Phonon dispersion curves and atomic mean square displacement for several fcc and bcc materials

The atomic mean square displacement (MSD) and the phonon dispersion curves (PDC's) of a number of face-centred cubic (fcc) and body-centred cubic (bcc) materials have been calclllated from the quasiharmonic (QH) theory, the lowest order (A2 ) perturbation theory (PT) and a recently proposed Green's function (GF) method by Shukla and Hiibschle. The latter method includes certain anharmonic effects to all orders of anharmonicity. In order to determine the effect of the range of the interatomic interaction upon the anharmonic contributions to the MSD we have carried out our calculations for a Lennard-Jones (L-J) solid in the nearest-neighbour (NN) and next-nearest neighbour (NNN) approximations. These results can be presented in dimensionless units but if the NN and NNN results are to be compared with each other they must be converted to that of a real solid. When this is done for Xe, the QH MSD for the NN and NNN approximations are found to differ from each other by about 2%. For the A2 and GF results this difference amounts to 8% and 7% respectively. For the NN case we have also compared our PT results, which have been calculated exactly, with PT results calculated using a frequency-shift approximation. We conclude that this frequency-shift approximation is a poor approximation. We have calculated the MSD of five alkali metals, five bcc transition metals and seven fcc transition metals. The model potentials we have used include the Morse, modified Morse, and Rydberg potentials. In general the results obtained from the Green's function method are in the best agreement with experiment. However, this improvement is mostly qualitative and the values of MSD calculated from the Green's function method are not in much better agreement with the experimental data than those calculated from the QH theory. We have calculated the phonon dispersion curves (PDC's) of Na and Cu, using the 4 parameter modified Morse potential. In the case of Na, our results for the PDC's are in poor agreement with experiment. In the case of eu, the agreement between the tlleory and experiment is much better and in addition the results for the PDC's calclliated from the GF method are in better agreement with experiment that those obtained from the QH theory.

Phonon spectra and thermodynamic properties of rare gas solids based on empirical and semi-empirical (ab initio) two-body potentials : a comparative study /

We study the phonon dispersion, cohesive and thermal properties of raxe gas solids Ne, Ar, Kr, and Xe, using a variety of potentials obtained from different approaches; such as, fitting to crystal properties, purely ab initio calculations for molecules and dimers or ab initio calculations for solid crystalline phase, a combination of ab initio calculations and fitting to either gas phase data or sohd state properties. We explore whether potentials derived with a certain approaxih have any obvious benefit over the others in reproducing the solid state properties. In particular, we study phonon dispersion, isothermal ajid adiabatic bulk moduli, thermal expansion, and elastic (shear) constants as a function of temperatiue. Anharmonic effects on thermal expansion, specific heat, and bulk moduli have been studied using A^ perturbation theory in the high temperature limit using the neaxest-neighbor central force (nncf) model as developed by Shukla and MacDonald [4]. In our study, we find that potentials based on fitting to the crystal properties have some advantage, particularly for Kr and Xe, in terms of reproducing the thermodynamic properties over an extended range of temperatiures, but agreement with the phonon frequencies with the measured values is not guaranteed. For the lighter element Ne, the LJ potential which is based on fitting to the gas phase data produces best results for the thermodynamic properties; however, the Eggenberger potential for Ne, where the potential is based on combining ab initio quantum chemical calculations and molecular dynamics simulations, produces results that have better agreement with the measured dispersion, and elastic (shear) values. For At, the Morse-type potential, which is based on M0ller-Plesset perturbation theory to fourth order (MP4) ab initio calculations, yields the best results for the thermodynamic properties, elastic (shear) constants, and the phonon dispersion curves.

