasma into the surrounding layers in the same quantity as they are formed. At pressures so low that the ions move under the influence of such plasma fields without colliding with gas molecules, the motion of the ions proceeds according to equations (2) and (3); in doing so it is necessary to take into account the fact that the different ions arise in different places.

The complete mathematical theory of the motion of ions under these conditions has been developed by Tonks and Langmuir.² It is extremely difficult to find a general solution for electrodes of any shape. The question can, however, be solved for the cases when the plasma is bounded by two parallel planes, or by cylindrical or spherical surfaces. Two cases were considered: at low pressures, when the ions move freely from the place of their formation to the boundary of the plasma; at high pressures, when they move co-

according to the laws of mobility, continually colliding with gas molecules.

In the present article we shall consider only the first case.

Assuming that the rate of ion formation \(s\) at any point is proportional to the electron density, one can express the principal properties of the plasma in terms of the parameter \(\lambda\), which may be defined as the number of ions formed in one second by each electron of the plasma. Tonks and Langmuir showed that, in order for the plasma field, determined by Boltzmann’s equation, to be able to draw off ions as they are formed, the following equation of plasma equilibrium must be satisfied:

\[ \lambda=\left(\frac{s_0}{a}\right)\left(\frac{2kT_e}{m_p}\right)^{\frac12}, \tag{17} \]

where \(a\) is the radius of the plasma in the case of a cylindrical plasma, or half the thickness of the plasma when the latter is bounded by two parallel planes. The factor \(s_0\) is equal to 0.7722 for a cylindrical plasma and 0.4046 for the case of parallel planes.

Substituting numerical values, we find:

\[ \lambda=\left(\frac{9957}{a}\right)\left(\frac{T_e}{M}\right)^{\frac12} \tag{18} \]

for a cylinder, and

\[ \lambda=\left(\frac{5217}{a}\right)\left(\frac{T_e}{M}\right)^{\frac12} \]

for parallel planes, where \(M\) is the molecular weight of the ion. For mercury vapor in a cylindrical plasma \((M=200)\) we obtain:

\[ \lambda=\left(\frac{703}{a}\right)T_e^{\frac12}. \tag{19} \]

The equation of plasma equilibrium thus requires a definite relation between \(\lambda\) and \(T_e\).

For a discharge taking place in a glass tube of radius \(1.6\ \mathrm{cm}\), in mercury vapor at room temperature and at a current of \(1\ \mathrm{A}\), \(T_e\) proved to be equal to \(29\,000^\circ\mathrm{K}\). Consequently, according to equation (19), \(\lambda\)—the number of ions formed in one second by each electron of the plasma—is equal to 75,000.

If the rate of ion formation \(s\), instead of being proportional to the electron density \(n_e\), is constant,

throughout the plasma, then the equation of plasma equilibrium takes the form:

\[ n_0=\left(\frac{s_a}{s_0}\right)\left(\frac{m_p}{2kT_e}\right)^{\frac12}. \tag{20} \]

In this case \(s_0\) is equal to 0.583 for a cylindrical plasma and 0.380 for a plasma bounded by parallel planes. In this equation \(n_0\) is the electron density at the center of the plasma. For the cylindrical case this equation reduces to

\[ n_0=1.33\cdot 10^{-4}\, s_a \left(\frac{M}{T_e}\right)^{\frac12} \text{ ions}\cdot \text{cm}^{-3}. \tag{21} \]

According to this theory, the ion velocities are very small at the center of a cylindrical plasma, while the radial components of their velocities gradually increase on approaching the walls. Thus the velocity distribution of the positive ions follows neither the Maxwell distribution nor the Boltzmann equation.

The amounts of energy acquired by the ions are determined by the plasma fields, which are formed owing to the motion of the electrons. The mean energies of the ions, however, are considerably smaller than the mean energies of the electrons.

Experiments in which ions flew through an aperture in a probe arranged in such a way\({}^{10}\) that it was possible to measure the ion velocities showed that the radial components of the ion velocities have a temperature approximately half that of the electrons. Therefore, approximately, we may assume that at the boundaries of the plasma the ions follow a Maxwellian distribution with temperature \(T_p=\frac{T_i}{2}\). However, it must be remembered that the ions enter the layer bounding the plasma while moving toward it from the center of the plasma, whereas electrons, which possess a true Maxwellian distribution, move in all directions, being reflected from the negatively charged regions in the layer.

