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Resonant ultrasound spectroscopy has been used to characterise strain coupling and relaxation behavior associated with magnetic/magnetoelectric phase transitions in GdMnO 3 , TbMnO 3 and TbMn 0.98 Fe 0.02 O 3 through their influence on elastic/anelastic properties. Acoustic attenuation ahead of the paramagnetic to colinear-sinusoidal incommensurate antiferromagnetic transition at ∼41 K correlates with anomalies in dielectric properties and is interpreted in terms of Debye-like freezing processes. A loss peak at ∼150 K is related to a steep increase in electrical conductivity with a polaron mechanism. The activation energy, E , of ≳0.04 eV from a loss peak at ∼80 K is consistent with the existence of a well-defined temperature interval in which the paramagnetic structure is stabilised by local, dynamic correlations of electric and magnetic polarisation that couple with strain and have relaxation times in the vicinity of ∼10 a , of ≳0.04 eV from a loss peak at ∼80 K is consistent with the existence of a well-defined temperature interval in which the paramagnetic structure is stabilised by local, dynamic correlations of electric and magnetic polarisation that couple with strain and have relaxation times in the vicinity of ∼10 -6 s. Comparison with previously published data for Sm 0.6 Y 0.4 MnO 3 confirms that this pattern may be typical for multiferroic orthorhombic R MnO 3 perovskites ( R = Gd, Tb, Dy). A frequency-dependent loss peak near 10 K observed for TbMnO 3 and TbMn 0.98 Fe 0.02 O 3 , but not for GdMnO 3 ⩾ ∼0.002 eV and is interpreted as freezing of some magnetoelastic component of the cycloid structure. Small anomalies in elastic properties associated with the incommensurate and cycloidal magnetic transitions confirm results from thermal expansion data that the magnetic order parameters have weak but significant coupling with strain. Even at strain magnitudes of ∼0.1-1‰, polaron-like strain effects are clearly important in defining the development and evolution of magnetoelectric properties in these materials. Strains associated with the cubic-orthorhombic transition due to the combined Jahn-Teller/octahedral tilting transition in the vicinity of 1500 K are 2-3 orders of magnitude greater. It is inevitable that ferroelastic twin walls due to this transition would have significantly different magnetoelectric properties from homogeneous domains due to magnetoelastic coupling with steep strain gradients.E a ⩾ ∼0.002 eV and is interpreted as freezing of some magnetoelastic component of the cycloid structure. Small anomalies in elastic properties associated with the incommensurate and cycloidal magnetic transitions confirm results from thermal expansion data that the magnetic order parameters have weak but significant coupling with strain. Even at strain magnitudes of ∼0.1-1‰, polaron-like strain effects are clearly important in defining the development and evolution of magnetoelectric properties in these materials. Strains associated with the cubic-orthorhombic transition due to the combined Jahn-Teller/octahedral tilting transition in the vicinity of 1500 K are 2-3 orders of magnitude greater. It is inevitable that ferroelastic twin walls due to this transition would have significantly different magnetoelectric properties from homogeneous domains due to magnetoelastic coupling with steep strain gradients., (Creative Commons Attribution license.)

Domain walls in Cu-Cl boracite develop as a consequence of an improper ferroelastic, improper ferroelectric transition, and have attracted close interest because some are conductive and all can be mechanically written and repositioned by application of an electric field. The phase transition and its associated dynamical properties have been analysed here from the perspective of strain and elasticity. Determination of spontaneous strains from published lattice parameter data has allowed the equilibrium long-range order parameter for F [Formula: see text]3c → Pca2 1 to be modelled simply as being close to the order-disorder limit. High acoustic loss in the cubic phase, revealed by resonant ultrasound spectroscopy, is consistent with the presence of dynamical microdomains of the orthorhombic structure with relaxation times in the vicinity of ∼10 -5 -10 -6  s. Low acoustic loss in the stability field of the orthorhombic structure signifies, on the other hand, that ferroelastic twin walls which develop as a consequence of the order-disorder process are immobile on this time scale. A Debye loss peak accompanied by ∼1% elastic stiffening at ∼40 K is indicative of some freezing of defects which couple with strain or of some more intrinsic freezing process. The activation energy of ⩾∼0.01-0.02 eV implies a mechanism which could involve strain relaxation clouds around local ferroelectric dipoles or freezing of polarons that determine the conductivity of twin walls.

Pyrrhotite, Fe 7 S 8 , is a commonly occurring carrier of magnetic remanence and has a low temperature transition, the Besnus transition, involving a change in spin state. Variations of the thermodynamic, magnetic and elastic properties through this transition at ∼33 K in a natural sample of 4C pyrrhotite have been tested against a group theoretical model for coupling between order parameters relating to Fe/vacancy ordering (irrep U (1/2,0,1/4)) and magnetic ordering (irreps mΓ4+and mΓ5+). Magnetoelastic coupling is weak but the pre-existing microstructure of ferroelastic and magnetic domains, that develop as a consequence of Fe/vacancy and ferrimagnetic ordering during slow cooling in nature ( 1 (1/2,0,1/4)) and magnetic ordering (irreps mΓ4+and mΓ5+). Magnetoelastic coupling is weak but the pre-existing microstructure of ferroelastic and magnetic domains, that develop as a consequence of Fe/vacancy and ferrimagnetic ordering during slow cooling in nature ( P 6 3 / mmc → C 2'/ c '), causes subtle changes in the low temperature transition ( C 2'/ c 1¯). The Besnus transition involves a rotation of magnetic moments out of the P 1¯). The Besnus transition involves a rotation of magnetic moments out of the a plane of the monoclinic structure, but it appears that the transition temperature might vary locally according to whether it is taking place within the pre-existing domain walls or in the domains that they separate. Evidence of metamagnetic transitions suggests that the magnetic field-temperature phase diagram will display some interesting diversity. Low temperature magnetic transitions in minerals of importance to the palaeomagnetism community have been used to identify the presence of magnetite and haematite in rocks and the Besnus transition is diagnostic of the existence of pyrrhotite, Fe c plane of the monoclinic structure, but it appears that the transition temperature might vary locally according to whether it is taking place within the pre-existing domain walls or in the domains that they separate. Evidence of metamagnetic transitions suggests that the magnetic field-temperature phase diagram will display some interesting diversity. Low temperature magnetic transitions in minerals of importance to the palaeomagnetism community have been used to identify the presence of magnetite and haematite in rocks and the Besnus transition is diagnostic of the existence of pyrrhotite, Fe 7 S 8 ., (Creative Commons Attribution license.)

