Soft Acoustic Mode Research Articles

Novel magnetoacoustic effects in heavy-Fermion systems in high magnetic fields.

We observed a strong softening of longitudinal acoustic modes in ${\mathrm{UPt}}_{3}$ and ${\mathrm{CeRu}}_{2}$${\mathrm{Si}}_{2}$ in high magnetic fields where the magnetic susceptibility exhibits an anomaly. The two phenomena can be correlated with a large electronic Gr\"uneisen parameter. Some shear modes exhibit oscillations in the same high-field and low-temperature range.

Acoustic symmetry and vibrational anharmonicity of rhombohedral Pb5Ge3O11 and Pb4.7Ba0.3Ge3O11.

Acoustic symmetry in crystals belonging to the rhombohedral 3\ifmmode\bar\else\textasciimacron\fi{} Laue group is discussed with special reference to the ultrasonic properties of ${\mathrm{Pb}}_{5}$${\mathrm{Ge}}_{3}$${\mathrm{O}}_{11}$. In this material the acoustic symmetry axes are so close to the crystallographic axes that its elastic behavior is essentially that of a 3\ifmmode\bar\else\textasciimacron\fi{}m Laue group crystal. Ultrasonic wave velocities in single crystals of ${\mathrm{Pb}}_{5}$${\mathrm{Ge}}_{3}$${\mathrm{O}}_{11}$ and ${\mathrm{Pb}}_{4.7}$${\mathrm{Ba}}_{0.3}$${\mathrm{Ge}}_{3}$${\mathrm{O}}_{11}$ have been measured at room temperature as a function of hydrostatic pressure and uniaxial stress. Results have been used to obtain the second-order elastic constants and their hydrostatic pressure derivatives. By treating the materials as if they had 3\ifmmode\bar\else\textasciimacron\fi{}m Laue symmetry, the 14 third-order elastic constants describing the third-order elasticity of crystals have also been determined. To quantify the vibrational anharmonicities of the long-wavelength acoustic modes, the mode Gr\uneisen parameters have been computed and are found to be positive. This finding indicates that there is no significant acoustic mode softening induced by incipient ferroelectric optic-phonon--acoustic-phonon interactions in these materials at room temperature, which is substantially below that at which the soft optic mode, ferroelectric phase transition takes place.

Structural phase transition in CeCu6

The structure of CeCu/sub 6/ has been studied at eight temperatures from 10K to 295K by time-of-flight neutron powder diffraction. The high temperature orthorhombic cell transforms to a monoclinic cell at 230K. Rietveld structural refinements at each temperature give details of the structural distortion associated with the transition. The space groups and the temperature dependence of the monoclinic strain are consistent with a second order transition driven by a soft acoustic phonon mode. However, small atom displacements perpendicular to the shear direction suggest that an optic mode may also be involved in the transition.

Structural phase transition in CeCu6

The structure of CeCu6 has been studied at eight temperatures from 10K to 295K by time-of-flight neutron powder diffraction. The high temperature orthorhombic cell transforms to a monoclinic cell at 230K. Rietveld structural refinements at each temperature give details of the structural distortion associated with the transition. The space groups and the temperture dependence of the monoclinic strain are consistent with a second order transition driven by a soft acoustic phonon mode. However, small atom displacements perpendicular to the shear direction suggest that an optic mode may also be involved in the transition.

Soft acoustic modes in trigonal crystals.

An analysis is given for the acoustic modes in trigonal crystals which go soft as the elastic constants change so as to make them mechanically unstable. The results are presented as part of a tabulation of the properties of soft modes for all crystal classes. A number of errors in the recent literature are pointed out.

Third-order elastic constants and acoustic-mode vibrational anharmonicity of cobalt

Abstract The third-order elastic constants of cobalt have been determined from measurements of the hydrostatic and uniaxial pressure dependences of the velocities of ultrasonic waves propagated through single crystals. The hydrostatic pressure dependences of the second-order elastic constants have also been obtained. Comparison between these experimental results and predicted values show that, not unexpectedly, a simple Lennard-Jones model employed by previous workers for the lattice potential does not give a realistic assessment of the elastic behaviour of cobalt under pressure. The volume dependence of cobalt upon pressure, i.e. the compression, is estimated by extrapolation up to high pressure of the ultrasonic wave velocity data. The lack of acoustic-mode softening establishes that the h.c.p. to f.c.c. transition in cobalt is not associated with ferroelastic behaviour. The anharmonicity of the long-wavelength acoustic modes is discussed in terms of mode Grüneisen parameters obtained in the elastic continuum approximation.

