Neutron Scattering

(000)
Figure 2 Schematic drawing of the vectors determining the position of an atom in the crystal lattice .
that the interaction potential (13 .2)
does not show the full translation symmetry of the
crystal lattice .
The local variation of the scattering length Splits the differential cross
section into two contributions
d2a d2( d2~
(1,S2(1,E' - (dS2dE' ) `~ rz + (d52dE')zT' c °'L .
The coherent contribution is determined by the mean scattering length, whereas the
incoherent contribution is given by the root mean square deviation to the averaged scattering
length .
In order to calculate the differential cross section in (13 .1) it is
necessary to know the
states in the sample or at least to parameterize them . 'The sum over the states Az in
(13 .1) may be transformed to the correlation function, in which one has to introduce the
parameterized Eigen-states of the system .
In order to achieve this transformation in case
of the phonons we consider the vibrations of a crystal in harmonic approximation .
- Description of lattice drynarrrics irr harrrronic approxirrration- A crystal consists of N
unit cells with ri atouts within each of theut, the equilibrium position of any atom is
given
by the position of the unit cell to which it belongs, 1,
and by the position of the atomic
site in the unit cell, d .
At a certain time the atout may be displaced from its equilibrium
position by u(l, d) .
The instantaneous position is hence given by Rl .d = l +d+u(l, d) (sec
figure 2) .
For simplification we consider first a lattice with only one atom in the unit cell, d = 0 ;
l describes then the equilibrium position of the atom . 'The interaction potential between
two atoms l and l', ~D (l, l'), may be expanded at the equilibrium position in teruts of
u(l)=u(l')=0 . The constant term does not give any contribution to the equations of
movement, it is relevant only for the total energy of the crystal structure .
Since at the
13-3