Static Critical Behavior Research Articles

Two-dimensional critical behaviour in antiferrodistortive Al ur 6(CIO 4) 3 and Ga ur 6(ClO 4) 3

Critical behaviour with dimensionality d = 2 has been observed for the 300 K antiferrodistortive phase transition in Al ur 6(ClO 4) 3 and Ga ur 6(ClO 4) 3 by means of the temperature dependence of the ESR parameter D. The systems exhibited d = 2 behaviour in the static critical behaviour for T< T c −40 K for T> T c + 40 K. From the ESR data including line width measurements the local order parameter relaxation rate ω 1 has been obtained for various temperatures above T c , with a lowest value of ω 1 = 150 MHz at T c + 15 K

The role of electrostrictive coupling in critical behavior of uniaxial ferroelectrics

Abstract The static and dynamic critical behavior of uniaxial ferroelectrics of TGS, TGSe-type has been studied using an lsing model at d=4 in which the order parameter is coupled to the lattice displacement field. Assuming strong elastic anisotropy, we have solved the renormalization group equations and investigated the stability of the free energy in the one-loop approximation. The discontinuities of the elastic constants and sound damping coefficients at the transition point have been calculated.

Nonlinear relaxation at first-order phase transitions: A Ginzburg-Landau theory including fluctuations

Kinetics of phase transitions are investigated in systems with nonconserved one-component order parameter (i.e., generalized time-dependent Ginzburg-Landau models ind dimensions). The correct static critical behavior as well as fluctuation effects on the kinetics are incorporated by a suitable adaptation of the theory on spinodal decomposition by Langer, Baron and Miller. Both the case of quenches from temperaturesT above to below the critical pointT c and the case of magnetic field changesΔH from positive to negative values are treated, and both time-dependent order parameter 〈m(τ)〉 and structure factorS(q, τ) are obtained numerically ford=2, 3. In the case of quenches atH=0, we find that 〈m(τ)〉≡0 andS(q, τ) —S(q, ∞)∝ exp(−τ1/2/7.2)\(\tilde S(q\tau ^{ - 1/2} )\), withS(q, ∞)∝q−2. In the case of field changes we find that forH not exceeding some critical valueH* the system is trapped in a metastable state with infinite lifetime. In contrast to the meanfield-spinodal, the susceptibility does not seem to diverge atH*. These results are compared with other treatments, in particular the Monte Carlo simulations of kinetic Ising models by Binder and Muller-Krumbhaar. While our theory describes some properties of the metastable states reasonably,H* distinctly exceeds the observed limit of metastability. We argue that the present theory does not take into account nucleation fluctuations, and also fails to describe correctly the domain growth in the late stages of the relaxation. Contrary to Langer et al. we suggest that “universality” holds for nonlinear relaxation and spinodal decomposition nearT c .

Critical behavior in anisotropic cubic systems with long-range interactions

Effects of the potential range of interaction to critical behaviors of anisotropic cubic systems are investigated by means of the Callan-Symanzik equations. As the static critical behavior the stability of fixed points, the critical exponents η C, γ C, φ C s and φ C c, and the equation of state are also investigated. As the dynamic critical behavior the dynamic critical exponent z φ is derived based on the time-dependent Ginzburg-Landau stochastic model. The two- and three-dimensional critical behaviors are discussed.

Static and dynamic critical behaviour of uniaxial ferroelectrics and the phase transition in TGS

AbstractUsing the Larkin‐Khmelnitskii theory for static critical behaviour it is shown how recent renormalization group calculations on static and dynamic phenomena in uniaxial ferroelectrics can be generalized to the whole critical region. The cross‐over between classical and asymptotic behaviour is described in terms of two additional non‐universal parameters b and t0 which are defined by microscopic quantities. The critical temperature dependence of the elastic and electrostrictive constants, the dynamic susceptibility, and the ultrasonic attenuation are treated as a ramification of the Larkin‐Khmelnitskii theory. The correlation parameter δ and the kinetic coefficient T exhibit no anomalous behaviour close to Tc. The ultrasonic attenuation above Tc is shown to be dominated by energy fluctuation. Contradictions in previous interpretations of the data of ferroelectric TGS are removed. At zero temperature Te0 = 4 × 10−7 s is found for the energy relaxation time and ξ0 = 3.7 and ξ∥o = 3.9 Å for the transverse and longitudinal correlation length, respectively. From b = 0.036 and t0 = 0.04 the size of the critical region of TGS is estimated to t = (T – Tc)/Tc ≈︁ 1.6 × 10−2 to 1.8 × 10−3 depending on the quantity which is considered, but the uncertainties in determing t0 are large.

