Phason Mode Research Articles

Optical study on incommensurate phase transitions of (C3H7NH3)2MnCl4.

The optical activity and birefringence of (${\mathrm{C}}_{3}$${\mathrm{H}}_{7}$${\mathrm{NH}}_{3}$${)}_{2}$${\mathrm{MnCl}}_{4}$ were measured with use of the high-accuracy universal polarimeter (HAUP) method. An important property manifested by the incommen- surate \ensuremath{\gamma} phase, i.e., reentrant phase transition, was studied in light of the knowledge that the optical gyration directly reflects the order parameter of the incommensurate modulation. Critical exponents \ensuremath{\beta} for the temperature variations of gyration in both high- and low-temperature regions of the \ensuremath{\gamma} phase were found to be nearly constant. Meanwhile, it is known that the values of critical exponents \ensuremath{\beta} of several ${\mathit{A}}_{2}$${\mathit{BX}}_{4}$-type incommensurate crystals are also in the same range \ensuremath{\sim}0.42--0.47, as determined by our recent HAUP experiments. These facts indicate that the incommensurately modulating waves of the \ensuremath{\gamma} phase appear as a helical phason mode and the \ensuremath{\gamma} phase reenters into a single phase accompanying two transitions of the same mechanism.

The mechanism of charge-density-wave pinning, excitations of the pinned condensate

The static and dynamic properties of the pinned charge-density-wave (CDW) is studied in the blue bronze, K 0.3MoO 3. High pressure suppresses the CDW order parameter and simultaneously the pinning energy decreases. Their relation favours weak pinning theories. The condensate may resonate around the equilibrium position but it can also be excited at low frequencies if screening by normal carriers allows internal deformations. The latter phason mode freezes out at low temperatures. This process is investigated by dielectric measurements.

Order parameter fluctuations in ferroelectric liquid crystals

Abstract The ferroelectric Sm C * phase1 provides a system which has gained an increasing amount of interest in the last few years. It represents an orientationally ordered liquid with a one-dimensional density modulation. The periodicity of the helicoidal orientational ordering is generally incommensurate to the periodicity of the density modulation. To describe this phase one introduces1 two two-component order parameters: the tilt of the director with respect to the normal to the smectic planes and the in-plane spontaneous polarization The order parameter fluctuation spectrum thus consists of four coupled tilt polarization modes2: two “hard” modes describe out-of phase polarization-tilt angle fluctuations and have relaxation frequencies in the 500 MHz range whereas the two in-phase polarization-tilt angle fluctuation modes occur below 1 MHz (Figure 1). They are commonly denoted as the soft (amplitudon) mode and the Goldstone (phason) mode.

Incommensurate phase transitions and optical activity.

Recent experiments with the high-accuracy universal polarimeter method have revealed optical activity of the incommensurate phases of ${\mathit{A}}_{2}$${\mathit{BX}}_{4}$ crystals. The origin of this important phenomenon is explained. A pure imaginary perturbation that takes place in the Hermitian dynamical matrix induces two circularly helical lattice waves. One of them is condensed in the same form in the incommensurate phase. This is also the physical picture of the existence of the phason mode. The symmetry of the crystal becomes ${\mathit{D}}_{\mathrm{\ensuremath{\infty}}}$, which permits optical activity. The theory is consistent with the Lifshitz condition and with our experimental results.

On some esr features of Ti2+relaxing paramagnetic defects in K2SeO4crystals with incommensurate phase

Abstract The TI2+ centers in K2SeO4 crystals have been shown to be the relaxing defects. The anomalous Tl2+ spectra temperature transformation have been treated in terms of long-range correlations induced by the soft or phason mode between the defects. The theoretical results are in a qualitative agreement with our ESR data and seem to be quite realistic.

Global universality in the Frenkel-Kontorova model.

The Frenkel-Kontorova model of a chain of atoms in an external periodic potential is studied. The phase diagram of this model contains infinitely many tongues of commensurate phases separated by gaps of incommensurate structures. The period of these structures is described by a devil's staircase (DS) function when the parameters which define the model are varied. The model exhibits a critical line above which the phason mode of the incommensurate phases is pinned and the DS is complete, while below the line the phason mode is unpinned and the DS is incomplete. We evaluate the critical line numerically and show that it has a fractal nature. The Hausdorff dimension ${D}_{0}$ and the spectrum of singularities f(\ensuremath{\alpha}) of the gaps along the critical line are calculated. The analysis is performed for several forms of the periodic potential. The resulting ${D}_{0}$ and f(\ensuremath{\alpha}) seem to be independent of the details of the potential with ${D}_{0}$=0.87\ifmmode\pm\else\textpm\fi{}0.02. It is interesting to note that ${D}_{0}$ is equal, within the numerical uncertainty, to the Hausdorff dimension corresponding to the critical line of dissipative systems, although the f(\ensuremath{\alpha}) of the two cases are found to be different.

