Gases in the Plasma State. I¹

R. Rompe and M. Steenbeck

VIII. Isothermal Plasma

a. Calculation of the Concentration of Components

Monatomic gas. Assuming the existence of complete thermal equilibrium, one can calculate the concentrations of the plasma components, i.e., the concentrations of excited atoms, ions, and electrons. The calculation of the concentration of light quanta is easily carried out for the case when the optical thickness of the plasma is so large that the radiation may be regarded as “black.” In this case the radiation density in the plasma is equal to the radiation density of an absolutely black body at the same temperature ³⁰⁹, ⁴⁰⁹, ⁵²⁷. Black radiation, however, is not a necessary condition of thermal equilibrium (see Section IX, a).

In order to calculate the concentrations, it is necessary to know, in addition to the temperature \(T\), the values of the excitation energy \(E_\nu\), the ionization potentials \(E^*\), and also the pressure \(p\) or the concentration of atoms \(N\). In addition, one can, of course, calculate by the well-known method from the Maxwellian velocity distribution the relative velocities or relative energies of atoms, ions, and electrons.

To calculate the number of excited atoms, one may, in accordance with statistical physics, use the Boltzmann distribution:

\[ N_\nu = N_0 \cdot \overline{g}_\nu \cdot e^{-\frac{E_\nu}{kT}}, \tag{8a, 1} \]

where \(N_\nu\) is the number of atoms excited to the state with energy \(E_\nu\) (measured with respect to the ground state), \(N_0\) is the number of atoms in the ground state, \(\overline{g}_\nu\) is the ratio of statistical weights, and \(k\) is Boltzmann’s constant. The quantity \(\overline{g}_\nu\) is equal to the ratio of the statistical weights of term \(\nu\) and of the ground state. These quantities are calculated for atoms with one or two valence electrons from the total angular momentum \(j\) of the term:

\[ g_\nu = 2j_\nu + 1. \tag{8a, 2} \]

The quantities \(j_\nu\) may be taken from the term scheme ⁵⁹, ¹⁸²; they are equal for the ground terms

¹ Ergebn. d. exa
Uspekhi fizich. nauk,

This equation is practically applicable as long as the proposition remains valid that the majority of atoms are in the ground state and that \(N_0\) may be taken equal to \(N\)—the total number of atoms. The limits of applicability are determined by two factors: first of all, at small \(\frac{E}{kT}\) there is a considerable number of atoms in the excited state; further, the number of atoms also decreases owing to ionization, so that:

the number of atoms \(+\) the number of ions \(=\) the number of atoms at low temperatures.

The number of ions or electrons can also be calculated, under the assumption that the number of excited atoms is negligibly small in comparison with the number of atoms in the ground state, from Saha’s formula \({}^{1)}\) \(^{105,119,397,426,452,527}\)

\[ \frac{c^2}{1-c^2} = \frac{(2\pi m)^{\frac{3}{2}}}{h^3 p} (kT)^{-\frac{5}{2}} \cdot e^{-\frac{E^*}{kT}}; \tag{8a,3} \]

here \(c\) is the degree of ionization, i.e. the ratio of the number of ions to the total number of atoms existing in the cold state, \(m\) is the mass of the electron, \(k\) the Boltzmann constant, \(T\) the temperature, \(E^*\) the ionization potential, \(h\) Planck’s constant, and \(p\) the pressure.

It follows from equation (8a, 3) that the number of ions may, at sufficiently high temperatures or low pressures, exceed the number of atoms; therefore the error made by neglecting ionization in computing the absolute number of excited atoms may be very considerable \(^{430}\).

To take account of the number of atoms excited to level \(i\), let us consider the probability \(A_i\) of the \(i\)-th term of the atom

\[ A_i = \frac{e^{\frac{E^*}{kT}}}{Z} g_i e^{-\frac{E_i}{kT}}, \tag{8a,4} \]

where \(g_i\) is the statistical weight of term \(i\). Since \(\frac{g_i}{g_0}=\overline{g_i}\), for the relative distribution of atoms over excited terms equation (8a, 4) expresses the same thing as (8a, 1).

The quantity \(Z\) in equation (8a, 4) is the sum over states; the latter

\[ Z = \sum_i g_i e^{-\frac{E_i}{kT}} \tag{8a,5} \]

can be calculated only if the summation is restricted to the lowest terms. Exact calculation is impossible because the sum diverges for large values of the summation index. To remove the difficulties in the calculation, a number of methods have been proposed \(^{28,135,154,214}\). In these studies it is assumed that, at a given density, each atom has a limited—

\({}^{1)}\) If atoms or molecules obey classical statistics, then the right-hand side of equation (8a, 3) must be multiplied by \(e=2.718\); cf. E. Schrödinger \(^{461}\); M. Planck (Sitzgsber. preuss. Akad. Wiss., Phys.-math. Kl., No. 19/24, 442, 1925).

large volume, because of which the existence of highly excited atoms is impossible (neighboring atoms prevent the existence of very large electron orbits). However, from investigations of line widths \(^{3,136,160}\) it is known that even when the path of an atomic electron encompasses several neighboring atoms, there is only an insignificant distortion of the atomic term; therefore the choice of the term at which the summation is cut off is somewhat arbitrary. The indicated consideration will make sense only in the case of very strong intermolecular fields, i.e., at high carrier concentrations (see VIII, e).

Nevertheless, for all cases encountered in practice it is possible to carry out the calculation of the sum of states up to \(i=5—7\), if, as Planck did \(^{410}\), one takes into account in calculating the relative probability of a state the decrease in the concentration of atoms due to incipient ionization. Then the generalized sum of states \(\tilde Z\) assumes the following form \(^{1)}\):

\[ \tilde Z= \left(\frac{(2\pi mkT)^{\frac{3}{2}}}{c^{2}N\cdot h^{3}}\right)^{c} \left(\frac{1}{1-c}\sum_{\nu} g_{\nu} e^{-\frac{E_{\nu}}{kT}}\right)^{1-c}. \tag{8a,6} \]

Here \(m\) is the mass of the electron, \(N\) is the concentration of atoms in the cold state, \(h=6.55\cdot 10^{-27}\); \(E_{\nu}\) is the energy of the term, negative with respect to the ionization energy, and \(c\) is the degree of ionization, which may be taken from equation (8a,3). In the case when, in ionization as well, the influence of excited terms must be taken into account, instead of equation (8a,3) one must adopt the following:

\[ \frac{c^{2}}{1-c^{2}}= \frac{g_{0}(2\pi m)^{\frac{3}{2}}(kT)^{\frac{5}{2}}e^{-\frac{E^{*}}{kT}}} {h^{3}p\left(\sum_{n} g_{n}e^{-\frac{E_{\nu}}{kT}}\right)} . \tag{8a,7} \]

The sum of states standing in the denominator is taken up to the value \(i\) of order 5.

It follows from (8a,7) that the degree of ionization is maximal when all atoms are in the ground state; in this case \(Z=\sum_n=g_0\). If there is a significant number of excited atoms, then \(\sum_n>g_0\), i.e., the degree of ionization decreases.

In this way it is possible to determine the quantities \(A_i\) with the accuracy required in practice, provided that the excitation potentials, the ionization potential, and the statistical weights of the terms are known. In Figs. 6–9 examples are given of calculations performed \(^{430}\) for mercury and hydrogen. From the curves presented in the figures it follows that at low pressures (\(1\) torr) the total number of excited atoms at temperatures somewhat above \(10\,000^\circ\) constitutes only a small fraction (for mercury one thousandth, and for hydrogen one hundred-thousandth) of the number of ions. At high pressures, for mercury at \(10^4\) torr, the number of excited atoms reaches \(10^0/0\) of the number of ions; the number of unexcited atoms

\(^{1)}\) For the case of classical statistics, into the first parentheses one must introduce the factor \(e=2.718\ldots\) (see the preceding footnote).

can become, at temperatures above \(5 \cdot 10^\circ\), smaller than the number of excited and ionized atoms.

In the same way one can calculate the excitation and ionization of an ion, etc. These calculations are of importance for the plasma of stellar atmospheres, considered in astrophysics, and form the basis for the classification of stars \(^{450,453,527}\). Under terrestrial conditions, similar cases may be expected in discharges at extremely low pressure, which is possible in high-frequency technology \(^{348}\).

In these calculations it is tacitly assumed that the plasma itself has no feedback effect on the atomic constants, i.e. on

Fig. 6. Mercury

\(p=1\) torr; relative probability \(A_i\) of terms:
\(0\)—\(1\,^1S_0\); \(1\)—\(2\,^3P_0\); \(2\)—\(2\,^3P_1\); \(3\)—\(2\,^3P_2\); \(B\)—all terms between 6.1 and 8.1 V.
\(C\)—all terms between 8.1 and 9.8 V; \(c\)—degree of ionization \(^{858}\)

Fig. 7. Mercury

\(p=10^4\) torr. For notation see Fig. 6 \(^{858}\).

the magnitudes of the excitation and ionization potentials. This proposition can be regarded only as conditional. As was shown in Section II, significant electric fields arise inside the plasma. They can reach a magnitude sufficient to influence the binding of the electron; above all this will be the case for an electron in a remote orbit, weakly attracted by the atomic residue. In addition, owing to the interaction of the individual components of the plasma—for example, the formation of molecules unstable in the normal state—the excitation energies may change substantially. There are direct and clear indications that in a number of cases the ionization potential differs from this value for an isolated atom and is, moreover, smaller than it. This will be discussed in more detail below (see VIII, e).

Diatomic gases. For a gas of diatomic molecules, in addition to the distribution over electronic levels and ionization, one must take into account the distribution over rotational and vibrational levels and the degree of dissociation.

An exact calculation allowing for the coexistence of individual components (as for an atomic gas) has not yet been carried out, although in principle it is possible. We shall confine ourselves to giving expressions for the relative probability of vibrational and rotational levels, and also for the degree of dissociation 134, 544, 563. The distribution of molecules over vibrational levels is the Boltzmann distribution:

\[ N_n = N_0 e^{-\frac{\omega_n}{T}}, \tag{8a, 8} \]

where \(\omega_n\) is the energy of the \(n\)-th level, \(N_0\) is the number of molecules in the lowest

Fig. 8. Relative probability \(A_i\) for \(H\)-terms at a pressure of 1 torr

Designations: see Fig. 9

Fig. 9. Relative probability \(A_i\) for \(H\)-terms at a pressure of \(10^4\) torr

\(0\) — normal state; \(1\) — principal quantum number \(n=2\); \(2\) — \(n=3\); \(3\) — \(n=4\); \(B\) — \(n=5+6\); \(C\) — \(n=7\) to \(\infty\); \(c\) — degree of ionization

vibrational level. The vibrational energy of a harmonic oscillator is

\[ \omega_n = \frac{hc}{2\pi}\,\omega_0\left(v+\frac{1}{2}\right);\qquad v=1,\,2,\,3,\ldots, \]

where \(\omega_0\) is the fundamental vibration of the molecule. The distribution over rotational levels is represented by the formula (\(\Sigma\)-term):

\[ N_j = 2(j+1)e^{-\frac{chB}{kT}j(j+1)},\qquad B=\frac{h}{8\pi^2 c\cdot J}, \tag{8a, 9} \]

\(j=1,\,2,\,3\ldots;\ J\) is the moment of inertia of the molecule.

