On the Ewald method and the quartic Helmholtz free energy of an anharmonic metallic crystal

Thesis (M. Sc.) - Brock University, 1978.

The application of the structural correlation method to determine the reaction coordinate for a putative ring closure reaction

The cr ystal structure of the compound 2-benzoylethylidene-3-(2,4- dibromophenyl)-2,3-dihydro-5-phenyl-l,3,4-thiadiazole* C23H16Br2NZOS (BRMEO) has been determined by using three dimensiona l x-ray diffraction data. The crys tal form is monoclinic, space group P21/c, a = 17.492(4), o -.t' 0 R 0 b =: 16.979(1), c = 14.962(1) A, "X. =o= 90 ',= 106.46(1) , z = 8, graphite-monochromatized Mo~ rad iation, Jl= 0.710J3~, D = 1.62g/cc and o D = 1.65g/cc. The data were col lected on ~ Nonius CAD-4 c diffractometer. The following atoms were made anisotropic: Br, S, N, 0, C7, and C14-C16 for each i ndependent molecu le ; the rest were left isotropic. For 3112 independent refl ec tions with F > 6G\F), R == 0.057. The compound has two independent molecules within the asymmetric unit. Two different conformers were observed which pack well together. /l The S---O interaction distances of 2.493(6) and 2 . 478(7) A were observed for molecules A and B respectively. These values are consistent with earlier findings for 2-benzoylmethylene-3-(2,4-dibromophenyl)- ~~ 2,3-dihydro-5-phenyl-l,3,4-thiadiazole C22H14Br2N20S (BRPHO) and 2-benzoylpropylidene-3-(2,4-dibromophenyl)-2,3-dihydroiii ,'r 5-phenyl-l,3,4-thiadiazole C24H18Br2N20S (BRPETO ) where S---O distances are l ess than the van der Waals (3.251\) but greater than those expected for () a single bond (1.50A). From the results and the literature it appears obvious that the energy/reaction coordinate pathway has a minimum between the end structures (the mono- and bicyclic compounds). * See reference (21) for nomenclature.

An anharmonic contribution to the Helmholtz free energy O(lambda 6)

The anharmonic contributions of order A6 to the Helmholtz free energy for a crystal in which every atom is on a site of inversion symmetry, have been evaluated The cor~esponding diagrams in the various orders of the perturbation theory have been presented The validity of the expressions given is for high temperatures. Numerical calculations for the diagrams which contribute to the free energy have been worked out for a nearest-n~ighbour central-force model of a facecentered cubic lattice in the high-temperature limit and in the leading term and the Ludwig approximations. The accuracy of the Ludwig approximation in evaluating the Brillouin-zone sums has been investigated. Expansion for all diagrams in the high-temperature limit has been carried out The contribution to the specific heat involves a linear as well as cubic term~ We have applied Lennard-Jones, Morse and Exponential 6 types of potentials. A comparison between the contribution to the free energy of order A6 to that of order A4 has been made.