Computer simulations of a distorted reciprocal lattice of an Al-Li-Cu single quasicrystal.

An Al-Li-Cu single quasicrystal was investigated using x-ray precession photography. The x-ray diffraction patterns show that the symmetries of the quasicrystal have an m3 point-group symmetry which deviates from the symmetry of a perfect icosahedron. The distortions of the reciprocal lattice are explained using the soft phason mode, and an explicit expression of the linear phason strain is given. Computer simulations of x-ray precession photographs taken along fivefold, threefold, and twofold axes agree well with the experiment.

Static and dynamic properties of incommensurate smectic-AIC liquid crystals.

We study the elasticity, topological defects, and hydrodynamics of the recently discovered incommensurate smectic (${A}_{\mathrm{IC}}$) phase, characterized by two collinear mass density waves of incommensurate spatial frequency. The low-energy long-wavelength excitations of the system can be described by a displacement field u(x) and a ``phason'' field w(x) associated, respectively, with collective and relative motion of the two constituent density waves. We formulate the elastic free energy in terms of these two variables and find that when w=0, its functional dependence on u is identical to that of a conventional smectic liquid crystal, while when u=0, its functional dependence on w is the same as that for the angle variable in a slightly anisotropic XY model. An arbitrariness in the definition of u and w allows a choice that eliminates all relevant couplings between them in the long-wavelength elastic energy. The topological defects of the system are dislocations with nonzero u and w components. We introduce a two-dimensional Burgers lattice for these dislocations, and compute the interaction between them. This has two parts: one arising from the u field that is short ranged and identical to the interaction between dislocations in an ordinary smectic liquid crystal, and one arising from the w field that is long ranged and identical to the logarithmic interaction between vortices in an XY model. The hydrodynamic modes of the ${A}_{\mathrm{IC}}$ include first- and second-sound modes whose direction-dependent velocities are identical to those in ordinary smectics. The sound attenuations have a different direction dependence, however. The breakdown of hydrodynamics found in conventional smectic liquid crystals, with three of the five viscosities diverging as 1/\ensuremath{\omega} at small frequencies \ensuremath{\omega}, occurs in these systems as well and is identical in all its details. In addition, there is a diffusive phason mode, not found in ordinary smectic liquid crystals, that leads to anomalously slow mechanical response analogous to that predicted in quasicrystals, but on a far more experimentally accessible time scale.

Noise and Shapiro step interference in the charge-density-wave conductor K 0.3MoO 3

Extremely thin (transverse dimension ∼0.2 μm) samples of the charge density wave (CDW) conductor K 0.3MoO 3 are found to display unusually coherent narrow-band noise spectra and well defined Shapiro step interference structure. The results suggest an intrinsic pinning potential with periodicity equal to the CDW wavelength. The Shapiro step interference appears to arise from a coupling of the external electric field to the low frequency dielectric relaxation mode rather than to the high frequency pinned phason mode.

A solvable model with a soft phason mode at the lock-in transition

A continuum one-dimensional model for the modulated structure characterised by the one-component order parameter is considered. It is shown that a special choice of the local potential with hard walls allows for an analytical calculation of the energy and the modulation period variation within the modulated phase. The analysis of the model yields a finite value of the modulation period at the lock-in transition, but also a softening of the whole phason branch. The lock-in transition is thus of the soft mode type, in contrast with the situation in analogous models considered so far.

Magnetic excitations in the longitudinally polarized antiferromagnetic phase of praseodymium.

In the ordered phase of Pr, which can be induced by the application of a uniaxial pressure, the length of the moments is modulated sinusoidally with a period incommensurate with the lattice. A comprehensive theory has been developed to describe the ground-state properties and the excitation spectrum of this antiferromagnetic phase. Numerical results, based on parameters determined from paramagnetic Pr, are compared with neutron scattering experiments. In order to account for linewidth effects several mechanisms which broaden the excitations are considered. Contributions due to the incommensurate modulation are included but are found to be small. To first order in the high-density (1/z) expansion, the effects of the single-site fluctuations vanish exponentially in the zero-temperature limit. The second-order contributions, derived in the present paper, are then the leading-order terms at low temperatures and are found to be important. Additionally, the scattering against electron-hole pair excitations of the conduction electrons is treated by incorporating it directly in the 1/z expansion. The theory yields a reasonably realistic description of the neutron scattering results. It can be concluded that the phason mode, near the ordering wave vector, is a damped diffusive mode. The principal remaining puzzle, the quasielastic peak not centered at the ordering wave vector, is tentatively ascribed to heavy-fermion behavior in Pr.

Field dependence of the phason absorption in K2SeO4: A Negative Result

Abstract Electric field dependence of the millimetre-wave absorption due to ir active phason mode was measured in K2SeO4. No changes have been detected for fields up to 6 kV/cm. The relative phason frequency change is therefore probably smaller than 1% for E — 1 kV/cm.

