Table of contents

A brief survey is given of the different theoretical approaches to the study of the low-density, high-temperature `classical' plasma and of the high-density, low-temperature `quantum' plasma. The random phase approximation, collective-variables method is shown to yield the same result as the collisionless Boltzmann equation combined with the self-consistent field method. Dynamic behaviour is studied in the classical plasma, with particular attention to the relationship between individual particle and collective behaviour. The frequency and wave number dependent longitudinal dielectric constant of the plasma is used to establish the many similarities in behaviour of quantum and classical plasmas in the random phase approximation; the validity of the random phase approximation is discussed. Applications of the theory are made to the ground state energy of the quantum plasma and to the explanation of the characteristic energy losses in solids. The generalizations required to take into account the ionic motion are indicated and the results are described.

The properties of a degenerate electron gas with a background of positive charge capable of propagating phonons are studied by means of a quasi-particle approach. The ground state of the system is pictured as a `vacuum' state, and any additional particles or holes together with their polarization clouds are thought of as quasi-particles. We treat separately the Coulomb repulsion and the effective electron-electron attraction due to phonon exchange. The specific heat, spin susceptibility and compressibility are determined in terms of the effective energy of interaction V_pp' of two quasi-particles. Considering only the lowest order process contributing to V_pp' reproduces known results for the high density limit Coulomb correction to the specific heat and spin susceptibility, but gives a new result for the phonon correction. To lowest order it is found that the electron-phonon interaction produces no change in the spin susceptibility, in contrast to general expectation. For real metallic densities higher order graphs must be included, however for this case it may be possible to choose V_pp' empirically.

A brief account is given of Tomonaga's method of introducing collective co-ordinates. This method is applied to a derivation of the dispersion relations for plasma oscillations. The result is different from the one obtained by Pines and Bohm and possible reasons for this difference are given. Finally we sketch Jepsen's method for deriving the hydrodynamic equations of motion.

The formalism of GLASSGOLD, HECKROTTE and WATSON for the expansion of the quantum-mechanical grand partition function has been applied to a gas of point charges. The expansion is shown to be closely related to Rayleigh-Schroedinger perturbation theory, and to give the results of MONTROLL and WARD in a simpler form. The theory has been extended to a multicomponent gas of fermions and bosons and the ring diagrams have been summed to give an approximate expression for the equation of state valid for the entire temperature range. The near-classical limit (high temperature and low density so that the gas is only slightly degenerate) is discussed in detail, and useful formulae for numerical computation of pressure and internal energy are derived from the general ring approximation to the equation of state. By expanding the chemical potential in powers of the coupling constant e², it is possible to eliminate the parametric dependence of pressure and density on the chemical potential and obtain the pressure as a function of density. The classical-limit Debye-Hückel results are obtained when h approaches 0. A general form of the screening length is obtained with the effective screening charge as z_ieθ_i, where θ_i is a measure of the degeneracy of the particle species i.

The contribution to the pressure from the simplest exchange interaction valid for all temperature is evaluated exactly.

The temperature and density range for which the near-classical-limit formulae are valid is discussed.

The interaction of a heavy mass positive plasma (mass M) with a light mass negative plasma (mass m) is examined in the random phase approximation. The problem is solved in this approximation by a normal mode analysis, leading to a dispersion relation which is analysed in detail. In the limit of low momentum transfer q, the results are identical to those of Bardeen (1937), Bohm and Staver (1952) and Bardeen and Pines (1955) corresponding to Coulomb interaction screened out with the Fermi-Thomas screening factor. Phonons arise with c = √(m/3M) v_f; the plasma frequency is √(4πne²/μ) where μ is the effective mass; v_f = the Fermi velocity; n = the electron density; c = the sound velocity. The phonon decays into electron-hole pairs with a relaxation time given by conventional perturbation theory using a screened electron-ion interaction. The screening arises in terms of a renormalized coupling constant, using a conventional field theoretic analysis.

