Gases in the Plasma State. I1

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

V. Plasma Oscillations

a. Electrostatic Oscillations of Electrons

Let us consider a quasineutral homogeneous plasma, in a unit volume of which there are \(N\) electrons and \(N\) ions. We shall assume at first that the charges of the plasma are at rest and that the pressure of the neutral gas is so low that the presence of uncharged particles may be neglected. Let the electrons be displaced from their initial positions by segments \(\mathbf{s}\); the segment \(\mathbf{s}\) is, in magnitude and direction, a function of the space coordinates: \(\mathbf{s}=\mathbf{s}(xyz)\). We further assume that the ions have remained in their places and that the ion charge is uniformly distributed in space. Such a displacement, involving only the electrons and leaving the ions immobile, may occur, for example, because a briefly acting field has been created in the plasma between two electrodes; in this case only the light electrons, but not the heavy ions, can be displaced from their places.

In a plasma that was quasineutral at all its points before the displacement of the electrons, spatial charges may appear after the displacement of the electrons; namely, in the volume element \(dx \cdot dy \cdot dz\) a positive volume charge is created if, upon displacement, a number of electrons greater than the number of electrons arriving in it leaves this element (or conversely). The density of the space charge that has thus arisen, for a homogeneous plasma (\(N=\mathrm{const}\)), will be

\[ \rho = e \cdot \operatorname{div}(N\mathbf{s}) = eN \operatorname{div}\mathbf{s}, \tag{5,1} \]

where \(e\) is the electron charge.

If the divergence of the displacement-vector field is zero, i.e. the field \(\mathbf{s}\) is purely vortical, then no space charges are formed. This becomes completely clear if one imagines that the electrons of a homogeneous plasma move uniformly along closed (vortical) filaments, for example along closed circles; in this case an equal number of electrons leaves and enters each volume, and nowhere is there formed an excess of the latter. An arbitrary vec-

the vector field \(\mathbf{s}(xyz)\) can be represented as the sum of a vortex field, free of sources, \(\mathbf{s}_r(xyz)\), and a potential field, free of vortices, \(\mathbf{s}_q(xyz)\). We shall now assume that in the sum \(\mathbf{s}(xyz)=\mathbf{s}_r(xyz)+\mathbf{s}_q(xyz)\) the vortex field \(\mathbf{s}_r(xyz)\) is from the very beginning equal to zero; in other words, by the symbol \(\mathbf{s}\) we shall henceforth mean the potential field \(\mathbf{s}_q(xyz)\).

The space charges whose density is given by (5,1) create an electric field \(\mathbf{E}\), which is related to \(\rho\) by the relation

\[ \operatorname{div}\mathbf{E}=4\pi\rho . \tag{5,2} \]

Substituting \(\rho\) according to (5,1), we obtain

\[ \operatorname{div}\mathbf{E}=4\pi eN\operatorname{div}\mathbf{s}. \tag{5,3} \]

Equation (5,3) can be integrated, since the vector field \(\mathbf{s}\), as well as the field \(\mathbf{E}\) of the space charges, have no vortices. The constant of integration may be put equal to zero, if only one assumes that the external electric field is absent, i.e. that when \(\mathbf{s}=0\) also \(\mathbf{E}=0\),

\[ \mathbf{E}=4\pi eN\mathbf{s}. \tag{5,4} \]

The electric field strength is parallel and proportional to the displacement \(\mathbf{s}\); on each electron (charge \(-e\)!) this field acts with a force \(-e\mathbf{E}\), proportional to the displacement of this electron from its initial position and directed antiparallel to the vector \(\mathbf{s}\); thus the force of the created field tends to return the electron to its initial position1. Consequently, in a homogeneous plasma the electrons are bound to their “position of rest” by a quasi-elastic force. The natural frequency can be calculated from the equation of motion

\[ \mathbf{K}=m\frac{d^2\mathbf{s}}{dt^2}=-e\mathbf{E}=-4\pi e^2N\mathbf{s}; \tag{5,5} \]

it is equal to

\[ \omega_0=2\pi\nu_0=\sqrt{\frac{4\pi e^2N}{m}}, \tag{5,6} \]

i.e. it does not depend on the kind of displacement (if the latter is free of vortices) and does not depend on the volume of the oscillating region. In the case, however, where the displacement field is vortical, no quasi-elastic binding forces and, consequently, no plasma oscillations arise. We shall make repeated use of this distinction in the section on the dielectric properties of plasma. For ordinary plasmas the natural frequency given by equation (5,6) approximately corresponds to decimeter and centimeter waves (\(\nu_0 \simeq 10^9\)—\(10^{11}\)). With this natural frequency the plasma electrons (I. Langmuir[^296], A. Tonks and I. Langmuir[^522]; also Thomson[^511], L. Tonks[^515], I. Kuntz[^275], J. J. Thomson[^509]) execute harmonic oscillations about the position

rest. One might have expected, by analogy with a compressible gas, the occurrence of traveling waves similar to sound waves; this, however, does not take place. On the contrary, at different points of the plasma there occur oscillations not connected with one another in phase. Such oscillations were considered much earlier by Reynolds in his model of independent pendulums1; the well-known discussions of group velocity, energy transfer, etc., also pertain to this problem (P. Zeliger and M. Steenbeck[^469], cf. L. Tonks[^514]). This will be discussed in more detail below.

In deriving (5.6) it was assumed that, before the displacement, all electrons were at rest. In reality this is not so. The electrostatic action of the space charge formed in the plasma owing to the displacement of electrons, in the form of large regions with a more or less uniform density distribution, does not depend on the motion of individual electrons inside the region—the electrons must merely not leave the boundaries of these regions. However, owing to the thermal motion of the electrons, small regions in which the displacement of the electrons should have created, say, an excess of electrons will in a short time be depleted of electrons (by diffusion), and the electron density in these small regions will become equal to the mean electron density. Thus, the causes producing quasi-elastic oscillations will be eliminated. If the region under consideration is so small (linear dimension in the direction \(x\), \(\Lambda_{-}\)) that an electron moving with the effective thermal velocity

\[ w_x=\sqrt{\frac{kT}{m}} \]

in the direction of the \(X\)-axis traverses this region in a time comparable with the reciprocal frequency of the plasma oscillations \(\frac{1}{\omega_0}\), then the space charges producing the oscillation field disperse very rapidly, and a more or less pronounced oscillation of the plasma is no longer possible. Therefore the extent of the plasma region \(\Lambda_{-}\), oscillating as a whole, must be considerably greater than the ratio \(\frac{w_x}{\omega_0}\):

\[ \Lambda_{-}>\frac{w_x}{\omega_0} = \frac{\sqrt{\frac{kT}{m}}}{\sqrt{\frac{4\pi e^2 N}{m}}} = \sqrt{\frac{kT}{4\pi e^2 N}} = D. \tag{5.7} \]

Thus, all linear dimensions of the oscillating region must be several times greater than the Debye length introduced in Section II (I. Langmuir\(^{296}\)). An oscillation arising for any reason will be damped owing to the divergence of the space charges that occurs because of the thermal motion of the electrons. From the preceding considerations it follows that the damping will be approximately proportional to \(\dfrac{D}{\Lambda_-}\). The energy of the plasma oscillations will pass into the energy of the thermal motion of the electrons. In this way the energy of the plasma oscillations is transferred to all the electrons of the system. On the other hand, it must be assumed that the energy of the plasma oscillations (partly or wholly; see below) is created at the expense of the electron temperature. A region of size \(\Lambda_-\) is an oscillator; the energy belonging to one degree of freedom is equal to \(\dfrac{1}{2} kT_-\), if the region is in thermal equilibrium with the electron temperature. This determines the field strength corresponding to the plasma oscillations for a given region of size \(\Lambda_-\), and at the same time also the energy carried away by electrons passing through the zone of oscillations in their thermal motion. Electrons flying with their thermal velocities through this region take away energy. In this way I. Langmuir\(^{296}\) calculated the exchange of energy between electrons; the result (for \(\Lambda_- = 2.8D\)) was given above in Section II. The significance of plasma oscillations for energy exchange between electrons had earlier been indicated by Penning\(^{403,405}\).

b. Allowance for electron pressure

The plasma electrons are not at rest, but move with a velocity corresponding to the electron temperature \(T_-\); therefore, when electrons are displaced from their positions in an initially homogeneous plasma, in those regions where, owing to the appearance of an excess electron density, a negative volume charge is formed, the electron pressure simultaneously increases, and conversely. This increase in pressure \(\Delta p\) may be represented as \(kT_- \cdot \Delta N_-\), where \(\Delta N_-\) is the increase in the electron concentration relative to the mean concentration \(N\). The local increase in the number of electrons due to displacements of the electrons of a homogeneous plasma, \(\Delta N_-\), is equal to \(-N\,\operatorname{div}\mathbf{s}\); therefore the excess electron pressure may be represented in the form

\[ \Delta p = - kT_- N \operatorname{div}\mathbf{s}. \tag{5,8} \]