Taking these facts into account,\({}^{5,9}\) it was shown that the ratio of the positive-ion current density to the electron-current density is, approximately, equal to

\[ \frac{I_e}{I_p}=\frac{1}{2}\left(\frac{T_e m_p}{T_p m}\right)^2 . \tag{22} \]

For mercury vapor this gives:

\[ \frac{I_e}{I_p}=429. \tag{23} \]

The average value of this ratio, obtained experimentally, was \(411 \pm 17\).

Using the measurements that determine \(I_e\) and \(T_e\), one can calculate the density of electrons and ions in the plasma according to the equation

\[ n_e = 4.03 \cdot 10^{13} \frac{I_e}{T_e^{\frac{1}{2}}}, \tag{24} \]

equivalent to equation (15).

Plasma Fields

Although the plasma potential is not entirely uniform, the potential drop from the center to the boundary of the layer is only \(5 \cdot 10^{-5} T_e\), and thus is usually less than \(1V\). Nevertheless, it is precisely these plasma fields that impart their velocities to the ions.

Near the center of the plasma there is a maximum of potential; in the remaining part of the plasma the potential changes, approximately, in proportion to the square of the distance from this central place where the potential has its maximum.

According to Boltzmann’s equation, applied to the plasma field, the electron density decreases only very slightly as this radial distance increases. The density of ions, however, is considerably more uniform and is not subject to the influence of the factors entering Boltzmann’s equation. As a result, near the walls of the tube the electron density decreases considerably more rapidly than the ion density, as a consequence of which a relatively large positive volume charge is created. This causes a rapid increase of the potential with increasing radial distance, as a consequence of which the electron density decreases practically to zero. The transition from the plasma to the layer, although continuous, takes place over a relatively small distance, so that for practical purposes we may regard these regions as sharply separated from one another.

Properties of the Layer of Positive Ions

If the glass walls of the tube containing the plasma had the same potential as the plasma itself, then the current of electrons to the walls would, according to equation (23), be hundreds of times greater than the current of positive ions.

In reality, in the steady state, the number of positive and negative charges reaching an insulated wall must be the same. Therefore,

to make these currents equal, the walls must become so negatively charged as to repel almost all electrons back into the plasma. As a result of this the walls become covered with a layer of positive ions. The potential drop in this layer can be calculated by means of Boltzmann’s equation (16).

In view of the fact that the temperature of the electron distribution is the same throughout the plasma, the current density overcoming the retarding field of the layer is proportional to the electron density \(n_e\), so that equation (16) may be written as follows:

\[ I = I_0 e^{\frac{Ve}{kT_e}} . \tag{25} \]

In the case of mercury vapor the density of the electron current \(I_e\) reaching the wall, equal to \(I_r\), is therefore, approximately, \(\frac{1}{420} I_0\), where \(I_0\) is the density of the electron current in the plasma near the boundary of the layer.

Hence the potential drop in the layer is

\[ V_0 = -\frac{6.06\,T_e}{11{,}600}. \tag{26} \]

Consequently, if \(T_e = 25\,000^\circ\), then the walls are, approximately, 13 V more negative than the boundary of the plasma.

The conditions within the layer of positive ions are very similar to those in an ordinary thermionic discharge.

The ions move radially from the plasma to the boundary of the layer, usually having energies of less than one volt. Within the layer, however, they are subjected to the action of strong electric fields, which rapidly accelerate their motion toward the walls.

Thus, it is clear that the space-charge equation (5) is applicable to the currents of positive ions flowing through these layers. If we do not wish to neglect the influence of the initial velocities with which the ions enter the layer, we may use equation (7). In this equation \(T\) may then be taken equal to \(\frac{1}{2}T_e\); \(V\) is the voltage drop in the layer, and \(I\) is equal to \(I_p\), the current density of the positive ions moving from the plasma, in amperes/cm\(^2\). We can thus, by equation (5) or (7), calculate the thickness of the layer. The greater the current density of the positive ions, the thinner the layer becomes.