At high-momentum transfer (q > 2p_f), the phonon is stable (i.e. the imaginary part of the dispersion relation as a function of complex variable vanishes) and the analytic character of the spectrum changes. This effect, as pointed out by Kohn (1959) in another context, can lead to determination of the Fermi surface in metals. Its pertinence to the present theory was pointed out by B. Knight.

Collective oscillations for an infinite system of fermions are discussed for the general case where the force between two particles depends on the internal variables as well as on their relative separation and momentum. Two theories are reviewed, one by Watson, Heckrotte and Glassgold, which is based on Sawada's treatment of the electron gas, and the other, a semiclassical treatment by Landau.

The characteristic electron energy loss spectra of aluminium and magnesium have been measured by analysing the energy distribution of 750, 1000, 1500 and 2020 eV electrons scattered by evaporated specimens through 90°. Measurements have also been made of the loss spectra of aluminium-magnesium alloys of unknown composition using a primary electron energy of 1500 eV. It was found that the loss spectra of both elements were similar in that they were composed entirely of combinations of two elementary energy losses, 10.3 and 15.3 eV in aluminium and 7.1 and 10.6 eV in magnesium. The alloy spectra were observed to consist of combinations of a loss that varied between 10.6 and 15.3 eV in different specimens, and a low-lying loss that varied between 7.1 and 10.3 eV. From measurements of the relative positions and intensities of the two fundamental losses in each element, the observed changes in position and intensity of the low-lying loss in very thin films of aluminium and the changes in position of the two fundamental losses in the alloys, it was concluded that the larger fundamental loss in each specimen was due to plasma excitation and that the smaller loss was a lowered plasma loss of the type proposed by Ritchie.

This paper, and the succeeding one, will concentrate on certain fields in which we have experience, which seem worthy of further exploration and development. The main topics dealt with will be: the significance of gas purity and the molecular gas problem; utilization of gross unresolved appearance of a plasma and the necessity for distinguishing between steady and oscillating plasmas, illustrated by reference to terminal conditions in moving striations; noise and turbulence; a possible quantal limitation imposed on time resolution by the Einstein coefficients for allowed and forbidden transitions; formulation of the excitation integral; further examples of spectroscopic techniques, including use of interference filters, study of potential in sheaths, and a check on the voltage amplitude of so-called plasma electron oscillations.

Subjects dealt with include the classification of probes; general remarks on metal (Langmuir probes, and the construction of probes for volume exploration; the Langmuir-Druyvesteyn method for determining isotropic distributions, its theory, the problem of variable reflection coefficients, and Boyd's development of techniques; exploration of oscillating plasmas; effects of negative ions on plasma fields and probe currents, and on the stability of discharges.

In working on the theory of electron beam devices such as travelling-wave tubes and klystrons, it has been necessary to consider a number of knotty problems, some at least of which occur in mathematical formulation of other plasma problems. Among these is the matter of multi-velocity flow. Here there are two mathematically equivalent approaches. In one, the electron flow is divided into streams according to the initial unperturbed velocities of the electrons; the variables are the densities and velocities in the various streams. In the other, velocity is regarded as a co-ordinate of phase space and the variable is the density in phase space. These approaches give different appreciations of the phenomena involved and lead to different mathematical difficulties, but are the same in content. Each demonstrates the non-existence of wave-type solutions in many infinite velocity distributions. Only a few specialized solutions of multi-velocity problems exist.

As different velocity co-ordinates can be used, so different spatial co-ordinates can be used. The Eulerian approach is common and leads to no difficulties if boundary conditions are met at fluctuating boundaries. In the case of thin beams, it is sometimes useful to use displacements from the mean position of the particles as variables. Some workers have made mistakes by mixing co-ordinate systems.

