On the Margulis constant for Kleinian groups, I curvature

The Margulis constant for Kleinian groups is the smallest constant $c$ such that for each discrete group $G$ and each point $x$ in the upper half space ${\bold H}^3$, the group generated by the elements in $G$ which move $x$ less than distance c is elementary. We take a first step towards determining this constant by proving that if $\langle f,g \rangle$ is nonelementary and discrete with $f$ parabolic or elliptic of order $n \geq 3$, then every point $x$ in ${\bold H}^3$ is moved at least distance $c$ by $f$ or $g$ where $c=.1829\ldots$. This bound is sharp

Soft Mode Dynamics Above and Below the Burns Temperature in the Relaxor Pb(Mg_1/3Nb_2/3)O_3

We report neutron inelastic scattering measurements of the lowest-energy transverse optic (TO) phonon branch in the relaxor Pb(Mg_1/3Nb_2/3)O_3 from 400 to 1100 K. Far above the Burns temperature T_d ~ 620 K we observe well-defined propagating TO modes at all wave vectors q, and a zone center TO mode that softens in a manner consistent with that of a ferroelectric soft mode. Below T_d the zone center TO mode is overdamped. This damping extends up to, but not above, the waterfall wave vector q_wf, which is a measure of the average size of the PNR.Comment: 4 pages, 4 figures; modified discussion of Fig. 3, shortened captions, added reference, corrected typos, accepted by Phys. Rev. Let

Neutron Diffuse Scattering from Polar Nanoregions in the Relaxor Pb(Mg1/3Nb2/3)O3

We have studied the neutron diffuse scattering in the relaxor PMN. The diffuse scattering appears around the Burns temperature (~620K), indicating its origin from the polar nanoregions (PNR). While the relative diffuse intensities are consistent with previous reports, they are entirely different from those of the lowest-energy TO phonon. Because of that, it has been considered that this TO mode could not be the ferroelectric soft mode. Recently, a neutron scattering study has unambiguously shown that the TO mode does soften on cooling. If the diffuse scattering in PMN originates from the soft mode condensation, then the atomic displacements must satisfy the center of mass condition. But, the atomic displacements determined from diffuse scattering intensities do not fulfill this condition. To resolve this contradiction, we propose a simple model in which the total atomic displacement consists of two components: $\delta_{CM}$ is created by the soft mode condensation, satisfying the center of mass condition, and, $\delta_{shift}$ represents a uniform displacement of the PNR along their polar direction relative to the surrounding (unpolarized) cubic matrix. Within this framework, we can successfully describe the neutron diffuse scattering intensities observed in PMN.Comment: 7 pages, 7 figures (Revised: 11-16-2001

Ferroelectric Dynamics in the Perovskite Relaxor PMN

We review results obtained from recent neutron scattering studies of the lead-oxide class of perovskite relaxors PMN and PZN. A ferroelectric soft mode has been identified in PMN at 1100 K that becomes overdamped near 620 K. This is the same temperature at which polar nanoregions (PNR) begin to form, denoted by Td, and suggests that a direct connection exists between the soft mode and the PNR. The appearance of diffuse scattering intensity at Td reported by Naberezhnov et al. lends further support to this picture. At lower temperature the soft mode in PMN reappears close to Tc = 213 K (defined only for E > Ec). These results are provocative because the dynamics below Tc are characteristic of an ordered ferroelectric state, yet they occur in a system that remains cubic on average at all temperatures. We discuss a coupled-mode model that successfully describes these data as well as those from earlier lattice dynamical studies of other perovskites such as BaTiO3.Comment: 10 pages, 7 figures, Conference - Fundamental Physics of Ferroelectrics 200

Soft Mode Anomalies in the Perovskite Relaxor Pb(Mg1/3Nb2/3)O3

Neutron inelastic scattering measurements of the polar TO phonon mode in the cubic relaxor Pb(Mg1/3Nb2/3)O3, at room temperature, reveal anomalous behavior similar to that recently observed in the Pb(Zn1/3Nb2/3)_{0.92}Ti_{0.08}O3 system in which the optic branch appears to drop precipitously into the acoustic branch at a finite value of the momentum transfer q = 0.20 1/Angstroms, measured from the zone center. By contrast, a recent neutron study showed that PMN exhibits a normal TO phonon dispersion at 800 K. We speculate this behavior is common to all relaxor materials and is the result of the presence of nanometer-scale polarized domains in the crystal that form below a temperature Td, which effectively prevent the propagation of long wavelength (q = 0) phonons.Comment: 5 pages, 4 figures To appear as an AIP Conference Proceedings Volume for the Aspen 2000 Winter Conference on the Fundamental Physics of Ferroelectric