Application of Probes to the Investigation of Discharges

If a portion of the glass wall surrounding the plasma is replaced by a metallic electrode or probe, then it is easy

one can measure the number of positive ions reaching the wall.^11 When this probe is made negative with respect to the plasma, the electrons are repelled, and a current of positive ions of density \(I_p\) flows to the probe. If the negative potential is sufficiently large, the electrons do not reach the probe at all. This occurs when the probe potential is \(-5\) or \(-10\) V higher relative to the surrounding plasma.

Experience shows that when all electrons are repelled, the current to a plane probe depends only very slightly on the applied voltage, and thus \(I_p\) can be measured accurately. Thus, for example, in the case of a flat square probe with side \(1.9\ \mathrm{cm}\), in contact with the wall of the tube, at a current of \(2\ \mathrm{A}\) in mercury vapor and a pressure of \(0.025\ \mathrm{mm}\), the voltage \(E\) and current \(i\) to the probe were as follows:

The slight variation of the current with voltage is explained by edge effects. When the voltage is changed, the thickness of the layer on the probe changes, whereas the layer on the surrounding glass walls remains unchanged. Therefore the boundary of the layer along the edges of the probe is curved. By means of the formulas derived^11 one can calculate corrections for edge effects for square and disk-shaped probes. This phenomenon can, however, be eliminated by surrounding the probe with a guard ring that is maintained at the same potential, and measuring the current only to the inner probe.

In the case of mercury vapor it was found that these positive-ion currents remain constant up to negative voltages of \(1200\ \mathrm{V}\).

If the potential of the probe is gradually raised (i.e., its negative potential is lowered) until it can no longer repel all the electrons of the plasma, the current to the probe does not remain constant, but decreases from the value corresponding to \(I_p\), because now both positive ions and electrons reach the probe. With a further increase of the potential the current passes through zero, changes its sign, and increases. From these increments of the electron current one can measure the density of the electron current \(I_e\) overcoming the negative potential of the probe.

From equation (25) it is seen that the logarithm \(I_e\) must depend linearly on the voltage \(V\), and that the slope of the straight line representing the dependence of \(\ln I_e\) on \(V\) is equal to

\[ \frac{e}{kT_e}=\frac{11606}{T_e}. \]

In experiments carried out under the most varied conditions, the graph of the dependence of \(\ln I_e\) on \(V\) was always (even with a thousandfold increase in \(I\)) a straight line. From these graphs it was established that, in almost all cases, the electrons in the plasma possess a Maxwellian velocity distribution; moreover, this distribution is maintained despite the fact that the walls continually absorb the fastest electrons, so that the disturbed equilibrium is restored with astonishing rapidity.^12

In the case of a discharge in mercury vapor, the electron temperatures \(T_e\), determined from the slope of these semilogarithmic straight lines, decrease noticeably with increasing pressure, but are almost independent of the discharge current. The mean values are as follows:

Property of the electron layer

If the potential of the probe is raised so much that it becomes equal to the potential of the plasma, then the layer of positive ions disappears, and the electron currents obtained by the probe will, on average, exceed by a factor of 400 (for mercury vapor) the currents of positive ions \(I_p\) obtained at high negative voltages. Up to this voltage the semilogarithmic curves of \(I_e\) are straight lines. If, however, the voltage on the probe is increased still further, then \(I_e\) tends to become constant, since all the electrons coming to it from the plasma reach the probe. The effect of increasing the voltage beyond this point appears in driving the positive ions back into the plasma and in forming at the probe a layer of electrons, within which only electron currents flow. Consequently, the thickness of the layer can be calculated from the space-charge equation (5) or (7). However, the voltage cannot increase very far without accelerating the electrons in the layer so much that additional ionization occurs; the electron layer will then be broken through, and a considerably larger current will be able to flow to the probe.

A characteristic feature of these current-voltage characteristics is that on the graph of \(\ln I_e\) against \(V\) there is a fairly sharp bend when the probe has the same potential as the plasma. The position of the bend makes it possible to determine accurately the potential of the space around the probe.

