X-ray diffraction of incommensurate structures in the soliton regime

In order to elucidate the sensitivity of X-ray diffraction to the structural differences in an incommensurate structure between the sinusoidal and soliton regimes, a general expression for the structure factor of a displacive incommensurate structure in an ideal soliton regime is obtained in terms of the so called “atomic scattering modulation factors”, which reduce the expression to a form analogous to that of a commensurate structure. Consequently an approximate relationship between the intensities of particular sets of satellites is determined and compared with that expected in the sinusoidal regime. It is also shown that an ideal soliton regime gives place to a particular extinction rule, and the corresponding average structure has the atomic positions splitted like a disordered structure. These results allow to explain some recently reported measurements in Rb₂ZnCl₄.     

Crystal structure of martensite and intermediate phases in Ni<Subscript>2</Subscript>MnGa studied by neutron diffraction

We have investigated crystal structures of martensite and intermediate phases in stoichiometric Ni₂MnGa. The neutron diffraction profile of the martensite phase measured at T = 4.2 K exhibits four satellites between [2 0 0]_P^* and [0 2 0]_P^* reflections (P stands for the parent phase) at incommensurate positions of [h 2-h 0]_P^* with h = 0.428, 0.863, 1.136 and 1.572. The profile of the intermediate phase measured at T = 210K exhibits two satellites between [2 0 0]_P^* and [0 2 0]_P^* reflections at incommensurate positions of [h 2-h 0]_P^* with h = 0.343 and 1.657. Although each satellite of the martensite phase moves toward its nearest fundamental reflection as temperature increases, that of the intermediate phase does not move significantly. On the contrary, the intensity of each satellite decreases significantly in the intermediate phase as temperature increases while not in the martensite phase. A synchrotron X-ray diffraction and a Rietveld analysis of the result reveal that, for both the phases, the displacement of atoms from the parent phase are represented by a sine wave whose propagation vector is parallel to [1 1 0]_P.     

Scattering of light in a ferroelectric with incommensurate phase: K2ZnCl4

We presented and analyzed spectra of Raman scattering in various phase states of K₂ZnCl₄ crystals, including the incommensurate phase. On the basis of this analysis we determined the parameters of modes associated with internal and external motions of ions. We found that the phase mode reveals itself in the Raman scattering spectra; this mode is responsible for the instability of the crystal lattice in the vicinity of the incommensurate phase transition.     

Nonlinear phenomena in A2MX4 type crystals with incommensurate phases

Investigations of the nonlinear phenomena in A₂MX₄ type crystals (A, organic cation or alkaline metal ion; M, metal ion; X, halogen ion) with incommensurate phases are very interesting from both a scientific and a practical point of view. In the crystals many interesting nonlinear phenomena are observed, e.g., thermal hysteresis of birefringence and absorption coefficient, piezooptic effect, electrooptic effect. Different anomalies (e.g., a discontinuity or change of sign of the briefringence coefficient, a distinct increase of the value of piezooptic and electrooptic coefficients) exist in the phase transition regions. They are especially distinct for a transition from the incommensurate phase to the commensurate phase. Some possibilities of detection and recognition of the existence of incommensurate phases by optical methods are suggested.     

Metastable states and defect density waves in modulated ferroelectrics

The influence of defects on the phase behaviour of Rb-doped K₂ZnCl₄-single crystals is studied using high-resolution gamma-ray diffractions in combination with in situ dielectric impedance spectroscopy. It is shown that the interaction between strain fields near discommensurations within the incommensurate phase and impurity ions lead to the formation of defect density waves. On rapid cooling, these are unable to relax and persist within the commensurate phase thus acting as an intrinsic memory. Hence, a metastable re-entrant incommensurate phase is observed on heating at temperatures well below the normal lock-in temperature that is found in equilibrium studies.     

Ferroelastic crystals and their nonlinear elastic properties observed in frequency range from gigahertz to milihertz

The experiments reported in this article have been performed to unify the results of earlier and recent determinations of parameters describing the elastic properties of Rb₄LiH₃(SO₄)₄, LiCsSO₄ (LCS) and KH₂PO₄ crystals measured by different experimental methods. The above crystals undergo a second-order phase transition, incommensurate phase transition (PT) and a first-order but close to second-order phase transition, respectively. To investigate the elastic properties of the crystals in the frequency range from 10^?1 to 10¹⁰ Hz, five experimental methods were applied: dynamic mechanical analysis, piezoelectric resonance, composite oscillator bar, ultrasonic wave propagation and Brillouin light scattering. The results of the experiments allowed identification of the processes contributing to the elastic response of the crystals investigated. Moreover, for the LCS crystal a model of the incommensurate PT was presented.     