Phonon spectra and thermal properties of some fcc metals using embedded-atom potentials /

By employing the embedded-atom potentials of Mei et ai.[l], we have calculated the dynamical matrices and phonon dispersion curves for six fee metals (Cu,Ag,Au,Ni,Pd and Pt). We have also investigated, within the quasiharmonic approximation, some other thermal properties of these metals which depend on the phonon density of states, such as the temperature dependence of lattice constant, coefficient of linear thermal expansion, isothermal and adiabatic bulk moduli, heat capacities at constant volume and constant pressure, Griineisen parameter and Debye temperature. The computed results are compared with the experimental findings wherever possible. The comparison shows a generally good agreement between the theoretical values and experimental data for all properties except the discrepancies of phonon frequencies and Debye temperature for Pd, Pt and Au. Further, we modify the parameters of this model for Pd and Pt and obtain the phonon dispersion curves which is in good agreement with experimental data.

Thermodynamic properties and Debye-Waller factor of fee materials

We have calculated the equation of state and the various thermodynamic properties of monatomic fcc crystals by minimizing the Helmholtz free energy derived in the high temperature limit for the quasiharmonic theory, QH, and the lowest-order (cubic and quartic), 'A2, anharmonic terms of the perturbation theory, PT. The total energy in each case is obtained by adding the static energy. The calculation of the thermal properties was carried out for a nearest-neighbour central-force model of the fcc lattice by means of the appropriate thermodynamic relations. We have calculated the lattice constant, the thermal expansion, the coefficient of volume expansion, the specific heat at constant volume and at constant pressure, the isothermal and adiabatic bulk moduli, and the Griineisen parameter, for the rare-gas solids Kr and Xe, and gold. Morse potential and modified Morse potential were each used to represent the atomic interaction for the three fcc materials. For most of the calculated thermodynamic properties from the QH theory, the results for Kr and Xe with the modified Morse potential show an improvement over the results for the Morse potential when compared with the experimental data. However, the results of the 'A 2 equation of state with the modified Morse potential are in good agreement with experiment only in the case of the specific heat at constant volume and at constant pressure. For Au we have calculated the lattice contribution from the QH and 'A 2 PT and the electronic contribution to the thermal properties. The electronic contribution was taken into account by using the free electron model. The results of the thermodynamic properties calculated with the modified Morse potential were similar to those obtained with the Morse potential. U sing the minimized equation of state we also calculated the Mossbauer recoilless fraction for Kr and Xe and the Debye-Waller factor (DWF) for Pb, AI, eu, Ag, and Au. The Mossbauer recoilless fraction was obtained for the above two potentials and Lennard-Jones potential. The L-J potential gives the best agreement with experiment for Kr. No experimental data exists for Xe. At low temperature the calculated DWF results for Pb, AI, and eu show a good agreement with experimental values, but at high temperature the experimental DWF results increase very rapidly. For Ag the computed values were below the expected results at all temperatures. The DWF results of the modified Morse potential for Pb, AI, eu and Ag were slightly better than those of the Morse potential. In the case of Au the calculated values were in poor agreement with experimental results. We have calculated the quasiharmonic phonon dispersion curves for Kr, Xe, eu, Ag, and Au. The calculated and experimental results of the frequencies agree quite well for all the materials except for Au where the longitudinal modes show serious discrepancies with the experimental results. In addition, the two lowest-order anharmonic contributions to the phonon frequency were derived using the Green's function method. The A 2 phonon dispersion curves have been calculated only for eu, and the results were similar to those of the QH dispersion curves. Finally, an expression for the Griineisen parameter "( has been derived from the anharmonic frequencies, and calculated for these materials. The "( results are comparable with those obtained from the thermodynamic definition.

Response curves of deterministic and probabilistic cellular automata in one and two dimensions

One of the most important problems in the theory of cellular automata (CA) is determining the proportion of cells in a specific state after a given number of time iterations. We approach this problem using patterns in preimage sets - that is, the set of blocks which iterate to the desired output. This allows us to construct a response curve - a relationship between the proportion of cells in state 1 after niterations as a function of the initial proportion. We derive response curve formulae for many two-dimensional deterministic CA rules with L-neighbourhood. For all remaining rules, we find experimental response curves. We also use preimage sets to classify surjective rules. In the last part of the thesis, we consider a special class of one-dimensional probabilistic CA rules. We find response surface formula for these rules and experimental response surfaces for all remaining rules.