Inelastic neutron scattering studies of collective modes in molten CsCl

The authors have measured the inelastic neutron scattering spectra of three samples of molten CsCl with different isotopic chlorine compositions (including the naturally occurring one) using an incident wavelength of 4.1 AA, and also of molten CsCl with natural Cl using 1.08 AA. The experiments were performed on the time-of-flight spectrometers IN6 and IN4 at the Institut Laue-Langevin. By a first-order subtraction they have substantially separated the contributions from mass (acoustic mode) and charge (optic mode) current correlations. A quasi-dispersion curve has been derived from the positions of the peaks in these functions. The acoustic and optic branches are well separated in energy over the measured region (1 AA-1<Q<4 AA-1) with relatively flat dispersion. The modes are strongly overdamped and although the energy of the low-Q optic mode corresponds to the crystal LO there is no evidence of a side peak in Sqq(Q, omega ). Thus the observed optic mode is not propagating, even at the lowest values of Q. Indeed the IN4 measurements show that Sqq(Q, omega ) is very broad (full width 18 meV) at low Q while Smm(Q, omega ) is much narrower. Small maxima in the widths of ST(Q, omega ) at Q approximately=1.7 AA-1 indicate the possible existence of a soft acoustic mode. Classical de Gennes narrowing may be observed at Q approximately=2.0 AA-1, corresponding to a change in diffusion mechanism. The statistical accuracy of the spectra was too low and the differences between scattering lengths too small to enable a second-order separation to be made using results from all three samples.

A Soft Acoustic Mode in the Ferroelastic Phase Transition of LaNbO4

An anisotropic softening of the acoustic mode in the ferroelastic phase transition of LaNbO 4 has been observed by the Brillouin scattering experiment. The transverse mode propagating along a direction deviated from the crystal axis by 23° in the a b -plane shows a complete softening at the transition temperature T 0 . The direction is parallel or normal to the ferroelastic domain boundary as predicted by the phenomenological analysis of the phase transition from 4/ m to 2/ m symmetry. Thc temperature dependence of the elastic susceptibility is well described by the classical indices γ=γ'=1 in both phases, suggesting a typical proper ferroelastic transition.

Soft acoustic subspaces at elastic phase transitions

The soft pure transverse acoustic modes and dimensionality of the soft acoustic subspace have been tabulated for all elastic phase transitions involving a homogeneous deformation of the crystal lattice in the absence of external fields.

A Neutron Scattering Study of Phonons in Per‐Deuterated S‐Triazine

AbstractThe lattice dynamics of a fully‐deuterated single crystal of s‐triazine is investigated using the technique of coherent inelastic neutron scattering. Detailed measurements of the temperature dependence of the dispersion of the soft acoustic mode associated with the ferroelastic phase transition are reported and compared with the qualitative behaviour of other acoustic modes. The results are in agreement with an analysis of the acoustic dynamical matrix for this material which predicts that in the high‐temperature phase the combination of elastic constants C2eff = 1/2(C11 − C12) C44 − C214 should become unstable on cooling towards the transition temperature. This effective elastic constant is found to have a temperature dependence of the form Ceff = C0(T − T0) with C0 = 3.404 × 106 Nm−2K−1 and T0 = 172.4 K. Measurements of some optic mode frequencies are also reported.

Phonon relaxation due to anharmonism and central peak in structural phase transitions. II. Soft optic modes in uniaxial ferroelectrics and soft acoustic modes in cubic crystals

AbstractIn the mode—mode coupling approximation the critical exponents (CE) are calculated for the dampings and frequency shifts of soft optic modes and acoustic phonons (in uniaxial ferroelectrics) and for the dampings of soft acoustic phonons (in cubic crystals) in the structural transition point. The CE are traced back to the anharmonic interaction (of all orders) of soft phonons with the intrinsic dynamic central peak of any nature.

Ferroelastic phase transition in BiVO4: IV. Relationships between spontaneous strain and acoustic properties

For pt.III see ibid., vol.16, p.5093 (1983). The spontaneous strain properties of the ferroelastic material BiVO4 are shown to be related to its acoustic properties. At the ferroelastic phase transition temperature, it is proved that a mutually perpendicular pair of pure transverse acoustic modes possess theoretically zero velocity. The propagation and displacement directions of these modes are collinear with the ferroelastic domain walls. The monoclinic ferroelastic strain in BiVO4 results from the freezing-in of the pure transverse displacements of these soft acoustic modes.