For molecules consisting of identical atoms, the degree of dissociation may be approximately represented by an equation analogous to (8a, 3) ^246:

\[ \frac{x^2}{1-x^2}p=\frac{(2\pi)^{\frac12}}{8}\,\frac{k^{\frac32}}{h} \left(1-e^{-\frac{Q_{\mathrm{diss}}}{kT}}\right) \frac{M^{\frac32}}{J}\cdot T^{\frac32}\cdot e^{-\frac{Q_{\mathrm{diss}}}{kT}}. \tag{8a, 10} \]

Here \(Q_{\mathrm{diss}}\) is the dissociation energy, \(M\) the mass of the atom, and \(J\) the moment of inertia of the molecule. In equation (8a, 10) the rotation of the molecule is taken into account, but the vibration of the nuclei is not. A rigorous calculation of the degree of dissociation is possible, taking into account both nuclear vibration and molecular rotation, by means of the generalized sum of states for a diatomic molecule given by Rive1.

The general character of the temperature dependence for the separate components will be approximately the same as in ionization. One should, however, expect a decrease in the number of molecules at much lower temperatures than is the case for ionization with the same characteristic temperature (the ionization formula contains the mass of the heavy particles).

Polyatomic gases. In these cases calculation is much more difficult and for the most part impossible, owing to the unsatisfactory data on the corresponding molecular constants. However, on the basis of what was said above about monatomic and diatomic gases, a qualitative discussion of problems concerning polyatomic molecules is possible.

b. Specific heat of a plasma

The specific heat of a gas at constant volume is equal to

\[ c_v=\frac{dE}{dT}; \tag{8b, 1} \]

here \(E\) is the energy of the gas and \(T\) its temperature. The specific heat of an ideal gas at constant pressure is calculated, as is well known, from the formula

\[ c_p=c_v+\frac{3}{2}Nk. \tag{8b, 2} \]

At low temperatures the heat capacity \(c_v\) of ideal gases is a constant, because the energy of the gas consists only of the kinetic energy of the atoms, and the latter is proportional to \(T\). In the case of a nonideal gas, for example a diatomic or triatomic one, the energy includes, at elevated temperature, the energies of vibration and rotation of the molecules. The specific heat of such a gas increases with temperature in a characteristic manner; the heat-capacity curve for diatomic molecules, constructed with the aid of definite spectroscopic molecular constants \(\omega_0, J, Q_{\mathrm{diss}}\) (see VIII, a), is in good agreement with experiment2.

By analogy with the excitation of internal degrees of freedom in a molecular gas, which leads to an increase in the specific heat

one can expect the same phenomenon also for an atomic gas, owing to the excitation or ionization of atoms at sufficiently high temperatures. Dissociation of a molecular gas creates an additional increase in the heat capacity. If one clearly keeps in mind that at low pressures sharp changes in the degree of ionization and degree of dissociation with temperature are possible already at relatively low temperatures, and that the energy liberated in each elementary process of ionization and dissociation is an enormous number of times greater than the vibrational and rotational energy of a molecule, then in these cases one may expect an extremely sharp increase in the heat capacity. Meglich, Riewe, and Rompe^363 established a considerable change in the heat-capacity curves when the internal degrees of freedom of the molecule are taken into account. For example, upon excitation of the vibration of a molecule the specific heat \(c_v\) increases from the value \(\frac{3}{2}k\) to \(\frac{5}{2}k\) and, with further increase of temperature, remains constant; allowance for ionization and dissociation shows an increase of the specific heat by amounts from \(k\) up to \(250\,\frac{k}{2}\), with subsequent returns to normal values, which, however, correspond to the increased number of particles (Fig. 10).

Fig. 10. Schematic representation of the temperature dependence of the heat capacity \(c_p\) for the first four stages of excitation of mercury atoms. \(n\) is a multiple of \(\frac{k}{2}\) (after^297).

This circumstance is not difficult to understand: excitation of internal degrees of freedom increases the energy of the gas at an unchanged number of particles, and therefore causes an increase in the specific heat; a decrease in the number of non-ionized or undissociated particles leads, as the temperature increases, to a decrease of the specific heat down to its former value.

The maximum occurs at degrees of ionization or dissociation of the order of 0.6.

In calculating the specific heat with allowance for ionization and dissociation, one starts from the formula:

\[ c = k \frac{d}{dT}\left(T^2 \frac{d}{dT}\ln Z\right), \tag{8в,3} \]

where \(Z\) is the partition sum. When ionization is taken into account, Planck’s formula (8a, b) is used for \(Z\); when dissociation is taken into account, Riewe’s formulas^427,428 are used.

c. Elementary processes in an isothermal plasma

The concentrations of plasma components, which were calculated in Section VIII, a, are quantities averaged over time; they are obtained the more accurately, the longer the observation time. We shall now turn to a detailed study of the structure of plasma. In doing so, it is necessary to consider the long-range interactions that occur between the various components of the plasma1. As is known, the individual constituent parts of a gas are in constant motion, and at certain moments in time the distances between two particles become very small—of the order of \(0.001\)–\(0.0001\) of the mean distance.

The forces of interaction of uncharged particles decrease with distance so rapidly that the action of neutral particles located at the mean distance may practically be set equal to zero. Force interaction between such particles begins only when the particles approach one another to a distance much smaller than the mean one. As a result of the interaction, the trajectories of the particles change, and exchange of energy and momentum takes place. Such an interaction is called an elastic collision, emphasizing the analogy with the collision of elastic spheres.

Such a representation of the interaction of neutral particles is possible because the repulsive forces[^207] acting between them arise only at distances of the order of an angstrom and decrease exponentially as the distance increases. The steepness of the decrease of the interaction curve makes it possible to speak of a definite extent of an atom or molecule and to introduce the concept of the “gas-kinetic radius” of elementary particles. This quantity is introduced into the calculation of the number of collisions as the radius of an elastic sphere when replacing the real case by a mechanical model. The described model of interaction does not take into account the weak attractive forces, the so-called van der Waals forces, which arise as a consequence of the polarizability of atoms and decrease in proportion to the seventh power of the distance[^328][^329]. Between homogeneous atoms there act van der Waals forces of a special kind—the so-called “dipole resonance forces”—which are proportional to \(\frac{1}{r^4}\). In collisions of charged particles the decisive forces are Coulomb forces, i.e. the electrical repulsion or attraction of charges. By means of a complicated averaging, in all these cases one can introduce an effective radius corresponding to the gas-kinetic one [see on this Section II (microfield)].

When particles interact with high relative energies, in addition to the transfer of energy and momentum, excitation or ionization and dissociation of an atom or molecule are possible. These collisions are called “inelastic,” since the kinetic energy of the particles is wholly or partly converted into the internal energy of the atom. The effective radius for such processes is determined by the relation

\[ \pi R^2 = \frac{Z}{N^2 v}, \tag{8c,1} \]

where \(N\) is the number of atoms, \(Z\) is the number of excitation or ionization events in 1 sec., and \(v\) is the relative velocity of the particles.

In what follows, to describe the interaction between the component parts of the plasma we shall use the quantity \(S\), denoting by this symbol the number of events (excitation or ionization) occurring on average in 1 sec. for each impacting and impacted particle. The dependence of \(S\) on \(R\) and \(v\) is given by the equality

\[ S=\int_{0}^{\infty}\pi R^{2}(v)N(v)v\,dv, \tag{8c,2} \]

where \(R(v)\) is the effective radius, depending on the relative velocity, and \(N(v)\) is the distribution function of the relative velocities of the particles \(^{1}\).

The total number of collisions, for example those in which atoms are transferred by the impact of an electron from the ground state to an excited state \(k\), is equal to

\[ Z_{0k}=N_{0}\cdot N_{-}\cdot S_{0k}, \tag{8c,3} \]

where \(N_{0}\) is the number of atoms in the normal state, \(N_{-}\) is the total number of electrons, \(S_{0k}=\int \pi R_{0k}^{2}N_{-}(v)v\,dv\), \(R_{0k}\) is the radius for excitation by electron impact, and \(N_{-}(v)\) is the distribution of electron velocities. Similar expressions may be written for excitation in collision with an atom, for ionization, etc. The number of effective collisions is always determined by the concentrations of the interacting particles, their relative velocities, and the effective radii for the corresponding process. The distribution of particle velocities in a plasma coincides exactly or approximately with the Maxwellian distribution; therefore the quantities that directly determine individual processes are \(S\), while the “excitation functions” \(R(v)\), determined at low densities and in a narrow energy interval, correspond to them only indirectly \(^{2}\). It may therefore happen that a process with smaller values of the excitation function, in comparison with others, occurs more often in the plasma owing to a more favorable distribution of the velocities of the impacting particles, which gives the quantity \(S\) a larger value.

The interaction of atoms, ions, and electrons leads to the continuous formation of excited atoms, ions, and electrons. In the stationary case of isothermal plasma, the particle concentrations are constant quantities depending only on the temperature; therefore, along with the interaction processes leading to the formation of particles, there must exist processes of their disappearance that exactly compensate this formation. We shall consider this circumstance using as an example the behavior in the plasma of excited atoms;

\(^{1}\) The value of \(S\) for inelastic impacts (excitation + ionization) in noble gases was calculated by Mierdel \(^{349}\); the effective radii of Mayer—Leibniz \(^{339}\) were used as a basis. Other effective radii, obtained experimentally, are given in works \(^{251}\) and \(^{119}\).

\(^{2}\) On excitation functions see V. Hanle and K. Parge \(^{203}\).

let the excited atoms be at level \(k\) with energy \(E_k\). There exists a large number of processes by means of which atoms pass to the \(k\)-level; there also occur processes removing atoms from level \(k\). The former include the following processes:

1. Collisions of atoms located at levels \(i<k,\ E_i<E_k\), with electrons; in this case the electron transfers to the atom the energy \(E_k-E_i>0\), owing to which the atom passes to the \(k\)-level. Such collisions are called collisions of the first kind; their number in 1 sec. is equal to

\[ \text{number of collisions of the first kind} = \sum_{0}^{k-1} N_i \cdot N_- \cdot S^{\mathrm I}_{ik}, \tag{8c, 4} \]

where \(N_i\) is the concentration of atoms situated at level \(i\), \(N_-\) is the number of electrons, and \(S^{\mathrm I}_{ik}\) is the number of effective impacts of an electron (per 1 sec. per atom) transferring the atom from state \(i\) to state \(k\).

1. Collisions of atoms located in states with greater energies \(E_j>E_k\), with electrons; in this case the electron receives energy from the atom, and the atom passes to level \(k\). The number of these processes in 1 sec. is equal to:

\[ \text{number of collisions of the second kind} = \sum_{k+1}^{\infty} N_j \cdot N_- \cdot S^{\mathrm{II}}_{kj}, \tag{8c, 5} \]

where \(N_j\) is the concentration of atoms in state \(j\), and \(S^{\mathrm{II}}_{kj}\) is the number of effective collisions of electrons with atoms in which the electrons acquire the energy \(E_j-E_k\) (per collision).

Such processes are called collisions of the second kind.

