An invariant expansion of the two‐body statistical correlation function of a fluid is proposed. This expansion does not depend on any particular reference frame used to define the orientation of the molecules, and therefore can be reduced to the expansions of the literature in a simple way. The new expansion permits a rather convenient way of including the effects of molecular symmetry into it. The expressions for a few thermodynamic properties in terms of this expansion are obtained. The equations for x‐ray, neutron, and light scattering are somewhat simpler using this expansion. The Ornstein—Zernike equation has a very convenient form, and is given in Fourier transformed form in terms of 6j angular recoupling coefficients.

Invariant Expansion for Two‐Body Correlations: Thermodynamic Functions, Scattering, and the Ornstein—Zernike Equation

Due to its technological importance, modeling of dendrite growth in pure metals and alloys remains a significant challenge in the field of materials science. In this review recent achievements in the dendrite modeling problem, using two distinct length scale approaches, are summarized. At the nanometer scale, molecular dynamics and Monte Carlo techniques have been developed to extract two important properties of the solid–liquid interface: the kinetic coefficient and the solid–liquid interfacial free energy. Perhaps more importantly the atomistic simulation methods are capable of accurately determining the small, yet crucially important, anisotropies of these parameters. At the mesoscopic scale, advances in phase field modeling have largely overcome the numerical problem associated with the large disparity in length scales typically found in dendrite growth. It is demonstrated that, when the atomistic and continuum level approaches are combined, accurate and parameter free predictions of dendrite growth velocities are possible. In addition, extensions of atomistic and phase field modeling to the case of binary alloys are described.

Atomistic and continuum modeling of dendritic solidification

A group-theoretic study is made of the degeneracies of the normal modes of vibration of a crystal and of the manner in which the polarization vectors describing these modes transform under the operations of the space group of the crystal. To describe the effects of the spatial symmetry operations a set of $3r$-dimensional matrices is constructed, where $r$ is the number of atoms in a primitive unit cell of the crystal, each of which commutes with the Fourier-transformed dynamical matrix for each value of the wave vector labeling the modes. These matrices are shown to provide a multiplier representation of the point group of the wave vector. The reduction of this representation yields the degeneracies (due to spatial symmetry) and transformation properties of the polarization vectors corresponding to a given wave vector, while the forms of the eigenvectors are obtained by projection operator techniques. For appropriate wave vectors, the consequences of time-reversal symmetry on the degeneracies and polarization vectors are investigated by introducing an anti-unitary matrix operator which commutes with the Fourier-transformed dynamical matrix. A criterion for the existence of extra degeneracies due to time-reversal symmetry is presented. The symmetries of lattice vibrations and selection rules for two-phonon absorption processes corresponding to several values of k in the first Brillouin zone of diamond are determined to illustrate the methods developed in this paper.

Symmetry Properties of the Normal Vibrations of a Crystal

Single-particle rotations in molecular crystals

Per me si va nella citt`a dolente Per me si va nell'eterno dolore Per me si va tra la perduta gente Dante, Inferno, III, 1–3 Experimental and theoretical investigations of optical absorption and emission spectra of alkali-halides doped with Tl+-like impurities are reviewed. The Jahn—Teller effect plays an important role. An extensive interpretation of the complex phenomena is presented and critically assessed.

The optical properties of thallium-like impurities in alkali-halide crystals

A review is given of some conventions and definitions required for the derivation of the irreducible representations of the space groups, and of a method to obtain lattice harmonics. These are given for all the irreducible representations of the simple cubic ($\mathrm{Pm}3m$), face-centered cubic ($\mathrm{Fm}3m$) and body-centered cubic ($\mathrm{Im}3m$) space groups for $l\ensuremath{\le}12$. The expansions are given in polar coordinates and care has been taken that different bases corresponding to the same representation span identical, rather than equivalent, representations, which are given in full. Moreover, all the different expansions listed in the tables are fully orthogonal.

Lattice Harmonics I. Cubic Groups

A general discussion of the treatment of the asymmorphic space groups is given, with special reference to the hexagonal close-packed lattice ($\frac{P{6}_{3}}{\mathrm{mmc}}$), for which the irreducible representations are given in full and the lattice harmonics listed for all $l$. The expansions possess the same properties as those given for the cubic groups in the preceding paper.

S. L. Altmann

Papers

Lattice Harmonics I. Cubic Groups

Lattice Harmonics II. Hexagonal Close-Packed Lattice

Semidirect Products and Point Groups