Exchange Interactions in Solid HELIUM-3 on a Parallel Computer.

We have developed a method of calculating the Gruneisen parameter, gamma equiv partialln J/ partial ln v (J - exchange splitting; v - molar volume), for energy splitting that are due to atoms exchanging lattice sites in fermionic systems for which the exchange splittings are small compared with the lattice kinetic and potential energies. The method involves making two separate path integral Monte Carlo calculations, one for a path (sequence of particle positions in time) that doesn't involve atoms exchanging lattice sites, and one for a path in which the atoms do exchange lattice sites. The difference between a quantity calculated for the two systems is related to the Gruneisen parameter. The central results of this thesis are firstly, that there is no significant variation of the Gruneisen parameters for two, three and four particle exchange ( gamma_2, gamma _3 and gamma_4, respectively) for solid ^3He, in going from 24 cm^3/mole, which is near melting, to 22 cm^3/mole. This result was extended to 20 cm^3/mole for gamma_2. Since no significant variation was observed, the values calculated for different molar volumes were combined to give best estimates for the Gruneisen constants, gamma_2 = 15.9 +/-.8, gamma _3 = 16.4 +/- 1.4 and gamma_4 = 13.8 +/- 1.5. The second result is that the magnitudes of the Gruneisen parameters are very similar for the three types of exchange despite their significantly different geometry. Some very limited runs were done for a system of hard spheres. It is interesting that for the hard sphere diameter chosen, the Gruneisen parameters are of the same order as for solid ^3He, gamma_2 = 21.7 +/-.8, gamma_3 = 16.8 +/- 2.1 and gamma_4 = 19.3 +/- 1.9 for the small test systems checked. The final result relates to ^3 He deposited on a graphite surface. The Gruneisen parameter for three particle exchange for a two-dimensional system with a triangular mesh is found to be consistent with gamma_3 = 50 +/- 5 for a range of nearest-neighbor, lattice spacings from 3.2 A to 3.6 A. This is inconsistent with experiment, indicating that three particle exchange confined to two dimensions does not account for the ferromagnetic properties of this system, as was claimed previously.