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.

Anharmonic Helmholtz free energy to O(lambdaâ ´) in the non- leading term approximation /|nby Leslie Wilk. -- 260 St. Catharines, Ont. : [s. n.],

Expressions for the anharmonic Helmholtz free energy contributions up to o( f ) ,valid for all temperatures, have been obtained using perturbation theory for a c r ystal in which every atom is on a site of inversion symmetry. Numerical calculations have been carried out in the high temperature limit and in the non-leading term approximation for a monatomic facecentred cubic crystal with nearest neighbour c entralforce interactions. The numbers obtained were seen to vary by a s much as 47% from thos e obtai.ned in the leading term approximati.on,indicating that the latter approximati on is not in general very good. The convergence to oct) of the perturbation series in the high temperature limit appears satisfactory.

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.

An application of cell-cluster theory to a rare gas crystal /|n[by] K. Westera. - 260 St. Catharines [Ont.] : Dept. of Physics, Brock University,

The Lennard-Jones Devonshire 1 (LJD) single particle theory for liquids is extended and applied to the anharmonic solid in a high temperature limit. The exact free energy for the crystal is expressed as a convergent series of terms involving larger and larger sets of contiguous particles called cell-clusters. The motions of all the particles within cell-clusters are correlated to each other and lead to non-trivial integrals of orders 3, 6, 9, ... 3N. For the first time the six dimensional integral has been calculated to high accuracy using a Lennard-Jones (6-12) pair interaction between nearest neighbours only for the f.c.c. lattice. The thermodynamic properties predicted by this model agree well with experimental results for solid Xenon.

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.

A treatment of atomic mean square displacement in higher order perturbation theory

A general derivation of the anharmonic coefficients for a periodic lattice invoking the special case of the central force interaction is presented. All of the contributions to mean square displacement (MSD) to order 14 perturbation theory are enumerated. A direct correspondance is found between the high temperature limit MSD and high temperature limit free energy contributions up to and including 0(14). This correspondance follows from the detailed derivation of some of the contributions to MSD. Numerical results are obtained for all the MSD contributions to 0(14) using the Lennard-Jones potential for the lattice constants and temperatures for which the Monte Carlo results were calculated by Heiser, Shukla and Cowley. The Peierls approximation is also employed in order to simplify the numerical evaluation of the MSD contributions. The numerical results indicate the convergence of the perturbation expansion up to 75% of the melting temperature of the solid (TM) for the exact calculation; however, a better agreement with the Monte Carlo results is not obtained when the total of all 14 contributions is added to the 12 perturbation theory results. Using Peierls approximation the expansion converges up to 45% of TM• The MSD contributions arising in the Green's function method of Shukla and Hubschle are derived and enumerated up to and including 0(18). The total MSD from these selected contributions is in excellent agreement with their results at all temperatures. Theoretical values of the recoilless fraction for krypton are calculated from the MSD contributions for both the Lennard-Jones and Aziz potentials. The agreement with experimental values is quite good.

On the formulation and the calculation of the harmonic contributions to the Debye-Waller factor in metals (sodium)

The algebraic expressions for the anharmonic contributions to the Debye-Waller factor up to 0(A ) and 0 L% ) £ where ^ is the scattering wave-vector] have been derived in a form suitable for cubic metals with small ion cores where the interatomic potential extends to many neighbours. This has been achieved in terms of various wave-vector dependent tensors, following the work of Shukla and Taylor (1974) on the cubic anharmonic Helmholtz free energy. The contribution to the various wave-vector dependent tensors from the coulomb and the electron-ion terms in the interatomic metallic potential has been obtained by the Ewald procedure. All the restricted multiple whole B r i l l o u i n zone (B.Z.) sums are reduced to single whole B.Z. sums by using the plane wave representation of the delta function. These single whole B.Z. sums are further reduced to the •%?? portion of the B.Z. following Shukla and Wilk (1974) and Shukla and Taylor (1974). Numerical calculations have been performed for sodium where the Born-Mayer term in the interatomic potential has been neglected because i t is small £ Vosko (1964)3 • *n o^er to compare our calculated results with the experimental results of Dawton (1937), we have also calculated the r a t io of the intensities at different temperatures for the lowest five reflections (110), (200), (220), (310) and (400) . Our calculated quasi-harmonic results agree reasonably well with the experimental results at temperatures (T) of the order of the Debye temperature ( 0 ). For T » © ^ 9 our calculated anharmonic results are found to be in good agreement with the experimental results.The anomalous terms in the Debye-Waller factor are found not to be negligible for certain reflections even for T ^ ©^ . At temperature T yy Op 9 where the temperature is of the order of the melting temperature (Xm) » "the anomalous terms are found to be important almost for all the f i ve reflections.