1. In the same way, the number of atoms in the \(k\)-state may be affected by collisions of the first and second kind of atoms with ions, with light quanta and with neutral atoms, and also by collisions of ions and electrons (recombination). For these processes one can write formulas analogous to those given above.

In addition to processes continuously increasing the number \(N_k\), there exist processes decreasing this quantity. We give formulas for those of them in which the atom collides with an electron. Collisions of the first kind transfer the atoms under consideration \(N_k\) to higher levels; the number of such collisions is

\[ N_k \cdot N_- \cdot S^{\mathrm I}_{kj}. \tag{8c, 6} \]

Collisions of the second kind transfer atoms to lower levels; the number of such collisions is

\[ N_k \cdot N_- \cdot S^{\mathrm{II}}_{ki}. \tag{8c, 7} \]

Similar relations can also be established for the remaining interactions.

Our requirement that \(N_k\), when observed over a long interval of time, remain constant means only that the number of newly formed atoms is equal to the number of atoms that have left state \(k\). The principle of detailed ...

equilibrium. This principle asserts that there occurs not only a total compensation of the atoms that have left the state \(k\) by atoms passing into this state, but also the balancing (on the average in time) of each of the processes removing an atom from the \(k\)-level by the same reverse process transferring the atom to the \(k\)-level.

Let us consider definite collisions of the first kind between electrons and atoms, namely those in which an electron with energy \(\varepsilon>|E_k-E_i|\), in colliding with an atom \(i\), creates an atom \(k\). The number of such impacts is equal to:

\[ N_i N_{-}(\varepsilon)\cdot S^{\mathrm I}_{ik}(\varepsilon). \tag{8c, 8} \]

Before such an impact there are an atom \(i\) and an electron with energy \(\varepsilon\); after the impact there are an atom \(k\) and an electron with energy \(\varepsilon-(E_k-E_i)\). The principle of detailed equilibrium requires equality of the number of impacts (8c, 8) to the number of impacts of the second kind of electrons possessing energies \(\varepsilon-(E_k-E_i)\) with atoms \(k\), i.e.

\[ N_k N_{-}(\varepsilon-E_k+E_i) S^{\mathrm{II}}_{ki}(\varepsilon-E_k+E_i) = N_i N_{-}(\varepsilon)\cdot S^{\mathrm I}_{ik}(\varepsilon). \tag{8c, 9} \]

As was indicated in Section VIII, a, the ratios \(\dfrac{N_k}{N_i}\) and \(\dfrac{n(\varepsilon)}{n(\varepsilon-E_k+E_i)}\) can be calculated; equation (8c, 9) therefore makes it possible to find the relation between the quantities \(S^{\mathrm I}\) and \(S^{\mathrm{II}}\), which determine impacts of the first and second kind. Of interest for a number of practical cases is the ratio of impacts of the first and second kind by electrons of all energies, which can readily be obtained from the principle of detailed equilibrium

\[ N_k\cdot N_{-} S^{\mathrm I}_{ik} = N_i N_{-} S^{\mathrm{II}}_{ki} \quad \text{or} \quad \frac{S^{\mathrm I}_{ik}}{S^{\mathrm{II}}_{ki}} = \frac{N_i}{N_k} = \frac{g_i}{g_k} e^{-\frac{E_i-E_k}{kT}} . \tag{8c, 10} \]

The principle of detailed equilibrium was rigorously proved by Boltzmann for the exchange of energies of translational motion. This proof may also be regarded as valid for the exchange of the kinetic and quantum forms of energy; the validity of the principle of detailed equilibrium for such cases was assumed by Klein and Rosseland \(^{243}\) (atom, electron), Einstein \(^{107}\) (atom, radiation), Milne \(^{351}\) (ionization by radiation), Fowler \(^{155}\) (ionization by electron impact).

The significance of the principle of detailed equilibrium for the study of plasma is extremely great. It makes it possible to decompose the large number of processes taking place into a series of cycles. The scheme for constructing such a cycle, or “complete mechanism” (Milne \(^{351}\)), is always one and the same: the number of acts creating the given state in the given manner is equated to the number of acts destroying this state by precisely the reverse route. We have just done this for interaction with electrons.

All cycles relating to the interaction of particles consist of two terms, namely—the one creating and the one destroying the given state. Cycles in which interaction with radiation participates consist of three terms. In this case there exist two terms destroying the given state; one of them is the reverse of the term creating this

state; the other is spontaneous destruction, namely, spontaneous emission of radiation \(^{107}\). If, for example, \(N\) atoms are in a radiation field of density \(\rho_\nu\) (\(\nu\) equal to the frequency of the resonance line), then \(N\rho_\nu B_1\) atoms pass into the excited state in the course of 1 sec. as a result of absorption. The number of atoms returning to the ground state owing to the reverse process, namely “induced emission” or “negative absorption,” is equal to \(N_i\rho_\nu B_2\), where \(N_i\) is the number of excited atoms, and \(B_2\) is the absorption coefficient of induced emission.

To this must also be added spontaneous emission \(N_i A\), where \(A\) is the transition probability, so that in the stationary case the equality holds

\[ N\rho_\nu B_1 = N_i A + N_i\rho_\nu B_2 . \tag{8c, 11} \]

Absorption and induced radiation of a quantized oscillator in a radiation field are analogous to the behavior of a classical oscillator with damping: the latter absorbs energy and radiates it by means of its own oscillations, the amplitude and phase of which are related in a definite way to the external radiation field [with the exception of very high densities \(^{1}\)].

The arguments set forth in this section are, of course, valid only for the case of thermal equilibrium. When there is a deviation from the equilibrium state in a nonisothermal plasma, it is impossible schematically to divide the elementary processes into separate cycles. We shall speak of this in more detail in Section IX, b.

d. Thermal conductivity of plasma

In the kinetic theory of gases the coefficient of thermal conductivity is derived from the so-called transport equation \(^{215}\). A gas is considered in which, owing to the transport of energy, a weak temperature gradient is formed. By \(N\) is denoted the concentration of atoms at the point \(x\). The following assumptions are then made:

1. At the point \(x\) thermal equilibrium obtains, i.e. the distribution of the velocities of the atoms and the mean energy are completely determined here by the temperature at the point \(x\). The temperature is a function of the coordinates.

2. It is assumed that proposition 1 is also valid for \(x+\lambda\), where \(\lambda\) is the mean free path of a gas atom, so that one may speak of \(T(x+\lambda)\); in this case \(T(x) \ne T(x+\lambda)\), namely it is greater or smaller, depending on the direction of energy transport.

These two propositions make it possible to construct an approximate theory of thermal conductivity, valid in the case when the temperature differences that arise can be represented as small distortions of strict thermal equilibrium (see also Sections IX, a and b).

\(^{1}\) If the number of absorbing atoms is large—of the order of 1000 in a cube with edge equal to the wavelength—then a phase relation arises between the incident and re-emitted radiations. A phenomenon arises analogous to reflection from metals (R. Wood \(^{539, 360}\); F. Weiskopf \(^{840a}\)).

In this way one calculates the amount of energy \(L\) flowing through unit area per unit time:

\[ L=\sum_{\varepsilon}\varepsilon\cdot N\cdot D_T\cdot \frac{dA_{\varepsilon}}{dT}\cdot \operatorname{grad} T,\quad \text{where } D_T=\frac{1}{3}\lambda\cdot \overline{v}. \tag{8d,1} \]

Here \(N\) is the total number of atoms, \(A_{\varepsilon}\) the relative probability of finding an atom in a state with energy \(\varepsilon\), and \(D_T\) the diffusion coefficient; it is assumed that the latter depends only weakly on \(T\).

The coefficient of thermal conductivity is, by definition,

\[ \sigma=\sum_{\varepsilon}\varepsilon\cdot N\cdot D_T\cdot \frac{dA_{\varepsilon}}{dT}. \tag{8d,2} \]

In the case where the transport of energy depends only on the exchange of kinetic energy, the classical coefficient of thermal conductivity is equal to\(^1\)

\[ \sigma_{kl}=\frac{3\cdot 2^{\frac{3}{2}}\cdot k^{\frac{3}{2}}\cdot T^{\frac{1}{2}}}{\pi^{\frac{5}{2}}\cdot d^2\cdot \sqrt{m}}, \tag{8d,3} \]

where \(d\) is the gas-kinetic diameter and \(m\) is the mass of the atom.

It had already been established by Nernst \({}^{371}\) that the transport of energy in a plasma is not limited to kinetic energy. If, for example, an ion is formed at the point \(x\) and recombination of the ion occurs at the point \(x+a\), this signifies transport of the ionization energy, i.e. an additional transport of energy in the direction of the \(X\)-axis. The same applies to the processes of formation and destruction of an excited atom, and also to the emission and absorption of a light quantum.

As was shown in Section VIII,a, one can find the probability \(A_{\varepsilon}\) for the individual components of the plasma at a given temperature. Therefore equation (8d,2) makes it possible to calculate, from \(\frac{dA_{\varepsilon}}{dT}\), the coefficients of thermal conductivity for other forms of plasma energy. Below we shall consider a whole series of such possibilities of nonclassical thermal conductivity, namely by means of diffusion of excited atoms and light quanta, by means of ions and electrons, and through dissociation of molecules. For calculating the thermal conductivity through ionization and dissociation, equations (8a,3) and (8a,10), respectively, are suitable, with the degrees of ionization and dissociation to be substituted instead of \(A_{\varepsilon}\).

We shall first take into account only the deepest levels (the upper levels of resonance lines) and assume that the concentration of excited atoms can be determined from equation (8a,1). Then

\[ A_{\varepsilon}=g_{\varepsilon}e^{-\frac{\varepsilon}{kT}}, \tag{8d,4} \]

where \(\varepsilon\) is the term energy.

\(^1\) In work \({}^{431}\), through an oversight, \(\frac{mv^2}{2}\) was equated to \(kT\), and not to \(\frac{3}{2}kT\); therefore the numerical coefficient in \(\sigma_{kl}\) there is incorrect.

For the coefficient of thermal conductivity we have

\[ \sigma=\varepsilon N D_T \frac{\varepsilon}{kT^2}A_\varepsilon =\left(\frac{\varepsilon}{kT}\right)^2\cdot k\cdot D_T\cdot N_\varepsilon \tag{8d, 5} \]

(\(N\) is the concentration of atoms).

In calculating \(D_T\) it must be taken into account that the mean free path for the diffusion of excitation energy is determined by the process of energy exchange (collisions of the second kind) between identical atoms \(^{1}\). The effective cross section for these collisions \(^{2}\) was calculated by Fursov and Vlasov \(^{161}\):

\[ \pi R_0^2=\frac{8\pi}{3}\cdot\frac{e^2}{m}\frac{f}{2\pi\nu_0}\cdot\frac{1}{v}. \tag{8d, 6} \]

Here \(e\) and \(m\) are the charge and mass of the electron, \(\nu_0\) is the natural frequency of the electron, \(v\) is the relative velocity, and \(f\) is the oscillator strength \(^{3}\).