Naturally, in considering electron oscillations, account must be taken not only of the electrostatic forces considered up to now, but also of the hydrodynamic force \(-\operatorname{grad} p = -\operatorname{grad}\Delta p\). This expression is equal to the force that would move a unit volume of electron gas in the direction from high to low pressure, if we regarded the electron gas as an ordinary molecular gas. The justification of this assumption is discussed below. Consequently, neglecting quadratic terms \(\left(\dfrac{\Delta N_-}{N} \ll 1\right)\), we can pre—

represent the pressure force for one electron in the form

$$ -\frac{1}{N}\operatorname{grad} p = kT_{-}\operatorname{grad}\operatorname{div}\mathbf{s}; $$

this force must be added to the electrostatic force \(-4\pi e^2 N\cdot \mathbf{s}\), which was discussed above [equation (5,5)]. Since always

$$ \operatorname{grad}\operatorname{div}\mathbf{s}=\operatorname{rot}\operatorname{rot}\mathbf{s}+\Delta\mathbf{s}, $$

then, still considering the displacement-vector field to be irrotational, i.e. putting \(\operatorname{rot}\mathbf{s}=0\), we obtain the total force acting on one electron in the form\(^1\)

$$ m\frac{\partial^2\mathbf{s}}{\partial t^2}=-4\pi e^2N\cdot \mathbf{s}+kT_{-}\Delta\mathbf{s} \tag{5,9} $$

or, for the one-dimensional case,

$$ m\frac{\partial^2\xi}{\partial t^2}=-4\pi Ne^2\cdot \xi+kT_{-}\frac{\partial^2\xi}{\partial x^2}, \tag{5,9a} $$

where \(\xi\) is the displacement in the direction \(x\).

Since in (5,9) and in (5,9a), along with the second derivatives with respect to coordinates, there are second derivatives with respect to time, the solutions of these equations give a traveling wave. This circumstance can be visualized if one takes into account that the pressure alone of the electron gas should lead to sound waves in the electron gas; the electrostatic coupling of the electrons with the remaining immobile ions, of course, leads to a modification of this “propagation of sound.”

If we denote the wavelength by \(2\pi\Lambda_{-}\), so that \(\Lambda_{-}\), as above, represents approximately the size of a region with a uniformly (to a certain degree) distributed volume charge, then the frequency \(\omega\) is written in the form (E. G. Linder \(^{319,320}\))

$$ \omega=\omega_0\sqrt{1+\frac{kT_{-}}{4\pi e^2N}\cdot\frac{1}{\Lambda_{-}^2}} =\omega_0\sqrt{1+\left(\frac{D}{\Lambda_{-}}\right)^2}, \tag{5,10} $$

where \(\omega_0\) is the natural frequency represented by equation (5,6).

One may arrive at formula (5,10) in an elementary way if, in (5,9a), one substitutes \(\xi=a\sin\left(\omega t+2\pi\frac{x}{\lambda}\right)=a\sin\left(\omega t+\frac{x}{\Lambda}\right)\) and finds \(\omega\). The phase velocity \(v_p\) is expressed by the equality

$$ v_p=\omega_0\sqrt{\Lambda^2+D^2}; \tag{5,11} $$

for the group velocity we have

$$ v_g=\frac{\omega_0D}{\sqrt{1+\left(\frac{\Lambda}{D}\right)^2}}. \tag{5,12} $$

\(^1\) A vortical displacement of the electrons of a homogeneous plasma nowhere creates an excess of electrons. In exactly the same way, a vortical displacement cannot create, anywhere, an excess electron pressure.

From the considerations set forth in the preceding section it follows that, for real oscillations of the plasma, \(\Lambda_{-}\) must be at least several times greater than \(D\); thus, the change of frequency due to the change of electron pressure is very insignificant (for \(\Lambda_{-}=10D\), only \(1/2\%\)!). It is quite plausible that the frequency increases with increasing electron pressure; the pressure drop, just like the electrostatic force, causes electrons to pass from places of higher concentration to places of lower concentration. One may say that the pressure increases the quasi-elastic electrostatic coupling. It is also quite understandable that the influence of pressure completely disappears at very large wavelengths; indeed, the pressure drop, owing to which the driving force arises, becomes the more gradual the larger \(\Lambda_{-}\) is. The fundamental significance of the electron pressure lies in the fact that it creates a traveling wave in the plasma and thereby acts on the independent oscillations of individual regions of the plasma (occurring under the action of the electrostatic force) in such a way that it couples the phases of the individual oscillations. True, the group velocity with which the energy is transported is, for \(\Lambda_{-}\gg D\), according to (5,12), very small.

Let us briefly discuss the question: can one, and if so under what conditions, ascribe a real physical meaning to pressure in plasma oscillations? The kinematic mechanism of compression oscillations in a gas is as follows: gas molecules fly out with their thermal velocities from a region \(A\), where at some instant the gas concentration is higher than in a region \(B\), and enter the region \(B\); as a result, owing to the transfer of momentum occurring through collisions, the whole aggregate of molecules is displaced on average in the direction of \(B\), where, thus, an increase of pressure will occur. The pressure in region \(A\) will decrease as long as the number of molecules flying out of \(A\) is greater than the number of molecules returning from \(B\). The process of pressure decrease in \(A\) continues even when the pressure \(B\) reaches the pressure in \(A\): the molecules returning from \(B\) to \(A\), whose number is now already large, traverse the path from \(B\) to \(A\) during a finite interval of time. Moreover, the returning molecules fly more slowly, since they have been reflected from the layer of retreating (i.e., moving from \(A\) to \(B\)) molecules (adiabatic cooling upon expansion). Thus pressure oscillations and traveling waves arise.

It follows from the above that the wavelengths must be considerably greater than the mean free paths of the gas molecules. Applied to the electron gas of the plasma, this means that the wavelength \(2\pi\Lambda_{-}\) must be considerably greater than the relaxation length \(s\) (the magnitude of the latter was discussed in Section II). In all cases encountered, the relaxation length \(s\) is much greater (for the most part by several orders of magnitude) than the Debye radius \(D\); therefore, all the more, \(\Lambda_{-}\gg D\), and this means that the influence of pressure on the frequency of the electron oscillations in the plasma may be neglected. As a consequence of the same inequality, the coupling between the oscillations of individual regions of the plasma becomes quite insignificant—the group velocity is practically equal to zero.

c. Oscillations of ions.

In section “a” we assumed that, when the electrons are displaced by segments \(s\), the inert ions remain at rest. Let us now investigate the case in which the ions move by segments \(\mathbf{s}=\mathbf{s}(xyz)\); we shall still assume that the field of the vector \(\mathbf{s}\) is irrotational. In this case, of course, one cannot assume that, as was accepted for the inverse case considered in “a,” the electrons remain at rest, forming a uniformly distributed negative charge. On the contrary, the electrons will be attracted by those regions in which the displacement of the ions produces an excess of positive electricity; since the light electrons will follow the attractive forces, part of the positive volume charge that has arisen will be neutralized by electrons drawn into these regions. If the electrons had no thermal velocities, this neutralization would be complete. The temperature of the electrons has a finite value, and therefore the differences in the concentrations of the electron gas will be smaller than the same differences for the ions, in complete analogy with what was said above concerning ambipolar diffusion1. The displacements of the ions occur considerably more slowly than the displacements of the electrons; therefore, for each distribution of potential formed by the instantaneous position of the ions, an equilibrium distribution of electrons has time to become established (according to Boltzmann’s law).

For simplicity we shall carry out the calculation for the one-dimensional case. Denoting by \(\xi\) the displacement of a positive ion in the direction \(x\), we may write the increment in the number of ions \(\Delta N_+\) relative to the constant mean value \(N\) in the form

\[ \Delta N_+ = -N \frac{\partial \xi}{\partial x}. \tag{5,13} \]

Let us denote the potential at the point \(x\) by \(U\) (this potential is determined relative to the potential at the same point of the plasma for a completely uniform distribution of electrons and ions, i.e., in the absence of any acting space charges); then, in accordance with the Boltzmann distribution, the concentration of electrons at this point is

\[ N+\Delta N_- = Ne^{\frac{eU}{kT_-}}, \]

or, for small displacements, and therefore also for small values of \(U\) (i.e. \(\frac{eU}{kT} \ll 1\)):

\[ \Delta N_- = N\left(e^{\frac{eU}{kT_-}}-1\right) \simeq N\frac{eU}{kT_-}. \tag{5,14} \]

For the density of the volume charge \(\rho\) we obtain

\[ \rho=e(\Delta N_+-\Delta N_-)=-Ne\left\{\frac{\partial \xi}{\partial x}+\frac{eU}{kT_-}\right\}. \]

Using Poisson’s equation \(\dfrac{\partial^2 U}{\partial x^2}=-4\pi\rho\), we have

\[ \frac{\partial^2 U}{\partial x^2} = 4\pi eN\left(\frac{\partial \xi}{\partial x}+\frac{eU}{kT_-}\right). \tag{5,15} \]

The force acting on an ion is equal to \(eE=-e\dfrac{\partial U}{\partial x}\); thus,

\[ M\frac{\partial^2 \xi}{\partial t^2}=-e\frac{\partial U}{\partial x}, \tag{5,16} \]

where \(M\) is the mass of the ion.

