Real-space Analysis Research Articles

Renormalization-group calculations for U(1)-symmetric spin systems and gauge theories

We present a real-space renormalization-group analysis for the U(1) spin model in two and three dimensions and for compact electrodynamics in three and four dimensions. The Hamiltonian formulation of the theory defined on a hypercubic spatial lattice is used. The nonperturbative renormalization-group transformations we derive give continuous phase transitions for the spin models, in two and three dimensions. In three dimensions the gap exponent agrees remarkably well with series-expansion results. For the gauge theory we find, in agreement with Polyakov, that compact electrodynamics in three dimensions confines for all values of the coupling constant. A calculation for the string tension shows that it vanishes with an essential singularity at zero coupling. In four dimensions for weak enough coupling confinement is lost and the theory shows a phase transition between a charge-confining and a free-photon phase.

The geometry of X-ray diffraction from extended-face single crystals

An analytical method for the description and analysis in real space of X-ray Bragg reflections from extended-face crystals is presented. The method is applied to discuss the stationary-crystal diffracted-beam images in both the symmetric and asymmetric reflection positions, to analyse the ω and ω:2θ scan modes and to derive the optimum coupling between the detector and crystal motions for an equatorial-plane diffractometer. In practice, the optimum scan mode would require the use of a receiving aperture inclined with respect to the diffraction plane, the angle of inclination being dependent on the diffractometer setting angles. It is concluded that the ω:2θ scan mode is most suitable for routine data collection. Experimental images of the diffracted beam illustrate clearly the validity and usefulness of the mathematical description.

On the jump in the superfluid density of helium films

The universality of the jump in the superfluid density of He films at Tc, predicted by Kosterlitz and Nelson (1977), is investigated when higher vortex configurations are taken into account. An extension of the real-space renormalisation group analysis carried out by these authors leads to a possible non-universality. However it is shown independently that this conclusion is incorrect and that universality must hold in an exact calculation.

Critical slowing down of energylike perturbation in the Glauber-Ising model

A real-space renormalization-group analysis of Glauber's equation of motion for a two-dimensional Ising system on a triangular lattice is carried out up to the second order of the cumulant approximations. The study of the relaxation of the energy gives the dynamic exponent ${\ensuremath{\Delta}}_{E}=2.09$.

Transmission Electron Microscopy: Direct Observation of Crystal Structure in Refractory Ceramics

Using high-resolution multibeam interference techniques in the transmission electron microscope, images have been obtained that make possible a real-space structure analysis of a beryllium-silicon-nitrogen compound. The results illustrate the usefulness of lattice imaging in the analysis of local crystal structure in these technologically promising ceramic materials.

Time-dependent real-space renormalisation group for the Ising system on a triangular lattice

A real-space renormalisation group analysis of Glauber's equation of motion for a two-dimensional Ising system on a triangular lattice is carried out up to the second order of the cumulant approximation. The calculations yield two dynamic modes, and a value z=2.2 is found for the dynamic exponent.