On the path integral formulation and the evaluation of quantum statistical averages

Four problems of physical interest have been solved in this thesis using the path integral formalism. Using the trigonometric expansion method of Burton and de Borde (1955), we found the kernel for two interacting one dimensional oscillators• The result is the same as one would obtain using a normal coordinate transformation, We next introduced the method of Papadopolous (1969), which is a systematic perturbation type method specifically geared to finding the partition function Z, or equivalently, the Helmholtz free energy F, of a system of interacting oscillators. We applied this method to the next three problems considered• First, by summing the perturbation expansion, we found F for a system of N interacting Einstein oscillators^ The result obtained is the same as the usual result obtained by Shukla and Muller (1972) • Next, we found F to 0(Xi)f where A is the usual Tan Hove ordering parameter* The results obtained are the same as those of Shukla and Oowley (1971), who have used a diagrammatic procedure, and did the necessary sums in Fourier space* We performed the work in temperature space• Finally, slightly modifying the method of Papadopolous, we found the finite temperature expressions for the Debyecaller factor in Bravais lattices, to 0(AZ) and u(/K/ j,where K is the scattering vector* The high temperature limit of the expressions obtained here, are in complete agreement with the classical results of Maradudin and Flinn (1963) .

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.

Mechanism of tocopherol transfer by human α-tocopherol transfer protein (α-hTTP)

Vitamin E is a well known fat soluble chain breaking antioxidant. It is a general tenn used to describe a family of eight stereoisomers of tocopherols. Selective retention of a-tocopherol in the human circulation system is regulated by the a -Tocopherol Transfer Protein (a-TIP). Using a fluorescently labelled a-tocopherol (NBD-a-Toc) synthesized in our laboratory, a fluorescence resonance energy transfer (FRET) assay was developed to monitor the kinetics of ligand transfer by a-hTTP in lipid vesicles. Preliminary results implied that NBD-a-Toe simply diffused from 6-His-a-hTTP to acceptor membranes since the kinetics of transfer were not responsive to a variety of conditions tested. After a series of trouble shooting experiments, we identified a minor contaminant, E coli. outer membrane porin F (OmpF) that co-purified with 6-His-a-hTTP from the metal affinity column as the source of the problem. In order to completely avoid OmpF contamination, a GST -a-hTTP fusion protein was purified from a glutathione agarose column followed by an on-column thrombin digestion to remove the GST tag. We then demonstrated that a-hTTP utilizes a collisional mechanism to deliver its ligand. Furthennore, a higher rate of a-tocopherol transfer to small unilamellar vesicles (SUV s) versus large unilamellar vesicles (LUV s) indicated that transfer is sensitive to membrane curvature. These findings suggest that ahTTP mediated a-Toc transfer is dominated by the hydrophobic nature of a-hTTP and the packing density of phospholipid head groups within acceptor membranes. Based on the calculated free energy change (dG) when a protein is transferred from water to the lipid bilayer, a model was generated to predict the orientation of a-hTTP when it interacts with lipid membranes. Guided by this model, several hydrophobic residues expected to penetrate deeply into the bilayer hydrophobic core, were mutated to either aspartate or alanine. Utilizing dual polarization interferometry and size exclusion vesicle binding assays, we identified the key residues for membrane binding to be F 165, F 169 and 1202. In addition, the rates of ligand transfer of the u-TTP mutants were directly correlated to their membrane binding capabilities, indicating that membrane binding was likely the rate limiting step in u-TTP mediated transfer of u-Toc. The propensity of u-TTP for highly curved membrane provides a connection to its colocalization with u-Toc in late endosomes.

Planar Topological Defects in Unconventional Superconductors

In this work, we consider the properties of planar topological defects in unconventional superconductors. Specifically, we calculate microscopically the interaction energy of domain walls separating degenerate ground states in a chiral p-wave fermionic superfluid. The interaction is mediated by the quasiparticles experiencing Andreev scattering at the domain walls. As a by-product, we derive a useful general expression for the free energy of an arbitrary nonuniform texture of the order parameter in terms of the quasiparticle scattering matrix. The thesis is structured as follows. We begin with a historical review of the theories of superconductivity (Sec. 1.1), which led the way to the celebrated Bardeen-Cooper- Schrieffer (BCS) theory (Sec. 1.3). Then we proceed to the treatment of superconductors with so-called "unconventional pairing" in Sec. 1.4, and in Sec. 1.5 we introduce the specific case of chiral p-wave superconductivity. After introducing in Sec. 2 the domain wall (DW) model that will be considered throughout the work, we derive the Bogoliubov-de Gennes (BdG) equations in Sec. 3.1, which determine the quasiparticle excitation spectrum for a nonuniform superconductor. In this work, we use the semiclassical (Andreev) approximation, and solve the Andreev equations (which are a particular case of the BdG equations) in Sec. 4 to determine the quasiparticle spectrum for both the single- and two-DW textures. The Andreev equations are derived in Sec. 3.2, and the formal properties of the Andreev scattering coefficients are discussed in the following subsection. In Sec. 5, we use the transfer matrix method to relate the interaction energy of the DWs to the scattering matrix of the Bogoliubov quasiparticles. This facilitates the derivation of an analytical expression for the interaction energy between the two DWs in Sec. 5.3. Finally, to illustrate the general applicability our method, we apply it in Sec. 6 to the interaction between phase solitons in a two-band s-wave superconductor.