Molecular conformation and craze fracture

A craze may fracture either by the breaking of covalent bonds or by a process of “viscous rupture” involving the movement of bulk material in the craze. In the latter case it is necessary that the majority of the chain ends of the polymer molecules passing through the craze-matrix interface terminate within the craze. We have therefore calculated the probability of a polystrene macromolecule spanning a thin craze and shown that for viscous rupture a craze in a normal commercial polystyrene must be something more than 40 nm thick. As the measured craze thicknesses have generally exceeded 100 nm, a viscous fracture process is clearly possible, though, of course, chain rupture is not excluded by the argument. More difficulties arise when fracture occurs within a specified region of the craze and the possibility of a bond fracture under these circumstances is briefly discussed.     

Long range 13C shift effects (δ and ϵ) in methylcyclohexanes

¹³C n.m.r. spectra of a number of methyl substituted cyclohexanes, some of them conformationally homogeneous, have been recorded in CDCl₃ and used to determine shift effects engendered by the introduction of methyl groups on carbon atoms remote from the site of substitution. Sizeable changes in shifts are found, including a substantial effect of an equatorial methyl group on an axial methyl group δ to it (+0.6₇ ppm, ‘δ_ea’). The effects reported are of consequence in investigations of conformational problems by ¹³C n.m.r. techniques.     

Reduction of silicon halides and alkoxides with diisobutylaluminum hydride. Stereochemistry-rate law correlations for the SNiSi and SN2Si mechanisms

Stereochemistry-rate law correlations are given for the reduction of R₃Si*X compounds with i-Bu₂ AlH in either and n-hexane.     

Partition function zeros for the two-dimensional Ising model

The distribution of complex temperature zeros of the partition function of the two-dimensional Ising model in the absence of a magnetic field is investigated. For anisotropic square and triangular lattices the distribution function is two-dimensional and satisfies a partial differential equation derived from a generalized scaling theory. Corresponding results for the isotropic square, triangular and honeycomb lattices are also presented.     

Magnetic percolation in iron-platinum alloys

Conclusions We have shown that the transition at 30.5% Fe is a percolation transition with the formation of clusters being correctly described by the accepted model, and that the Mössbauer spectra of a 26.5 at% Fe alloy show evidence of a SDW. Such a SDW with wave vector parallel to a cube axis could provide a preferred direction in an otherwise cubic crystal.     

Theory of infra-red and optical spectra of antiferromagnets

The contributions of magnons to the optical properties of antiferromagnets having the rutile structure are discussed. The properties considered are electric-dipole active two-magnon absorption in the infra-red, and magnon sidebands on sharp-line exciton transitions in the visible. The discussion is based on a thorough treatment of the properties of excitons and magnons in the antiferromagnetically ordered state. The site-group and space-group symmetries of the magnetic excitations are derived and the selection rules for electric and magnetic dipole transitions are determined. The occurrence of magnetic Davydov splittings of the excitons is investigated, and their symmetry properties throughout the Brillouin zone are derived. The functional dependences of exciton energy on wave vector are calculated. Applications of the theory are made to experimental results on excitons and magnons in MnF₂, FeF₂ and CoF₂.

The possible mechanisms for two-magnon and magnon-sideband absorption are discussed, and the influence of crystal symmetry on these mechanisms is described. The two-magnon state responsible for electric-dipole absorption is identified and selection rules for electric-dipole activity are presented. A spin Hamiltonian for the two-magnon process is set up and used to derive expressions for absorption coefficients for electric vector parallel and perpendicular to the crystal c-axis. Comparison with experiment for MnF₂ yields numerical values for the parameters of the basic coupling mechanism. The exciton-magnon states which give rise to magnon-sideband absorption are explicitly constructed and electric-dipole selection rules are derived for all possible types of sideband. Spin Hamiltonians for the various magnonsideband absorption processes are presented and used to derive expressions for sideband shapes. The results are applied to the experimental spectra for MnF₂ and FeF₂ and the sideband shapes in MnF₂ are calculated numerically. The sideband shapes observable in emission spectra are also briefly discussed.     

Sparsity Inducing Prior Distributions for Correlation Matrices of Longitudinal Data

For longitudinal data, the modeling of a correlation matrix ?R can be a difficult statistical task due to both the positive definite and the unit diagonal constraints. Because the number of parameters increases quadratically in the dimension, it is often useful to consider a sparse parameterization. We introduce a pair of prior distributions on the set of correlation matrices for longitudinal data through the partial autocorrelations (PACs), which vary independently over (?1,1). The first prior shrinks each of the PACs toward zero with increasingly aggressive shrinkage in lag. The second prior (a selection prior) is a mixture of a zero point mass and a continuous component for each PAC, allowing for a sparse representation. The structure implied under our priors is readily interpretable for time-ordered responses because each zero PAC implies a conditional independence relationship in the distribution of the data. Selection priors on the PACs provide a computationally attractive alternative to selection on the elements of ?R or ?R^{? 1} for ordered data. These priors allow for data-dependent shrinkage/selection under an intuitive parameterization in an unconstrained setting. The proposed priors are compared to standard methods through a simulation study and illustrated using a multivariate probit data example. Supplemental materials for this article (appendix, data, and R code) are available online.