Theory of the Positive Column at Low Pressures

Let us apply these views to the investigation of typical properties of discharges at low pressures, when, for example, a current of 0.1 A or more passes through a cylindrical tube filled with neon or mercury vapor.

Every discharge is determined by a large number of variables. We may take some of them as independent variables and others as dependent variables. Thus, for example, if a discharge occurs at a given current in a gas at certain pressures and temperature, in a tube of a given diameter, then we may regard these variables as independent. By fixing their values, we thereby uniquely determine the parameters of the discharge. We must calculate the following five dependent variables: the electron density \(n_0\) on the axis of the tube; the electron temperature \(T_e\); the current density of positive ions \(I_p\) reaching the walls of the tube; the number of ions \(\lambda\) produced by one electron in one second; and the potential gradient \(\dfrac{dV}{dx}\) along the axis of the tube. To determine these five unknown quantities we need five equations. On the basis of the considerations set forth above it can be seen that, at least formally, we can give the five equations necessary for determining these unknowns. These five equations are as follows.

1) The plasma equilibrium equation [(equation (19) or (22)].

2) The ion-current equation. The total ion current reaching the walls is equal to the total number of ions produced in the plasma, since recombination is negligible. This equation may be expressed as

\[ I_p = h_0 e a n_0 \lambda, \tag{27} \]

where \(h_0\) is numerically equal to 0.350 for a cylindrical plasma, when at every point the number of ions produced is proportional to the electron density.

3) The ion-production equation. Killian, and also Tonks and Langmuir\(^9\), showed that the production of ions in a mercury discharge occurs as a result of ionization by fast electrons, which are always present in the ordinary Maxwellian distribution with temperature \(T_e\).

In this way equations were obtained for calculating the rate of ion production \(\lambda\).

4) The mobility equation. Langevin’s mobility equation gives, for the mean directed velocity, an expression into which enter the gas pressure, the potential gradient, and the electron temperature. In this way there is obtained

equation into which all these quantities enter together with the electron density \(n_0\).

5) Energy-balance equation. The total amount of energy supplied to the discharge must be equal to the total losses. In the case of a discharge in mercury vapor, about one third of the energy goes into the formation of positive ions and electrons, recombining at the walls. The remaining energy probably, for the most part, leaves the discharge in the form of radiation, part of which passes through the glass, while a considerable amount of ultraviolet light is not transmitted by the glass. At present the factors entering into the energy-balance equation are not yet fully known.

With the aid of these five equations it is theoretically possible to determine all the parameters of the discharge and thus, given our present knowledge of these phenomena, to explain the principal properties of the discharge.

Ratio of the Directed Current to the Random Current \(\dfrac{I_d}{I_e}\)

From the theory of the low-pressure discharge set forth above, it is clear that the ratio of the directed current to the random current, under any given conditions, has a definite value, and that this ratio is constant along the entire length of the discharge.

If the discharge takes place in a tube with different cross sections, then the density of the directed current must vary inversely proportionally to the cross section of the tube, since the total directed current must remain constant.

The same changes must occur with the random current \(I_e\). Thus, at neighboring points of a region where a sharp change in the diameter of the tube occurs, there will be different \(n_e\), which will cause, according to Boltzmann’s equation, the appearance of a corresponding potential difference. If, at the cathode end of the constriction of the tube, the density of the random electron current moving toward the constriction is insufficient to supply the directed current to this region, then near the constriction there will be an increase in the potential gradient, whereby the electrons will be accelerated to such velocities that they can form new ions. As a result, a double layer arises in the discharge, in which the potential drop approximately corresponds to the ionization potential. This will cause the appearance of a glow resembling the glow of the head of the positive column in an ordinary low-pressure discharge.