Taking the partial derivative with respect to \(x\) of (5,15) and substituting the values of \(\dfrac{\partial^3 U}{\partial x^3}\) and \(\dfrac{\partial U}{\partial x}\) from (5,16) [having first differentiated (5,16) twice with respect to \(x\)], we obtain the following differential equation for the oscillation of ions:

\[ \frac{\partial^2}{\partial x^2} \left( \ddot{\xi}+\frac{4\pi Ne^2}{M}\xi \right) - \frac{4\pi Ne^2}{kT_-}\ddot{\xi} = 0. \tag{5,17} \]

Here partial differentiation with respect to time is denoted by dots over the letters. Since in the equation, along with the second derivative with respect to time, the second derivative with respect to the coordinate again enters linearly, (5,17) is the equation of a traveling wave. If the wavelength is equal to \(2\pi\Lambda_+\), where \(\Lambda_+\) denotes the approximate size of a region of more or less homogeneous displacement of positive ions, then the frequency is equal to

\[ \omega_+ = \sqrt{ \frac{4\pi Ne^2} {M\left(1+\dfrac{4\pi Ne^2}{kT_-}\Lambda_+^2\right)} } = \sqrt{\frac{4\pi Ne^2}{M}}\cdot \sqrt{ \frac{1} {1+\left(\dfrac{\Lambda_+}{D}\right)^2} }. \tag{5,18} \]

Formula (5,18) can be derived by substituting, as before, into the differential equation the quantity:
\[ \xi=a\sin\left(\omega t+\frac{x}{\Lambda_+}\right). \]
The phase velocity of the wave \(v_p\) is equal to \(\lambda\Lambda=\omega\Lambda_+\);

\[ v_p= \sqrt{\frac{kT_-}{M}}\cdot \sqrt{ \frac{1} {1+\left(\dfrac{D}{\Lambda_+}\right)^2} }; \tag{5,19} \]

the group velocity is found from the relation

\[ v_g=v_p-\lambda\frac{dv_p}{d\Lambda}; \]

it is equal to

\[ v_g= \sqrt{\frac{kT_-}{M}}\cdot \sqrt{ \frac{1} {\left\{1+\left(\dfrac{D}{\Lambda_+}\right)^2\right\}^3} }; \tag{5,20} \]

If the size of the oscillating region is very small, i.e. \(\Lambda_+\ll D\), then the frequency (5,18) coincides with that calculated above [equation (5,6)]

with the plasma electrons’ own frequency—of course, with \(m\) replaced by \(M\). This corresponds to the assumption used in deriving (5,6), if the role of the ions is played by the electrons, and conversely, i.e., in other words, to the assumption that the electron density is independent of the displacement of the ions. Indeed, a very small region of positive space charge has little noticeable effect on the concentration of electrons in the plasma; therefore the latter may remain uniform, which explains the upper limiting value of the frequency \(\omega_+\). It should be mentioned that for ion oscillations one may take \(\Lambda_+\) to be smaller than \(D\), in contrast to what we had above for electron oscillations. In any case this is valid when the mean kinetic energy of the ions is much less than the kinetic energy of the electrons (i.e., the “ion temperature” \(\ll\) the temperature of the electrons, a nonisothermal plasma). Then, despite the low frequency of ion oscillations, the time of one oscillation is not sufficient for the random motion of the ions to cause a significant number of them to leave the oscillating region. As the size of the oscillating region increases, i.e., as the wavelengths increase to \(\Lambda_+ \gg D\), the group and phase velocities reach the value

\[ \sqrt{\frac{kT}{M}}, \]

independent of the concentration (!). This value corresponds to the velocity of sound in a gas whose molecules have the mass of an ion and the temperature of the electrons, if the adiabatic cooling in the expansion of compressed regions is not taken into account. For an intuitive picture one must take the following into account: large regions of increased ion concentration become “saturated” with electrons to such an extent that the space charges compensate one another; therefore for \(\Lambda_+ \gg D\) the charge \(e\) drops out of the formula. Consequently, only compression waves are formed, and the driving force is the electron pressure, which acts on the ions through the electrostatic coupling between charges (exactly as in ambipolar diffusion). In the dynamical sense the behavior of such a compressed region of plasma fully corresponds to the behavior of a gas under a pressure equal to the electron pressure, with particles whose mass is equal to the mass of the ions\(^{1}\). The decrease in temperature due to adiabatic expansion need not be taken into account, since an intensive exchange of energy takes place among the plasma electrons. In an isothermal plasma the pressure of the ion gas is equal to the pressure of the electron gas; the “speed of sound” in this case exceeds (5,19) or (5,20) for \(\Lambda_+ \gg D\) by a factor of \(\sqrt{2}\). This corresponds to the fact that the “mean molecular weight” of a mixture of equal numbers of ions and electrons is half the corresponding value for ions alone.

In contrast to electron oscillations, in which the pressure of the electrons, and together with it the occurrence of a traveling wave,

\(^{1}\) If, in equations (5,13) and the following ones, one takes into account friction in the neutral gas, then at sufficiently strong damping an aperiodic equalization of pressure is obtained. In the limiting case of large friction we arrive at the equations of ambipolar diffusion.

plays only a secondary role, oscillations of the ions can cause a sufficiently intense exchange of energy between the zones of the plasma. Possible natural frequencies, beginning with very high values

\[ \omega=\sqrt{\frac{4\pi e^2N}{M}} \]

(for ordinary plasmas corresponding to 10–100-meter waves) and ending approximately with the natural acoustic tones of the vessel in which the discharge occurs. The conditions for the occurrence of plasma oscillations have not yet been studied in sufficiently detailed fashion. Some, however very meager, experimental facts will be discussed in the following section.

We have throughout considered the case of a homogeneous plasma (\(N=\mathrm{const}\)). If \(N\) depends on position—a case very important for practice—the relations become unusually complicated; one can point only to the works of L. Tonks \(^{515,516}\), and we cannot dwell here on the content of these investigations.

d. The occurrence of plasma oscillations

The first observation of clearly expressed, spontaneously arising natural oscillations of the plasma of a gas discharge was evidently made by Penning \(^{403,405}\). In a Lecher system connected to electrodes, sharp maxima and minima corresponding to wavelengths of 50–100 cm were observed. Such an order of magnitude was to be expected for natural electron oscillations. Tonks and Langmuir \(^{522}\) established the presence in the plasma of a gas discharge of oscillations with wavelengths from 25 to 80 cm, by creating a capacitive coupling by means of external electrodes between the plasma and a Lecher system; the experimentally found frequencies agreed approximately with equation (5,6); the experiment gave the correct dependence on the electron density. With approximately the same experimental arrangement M. Steenbeck showed agreement, to an accuracy of up to 15%, between wavelengths found experimentally and calculated from (5,6). This accuracy in all probability exceeds the accuracy of measurements made by means of probes. A number of further works will be discussed below in the section on the dielectric properties of plasma. The existence of regular free oscillations of electrons, emitted by the plasma, of the type corresponding to equation (5,6), cannot be called into question.

With regard to ion oscillations we cannot make so categorical a judgment. The reason for this circumstance is partly that the frequencies of ion oscillations are distributed over a very broad interval and can only with great difficulty be distinguished from the statistical noise of the discharge. In almost any investigation of a gas discharge one can find some frequencies (see, for example, L. Bloch \(^{35}\)). Thus, for example, Webb and Pardue \(^{536,537}\) found frequencies from several \(10^2\ \mathrm{sec}^{-1}\) to several \(10^5\ \mathrm{sec}^{-1}\); within this entire interval there are frequencies that could belong to ion oscillations.

The occurrence of clearly expressed natural oscillations is indicated in Gerber’s work \(^{169}\); the author found, for an arc, obvious, very sharply expressed anomalies (of the type of anomalies in dispersion) in the absorption of frequencies of the order of \(10^5\)–\(10^6\ \mathrm{sec}^{-1}\). It is possible that these anomalies

are explained by resonance with the natural oscillations of the ions1. For more precise conclusions, the available experimental material is still not sufficiently varied and satisfactory.

Langmuir posed the question of why, properly speaking, it is possible to detect the presence of oscillations of plasma electrons by exciting the Lecher system from the discharge plasma. If the energy of the oscillation is in thermal equilibrium with the electron gas, then to the Lecher system there can be transferred an oscillation energy of the order of $kT$; however, despite the high temperature of the electrons, this quantity is too small to be observable. In reality this energy is observed; it is therefore necessary to assume that under some circumstances electronic oscillations arise which draw their energy not from a statistically random differential effect in a disordered plasma, but from some factor supplying energy systematically (systematically driving the electrons). Penning already pointed to the evident influence of the electrodes. It consists, possibly, in the following: electrons arrive one after another at the wall bounding the plasma; upon reaching the electrodes, the electrons pass into the external circuit; upon reaching an insulated wall, the electrons recombine with ions arriving at this wall. Suppose that a large number of electrons in the region of the plasma adjoining the wall oscillates uniformly in a direction perpendicular to it; the influx of electrons to the wall is greatest when the motion of the electrons toward it reaches a maximum. The periodically varying influx of electrons to the wall creates oscillations of potential on it; the phase of the latter is such that the oscillation of the electrons in the plasma is “inflated”2. This is not difficult to show by calculation. We have indicated the existence of a cause capable of ordering the oscillations of the plasma; it remains unclear whether this cause is the only and the most important one.

