Critical Power Law Research Articles

Magnetic anisotropy in quasi-2D Co(SCN) 2( n-C 3H 7OH) 2

The title system exhibits substantial magnetic anisotropy below 50 K, with only one principal crystal susceptibility of Curie-Weiss form and characterized by a large effective g′ = 6.14. One of the other principal susceptibilities appears to diverge at lower temperatures, and conforms to a critical power law with γ = 1.75.

Phase transition at 434 K, independent strain coupling in second transition at 400 K, and thermal expansivity in ferroelastic K2TeBr6

X-ray diffraction and specific heat measurements have identified two independent strain-coupled processes in a broad phase transition at about 400 K and a normal lambda-type phase transition at 434 K in K2TeBr6 between 20 K and decomposition at 590 K. Phase I is cubic Fm3m with four K2TeBr6 per unit cell, a435=10.7456(14) Å, and expansion coefficients to third-order of α=0.134(3)×10−3 Å K−1, β=0.198(17)×10−6 Å K−2, and γ=0.162(27)×10−9 Å K−3: it transforms on cooling to phase II at 434 K with an entropy change of 1.3 J mol−1 K−1. Phase II is tetragonal, P4/mnc, with two K2TeBr6 per unit cell, a410=7.5755(10) and c410=10.7702(22) Å, with expansion coefficients to third-order for the tetragonal a axis identical to those of the cubic a axis, and linear coefficient αc=−0.103(12) ×10−3 Å K−1: it transforms on cooling to phase III at about 400 K with an entropy change of 2.9 J mol−1 K−1. Phase III is monoclinic, P21/n, with two K2TeBr6 per unit cell, a295=7.4908(10), b 295=7.5492(7), c295=10.6984(11) Å, β 295=90.307(6)°, with monoclinic a axis expansion coefficients to third-order identical to those of the cubic a axis, linear coefficient αb =−0.308(6)×10−4 Å K−1, second-order coefficients αc=0.711(14) ×10−4 Å K−1, βc =0.290(99)×10−7 Å K−2. The monoclinic angle follows a Landau critical power law with β−90∝(400−T)1/2, whereas a−b∝(359−T). Group theoretical analysis of the phase II symmetry operators predicts two independent soft Eg modes for the K and Br atom special positions at the transition to phase III, leading to independent proportionality constants for the thermal variation of β and a−b, as observed. By contrast, the transition from phase I to phase II has only a single soft A2g mode, leaving the K atom special position unchanged. A simple model predicts 1.52 J mol−1 K−1 for the phase transition at 434 K and 2.64 J mol−1 K−1 for that at 400 K. Dynamical short-range tetragonal order may set in above 359 K. Ferroelastic transformation in phase III at room temperature is shown to be accompanied by atomic displacements no larger than about 0.6 Å as the a and b axes are exchanged under applied compressive stress.

ON THE SOLVABILITY OF QUASILINEAR ELLIPTIC EQUATIONS OF ARBITRARY ORDER

Quasilinear elliptic equations of arbitrary order 2$ SRC=http://ej.iop.org/images/0025-5734/45/2/A07/tex_sm_2598_img1.gif/> with smooth linear operators under general linear boundary conditions are considered in the space , 1$ SRC=http://ej.iop.org/images/0025-5734/45/2/A07/tex_sm_2598_img3.gif/>.Theorems are given presenting a priori estimates of in terms of with some , , and in terms of .For these cases, a critical power law growth is obtained for the nonlinear operator relative to the relevant derivatives. Counterexamples are constructed to show that this critical characteristic growth law cannot be improved without additional assumptions.On the basis of this theory, existence theorems for certain quasilinear elliptic problems are established under the condition that there exists an a priori estimate for (in the appropriate family of such problems). An existence theorem is also obtained for the solvability of the Dirichlet boundary value problem for some quasilinear elliptic equations of arbitrary order.Examples are given.Bibliography: 11 titles.

Surface tension and energy of a classical liquid-vapour interface

Rigorous general expressions for the surface tension σ and the surface energy per unit area e are derived in the form of three-fold integrals. In the approximation n 2(r 1, r 2) = n(z 1)n(z 2)g(r 12, n) we obtain the following results: (i) Both σ and e are proportional to (n 1 - n v)2. (ii) The expressions for σ and e are formally reduced to a single integral, with integrands determined in terms of the density profile n(z). (iii) Explicit expressions are given for an exponential density variation. (iv) In the limit of a density variation which is slow on the scale of the molecular diameter, we derive the general expressions σ = A(n 1 - n v)2/λ, e = B(n 1 - n v)2λ from the microscopic theory (λ is a measure of the surface thickness). The same forms for σ and e follow from (iii), with explicit expressions for A and B. These forms for σ and e are shown to be very good approximations even well away from the critical point. It is argued that the critical power laws have the same range of validity. (v) The crit...

Microscopic 1/n expansion for critical dynamics and hydrodynamics near displacive phase transitions

The 1/n expansion is used in order to study long-time correlations near displacive phase transitions. A lattice-dynamical Hamilton operator with O(n) symmetry in the local-normal-coordinate space is considered. The model has microscopically defined dynamics via a kinetic energy term. Since the scattering rate of phonons is of O(1/n), correlations for time and space distances of O(n) are studied. Hydrodynamic modes such as second sound or heat diffusion as well as critical power law are found by microscopic calculation of an energy-density response function above Tc.