Magnetoelastic coupling associated with the hexagonal-monoclinic transition in a natural sample of the mineral pyrrhotite, Fe 7 S 8 , has been analysed in terms of separate coupling of spontaneous strains with two discrete order parameters, q v for magnetic ordering. Coupling of the two order parameters separately with strain gives rise to two terms for coupling between them, q m qm2qv8, and a pattern of evolution in which λ qm2qv2and λ discontinuously through a single transition point. The transition is ferrimagnetic and ferroelastic but the relatively slow relaxation rate for Fe/vacancy ordering, in comparison with magnetic ordering, results in elastic and anelastic properties which are quite different from those observed in other ferroic or multiferroic materials with two instabilities. Instead of classical elastic softening, there is stiffening of the elastic constants which scales withqm2andqv2. Instead of the normal pattern of acoustic loss associated with the mobility and subsequent freezing for ferroelastic twin walls, the loss is consistently low throughout the temperature range 300 K-875 K.q v varies continuously and q m discontinuously through a single transition point. The transition is ferrimagnetic and ferroelastic but the relatively slow relaxation rate for Fe/vacancy ordering, in comparison with magnetic ordering, results in elastic and anelastic properties which are quite different from those observed in other ferroic or multiferroic materials with two instabilities. Instead of classical elastic softening, there is stiffening of the elastic constants which scales withqm2andqv2. Instead of the normal pattern of acoustic loss associated with the mobility and subsequent freezing for ferroelastic twin walls, the loss is consistently low throughout the temperature range 300 K-875 K., (Creative Commons Attribution license.)

The transition sequence in the Heusler alloy Ni 50 Mn 34 In 8 Ga 8 has been determined from measurements of elasticity, heat flow and magnetism to be paramagnetic austenite → paramagnetic martensite → ferromagnetic martensite at ∼335 and ∼260 K, respectively, during cooling. The overall pattern of elastic stiffening/softening and acoustic loss is typical of a system with bilinear coupling between symmetry breaking strain and the driving structural/electronic order parameter, and a temperature interval below the transition point in which ferroelastic twin walls remain mobile under the influence of external stress. Divergence between zero-field-cooling and field-cooling determinations of DC magnetisation below ∼220 K indicates that a frustrated magnetic glass develops in the ferromagnetic martensite. An AC magnetic anomaly which shows Vogel-Fulcher dynamics in the vicinity of ∼160 K is evidence of a further glassy freezing process. This coincides with an acoustic loss peak and slight elastic stiffening that is typical of the outcome of freezing of ferroelastic twin walls. The results suggest that local strain variations associated with the ferroelastic twin walls couple with local moments to induce glassy magnetic behaviour., (Creative Commons Attribution license.)

The hypothesis that strain has a permeating influence on ferroelastic, magnetic and superconducting transitions in 122 iron pnictides has been tested by investigating variations of the elastic and anelastic properties of a single crystal of Ba(Fe 0.957 Co 0.043 ) 2 As 2 by resonant ultrasound spectroscopy as a function of temperature and externally applied magnetic field. Non-linear softening and stiffening of C 66 in the stability fields of both the tetragonal and orthorhombic structures has been found to conform quantitatively to the Landau expansion for a pseudoproper ferroelastic transition which is second order in character. The only exception is that the transition occurs at a temperature (T S   ≈  69 K) ~10 K above the temperature at which C 66 would extrapolate to zero ([Formula: see text]  ≈  59 K). An absence of anomalies associated with antiferromagnetic ordering below T N   ≈  60 K implies that coupling of the magnetic order parameter with shear strain is weak. It is concluded that linear-quadratic coupling between the structural/electronic and antiferromagnetic order parameters is suppressed due to the effects of local heterogeneous strain fields arising from the substitution of Fe by Co. An acoustic loss peak at ~50-55 K is attributed to the influence of mobile ferroelastic twin walls that become pinned by a thermally activated process involving polaronic defects. Softening of C 66 by up to ~6% below the normal-superconducting transition at T c   ≈  13 K demonstrates an effective coupling of the shear strain with the order parameter for the superconducting transition which arises indirectly as a consequence of unfavourable coupling of the superconducting order parameter with the ferroelastic order parameter. Ba(Fe 0.957 Co 0.043 ) 2 As 2 is representative of 122 pnictides as forming a class of multiferroic superconductors in which elastic strain relaxations underpin almost all aspects of coupling between the structural, magnetic and superconducting order parameters and of dynamic properties of the transformation microstructures they contain.

The elastic and anelastic properties of a single crystal of Co-doped pnictide Ba(Fe 0.957 Co 0.043 ) 2 As 2 have been determined by resonant ultrasound spectroscopy in the frequency range 10-500 kHz, both as a function of temperature through the normal-superconducting transition (T c   ≈  12.5 K) and as a function of applied magnetic field up to 12.5 T. Correlation with thermal expansion, electrical resistivity, heat capacity, DC and AC magnetic data from crystals taken from the same synthetic batch has revealed the permeating influence of strain on coupling between order parameters for the ferroelastic (Q E ) and superconducting (Q SC ) transitions and on the freezing/relaxation behaviour of vortices. Elastic softening through T c in zero field can be understood in terms of classical coupling of the order parameter with the shear strain e 6 , λe 6 [Formula: see text], which means that there must be a common strain mechanism for coupling of the form λ [Formula: see text] Q E . At fields of ~5 T and above, this softening is masked by Debye-like stiffening and acoustic loss processes due to vortex freezing. The first loss peak may be associated with the establishment of superconductivity on ferroelastic twin walls ahead of the matrix and the second is due to the vortex liquid-vortex glass transition. Strain contrast between vortex cores and the superconducting matrix will contribute significantly to interactions of vortices both with each other and with the underlying crystal structure. These interactions imply that iron-pnictides represent a class of multiferroic superconductors in which strain-mediated coupling occurs between the multiferroic properties (ferroelasticity, antiferromagnetism) and superconductivity.

A list of potential (multi-) ferroic systems is derived based on the idea that structure gradients can generate ferroic distortions. Structure gradients are restricted here to structural modulations, which are commonly observed in natural minerals. These minerals contain transition metals and are prone to Jahn-Teller distortions and magnetic instabilities. The list contains mineral groups, which are often available in mineralogical collections.