Critical behavior of pure and site-random two-dimensional antiferromagnets

Quasielastic neutron scattering studies of the static critical behavior in the two-dimensional antiferromagnets ${\mathrm{K}}_{2}$Ni${\mathrm{F}}_{4}$, ${\mathrm{K}}_{2}$Mn${\mathrm{F}}_{4}$, and ${\mathrm{Rb}}_{2}$${\mathrm{Mn}}_{0.5}$${\mathrm{Ni}}_{0.5}$${\mathrm{F}}_{4}$ are reported. For $T&lt;0.95{T}_{N}$ the diffuse scattering arises principally from the noncritical transverse susceptibility ${\ensuremath{\chi}}^{\ensuremath{\perp}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{Q}})$. In all three materials ${\ensuremath{\chi}}^{\ensuremath{\perp}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{Q}})$ is found to be only weakly temperature dependent for $T&lt;{T}_{N}$ with a half width, ${\ensuremath{\kappa}}^{\ensuremath{\perp}}$, consistent with spin-wave theory. For $|1\ensuremath{-}\frac{T}{{T}_{N}}|&lt;0.05$. the overall scattering is dominated by the critical Ising component ${\ensuremath{\chi}}^{\ensuremath{\parallel}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{Q}})$. The total scattering is proportional to ${\ensuremath{\chi}}^{\ensuremath{\parallel}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{Q}})+{\ensuremath{\chi}}^{\ensuremath{\perp}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{Q}})$ so that, with an appropriate correction for ${\ensuremath{\chi}}^{\ensuremath{\perp}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{Q}})$, the detailed critical behavior for ${\ensuremath{\chi}}^{\ensuremath{\parallel}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{Q}})$ may be determined. For the reduced temperature range $0.008&lt;\frac{T}{{T}_{N}}\ensuremath{-}1&lt;0.15$ one finds in all three materials $v=0.9\ifmmode\pm\else\textpm\fi{}0.1$, $\ensuremath{\gamma}=1.6\ifmmode\pm\else\textpm\fi{}0.15$, and from the scaling relation $\ensuremath{\gamma}=\ensuremath{\nu}(2\ensuremath{-}\ensuremath{\eta})$, $\ensuremath{\eta}=0.2\ifmmode\pm\else\textpm\fi{}0.05$. For $T&lt;{T}_{N}$ one finds $\ensuremath{\beta}=0.15\ifmmode\pm\else\textpm\fi{}0.015$ in the three systems: Finally, in ${\mathrm{K}}_{2}$Ni${\mathrm{F}}_{4}$ for $T&lt;{T}_{N}$, ${\ensuremath{\chi}}^{\ensuremath{\parallel}}(0)$ and ${\ensuremath{\kappa}}^{\ensuremath{\parallel}}$ are consistent with two-dimensional Ising behavior with exponents ${\ensuremath{\gamma}}^{\ensuremath{'}}=1.75$, ${\ensuremath{\nu}}^{\ensuremath{'}}=1$; further ${\ensuremath{\chi}}^{\ensuremath{\parallel}}(+|\ensuremath{\epsilon}|\ensuremath{\simeq}(50\ifmmode\pm\else\textpm\fi{}10){\ensuremath{\chi}}^{\ensuremath{\parallel}}(\ensuremath{-}|\ensuremath{\epsilon}|)$ compared with the Ising asymmetry factor of 37. These results thence demonstrate that the site-random and pure systems have identical critical behavior in agreement with current theory. Further, the critical behavior is close to that of the two-dimensional Ising model, although there are small differences assumedly due to the fact that the experiments do not probe the true asymptotic region. Finally a number of inconsistencies in earlier experiments are resolved.