Devil's staircase and order without periodicity in classical condensed matter

The existence of incommensurate structures proves that a crystalline ordering is not always the most stable one for non-quantum matter. Some properties of structures which are obtained by minimizing a free energy are investigated in the Frenkel-Kontorova and related models. It is shown that an incommensurate structure can be either quasi-sinusoidal with a phason mode or built out of a sequence of equidistant defects (discommensurations) which are locked to the lattice by the Peierls force. In that situation the variation of the commensurability ratio with physical parameters forms a « complete devil's staircase » with interesting physical consequences. Some general results for all structures which minimize a free energy are given. In addition to the known crystal and incommensurate structures, the existence of a new class of structures which have local order at all scale is predicted. Properties of the new class are described in physical terms and possible applications to certain amorphous or non-stoichiometric compounds are discussed.

Phase Mode in Fluctuating Charge-Density-Wave State of One-Dimensional Systems

The lattice vibration of a one-dimensional incommensurate lattice is investigated at finite temperature by using the dynamical matrix approach. We consider the temperature region where the short ranse order or a charge density wave is well develored. By a transformation of coordinates, the dynamical matrix is separated into two terms, the dominant term or which coincides with that or the state with long range order. Using the perturbation theory it is found that the spectrum of the phase mode has a finite gap. It is shown that the phason mode appearing in the dynamical structure factor has the gap and broadening.

Evidence for soft modes at [formula omitted] ≠ 0 in the far-infrared spectrum of Rb 2ZnBr 4

The possibility to observe optically inactive and q → ≠ 0 ≠ 0 soft modes in an indirect way by using far-infrared techniques is investigated. Using the fact that, due to anharmonicity, all modes of lattice vibrations are coupled with each other, the softening of one particular mode can be seen indirectly in the damping of other modes. By applying this idea to the modulated structure Rb 2ZnBr 4, the infrared inactive soft mode at the normal-incommensurable transition, the soft phason mode at the lock-in transition as well as a soft mode behaviour at an intermediate temperature have been observed.

Pseudopotential Formalism for Structural Energy and Dynamical Motions in Quasi One-Dimensional Conductors

The structural energy and the dynamical motions in quasi one-dimensional (1-D) conductors are investigated microscopically from the view point of the pseudo-potential model for such systems developed by the author. When the 1-D Peierls instability is well-developed, the ±2 k F Fourier components of the pseudopotential is absorbed into the unperturbed Hamiltonian. The stabilizing energy of the Peierls state is quartic with respect to the quasi long-range order (QLRO) parameter. Explicit expressions are given for the structural energy including the commensurability energy. Including the effect of the QLRO into the phonon Hamiltonian, the soft phonon (phason mode) and the phonon with an energy gap (amplitude mode) are naturally derived at | q |=2 k F = Q . The phason dispersion near ± Q is derived and compared with the result derived by the conventional method.

Far infrared and millimetre dielectric response of incommensurate and ferroelectric K2SeO4

Dielectric response of K2SeO4 in the spectral region 5–460cm−1 was determined using transmissivity and reflectivity measurements as a function of temperature between 80 and 300K. The spectral features above 20cm−1 are interpreted using results of lattice vibrational analysis in three known commensurate phases. The low-frequency dielectric anomaly in the incommensurate phase can be roughly described by critical slowing-down of a Debye relaxation given rise to by the overdamped infrared active phason mode which softens at the incommensurate-commensurate transition.

One- to Three-Dimensional Crossover in the Lattice Dynamics ofHg3−δAsF6: Low-Temperature Specific Heat

The temperature dependence (0.35 K to 6 K) of the low-temperature specific heat of the linear chain mercury compound, ${\mathrm{Hg}}_{3\ensuremath{-}\ensuremath{\delta}}\mathrm{As}{\mathrm{F}}_{6}$ is reported. The one- to three-dimensional crossover in the low-temperature lattice specific heat due to the phase-ordered incommensurate Hg chains is observed. The results are analyzed with use of Tarasov's phenomenological theory for the specific heat of the weakly interacting one-dimensional chains. The specific-heat results yield a value for the phason mode energy, $\frac{\ensuremath{\Delta}}{{k}_{\mathrm{B}}}=1.75$ K.

Infrared and Raman activity of soft modes in the incommensurate structure

Low-frequency normal modes of the incommensurate structure are discussed using the example of K2SeO4. The amplitudon and phason branches are calculated and their activity in infrared and Raman spectra is determined. The observed static dielectric anomaly is explained by softening of the infrared-active phason mode when the wavevector of the incommensurate wave approaches its commensurate value.

Charge density waves effects on the phonon modes of 1T-TaS 2

The effects of lattice distortion driven charge density waves on the 1T polytype of TaS 2 are reported. The results of a raman study of this transition metal dichalcogenide show features consistent with results previously reported using resistivity and electron diffraction techniques. The possibility of the presence of a phason mode in the raman spectra is suggested and the temperature dependence of this mode is fitted to the expected functional form.