The phase-transition order parameter, fluctuations, and nanoclusters in benzil crystals

Bragg and diffuse X-ray reflections from the M points of the Brillouin zone boundary of a benzil crystal paraphrase, the integral intensity of which is associated with the order parameter and its fluctuations, are studied. Original data on the temperature behavior of the order parameter and diffuse scattering are obtained. The relevant critical exponents are determined. A conclusion is drawn as to the proximity of the phase transition in these crystals to the tricritical point. A phase transition model is discussed.     

Temperature dependence of the order parameter and of diffuse scattering in the Hg<Subscript>2</Subscript>Cl<Subscript>2</Subscript> model ferroelastics

The fundamental and diffuse x-ray reflections from Brillouin zone-edge X points of the paraelastic phase of Hg₂Cl₂ crystals (whose integrated intensity is related to the order parameter and its fluctuations) were studied. Information was obtained on the temperature dependence of the order parameter and of diffuse scattering, and the critical exponents were determined. The conclusion is drawn that the ferroelastic phase transition in these crystals is close to the tricritical point.     

Phase transition effects in polycrystalline Hg<Subscript>2</Subscript>Br<Subscript>2</Subscript> samples

The effects accompanying the ferroelastic phase transition in Hg₂Br₂ polycrystalline samples are compared in an x-ray diffraction study with similar effects observed to occur in Hg₂Br₂ single crystals. In particular, an analysis is made of the “orthorhombic” splitting of the basal plane reflections and the behavior with temperature of the Bragg and diffuse reflections from the X points of the Brillouin zone, which characterize the behavior of the order parameter and its fluctuations, respectively. Polycrystalline samples exhibit strong smearing of the phase transition effects originating from the existence of damaged surface layers and elastic and plastic strain fields which induce order parameter fluctuations over a wide temperature range.     

Order parameter and its fluctuations in Hg<Subscript>2</Subscript>Br<Subscript>2</Subscript> model ferroelastics

The Bragg and diffuse reflections from different X points of the Brillouin zone boundary of the Hg₂Br₂ crystal in the paraphase are studied. The intensities of these reflections are related to the order parameter and its fluctuations. The temperature dependence of the order parameter and its fluctuations is analyzed, and the values of the critical exponents indicating closeness of the phase transition in these crystals to the tricritical point are determined.     

The influence of chromium impurity centers on the critical properties of a weakly polar ferroelectric Li<Subscript>2</Subscript>Ge<Subscript>7</Subscript>O<Subscript>15</Subscript>

The Cr³⁺ EPR spectra of Li₂Ge₇O₁₅ (LGO) crystals are analyzed in the temperature range of the ferroelectric phase transition. The temperature dependence of the local order parameter is determined from the measured splittings of the EPR lines in the polar phase. The experimental critical exponent of the order parameter β=0.31 in the range from the phase transition temperature T _C to (T _C -T) ～ 40 K corresponds to the critical exponent of the three-dimensional Ising model. Analysis of the available data demonstrates that, away from the phase transition temperature T _C , the macroscopic and local properties of LGO crystals are characterized by a crossover from the fluctuation behavior to the classical behavior described in terms of the mean-field theory. The temperature dependence of the local order parameter for LGO: Cr crystals does not exhibit a crossover from the Ising behavior (β=0.31) to the classical behavior (β=0.5). This is explained by the defect nature of Cr³⁺ impurity centers, which weaken the spatial correlations in the LGO host crystal. The specific features of the critical properties of LGO: Cr³⁺ crystals are discussed within a microscopic model of structural phase transitions.     

The order parameter, its fluctuations, and the debye-waller factor of Hg2Cl2 model Ferroelastics

Data on the temperature behavior of the order parameter and its fluctuations were obtained by studying the X-ray reflexes and acoustic soft modes generated from X points on the boundary of the Brillouin zone in the paraphase of Hg₂Cl₂ crystals induced by phase transition and the doubling of unit cells. The corresponding critical exponents, the values of which coincide within the experimental error and testify to how close the phase transition in these crystals is to the tricritical point, are determined by means of X-ray structural analysis and spectroscopic investigations.     

Phase transition, order parameter, and their manifestation in optical spectra of Hg2Cl2

This paper reports on a high-precision study of the effects induced by the phase transition in Raman spectra of Hg₂Cl₂ crystals. The odd (acoustic and IR-active) phonons, including the soft modes, from the Brillouin zone (BZ) X points of the paraphase of these crystals, which are induced by the phase transition, unit cell doubling, and the X ?? ?? BZ folding, have been investigated both theoretically and experimentally. The temperature dependences of the intensities of these phonons, complemented by a theoretical analysis, have yielded the first information on the behavior of the transition order parameter. The corresponding critical exponents have been derived, which correlate in magnitude with the data obtained from X-ray diffraction measurements and, thus, imply the closeness of the phase transition in these crystals to the tricritical point.     