Vibrational anharmonicity of oxide and fluoride scheelites

Lattice parameter measurements of LiYF4 and LiY0.5Tb0.5F4 made between -100 degrees C and +550 degrees C show that the anisotropy of the thermal expansion ( alpha 11> alpha 33) of these scheelite structure fluorides has the opposite sense to oxide scheelites ( alpha 33> alpha 11). This, together with corresponding differences found between the anisotropy of linear compressibility and elastic stiffness constants, can be understood if the oxides, but not the fluorides, are considered to show a tendency towards layer-like behaviour. Pulse echo overlap measurements of ultrasonic wave velocities have been used to obtain the hydrostatic pressure derivatives delta CIJ/ delta P of the elastic constants of CaWO4. Since the acoustic symmetry axes do not shift appreciably with pressure, the pressure derivatives delta CIJ/ delta P, like the elastic constants CIJ themselves, can be transformed (with useful simplification) to reference bases comprised of acoustic symmetry axes (( kappa and kappa + pi /2) or ( gamma and gamma + pi /2)) and the Z axis. The acoustic mode Gruneisen gammas in the long-wavelength limit obtained for CaWO4 show that shear acoustic mode softening occurs under pressure along the gamma and gamma + pi /2 acoustic symmetry axes, in contrast to that for LiYF4 and LiY0.5Tb0.5F4 which takes place along the other type ( kappa and kappa + pi /2) of acoustic symmetry axes in the (001) plane.

Vibrational anharmonicity and the elastic phase transition of indium-thallium alloys

The anharmonic behaviour of the long wavelength acoustic modes in In-Tl alloys in the vicinity of the elastic, f. c. c. to f. c. t. phase transition has been determined from the composition, temperature, hydrostatic and uniaxial pressure dependences of ultrasound wave velocities in single crystals containing between 10 at. % and 30 at. %TI. The hydrostatic pressure derivatives show that the soft acoustic mode of f. c. c. alloys softens while that of the f. c. t. alloys stiffens with pressure. An empirical equation of state has been constructed for the f. c. c. alloys. A large negative mode Grüneisen parameter found for the soft transverse acoustic phonon (wave vector &lt;110&gt;, polarization &lt;11̄0&gt;) for f. c. c. alloys near the transition reflects the marked vibrational anharmonicity of the soft mode. The pressure effects on ultrasound wave velocity have been used to determine sets of independent, room temperature third order elastic constants for several alloy compositions. The third order invariants in the elastic strain energy have been evaluated to assess quantitatively the applicability of Landau theory, in conjunction with the soft mode concept, to an elastic phase transition. The relative magnitudes of the third order invariants conform to the nearly second order character of the phase transition. However, the first order character has been established by a marked hysteresis in ultrasonic wave velocities as crystals are cycled by temperature or pressure through the transition. There is a third order invariant in the order parameter: Landau theory predicts correctly the first order nature of the transition.

Elastic constant measurements of rare earth transition metal compounds

The availability of single crystals of some of the rare earth transition metal compounds has allowed more detailed studies of the elastic and magnetoelastic effects in these materials [1]. Recent developments in this field will be reviewed with particular emphasis on the RCo 2, RFe 2 and RCo 5 compounds. The very large magnetostrictive effects observed in the former two series are reflected in anomalous changes in their elastic properties. Measurements on two compounds will be described in detail. The elastic constants of the cubic Laves phase compound HoCo 2 have been measured as a function of temperature and magnetic field covering both the initial ferrimagnetic ordering at 78 K and the rotation of the magnetic easy direction from [100] to [110] below 16 K [2]. Dramatic softening of acoustic modes is observed in both of these temperature regions. Preliminary measurements will also be presented of the temperature dependence of the elastic constants of NdCo 5, where there is particular interest in the rotation of the easy direction of magnetisation. NdCo 5 which is hexagonal has a magnetic easy hexagonal c-axis at room temperature. Below 290 K the moments move away from the c-axis until at 240 K they are confined to the basal plane. An inspection of earlier work on Gd would indicate strong magnetoelastic interactions in this region.