It is interesting that in all observed cases the radiation of the oscillating plasma electrons lies in the decimeter-wave region, whereas according to equation (5.6), at sufficiently large electron concentrations, oscillations are possible with frequencies corresponding to centimeter and millimeter waves. For technical reasons the latter would be especially desirable. The fact that spontaneous plasma oscillations with large energy occur in the decimeter region is perhaps explained by the relatively small relative damping in these cases. Plasma oscillations will naturally be the more intense, the smaller the number of interfering collisions to which an electron is subjected during the period of oscil-

decay. Therefore slow oscillations of the electrons are excluded, since for them the probability is very high that an electron will collide with a gas molecule or with the wall of the vessel in which the discharge takes place. Very rapid oscillations of the electrons, for whose existence large densities of electrons and ions are necessary, lead to an increase in the number of scatterings of the oscillating electrons by positive ions—the latter in reality do not constitute a charge uniformly distributed in space, as was assumed in the derivation of (5,6). The number of collisions with positive ions occurring per unit time increases in proportion to the ion concentration (see Sec. III); on the other hand, the time of one oscillation decreases only in proportion to the square root of the concentration; therefore the probability that an electron will be excluded from the oscillation by collision with a positive ion is the greater, the higher the ion concentration, i.e., the greater the natural frequency of the plasma oscillations. For this reason the plasma oscillations observed with Lecher’s system cease as soon as the density of the neutral gas, or the discharge current (together with which the electron concentration increases), is greatly increased. A rough calculation shows that the action of impacts, owing to which plasma oscillations are damped, is minimal in the region of decimeter waves.

In conclusion we shall briefly describe two optical examples. Klumb\(^{244}\) found in hydrogen plasma an especially strong absorption of waves with wavelengths of 3.9 and 28 cm; he explained this fact by reducing the observed absorptions to three electronic transitions with a small difference in energy in the term scheme of the hydrogen atom; these transitions should have been detected spectroscopically for waves of 2.74, 9.25, and 24.75 cm. If Klumb’s explanation is correct, then the indicated absorption has nothing in common with the properties of plasma discussed in this section. Equation (5,6) was applied by M. Steenbeck\(^{495}\) to the “plasma” of a metal (in which the electrons move freely among a positive volume charge distributed in the form of a lattice; the “plasma” of a metal is quasineutral). Direct use of (5,6) leads to “plasma oscillations” corresponding to ultraviolet wavelengths, namely:

\[ \lambda = 431 \sqrt{\frac{A}{q}}\,\text{\AA}. \tag{5,21} \]

Here \(A\) is the volume of the atom, \(q\) the number of conduction electrons detached from each atom. If such a calculation, in which the classical equations are applied to a pronounced electron gas, is at all possible, then at wavelengths corresponding to (5,21) anomalies in the absorption and reflection of light should be observed. One might expect that the vortex field of an unperturbed light wave should cause only vortex displacements of electrons, which cannot lead to plasma oscillations. However, a more careful analysis shows that the vortex field of the electric vector of the incident wave creates a displacement field practically free of vortices, provided only that the electric vector is perpendicu-

Table 6
Plasma oscillations

$^{1}$ Calculated for the electron path length at 1 torr and 273° K: for Hg — $3\cdot 10^{-3}\ \mathrm{cm}$, for Ne — $2\cdot 10^{-4}\ \mathrm{cm}$, for $\mathrm{N}_2$ — $2\cdot 10^{-2}\ \mathrm{cm}$.

$^{2}$ Number of conduction electrons per atom, $q=1$.

is perpendicular to the metal surface; this fact is connected with the strong absorption of light waves in the metal. If, however, the electric vector is parallel to the surface, as, for example, in normal incidence of the ray, then only vortical displacements of the plasma electrons take place, and therefore oscillations in the plasma are not excited. A number of older experimental works indicate anomalies in the reflection and absorption of light in metals lying in the ultraviolet region; the arguments given above find confirmation in these works.

The most recent measurements by Smakula^484 raised the question of the validity of these ideas, since the indicated anomalies were not observed by this investigator; it is true that Smakula’s measurements were carried out for small angles of incidence.

VI. DIELECTRIC AND MAGNETIC PROPERTIES OF PLASMA

Plasma possesses dielectric and magnetic properties for two reasons. First, neutral molecules, excited atoms, ions, and excited ions possess intrinsic or induced electric and magnetic moments (electrons have only a magnetic moment); a more or less regular orientation of these moments in an external field leads to the same phenomena as in a neutral gas. Plasma differs from its neutral gas by the magnitude of the atomic dipoles, which change in the excited state. This influence, however, will not be discussed further. Secondly, dielectric properties are acquired by plasma through the displacement of free charges; magnetic properties are acquired by it owing to the curvature of the paths of charges in an external magnetic field. These causes of the appearance of dielectric and magnetic properties are specific to plasma and, of course, do not exist for a neutral gas. Only these features of plasma as a system with dielectric and magnetic properties will be discussed below, since the influence of these factors on the properties of interest to us, under suitable conditions, considerably exceeds the influence of atomic dipole moments.

a. Dielectric properties

The dielectric “constant” of plasma \(\varepsilon\) is defined in the usual way as

\[ \varepsilon = 1 + 4\pi\chi, \tag{6,1} \]

where \(\chi\) is the sum of the dipole moments per unit volume in an external field equal to unity. If the dipole moment of a charge in a field \(E\) is denoted by \(M\), then

\[ \chi = N\frac{M}{E} = N\frac{er}{E}, \tag{6,2} \]

where \(r\) is the displacement of the charge \(e\) under the action of the field \(E\). If the plasma consists of different charge carriers with concentrations \(N_1, N_2, \ldots, N_i\), then

\[ \chi = \frac{1}{E}\sum_i N_i M_i . \]

The free charges of a plasma cannot be regarded as charges with a firm quasielastic bond; therefore the moment \(\mathbf{M}\) is not determined only by the field strength \(\mathbf{E}\).

If the plasma is in a constant field, then the charges are displaced in this field over distances that increase with time to any magnitude. Thus, for the frequency \(\omega=2\pi\nu=0\) it is impossible to determine a finite moment \(\mathbf{M}\), and consequently also the dielectric constant; as in the case of electrolytes, the electrical conductivity covers all the dielectric properties. If, however, a variable field \(\mathbf{E}=\mathbf{E}_0\sin\omega t\) acts on a free charge \(e\) with mass \(m\), then the displacement of the charge \(s\), calculated from the equation of motion

\[ m\frac{d^2s}{dt^2}=eE_0\sin\omega t, \]

is the oscillation

\[ s=-\frac{e}{m\omega^2}E_0\sin\omega t. \]

The induced moment

\[ \mathbf{M}=es=-\frac{e^2}{m\omega^2}\mathbf{E} \]

is proportional to the field strength; the coefficient of proportionality is negative. The dielectric constant, calculated from (6,1) and (6,2),

\[ \varepsilon = 1-\frac{\dfrac{4\pi Ne^2}{m}}{\omega^2} = 1-\frac{\omega_0^2}{\omega^2} \tag{6,3} \]

(Eccles’ relations\(^{102}\) \(^{1}\)) has, for \(\omega>\omega_0\), values between 1 and 0 and depends sharply on the frequency\(^{2}\). Because of the mass standing in the denominator of expression (6,3), the photoelectric properties of the plasma are determined predominantly by the free electrons of the plasma. The term \(\dfrac{Ne^2}{m}\) can assume, for ions, a value comparable with \(Ne^2/m\) for electrons only in the case (G. Gubau\(^{176}\)) where the concentration of ions is much greater than the concentration of electrons; the latter is possible only in the presence of a large number of negative ions (the condition of quasineutrality). The charge \(e\) enters (6,3) squared, and therefore the actions of charges of different signs are added. We shall arrive at expression (6,3) also if we take into account that the motion of electrons in a variable electric field is superposed on disordered thermal motion. The same can be said also with respect to the other conclusions, except those in which damping is taken into account. Only in the latter cases is it necessary to take account of the thermal motion of the electrons.

If a plasma with sufficiently small \(N\), and consequently with sufficiently small \(\omega_0^2\), is placed between the plates of a capacitor, then, according to equation (6,3), the charging current of the capacitor will decrease. This is visually explained by the fact that the currents produced by freely oscillating charges, being purely reactive in phase, weaken the capacitive charging current as a result of a phase shift; the plasma charges, possessing inertia, follow the oscillations of the field with a delay. The voltage, ap-

\(^{1}\) See also \(^{455,456}\).