Dielectric properties of glycine phosphite crystals in the model of a phase transition with inclusion of high-order invariants

Dielectric anomalies in the vicinity of the ferroelectric phase transition in nominally pure glycine phosphite (GPI) crystals and glycine phosphite crystals containing 2 mol % glycine phosphate (GP) are studied. It is revealed that the impurity-induced internal macroscopic polarization observed for GPI-GP crystals brings about smearing of the dielectric anomalies in directions both parallel and perpendicular to the axis of spontaneous polarization. The ferroelectric phase of the GPI and GPI-GP crystals is characterized by an unusual variation in the inverse permittivity in the Z direction perpendicular to the Y axis of spontaneous polarization. The temperature dependence of the inverse permittivity is described by a power expression (T _c ? T) ⁿ with an exponent n larger than unity. The experimental data are analyzed in terms of the proposed thermodynamic model with two order parameters, namely, the displacement parameter η and the order-disorder parameter P, which have different physical natures but the same symmetry and allow for coupling invariants of the ηP and η³ P types, as well as for the built-in polarization in the case of GPI-GP crystals. The experimental and theoretical dependences are in good agreement. The coefficients of bilinear and nonlinear coupling between the order parameters are determined. It is shown that the phase transition in the crystals occurs in the vicinity of the tricritical point and that the unusual behavior of the permittivity with a variation in the temperature is explained by the contribution from high-order invariants of coupling.     

Diffuse X-ray scattering and nanoclusters in the Hg2Br2 model ferroelastics

Diffuse maxima are revealed and studied for the first time in the x-ray scattering patterns of Hg₂Br₂ single crystals. These maxima originate from clusters of the orthorhombic phase due to the phase transition in the paraphase tetragonal matrix. The nucleation and growth of clusters are initiated by spatiotemporal fluctuations of the order parameter corresponding to the soft transverse acoustic mode at X points of the Brillouin zone boundary. Information on the behavior of the susceptibility and correlation length with variations in temperature and on the shape and anisotropy of the clusters is obtained, and the critical exponents are determined.     

Diffuse x-ray scattering in the Hg2I2 model incipient ferroelastics

This paper reports the first detection and study of diffuse maxima in x-ray scattering in Hg₂I₂ incipient ferroelastics. These maxima originate from the formation of clusters of the incipient orthorhombic phase in the paraelastic tetragonal matrix. The nucleation and growth of the clusters are caused by spatial and temporal fluctuations of the order parameter (which correspond to the TA soft mode at the X point of the Brillouin zone edge) and are induced by the incipient phase transition. Information is obtained on the temperature behavior of the susceptibility and correlation length and on the shape and anisotropy of clusters, and the critical indices are determined.     

X-ray study of microcrystalline Hg<Subscript>2</Subscript><Emphasis Type="Italic">Hal</Emphasis><Subscript>2</Subscript> ferroelastics

The rhombic splitting of basal plane reflections and the thermal behavior of fundamental and diffuse reflections from X-points of the Brillouin zone characterizing the behavior of an order parameter and its fluctuations, respectively, were studied in polycrystalline Hg₂Cl₂ and Hg₂Br₂ samples. In the case of polycrystalline samples, a strong broadening of phase transition effects was observed, due to damaged surface layers and elastic and plastic stress fields inducing order parameter fluctuations over a wide temperature range.     

Computational investigations of inhomogeneous elastic effects near phase transitions

Abstract

We present different approaches to computer simulations of the order-disorder phase transition in thiocyanate crystals. In these systems the order parameter is coupled to the strain by coupling of the type ?²ε. This coupling plays the dominant role in the behavior of the system below T _c. Molecular Dynamics and Monte Carlo techniques give the proper temperature dependence of the order parameter and the behavior of the fluctuations similar to that observed experimentally. The ab initio calculations allow us to render properly the structural properties of crystals like KSCN and additionally can become the basis for more realistic phenomenological models.     

On the mechanism of the α-β phase transformation of quartz

With a simple geometrical model the atomic displacements at the transition from α- to β -quartz and the eigenvector of the soft mode at the Γ -point are related to a single (order-) parameter. The model is refined by taking into account possible distortions of the SiO₄ tetrahedra. These refinements, together with considerations on the temperature dependence of the order parameter, suggest an increase of π-bond order in the Si-O bond with increasing temperature as the driving mechanism for the phase transition. The model also offers a qualitative understanding of the appearance of diffuse X-ray scattering in the Γ-M direction.     

Spectroscopic study of the behavior of the order parameter in model ferroelastic crystals of Hg<Subscript>2</Subscript>Cl<Subscript>2</Subscript>

Information is obtained about the temperature behavior of the order parameter of a phase transition by theoretical and experimental investigation of odd (acoustic and IR-active) phonons that appear in the Raman scattering spectra from the X points of the Brillouin zone (BZ) boundary in the paraphase of Hg₂Cl₂ crystals and are induced by the phase transition, unit-cell doubling, and the X → Γ folding in the BZ. The relevant critical exponents are determined, whose values are in agreement with the results of X-ray diffraction measurements and, within the Landau phenomenological theory of phase transitions, indicate that the phase transition in these crystals is close to the tricritical point.     