For levels with \(f\simeq 1\), experiment and calculation \(^{437}\) give, in good agreement, cross sections of the order of \(10^{-12}\ \mathrm{cm}^2\). The gas-kinetic cross sections determining the diffusion of kinetic energy are of the order of \(10^{-15}\)—\(10^{-16}\ \mathrm{cm}^2\); therefore the diffusion of excitation energy proceeds correspondingly many times more slowly than the diffusion of kinetic energy. A calculation \(^{439}\) carried out for mercury plasma with an atom concentration of \(1.85\cdot 10^{19}\ \mathrm{cm}^{-3}\) and a temperature of \(8000^\circ\) gave, for the transport of energy by diffusion of the excitation energy of the term \(2^1P_1\) (the upper term of the resonance line 1848 Å), with \(f=1.3\), the value

\[ \sigma_{2^1P_1}<3\cdot 10^{-9}\ W/\mathrm{cm}\cdot\mathrm{degree}, \]

and for the term \(2^3P_{210}\), for which \(f=0.025\), the value

\[ \sigma_{2^3P_{210}}<2\cdot 10^{-7}\ W/\mathrm{cm}\cdot\mathrm{degree}, \]

whereas under the same conditions \(\sigma_{k1}\simeq 10^{-4}\ W/\mathrm{cm}\cdot\mathrm{degree}\).

We see that terms with still smaller values of \(f\), i.e. of metastable character, can give a significant increase in thermal conductivity. This can be expected, however, only at extremely low pressures, since the lifetime of a metastable level in a plasma is determined by the number of collisions.

The thermal conductivity of excited terms depends on the relative number of excited atoms; the latter (see Figs. 7 and 9) increases somewhat with increasing pressure and increases strongly with

\(^{1}\) Cf. A. Mitchell and M. Zemansky \(^{355}\).

\(^{2}\) Weizel (reported at the meeting devoted to the question of line width, Bonn, 28 November 1938) and Huston \(^{225}\) suppose that the “optically effective” cross section for collisions of identical atoms, calculated by Weisskopf \(^{541-543}\), is identical to the cross section for collisions of the second kind. It differs from equation (8d, 6) only by the factor \(4/3\).

\(^{3}\) The term oscillator strength means the following: according to the classical theory, the number of electrons participating in one electronic vibration must always be an integer, i.e. 1, 2, 3, ... . According to quantum theory, decimal fractions are also possible, i.e. 0.5 or 0.9, 1.3, etc. A summary of the known values of \(f\) may be found in the book by Knoll, Ollendorff, and Rompe \(^{246}\). See also the note on p. 338.

lowering of the temperature. Classical thermal conductivity does not depend on pressure and varies proportionally to the square root of the temperature. For the density \(1.85\cdot 10^{19}\) given above, the thermal conductivity of the \(2^3P_{210}\)-term of the mercury atom at \(15\,000^\circ\) would have the same order of magnitude as the classical thermal conductivity.

The diffusion of light quanta is determined by the quantity \(A_{\varepsilon}\), the same one that was used above. The two processes differ only in the mechanism of transport: in the case of diffusion of excitation energy, a mechanism of exchange of excitation energy without participation of radiation is assumed; on the other hand, in the case of diffusion of light quanta the transport takes place by emission and absorption of radiation. Therefore here one must use a different diffusion coefficient. Since the time taken by a light quantum to traverse the average distance between atoms may be regarded as infinitely small, it is expedient to define as the flight velocity the quantity \(\dfrac{\lambda}{\tau}\), where \(\lambda\) is the mean free path, and \(\tau\) is the mean time of residence\(^1\) of the light quantum in the state of absorption by an atom. We shall put the mean free path equal to

\[ \frac{1}{N\cdot q_{\mathrm{opt}}}, \]

where \(N\) is the concentration of atoms, and \(q_{\mathrm{opt}}\) is the effective cross section for absorption of a light quantum. In the theory of dispersion it is shown that a bound electron absorbs incident radiation in the same amount as a black disk of area \(\pi\dfrac{c^2}{\nu^2}\), where \(\nu\) is the electron’s proper frequency\(^2\). This is true in the case where absorption takes place only within the natural width of the line, caused by damping of the radiation. If, however, there is an additional broadening of the absorption line owing to the Doppler effect or to damping caused by collisions\(^3\), then the effective cross section is reduced in the ratio \(\gamma:\Gamma\), where \(\gamma\) is the natural width and \(\Gamma\) the actual width of the line.

Thus for the coefficient of thermal conductivity through diffusion of radiation we have:

\[ \sigma_{\mathrm{opt}} = \left(\frac{\varepsilon}{kT}\right)^2 \cdot k\cdot \frac{1}{3}\cdot \frac{\lambda^2}{\tau}\cdot N\cdot A_{\varepsilon} = \left(\frac{\varepsilon}{kT}\right)^2 \cdot k\cdot \frac{1}{3}\cdot \frac{1}{N\cdot q_{\mathrm{opt}}^{\,2}}\cdot \frac{1}{\tau}\cdot A_{\varepsilon}. \tag{8d, 7} \]

From this one obtains the ratio of the thermal conductivity due to collisions of the second kind of excited atoms to the thermal conductivity due to radiation:

\[ \frac{\sigma_{\mathrm{возб}}}{\sigma_{\mathrm{opt}}} = \frac{N\cdot q_{\mathrm{opt}}^{\,2}\,\bar v}{q_{\mathrm{уэ}}}. \tag{8d, 7'} \]

\(^1\) The mean residence time is connected with the transition probability \(A\) (see Section VIII, c) by the simple relation \(\tau=\dfrac{1}{A}\); \(A\) is proportional to the oscillator strength \(f\). See M. Born\({}^{41}\).

\(^2\) If atomic electrons are meant, then this quantity must be multiplied by the oscillator strength \(f\).

\(^3\) Cf. \({}^{46_2}\).

If one assumes that the broadening of the line due to the Doppler effect is small in comparison with the width determined by collisions (the latter for homogeneous atoms is proportional to \(f\)), then \(q_{\mathrm{opt}}\sim f\), \(\tau\sim \dfrac{1}{f}\), \(q_{y0}\sim f\), so that in this case the ratio does not depend on \(f\); thus it has the same value, for example, for the \(2^1P_1\) and \(2^3P_{210}\) levels of mercury. In those cases in which \(\Gamma\) cannot be regarded as proportional to \(f\), i.e., at low densities, where broadening due to the Doppler effect predominates, \(\dfrac{\sigma_{\mathrm{exc}}}{\sigma_{\mathrm{opt}}}\approx f\). Under the condition \(N=1.85\cdot 10^{19}\) atoms per \(1\ \mathrm{cm}^3\) and \(8000^\circ\) (Hg) \(\dfrac{\sigma_{\mathrm{exc}}}{\sigma_{\mathrm{opt}}}\approx 1\); at higher densities collisions predominate, at lower densities—radiation. These considerations are valid, of course, only for resonance lines. The error made by neglecting higher terms for temperatures attainable under terrestrial conditions is small, since the concentration of the higher terms is much lower than the concentration of the upper levels of the resonance lines. On the other hand, the radiation of the higher terms is so little absorbed that the radiation leaves the plasma practically without diffusion.

The thermal conductivity that appears as a result of the presence of ionization or dissociation obeys the following equation \(^{431}\) [see note \(^{1}\) on p. 471]:

\[ \sigma=\varepsilon ND\frac{\partial c}{\partial T} =\frac{\varepsilon NDpc}{2T(p+G)} \left(\frac{5}{2}+\frac{\theta_j}{T}-\frac{Z'_i}{Z_i}\right). \tag{8d, 8} \]

Here \(\varepsilon\) is the energy of ionization or dissociation, \(N\) the concentration of atoms or molecules in the cold state, \(D\) the diffusion coefficient, \(c\) the degree of ionization or dissociation, \(G=\dfrac{c^2}{1-c^2}p\), \(p\) the pressure, \(\theta_j=\dfrac{\varepsilon}{k}\), \(Z_i\) the sum of states for the internal degrees of freedom (excitation or vibration and rotation), and \(Z'_i=\dfrac{\partial Z_i}{\partial T}\).

One should dwell especially on the magnitude \(D\). For the diffusion of atoms formed by dissociation one may use the gas-kinetic diameter; in doing so, one must take into account the coexistence of atoms and molecules. With regard to the diffusion of ions and electrons, one may assume that it is determined by the diffusion constant of the more slowly moving ions. The effective cross section for ions is the cross section for charge exchange, which differs by no more than one order of magnitude from the gas-kinetic one \(^{386}\).

In Figs. 11–14 examples are given of calculations of the thermal conductivity of ions and dissociated molecules. They all lead to curves with a sharp maximum, which shifts toward higher temperatures and broadens as the pressure increases. Qualitatively, this behavior is not difficult to understand: at low temperatures the degree of ionization is small and its influence may be neglected. At high temperatures the degree of ionization is close to \(100\%\), so that changes

the degrees of ionization become small with temperature, and the thermal conductivity will again be classical; in this case, of course, the increased number of particles must be taken into account. The largest of the thermal-conductivity coefficients encountered approach the values of \(\sigma\) for metals; for example, \(\mathrm{H}_2 \leftrightarrow 2\mathrm{H}\) at \(1\) torr and \(3000^\circ\) gives \(0.37\)

Fig. 11. Thermal conductivity due to dissociation of \(\mathrm{H}_2\) \(^{431}\)

Fig. 12. Thermal conductivity due to dissociation of \(\mathrm{O}_2\) \(^{431}\)

\(\mathrm{W}/\mathrm{cm}\cdot\mathrm{deg}\), whereas for Al we have \(2.1\), and for Fe \(0.6\ \mathrm{W}/\mathrm{cm}\cdot\mathrm{deg}\). Similar nonclassical thermal conductivity has been observed experimentally in \(\mathrm{H}_2\) and \(\mathrm{J}_2\); the experimental data were used to calculate the dissociation energy \(^{232, 289-291}\). The existence of ion thermal conductivity in gas-discharge plasma is indicated by some observations on a high-pressure mercury discharge \(^{439}\).

e. The influence of plasma on the properties of atoms

The properties of an atom situated in a plasma undergo changes. The latter may be a consequence of interaction with other compo-

nents of the plasma or by the influence of the internal field of the plasma. The typical changes in the properties of plasma atoms are the following:

Broadening of lines as a result of electron or ion impacts; formation of molecules in interaction with excited atoms; lowering of the ionization potential; disappearance of the higher members of series1; appearance of forbidden lines; appearance in emission of a recombination continuous spectrum. In addition, there arises a special continuous spectrum characteristic of plasma, having something in common with the continuous spectrum of solid conducting bodies.

Fig. 13. Thermal conductivity due to ionization of Na⁴³¹

Other phenomena similar to those listed above—the formation of van der Waals molecules, broadening of lines due to high density, etc.—are not typical of plasma, since they are also observed in gases at low temperatures.

Broadening due to ion and electron impacts. If an electron or ion approaches an excited atom, then the latter, owing to the action of the Coulomb field of the electron (ion), changes the frequency of emission. The time during which an appreciable change of frequency occurs is of the order of \(10^{-13}\) sec.; therefore such a short-time change of frequency may be regarded as a phase jump in the emission of the atom; here the interval of time between two such jumps is large in comparison with the duration of the “collision”⁴⁶²2.

