Quantum Mechanical Second Virial Coefficient of a Lennard-Jones Gas. Helium

Abstract

The quantum-mechanical second virial coefficients of Lennard-Jones 3He and 4He gases with the De Boer parameters have been obtained over the complete temperature range from near absolute zero to the classical region. A formalism separating the virial into direct (Boltzmann) and exchange (spin and quantum statistics) contributions has been employed. The calculation is based on phase shifts except at the very highest temperatures where a Wigner–Kirkwood method has been used. Examination of the exchange term shows in detail the rapid suppression of the statistical effects with rising temperature, their contribution dropping to less than 0.001 cm3 by 7°K (4He). Comparison of the high-temperature (Boltzmann) results with those obtained by a third-order Wigner–Kirkwood expansion shows excellent agreement down to about 50°K for 4He and 60°K for 3He. The Wigner–Kirkwood expansion is shown to be unsuitable for determining the behavior of the exchange terms. Finally, results are compared with the available experimental data.

Quantum second virial coefficient calculation for the 4He dimer on a recent potential

“…This term is the most important one in establishing the second virial coefficient for the helium system. 22 …”

“…The classical second virial coefficient is not appropriate to describe the helium equation of state at temperatures lower than 100 K. 22 Instead, the exact quantum second virial coefficient expression relating the phase shift and the energy of bound state is used for an accurate system description. 6,23,24 The molar virial expansion is represented in the form (3) in which e is the He 2 binding energy for zero angular momentum, k B is Boltzmann's constant and (4) with  = 2.556 Å.…”

“…The results obtained in the present work will be compared to experimental data for the helium-helium system. [22][23][24] …”

“…[16][17][18][19][20] The most recent potential has not been tested against thermodynamic and transport property data, 21 such as the quantum second virial coefficients. In the present work, a study of this recent potential was conducted using second virial coefficient data at low temperature.The quantum virial coefficient can be determined by evaluating the quantum ideal gas term, the scattering phase shift dependence on angular momentum [22][23][24][25][26][27][28][29] and the bound states by Levinson's theorem. [25][26] This low-temperature second virial coefficient calculation is an important step to test the quality of the potential under consideration.…”

Erratum

“…Calculation of the second and third pressure virial coefficients In this paper two alternative ways were used to calculate the second virial coefficient of naturally occurring neon as a function of temperature T. In the first variant B(T) is determined like that of a mixture composed of the three neon isotopes: whereas the second virial coefficients B ij are evaluated fully quantum-mechanically for the different statistics using two contributions B direct and B exch [8]. In the Boltzmann statistics (B) the second virial coefficient is given as…”