By Dr. Bruno Dorner (auth.)
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Extra resources for Coherent inelastic neutron scaterring in lattice dynamics
All unshaded areas in Fig. 25a correspond to f i l l e d electron states below an energy gap along the boundary of the second B r i l l o u i n zone. As Cd has two conduction (perhaps better, valence) electrons per atom and two atoms per u n i t c e l l , there are 4 electrons per u n i t cell in real space. The volume of t h e i r states in reciprocal space is equal to the volume of two B r i l l o u i n zones, because two electrons (one spin up, one spin down) per u n i t cell could j u s t f i l l the f i r s t B r i l l o u i n zone.
From the same arguments we find that the optic mode is v i s i b l e proportional to (b+b) 2 at (001). Although the Bragg intensity at (001) is extinct, the B r i l l o u i n zone boundary is at (0 0 I / 2 ) . Lattice dynamical calculations for the longitudinal modes in c direction with Q llq show that in B r i l l o u i n zones (00~) with even ~, only the LA branch is v i s i b l e while the LO branch is extinct, and vice versa for odd ~. Apparently the l a t t i c e dynamical calculation for the [00~] direction reflects only one atom per unit c e l l .
This gives the free molecule frequencies mM which can be checked against the high-frequency internal modes in the solid. Then one assumes that the eigenvectors ~M of the internal free molecule modes and the internal force f i e l d do not change introducing the i n t e r molecular potential. In harmonic approximation the dynamical problem is then reduced to solve the following secular equation: (57) Here ~ and v label molecules in the unit c e l l , and ~ and m the 144 normal modes with frequencies m(q).