Neutron Scattering

s
- ' f
N .mwn .I )oo K L .~um..un . .JOOK
11001 au.a~o . I I I l01 lJfOWn
a I
3 ~ I
. . [ .
F
Sr
I_r
I I
r I I 1 I 1 i
0 : 0+ Os I I I 0 0! 10
0 0 : 0+ On ot l0
Figure 4 Phonon dispersion of Al along the [100] and [011]-directions (from reference [1]) .
The extension to a System with n Atoms in the primitive Gell can be easily donc, one
gets for each q an Eigen-value problem in 3n dimensions .
The Eigen-vectors have the
dimension ara and the dynamical matrix is 3ra * 3ra dimensional .
For each q
one finds
hence 3n Eigen-modes, therefore the complete phonon dispersion consists of 3n branches,
three of which have zero frequency for q ---> 0 . The latter branches are called acoustic
rince they are associated `vith the propagation of round .
The figure 4 shows the t ypical
presentation ùj against wave-vector q for the phonon dispersion of aluminum .
The plane waves according to the Eigen-frequencies and to the Eigen-vectors to each
of the allowed (1-values form a.
complete set of fonctions for the displacements of the N
tunes n atours in the crystal .
Any distortion can be represented as a linear combination
of these plane waves . Obviously it is rather favorable to use this set for calculating the
differential cross section . However, for this purpose it is necessary to couvert the lattice
vibrations into quantum mechanics, the corresponding quasi-particle being the phonon .
A
one-phonon protes is shown in the schematic figure 5 .
The momentum transfer Q = k-k'
consists of the rum of a reciprocal lattice vector T and the wave-vector q, which lies within
the firnt Brillouin-zone .
Only q determines the wave vector of the contributing phonons,
however Q determines the differential cross section, i .e . the intensity .
The exact scattering may be deduced from the correlation fonction .
In the following
we only avant to discos the meaning of the different terms in the cross section
13-6