Static and dynamic critical behavior at orientational phase transitions in molecular crystals

Critical behavior at orientational order-disorder transitions in plastic molecular crystals is discussed. Primary emphasis is placed on an analysis of the coherent elastic and inelastic neutron cross sections. It is shown that in molecular crystals, where the multipolar operators describing the orientational interaction between pairs of molecules have several components, it is necessary to take into account the full vector character of the order parameter when calculating static and dynamic scattering laws. In the case of the I-II transition in C${\mathrm{D}}_{4}$, the recent experimental neutron-scattering results of Press et al. are well understood on the basis of a molecular-field-type analysis of the James and Keenan Hamiltonian.

Classical displacive limit of ann-component model for structural phase transitions

The static critical behavior of an $n$-component model at the displacive limit is investigated by means of classical mechanics. In the large-$n$ limit, critical exponents and crossover behavior are found by explicit calculation, whereas renormalization-group and scaling relations are used to treat the case of general $n$ exactly for arbitrary dimension. The exponents at the displacive limit, i.e., for ${T}_{c}=0$, are classical for $dg2$. They are independent of $n$ and differ from those of ${T}_{c}g0$ for $dl4$.

Static and dynamic critical behaviour of displacive phase transition at Tcin 1/n expansion

Quantum effects on the static critical behaviour of an isotropic n-vector model undergoing a displacement-type phase transition are studied at Tc and to O(1/n). For Tc not=0 there is no quantum effect on eta , and for Tc=0 one gets eta =0 for d>3, and for 2<d<3 eta is due to quantum fluctuations, implying that it assumes a different value than in the classical case. The dynamical behaviour of the system violates the dynamic scaling hypothesis.

Critical Magnetic Scattering inK2NiF4

A detailed study of the static and dynamic critical behavior of the planar antiferromagnet ${\mathrm{K}}_{2}$Ni${\mathrm{F}}_{4}$ is reported. The dynamic experiments reveal several unusual features. Magnons over most of the Brillouin zone are observed to persist well above ${T}_{N}$. The magnons ultimately become unobservable because of lifetime effects rather than any appreciable renormalization of the real part of the energy. The spin-wave gap remains sharp right up to the phase transition and, in fact, it seems to renormalize like the sublattice magnetization at all temperatures. The transverse excitations for $q\ensuremath{\ne}0$ in general exhibit no explicit recognition of the phase transition and even at $q=0$ there is no apparent variation through ${T}_{N}$ of the integrated intensity. The function ${\mathcal{S}}^{\ensuremath{\parallel}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}},\ensuremath{\omega})$, which manifests the expected critical behavior, is found to be very narrow in $q$, $\ensuremath{\omega}$ space. For $Tl{T}_{N}$ there is no measurable inelasticity of the critical fluctuations at $q=0$. For $Tg{T}_{N}$, ${\mathcal{S}}^{\ensuremath{\parallel}}(0,\ensuremath{\omega})$ has a finite width which goes to zero as $T\ensuremath{\rightarrow}{T}_{N}^{+}$ thus exhibiting the critical slowing down first predicted by Van Hove. Quasielastic measurements of ${\ensuremath{\chi}}^{\ensuremath{\parallel}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}})$ have been carried out for $T\ensuremath{\ge}{T}_{N}$. These experiments yield the critical exponents $\ensuremath{\eta}=0.4\ifmmode\pm\else\textpm\fi{}0.1$ measured at ${T}_{N}$ and $\ensuremath{\nu}=0.57\ifmmode\pm\else\textpm\fi{}0.05$, $\ensuremath{\gamma}=1.0\ifmmode\pm\else\textpm\fi{}0.1$ for $Tg{T}_{N}$. The value for $\ensuremath{\eta}$ is similar to that for the two-dimensional Ising model, $\ensuremath{\eta}=0.25$, but $\ensuremath{\nu}$, $\ensuremath{\gamma}$ differ considerably from the Ising-model values of 1, $\frac{7}{4}$. This latter result is not in agreement with current theory.