Sigurd Yves Larsen

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

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 experime...

Exchange and Direct Second Virial Coefficients for Hard Spheres

We have separated the quantum‐mechanical second virial coefficient for hard spheres B into two terms. The first represents the contribution of a Boltzmann gas, and the second is an exchange term embodying the effects of quantum statistics. Numerical computation of B to high precision then allows us to analyze the temperature dependence of the exchange term, which is found to decrease exponentially with temperature, and to determine the asymptotic expansion of the Boltzmann term at high temperatures.

On the Validity of the Lorentz—Lorenz Equation Near the Critical Point

Corrections to the Lorentz—Lorenz equation for the refractive index of a fluid, due to long‐range density fluctuations, have been evaluated near the critical point, using Yvons statistical‐mechanical theory of the refractive index, together with the Ornstein—Zernicke asymptotic form of the two‐particle correlation function. For argon, with experimentally reasonable values for the numerical parameters, the correction is smaller than one part in 104.

KANTBP: A program for computing energy levels, reaction matrix and radial wave functions in the coupled-channel hyperspherical adiabatic approach

A FORTRAN 77 program is presented which calculates energy values, reaction matrix and corresponding radial wave functions in a coupledchannel approximation of the hyperspherical adiabatic approach. In this approach, a multi-dimensional Schrodinger equation is reduced to a system of the coupled second-order ordinary differential equations on the finite interval with homogeneous boundary conditions of the third type. The resulting system of radial equations which contains the potential matrix elements and first-derivative coupling terms is solved using high-order accuracy approximations of the finite-element method. As a test desk, the program is applied to the calculation of the energy values and reaction matrix for an exactly solvable 2D-model of three identical particles on a line with pair zero-range potentials. Program summary

On the Quantum‐Mechanical Pair‐Correlation Function of 4He Gas at Low Temperatures

The density‐independent part of the direct‐ and exchange‐correlation functions for 4He, using the de Boer and Michels Lennard‐Jones potential, have been calculated at 2°, 1°, 0.5°, and 0.1°K. The necessary wavefunctions were obtained by numerical integration of the Schrodinger equation. The peak of the direct‐correlation function is much less than its classical value and occurs at a slightly larger radius. The tail (large r) agrees very well with the form given by the Wigner—Kirkwood expansion. The exchange‐correlation function which represents the entire effect of the spin and statistics of the system vanishes very rapidly with increasing temperatures.

Quantum‐Mechanical Pair‐Correlation Function of Hard Spheres

The density‐independent part of the quantum‐mechanical direct and exchange pair‐correlation functions for hard spheres has been calculated for λT / σ = 1.4, 2, 2.93761, 4, 6, 8, and 10. In addition, points were obtained very near the surface of the sphere for λT / σ = 0.025 and 0.05. The results delineate the approach of the direct (Boltzmann) correlation function to the classical limit and emphasize the rapid disappearance of the effects of statistics with increasing temperature. These features are explained using Wiener (path) integrals. Analytical expressions valid near the surface of the sphere are derived and compare well with numerical results. Virial coefficients obtained from the correlation functions are in satisfactory agreement with very precise value calculated by Boyd, Larsen, and Kilpatrick.

The Ramsauer–Townsend effect in molecular systems of electron‐spin‐polarized hydrogen and helium and their isotopes

From arguments based on the De Boer quantum parameter and accurate phase shift calculations, we predict that the Ramsauer–Townsend effect occurs in numerous molecular systems involving electron‐spin‐polarized hydrogen and helium and their isotopes. Our work raises prospects of more precise determination of the two‐body potential parameters of these systems as well as exciting quantum behavior in their macroscopic states.

Thermodynamics of the Rigid Rotor at High Temperature

It is shown that Qe, the partition function for a rigid rotor summed over even levels and Qo, summed over odd levels, have exactly the same asymptotic (power series in σ=ℏ2/2IkT) expansion. No information as to differences in thermodynamic properties due to spin and statistics can be obtained from this expansion. The exchange partition function, Qe—Qo, is calculated directly and used to give simple expressions for the differences in thermodynamic properties between the para, ortho, and equilibrium cases.

Effect of a spurious component of the potential in Faddeev-type equations

In the integro-differential-equation approach (IDEA) and its associated hyperspherical-harmonic expansion method (HHEM) terms appear involving the potential for a single pair of particles. This introduces a spurious component in the potential that is eliminated in the Schrödinger equation by summing over all pairs. We calculate the contribution of this spurious potential in the solution of the HHEM for an even number of bosons in the ground state, using the Afnan-Tang S3 and Malfliet-Tjon MT5 interactions.

A Set of Hyperspherical Harmonics Especially Suited for Three-Body Collisions in a Plane

We obtain a set of four-dimensional hyperspherical harmonics in closed form. These harmonics are not only quantized with respect to the rotation group (O2), but are an irreducible basis for the permutation groupS3. An additional symmetry is found which allows us to write hyperspherical harmonics classified with respect to a 12 element groupS3×i×O2. We give a set of three mutually commuting operators whose eigenvalues uniquely characterize each spherical harmonic with respect to degree, symmetry, and angular momentum in the plane.