Resonant Ultrasound Spectroscopy has been used to characterize elastic and anelastic anomalies in a polycrystalline sample of multiferroic Pb(Fe(0.5)Nb(0.5))O(3) (PFN). Elastic softening begins at ~550 K, which is close to the Burns temperature marking the development of dynamical polar nanoregions. A small increase in acoustic loss at ~425 K coincides with the value of T(*) reported for polar nanoregions starting to acquire a static or quasi-static component. Softening of the shear modulus by ~30-35% through ~395-320 K, together with a peak in acoustic loss, is due to classical strain/order parameter coupling through the cubic → tetragonal → monoclinic transition sequence of ferroelectric/ferroelastic transitions. A plateau of high acoustic loss below ~320 K is due to the mobility under stress of a ferroelastic microstructure but, instead of the typical effects of freezing of twin wall motion at some low temperature, there is a steady decrease in loss and increase in elastic stiffness below ~85 K. This is attributed to freezing of a succession of strain-coupled defects with a range of relaxation times and is consistent with a report in the literature that PFN develops a tweed microstructure over a wide temperature interval. No overt anomaly was observed near the expected Néel point, ~145 K, consistent with weak/absent spin/lattice coupling but heat capacity measurements showed that the antiferromagnetic transition is actually smeared out or suppressed. Instead, the sample is weakly ferromagnetic up to ~560 K, though it has not been possible to exclude definitively the possibility that this could be due to some magnetic impurity. Overall, evidence from the RUS data is of a permeating influence of static and dynamic strain relaxation effects which are attributed to local strain heterogeneity on a mesoscopic length scale. These, in turn, must have a role in determining the magnetic properties and multiferroic character of PFN.

Resonant ultrasound spectroscopy (RUS) provides a window on the pervasive influence of strain coupling at phase transitions in perovskites through determination of elastic and anelastic relaxations across wide temperature intervals and with the application of external fields. In particular, large variations of elastic constants occur at structural, ferroelectric and electronic transitions and, because of the relatively long interaction length provided by strain fields in a crystal, Landau theory provides an effective formal framework for characterizing their form and magnitude. At the same time, the Debye equations provide a robust description of dynamic relaxational processes involving the mobility of defects which are coupled with strain. Improper ferroelastic transitions driven by octahedral tilting in KMnF3, LaAlO3, (Ca,Sr)TiO3, Sr(Ti,Zr)O3 and BaCeO3 are accompanied by elastic softening of tens of % and characteristic patterns of acoustic loss due to the mobility of twin walls. RUS data for ferroelectrics and ferroelectric relaxors, including BaTiO3, (K,Na)NbO3,Pb(Mg1/3Nb2/3)O3 (PMN), Pb(Sc1/2Ta1/2)O3 (PST), (Pb(Zn1/3Nb2/3)O3)0.955(PbTiO3)0.045 (PZN-PT) and (Pb(In1/2Nb1/2)O3)0.26(Pb(Mg1/3Nb2/3)O3)0.44(PbTiO3)0.30 (PIN-PMN-PT) show similar patterns of softening and attenuation but also have precursor softening associated with the development of polar nano regions. Defect-induced ferroelectricity occurs in KTaO3, without the development of long range ordering. By way of contrast, spin-lattice coupling is much more variable in strength, as reflected in a greater range of softening behaviour for Pr0.48Ca0.52MnO3 and Sm0.6Y0.4MnO3 as well as for the multiferroic perovskites EuTiO3,BiFeO3, Bi0.9Sm0.1FeO3, Bi0.9Nd0.1FeO3, (BiFeO3)0.64(CaFeO2.5)0.36, (Pb(Fe0.5Ti0.5)O3)0.4(Pb(Zr0.53Ti0.47)O3)0.6. A characteristic feature of transitions in which there is a significant Jahn-Teller component is softening as the transition point is approached from above, as illustrated by PrAlO3, and this is suppressed by application of an external magnetic field in the colossal magnetoresistive manganite Pr0.48Ca0.52MnO3 or by reducing grain size in La0.5Ca0.5MnO3. Spin state transitions for Co(3+) in LaCoO3, NdCoO3 and GdCoO3 produce changes in the shear modulus that scale with a spin state order parameter, which is itself coupled with the order parameter(s) for octahedral tilting in a linear-quadratic manner. A new class of phase transitions in perovskites, due to orientational or conformational ordering of organic molecules on the crystallographic A-site of metal organic frameworks, is illustrated for [(CH3)2NH2]Co(HCOO)3 and [(CH2)3NH2]Mn(HCOO)3 which also display elastic and anelastic anomalies due to the influence of intrinsic and extrinsic strain relaxation behaviour.

A simulation study of negative thermal expansion in Y2W3O12 was carried out using calculations of phonon dispersion curves through the application of density functional perturbation theory. The mode eigenvectors were mapped onto flexibility models and results compared with calculations of the mode Grüneisen parameters. It was found that many lower-frequency phonons contribute to negative thermal expansion in Y2W3O12, all of which can be described in terms of rotations of effectively rigid WO4 tetrahedra and Y-O rods. The results are strikingly different from previous phonon studies of higher-symmetry materials that show negative thermal expansion.

Resonant ultrasound spectroscopy has been used to monitor variations in the elastic and anelastic behaviour of polycrystalline CoF2 through the temperature interval 10-290 K and in the frequency range ∼0.4-2 MHz. Marked softening, particularly of the shear modulus, and a peak in attenuation occur as the Néel point (TN=39 K) is approached from both high and low temperatures. Although the effective thermodynamic behaviour can be represented semiquantitatively with a Bragg-Williams model for a system with spin 1/2, the magnetoelastic coupling follows a pattern which is closely analogous to that of a Landau tricritical transition which is co-elastic in character. Analysis of lattice parameter data from the literature confirms that linear spontaneous strains scale with the square of the magnetic order parameter and combine to give effective shear and volume strains on the order of 1‰. Softening of the shear modulus at T>TN is attributed to coupling of acoustic modes with dynamical local ordering of spins and can be represented by a Vogel-Fulcher expression. At T

The evolution of ferroelastic microstructures under external shear is determined by large-scale molecular dynamics simulations in two and three dimensions. Ferroelastic pattern formation was found to be almost identical in two and three dimensions, with only the ferroelastic transition temperature changing. The twin patterns generated by shear deformation depend strongly on temperature, with high wall densities nucleating under optimized temperature conditions. The dynamical tweed and mobile kink movement inside the twin walls is continuous and thermally activated at high temperatures, and becomes jerky and athermal at low temperatures. With decreasing temperature, the statistical distributions of dynamical tweed and kinks vary from a Vogel-Fulcher law P(E)~exp-(E/(T-TVF)) to an athermal power-law distribution P(E)~E-E. During the yield event, the nucleation of needles and kinks is always jerky, and the energy of the jerks is power-law distributed. Low-temperature yield proceeds via one large avalanche. With increasing temperature, the large avalanche is thermally broken up into a multitude of small segments. The power-law exponents reflect the changes in temperature, even in the athermal regime.

By exploring a simple model of a negative thermal expansion (NTE) system, we introduce a phenomenological expression to describe the temperature dependence of the pressure-induced softening in NTE structures.