\(^{2}\) Found experimentally by van der Pol\(^{411}\). See also Bergmann and Döring\(^{30}\).

driven by the motion of the plasma charges, opposite to the external one. If this inductive current of the charges is greater than the capacitive current of the capacitor, then the whole system acts as an inductance, i.e., as a negative capacitor. Such is the meaning of the negative dielectric constant possible according to equation (6.3) at large \(\omega_0\). The numerator in (6.3) must have the dimension of the square of a frequency; the fact that it happens\(^{1}\) to be equal to the square of the natural frequency of the plasma electrons [compare equation (5.6)] does not mean that the calculation was carried out under the assumption of a quasi-elastic coupling (with this frequency) of the electrons with the equilibrium position; in the derivation, the laws for free charges were used. If, in addition, the quasi-elastic force is taken into account, then for the dielectric constant one obtains the expression

\[ \varepsilon = 1 - \frac{\omega_0^2}{\omega^2 - \omega_0^2}, \tag{6.4} \]

which for small \(N\) practically coincides with (6.3). Which of the two formulas—(6.3) or (6.4)—should be used in any given case depends, according to what was set forth in Section V, on the character of the displacement of the electrons. If the latter is free from sources, then (6.3) is valid; for a vortex-free field one must use (6.4). In the general case, however, an unambiguous definition of the dielectric constant is impossible even for given \(N\) and \(\omega\).

The appearance of resonance phenomena in some device with a capacitor filled with plasma\(^{11, 13, 14, 188–190}\) still does not mean that a true resonance arises in the plasma, corresponding approximately to equation (6.4).\(^{2}\) Rather, such a resonance will occur owing to the joint action of an inductive capacitor with plasma (\(\varepsilon < 0\)) and some parasitic capacitance of the device used. Appleton\(^{11, 13, 14}\) and Segrist\(^{477}\) believe that the occurrence of a true plasma resonance is possible only in the case of a nonuniform plasma; the authors of the present article believe that oscillations are also possible for a uniform plasma (see above, Section V). We shall discuss the possibility of exciting such oscillations in the ionosphere.

For the propagation of electromagnetic waves in the earth’s ionosphere, the dielectric constant of the ionized gas is of great importance. If an electromagnetic wave of frequency \(\omega\) is incident from a nonionized gas on the plane boundary of a homogeneous plasma having \(N\) electrons per unit volume, then \(\left(\text{we take } \omega_0^2 = \frac{4\pi N e^2}{m} < \omega^2\right)\) in any case the dielectric constant \(\varepsilon < 1\), although it is positive. Thus, the transition of a wave from a nonionized to an ionized gas corresponds to a transition into an optically less dense medium (between the refractive index \(n\) and the dielectric constant \(\varepsilon\), as is well known, the relation \(n^2 = \varepsilon\) holds). Therefore, upon

\(^{1}\) The appearance of the term \(\dfrac{N e^2}{m}\) is not accidental in the sense that the combination \(N, e, m\), having the dimension \(\omega^2\), must have precisely this form.

\(^{2}\) P. O. Pedersen\(^{399}\); see also the detailed review of K. K. Darrow\(^{70}\), D. Rayner\(^{451}\).

for angles of incidence \(\alpha\) greater than \(\arcsin \sqrt{\varepsilon}\) total internal reflection is possible. Consequently, for \(\varepsilon = 0\) the wave is completely reflected even at normal incidence, in any case when the thickness of the plasma layer is large in comparison with the wavelength1. This reflection of short waves is indeed observed in wireless telegraphy[^90]. If the wave penetrates into the plasma \((\varepsilon > 0)\), then the phase velocity \(v_p > c\); however, the group velocity \(v_g\) is always less than \(c\). An electromagnetic wave whose electric field is free from sources produces only purely vortical displacements of the electrons (in any case, if the wave is little absorbed along a path \(\lambda\); for the opposite case of absorption in metals see above, p. 321); therefore, in calculating \(\varepsilon\) one need not take into account the coupling force, i.e. \(\varepsilon\) is determined by formula (6, 3)2. If, on the contrary, the dielectric constant is calculated for a plasma situated in the field of a capacitor, and if measures are taken to prevent the oscillating charges from reaching the electrodes (for this purpose one may, for example, impose on the electrodes a preliminary constant potential, negative with respect to the plasma), then one must use formula (6,4), since in this potential field the plasma coupling force acts3.

If the concentration \(N\) of electrons in the plasma is so large that \(\omega'^2 \geq \omega^2\), then the passage of electromagnetic radiation through it is excluded4. For phenomena in the ionosphere this case is of no significance—

GASES IN THE PLASMA STATE

… since the penetrating electromagnetic wave is already completely reflected from the less ionized regions. However, it is quite possible to investigate such a plasma between the plates of a capacitor. The course of the dispersion is in the main consistent with the predictions of formula (6.4) [B. Sigriste 477], but is to a considerable extent blurred by the entirely negligible damping and by the inevitably arising inhomogeneities of the plasma.

The damping of electronic oscillations occurs for various reasons. One of them consists in the radiation of electromagnetic waves by the oscillating electrons (dipole radiation). Much more important—for frequencies of hertzian waves by several orders of magnitude—are the losses of energy of the oscillating electrons through collisions with neutral gas molecules. An elastic collision of a gas molecule with an electron does not take energy away from the latter, but only changes the direction of its motion; this means, however, a complete loss of the oscillatory energy of the given electron, since it passes into the energy of disordered thermal motion1. The number $\nu$ of collisions per unit time is determined mainly by the thermal velocity $w$ and the mean free path $\lambda$, namely $\nu = \frac{w}{\lambda}$.

If the energy (kinetic and potential) of an oscillating electron is equal to $m v_s^2$, where $v_s$ is the effective velocity of its harmonic oscillation, which in the case of weak fields (!) may be taken as small in comparison with the thermal velocity $w$, then the loss of energy through collisions2 per unit time per electron is equal to $\nu \cdot m v_s^2$. Formally these energy losses of the oscillation may be regarded as losses due to friction in a continuous medium; for this it is necessary to introduce a frictional force proportional to the velocity, $-\rho \cdot v$. Then the losses of energy due to friction per unit time are equal to $\rho v_s^2$. We may therefore use the known equations of oscillation of a material point under the action of a variable external force in a medium with friction in order to determine the magnitude of the dipoles actually formed in the plasma; here we shall put the Newtonian coefficient of friction equal to3

\[ \rho = \nu \cdot m = \frac{w \cdot m}{\lambda}. \tag{6.5} \]

A well-known elementary calculation leads to a complex dielectric constant, i.e. to the damping of electromagnetic waves; in the experiment with a condenser this reduces to the parallel connection of an equivalent resistance. The complex dielectric constant has the form

\[ \varepsilon = 1 - \frac{\omega_0^2}{\omega^2 - i\omega \rho}, \tag{6,6} \]

if one does not take into account the binding force of the plasma [compare (6,3)], and

\[ \varepsilon = 1 - \frac{\omega_0^2}{\omega^2 - \omega_0^2 - i\omega \rho} \tag{6,7} \]

when this force is taken into account [compare equation (6,4)]. Equations (6,5), as well as (6,6) and (6,7), are important in that they make it possible, on the basis of measurements of the intensity and “fullness” of the reflected Hertzian waves, to determine \(\rho\), and consequently also the gas density \((\lambda l)\) in the ionosphere. Agreement is achieved as to order of magnitude.

Let us also point out the possibility of another frictional force. In discussing the question of the relaxation length it was noted that, owing to the asymmetry of the Debye cloud of charges surrounding a charge in a plasma, a retarding relaxation force arises, acting on this charge. The formation of a cloud around an oscillating charge requires time; therefore, between the force that has arisen and the velocity of oscillation there will be a phase shift. If the number of neutral molecules in the plasma is relatively large \((\lambda < D)\), then the time of formation of the Debye layer around a plasma charge may be calculated from the coefficient of friction for ions. The relaxation force calculated in this way is in complete agreement with the frequency-dependent retarding force for strong electrolytes1.

The resulting damping (complexity of the frequency) may be neglected for all phenomena occurring in the ionosphere (K. F. Niessen \(^{375}\)), both with respect to the loss of energy and with respect to the amplitude of the moments of the dipoles that are formed. Such neglect, in all probability, cannot be made for laboratory experiments.

Of special interest is the propagation of Hertzian waves in a plasma situated in a homogeneous magnetic field. In this case there arise effects analogous to the classical Zeeman effect; here, as in the Zeeman effect, it becomes possible to determine the quantity of electricity of the oscillating plasma charges, as a result of which identification of these charges with free electrons is possible (G. Gutton \(^{186,187}\)). It should be expected that the application of a magnetic field makes the plasma optically inhomogeneous, similar to a birefringent crystal. When considering the general case—the magnetic field and the direction of the ray forming an arbitrary angle with one another—very complicated relations arise. For clarity we shall confine ourselves to calculating two simple cases, namely: 1) a longitudinal mag-

netic field, i.e., the field-strength vector and the ray are parallel; 2) a transverse field, i.e., the field-strength vector is perpendicular to the ray. We shall also disregard any friction of the electrons, thereby avoiding complex quantities; the inclusion of damping, however, is quite elementary and gives nothing new.

The action of the electrodynamic force on a freely flying electron consists, as is known, in curving its path into circular motion without change of speed; here the axis of the circle coincides with the direction of the magnetic field, and the angular frequency of the motion is independent of the speed and is equal to

\[ \omega_H=\frac{eH}{mc}; \tag{6,8} \]

here, as always, \(e\) is in electrostatic units.