Fig. 14. Thermal conductivity due to ionization of Hg⁴³¹

Unsöld⁵²⁸, with the aid of Fourier analysis, obtained a line broadening with a dispersion distribution. The half-width of the line is given by:

\[ \delta=\frac{1}{2\pi c}\cdot 2^{\frac{3}{2}}\cdot \pi^{\frac{13}{6}} C^{\frac{2}{3}}(kT)^{\frac{1}{6}} \left(\frac{1}{M}+\frac{1}{m}\right)^{\frac{1}{6}}N_- . \tag{8e, 1} \]

Here \(N\) is the number of electrons or ions per \(1\ \mathrm{cm}^3\), \(M\) is the mass of the radiating atom, \(m\) is the electron mass, \(k\) is Boltzmann’s constant, and \(C\) is the constant of the quadratic Stark effect. If in this formula the ion mass is substituted for \(m\), then, owing to the larger ion mass, the broadening will be correspondingly smaller. For example, for mercury it is 7.5 times smaller than the broadening by electron impact.

It follows from equation \((8e, 1)\) that the broadening of different terms in a plasma of given temperature (electron concentration) is determined mainly by the value of the constant \(C\). The latter is the larger, the more hydrogen-like the terms are, i.e., the larger the principal quantum number of the term and, for a given principal quantum number, the larger the orbital angular momentum of the term. Such broadening, for example, for the \(3D\)-term of mercury at an electron concentration of \(3.6\cdot 10^{17}\), is, in accordance with equation \((8e, 1)\), \(30\ \mathrm{cm}^{-1}\) 438, i.e., it exceeds the natural line width by about a factor of 30001.

Unstable molecules. In the excited state most atoms are capable of forming diatomic molecules both with an atom of the same kind and with a foreign atom. In a plasma one observes spectra of compounds unknown to chemistry, for example, \(\mathrm{Hg}_2\) 199, 201, 406, 419, 420, 459, 533, 551—554, \(\mathrm{He}_2\) 210, 221, 222, 526, 544, 545, \(\mathrm{HgHe}\), \(\mathrm{HgAr}\) 270—271, 379—381, 413, 414 (cf. also 145).

The formation of a molecule from one excited and one unexcited atom can occur only if the heat of combination liberated is radiated away or is transferred to a third partner in a so-called triple collision 159. In most cases an excited molecule is formed, which by radiation passes into the unstable ground state of a molecule consisting of unexcited atoms. The lifetime of such a molecule is a quantity of the order of the mean time \(\tau\) of residence in the excited state. A much larger number of “transient molecules” are formed in the collision of excited and unexcited atoms during the time of impact \(\left(\dfrac{2R}{\bar v},\ \text{see above}\right)\). The concentration of such molecules is given by the following expression:

number of transient molecules in \(1\ \mathrm{cm}^3\) = number of impacts \(\times\) time of impact =

\[ = N_a \cdot N \cdot \pi R^2 \cdot \bar v \cdot \frac{2R}{\bar v} = N_a N \pi R^3; \tag{8e, 2} \]

here \(N_a\) is the number of excited atoms, \(N\) is the number of unexcited atoms. The magnitude of the effective radius \(R\) for collisions with the formation—

GASES IN THE PLASMA STATE

the formation of molecules of this kind must lie between the values of the gas-kinetic radius and the radius for collisions of the second kind. The ratio of the number of molecules to the number of excited atoms mostly lies in the range \(10^{-4}\)—\(10^{-10}\); therefore the detection of transient molecules by spectroscopic means is hardly possible.

It is necessary to emphasize that excited and unexcited atoms can form a stable compound. This occurs only in a triple collision. Unfortunately, our knowledge of such “chemical synthesis” in a plasma is very scanty\(^{1}\).

The continuous spectrum of a plasma. The formation of unstable molecules leads to the emission of continuous bands\(^{145}\), most often adjoining intense atomic lines. Free electrons present in the plasma and responsible for its electrical conductivity may, in various ways, give rise to continuous absorption and emission\(^{146}\).

In this section we shall dwell on the continuous spectrum associated with the presence of electrons in the plasma. First of all, an electron can directly take energy from the radiation field—the acceleration of the electron in the electric field of the incident radiation. This process, called “free-free” absorption, corresponds completely to the process of absorption by the electrons of a metal in the theory\(^{2}\) of Riecke—Drude\(^{37, 92, 94, 330, 424, 425}\), which treats the electrons of a metal as a nondegenerate electron gas. Secondly, an electron can be captured by an ion; in this case the kinetic energy of the electron and the recombination energy, equal to the ionization potential of the given term, pass into radiation. A band arises, sharply bounded on the long-wavelength side; this is “free-bound” emission. The corresponding absorption process consists in the photoionization of the atom.

The third process relates to the intermediate state of electrons. As we shall show in the next section, electrons which are at high levels, less than 1 V from the series limit, may be regarded as “quasi-free.” Such electrons may be acted upon both by the microfield of the plasma—which for such trajectories has approximately the same strength as the central field of the atomic core—and by electron and ion impacts; therefore an unambiguous association of the given electron with a definite atom becomes impossible. This state of the electrons is intermediate between their state in a gas and in a solid. Very close

\(^{1}\) The experimental material concerning the indicated problem is very large; however, there is no possibility of establishing sufficiently clearly the conditions of the experiment, as is necessary for making any theoretical conclusions.

From the older literature we shall indicate first of all\(^{513}\); in addition \(^{44-55, 137-144, 184, 220, 312-317, 547}\). In papers \(^{116}\) and \(^{117}\) a statistical theory of chemical reactions in a discharge is presented, based on considerations analogous to those set forth by us in Sections VIII, C and IX, b.

\(^{2}\) The correspondence between plasma and a “classical” metal also applies to conductivity; the mobility of electrons in a metal obeys a law analogous to the Langevin law (see Section III).

from the boundary of the series, below it, something like an electron band begins to form; here, of course, owing to the high temperature the band does not have a definite width, and the fluctuations of the energy interval are of order \(kT\). According to experimental data, such blurring of atomic terms occurs in a plasma at small atom concentrations and is of order \(0.01\ \mathrm{V}\) for \(10^{16}\ \mathrm{cm}^{-3}\) and \(1\ \mathrm{V}\) for \(10^{19}\ \mathrm{cm}^{-3}\). These conditions correspond to typical discharges at low and high pressures.

The intensity of the continuous spectrum of radiation or absorption was calculated by Unsöld \(^{529}\), who treated the high terms as hydrogen-like. For the absorption coefficient of the continuous spectrum \(k_\nu\), for all three types of electron behavior, one obtains the expression

\[ k_\nu=\gamma\cdot\frac{16\pi^2}{3\sqrt{3}}\cdot\frac{e^6\cdot Z_{\mathrm{eff}}^2\varphi\phi}{c\cdot h\cdot k^2}\cdot\frac{e^{-U_i}}{T^2}\cdot\frac{e^{u+\Delta u}}{u^3}. \tag{8e, 3} \]

Here \(\gamma\) is a weight factor, \(Z_{\mathrm{eff}}\) is the effective nuclear charge, \(U_i=\dfrac{E^*}{kT}\), \(E^*\) is the ionization potential, \(u=\dfrac{h\nu}{kT}\), \(\Delta u=\dfrac{\Delta E}{kT}\), and \(\Delta E\) is the energy width of the blurring.

This total absorption coefficient is divided into its component parts as follows:

\[ \begin{array}{ccc} \text{Free-} & \text{Free-} & \text{Blurring} \\ \text{free} & \text{bound} & \\ 1 & e^u-1 & e^{u+\Delta u}-e^u \end{array} \left\} \right. \tag{8e, 4} \]

Here the part corresponding to the blurring includes the transition from the blurred part of the spectrum both into a discrete term (“quasi-free-bound”) and into the continuous part (“quasi-free-free”). The introduction of blurring also includes the influence on high terms of the Stark effect in the plasma field \(^{3,21,24,25,266,274,470}\) and the appearance of forbidden members of series.

From (8e, 4) it is evident that, for the visible spectrum at \(4000\ \text{\AA}\) and a temperature of \(6000^\circ\) (\(U\) approximately equal to 6), free-free absorption is entirely insignificant in comparison with free-bound radiation, and that blurring for \(\Delta E\simeq 1\), i.e. at \(T=6000^\circ\), about \(0.5\ \mathrm{V}\), is the principal part of the absorption.

An analogous formula can be written for emission, referred to unit volume \((\mathrm{cm}^3)\), unit frequency, and solid angle \(4\pi\):

\[ 4\pi\varepsilon_\nu =\gamma\cdot\frac{128\pi^3}{3\sqrt{3}}\cdot \left(\frac{e^2}{hc}\right)^3\cdot Z_{\mathrm{eff}}^2 e^{-u_1+\Delta u}\cdot p. \tag{8e, 5} \]

Thus, the emission extending from long wavelengths to the boundary of the principal series does not depend on the frequency.

Similar continuous emission spectra are observed chiefly at high pressures \(^{1)}\). Independence of the frequency is observed—

\(^{1)}\) On continuous spectra in stationary discharges see \(^{111,112,168,170,200,227,241,253,365,440,446}\), in nonstationary ones (sparks, etc.) \(^{5,7,8,68,103,149,150,151,202,253,308,336,457,486,501,561,562,564}\). Review \(^{145}\).

is not given accurately; continuous spectra possess a structure indicating that the resulting radiation is more correctly conceived as a superposition of different continuous spectral regions. For example, in Hg plasma the ratio of the emissions of two different spectral regions is different from unity; it does not, however, change when the temperature and pressure are varied. The relative dependence on temperature and pressure was established experimentally440. The absolute values of the intensity, calculated from (8e, 5), are too small529; however, it is probable that this discrepancy is partly explained by the too small value that was adopted for the plasma temperature440.

Along with the characteristic absorption of a plasma, there is absorption caused by the oscillations of free electrons and ions; the latter we discussed in Section V.

Change of the ionization potential in a plasma. The change of the ionization potential in a plasma is closely connected with the smearing of the spectrum at the series limit, considered in the preceding section. Indeed, an electron which cannot be bound in a definite way to a given atom is acted upon by the electric field and takes part in electrical conduction. The number of electrons calculated for a given plasma temperature from Saha’s formula proves to be too small. The deficit can be compensated by a corresponding lowering of the ionization potential.

A decrease of the ionization potential in an arc was observed by Mankopf340, 341. Broadening of terms sensitive to the Stark effect, by means of electron impacts, made it possible for Rompe and Schulz438 to establish in the plasma of a mercury discharge at 20 atm a value of the electron density corresponding to a decrease of the ionization potential by 1–1.5 V. The same lowering of the ionization potential is implied by the dependence of the intensity of the continuous spectrum of the same mercury plasma on the current strength according to equation (8e, 5)440.

The observed decrease of the ionization potential is evidently composed of various effects that depend on one another only in part. Their contributions to the total action can be measured independently; however, an estimate of the separate effects can hardly be made with the means known at present.

The first effect is the “statistical” decrease of the ionization potential caused by the presence of the atom in the microfield of the plasma. We already considered this phenomenon in Section IV and established that the average decrease is very small, as is illustrated by the figures of Table 5. For plasma VII it is equal to \(3 \cdot 10^{-2}\) V, whereas the experimental values, according to440 and438, are 1.2 V. As was already mentioned in Section IV, the decrease of the ionization potential that occurs is due to fluctuations of the potential in the immediate vicinity of the atom \((r < D)\); as the order of magnitude of these fluctuations one may take \(kT\) \((kT_{-}\) for a nonisothermal plasma).

