Anisotropic softening of dispersion branches in quartz approaching the α-β phase transformation

Abstract

Diffuse streaks have been found in x-ray scattering especially for the [ξ00] directions by H. ArnoId¹ and by R. Comes et al². Preliminary inelastic neutron scattering near the α-β phase transition by K. W. Bauer et a1.³ has shown that these diffuse streaks are due to an anisotropic softening of the phonon branches. Now we have mapped branches in [ξ00] (Г-M) and in M-K directions for energies below 7 THz. In this region there are three optic and three acoustic branches. The highest branch is that which at the Г-point connects to the soft mode (A₁; at room temperature 6.3 THz). This soft mode has the eigenvector of the displacement at the α-β phase transformation.⁴ This optic branch shows a temperature dependence for 0 ⩽ q < 0.3, whereas for 0.2 < q ⩽ 0.5 (M-point) the two lowest acoustic branches soften. The other branches remain rather unchanged. The dispersion of the temperature dependent modes is measured at (T_o-T) = 555; 280; 130.5; 60.5; 30.5; 18.5, 7.9, 2.5, -8.5. Along the [ξ00] direction the softening clearly produces a deep valley in the dispersion surface, which explains the diffuse streaks seen in x-ray scattering. Model calculations by T. H. K. Barron and C. C. Huang⁵ describe the room temperature dispersion curves sufficiently well. For the lowest branch at the M-point we measured the intensity at room temperature at 56 different positions in reciprocal space and found very good agreement with the dynamical structure factors calculated from the eigenvector of the model by Barron and Huang. The frequency of the lowest branch of M (T₂-mode) has the same temperature dependence as the lattice expansion ΔL.H. Grimm⁶ has shown that the lattice expansion is proportional to the order parameter squared; the order parameter δ being a rotation of the SiO₄- tetrahedra. Including results of U. T. Höchli and J. F. Scott⁷ one finds with T_o - T_c ≊ 10°C. This is the same analytical form as used for Tb₂(MoO₄)₃ by B. Dorner et al, Applying a formula δ ∼ (T*_c - T)^β where T*_c T_o, we find β = ⅙ for quartz as well as for Tb₂(MoO₄)₃