At the anodic end of the constriction, the electron density in the narrow part is several times greater than the density in the adjacent wide part. Consequently, according to Boltzmann’s equation, a retarding field of several volts is formed, corresponding to this difference in densities. As a result of the excess in this region of fast electrons, in a quantity greater than is needed to preserve the constant ratio

\[ \frac{I_d}{I_e}, \]

there appears a tendency for the electron temperature to decrease; ionization begins to occur on a smaller scale, the glow diminishes, and this region appears as a dark space. This may be regarded as entirely analogous to Faraday’s dark space in ordinary discharges. Indeed, such a dark space can easily be created artificially by means of electron emission, by introducing into the plasma a negatively charged incandescent filament. Faraday’s dark space may therefore be considered as a region in which the ratio

\[ \frac{I_d}{I_e} \]

is smaller than the normal value required to maintain the positive column.

The cathode, or Crookes, dark space in an ordinary discharge at low pressure is simply a layer of positive ions into which, owing to the negatively charged cathode, electrons from the plasma cannot penetrate.

The space-charge equations are applicable, however, to the cathode dark space only as a very rough approximation, since not all ions come from the boundary of the dark space, but many of them are formed within the layer itself by fast electrons flying out from the cathode. The circumstance that these electrons carry a considerable part of the current also hinders the exact application of the space-charge equation.

The cathode glow, owing to the presence of many fast electrons flying out from the cathode, is a region where the electron density and temperature \(T_e\) are very large. As a consequence of this excessive ionization, the ratio

\[ \frac{I_d}{I_e} \]

is very small, and therefore the cathode glow is accompanied by a dark Faraday space. When this ratio falls below the normal value, ions are no longer formed in a quantity satisfying the plasma-equilibrium equation. As a result, at the beginning of the positive column a field is formed which accelerates the electrons and produces the required number of ions.

If more ions are formed than is required to satis-

integration of the given equations, a layered discharge arises, and we obtain a succession of Faraday dark spaces alternating with “heads” of positive columns. If, however, the conditions are such that, owing to the double layer, the number of ions formed at the beginning of the positive column better satisfies the requirements of the equations, then a uniform positive column is formed. If the anode is of such large dimensions that its area, multiplied by \(I_e\), is greater than the directed current, then there is usually a small negative anode fall, of the order of one volt. If, on the other hand, the anode is very small, so that the random current falling upon it is insufficient to supply the current required by the external circuit, then we have conditions similar to those which exist at the beginning of striations, since a double layer arises and, within the first plasma, a second is formed. In this case a spherical anode glow appears at the anode.

Thus considerations concerning the magnitude of the ratio

\[ \frac{I_d}{I_e} \]

help to clarify, at least from the qualitative side, the most important properties of discharges at low pressures.

REFERENCES

1. I. Langmuir, Phys. Rev., 2, 450, 1913.
2. I. Langmuir and K. T. Compton, Rev. of Modern Phys., 3, 191—257, 1931; see p. 237 ff.
3. I. Langmuir, Phys. Rev., 21, 419, 1923; see also reference 2, p. 239 ff.
4. I. Langmuir and K. T. Blodgett, Phys. Rev., 22, 347, 1923; 23, 49, 1924; see also reference 2, p. 245 ff., where graphs of the functions \(\alpha\) and \(\beta\) entering into the formula are given.
5. I. Langmuir, Phys. Rev., 33, 954, 1929; see p. 961.
6. K. T. Compton and I. Langmuir, Rev. of Modern Phys., 2, 123—242, 1930; see Table 2, p. 127 (for the Russian translation see Uspekhi Fizicheskikh Nauk, XI, 1, 2, and 3, 1931).
7. I. Langmuir, Proc. Nat. Acad. Sci., 14, 627, 1928.
8. L. Tonks and I. Langmuir, Phys. Rev., 33, 195, 1929; see reference 6, p. 239.
9. L. Tonks and I. Langmuir, Phys. Rev., 34, 876—922, 1929.
10. L. Tonks, H. Mott-Smith, I. Langmuir, Phys. Rev., 28, 104, 1926; see pp. 120—123.
11. I. Langmuir and H. M. Mott-Smith, G. E. Rev., 27, 449, 538, 616, 762, 1924.
12. I. Langmuir, Phys. Rev., 26, 585, 1925; see pp. 586—592; for more detail see reference 9.
13. T. J. Killian, Phys. Rev., 35, 1288, 1930.