Hugh E. DeWitt

Strongly coupled plasma physics

A NATO Advanced Research Workshop on Strongly Coupled Plasma Physics was held on the Santa Cruz Campus of the University of California, from August 4 through August 9, 1986. It was attended by 80 participants from 13 countries, 45 of whom were invited speakers. The present volume contains the texts of the invited talks and many of the contributed papers. The relative length of each text is roughly proportional to the length of the workshop presentation. The aim of the workshop was to bring together leading researchers from a number of related disciplines in which strong Coulomb interactions play a dominant role. Compared to the 1977 meeting in Orleans-la-Source, France and the 1982 meeting in Les-Houches, France, it is apparent that the field of strongly coupled plasmas has expanded greatly and has become a very significant field of physics with a wide range of applications. This workshop had a far greater participation of experimental researchers than did the previous two, and some confrontations of real experiments with theoretical calculations occurred. In the two earlier meetings the theoretical presentations were dominated by numerical simulations of static and dynamic properties of various strongly coupled plasmas. The dearth of experiments in the 1970s is now replaced by some very good experimental efforts.

Evaluation of the Quantum‐Mechanical Ring Sum with Boltzmann Statistics

A procedure is given for the evaluation of the quantum‐mechanical ring sum at finite temperature. The method is used for the evaluation of the quantum corrections to the classical Debye‐Huckel free energy for an electron gas obeying Boltzmann statistics. The ring sum is shown to be of the form (βe2/πλD)P(γ), where γ=ƛ/λD,ƛ=ℏ/(2mkT)1/2, and λD is the Debye screening length. The quantum effects for finite γ are due only to the operation of the uncertainty principle. The function P(γ) decreases monotonically from the classical value π/3, and the form is shown to be P(γ)=(π/3)[1+ ∑ n=2anγ2(n−1)]3/2− ∑ n=2bnγ2n−3. The coefficients an and bn are evaluated exactly for small n and asymptotically for large n. The two series converge for γ2 < γc2 = 2.042 ….For γ ≫ γc the function P(γ) is also evaluated as an asymptotic expansion in inverse powers of γ1/2. Thus the low‐temperature correlation energy of distinguishable electrons is obtained in random phase approximation. At zero temperature, the result is the same as...

Statistical Mechanics of High‐Temperature Quantum Plasmas Beyond the Ring Approximation

A quantum mechanical perturbation expansion of the partition function is used to evaluate the free energy of the electron gas and multicomponent plasmas to logarithmic accuracy in the particle number density, thus including the next important contribution beyond the ring approximation. The quantum generalization of Abes work on the classical electron gas is given for the ladder interactions with the dynamic screened Coulomb potential, and each ladder is shown to be separately finite because of the finite size of the wave packets describing point electrons [of the order of the thermal de Broglie wavelength ƛ = h(β/2m)1/2]. The results show that quantum effects due to the uncertainty principle persist at high temperature, and that when kT > Ryd plasmas are quantum systems, rather than classical, because ƛ is greater than the average distance of closest approach, βe2. Results are also obtained for the Wigner‐Kirkwood wave mechanical diffraction corrections to the classical electron‐gas free energy which ar...

Low density expansion of the equation of state for a quantum electron gas

Abstract Using the method of thermodynamic Green functions, a low density expansion of the equation of state for a quantum electron gas is derived up to order ( ne 2 ) 5/2 . This result is obtained staring from a generalized Montroll-Ward formula.

ANALYTIC PROPERTIES OF THE QUANTUM CORRECTIONS TO THE SECOND VIRIAL COEFFICIENT

The usefulness of the perturbation expansion and the Wigner‐Kirkwood expansion of the quantum‐mechanical partition function is discussed for various interaction potentials. It is shown that, contrary to what is expected from the Wigner‐Kirkwood expansion, quantum‐mechanical diffraction corrections at high temperature to the classical partition function may involve nonanalytic forms of ℏ2. This occurs when the second‐order perturbation term is finite in the classical limit, and the interaction potential has a cusp or singularity in any derivative. The second‐order perturbation term is evaluated exactly for the exponential, screened Coulomb, and square barrier potentials, and the nonanalytic form (ℏ2)½ is found. For potentials more singular than 1/r at the origin, the diffraction corrections are analytic in ℏ2.A new method of deriving the Wigner‐Kirkwood expansion from the perturbation expansion is given. The method allows one to subtract off any order of the perturbation expansion which may be evaluated se...

Derivation of the one component plasma fluid equation of state in strong coupling

Abstract A variational calculation of the one component plasma energy using the hard sphere Percus-Yevick g(r) and the virial entropy gives U/NkT = aΓ + bΓ 1 4 + c + d/Γ 1 4 + ... in agreement with the empirical fit to Monte Carlo data.

Equation of state of the weakly degenerate one-component plasma

Starting from a general quantum statistical formula for the pressure in terms of thermodynamic Greens functions, different contributions to the equation of state (EOS) for the one component plasma (OCP) are given analytically and by numerical calculations. Exact results for the EOS of a quantum electron gas are presented in the shape of a low density expansion up to the order (ne2)52 including ladder type contributions and “beyond Montroll-Ward” terms.

Phase shift contributions to the partition function of a system of interacting particles

Abstract A complete internal partition function requires scattering (phase shift) contributions in addition to the usual bound state sum. This requirement is applied to the static screened Coulomb (Yukawa) potential to demonstrate phase shift effects.

Electrical conductivity of dense plasmas

Abstract A Chapman-Enskog-type solution to the kinetic equation of Gould and DeWitt has been used to calculate the electrical conductivity of argon and xenon plasmas. Reasonable agreement with experiment is obtained.

Evaluation of Some Fermi‐Dirac Integrals

The evaluation of Fermi‐Dirac integrals is discussed for cases in which the Sommerfeld method fails. Such cases occur when the integrand has a singularity at the Fermi surface and when the integrand is a rapidly oscillating function. As examples, the first‐order exchange integral for electrons and the free‐energy integral of the noninteracting electron gas in a magnetic field are evaluated. The method uses a contour‐integral representation of the Fermi function (previously mentioned by Dingle), supplemented by Mittag‐Leffler type expansions.