A rough estimate can be made on the basis of the following considerations. If the intensities of the central field and of the microfield are

quantities are of the same order, then in practice the electron may be regarded as free (⁹⁶ and Section II), i.e.,

\[ \frac{e}{r^{2}}=20e\cdot N^{\frac{2}{3}}; \tag{8e, 6} \]

here \(e\) is the electron charge, \(r\) is the distance of the valence electron from the atomic residue in centimeters, and \(N\) is the electron concentration.

On the other hand, the radius of the Bohr orbit is

\[ r_a=0.528n^{2}\cdot 10^{-8}\ \text{cm}, \]

where \(n\) is the principal quantum number of the H atom. It follows that all electronic states with principal quantum numbers \(n\) greater than

\[ n=0.7\cdot 10^{4}N^{-\frac{1}{6}}, \tag{8e, 7} \]

may be regarded as quasifree.

Thus, at an electron concentration of the order of \(10^{18}\ \text{cm}^{-3}\) there are no terms with quantum number greater than 7. And indeed, in mercury plasma, where such densities are attainable, lines of terms with quantum numbers greater than 5 are practically not observed ⁴³⁸, ⁴⁴⁷. The difference between the ionization potential and the excitation potential of the term with quantum number 6 is a quantity of the order of 0.5 V; the decrease observed experimentally has a value of about 1 V; thus, the influence of fluctuations of the microfield strength on the ionization potential is quite possible. Be that as it may, on the basis of a rough estimate such as that given above, it cannot be said whether the overall lowering reduces to the indicated effect.

The magnitude of the duration of action of the plasma field is of the order of the time required for an electron to traverse a path equal to the mean distance between plasma electrons. At an electron concentration of \(10^{18}\ \text{cm}^{-3}\) and their temperature \(\simeq 1\ \text{V}\simeq 10^{8}\ \text{cm/sec}\), we obtain the value \(\tau\simeq 10^{-14}\) sec.; therefore the action of the plasma field can also be reduced to a broadening of the higher terms as a result of collisions¹).

In conclusion it should be mentioned that a further cause of the lowering of the ionization potential may be the so-called preionization ²⁸³, ⁸⁴, i.e., the transition of electrons of atoms into the region of free electrons without radiation by means of the quantum-mechanical tunnel effect. For ordinary plasmas, however, there is no reason to expect any significant influence of this effect.

IX. NON-ISOTHERMAL PLASMA

As indicated in Section VII, a completely or almost isothermal plasma can be realized experimentally, for example, by heating a gas in a furnace to several thousand degrees; in this case the deter—

¹) Cf. Section II and reference ²) on p. 483. As in the case of line broadening, so also in the case of the lowering of the ionization potential, the influence of collisions is greater than the “static” action (see also the note on p. 484).

the furnace through which unhindered escape of the plasma radiation is possible must be small in comparison with the surface of the furnace (a black body). If this condition is not fulfilled, then the plasma, owing to its radiating ability, acts as a transmitter of energy flux and, strictly speaking, such a state can no longer be regarded as thermally equilibrium. The same is true when energy is supplied not by means of “thermal contact,” i.e., by radiation or by transfer of kinetic energy, but by acceleration of the plasma electrons by an external electric field.

The behavior of plasma in such cases constitutes a complex problem of heat conduction; an exhaustive solution of such problems by the known methods of gas theory leads to reasonable results only in rare cases (very large concentrations of atoms, high-pressure plasmas). One of the reasons for the inapplicability of the basic propositions of the theory of heat conduction is the hypothesis of “local thermal equilibrium” mentioned in Section VIII, d; another is the circumstance that part of the energy leaves the plasma without participating in the diffusion process. The latter occurs, for example, in the case of non-reabsorbed radiation, when the concepts of “mean free path” or “diffusion coefficient” lose their meaning. However, it is certainly possible to construct a thermodynamically rigorous theory of heat conduction that does not make use of the limitations just named (\(^{106}\); cf. also \(^{215}\)). Unfortunately, in this direction there are as yet no works on which it would be worthwhile to dwell here.

Despite the absence of a quantitative theory, a fairly detailed qualitative consideration of nonisothermal plasma is possible.

a. The concept of “temperature” in a nonisothermal plasma

Thermodynamics has established the concept of temperature for the case of thermal equilibrium. The introduction of this concept requires the adoption of certain assumptions concerning distribution functions, concerning the spatial and temporal distribution of particles, and also concerning the distribution of the individual forms of energy (the law of equipartition).

Let us first consider how it can be established whether a plasma is in a state of thermal equilibrium. This question can, of course, be resolved only by measurements. Measurement of the temperature of some system indicates only the mean value of the energy of a definite form \(^{29,408}\). Using an optical method, for example a pyrometric one, we obtain a number showing that temperature of a black body at which, in the measured spectral region, it gives the same radiation as the system under investigation. We obtain the “black temperature” of a spectral region and, consequently, under certain additional assumptions about the mechanism of radiation, the mean energy of certain atomic terms. If a measurement is made of the Doppler width of a line, then a value of the mean energy of motion is obtained. Measurement by means of

of a contact thermometer indicates (in the case where the incident radiation is completely reflected) a temperature related in a definite way, depending on the conditions of heat transfer, to the kinetic energy. If the thermometer absorbs the radiation, then the measurement gives an average of the values of the radiation energy and the energy of motion. We shall not discuss here the accuracy or even the possibility of temperature measurements in gases (these questions are considered in works 29, 125–127, 130, 209, 245, 272, 309, 310, 408, 502, as well as in the works cited in Section VII), and shall dwell only on the following.

On the basis of a temperature measurement carried out by any one method, one cannot decide whether the system is in thermal equilibrium or not. If, however, by measuring the temperature by several methods associated with different forms of energy, we arrive at one and the same value, then the plasma under consideration is in strict thermal equilibrium. In the case where such an experiment leads to different figures, we cannot speak of thermal equilibrium, since there is no temperature in the thermodynamic sense of the word that could be assigned to the entire plasma. Nevertheless, it is customary to call the figures obtained by different measurements “temperatures” even in the case of their disagreement. In this sense one speaks of the electron temperature, the radiation temperature, the ionization temperature, and the gas temperature. These quantities correspond to the distributions of velocities of electrons or atoms, the degree of ionization, or the relative number of atoms located at a given level, for a system in equilibrium. Therefore one cannot object to the use of the word “temperature” in the indicated sense; otherwise, if one wished to be consistent, one would have to reject such established terms as “black temperature,” “color temperature”^309 of a light source, or radiation temperature^409.

As was shown in Section I, the deviations of a plasma from the state of equilibrium are different for different forms of energy.

As the basis for the practical determination of thermal equilibrium we shall take the law of uniform distribution. Wishing to have a judgment about individual forms of energy, for example, the energy of motion of electrons or atoms, we can, in addition, determine the temperature of these forms of energy from the Maxwellian distribution in probe measurements (see Section I) or in determining the Doppler broadening of spectral lines.

It is possible, however, to invoke an even more indirect criterion—the principle of detailed equilibrium (cf. VIII, c). If it is established that the principle of detailed equilibrium holds in the interaction of two forms of energy, then this means that they are in equilibrium with respect to each other. If, as we shall show in Section IX, b, the principle of detailed equilibrium is fulfilled for the electron gas and atomic terms, then this means that the electrons and the atomic terms have reached a state of mutual equilibrium. At the same time, of course, such a situation does not hold for, say, the interaction of electrons and the energy of motion of atoms.

When the principle of detailed equilibrium is satisfied for any forms of energy, one may speak of strict thermal equilibrium. The radiation density of a plasma may be considerably less than the radiation density of an absolutely black body at the same temperature, and this does not disturb the state of equilibrium\(^{409}\) ¹). It is, of course, impossible to regard the thermal equilibrium of a plasma as impossible on the grounds that its spectrum is line-like, i.e. differs sharply from “black” radiation.

b. Elementary processes in a nonisothermal plasma

Let us return to the considerations set forth in Section VIII, d. The process of transfer of energy by the electron gas to atoms and ions consists mainly in collisions of electrons, which have received their energy from the electric field, with atoms. In this process (Section VIII, c) either the kinetic energy of the atom is increased as a result of an elastic impact, or an excited atom or a pair of charges is formed by an inelastic impact. In the stationary state of an isothermal plasma, both the mean kinetic energy of the atoms and the number of excited atoms and ions have definite values determined by the plasma temperature; this is not the case for a nonisothermal plasma. Hence it follows that the principle of detailed equilibrium, which regulates the processes of creation and destruction of the components of the plasma, is not of unlimited applicability to a nonisothermal plasma.

Let us consider a simplified model of a nonisothermal plasma. It consists of electrons, atoms, and ions.

As was shown in Section II, interaction with the microfield always makes it possible to assume for the electrons a Maxwellian distribution of velocities, to which one may assign a temperature \(T_{-}\). The external field maintains this temperature \(T_{-}\). The direct transfer of energy from the field to the ions may be neglected. We shall further assume that exchange between the various forms of energy of atoms or ions is negligible in comparison with the energy exchange in which electrons participate. This means, for example, that excited atoms are formed only through electrons, and not through atomic or ionic impacts. The energy transferred to the atomic and ionic gas leaves the plasma by means of radiation or thermal conduction of any kind [we include here also the diffusion of excited atoms and ions (see VIII, d), and also, necessarily, ambipolar diffusion (see I)].

Let us consider an atomic level \(k\), which participates in the removal of plasma energy by converting excitation energy into radiation. The energy emitted in 1 sec. by atoms situated on the \(k\)-th level,

¹) The absorption coefficient of an approximately isothermal plasma can be determined, for a given temperature, from the ratio of the radiation density of the plasma to that of a black body. Its magnitude lies within the limits from 0.1 to 0.001.

is equal to

\[ N_k \sum_i A_{ki} h \nu_{ki}. \]

Here \(N_k\) is the concentration of atoms, \(A_{ki}\) is the transition probability, and \(\nu_{ki}\) are the frequencies of the lines emitted by the \(k\)-atoms. For simplicity we shall assume that the \(k\)-atoms emit a single line, namely the one corresponding to the transition to the ground level 0. The level \(k\) arises by means of an electronic impact from the principal level. The number of such transitions in 1 sec. is

\[ N_0 N_- S^I_{10}. \]

In the stationary case

\[ h\nu_{k0}\cdot N_0N_-S^I_{10}=N_kA_{k0}h\nu_{k0}. \tag{9b, 1} \]

This relation indicates that the energy transferred by the electrons to the terms \(k\) is radiated without residue. When \(N_-\) is increased, the value of \(N_k\) may become arbitrary. The temperature that corresponds to the concentration of atoms excited to the level \(k\), \(N_k\) (see IX, a), is determined by the ratio \(N_k:N_0\); it may reach a value greater than the electron temperature. Hence one may conclude (if one accepts the validity of the second law of thermodynamics for stationary, but thermally nonequilibrium processes) that relation (9b, 1) is incomplete, namely that in its right-hand side there is lacking a term depending on the electron concentration. In the sense of the considerations of Sec. VIII, c, this missing term may be interpreted as electronic impacts of the second kind. Equation (9b 1) must then be replaced by the following\(^{99,137,258}\):

\[ N_0N_-S^I_{10}=A_kA_{k0}+N_kN_-S^{II}_{10}. \tag{9b, 2} \]

Experience shows that the second law of thermodynamics is applicable to a nonisothermal plasma. Kopfermann and Ladenburg\(^{256,257}\) showed that, when the electron concentration is increased (and the electron temperature is kept constant), the concentration of excited atoms in a nonisothermal low-pressure discharge plasma does not increase without bound, but tends to a limit which can be expressed by the approximate equality

\[ \frac{N_k}{N_0}\simeq e^{-\frac{h\nu_{k0}}{kT_-}}. \]

This result was confirmed by a number of other authors\(^{264}\), who measured the concentration of excited atoms both directly and indirectly\(^{441}\). This result is of the greatest importance for understanding the processes occurring in a nonisothermal plasma, since it permits one to apply the approximate principle of detailed equilibrium in the form (9b, 2)\(^{1}\).