In dealing with power flow in electron beams, one can either replace the actual physical system with a linear system and find an expression for power in this system, or try to deal with the power flow in the true non-linear system. The former alternative is much simpler. Kinetic power, as well as electromagnetic power, is important in considering the orthogonality of wave-type components.

In solving an actual physical problem, one can either assemble various wave-type components so as to satisfy the boundary conditions, or solve the problem by transform or perhaps by other means.

Components of the solution must be regarded as merely a part of the solution of an actual problem, but they can sometimes be given a reasonable physical interpretation. Thus, one finds waves with positive and negative powers. When two unattenuated waves having powers of the same sign are coupled together, one observes beats. When two unattenuated waves having powers of opposite signs are coupled together, one observes growing and decaying waves. Growing waves can also be produced when a moving discontinuity couples two unattenuated waves together (parametric amplification).

During ohmic heating of plasma confined in a Stellarator it is observed that the plasma moves across the magnetic confining field at a much higher average velocity than is predicted by classical considerations. An attempt has been made to compare the observed velocity with the calculated drifts that should result from the measured randomly varying electric fields and density gradients present in the plasma. It is found that the random quantities, as measured by double probes, are of sufficient intensity to give order-of-magnitude agreement between observed and calculated velocities. A new hypothesis is put forward to explain the origin of these high-amplitude fluctuations, a hypothesis which might be applicable to gas discharges generally. It is assumed that the non-Maxwellian electron distribution resulting from the applied electric field excites large amplitude, coherent plasma oscillations near the plasma frequency. These large amplitude oscillations act as the `pump' in the non-linear process known as parametric amplification. Thermal fluctuations, normally present with small amplitudes in quiescent plasmas, are amplified over a wide band of frequencies, both above and below the pump frequency, by the resulting parametric amplifier action of the plasma. This results in high-amplitude noise which might produce enhanced diffusion and rapid thermalization effects.

Experiments with ion cyclotron heating of a deuterium plasma confined in the B-65 stellarator are reported. Single-particle type behaviour is evidenced by the production of neutrons, ascribed to deuterons accelerated in the low density region beneath the induction coil but at the outside of the plasma, at a value B₀ of confining magnetic field corresponding to single particle resonance. Ion cyclotron wave behaviour is indicated by the emission of light from deuterium and from several times ionized impurity elements, the light intensity peaking at a value of the magnetic field about 10 per cent larger than B₀. Maximum absorption of the ion cyclotron wave energy is calculated to occur in a cylindrical shell where the ion density is about 10¹³ cm^-3, and qualitative support is given experimentally by the emitted light profile. Complete ionization of the deuterium plasma may be obtained using r.f. induction heating alone, with the highest electron temperatures occurring for magnetic fields corresponding to ion cyclotron wave resonance. Operation of the B-65 machine as a magnetic mirror confinement device with ion cyclotron heating led to neutron production, induction coil loading, emission of deuterium and impurity light similar to stellarator operation.

Interaction between a slow space charge wave which travels along a cylindrical plasma column and an electron beam which passes down the axis of the column is demonstrated experimentally. A spatially growing wave exists when the velocity of the beam is approximately equal to the velocity of the unperturbed wave.

In their Geneva paper, Trubnikov and Kudryavtsev calculated the cyclotron radiation from a hot plasma. In doing this, the approximation was made that the individual particles radiated as though they were in a vacuum. We have investigated this approximation by calculating the absorption length directly from the Boltzmann equation and we find that indeed this approximation is correct whenever ω_p²/Σ² << n² where n is the harmonic number of the radiation in question, ω_p is the plasma frequency, and Σ is the cyclotron frequency. For a contained plasma, the left-hand side of this inequality is of the order of magnitude of one and thus the inequality is well satisfied for the dominant radiation from a plasma at high temperature.

The physical reason for this inequality can be investigated by solving the test-charge problem to find the transverse response current of the plasma to the motion of an electron. The result is that the transverse organizing length in the plasma is c/ω_p, and thus one may picture an electron in its Larmor orbit as being surrounded by a co-moving current cloud of radius c/ω_p. Classical electrodynamics then leads to the above inequality.