Let us first consider the case of a longitudinal field.

From symmetry considerations it is evident that especially simple relations hold for a circularly polarized wave, in which the electric and magnetic vectors rotate with the frequency of light around the direction of the ray as an axis, without change of amplitude. We shall call the direction of rotation of the field vectors right or left from the point of view of an observer looking along the direction of propagation of the light.

Under the action of such a wave, in the absence of a magnetic field, a free electron moves in a circle, and the frequency and direction of the electron’s motion coincide with the frequency and direction of rotation of the wave. This displacement field is proportional in magnitude to, and coincides in direction with, the vortical electric field of the wave; therefore the displacement field is also vortical, i.e., its divergence is equal to zero. Our calculation is thus analogous to the derivation of equation (6,3) for free electrons: there is no binding force of the plasma. Let us denote the electric field strength by \(E\), and the frequency by \(\omega\). The radius of the circle can be computed from the condition of equality of the oppositely directed forces—the centrifugal force \(mr\omega^2\) and the electric force \(eE\). We obtain

\[ r=-\frac{eE}{m\omega^2}. \]

In accordance with equations (6,1) and (6,2), the dielectric constant of a plasma with \(N\) charges per unit volume is equal to

\[ \varepsilon=1+4\pi\chi=1+4\pi N\cdot\frac{er}{E} =1-\frac{4\pi Ne^2}{m\omega^2}=1-\frac{\omega_0^2}{\omega^2}, \]

which coincides with (6,3).

In the presence of a longitudinal magnetic field, the centrifugal force is balanced not only by the electric force of the wave field, but also by the electrodynamic force \(\frac{evH}{c}\). The latter weakens or strengthens the action of the electric field strength of the wave depending on the direction of rotation of the wave (electron). From the condition

\[ -mr\omega^2=eE\pm\frac{evH}{c}=eE\pm\frac{er\omega H}{c} \]

we obtain the following expression for the dielectric constant:

\[ \varepsilon = 1-\frac{\omega_0^2}{\omega^2 \pm \omega \omega_H}. \tag{6,9} \]

The plus sign \((+)\) occurs in the case when the direction of rotation of the wave coincides with the direction of rotation of the free electron in the magnetic field; the minus sign \((-)\), when these directions are opposite.

A linearly polarized wave may be represented as two waves circularly polarized in opposite directions. If a linearly polarized wave passes through a homogeneous plasma in the direction of the magnetic field, then the dielectric constants, and consequently the refractive indices and propagation velocities of the two components of the linearly polarized wave, are different. On emerging from the plasma the circularly polarized waves will lag behind one another in phase and therefore will combine not into a linearly polarized wave but, in the general case, into an elliptically polarized wave. In this case the axis of the polarization ellipse will be inclined with respect to the plane of polarization of the linear light incident on the plasma. This “Faraday magneto-optical effect” was observed, for example, by P. Keck \(^{236}\); the experimental data are in good agreement with the theory.

In considering the passage of an electromagnetic wave in a direction perpendicular to the magnetic field, it is likewise convenient to start from the behavior of waves polarized in a definite way, and then to represent the incident wave as a combination of these polarized waves.

The direction of the ray and the plane of the circular path of an electron moving in a magnetic field coincide; when viewed along the direction of the ray, the motion of the electron appears as a linear oscillation (an oscillating dipole). It is therefore expedient first of all to consider the action on the plasma of two linearly polarized waves, in which the electric vector oscillates: a) in the direction of the magnetic field, b) in a direction perpendicular to the vector of the magnetic field and parallel to the direction of the apparent oscillation of the electron. Case a) differs in no way from the case of absence of a magnetic field, since on an electron oscillating in the direction of the magnetic field no electrodynamic force acts from this field. For this “ordinary” ray equation (6,3) is valid.

The electric vector of a wave oscillating perpendicular to the magnetic field imparts to the plasma electrons periodic velocities perpendicular to the vector of the magnetic field; in this case electrodynamic forces act on the electrons. These forces are perpendicular to the vector of the magnetic field and to the vector of the electric field of the wave, i.e. longitudinal, in other words, parallel to the direction of the ray. The action of these forces consists in a displacement of the electrons alternately in the direction of and against the direction of the ray: a compression field arises in the electron gas, with alternating zones of increased and decreased electron density located at a distance equal to half-

wavelength. For an infinite plane wave the displacement field that has arisen is a purely vortex-free field. Therefore, for longitudinal displacements, and only for them, it is necessary to take into account the quasi-elastic binding forces of the plasma \(^{1}\) [see above, Section V, equation (5.5)]. In this case the natural oscillations of the plasma electrons are excited directly, despite the fact that the primary exciting electric field of the wave is free of sources.

Let the direction of propagation of the wave coincide with the \(Z\)-axis; let the magnetic field, whose absolute magnitude is \(H\), be parallel to the \(Y\)-axis. The wave is linearly polarized, so that the electric vector \(\mathbf E\) oscillates along the \(X\)-axis. Denoting differentiation with respect to time by dots above the letters, we write the equations of motion of the electron in the form

\[ m\ddot{x}=eE_x \pm \frac{eH}{c}\dot{z} =\text{field force}+\text{electrodynamic force}, \tag{6,10} \]

\[ m\ddot{z}=-m\omega_0^2 z \mp \frac{eH}{c}\dot{x} =\text{quasi-elastic binding force of the plasma}+\text{electrodynamic force}. \tag{6,11} \]

The sign \((+)\) or \((-)\) is taken depending on the direction of the magnetic field. Assuming that the motion is purely periodic with frequency \(\omega\), put \(x=\xi e^{i\omega t}\) and \(z=\zeta e^{i\omega t}\). Differentiating and solving the equations with respect to \(x\), we obtain

\[ x=-\frac{e}{m}E_x\, \frac{1-\dfrac{\omega_0^2}{\omega^2}} {\left(1-\dfrac{\omega_0^2}{\omega^2}\right)^2-\dfrac{\omega_H^2}{\omega^2}} \cdot \frac{1}{\omega^2}. \]

For the dielectric constant we have

\[ \varepsilon =1-\frac{\omega_0^2}{\omega^2}\cdot \frac{1-\dfrac{\omega_0^2}{\omega^2}} {1-\dfrac{\omega_0^2+\omega_H^2}{\omega^2}}; \tag{6,12} \]

here the solution is unique, despite the two signs in equations (6,10) and (6,11). Thus, the refractive index of this “extraordinary” ray differs from the refractive index for the “ordinary” ray.

Thus, a longitudinal magnetic field decomposes the wave into two waves circularly polarized in opposite directions; a transverse field decomposes the wave into two waves linearly polarized in mutually perpendicular directions. The refractive indices of these waves differ from one another.

In the same way the general case may be considered—the direction of the ray makes an arbitrary angle with the vector of the magnetic field.

\(^{1}\) On the natural frequencies of a plasma in a magnetic field, see also \(^{230}\).

Nor do any fundamental difficulties arise in taking account of the friction $\rho$ [see equation (6.5) above]. The calculation for the general case leads to the splitting of the incident wave into two waves polarized elliptically, in such a way that the major axes of the ellipses are mutually perpendicular and the ratio of the axes is the same for both waves.

The complex dielectric constant is equal to:

\[ \varepsilon = 1-\frac{\omega_0^2}{\omega^2} \left\{ 1-\frac{i\nu}{\omega} - \frac{\dfrac{\omega_T^2}{\omega^2}} {2\left(1-\dfrac{\omega_0^2}{\omega^2}-i\dfrac{\nu}{\omega}\right)} \left[ 1 \pm \sqrt{ 1+ 4\frac{\omega_L^2\omega^2}{\omega_T^4} \left( 1-\frac{\omega_0^2}{\omega^2} -i\frac{\nu}{\omega} \right)^2 } \right] \right\}^{-1}; \tag{6.13} \]

here

\[ \omega_T=\frac{eH_T}{mc}; \qquad \omega_0=\sqrt{\frac{4\pi Ne^2}{m}}; \]

\[ \omega_L=\frac{eH_L}{mc}; \qquad \nu=\frac{w}{\lambda}. \]

$H_T$ and $H_L$ are the transverse and longitudinal components of the magnetic field. The remaining notation has its usual meaning. The cases we have calculated are contained in equation (6.13). The smaller refractive index for $\omega>\omega_H$ corresponds to the ray which, in the limiting case of a transverse field, coincides with the extraordinary ray. Equation (6.13) agrees in essentials with the formula of Appleton–Hartree $^{9,204}$ (see also D. Breit $^{43}$); the latter, however, was derived by another method and takes the polarization factor into account. K. D. Darwin $^{72}$ points out the legitimacy of neglecting the polarization factor.

The direction of rotation of the elliptically polarized rays is easy to determine by passing to the case of a longitudinal field. The presence of two refractive indices makes the plasma birefringent. Any incident wave can be represented as a combination of right- and left-rotating elliptically polarized waves; therefore the description of wave propagation given by us is suitable for all possible cases.