Starting from the second law, we can draw the following conclusions regarding the behavior of our simplified model of a nonisothermal plasma.

\(^{1}\) The considerations set forth in this section have recently been used in astrophysics to elucidate the mechanism of radiation in nebulae and in the outer atmospheres of stars; see K. Wurm, Z. Astrophys., 14, 321, 1937; O. Struve and K. Wurm, Astroph. J., 88, 84, 1938.

1. A form of energy that takes part in the transfer of energy to the outside—both directly and with the aid of intermediate processes—will acquire the electron temperature, provided only that it (the energy) is in interaction with the electrons. Thus the number of metastable terms corresponds to the electron temperature. With good thermal insulation, the energies of motion of atoms and ions correspond to the temperature \(T_{-}\), or, if recombination at the boundaries is hindered, the degree of ionization is calculated from \(T_{-}\) by means of Saha’s formula.

2. If some form of energy takes part in the transfer of energy, then its temperature becomes lower than the electron temperature and, moreover, differs from the latter the more, the weaker the coupling of the form of energy under consideration with the electrons and the stronger the coupling with the external medium.

In the case of electron concentrations so small that collisions of the second kind may be neglected, the temperature of the term is determined by the expression

\[ kT_{k0}=\frac{eE_{k0}}{\ln A_{k0}-\ln N_{-}S^{\mathrm{I}}_{k0}} . \tag{9b, 3} \]

The greater the coupling with the electrons, i.e., the greater the effective number of collisions \(N_{-}S^{\mathrm{I}}_{k0}\), the higher \(T_{k0}\). Conversely, as the transition probability \(A_{k0}\) increases, \(T_{k0}\) decreases.

In the same way, for our model one can compute the temperature of all existing forms of energy. One obtains as many different temperatures as there are forms of energy; in this connection, individual electronic terms must be regarded as different forms of energy, unless the difference in energies is \(\ll kT_{-}\) (the relative numbers of atoms excited to levels with small energy differences correspond to their statistical weights\(^{257}\)).

1. With increasing \(N_{-}\), if \(T_{-}\) is constant, the temperatures of all forms of energy tend to \(T_{-}\). The degree of approximation is determined by an equality of the type (9b, 2), taking collisions of the second kind into account; equation (9b, 3) is replaced by the following:

\[ kT_{k0}=\frac{eE_{k0}}{\ln\left(A_{k0}+N_{-}S^{\mathrm{II}}_{k0}\right)-\ln N_{-}S^{\mathrm{I}}_{k0}} . \tag{9b, 4} \]

As \(N_{-}\) increases, the plasma approaches ever more closely the state of thermal equilibrium with the electron gas. The attainment of the equilibrium state is determined, according to (9b, 4), by the quantities \(A_{k0}\) and \(S^{\mathrm{I,II}}_{k0}\).

The transition probabilities are atomic constants related to the \(f_{k0}\)-numbers (see p. 479) and to the mean lifetimes of the terms \(\tau_{k0}\) by the following relation:

\[ A_{k0}=\frac{8\pi e^{2}}{3mc^{3}}\nu^{2}_{k0}\cdot f_{k0}=\frac{1}{\tau_{k0}};\qquad \nu_{k0}=\frac{E_{k0}}{h}. \tag{9b, 5} \]

Reabsorption by the plasma of a light quantum emitted by an atom acts as a decrease in the number \(A\) or an increase in the mean lifetime of the emitting atom.

Milne352 showed that, as a measure of the influence of reabsorption, one should introduce the “effective” transition probability \(A_{\mathrm{eff}}\). This quantity, unlike \(A_{k0}\) of equation (8b, 5), depends on the pressure; namely, as the pressure increases it decreases by several orders of magnitude from the value \(A_{\mathrm{eff}} = A_{k0}\), and then, owing to the ensuing broadening of the line, again increases up to the value \(A_{k0}\)(355, as well as 205, 237, 238, 535, 538, 549, 571). The range of concentrations within which this phenomenon is observed lies between \(10^{14}\) and \(10^{16}\ \mathrm{cm}^{-3}\); here what is meant are atoms capable of absorption. Thus, for excited atoms this range lies, correspondingly, at a higher concentration of all atoms. For mercury plasma at \(6000^\circ\), which emits almost exclusively lines originating from the \(2^3S\)- and \(3^3D\)-levels, this range lies between \(10^{17}\) and \(10^{20}\) atoms per \(1\ \mathrm{cm}^3\), in agreement with experimental measurements1).

The values \(A_{k0}\) and \(A_{\mathrm{eff}}\) have an order of magnitude from \(10^9\) to \(10^4\). The quantities \(S_{k0}\) (see Section VIII, c) may be represented approximately as the product of the mean electron velocity by the mean effective cross section for excitation. For an electron velocity of \(10^8\ \mathrm{cm/sec}\) (corresponding approximately to \(1\ \mathrm{V}\)) and a mean effective excitation cross section of \(10^{-16}\ \mathrm{cm}^2\), \(S_{k0} \simeq 10^{-8}\ \mathrm{cm}^3/\mathrm{sec}\).

Thus, at electron concentrations of \(10^{12}\ \mathrm{cm}^{-3}\) and higher, we approach the region of equilibrium. Such electron concentrations are attainable at atom concentrations of \(10^{12}\ \mathrm{cm}^{-3}\) and higher. In accordance with the data in Section VIII, c, when the quantity \(S\) is defined, the effective cross sections for the transfer of kinetic energy have a magnitude of the order of \(10^{-20}\ \mathrm{cm}^2\). Thus, the energy of motion of the atoms comes into equilibrium with the electrons at concentrations of the latter approximately \(10^4\) times greater, i.e. at atom concentrations of \(10^{19}\ \mathrm{cm}^{-3}\). The effective cross section of ionization by electron impact must have dimensions \(10^{-18}\)—\(10^{-19}\ \mathrm{cm}^2\), so that the saturation phenomenon is noticeable in the region of atomic concentrations \(10^{17}\)—\(10^{18}\ \mathrm{cm}^{-3}\). These data indicate a difference between low-pressure plasma—the saturation phenomenon only for atomic terms—and high-pressure plasma—the saturation phenomenon for all forms of energy. On the basis of the works of Mankopf341, Witte555, Hermann223, and Ellenbaas110—114, one may confidently assert the existence of local thermal equilibrium2) at atomic concentrations above \(10^{19}\ \mathrm{cm}^{-3}\).

1. The energy flux leaving the plasma in one or another form of energy may reach considerable magnitudes, despite—

1) V. A. Fabrikant and F. Butaeva133, R. Rompe and V. Tuře437, 443. In an Hg discharge, reabsorption is noticeable at a pressure of the order of \(1\ \mathrm{atm}\), and it may be neglected at pressures above \(20\ \mathrm{atm}\).

The inverse increase in the transition probability with increasing line width may be visualized as a decrease in the probability that two atoms emit at exactly one and the same frequency.

2) It is better to speak of local thermal equilibrium, because in the discharge plasmas investigated in the cited works there was a strong temperature gradient outward.

GASES IN THE PLASMA STATE

... to an approximate equilibrium with the electron gas. Let us consider, for example, again the excitation of an atomic term; in this case let equation (9b, 2) be satisfied

\[ N_0 N_- S^{\mathrm I}_{01}=N_1 A_{10}+N_1 N_- S^{\mathrm{II}}_{10}. \tag{9b, 2} \]

Our equilibrium condition is fulfilled if

\[ N_0 N_- S^{\mathrm I}_{01}\simeq N_1 N_- S^{\mathrm{II}}_{01}\gg N_1 A_{10}, \]

then also

\[ N_1\simeq N_0 e^{-\frac{E_{10}}{kT}} . \]

The radiation \(\sigma\) of a definite term is given by the expression

\[ \sigma=h\nu_{10}\cdot \bar g_{10}\cdot A_{10}\cdot N_0\cdot e^{-\frac{h\nu_{10}}{kT}} . \tag{9b, 6} \]

Equation (9b, 6), which represents Wien’s law for the case of small absorption coefficients, is valid when reabsorption in the plasma itself may be neglected (see above). If, as an example, we consider the relations taking place in a mercury plasma of very high pressure, namely \(N_- \simeq 10^{18}\ \mathrm{cm}^{-3}\), \(S^{\mathrm I}\simeq 10^{-8}\ \mathrm{cm}^3/\mathrm{sec}\), \(N_0\simeq 10^{19}\ \mathrm{cm}^{-3}\), then the left-hand side of equation (9b, 2) is approximately equal to \(10^{-29}\). This means that in \(1\ \mathrm{sec}\) \(10^{29}\) excited atoms capable of radiation are formed. The term \(N_1A_{10}\), characterizing the radiation, is equal to \(10^{23}\), if one puts \(\dfrac{N_1}{N_0}\simeq 10^{-4}\) and \(A_{10}\simeq 10^8\). Consequently, for every \(10^6\) exciting collisions there is one act of radiation. Hence it is seen that the loss of energy by radiation disturbs the state of equilibrium only quite insignificantly, and that the number of collisions of the second kind is practically equal to the number of collisions of the first kind. Taking for the mean energy of the emitted quantum the value \(0.5\cdot 10^{-19}\ \mathrm W/\mathrm{sec}\), we obtain for the energy flux from \(1\ \mathrm{cm}^3\) a value of the order of \(5\cdot 10^3\ \mathrm W\). The total (over the whole spectrum) energy flux observed under the indicated conditions \(^{108,109}\) has a value of approximately \(20\cdot 10^3\ \mathrm W/\mathrm{sec}\). Our approximate estimate thus leads to quite acceptable values.

With continuous loss of energy, the temperature of the plasma electrons must be, even if only slightly, higher than the temperature of the gas. Mankopf and Bitter \(^{341,555}\) estimate this difference at \(20\)—\(100^\circ\) when the electron temperature is \(6000^\circ\). Approximately the same value, namely the ratio gas temperature : electron temperature \(=0.97\), is obtained by Sommermeyer \(^{488}\), who calculated this figure using Kravath’s formula \(^{66}\) for the power transmitted by the electron gas to the atomic gas with allowance for reverse processes. In addition, it is accepted that the heating of the gas in a high-pressure plasma in the case of a monatomic gas is due mainly to elastic collisions of electrons with gas atoms \(^{1}\).

