An analysis of decoupling procedures in generating thermal Green's functions of O([lambda]â ´) by the Zubarev equation of motion method

The Zubarev equation of motion method has been applied to an anharmonic crystal of O( ,,4). All possible decoupling schemes have been interpreted in order to determine finite temperature expressions for the one phonon Green's function (and self energy) to 0()\4) for a crystal in which every atom is on a site of inversion symmetry. In order to provide a check of these results, the Helmholtz free energy expressions derived from the self energy expressions, have been shown to agree in the high temperature limit with the results obtained from the diagrammatic method. Expressions for the correlation functions that are related to the mean square displacement have been derived to 0(1\4) in the high temperature limit.

Application of [Lambda] to the fourth perturbation theory in calculating the equation of state of rare gas solids and fcc metals

We have calculated the thermodynamic properties of monatomic fcc crystals from the high temperature limit of the Helmholtz free energy. This equation of state included the static and vibrational energy components. The latter contribution was calculated to order A4 of perturbation theory, for a range of crystal volumes, in which a nearest neighbour central force model was used. We have calculated the lattice constant, the coefficient of volume expansion, the specific heat at constant volume and at constant pressure, the adiabatic and the isothermal bulk modulus, and the Gruneisen parameter, for two of the rare gas solids, Xe and Kr, and for the fcc metals Cu, Ag, Au, Al, and Pb. The LennardJones and the Morse potential were each used to represent the atomic interactions for the rare gas solids, and only the Morse potential was used for the fcc metals. The thermodynamic properties obtained from the A4 equation of state with the Lennard-Jones potential, seem to be in reasonable agreement with experiment for temperatures up to about threequarters of the melting temperature. However, for the higher temperatures, the results are less than satisfactory. For Xe and Kr, the thermodynamic properties calculated from the A2 equation of state with the Morse potential, are qualitatively similar to the A 2 results obtained with the Lennard-Jones potential, however, the properties obtained from the A4 equation of state are in good agreement with experiment, since the contribution from the A4 terms seem to be small. The lattice contribution to the thermal properties of the fcc metals was calculated from the A4 equation of state, and these results produced a slight improvement over the properties calculated from the A2 equation of state. In order to compare the calculated specific heats and bulk moduli results with experiment~ the electronic contribution to thermal properties was taken into account~ by using the free electron model. We found that the results varied significantly with the value chosen for the number of free electrons per atom.

On the equation of state and atomic mean square displacement of crystals

We have presented a Green's function method for the calculation of the atomic mean square displacement (MSD) for an anharmonic Hamil toni an . This method effectively sums a whole class of anharmonic contributions to MSD in the perturbation expansion in the high temperature limit. Using this formalism we have calculated the MSD for a nearest neighbour fcc Lennard Jones solid. The results show an improvement over the lowest order perturbation theory results, the difference with Monte Carlo calculations at temperatures close to melting is reduced from 11% to 3%. We also calculated the MSD for the Alkali metals Nat K/ Cs where a sixth neighbour interaction potential derived from the pseudopotential theory was employed in the calculations. The MSD by this method increases by 2.5% to 3.5% over the respective perturbation theory results. The MSD was calculated for Aluminum where different pseudopotential functions and a phenomenological Morse potential were used. The results show that the pseudopotentials provide better agreement with experimental data than the Morse potential. An excellent agreement with experiment over the whole temperature range is achieved with the Harrison modified point-ion pseudopotential with Hubbard-Sham screening function. We have calculated the thermodynamic properties of solid Kr by minimizing the total energy consisting of static and vibrational components, employing different schemes: The quasiharmonic theory (QH), ).2 and).4 perturbation theory, all terms up to 0 ().4) of the improved self consistent phonon theory (ISC), the ring diagrams up to o ().4) (RING), the iteration scheme (ITER) derived from the Greens's function method and a scheme consisting of ITER plus the remaining contributions of 0 ().4) which are not included in ITER which we call E(FULL). We have calculated the lattice constant, the volume expansion, the isothermal and adiabatic bulk modulus, the specific heat at constant volume and at constant pressure, and the Gruneisen parameter from two different potential functions: Lennard-Jones and Aziz. The Aziz potential gives generally a better agreement with experimental data than the LJ potential for the QH, ).2, ).4 and E(FULL) schemes. When only a partial sum of the).4 diagrams is used in the calculations (e.g. RING and ISC) the LJ results are in better agreement with experiment. The iteration scheme brings a definitive improvement over the).2 PT for both potentials.

Some Travelling Wave Solutions of KdV-Burgers Equation

In this paper we study the extended Tanh method to obtain some exact solutions of KdV-Burgers equation. The principle of the Tanh method has been explained and then apply to the nonlinear KdV- Burgers evolution equation. A finnite power series in tanh is considered as an ansatz and the symbolic computational system is used to obtain solution of that nonlinear evolution equation. The obtained solutions are all travelling wave solutions.

Memo regarding the drainage of marsh lands

Memo regarding the drainage of marsh lands (1 page, handwritten), Jan. 15, 1855.

Memorandum respecting the marsh drainage

Memorandum respecting the marsh drainage (2 ½ pages, handwritten), July 7, 1855.

Report by Jacob Misner on setting contracts for deepening and clearing ditches and estimates of quantities and costs of marsh drainage

Report by Jacob Misner on setting contracts for deepening and clearing ditches and estimates of quantities and costs of marsh drainage (3 ½ pages, handwritten). This is marked as a copy, July 14, 1855.

Estimate of amount of excavation done by A. Cook on the ditch for marsh lands drainage

Estimate of amount of excavation done by A. Cook on the ditch for marsh lands drainage, Aug. 29, 1855.

Estimate of work done on marsh drainage by A. Cook, signed by S.D. Woodruff

Estimate of work done on marsh drainage by A. Cook, signed by S.D. Woodruff, Aug. 31, 1855.

Estimate of work done on marsh lands drainage by Alexander Cook, signed by S.D. Woodruff

Estimate of work done on marsh lands drainage by Alexander Cook, signed by S.D. Woodruff, Nov. 10, 1855.

Estimate of work done on marsh lands drainage by A. Cook and A. Mains and an estimate of time on the marsh lands survey from September to November

Estimate of work done on marsh lands drainage by A. Cook and A. Mains and an estimate of time on the marsh lands survey from September to November by Fred Holmes, Thomas Baird and Joseph Simpson. This is signed by S.D. Woodruff, Nov. 29, 1855.

Estimate of work done on marsh lands drainage by Alexander Cook and Andrew Mains

Estimate of work done on marsh lands drainage by Alexander Cook and Andrew Mains. This is signed by S.D. Woodruff. Also on this sheet is pay roll for engineer services for the months of September, October and November, 1855. This was paid to Frederick Holmes, Thomas Baird, Joseph Simpson and Len [illegible name], Nov.30, 1855.

Estimates for marsh drainage sent to Dexter Deverardo from S.D. Woodruff for Alexander Cook and Andrew Mains

Estimates for marsh drainage sent to Dexter Deverardo from S.D. Woodruff for Alexander Cook and Andrew Mains, Nov. 30, 1855.

Estimate of work done on marsh drainage by Alexander Cook and Andrew Mains

Estimate of work done on marsh drainage by Alexander Cook and Andrew Mains, Nov. 1855.

Engineer estimate (5 ½ pages, handwritten) in the marsh lands drainage presented to the Chamber and Councilors of the County of Welland in council assembled

Engineer estimate (5 ½ pages, handwritten) in the marsh lands drainage presented to the Chamber and Councilors of the County of Welland in council assembled. This report is signed by S.D. Woodruff, June 20, 1857.