The cyclotron emission from a hot, completely ionized, magnetically-confined plasma has been estimated by computing the absorption of an incident plane wave. The harmonics of the fundamental cyclotron frequency, which are emitted perpendicular to the magnetic field direction, were summed over and also treated individually. Because of the Doppler effect and the relativistic variation in mass the behaviour of the electrons and the electro-magnetic properties of the medium are functions of electron velocity. Assuming a Maxwell-Boltzmann distribution in electron velocity, the polarization of the plasma was computed by integrating over electron velocity. Solution of Maxwell's equations yielded the optical constants of the plasma and thereby the absorption of an incident wave. By invoking Kirchhoff's relation, the emission of the plasma was then determined.

The concept of conductivity in a hot plasma is reviewed and criticized. In particular, it is shown that even for arbitrarily small field strengths, a proportionality constant (tensor) does not in general exist between a frequency component of the field strength and the corresponding component of current density. Rather, insisting upon such a proportionality for a homogeneous plasma, defines a complete set of spatial field configurations (eigenfunctions of the conductivity). Each of these eigenfunctions separately satisfies Maxwell's equations. However, for an inhomogeneous plasma, even this is not possible since the eigenfunctions of conductivity are not separately solutions of Maxwell's equations.

It is possible, however, to define an intrinsic high-frequency property of an inhomogeneous plasma. This does not relate current density to field strength but, rather relates the field strength produced by the current density to the field strength causing the current density. Both Maxwell's equations and the particle dynamics are represented in this new property of the inhomogeneous plasma. Equating the produced and causing field strengths again defines eigenfunctions in terms of which the solution for arbitrary boundary conditions can be found. Examples will be given.

A short discussion is presented of some experimental work in the field of plasma physics in progress at CERN.

It has been shown theoretically that a charged particle can be bound in stable orbits about the axis of a circular wave guide by the fields of the TE₀₁ mode at cut-off (Weibel, 1959). This result was verified experimentally. The apparatus consists of a 20-cm long section of a circular wave guide - properly terminated - into which is fed radio-frequency power from a magnetron at f = 9.29 kMc/s in 2 μsec pulses. A uniform TE₀₁ field is excited over the entire 20-cm length of the guide. An electron beam which is injected axially at one end of the cavity diverges strongly due to coulomb repulsion if no radio-frequency power is applied. During the radio-frequency pulse, however, the beam is confined to the axis and 90 per cent of the current is collected by an axial probe of 0.3-cm diameter which is located at the other end of the guide. Between pulses when no radio frequency is applied, no measurable current reaches the probe. In its trajectory from the gun to the probe each electron spends about 170 radio-frequency periods in the focusing field.

This paper will give a survey of some experiments on the kMc/s oscillations, believed mainly longitudinal, generated when electron beams traverse a plasma, touching on the evidence they provide for the mechanism of collapse of directed motion and its bearing on astrophysical and heavy-current problems and the probable importance of electron concentration gradients evidenced from Wehner's and our experiments.

Collective interactions can provide a rapid mechanism for restoring grossly non-Maxwellian distributions to near-Maxwellian in that electrodynamic instabilities will build up to high level within a few plasma periods. Computation of the further development of the instabilities in the non-linear regime shows an approach to statistical randomness.

Some forms of electron flow are unequivocally unstable in the sense that perturbations grow with time. Excessive currents between grids or through tubes in the presence or absence of neutralizing positive charge provide examples. Such instabilities are not associated with wave components which grow with distance for real frequencies. Thin beams in magnetic fields also show instability under the action of space charge forces. Tubes sometimes exhibit gain in the absence of any wave component which exhibits spatial growth for real frequencies.