Brillouin scattering investigation of paraelectric KH2PO4near the tricritical point

A high-resolution Brillouin scattering study of the soft acoustic mode of K${\mathrm{H}}_{2}$P${\mathrm{O}}_{4}$ in the hydrostatic-pressure range 0-4 kbar is presented. All spectra could be fitted closely by the usual mean-field coupled-mode model. From those fits, the pressure dependence of the ferroelectric-mode relaxation rate is derived. At pressures near and above the tricritical point, the transition can be approached so closely that the Brillouin spectrum becomes strongly overdamped and very narrow (\ensuremath{\sim} 50 MHz). Under those conditions, the elastic central peak of even the best available crystal becomes a dominant spectral feature, while the maximum dielectric constant is close to ${10}^{6}$.

Third-order elastic constants and acoustic mode vibrational anharmonicity of bismuth

Abstract The third-order elastic constants of cobalt have been determined from measurements of the hydrostatic and uniaxial pressure dependences of the velocities of ultrasonic waves propagated through single crystals. The hydrostatic pressure dependences of the second-order elastic constants have also been obtained. Comparison between these experimental results and predicted values show that, not unexpectedly, a simple Lennard-Jones model employed by previous workers for the lattice potential does not give a realistic assessment of the elastic behaviour of cobalt under pressure. The volume dependence of cobalt upon pressure, i.e. the compression, is estimated by extrapolation up to high pressure of the ultrasonic wave velocity data. The lack of acoustic-mode softening establishes that the h.c.p. to f.c.c. transition in cobalt is not associated with ferroelastic behaviour. The anharmonicity of the long-wavelength acoustic modes is discussed in terms of mode Gruneisen parameters obtained in the elastic con...

Cystalline modifications and structural phase transitions of NaOH and NaOD

Structural work on the three modifications of sodium hydroxide is reviewed. The monoclinic and cubic modifications were determined with neutron and X-ray diffraction, respectively. The phase-transition temperatures were determined by specific-heat measurements. The cubic to monoclinic transition is a first-order transition with a freezing of the rotational motion of the OH (and OD) groups. The monoclinic axes a, b and c* tend to be oriented parallel to the original cubic directions [{\bar 1}{\bar 1}2], [1{\bar 1}0] and [ 111], respectively. The orthorhombic to monoclinic transition is a nearly continuous displacive phase transition with a soft acoustic shear mode. The order parameter is the homogeneous shear of the crystal in the a direction. Its temperature dependence is described within Landau theory.

Ferroelastic and ferroelectric phase transition in a molecular crystal : tanane - 2. — Phenomenological model and piezoelectric resonance study of the soft acoustic mode

Ferroelastic and ferroelectric phase transition in a molecular crystal : tanane - 2. — Phenomenological model and piezoelectric resonance study of the soft acoustic mode

Application of the Landau Theory to Elastic Phase Transitions

At an elastic phase transition, a crystal structure change takes place by a homogeneous lattice deformation, the associated soft mode being a zone centre acoustic phonon. As such a transition is approached, the mode softening is manifested as a reduction in the second order elastic constant which acts as the quadratic invariant and so the higher order terms become increasingly important: a pure elastic transition will be dominated by vibrational anharmonicity of the soft acoustic mode. For a first order, elastic transition, a quantitative understanding of this anharmonic behaviour must rest upon measurements of the third order elastic constants which constitute the cubic invariant in the order parameter (which is a strain tensor component). To obtain the forms of these physically significant invariants in the Landau free energy expansion up to third order in the strain eigenvectors, the elastic free energy is transformed from finite strain space to the irreducible strain tensor space keeping the free energy invariant. In-Tl alloys undergo a f.c.c.-f.c.t. elastic phase transition whose order has been difficult to establish experimentally, although the Landau theory suggests that the transition should be first order because there is a cubic invariant of the order parameter. These alloys provide a suitable vehicle for a quantitative asessment of the applicability of the Landau theory to a first order elastic phase transition in which the atomic displacements are small. The transition is accompanied by softening of the N[110], polarisation U[1-10] zone centre acoustic phonons: the second order invariant ½(C11 - C12) goes to zero at Tc. The third order invariants obtained from third order elastic constant measurements made on In-Tl alloys are discussed. The hydrostatic pressure derivative ∂(½(C11 - C12))/∂P is negative for those f.c.c. alloys which undergo the transition. As suggested by the Landau theory, the third order invariant (1/8)(C111 - 3C112 + 2C123) of the order parameter is small, in agreement with the near second order character of the transition: there is only a slight discontinuity in the strain at Tc. The first order character of the transition is evidenced further by a marked hysteresis in ultrasonic wave velocities as crystals are taken through the transition by pressure or temperature cycles.