Fluctuation-induced broadening of Mn2+ EPR lines in the incommensurate phase of Rb2ZnCl4 crystals

The EPR spectra of Mn²⁺ ions in Rb₂ZnCl₄ crystals is investigated in the vicinity of the transition from the paraelectric phase to an incommensurate modulated phase. When these crystals are cooled below the transition temperature T _i =304 K, a splitting of the resonance lines is observed in the singular spectrum. A one-harmonic model is used to discuss the contributions that fluctuations in the amplitude and phase of the incommensurate displacement wave make to the local width of the singular spectra. It is shown that anomalies in the local width of the low-temperature singular peaks observed in the vicinity of T _i are caused by amplitude fluctuations. Fiz. Tverd. Tela (St. Petersburg) 41, 1668–1674 (September 1999)     

Influence of crystal synthesis on the incommensurate phase and lock-in transition of Rb2ZnCl4 crystals

A comparative study of several crystals of Rb₂ZnCl₄, obtained by different crystal growing methods, has allowed us to determine the influence of growth defects on the incommensurate phase and on the lock-in transition of these samples. X-ray diffraction has allowed us to complete previous dielectric measurements realized on the same samples and to relate the crystalline quality to the evolution of the modulation as a function of the temperature. The principal influence of an increasing defect density seems to be a stronger pinning of the modulated phase and this induces a lower lock-in temperature and a wider hysteresis.     

Bi2Sr2CaCu2Oy单晶调制结构的X射线衍射研究

用Ｘ射线转靶衍射仪，详细地观察和分析了Bi₂Sr₂CaCu₂O_y单晶的调制结构．指出布喇格反射的卫星峰是沿ｃ^*拉长的，与Ｂ心正交对称结构不相符的宽化峰正是由于这种拉长的结果，且导致卫星峰强度的不对称性．Ｂｉ-Ｏ层和钙钛矿层晶格失配形成的超晶格为非正交而是单斜的，并可能存在超晶格孪晶     

Low temperature phase transitions and magnetic structure of PrB6

Specific heat, thermal expansion and electrical resistivity measurements on PrB₆ single crystals show that there are two low temperature phase transitions at 6.9 K and 4.2 K, respectively, the latter temperature varying somewhat among different crystals. Neutron diffraction measurements were made on both single and polycrystalline samples of PrB₆. The neutron data indicate a spontaneous incommensurate magnetic ordering at 6.9 K with Q = (0.23, 0.23, 0.5) 2π/a_o. At 4.2 K a commensurate magnetic phase is seen with Q = (0.25, 0.25, 0.5) 2π/a_o coexisting with the incommensurate phase At 1.74 K, only the commensurate phase remains. A model is proposed for the commensurate antiferromagnetic structure and a profile analysis based on that model yields a magnetic moment of 1.77 Bohr magnetons per praseodymium ion at 1.74 K.     

The aging effect within incommensurate phase in TlGaSe2

The effects of physical aging on the dielectric constant of TlGaSe₂ layered ferroelectric crystal were investigated. Isothermal aging processes were done within incommensurate phase in TlGaSe₂ ferroelectric single crystals when slow cooling was interrupted for a time at a constant temperature. The dielectric constant exhibited an unusual behaviour: it decreased exponentially with time. The results can be interpreted taking into account the interaction of the structure of the incommensurate phase with crystal fields of defects randomly distributed in the crystal structure. The measurements show that there are two relaxation mechanisms associated with two relaxation sites in mentioned temperature interval. This means that two different types of incommensurate phases occur in mentioned temperature interval.     

Pinning effect on microwave dielectric properties and soft mode in TlInS2 and TlGaSe2 ferroelectrics

This paper presents the results of an investigation of microwave dielectric dispersion in the proper semiconductive ferroelectrics TIInS₂ and TIGaSe₂ with an incommensurate structure modulation. In these crystals there is a strongly overdamped soft ferroelectric mode, whose frequency in the vicinity of the phase transitions drops to the millimetre wave region and causes dielectric microwave dispersion plus a high contribution to the static dielectric permittivity. Within the incommensurate phase crystal defects, such as impurities, cause pinning of the soft mode. Because of this pinning effect the phason frequency increases. Pinning also changes the dynamical dielectric properties and the contribution of the phason and amplitudon to the static permittivity.     

Symmetry determination of commensurate and incommensurate phases of K2SeO4 by EPR

From a reinterpretation of EPR results on γ-irradiated K₂SeO₄ crystals, previously reported by Aiki et al., the space group symmetry for commensurate (ferroelectric) phase of K₂SeO₄ is determined to be Pna2₁. The EPR results support the modulation of polarization as the reason for disappearance of ferroelectricity in incommensurate phase.     

Manifestation of Spatial Modulation in the Vibration Spectra of (N(CH3)4)2CuCl4 Crystals

The specific manifestations of the spatial modulation in the vibration spectra of (N(CH₃)₄)₂CuCl₄ crystals have been investigated. It has been shown that because of the narrow temperature range of existence of the incommensurate phase and the complexity of the spectra being analyzed, reliable resolution of additional modes arising under the influence of both incommensurate and commensurate modulation is only possible using a computer approximation of the spectra obtained. In accordance with the theoretical models considered, additional modes have been observed mainly in the region of low-frequency skeletal vibrations exhibiting a marked dispersion.     

