Neutron Scattering

Similar branches may induce larger gaps, they are called to strougly interact .
13 .2 .2 Model calculations
The examples discussed above are characterized through their simplicity, which permits
to analyze the polarization patterns without detailed calculations .
one has to deal with systems of more than 10 atoms per primitive cell,
In actual topics
oue may identify
then the character of the phonon modes only with the aid of predictions of model
calculations .
As discussed above the lattice dynamics is described by the 3n-dimensional Eigen-value
problem with the dynamcal matrix
w 2e = De (13 .16)
D,a(d, (f ) =
1
1772~772~,)1/2
(D~,3 (Od,i' (f )CXP(i(Ii') (13 .17) .
(Od, l'(/') denote the force constants between the atoms d and d' in by l shifted cells .
The determination of the force constants represents the real problem in lattice dynamics,
in particular the question which constants are relevant .
Already by symmetry, the nine parameters per atom-pair are significantly reduced .
Furthermore, one may reduce the analysis to the cloner neighbors, as far as Do long range
force is involved .
'The next step consists in the development of potentials, from which one
may obtain the force constants for many pairs inducing only a few free parameters .
Frequently it is sufficient to consider axial-symmetric potentials with
V(r), i .e . the potential depends only on the distance between the two atoms . Such a
potential yields only two force constants, a
radial and a transversal one
a 2 V
(D x(ld, l'd') = a'-To (13 .18),
r
(P1,(Id' 'd') = l l r
ar ~'-TO (13 .19) .
In the mont simple model one may only introduce these radial and transverse force
constants for the close neighbors (Born-von-Kàrmàn Model) .
However, extending more
and more Shells will rapidly increase the number of parameters .
In particular the lattice
13-15