Neutron Scattering

esults from the quantum mechanics of a single harmonie oscillator :
the square of the
amplitude is proportional to W7 \4~ .
The terni (13 .10 .c) results from the Bose-statistics of the singulär mode with frequency
w j (q) , the occupation number is given by the Bose-function :
n(w~(q)) - exp( tLkl
(~) ~ ) 1
13 .11
ri,(wj (_q» tends for T --> 0 towards zero, the phonons are frozen . For T --> oc, n(wj(q»
approaches ( r kT"»,
i .e . the classical relation . The terni (13 .10 .c) indicates that the
Bose-function is increased by +1 only in case of phonon creation . At low temperature
where n(wj(q)) is
close to zero, the phonon may be observed only in the creation mode,
the cross section for the annihilation process becomes vanishing since the phonon states
are no longer occupied . At finite temperature the Bose-statistics further simplifies the
observation of phonons with low frequencies .
The complex term in (13 .10 .a) is given by the dynamical structure factor :
C j (q, Q) = ~- bd - exp( - I1'd(Q) + z(2d d) * (Q e (q)) (13 .12),
d rri:d
where the sum is
extending to the atoms, numbered by the scalar d, in the primitive
cell .
They have the mass md scattering length bd and a three dimensional polarization
vector ed(q), the equilibrium position in the unit cell is given by dd .
Without the last
part equation (13 .12) corresponds to the elastic structure factor, in particular one finds
the same Debye- M'aller -factor exp(-Wd(Q » .
'T'he whole term in the exponential function
determines whether the interference between atoms is constructive or destructive .
In the
dynamical structure factor there is
in addition to the elastic one the polarization term
(last parenthesis) ;
for instance only those atoms may contribute whose polarization vector,
possesses a component parallel to Q.
Since Q enters this term directly, the structure
factor increases `vith ~QI ; the intensity is proportional to Q2 . The interference and the
polarization term in (13 .12) cannot be separated in genoral, therefore one has to calculate
the structure factors precisely.
In addition to the interpretation of the frequency data the prediction of the intensity
is an important out-coming of model calculations .
Only with the help of these predictions
a neutron scattering experiment aiming at the lattice dynamics of a complex material can
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