Thermal hopping ofμ + between “FμF” centres in NaF

Strongly hydrogen-bonded diamagnetic “FμF” centres are formed by theμ ⁺ in a wide variety of fluoride crystals. Hydrogen atoms are expected to form similar “FHF” complexes. Through the “motional narrowing” of the zero-field muon relaxation function in NaF, we have observed an Arrhenius temperature dependence of the dissociation rate of theFμF complex, yielding a binding energy of 1700 (200) K for theμ ⁺ in theFμF centre.     

Determination of very slow μ+ hop rates in Cu by LLF-μSRhop rates in Cu by LLF-μSR

Muon spin relaxation in low (weak) longitudinal magnetic field (LLF-μSR) provides a means of independently determining the static dipolar width Δ characterizing the μ⁺ lattice site and the correlation time τ_c for μ⁺ hopping, in a manner that is nearly model-independent for τ_c and especially accurate in the near-static limit (τ_c>τ_μ). The advantages of this method are illustrated by its application to muon hopping in Cu near the τ_c maximum around 50 K.     

Thermally activated muonium formation in Al2O3 and BaF2

Direct evidence is presented for thermally activated muonium formation in Al₂O₃ and BaF₂. In each case the signature of the transition is a transverse field depolarization rate of the prompt diamagnetic component which is much too large to be attributable to muon-nuclear dipolar couplings. Longitudinal field decoupling measurements in Al₂O₃ support the interpretation that the initial diamagnetic fraction converts into a state with a stongly coupled electron —-i.e., muonium.     

The effect of a crystalline electric field on the magnetic transition temperatures of rare-earth rhodium borides

A simple molecular field model is presented for the prediction of magnetic transition temperatures of rare-earth (RE) compounds when crystalline electric field (CEF) splittings are significant. The model is applied to the RERh₄B₄ (RE=Gd?Tm) series, using what is known about the crystal field in these materials.     

Cubics and negative curvature

Riemannian cubics are curves that generalise cubic polynomials to arbitrary Riemannian manifolds, in the same way that geodesics generalise straight lines. Considering that geodesics can be extended indefinitely in any complete manifold, we ask whether Riemannian cubics can also be extended indefinitely. We find that there are always exceptions in Riemannian manifolds with strictly negative sectional curvature. On the other hand, we show that Riemannian cubics can always be extended in complete locally symmetric Riemannian manifolds of non-negative curvature.     

Stacking faults in ultra-thin films of silver on Al(111) investigated by medium energy ion scattering

Medium energy ion scattering has been used to investigate depositions of 0.2, 1.4, 3.5 and 4.8 ML of silver onto Al(111). Energy profiles indicate alloying to the extent that aluminium is still visible after the deposition of 4.8 ML. From assessments of the visibility, blocking dips and fits using VEGAS simulations it is shown that the first two layers continue the fcc stacking but after that hcp and fcc twin-type stacking faults occur. The 1.4 ML structure is consistent with a mixed structure of 85% fcc and 15% hcp indicating that some silver occupies a third layer. The blocking curve from the structure formed by 3.5 ML equivalent deposition can be simulated by 56% fcc, 32% hcp and 12% fcc twin and that from 4.8 ML by 59% fcc, 23% hcp and 18% fcc twin. This provides direct evidence of the incidence of hcp stacking when silver is deposited onto Al(111) in the range between 2 and 5 ML.     

Computational Method for Time-Optimal Switching Control

An efficient algorithm, called the time-optimal switching (TOS) algorithm, is proposed for the time-optimal switching control of nonlinear systems with a single control input. The problem is formulated in the arc times space, arc times being the durations of the arcs. A feasible switching control, or as a special case bang-bang control, is found using the STC method previously developed by the authors to get from an initial point to a target point with a given number of switchings. Then, by means of constrained optimization techniques, the cost being considered as the summation of the arc times, a minimum-time switching control solution is obtained. Example applications of the TOS algorithm involving second-order and third-order systems are presented. Comparisons are made with a well-known general optimal control software package to demonstrate the efficiency of the algorithm.     

Bézier curves and interpolation in Riemannian manifolds

In a connected Riemannian manifold, generalised Bézier curves are C^∞ curves defined by a generalisation, in which line segments are replaced by minimal geodesics, of the classical de Casteljau algorithm. As in Euclidean space, these curves join their first and last control points. We compute the endpoint velocities and (covariant) accelerations of a generalised Bézier curve of arbitrary degree and use the formulae to express the curve's control points in terms of these quantities. These results allow generalised Bézier curves to be pieced together into C² splines, and thereby allow C² interpolation of a sequence of data points. For the case of uniform splines in symmetric spaces, we show that C² continuity is equivalent to a simple relationship, involving the global symmetries at knot points, between the control points of neighbouring curve segments. We also present some examples in hyperbolic 2-space.     

The physics of diamond

The accurate measurement of the thermal conductivity of metals still poses considerable experimental difficulty. The reasons for this, and the main ways in which the problem has been approached, are summarized. A detailed examination of the development in technique and methods of measurement shows that while most of the steady-state methods introduced over 50 years ago are still being improved, the emphasis in the last few years has been on the development of methods which give more rapid results, particularly at high temperatures.