Magnetic phase transitions in RCo2 Laves phases with R as a rare earth element are accompanied by changes in crystallographic space group. For purely structural transitions they would be described as improper ferroelastic and therefore fulfil the condition for multiferroic phase transitions in combining two out of three properties, ferro/antiferromagnetism, ferroelectricity and ferroelasticity. Here lattice parameter data from the literature and new measurements of elastic and anelastic properties, by resonant ultrasound spectroscopy, for NdCo2 and ErCo2 have been analysed from this perspective. The temperature dependence of symmetry-breaking shear strains is consistent with the cubic ↔ tetragonal transition in NdCo2 being close to tricritical in character and the cubic ↔ rhombohedral transition in ErCo2 being first order. Elastic softening and acoustic loss within the stability ranges of the ferroelastic phases can be understood in terms of a combination of intrinsic softening due to strain/order parameter coupling and ferroelastic twin-wall motion. Softening ahead of the transitions does not fit with standard macroscopic descriptions of dynamic effects from other systems but, rather, in the case of NdCo2, might be attributed to the involvement of a second zone centre order parameter related to a separate instability driven by cooperative Jahn-Teller distortions. In ErCo2, acoustic loss in the temperature interval above the transition point is discussed in terms of a possible tweed microstructure associated with strain coupling to local magnetic ordering. The overall multiferroic behaviour can be understood in terms of a single magnetic order parameter (irrep mΓ+4 of magnetic space group Fd3m1') which couples with a structural order parameter (irrep Γ+3 or Γ+5). The coupling is linear/quadratic which, in the case of two separate instabilities, causes them to combine in a single multiferroic phase transition.

Resonant ultrasound spectroscopy has been used to measure the elastic and anelastic behaviour through known structural and magnetic phase transitions in single crystal hexagonal YMnO3. Anomalous elastic behaviour is observed at the high temperature structural transition at ∼1260 K, with a discontinuity in the elastic constants and nonlinear recovery below Tc, consistent with [Formula: see text] coupling. There is no change in dissipation associated with this high temperature transition, and no evidence in the elastic or anelastic behaviour for any secondary transition at ∼920 K, thus supporting the thesis of a single high temperature transformation. Elastic stiffening is observed on cooling through TN, in accordance with previous studies, and the excess elastic constant appears to scale with the square of the magnetic order parameter. The strains incurred at TN are a factor of ∼20 smaller than those at the structural transition, implying very weak [Formula: see text] coupling and a dominant contribution to the variation in the elastic constants from [Formula: see text]. The increased acoustic dissipation above TN is consistent with an order-disorder process.

Elastic properties of the ferroelastic compound Pb3(PO4)2 were investigated using resonant ultrasonic spectroscopy. Results show softening of the mechanical resonance frequencies at the D3m → C2/c ferroelastic transition temperature Ttrans = 453.6 K with no noticeable frequency dispersion. The reduction of resonance frequencies corresponds to 25% softening of the effective elastic constants at Ttrans relative to the value at 700 K. The data analysis indicates that the elastic precursor softening is driven by a displacive soft mode that is coupled to the order-disorder movements of Pb atoms around the rhombohedral threefold axis, which gives rise to local monoclinicity in the paraelastic phase. Finally, resonant piezoelectric spectroscopy (RPS) is used to determine if microstructures are polar in the cubic phase. RPS measurements find no evidence of piezoelectric signals in Pb3(PO4)2, confirming that the possible polar behavior detected using second harmonic generation is due to crystal imperfections.

Effects of radiation damage and thermal annealing on the crystal structure of natural titanite (CaTiSiO(5)) were studied using Raman spectroscopy. The results show that well crystallized natural titanites generally have the P2(1)/a structure at the unit cell level, in contrast to the A2/a symmetry reported previously. Radiation caused by naturally incorporated impurities (such as U and Th) leads to structural damage and amorphization in titanite, as evidenced by a significant loss of band intensity, spectral broadening and frequency shifts. Additional bands (e.g. near 574 and 650 cm(-1)) occur in weakly or partially metamict titanite due to the formation of an intermediate phase (with the A2/a symmetry). Raman spectra of titanite thermal glasses showed features different from those of metamict titanite, especially in the Ti-O and Si-O stretching regions. The effect of thermal annealing is strongly affected by the initial degrees of damage that the sample experienced. Weakly damaged titanite samples showed that annealing leads to a structural recovery, and the spectral patterns of these recovered crystals are consistent with the P2(1)/a symmetry. Highly damaged titanite starts to recrystallize into an A2/a phase near 700-800 K, and additional structural modification occurs when annealed at 1300-1400 K, which involves significant change in broad Ti-O features. However, in terms of bandwidths, the metamict samples are far from fully recovered even on being annealed at 1300-1400 K.

Hematite, Fe(2)O(3), provides in principle a model system for multiferroic (ferromagnetic/ferroelastic) behavior at low levels of strain coupling. The elastic and anelastic behavior associated with magnetic phase transitions in a natural polycrystalline sample have therefore been studied by resonant ultrasound spectroscopy (RUS) in the temperature range from 11 to 1072 K. Small changes in softening and attenuation are interpreted in terms of weak but significant coupling of symmetry-breaking and non-symmetry-breaking strains with magnetic order parameters in the structural sequence R3(¯)c1'-->C2/c-->R3(¯)c. The R3(¯)c1'-->C2/c transition at T(N) = 946 ± 1 K is an example of a multiferroic transition which has both ferromagnetic (from canting of antiferromagnetically ordered spin moments) and ferroelastic (rhombohedral → monoclinic) character. By analogy with the improper ferroelastic transition in Pb(3)(PO(4))(2), W and W' ferroelastic twin walls which are also 60° and 120° magnetic domain walls should develop. These have been tentatively identified from microstructures reported in the literature. The very low attenuation in the stability field of the C2/c structure in the polycrystalline sample used in the present study, in comparison with the strong acoustic dissipation reported for single crystal samples, implies, however, that the individual grains each consist of a single ferroelastic domain or that the twin walls are strongly pinned by grain boundaries. This absence of attenuation allows an intrinsic loss mechanism associated with the transition point to be seen and interpreted in terms of local coupling of shear strains with fluctuations which have relaxation times in the vicinity of ~10(-8) s. The first order C2/c-->R3(¯)c (Morin) transition occurs through a temperature interval of coexisting phases but the absence of an acoustic loss peak suggests that the relaxation time for interface motion is short in comparison with the time scale of the applied stress (at ~0.1-1 MHz). Below the Morin transition a pattern of attenuation which resembles that seen below ferroelastic transitions has been found, even though the ideal low temperature structure cannot contain ferroelastic twins. This loss behavior is tentatively ascribed to the presence of local ferromagnetically ordered defect regions which are coupled locally to shear strains.

We report the simulation of alpha recoil cascades in zirconolite (CaZrTi(2)O(7)) using the molecular dynamics method under a variety of simulation conditions. Interatomic potentials have been developed through fitting to crystalline and melted structures and their energy surfaces obtained from ab initio calculations. Single and double cascades have been modelled, showing that damage recovery in damaged crystalline material is greatly impaired when there are multiple events. The calcium ions are significantly easier to displace than other ions, which is consistent with a recent ab initio study of defect energies.

