A. Brooks Harris

Molecular Chirality and Chiral Parameter

The fundamental issues of symmetry related to chirality are discussed and applied to simple situations relevant to liquid crystals. We show that any chiral measure of a geometric object is a pseudoscalar (invariant under proper rotations but changing sign under improper rotations) and must involve three-point correlations which only come into play when the molecule has at least four atoms. In general, a molecule is characterized by an infinite set of chiral parameters. We illustrate the fact that these parameters can have differing signs and can vanish at different points as a molecule is continuously deformed into its mirror image. From this it is concluded that handedness is not an absolute concept but depends on the property being observed. Within a simplified model of classical interactions, we identify the chiral parameter of the constituent molecules which determines the macroscopic pitch of cholesterics.

Self-avoiding walks on random lattices

The properties of self-avoiding walks on dilute lattices are studied, both directly and using the replica formalism. It is shown that dilution does not affect the exponents and careful use of the Haris criterion also leads to this conclusion.

Symmetry Analysis for the Ruddlesden-Popper Systems Ca 3 Mn 2 O 7 and Ca 3 Ti 2 O 7

We perform a symmetry analysis of the zero-temperature instabilities of the tetragonal phase of Ca3Mn2O7 and Ca3Ti2O7 which is stable at high temperature. We introduce order parameters to characterize each of the possible lattice distortions in order to construct a Landau free energy which elucidates the proposed group-subgroup relations for structural transitions in these systems. We include the coupling between the unstable distortion modes and the macroscopic strain tensor. We also analyze the symmetry of the dominantly antiferromagnetic ordering which allows weak ferromagnetism. We show that in this phase the weak ferromagnetic moment and the spontaneous ferroelectric polarization are coupled, so that rotating one of these ordering by applying an external electric or magnetic field one can rotate the other ordering. We discuss the number of different domains (including phase domains) which exist in each of the phases and indicate how these may be observed.

Flory approximant for self-avoiding walks on fractals

A Flory approximant for the exponent describing the end-to-end distance of a self-avoiding walk (SAW) on fractals is derived. The approximant involves the fractal dimensionalities of the backbone and of the minimal path, and the exponent describing the resistance of the fractal. The approximant yields values which are very close to those available from exact and numerical calculations.

Delocalization Transition in Two-Dimensional Quantum Percolation

The average transmission coefficient between two terminals in the quantum percolation model was studied using low-concentration series. In dimensions d ≥ 3 we identify Anderson delocalization transitions, at thresholds consistent with (and more accurate than) previous studies. In d = 2 our new series behave much better than earlier ones, and we find clear evidence for a transition at a finite concentration. Several scenarios for the new transition, and for the nature of the states above it, are discussed.

Librational anharmonicity in solid α-N2

Abstract The effects of librational anharmonicity in solid α-N 2 are studied. Anharmonic frequencies are obtained using a boson representation for torsional motion. Two-libron and (one stretch + one libron) processes observed via Raman spectroscopy are discussed. Cubic libron-phonon interactions are shown to account for anomalies in the widths of the optical phonons as observed in infrared absorption.

Superlocalization, correlations and random walks on fractals

Abstract Upper and lower bounds are found for the anomalous decay exponent a of localized wave functions and of various two-point correlation functions on typical fractal configurations. The decay of the “superlocalized” wave functions is used to evaluate the decay of the probability distribution of a random walk, which is then used to obtain a Flory-like expression for self-avoiding walks. Emphasis is placed on the differences between “typical” and average quantities.

Bounds for Certain Thermodynamic Averages

Upper and lower bounds for thermodynamic averages of the form 〈{A, A†}〉 are presented.Upper and lower bounds for thermodynamic averages of the form 〈{A, A†}〉 are presented.

What does Landau theory tell us about the orientational state of fullerenes

Various orientation transitions in the fullerenes are described within the Landau theory of phase transitions. Among these we review the description of the orientational ordering transition from an fcc to an sc unit cell which occurs in undoped C60 at a temperature T≈260 K. We also discuss the symmetry of the orientational glass transition which is found via elastic neutron scattering at T≈90 K. We comment on the seemingly spurious transition (so called “2-a transition”) which was claimed to double the size of the unit cell from simple cubic to fcc with eight molecules per unit cell. We also discuss the orientational state of the doped systems M3C60, where M denotes an alkali metal. In this system the C60 molecules are restricted to two “standard” orientations. The resulting Ising model has antiferromagnetic nearest neighbor interactions and nonnegligible four-spin interactions whose properties have recently been discussed. The Landau description of the orientational ordering in C70 is also presented. Here we include a description of the elastic distortion which accompanies orientational ordering.

Magnetic Structure of the Jahn-Teller System LaTiO 3

We investigate the effect of the experimentally observed Jahn-Teller distortion of the oxygen octahedra in LaTiO3 on the magnetic exchange. We present a localized model for the effective hopping between nearest-neighbor Ti ions and the intra-site Coulomb interactions, based on a non-degenerate orbital ground state due to the static crystal field. The latter corresponds to an orbital order which recently has been confirmed experimentally. Using perturbation theory we calculate, in addition to the Heisenberg coupling, antisymmetric and symmetric anisotropy terms of the superexchange spin Hamiltonian which are caused by the spin-orbit interaction. Employing our superexchange Hamiltonian, we deduce the magnetic order at low temperatures which is found to be in good agreement with experiment.