Analysis of formula (6.13) in its general form $^{39,153,175,354,504,505}$ is extremely complicated because of the multitude of independent parameters $(\omega,\omega_0,\omega_T,\omega_L$ and $\nu)$; we shall therefore not dwell on it. The great significance of the theory set forth lies in explaining the phenomena of propagation of Hertzian waves in the ionosphere with allowance for the Earth’s magnetic field. The boundary between short waves and ordinary radio waves is determined by the values of $\omega_H$ and $\omega_0$ for the ionosphere. The phenomena are made extremely complicated by the fact that the ionosphere consists of different layers with different concentrations $N$ and collision numbers $\nu$. Nevertheless, it may be asserted, as it were, that the theory of the dielectric properties of plasma set forth explains the basic character of the phenomena occurring in the ionosphere.

The calculation of wave propagation in inhomogeneous media, to which the ionosphere belongs, is very complicated. Approximate solutions can be obtained by decomposing the ionosphere into a number of parallel homogeneous layers (see, for example, K. Försterling and G. Lassen^153). In this way the echo phenomena for Hertzian waves have been elucidated in sufficient detail (see, for example,^9a,10,12,15,16). These questions are treated in greater detail in the special literature (see, for example, the detailed survey by Dellinger^90, as well as works^59,172,183,357,402,573–575,579).

We shall mention only that, in complete agreement with theory, from the measurement of the degree of polarization of the reflected wave a value of 0.42 gauss was found for the magnitude of the earth’s magnetic field at an altitude of 200 km. In the lower layers of the ionosphere, where, naturally, the probability of the occurrence of negative ions is especially great, it is apparently no longer possible to neglect the dielectric action of ions in comparison with the action of electrons (D. Goubau^176). The action of negative ions is also assumed in laboratory experiments (V. P. Mikha^350). We shall not, however, dwell on these questions, in order not to go beyond the scope of the subject of this article.

b. Magnetic properties

As was already noted at the beginning of this section, only those magnetic properties of plasma will be considered here which reduce to the curvature of the paths of the flying plasma charges under the action of an external magnetic field; phenomena associated with a change in the direction of existing and induced dipole moments of the individual components of the plasma are not considered. The curvature of the electron paths is so much more considerable than the curvature of the ion paths that the magnetic properties of plasma reduce exclusively to the action of the magnetic field on the trajectories of the electrons.

If a plasma is in a magnetic field, the charges of the plasma describe circumferences or spiral lines; these paths may be regarded as contours traversed by a current. The dipole moments of these currents, irrespective of the sign of the moving charge, are directed in such a way that the plasma must acquire diamagnetic properties. At low pressures, and consequently for large mean free paths or in strong magnetic fields, the paths traversed by the charges are essentially complete circumferences. If the mean free path is comparable in magnitude with, or less than, the radius of the trajectory, then the surface of the magnetic sheet determining the magnitude of the dipole moment is bounded by two circular arcs of length \(\lambda\).

A box filled with a homogeneous isothermal plasma, with walls at the plasma temperature, possesses diamagnetic properties if only the moments of the paths described by the charges mutually compensate (see Fig. 4, motion clockwise).

However, as was shown by Bohr^37, along the edges of the region bounding the plasma there arises a current in the opposite direction; this current is composed of segments of circular paths cut off by the edges of the region (the paths drawn in bold together constitute a counterclockwise current,

Fig. 4). Owing to the conditions of thermal equilibrium, the electrons describing such “cut-off” circles are repelled from the walls into the plasma.

If—under thermal equilibrium—the electron concentration at the edges is equal to the electron concentration in the middle of the plasma, then the diamagnetic moments of the trajectories inside the plasma are balanced, from the classical point of view, by the paramagnetic moment of the edge current. The absence of a magnetic moment in the given case is a consequence of the general proposition: classical mechanics cannot explain the magnetic properties of bodies that are in a state of thermal equilibrium (H. A. Lorentz^333). The trajectories of the electrons of a gas enclosed in a box and heated to such high temperatures that it becomes ionized, i.e. turns into a plasma, do not give the plasma diamagnetic properties; in exactly the same way, the motion of conduction electrons in a piece of metal does not make it diamagnetic^1).

Fig. 4. Trajectories with diamagnetic moments; the paramagnetic edge current is clearly drawn

If the plasma is not in an equilibrium state and, for example, the charges reaching the wall from the plasma are not reflected but are absorbed by the latter, then no edge paramagnetic current arises. The diamagnetic moments of the internal trajectories will not be compensated—the plasma must possess sharply expressed diamagnetic properties. In order for the process to be stationary, the ions absorbed by the wall must be replaced by ions newly formed inside the plasma (for example, as a result of ionization during the discharge). Such a case occurs in the positive column of a low-pressure discharge taking place in a long cylindrical glass tube in the presence of an axial magnetic field. The glass wall takes upon itself all, or in any case the greater part, of the charges flowing to it from the plasma (W. Schottky^473,474; W. Schottky and I. Issendorff^475), and thereby prevents the occurrence of a paramagnetic edge current. And indeed, such a plasma possesses strong diamagnetic properties (M. Steenbeck^498). As the magnetic field is increased, the radius of the electron trajectory, and with it the area of the contours of the currents formed, become smaller, and the diamagnetic moment begins to decrease.

^1) More formally, but at the same time much more generally, the state of affairs may be described as follows. The curvature of the paths of charges flying in the plasma and situated in a constant magnetic field does not alter the magnetic properties of the gas, since the magnetic field does not transfer energy to the plasma charges. This is how matters stand from the classical point of view. According to quantum theory, however, the Larmor motion of plasma charges must be quantized. The quantization of the energy leads to a dependence of the energy on the magnetic field. This change leads to the formula \(E - E_0 = \frac{1}{2}\chi H^2\), and consequently to a finite diamagnetic susceptibility \(\chi\) of the electronic motion.

The diamagnetism of plasma decreases as the field strength increases; therefore, in differential terms, the plasma behaves as a paramagnet, while remaining diamagnetic in absolute magnitude (Fig. 5). In weak fields the radius of the trajectory is large in comparison with the mean free path (relaxation length) of the electrons; here there is a linear increase of the magnetic moment as the external field increases1. Qualitative agreement between experiment and theory has been achieved (Fig. 5). From such measurements one may draw conclusions about the characteristic constants of the plasma; however, a sufficiently satisfactory quantitative theory is lacking.

Fig. 5. Dependence of the density of the magnetic moment \(M\) in mercury plasma on the field strength \(H^{498}\)

The experiment described above can, however, also be explained in another way, though one reducible to the explanation just given. There exists a flow of charges of both signs from the plasma toward the walls. This radial flow is deflected by the external magnetic field in the tangential direction; in this process charges of different signs are deflected in opposite directions.

The tangential components of the motion of the charges may be represented as circular currents producing an axially directed magnetic moment. This moment, which is the sum of the moments formed by charges of both signs, gives rise to the induced diamagnetism of the plasma. This representation clearly shows that the diamagnetic properties of the plasma arise in its nonequilibrium state and are connected with a stationary radial diffusion current.

Such an interpretation of the phenomenon brings the effect under consideration closer to the Corbino effect2, in which a radial electric current in a circular cylinder, under the action of an axial magnetic field, creates a tangential Hall voltage; the latter, in turn, causes the appearance of a circular current and a magnetic moment.

It is quite natural to try to measure directly the Hall voltage in a plasma through which a current flows; this was done in many older works (G. A. Wilson \(^{548, 549}\), D. D. Thomson \(^{512}\), E. Marx \(^{342}\), G. A. Wilson \(^{550}\), D. S. Watt \(^{534}\), P. E. Boucher \(^{42}\); see also G. Sirk \(^{478}\)). Although the Hall voltage in a plasma should be very large (the electron concentration of a plasma is much smaller than that for metals; therefore, at equal current densities, the directed velocities of the electrons in the case of a plasma will be much greater), the disturbances in measuring the voltage are so great that the results obtained are very unsatisfactory.

The cause of the disturbances lies, first, in the disproportionately large value of the insufficiently constant probe—plasma “contact potential” (see Section I), and, second, in the strong influence of the imposed magnetic field on the discharge current and, consequently, on the plasma itself. Therefore most measurements have been carried out not on the plasma of a gas discharge, but on the plasma in the hot gas of a flame, which is formed independently of the passage of current and therefore is affected by the magnetic field only to an insignificant degree. New investigations of the motion of charge carriers in a magnetic field, especially of such a direct influence of the magnetic field on the current as occurs in the Hall effect (L. Tonks and W. P. Allis \(^{521}\), W. P. Allis and H. P. Allen \(^{2}\), A. Slutskin \(^{483}\)), may, with greater success than hitherto, serve to deepen our knowledge of plasma.

Up to now we have spoken of plasma diamagnetism, possible from the classical point of view only for a nonequilibrium state. These diamagnetic properties, if they occur, are considerably greater than the diamagnetism possible in an equilibrium state from the point of view of quantum theory. This is explained by the fact that the classical moment corresponds to a current loop whose length may be of the order of the mean free path. Quantum-theoretical effects give, for a charge, moments of the order of the Bohr magneton

\[ \frac{e h}{4 m c}, \]

which corresponds to the moment of a current loop of much smaller magnitude, namely of the order of the dimensions of an atom.