There is a large class of amplifiers whose operation can be explained in terms of spatially growing waves. The physical mechanism can be variously described as instability in a moving co-ordinate system, as a result of a negative dielectric constant at the frequency of operation which causes electrons to attract rather than to repel one another, as a result of coupling between positive energy and negative energy waves, and in other terms.

In the easitron, an inductive wall results in growing waves. In the resistive wall amplifier, the abstraction of power from negative energy waves results in growth. In the travelling-wave tube amplifier, a positive power circuit wave is coupled to a negative power space charge wave. In the double stream amplifier, the negative power space charge wave of the faster stream is coupled to the positive power space charge wave of the slower stream. In velocity jump and rippled stream amplifiers, periodic discontinuities couple the positive power and negative power waves of the same stream. In wave-type parametric amplifiers, moving periodic discontinuities couple two unattenuated waves.

The explanations listed are convenient in connexion with the particular device, rather than unique.

The general dispersion relation is derived for small amplitude waves in a fully-ionized plasma in an external magnetic field. This derivation is based on the Vlasov equations (Boltzmann equations without collision terms for the electrons and ions plus Maxwell's equations). The dispersion relation involve integrals over the zero-th order velocity distributions. It is found that for sufficiently anisotropic velocity distributions waves exist which have exponentially growing amplitudes. A number of special cases are discussed. It is shown that streams of charged particles passing through a plasma may excite either longitudinal or transverse waves. Other instabilities exist when the distributions are such that the particles have their velocity vectors perpendicular to the magnetic field.

It is pointed out that interpenetration of two plasma streams along a uniform magnetic field is prevented by the hose instability. Thus the shock thickness resulting from supersonic streaming of a tenuous plasma along a uniform magnetic field may be very much less than the collision length. We suggest that an actual stationary shock front under these conditions is formed by the hose interaction between the incoming supersonic stream and a precursor, consisting of ions evaporating forward from the non-linear disordering which results when the hose instability reaches large amplitude. Treating the growth of the hose instability with the WKB approximation applied to the linearized equations, we show that the scale of the shock front is characterized by the ion and electron Larmor radii, and that the density in the precursor relaxes to a small but non-vanishing value, proportional to one over the square of the Mach number, infinitely far ahead of the shock.

Of prime importance in plasma investigations is the knowledge of the interaction of a bounded plasma with external magnetic fields. The present paper concerns itself with the general properties of the radiation due to plasma oscillations of bounded plasmas of single geometries situated in a strong magnetic field. The special problems of transmission, reflection and scattering due to plasma slabs and columns are also discussed as well as the problems of the response of the plasma to near fields in the vicinity of its surface. The effects of thermal spread in the particle velocity distribution are included in the presentation.

The effect of non-linear terms in the dynamical equations governing wave propagation in plasmas may be analysed by a perturbation procedure which is acceptable for amplitudes which are not too large. The Hamiltonian describing the complete system is separated into two parts: the quadratic part which yields the linearized equations and the non-linear part. The quadratic part may be eliminated by a normal-mode analysis, the `normal modes' comprising travelling waves. The non-linear part then results in interaction between these waves.

Two theorems concerning wave interaction are proved. The first relates energy-transfer between a group of interacting waves to the frequency of these waves. These `action-transfer relations' lead to the Manley-Rowe relations for steady-state or quasi-steady-state configurations. The second theorem relates the frequency-displacements of a group of interacting waves to the energies of these waves.

The properties of electron plasmas undergoing longitudinal oscillations are re-examined in the light of the preceding theorems. Interaction terms may be classed as `coherent' and `incoherent': the former do not result in energy transfer but only frequency displacement which may be characterized by a dispersion relation. The second group leads to transfer of energy between waves and hence to spectral decay.

The interaction between longitudinal (electrostatic) and transverse (electromagnetic) waves in plasmas is considered and it is shown that in a uniform plasma in the absence of magnetic fields, the dominant interaction couples two longitudinal waves with one transverse wave. Hence one would anticipate that the dominant non-linear mechanism for radiation from excited plasmas leads to emission at twice the plasma frequency.