\(^{1}\) On the influence of elastic collisions of electrons with ions, see note \(^{2}\) on p. 206.

What was said above is valid not only for plasma at very high pressures, but also for low-pressure plasma. In the general case, however, the establishment of equilibrium between the energy of motion and the electron temperature does not occur.

The absolute magnitude of the energy flux existing in a plasma does not make it possible to judge the thermodynamic state of the plasma; in the general case of small energy fluxes (low electron concentration!) the deviations from the state of equilibrium are large; at large energy fluxes the deviations decrease. The ratio of the individual terms of a balance equation of the type (9b, 2) may serve as a measure of the degree of approximation to the state of equilibrium. If the power transmitted by electrons, by means of collisions of the first kind, to the atomic gas (“total power”) is a quantity of the same order as the power transmitted to the electron gas in the form of collisions of the second kind (“accumulated power”), then a state close to thermal equilibrium obtains. The power leaving the plasma (“effective power”) may nevertheless assume very large values as the difference of two very large numbers of the same order of magnitude.

c. Energy balance in a nonisothermal plasma

The energy transmitted to the electrons by the electric field must be equal to the sum of the energies of all forms leaving the plasma. In the case where the form of energy under consideration is radiated to a much greater extent than it is transmitted to the electron gas, one may take the power transmitted by the electrons to the atomic gas as equal to the power leaving the plasma, as a result of which the calculation is greatly simplified. Such a neglect may be made at atomic concentrations less than \(10^{17}\ \mathrm{cm}^{-3}\), and degrees of ionization less than \(10^{-2}\) for the kinetic energy of atoms and ionization; in the case of small electron concentrations this neglect is valid for the total power leaving the plasma \(^{349}\).

The power given up outward by a nonisothermal plasma is divided into three parts: transfer of kinetic energy, i.e. classical thermal conductivity, radiation of excitation energy, and transfer of ionization energy by means of ambipolar diffusion or ionic thermal conductivity.

The power received by the electrons of a nonisothermal plasma from the electric field may be represented, according to Section III, in the form

\[ N_e b_e E^2, \]

where \(b_e\) is the mobility of the electrons, and \(E\) is the electric-field strength. Consequently,

\[ N_e \cdot b_e \cdot E^2 = L_{\mathrm{изл}} + L_{\mathrm{ион}} + L_{\mathrm{кин}} . \tag{9c, 1} \]

\(L_{\mathrm{ион}}\) appears at low pressures as the heat of recombination of ions and electrons recombining on the wall; \(L_{\mathrm{кин}}\) appears as heating of the gas caused mainly by elastic collisions—

Table 7 (for VIII, c and IX, b)

	\(v_{-}\), cm/sec	Number of excitation events per 1 sec. per \(1\ \mathrm{cm}^{3}\) \(S^{1}\))	Number of excitation events per 1 sec. per \(1\ \mathrm{cm}^{3}\) \(S^{1}\))	Number of ionization events per 1 sec. per \(1\ \mathrm{cm}^{3}\) \(S^{2}\))	Number of ionization events per 1 sec. per \(1\ \mathrm{cm}^{3}\) \(S^{2}\))	Number of spontaneous emission events per 1 sec. per \(1\ \mathrm{cm}^{3}\)	Number of acts of free recombination per 1 sec. per \(1\ \mathrm{cm}^{3}\) \(S^{3}\))	Number of acts of free recombination per 1 sec. per \(1\ \mathrm{cm}^{3}\) \(S^{3}\))
I	\(\approx 10^{8}\)	\(10^{-8}\)	\(1.8\cdot 10^{16}\)	\(10^{-10}\)	\(1.8\cdot 10^{14}\)	\(1.8\cdot 10^{16}\)	\(10^{-13}\)	\(10^{7}\)
II	\(\approx 5\cdot 10^{6}\)	\(5\cdot 10^{-10}\)	\(1.8\cdot 10^{11}\)	\(5\cdot 10^{-12}\)	\(1.8\cdot 10^{9}\)	\(1.8\cdot 10^{11}\)	\(5\cdot 10^{-15}\)	\(5\cdot 10^{4}\)
III	\(\approx 6\cdot 10^{7}\)	\(\cdot 1-\)	\(1.1\cdot 10^{20}\)	\(6\cdot 10^{-11}\)	\(1.1\cdot 10^{18}\)	\(10^{20}\) \(^{5}\))	\(6\cdot 10^{-14}\)	\(6\cdot 10^{12}\)
IV	\(\approx 9\cdot 10^{7}\)	\(9\cdot 10^{-9}\)	\(10^{21}\)	\(9\cdot 10^{-11}\)	\(10^{19}\)	up to \(10^{21}\) \(^{5}\))	\(9\cdot 10^{-14}\)	\(2.3\cdot 10^{12}\)
V	\(\approx 4\cdot 10^{7}\)	\(4\cdot 10^{-9}\)	\(2\cdot 10^{24}\)	\(4\cdot 10^{-11}\)	\(2\cdot 10^{22}\)	\(10^{22}\)	\(4\cdot 15^{-14}\)	\(4\cdot 10^{14}\)
VI	\(\approx 4.5\cdot 10^{7}\)	\(4\cdot 10^{-9}\)	\(2\cdot 10^{24}\)	\(4\cdot 10^{-11}\)	\(2\cdot 10^{22}\)	\(10^{22}\)	\(4\cdot 10^{-14}\)	\(4\cdot 10^{16}\)
VII	\(\approx 4.5\cdot 10^{7}\)	\(4.5\cdot 10^{-9}\)	\(2\cdot 10^{26}\)	\(4.5\cdot 10^{-11}\)	\(2\cdot 10^{24}\)	\(10^{23}\)	\(4.5\cdot 10^{-14}\)	\(4.5\cdot 10^{18}\)
VIII	\(\approx 5\cdot 10^{7}\)	\(5\cdot\ 0-\)	\(10^{28}\)	\(5\cdot 10^{-11}\)	\(10^{26}\)	\(10^{24}\)	\(5\cdot 10^{-14}\)	\(5\cdot 10^{20}\)

1) The value of the effective cross section for excitation is taken to be \(10^{-16}\ \mathrm{cm}^{2}\).

2) The effective cross section for ionization is \(10^{-18}\ \mathrm{cm}^{2}\).

3) The effective cross section for free recombination is \(10^{-21}\ \mathrm{cm}^{2}\).

4) At very low electron velocities the effective cross section will be greater than the adopted value.

5) The observed quantities are smaller because of reabsorption.

by collisions of electrons with atoms^487. The energy balance in a low-pressure plasma was investigated mainly by Zommermeyer (Ne^487,488), Druyvesteyn (Na^95,98,100), and Möller (Cs^358). A sharp dependence was found of the relative magnitude of the three terms on the right-hand side of equation (9c, 1) on the discharge parameters: pressure, tube diameter, and current strength. At

Table 8

	Number of atoms in the normal state, in \(1\ \mathrm{cm}^3\)	Number of excited atoms in \(1\ \mathrm{cm}^3\)	Number of ions in \(1\ \mathrm{cm}^3\)	Number of dissociated atoms in \(1\ \mathrm{cm}^3\)	Ionization temperature	Mean radiation per \(1\ \mathrm{cm}^3\), in watts^1)
I	\(1.8\cdot 10^{14}\)	up to \(\simeq 10^8\)^2)	\(1\cdot 10^{10}\)	\(\simeq 0\)	3000	up to \(\simeq 10^{-3}\)
II	\(\leqslant 3.6\cdot 10^{14}\)	\(\simeq 10^3\)	\(\leqslant 10^6\)	\(\simeq 0\)	2500	\(10^{-8}\)
III	\(1.8\cdot 10^{15}\)	\(\simeq 10^{12}\)	\(1\cdot 10^{13}\)	\(\simeq 0\)	6500	\(10^2\)
IV	\(3.7\cdot 10^{16}\) to \(1.8\cdot 10^{17}\)	up to \(\simeq 10^{12}\)^2)	\(5\cdot 10^{12}\)	\(\simeq 0\)	7500	up to \(\simeq 100\)^3)
V	\(5\cdot 10^{18}\)	\(\simeq 10^{14}\)	\(1\cdot 10^{14}\)	\(\simeq 5\cdot 10^{18}\)	\(\simeq 6500\)	\(\simeq 400\)
VI	\(5\cdot 10^{18}\)	\(\simeq 10^{14}\)	\(1\cdot 10^{15}\)	\(\simeq 5\cdot 10^{18}\)	\(\simeq 6800\)	\(\simeq 500\)
VII	\(5\cdot 10^{19}\)	\(\simeq 10^{15}\)	\(1\cdot 10^{16}\)	\(\simeq 5\cdot 10^{18}\)	\(\simeq 7500\)	\(\simeq 5000\)
VIII	\(5\cdot 10^{20}\)	\(\simeq 10^{16}\)	\(1\cdot 10^{17}\)	\(\simeq 5\cdot 10^{18}\)	\(\simeq 8000\)	\(\simeq 50\,000\)

^1) According to equation (9b, 6).
^2) In the case where equilibrium with the electron temperature is established (see IX, b).
^3) The observed values are 100–1000 times smaller owing to the decrease of \(A\), caused by reabsorption (see IX, b).

constant tube diameter \(L_{\text{ion}}\) prevails if the pressure is small; at higher pressure \(L_{\text{kin}}\) prevails. The influence of the current strength, which determines the discharge characteristic, was discussed by Rompe and Schen^441,442.

For pressures from 1 to 8 atm in Hg, Elenbaas^111–113 proposes the following experimental formula for the total radiation:

\[ \mathfrak{S}=f(L-A), \tag{9c, 2} \]

where \(A\) is the fraction of energy leaving the plasma by classical thermal conduction; \(L\) is the total energy given up by the plasma, and \(f\) is a constant which, for example, for mercury is equal to 0.72. Recently it was pointed out^439 that (9c, 2) can also be written in the form

\[ \mathfrak{S}=L-(1-f)L-fA; \]

the term \((1-f)L\) has the meaning of that part of the energy which leaves the plasma by ionic thermal conduction. The fact that the energy balance proceeds according to (9c, 3) is also indicated by a number of further observations; it may be accepted that, for high pressures as well, a three-term formula of the type (9c, 1) is suitable. The term \((1-f)L\) decreases with increasing pressure and constant temperature^111.

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1. The duration of oscillations of the microfield (see Section II) is a quantity of the same order as the collision time (\(10^{-12}\)—\(10^{-15}\) sec.); therefore we believe that the influence of electrons and ions on line width can be described with greater success by the “impact damping theory” than by a “statistical” theory. Such a theory was constructed by Holtsmark 217—219; in it the assumption of very slowly varying fields was taken as the basis. The statistical theory can give an adequate description of the phenomena only for the case of a non-quasineutral plasma with a considerable excess of ions. ↩↩↩↩

2. The duration of a collision \(\tau\) is given by the time during which the colliding particles interact. In order of magnitude it is equal to \(2\frac{R}{v}\), where \(R\) is the effective radius (see Section VIII, c) and \(v\) is the relative velocity. Usually the values of \(\tau\) lie between \(10^{-12}\) and \(10^{-15}\) sec. ↩↩