Thermal hysteresis and memory effect in the ferroelectric incommensurate tetramethylammonium tetrachloro cobaltate

Dielectric susceptibility measurements checking thermal hysteresis and memory effects in (N(CH₃)₄)₂CoCl₄ crystals in the temperature range 273–298K are reported. The differences between the values measured in the heating and cooling curves include a definite difference between the dielectric susceptibility maxima when entering and exiting the ferroelectric phase. Thermal hysteresis loops constructed in one of the incommensurate phases differ Considerably from those previously reported in Rb₂ZnCl₄. Characteristic memory effects could also be observed with the same order of magnitude as those found in other incommensurate materials.     

Propagation of light in Rb2ZnCl4-type crystals with incommensurate modulation

Wave equation for optical waves in the incommensurate phase of Rb₂ZnCl₄-type crystals is solved taking into account spatial dependence of optical permittivities due to incommensurate modulation of the lattice. Using the procedure originally proposed by Golovko and Levanyuk it is shown that for the wave vectors parallel to the orthorhombic axes: a) the waves are not purely transverse, b) their amplitudes are spatially modulated and c) their velocities are changed so that corrections to refractive indices exist due to permittivity modulation. Data on birefringence of Rb₂ZnCl₄ are analysed and it is shown that these corrections are responsible for the jump-like changes of refractive indices at the incommensurate-ferroelectric phase transition, together with the quasi-discontinuous change of the order parameter modulus. The spatial variance of the wave amplitudes may be responsible for the possibility of SHG in the incommensurate phase whose averaged point group is centrosymmetric.The authors gratefully acknowledge helpful discussions with Dr. V. Dvoák.     

Incommensurate cooperative tilting modes of MnF6 octahedra relatedto the structural phasetransitions in BaMnF4

The structural phase transitions of the layer compound BaMnF₄ were studied by high-resolution X-ray diffraction using synchrotron radiation. The intensities and profiles of two kinds of superlattice reflections having incommensurate reduced wave vectors q ₁= (～ ± 1/5,0,0) _p and q ₂ =(～ ± 2/5,1/2,1/2)_p, respectively, were measured as a function of temperature from 25 K to 280 K. These temperature dependencies show that incommensurate structural phase transitions of second order occur at 234 K and 244 K. These structural phase transitions are interpreted as successive condensations of a folding-screen-like incommensurate plane-distortion mode and a commensurate anti-ferro-distortive tilting mode of the MnF₆ octahedra around the primitive a₀ - and b₀ -axes, when cooled down. It is also found that there is another structural phase transition at about 45 K related to a precursor structural distortion for the antiferromagnetic transition occurring at about 26 K.     

Existence of an incommensurate phase in a Li2B4O7 crystal

The x-ray diffraction spectra of Li₂B₄O₇ single crystals are investigated in the temperature range 80–300 K, and the lattice parameter c is determined in the same temperature range in the presence of a periodically varying temperature field. An incommensurate phase is not observed anywhere in the temperature range investigated, regardless of whether the crystals are subjected to a periodic temperature field. Fiz. Tverd. Tela (St. Petersburg) 39, 1461–1463 (August 1997)     

Low-temperature phase transition in Ba2TiGe2O8

Previous work on Ba₂TiGe₂O₈ crystals has shown an unusual low-temperature (～ 223 K on cooling, ～ 273 K on heating) phase transition. Precession x-ray photographs on Ba₂TiGe₂O₈ single crystals show an incommensurate modulation along b*, and, for the first time, also along a*. Single crystal intensity data confirm the average structure in space group Cmm2. There is positional disorder in the pyrogermanate groups, and this is the probable cause for the modulated structure. The low-temperature phase transition is proposed to be a lock-in transition, with the modulation along a* locking in at a value of 1/3. Several properties, as well as other unusual features of the low-temperature phase transition, are discussed in light of the proposed lock-in transition. Domain studies show that the ferroelastic domains are unstable in the low-temperature phase.     

Problems in polarized light microscopy observation of birefringence of calcium pyrophosphate dihydrate crystals

To reconsider the problems arising from the use of the phase plate as a test plate inserted into a polarized light microscope system for the analysis of triclinic calcium pyrophosphate dihydrate (t-CPPD) crystals, or Ca₂P₂O₇·2H₂O in the synovial fluid of arthritis patients, we made the polarized light microscopy observations using a phase plate with a retardation of 530 nm for the synthesized t-CPPD crystals well-characterized by X-ray powder pattern indexing and single crystal X-ray diffraction measurements. The microscopy observations were made of crystals of different sizes, thicknesses and shapes. The retardation was assessed using the interference color chart at four extinction and diagonal positions both with and without the test plate. The addition and subtraction states produced by superimposing the retardations of two anisotropic materials, that is, the t-CPPD crystal and the 530 nm phase plate, were deduced from the interference color change by inserting the test plate at four diagonal positions. When the color change of a crystal at a diagonal position resulting from 90-degree rotation exhibits no clear birefringence, the interference color chart was shown to be useless. We suggested the use of a compensator whose retardation can be changed to obtain an accurate value for the retardation of the crystal.