Experiments designed to study the production of a steady-state plasma column by microwave cavity means are described. At low plasma densities electrons are heated by cyclotron resonance in crossed microwave electric and static magnetic fields. Phenomena associated with large energies which the electrons possess near cyclotron resonance are discussed. Large plasma densities are achieved by resonating the plasma column by suitably varying the static magnetic field, the microwave frequency and the input power. In this manner, densities of the order of 10¹² cm^-3 are obtained at a neutral gas pressure in the micron range.

After a short presentation of the general theory of irreversible processes established by Prigogine and Balescu, we discuss in some detail its application to the problem of long range forces in a plasma. Our diagram technique permits an unambiguous choice of the main contributions to the distribution function. The result is an equation describing the evolution of the distribution function of momenta, which takes rigorously into account the many-body collisions. The latter results from a summation of an infinite number of diagrams, the structure of which is very simple. The successive diagrams involve more and more particles interacting simultaneously.

The equation has a formal similarity with the usual Fokker-Planck equation, but the Coulomb interaction between two particles is replaced by an effective interaction, which depends on the distribution of all the other particles of the medium. At very low temperatures this effective interaction reduces to a screened Coulomb-Debye potential.

The relation of our approach to the previous theories of plasmas (Fokker-Planck equation, Vlasov equation, random phase approximation) can be discussed very clearly by means of the diagrams.

In Part 1 we discuss the present state of the theory of the propagation of small amplitude disturbances in monatomic gases and plasmas. The analysis is based on kinetic equations for one-particle velocity distributions involving self-consistent field forces and Boltzmann-like collision terms. In Part 2 a microscopic formulation of the classical N-body problem is developed. The theory is used to analyse the meaning of earlier microscopic approaches, and carried to the point where the problem of Part 1 can be treated in a more fundamental way.

The motion of a charged particle, including the effect of its self field is treated in the frame of classical mechanics by the Liouville method, developed recently by the author and his co-workers mainly for statistical mechanical problems. The author summarizes the results which he and Leaf has obtained in this way for the following cases:

1. free particle in vacuum

2. free particle in a black body

3. harmonic oscillator.

In every case the partial differential equation giving the evolution of the particle phase distribution function is established.

A review is given of some applications of thermodynamics of irreversible processes to systems in the presence of an electromagnetic field. The conservation laws of mass, momentum and energy and the entropy balance equation are developed for mixtures without and with electric and magnetic polarization, and also for systems in which the momentum transfer between some of their components is inhibited.

Starting from the Liouville equation, a chain of equations is obtained by integrating out the co-ordinates of all but one, two-etc. particles. One particle is singled out. All other particles are considered to be initially in thermal equilibrium. For the time evolution of the distribution function of the `singled out' particle an equation is obtained whose asymptotic form is of the usual Fokker-Planck type. It is characterized by a frictional drag force that slows the particle down and a fluctuation tensor that speeds it up and produces diffusion in velocity space. The objective of the calculation is to determine these quantities for a plasma consisting of electrons and protons in a constant external magnetic field.

The chain of equations contains two dimensionless parameters λ = ω_c/ω_p and g = l/nL_D³. A solution for the s-body correlation function is obtained in the form f_s = f_s⁽⁰⁾(λ) + gf_s⁽¹⁾(λ) + etc. f_s⁽⁰⁾ and f_s⁽¹⁾ have been determined to all orders of λ.