As we know from atomic spectra and a number of other phenomena, electrons have a magnetic moment of this magnitude. Landau \(^{286}\) and L. Rosenfeld \(^{412}\) showed that this intrinsic moment leads to paramagnetism greater than the quantum-theoretical diamagnetism of motion along an orbit (see the footnote on p. 335); in thermal equilibrium there remains, per electron, a magnetic moment equal to \(^{2}/_{3}\) of a Bohr magneton. Thus the plasma as a whole should, in a state of equilibrium, possess paramagnetic properties. It is possible, however, that in this state diamagnetic properties are present to a certain extent, as occurs in the cases of bismuth and other metals.

The possibility of observing the quantum-theoretical magnetic properties of a plasma in a state of equilibrium remains unclear: these properties

can overlap classical effects, larger by an order of magnitude, as soon as the plasma deviates slightly from the state of complete equilibrium.

VII. ACCOUNT OF THE INDIVIDUAL PROPERTIES OF ATOMS

We shall now dwell on those properties of a plasma which are explained by the individual features of its atoms. As was indicated in Section I, a plasma is a gas containing, in significant concentration, in addition to neutral atoms or molecules, also excited atoms, ions, and electrons, and also possessing a certain radiation density. There are numerous possibilities for realizing such a gaseous state (see Section I on this); most simply it arises when any gas is heated to high temperatures—practically to several thousand degrees. A plasma is therefore such a state of a gas in which it possesses an energy greater than that of an ordinary gas. Accordingly, the state of a plasma can be preserved only if the possibility of energy losses is eliminated, or if there is a constant influx of energy. The first case occurs when a gas is heated in a furnace closed on all sides, the second when the plasma of a gas discharge is realized.

It is not difficult to trace the process of transition, with increasing temperature, of an ordinary gas (such as exists, say, at room temperature) into a plasma possessing entirely new properties. Let us consider, for simplicity, an atomic gas; let its excitation energies be \(E_1, \ldots, E_k\) and its ionization energy \(E^*\). As long as \(kT \ll E_1, \ldots, E_k\), in collisions between atoms there occurs only an exchange of kinetic energy. As soon as the value \(kT\) becomes comparable with the quantities \(E_1, \ldots, E_k\), excited atoms are formed. When \(kT\) becomes comparable with \((E^* - kT \ln p)\), where \(p\) is the pressure, then ionization of atoms begins, i.e. the formation of charged particles. As a consequence of the excitation of atoms, emission of radiation also arises. In the case of a more complex, non-monatomic gas, for example a diatomic one, a larger number of possible interactions takes place.

In the following exposition we shall consider a plasma as a gas heated to high temperatures; first, in Section VIII, the case of complete thermal equilibrium in such a gas will be discussed, i.e. the so-called isothermal plasma. In the last section of this review the properties of a nonisothermal plasma will be considered.

Taking account of individual properties provides further experimental possibilities for determining quantities characterizing the properties of a plasma, for example, the concentrations of excited atoms, ions, and also the temperatures of different forms of energy (see Section IX, a). All these experiments belong to the domain of optics.

Quantitative measurements of absorption make it possible to determine the absolute number of absorbing atoms in a given layer 179, 180, 259, 264, 276, 277, 278, 422, 464, 540, 572; for this it is necessary

calculate from atomic constants or determine experimentally the absorption coefficient. In such absorption measurements, the plasma’s own emission proves to be an interference; the influence of the latter can be allowed for or eliminated by special optical techniques \(^{179}\). The concentration of metastable atoms in the plasma of discharges at low pressures can be determined by measuring \(^{91,346,347}\) the absorption in an alternating-current discharge during the so-called “dark pause” (“Dunkelpause”) \(^{132}\). The most elegant solution is provided by the “modulated-light” method, in which modulated light is absorbed in a stationary plasma, and then, with the aid of a photoelement, the plasma’s own glow (constant light) is separated electrically from the absorbed light of the source (alternating light) \(^{226,264}\).

Anomalous dispersion. According to the theory of dispersion, the refractive index increases sharply on passing from the absorption band to longer wavelengths and decreases sharply on passing to shorter wavelengths. In this case there is a quantitative relation between the change in the refractive index and the quantity \(\mathfrak{N}^{41,355,527}\). \(\mathfrak{N}\) is connected with the concentrations of atoms of the upper \((N_k)\) and lower \((N_i)\) terms of the absorption line by the following relation \(^{276,277,279,280}\):

\[ \mathfrak{N}=N_i f_{ki}\left(1-\frac{N_k}{N_i}\cdot\frac{g_i}{g_k}\right); \tag{7,1} \]

here \(g_i\) and \(g_k\) are the statistical weights of the terms (see VIII, a), and \(f\) is the oscillator strength (see VIII, c).

Measurement of the refractive-index curve is carried out for the most part with the aid of Jamin and Rozhdestvenskii interferometers \(^{444,445}\).

The most reliable measurements of the concentration of excited atoms have been made by the method of anomalous dispersion.

Measurements in the emission spectrum. In an optically thin layer (see Section IX, b) the emission \(J_{nm}\) per \(1\ \mathrm{cm}^3\) of \(N_n\) excited atoms is given by the formula

\[ J_{nm}=N_n A_{nm}\cdot h\nu_{nm}; \tag{7,2} \]

here \(n\) is the initial term and \(m\) the final term of the emission, \(A_{nm}\) is the transition probability, and \(\nu_{nm}\) is the emitted frequency.

Thus, measurement of the absolute intensity of a line, for a known value of \(A_{nm}\) \(^{1)}\), makes it possible to determine the concentration of excited atoms. Estimation of the absolute intensity of an individual spectral line is extremely complicated; therefore such measurements have been made rather rarely \(^{111,112,191–193,208,267–269,338,391,447,565}\).

It is somewhat simpler to measure the ratio of two or several lines. In the case of an isothermal plasma it is valid

\(^{1)}\) A summary of the values of \(A_{nm}\) known at present may be found in works \(^{246,355,527}\).

relation (see VIII, a):

\[ \frac{J_{nm}}{J_{kr}} = \frac{N_n}{N_k}\cdot \frac{A_{nm}}{A_{kr}} \frac{\nu_{nm}}{\nu_{kr}} = \frac{g_n}{g_k} e^{-\frac{E_n-E_k}{kT}} \frac{A_{nm}}{A_{kr}}\cdot \frac{\nu_{nm}}{\nu_{kr}}, \tag{7,3} \]

where \(E_n\) and \(E_k\) are the excitation energies of the initial terms of emission. Equation (7,3) makes it possible to set up an experiment for measuring the temperature of an isothermal plasma.

The meaning of the quantity “\(T\)” in the case of a nonisothermal plasma will be discussed in Section IX, a.

Such measurements in the emission spectrum are successful when reabsorption is avoided; otherwise the \(A_{nm}\) vary in a very complicated manner \(^{278,\ 355,\ 377}\) (see Section IX, b).

With the aid of these experiments—mainly measurements of the course of the intensities of rotational lines of molecular spectra (VIII, a)—a large number of determinations of the plasma temperature have been made \(^{224,\ 321,\ 322,\ 385,\ 387-390,\ 392-395,\ 546}\).

(To be concluded in the next issue)

1. The area formed by two arcs of length \(\lambda\) and with radius of curvature \(R\) increases for \(\lambda \ll R\), if both arcs become more convex as \(R\) decreases, i.e., as \(H\) increases. ↩↩↩↩↩↩↩↩↩

2. An essential difference from the Corbino effect consists in the fact that in the latter, as in the Hall effect, the currents of unlike charges weaken one another; thus, in the Corbino effect a differential action is perceived, whereas in the case of the plasma under consideration the total action of the charges is perceived. Ultimately this reduces to the fact that in an electric current both negative and positive charges participate, moving in opposite directions; in our case, however, there is a nonelectric ambipolar diffusion flow, and charges of both signs move in one and the same direction. ↩↩↩↩

3. The magnitude of the numerical factor (here 1) is somewhat different from unity. This follows from the close connection between $\rho$ and the mobility of electrons.

   At low frequencies (constant fields) the field $\mathbf{E}$ acts on the charge with the force $e\mathbf{E}$ and produces a velocity $b\mathbf{E}$; the arising frictional force $\rho b\mathbf{E}$ is equal in magnitude to the driving force $e\mathbf{E}$. From $e\mathbf{E} = \rho b\mathbf{E}$ it follows that $\rho = \frac{e}{b}$. Our conclusion shows that $\rho$ of formula (6.5) differs only insignificantly from $\rho$, determined by formula (3.3) for $b$ ($z = 1$ instead of $0.75$!). ↩↩

4. According to Keck’s data[^236], a wave \(4\ \mathrm{cm}\) long does not pass, at large electron concentrations \((N > \simeq 6\cdot 10^{11}\ \mathrm{cm}^{-3})\), through a plasma layer \(20\ \mathrm{cm}\) thick. ↩