The dynamic mechanical properties of quartz have been studied as a function of temperature across the α-β phase transition and in the vicinity of the incommensurate (IC) phase on cooling from the β phase. The mechanical response of the IC phase shows strong anelasticity for measurement of Young's modulus (closely related to C(11) in our geometry) with modulated stress driven at 1 Hz. The dynamic shear modulus does not show similar strong effects in its imaginary component, although a very weak anomaly is barely detectable in the real part of the modulus. Our results indicate that the incommensurate microstructures within the quartz transition interval are susceptible to dilatational stress with relaxation times around 1 s.

The elastic and anelastic properties of three different samples of Fe(x)O have been determined in the frequency range 0.1-2 MHz by resonant ultrasound spectroscopy and in the range 0.1-50 Hz by dynamic mechanical analysis in order to characterize ferroelastic aspects of the magnetic ordering transition at T(N) ~ 195 K. No evidence was found of separate structural and magnetic transitions but softening of the shear modulus was consistent with the involvement of bilinear coupling, λe(4)q, between a symmetry-breaking strain, e(4), and a structural order parameter, q. Unlike a purely ferroelastic transition, however, C(44) does not go to zero at the critical temperature, T*(c), due to the intervention of the magnetic ordering at a higher temperature. The overall pattern of behaviour is nevertheless consistent with what would be expected for a system with separate structural and magnetic instabilities, linear-quadratic coupling between the structural (q) and magnetic (m) driving order parameters, λqm(2), and T(N) > T*(c). Comparison with data from the literature appears to confirm the same pattern in MnO and NiO, with a smaller difference between T(N) and T*(c) in the former and a larger difference in the latter. Strong attenuation of acoustic resonances at high frequencies and a familiar pattern of attenuation at low frequencies suggest that twin walls in the rhombohedral phase have typical ferroelastic properties. Acoustic dissipation in the stability field of the cubic phase is tentatively attributed to anelastic relaxations of the defect ordered structure of non-stoichiometric wüstite or of the interface between local regions of wüstite and magnetite, with a rate controlling step determined by the diffusion of iron.

The combined structural and antiferromagnetic phase transition in wüstite, Fe(1-x)O, at the Néel temperature, T(N) ~ 195 K, is treated in terms of coupling between two macroscopic order parameters related to separate discrete instabilities. These each couple with a rhombohedral shear strain to give an indirect mechanism of linear (structural)-quadratic (magnetic) coupling between them. Based on patterns of lattice parameter and magnetic ordering data from the literature, it appears that the overall behaviour matches the general pattern of linear-quadratic coupling for a system with two rather similar instability temperatures. At low pressures, the magnetic instability occurs at a higher temperature than the structural instability but the coupling results in a single phase transition dominated by the influence of the magnetic order parameter. For Fe(0.99)O the magnetic order parameter, m, varies with temperature as ~m(8) ∝ (T(N) - T), while for Fe(0.94)O the temperature dependence is of ~m(4) or ~m(2). It is proposed that, with increasing pressure, there is a crossover of instability temperatures such that the structural instability occurs first at pressures above ~13 GPa. This would be expected to give rise to a first-order phase transition, as appears to occur, but magnetic ordering would still occur simultaneously if the coupling is sufficiently strong. Symmetry analysis and comparison with the behaviour of MnO shows that there are a number of different possible magnetically ordered structures which could be stabilized by pressure or non-hydrostatic stress to give topologically rich phase diagrams.

Elastic and anelastic behaviour of single crystal and ceramic samples of Pb(Mg(1/3)Nb(2/3))O(3) has been investigated at frequencies of ~0.1-1.2 MHz through the temperature interval 10-800 K by resonant ultrasound spectroscopy (RUS). Comparison with data from the literature shows that softening of the shear modulus between the Burns temperature and the freezing interval is independent of frequency. The softening is attributed to coupling between acoustic modes and the relaxation mode(s) responsible for central peaks in Raman and neutron scattering spectra below the Burns temperature, and can be described with Vogel-Fulcher parameters. Shear elastic compliance and dielectric permittivity show similar patterns of temperature dependence through the freezing interval, demonstrating strong coupling between ferroelectric polarization and strain such that the response to applied stress is more or less the same as the response to an applied electric field, with a frequency dependence consistent with Vogel-Fulcher-like freezing in both cases. Differences in detail show, however, that shearing induces flipping between different twin orientations, in comparison with the influence of an electric field, which induces 180° flipping: the activation energy barrier for the former appears to be higher than for the latter. Below the freezing interval, the anelastic loss also has a similar pattern of evolution to the dielectric loss, signifying again that essentially the same mechanism is involved in the freezing process. Overall softening at low temperatures is attributed to the contributions of strain relaxations due to coupling with the local ferroelectric order parameter and of coupling between acoustic modes and continuing relaxational modes of the polar nanostructure. Dissipation is attributed to movement of boundaries between PNRs or between correlated clusters of PNRs. Overall, strain coupling is fundamental to the development of the characteristic strain, dielectric and elastic properties of relaxors.

The structural evolution of Pb(Mg(1/3)Nb(2/3))O(3) (PMN) has been reviewed in terms of characteristic temperatures, length scales and timescales, with a view to considering the overall relaxor behaviour from the perspectives of strain and elasticity. A conventional analysis of lattice parameter data in terms of spontaneous strain and strain/order parameter coupling shows that even though a normal phase transition does not occur the relaxor ordering process is accompanied by a significant volume strain which follows the pattern of a static order parameter evolving according to that expected for a tricritical phase transition with T(c) ≈ 350 K. This matches the evolution of the intensity of the elastic central peak in neutron scattering spectra, and reflects the development of static (or quasistatic) polar nanoregions (PNRs) as if by a mean-field phase transition. Use of a Landau free energy expansion, which includes Γ4(-) order parameter components to describe ferroelectric contributions and an R1(+) order parameter to describe cation ordering together with their formal coupling with strain, then allows the pattern of elastic softening expected for a cubic → rhombohedral phase transition to be anticipated. The extent to which observed softening differs from this static mean-field pattern serves to highlight the additional roles of local heterogeneity and relaxation dynamics in determining the relaxor properties of PMN.

The coupling between order parameters in systems with several instabilities has been analysed within Landau theory. The dominant term considered in this paper is linear in one order parameter and quadratic in the second order parameter ~QP²; other coupling terms have been treated previously. Typical examples for Q are proper or pseudo-proper ferroelastic instabilities, while P might be octahedral tilting in a perovskite, (anti-)ferromagnetic ordering or (anti-)ferroelectric soft modes. Coupling of this type is common in fluorites, Verwey transitions, Jahn-Teller systems, pnictide superconductors, etc. Analytical solutions and characteristic phase diagrams of the stable configurations are compiled. The coupling can lead to stepwise phase transitions even when the uncoupled systems would show continuous transitions. Mixed phases are common, so that many 'intermediate phases' described in the literature may be the result of this linear-quadratic coupling.

