Intermediate Lattice Coupling Theory of the <111> Paraelastic and Paraelectric Tunneling System (XY8)

Abstract

We have investigated the effect of the “phonon dressing” on the energy splittings and the lattice relaxation rates of the paraelastic and paraelectric tunneling system (XY8) (1) in a host lattice of cubic symmetry. In paraelastic and paraelectric systems the dipole-phonon interaction is generally very strong. The case is considered when the small-polaron binding energy Eb of the oriented dipole is comparable with the tunneling matrix elements. The symmetry group of the dipole-lattice hamiltonian \(\hat H\) is Oh. It has a subgroup D2h which is Abelian group and has therefore only one-dimensional irreducible representations. As the symmetry group of \(\hat H\) contains an Abelian group one can exploit this fact in a similar way as this is done in the perfect crystal physics where the group of lattice translations is also Abelian. First, new normal coordinates for the displacements of ions which transform as the irreducible representations of the D2h group are introduced. The phonon annihilation operators a(m,λ) for the new normal modes are the following linear combinations of the annihilation operators a(\(\hat G\)λ) for the plane wave modes: $$a\left( {m,\lambda } \right) = {8^{ - 1/2}}\sum\limits_G {{x_m}\left( G \right)a\left( {\hat G\lambda } \right).} $$ (1) Here λ denotes the wave vector and the polarization of the lattice plane wave in the first octant of the Brillouin zone. G are symmetry operations of the D2h group. m = 1−8 specifies the one-dimensional representations and Xm(G) = ± 1 are group characters.