The frictional drag consists of a part due to collisions and a part due to plasma wave emission. When λ=1 the modification of the collisional part due to the magnetic field is negligible. There is a significant change in the properties of plasma waves of wavelength greater than the Larmor radius which modifies the force due to plasma wave emission. When λ=1 the force due to plasma wave emission disappears. The collisional force is altered to the extent that the maximum impact parameter is sometimes the Larmor radius instead of the Debye length, or something in between. A more interesting modification obtains for the particular case of a slow ion moving perpendicular to the field. It is due to repeated collisions with fast electrons. This collisional force is of a qualitatively different form, but the quantitative modification is not large. In addition to the drag force anti-parallel to the velocity of the particle, there is a collisional force anti-parallel to the Lorentz force. This force arises because the particle and its `shield cloud' are spiralling about field lines. The force on the particle is equal and opposite to the centripetal force acting on the `shield cloud.' It is much smaller than the Lorentz force.

The main result is that to the lowest order in g, the frictional drag and fluctuation tensor are slowly varying functions of λ.

Utilizing the Montroll-Ward approach to quantum statistics, generalized to many components, we seek an equation of state for a high-temperature low-density plasma. We propose a classification of diagrams analogous to that used by Meeron and by Friedman in classical statistical mechanics. In the resulting expansion, the leading term is the ideal gas contribution plus the lowest-order exchange contribution plus the ring contribution, the latter representing the effect upon the pressure of collective motions of a completely ionized plasma. The next term appears to represent a contribution due to modifications of free particle motion due to the interaction of the single particles with the rest of the plasma. The third term represents the contribution of two-particle states, both of positive energy between all particle pairs, and negative energy (bound states) between pairs of opposite charge sign. The third term also contains parts representing modifications of the motion of the pair of particles due to interaction of the individual particles with the rest of the plasma. Higher terms represent, successively, contributions of three-, four-, etc., particle states, again including modifications of the single-particle motions. The classification automatically eliminates the ultraviolet catastrophe which ordinarily arises in the treatment of Coulomb force bound states in statistical mechanics, since, in effect, it uses a screened Coulomb potential instead of the ordinary potential. In addition, the short range divergence, which occurs in classical theory because of the `fall' of the electron to the nucleus, does not arise, being prevented by the uncertainty principle.

The equations of the Debye-Hückel theory, modified to include quantum statistics, are discussed. It is found that the non-linear equations used by Cowan and Kirkwood are not unique, and that the non-linear theory can be formulated in different ways to give different answers. The linearized equations of these alternative formulations are discussed, and the correct form of the linearized theory is established. From the linear theory, the Helmholtz free energy of a slightly degenerate plasma is derived and, from this result, useful formulae in the near-classical limit are obtained for the pressure and internal energy.

A model for a partially-ionized, partially-dissociated plasma has been formulated using known theoretical concepts to describe both bound and free electron states, internal molecular degrees of freedom and Coulomb interactions. It has been applied to a system of particles arising from the hydrogen molecule. The Coulomb interaction is treated in the classical Debye approximation. However, a distance of closest approach between ions and electrons depending on the kinetic energy of the electrons is included to avoid the short range divergence of the Coulomb potential. The kinetic energy of the free electrons is calculated from the partition function for a perfect Fermi gas. The vibrational and rotational motion is treated in the harmonic oscillator and rigid rotor approximation with the number of energy levels counted for a given electronic state depending on the dissociation energy of the state. A volume dependence of the bound electronic energy eigenvalues is included by considering the effect of surrounding particles as a confinement of a given particle to a spherical box of variable size. For the counting of the bound electronic states, a given state is bound until its energy increases to zero due to confinement. From the partition function for the entire system, the free energy is calculated. By a minimization of the free energy of the system, the equilibrium composition as a function of temperature and volume is obtained. Then not only can thermodynamic quantities be calculated, but it is believed that a reasonable approximation to the correct balance of molecular, ionic and free electonic states is achieved over a wide range of V-T space. Consequently, regions where incomplete ionization and dissociation are important are delineated. In addition, for different regions of V-T space, the relative contributions of charged particle interactions, non-classical behaviour of electrons, internal degrees of freedom and translation to the total energy of the system can be determined.