Calcium stannate perovskite (CaSnO(3)) has been studied by Raman spectroscopy at two excitation wavelengths (514.5 and 632.8 nm). No phase transition was observed. Rather, the thermal evolution of the Raman lines showed a high degree of harmonicity with small Grüneisen parameters and thermal line broadening following Γ=Acothθ/T, where the quantum temperature θ is determined by the phonon branch without further coupling with other degrees of freedom. The geometrical nature of phonon lines has been identified. High-temperature powder x-ray diffraction measurements provide thermal expansion coefficients of α(x)=13.9 × 10(-6) K(-1), α(y)=2.7 × 10(-6) K(-1), α(z)=14.3 × 10(-6) K(-1). The strongly quasi-harmonic behaviour observed and the lack of any indication of instability with respect to the post-perovskite structure points to the strongly first-order character of the reported perovskite to post-perovskite phase transition in this material, which appears to behave as a very good analogue to MgSiO(3) in the Earth's interior., (© 2011 IOP Publishing Ltd)

High pressure measurements of the ferroelastic phase transition of SrTiO₃ (Guennou et al 2010 Phys. Rev. B 81 054115) showed a linear pressure dependence of the transition temperature between the cubic and tetragonal phase. Furthermore, the pressure induced transition becomes second order while the temperature dependent transition is near a tricritical point. The phase transition mechanism is characterized by the elongation and tilt of the TiO₆ octahedra in the tetragonal phase, which leads to strongly nonlinear couplings between the structural order parameter, the volume strain and the applied pressure. The phase diagram is derived from the Clausius-Clapeyron relationship and is directly related to a pressure dependent Landau potential. The nonlinearities of the pressure dependent strains lead to an increase of the fourth order Landau coefficient with increasing pressure and, hence, to a tricritical-second order crossover. This behaviour is reminiscent of the doping related crossover in isostructural KMnF₃.

The Pnma incommensurate phase transition in the perovskite Pr(0.48)Ca(0.52)MnO(3) at ∼ 235 K is accompanied by shear strains of up to ∼ 2.5% (from neutron diffraction) and changes in the shear modulus of up to ∼ 40% (from resonant ultrasound spectroscopy, RUS), indicating strong coupling between the structural order parameter and strain. In contrast, the antiferromagnetic (AFM) ordering transition at ∼ 180 K displays no detectable static strain, implying that there is either no coupling or only very weak coupling between the magnetic order parameter and strain. Conventional analysis of RUS data, based on measurements of resonance peak frequencies and peak widths, also failed to detect any anomaly in elastic or anelastic properties due to the AFM ordering. A new approach to the analysis the RUS data, based on autocorrelation and convolution of the entire spectra, however, has revealed that the AFM order does indeed affect the elastic behaviour in an unexpected manner. The new analysis shows, firstly, that dynamical fluctuations of the charge density ordering at T > T(c) = 237 K lead to an increase of the RUS amplitude and of the spectral convolution function. Secondly, fluctuations and convolution effects peak at the transition point and decrease with decreasing temperatures. Below 180 K the stripe structure is essentially static. Finally, AFM ordering leads to an increase in the damping of the elastic resonances.

The temperature dependence of the crystal structures of the low-temperature α-phase and the intermediate-temperature β-phase of crystalline malononitrile, CH(2)(CN)(2), have been determined using Rietveld refinement on neutron powder diffraction data. The results confirm that the α-phase has the same structure as the high-temperature γ-phase, and the refined structure of the β-phase is in agreement with previous structure determination using single-crystal x-ray diffraction. The variations of the structure with temperature due to the phase transitions are identified and analysed in terms of molecular rotations and spontaneous strains.

Ferroelectric hysteresis measurements on ceramic lawsonite show a temperature dependence of the remanent polarization P(r) = P(o)Θ(s)(cothΘ(s)/T - cothΘ(s)/T(c)) ∼ Q(2), Θ(s) = 26 K, where Q is the thermodynamic order parameter of the phase transition Pmcn-P 2(1)cn. This almost linear temperature evolution of P(r) proves the improper nature of ferroelectricity in lawsonite. The Curie temperature is T(c) = 124 K. The phase transition is strictly continuous, with a weak conjugated field near the transition point, and hydrogen ordering is discussed as the primary driving mechanism.

Advantage is taken of the wealth of experimental data relating to the evolution with temperature of spin states of Co(3+) in LaCoO₃ in order to undertake a detailed investigation of the mechanisms by which changes in electronic structure can influence strain, and elastic and anelastic relaxations in perovskites. The macroscopic strain accompanying changes in the spin state in LaCoO₃ is predominantly a volume strain arising simply from the change in effective ionic radius of the Co(3+) ions. This acts to renormalize the octahedral tilting transition temperature in a manner that is easily understood in terms of coupling between the tilt and spin order parameters. Results from resonant ultrasound spectroscopy at high frequencies (0.1-1.5 MHz) reveal stiffening of the shear modulus which scales qualitatively with a spin order parameter defined in terms of changing Co-O bond lengths. From this finding, in combination with results from dynamic mechanical analysis at low frequencies (0.1-50 Hz) and data from the literature, four distinctive anelastic relaxation mechanisms are identified. The relaxation times of these are displayed on an anelasticity map and are tentatively related to spin-spin relaxation, spin-lattice relaxation, migration of twin walls and migration of magnetic polarons. The effective activation energy for the freezing of twin wall motion below ~590 K at low frequencies was found to be 182 ± 21 kJ mol(-1) (1.9 ± 0.2 eV) which is attributed to pinning by pairs of oxygen vacancies, though the local mechanisms appear to have a spread of relaxation times. It seems inevitable that twin walls due to octahedral tilting must have quite different characteristics from the matrix in terms of local spin configurations of Co(3+). A hysteresis in the elastic properties at high temperatures further emphasizes the importance of oxygen content in controlling the properties of LaCoO₃.

