Neutron Scattering

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Lambda lattice parameter
Figure 3 Scheme of the displacernents in a transversal plane wave .
=1/.olr~_ D~(L, o)exp(-igl)e 3 (13 .7) .
1,3
Here we use that the force constants depend only on the distance in real space (l - l')
and not on the particular values of l and l' . With (13 .7) one gets an easily solvable system ;
the original complexity of the problem is transferred to the number of these 3-dimensional
probem .
For a complote solution of the crystal dynamics one would need to solve the
problem (13 .7) for Bach of the N-10 23 allowed q-values . In reality, however, one han to
analyze only a few different q-values . In general it is sufficient to study only q-values
within the firnt Brillouin-zone .
M,e may define the dynamical matrix as
D~,~(q) = 1/~~D~,/~(l,o)e~p(-Zql) (13 .8)
which allows to rewrite the equations (13 .7) in matrix form
w,e = D e (13 .9) .
The system (13 .9) is just a these-dimensional Eigen-value problem .
For fixed q one han to determine the these Eigen-vectors ej , j=1,2,3, together with
the these corresponding Eigen-values, wj (q) . This may be achieved with the standard
numerical techniques . The dependence of the Eigen-frequencies on the wave-vector, wj (q)
with j = l, 2, 3, is called the dispersion relation .
13-5