Quantum Corrections to the Square-Well Classical Second Virial Coefficient

1969 ◽  
Vol 186 (1) ◽  
pp. 262-264 ◽  
Author(s):  
Trygve S. Nilsen
Keyword(s):  
Virial Coefficient ◽  
Second Virial Coefficient ◽  
Quantum Corrections ◽  
Square Well

The analytical evaluation of the first-order density cluster integral in the radial distribution function

1970 ◽  
Vol 320 (1542) ◽  
pp. 323-348 ◽  
Keyword(s):  
Hard Sphere ◽  
Virial Coefficient ◽  
Second Virial Coefficient ◽  
Scattering Data ◽  
X Ray ◽  
Lennard Jones ◽  
X Ray Scattering ◽  
Cluster Integrals ◽  
Square Well ◽  
Three Body

It is shown how to evaluate the two-body, and three-body cluster integrals, ɳ 3 , ɳ * 3 , β 3 , β * 3 (equations (1.1) to (1.4)) for the hard-sphere, square-well and Lennard-Jones ( v :½ v ) potentials; the three-body potential used is the dipole-dipole-dipole potential of Axilrod & Teller. Explicit expressions are presented for the integrals ɳ * 3 , β * 3 using the above potentials; in the case of the first integral, its values for both small and large values of the separation distance are also given, for the Lennard-Jones ( v :½ v ) potential. Similar considerations have been carried out for ɳ 3 and β 3 , except that explicit expressions for the hard-sphere, and square-well potentials are not given, since these had been done before by other authors. The intermediate expressions for the four cluster integrals, are in terms of single integrals, and such expressions are valid for any continuous potential. Numerical results based on some of the expressions in this paper are compared with the results of numerical evaluation of the above integrals by other authors, and the agreement is seen to be good. Making use of the Mikolaj-Pings relation, the above results are used to obtain relationships between the second virial coefficient, and X-ray scattering data, as well as a means of deducing the pair potential at large separations, directly from a knowledge of X-ray scattering data, and the second virial coefficient.

Second Virial Coefficient Calculations for Square-Well Chain Molecules

1994 ◽  
Vol 27 (10) ◽  
pp. 2744-2756 ◽  
Author(s):  
John M. Wichert ◽  
Carol K. Hall
Keyword(s):  
Virial Coefficient ◽  
Second Virial Coefficient ◽  
Chain Molecules ◽  
Square Well

STICKY SPHERES IN QUANTUM MECHANICS

1994 ◽  
Vol 06 (05a) ◽  
pp. 947-975 ◽  
Author(s):  
M. D. PENROSE ◽  
O. PENROSE ◽  
G. STELL
Keyword(s):  
Quantum Mechanics ◽  
Brownian Motion ◽  
Gaseous Phase ◽  
Virial Coefficient ◽  
Hard Spheres ◽  
Second Virial Coefficient ◽  
Dimensional System ◽  
3 Dimensional ◽  
Fermi Statistics ◽  
Square Well

For a 3-dimensional system of hard spheres of diameter D and mass m with an added attractive square-well two-body interaction of width a and depth ε, let BD, a denote the quantum second virial coefficient. Let BD denote the quantum second virial coefficient for hard spheres of diameter D without the added attractive interaction. We show that in the limit a → 0 at constant α: = ℰma2/(2ħ2) with α < π2/8, [Formula: see text] The result is true equally for Boltzmann, Bose and Fermi statistics. The method of proof uses the mathematics of Brownian motion. For α > π2/8, we argue that the gaseous phase disappears in the limit a → 0, so that the second virial coefficient becomes irrelevant.

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