Anelastic loss mechanisms associated with phase transitions in SrZr(1-x)Ti(x)O(3) perovskites (x = 0.375, 0.450, 0.550, 0.775) have been investigated by dynamic mechanical analysis between 128 and 723 K at frequencies of 0.1-50 Hz. Distinctive patterns of changes in the elastic moduli due to octahedral tilting transitions correlate closely with data for the shear modulus obtained previously by resonant ultrasound spectroscopy at high frequencies (∼0.5 MHz). The I4/mcm <--> Imma transition is first order and has a characteristic minimum in the shear modulus and Young's modulus. For x = 0.450 and 0.550, a dissipation peak occurs at the transition temperature, the maximum of which varies with frequency according to a power law relationship of the form tanδ = Af(n), with n≈ - 0.3. Debye-like dissipation peaks in the stability field of the Imma structure at x = 0.375 have a frequency and temperature dependence consistent with twin wall pinning by defects with an activation energy ∼ 184 kJ mol(-1). These results indicate that there is diversity of pinning and relaxation processes for transformation twin walls and interfaces in different perovskites with I4/mcm, Imma and Pnma structures.

We address the question of whether the structural disorder in the high-temperature β-cristobalite phase of silica can be explained on the basis of domain models using a new technique of phonon-based computer simulations of the pair distribution function. None of the domain models give as good an agreement to experimental data as previously reported atomic configurations derived from a reverse Monte Carlo analysis that are consistent with the rigid unit mode model.

We present ab initio calculations of the longitudinal flexoelectricity for BaTiO(3) and SrTiO(3) using a direct approach. The calculated value for SrTiO(3) agrees with recently reported measurements. For BaTiO(3), however, the theoretical values are smaller than the measured ones; possible reasons for the discrepancy are discussed.

The phase transition in ferroelastic Pb(3)(PO(4))(2)-Pb(3)(AsO(4))(2) mixed crystals shows typical multiscaling behaviour with two relevant length and timescales. One length scale is macroscopic and shows uniform, weakly first-order phase transitions between a rhombohedral paraphase (R3m) and a monoclinic ferroelastic phase (C 2/c). The second length scale is on the level of tetrahedral complexes which display monoclinic distortions at temperatures well above the macroscopic transition point. For instance, in Pb(3)(P(0.43)As(0.57)O(4))(2) the AsO(4) polyhedra show static deformation up to ∼60 K above the deformation of the PO(4) tetrahedra. The two timescales are either short compared with the time of observation, namely the dynamic reorientation of the PO(4) tetrahedral distortion, or very long. The long timescale refers then to the (quasi)static distortion of the AsO(4) tetrahedra which persists at T>T(c). These distortions appear to be uncorrelated or only weakly correlated and their random field leads to an order/disorder aspect of the phase transition which remains, on a phonon timescale, essentially displacive in character.

Resonant ultrasound spectroscopy has been used to characterize elastic softening and anelastic dissipation processes associated with the Pm3m <--> R3c transition in single crystal and ceramic samples of LaAlO(3). Softening of the cubic structure ahead of the transition point is not accompanied by an increase in dissipation but follows different temperature dependences for the bulk modulus, (1/3)(C(11) + C(12)), and the shear components, (1/2)(C(11) + C(12)) and C(44), as if the tilting instability contains two slightly different critical temperatures. The transition itself is marked by the complete disappearance of resonance peaks (superattenuation), which then reappear below ∼700 K in spectra from single crystals. Comparisons with low frequency, high stress data from the literature indicate that the dissipation is not due to macroscopic displacement of needle twins. An alternative mechanism, local bowing of twin walls under low dynamic stress, is postulated. Pinning of the walls with respect to this displacement process occurs below ∼350 K. Anelasticity maps, analogous to plastic deformation mechanism maps, are proposed to display dispersion relations and temperature/frequency/stress fields for different twin wall related dissipation mechanisms. These allow comparisons to be made of anelastic loss mechanisms under mechanical stress with elastic behaviour observed by means of Brillouin scattering at high frequencies which might also be related to microstructure.

Resonant ultrasound spectroscopy has been used to characterize elastic softening and acoustic dissipation behaviour in single crystal and ceramic samples of LaAlO(3) between 10 and 300 K. For the twinned R3c single crystals, average values of the cubic elastic moduli (1/2)(C(11) - C(12)) and C(44) were followed while the ceramic sample provided data for the bulk and shear moduli. A Debye-like dissipation peak occurs in the vicinity of 250 K, from which an activation energy of 43 ± 6 kJ mol(-1) has been obtained. The mechanism for this is not known, but it is associated with C(44) and therefore could be related in some way to the cubic <--> rhombohedral transition at ∼817 K. Slight softening in the temperature interval ~220 --> 70 K of resonance peaks determined by shear elastic moduli hints at an incipient E(g) ferroelastic instability in LaAlO(3). The softening interval ends with a further dissipation peak at ∼60 K, the origin of which is discussed in terms of freezing of atomic motions of La and/or Al away from their high symmetry positions in the R3c structure. LaAlO(3) thus shows evidence of incipient structural instability at low temperatures which is potentially analogous with the phenomenologically rich behaviour of SrTiO(3).

Values for all six independent components of the 3m elastic modulus tensor of LaAlO(3) perovskite are reported. These were determined by means of Brillouin scattering measurements of acoustic velocities in single crystal plates cut parallel to (110) and (100), as defined with respect to the cubic parent structure, and by pure-mode longitudinal and transverse sound velocity measurements along [100], [110] and [111] directions using GHz pulse-echo ultrasonics. The crystals contained intimate intergrowths of twins arising from the Pm3m <--> R3c transition at higher temperature but, in combination with a careful analysis of twin orientation relationships, the two sets of data have allowed a unique solution to be obtained for individual twin components. The new data set represents an important contribution to the characterization of LaAlO(3) single crystals which are widely used as the substrate for a plethora of different thin films with technological applications.

Brillouin spectra have been collected in situ at temperatures up to ∼1000 K for different crystallographic directions from two single crystal plates of LaAlO(3) perovskite. Elastic moduli derived from these, together with heat capacity, spontaneous strain and Raman data from the literature, have then been used to calibrate the coefficients in a classical Landau free energy expansion for the second order [Formula: see text] phase transition at T(c) = 817 K. The static strain/order parameter coupling model provides a quantitative description of elastic softening between room temperature and ∼700 K, but from ∼700 K up to T(c) additional elastic softening correlates with the development of a central peak in the Brillouin spectra. The presence of quasi-elastic scattering, which reaches maximum intensity ∼5-15 K below T(c), implies a strong dynamical component to the phase transition. Relaxation times estimated from the width of the central peak are of the order of ∼10-100 ps and appear to be more or less constant between ∼700 and 800 K, which is consistent with an intrinsic origin associated with phonon density fluctuations. Central peak width variations and an irregular pattern of acoustic velocity variations in a 20 K temperature interval below T(c) are interpreted in terms of flipping of clusters of tilted octahedra between different ⟨111⟩, ⟨011⟩ and ⟨001⟩ tilt axes. The additional softening beyond that expected from the classical strain/order parameter coupling model can be understood in terms of coupling of acoustic modes with the central peak mode(s).