Two-dimensional and layered structures in the discrete φ4 model

The properties of phase transitions in two-dimensional and layered systems are investigated on the basis of a discrete φ⁴ model by numerical and analytical methods. The only parameter a of the discrete φ⁴ model determines the behavior of the system and makes it possible to investigate phase transitions ranging from transitions of the displacement type (a → +0) to order-disorder type (a → +∞). The behavior of a two-dimensional system is investigated in a wide range of values of the parameter a. The temperature dependences of the squared order parameter η²(T) and the phase transition temperature T _c as a function of the thickness N of the system are obtained for three characteristic values of the parameter a using the Monte Carlo method. The properties of phase transitions in the discrete φ⁴ model are investigated on the basis of the mean-field approximation and the independent-mode approximation. The results obtained in the numerical experiments are compared with the analytical approximations. It is shown that the mean-field approximation qualitatively describes the behavior of the phase-transition temperature T _c as a function of the thickness N of the system for a wide range of values of the parameter a, and the independent-mode approximation describes quantitatively, to within 5%, the results of the numerical simulation for small values of a.     

Acoustic properties of the disordered relaxor ferroelectric PbSc1/2Ta1/2O3

The behavior of acoustic phonons in crystals of a relaxor ferroelectric, namely, the lead scandium tantalate PbSc_1/2Ta_1/2O₃ (PST), is studied in the vicinity of the diffuse phase transition. The behavior of longitudinal and transverse acoustic phonons in a PST single crystal is examined using Brillouin scattering. The phonon subsystem is found to behave anomalously in the vicinity of T = 297 K, which can probably be assigned to the existence of a phase transition. Analysis of the results obtained yields the values of the elastic moduli C₁₁, C₁₂, and C₄₄ for the cubic phase of the crystal over a wide temperature range.     

Acoustic and dielectric anomalies in the temperature range of a ferroelectric phase transition in glycine phosphite crystals

The acoustical and dielectric properties of glycine phosphite crystals are investigated in the temperature range of a ferroelectric phase transition. The acoustic anomalies for longitudinal waves along the X, Y, and Z crystallographic axes (where Y is the spontaneous polarization axis) are analyzed in the framework of the Landau theory. It is shown that pronounced acoustic anomalies of the velocity can be quantitatively described within the pseudoproper ferroelectric phase transition model with due regard for the long-range dipole-dipole interaction. For longitudinal acoustic waves propagating along the polar crystal axis, the striction anomaly of the velocity is only partly suppressed by the long-range dipole-dipole forces and an abrupt jump in the velocity is observed in the vicinity of the phase transition. The temperature coefficients of the velocity in the paraphase are determined. The striction contribution and the contributions biquadratic in the order parameter and in the strain to the velocity anomaly are separated.     

A neutron diffraction determination of the critical exponent β for the n = 4 system holmium

The temperature dependence of the order parameter associated with the magnetic phase transition in Ho near 131 K has been measured by neutron diffraction. The results show a power law behavior for the order parameter with an exponent of β = 0.39 (+0.04, ?0.03) for the reduced temperature range 0.007 ? ? ? 0.3. This value is in agreement with recent predictions for an n = 4 vector model studied in the renormalization group technique.     

Fluctuation shift of the nematic–isotropic phase transition temperature

A macroscopic counterpart to the microscopic mechanism of the straightening dimer mesogens conformations, proposed recently by S.M. Saliti, M.G. Tamba, S.N. Sprunt, C. Welch, G.H. Mehl, A. Jakli, and J.T. Gleeson [Phys. Rev. Lett. 116, 217801 (2016)] to explain their experimental observation of the unprecedentedly large shift of the nematic–isotropic transition temperature is discussed. The proposed interpretation is based on singular longitudinal fluctuations of the nematic order parameter. Since these fluctuations are governed by the Goldstone director fluctuations, they exist only in the nematic state. External magnetic field suppresses the singular longitudinal fluctuations of the order parameter (similarly as is the case for the transverse director fluctuations, although with a different scaling over the magnetic field). The reduction of the fluctuations changes the equilibrium value of the magnitude of the order parameter in the nematic state. Therefore, it leads to additional (with respect to the mean field contribution) fluctuation shift of the nematic–isotropic transition temperature. Our mechanism works for any nematic liquid crystals, however the magnitude of the fluctuation shift increases with decrease in the Frank elastic moduli. Since some of these moduli supposed to be anomalously small for so-called bent-core or dimer nematic liquid crystals, just these liquid crystals are promising candidates for the observation of the predicted fluctuation shift of the phase transition temperature.     

Mechanism of the polarization response in the relaxor state of lead scandotantalate single crystals with different levels of ion ordering

The dielectric properties and optical transmission of stoichiometric lead scandotantalate (PST) single crystals in strong electric fields was studied above the temperature of the spontaneous ferroelectric phase transition (T_sp). It is shown that the mechanism of polarization response directly above T_sp is related to induced polarization effects and macrohysteretic behavior only in ac fields above 5 kV/cm. Ananalysis of reciprocal dielectric permittivity carried out over a broad temperature range far above the temperature at which the dielectric permittivity passes through a maximum revealed that specific features of the relaxor behavior manifest themselves up to 400°C even in highly ordered PST crystals.