We propose that the properties of a glass transition can be understood on the basis of elastic waves. Elastic waves originating from atomic jumps in a liquid propagate local expansion due to the anharmonicity of the interatomic potential. This creates dynamic compressive stress, which increases the activation barrier for other events in a liquid. The non-trivial point is that the range of propagation of high-frequency elastic waves, d(el), increases with liquid relaxation time τ. A self-consistent calculation shows that this increase gives the Vogel-Fulcher-Tammann (VFT) law. In the proposed theory, we discuss the origin of two dynamic crossovers in a liquid: (1) the crossover from exponential to non-exponential and from Arrhenius to VFT relaxation at high temperature and (2) the crossover from the VFT to a more Arrhenius-like relaxation at low temperature. The corresponding values of τ at the two crossovers are in quantitative parameter-free agreement with experiments. The origin of the second crossover allows us to reconcile the ongoing controversy surrounding the possible divergence of τ. The crossover to Arrhenius relaxation universally takes place when d(el) reaches system size, thus avoiding divergence and associated theoretical complications such as identifying the nature of the phase transition and the second phase itself. Finally, we discuss the effect of volume on τ and the origin of liquid fragility.

The ferroelastic phase transition and shape memory effect in La-modified lead zirconate titanate ferroelectric ceramics are demonstrated directly through the temperature-dependent macroscopic recoverable strain measured in a three-point bending configuration. X-ray diffraction measurements reveal that non- 180° domain switching occurs in the mechanically poled sample in two different ways at the bottom and top of the sample which have been under tensile and compressive stresses, respectively. The calculated fraction of non- 180° switched domains in the poled sample increases nonlinearly with the applied force and shows a saturation trend, which is consistent with the nonlinear behavior of the remnant strain. This study confirms that the mechanical stress applied upon cooling ferroelectric ceramics from the paraelectric to the ferroelectric phase can easily activate ferroelastic domain switching and give rise to preferred domain orientation and consequent macroscopic remnant strain which results in a history effect and shape memory effect via the ferroelastic phase transition.

The sequence of phase transitions due to octahedral tilting across the Sr(Zr,Ti)O(3) solid solution series has been investigated by resonant ultrasound spectroscopy at high and low temperatures using ceramic samples. The elastic behaviour associated with phase transitions as a function of composition in Sr(Zr,Ti)O(3) at room temperature is proposed to be analogous to that as a function of temperature in SrZrO(3), with the [Formula: see text] transition at SrZr(0.57)Ti(0.43)O(3), [Formula: see text] at SrZr(0.35)Ti(0.65)O(3), and [Formula: see text] at SrZr(0.05)Ti(0.95)O(3). Changes in elastic constants and acoustic dissipation with temperature have been analysed for samples across the compositional range. The intermediate phases, I4/mcm and what is assumed to be Imma, appear to have stability fields across the full compositional range and both show large dissipation effects, most probably due to twin wall mobility. In contrast, the Zr-rich Pnma phase, which should contain transformation twin walls, is an unexpectedly stiff and non-dissipating material, similar to the high temperature and/or Ti-rich [Formula: see text] phase. In the case of Pnma, this is attributed to coupling between the two order parameters, which could impede relaxation responses to an applied stress. The [Formula: see text] structure is a classically stiff cubic perovskite and no transformation-related dissipation processes are expected.

Elastic behaviour associated with the hierarchy of tilting transitions in SrZrO(3) has been examined using resonant ultrasound spectroscopy on a ceramic sample at temperatures between 153 and 1531 K. Changes in slope of the evolution of resonance frequencies with temperature indicate that phase transitions occur at 1038 K ([Formula: see text]), 1122 K ([Formula: see text]), and 1367 K ([Formula: see text]). Strain analysis of previously recorded neutron diffraction data shows that the [Formula: see text] and [Formula: see text] transitions are close to tricritical in character, and that [Formula: see text] is first order. Deviations from the form of the elastic behaviour predicted by Landau theory are found. In particular, elastic softening in the vicinity of the [Formula: see text] transition suggests that local dynamical fluctuations between individual tilt systems occur, rather than a discontinuous switch from one phase to another. Determinations of the mechanical quality factor, Q, show that SrZrO(3) in the [Formula: see text] phase is a classically high-Q (i.e. non-dissipating) cubic material. I4/mcm and Imma phases both have much greater dissipation (low Q), which is tentatively attributed to the mobility of twin walls. The room temperature Pnma phase is unexpectedly much stiffer than both I4/mcm and Imma phases and has high Q. It appears that when two separate tilt systems operate, as in Pnma, they can interact to reduce strain/order parameter relaxations.

The Landau potential of the martensitic phase transformation in Ni(46.8)Ti(50)Fe(3.2) was determined using high resolution x-ray diffraction to measure the spontaneous strain and calorimetric measurements to determine the excess specific heat of the R phase. The spontaneous strain is proportional to the square of the order parameter which is tested by the relation of the excess entropy and the order parameter. The parameters of the Landau free energy were determined by fitting the temperature evolution of the order parameter and using the scaling between the excess entropy and the order parameter. The double well potential at absolute zero temperature was calculated and the interface energy and domain wall thickness were estimated.

Recent developments in and around the SIESTA method of first-principles simulation of condensed matter are described and reviewed, with emphasis on (i) the applicability of the method for large and varied systems, (ii) efficient basis sets for the standards of accuracy of density-functional methods, (iii) new implementations, and (iv) extensions beyond ground-state calculations.

A real-space atomistic refinement approach to the analysis of experimental electron diffraction patterns is described. The method employs the reverse Monte Carlo algorithm to produce atomistic configurations capable of qualitatively reproducing diffuse electron scattering patterns. Its implementation in the program EDRMC is described in detail, together with a number of additional constraints/restraints that can be used to guide the refinement process. In particular, appropriate restraints ensure the individual atomic displacements introduced to model the diffuse scattering patterns are simultaneously consistent with the known average structure. The approach is then used to interpret electron diffraction patterns measured for Bi(2)Ru(2)O(7-δ). The diffuse scattering patterns observed are shown to arise primarily from concerted translations of Bi atoms. These translations can be interpreted in terms of rotations of [O(')Bi(4) ] tetrahedra correlated along the [Formula: see text] crystal axes and uncorrelated along orthogonal directions.

We use neutron total scattering measurements with reverse Monte Carlo analysis methods incorporating an atom-swapping algorithm to identify the short-range Ca/Sr cation ordering within the Ca(x)Sr(1-x)TiO(3) solid solution (compositions x = 0.2,0.5,0.8). Our results show that nearest-neighbour pairs have a strong tendency for unlike Ca/Sr first-neighbour coordination in the x = 0.2 and 0.5 cases. In the x = 0.5 case the Ca/Sr ordering results in a structure with space group P 2(1)nm. In contrast, there is much less short-range cation ordering in the x = 0.8 case.