Neutron Scattering

equilibrium position the atom is in rest, terms of first order are not allowed . In harmonic
approximation one assumes the expansion till the second order to be suficient . This
assumption is essential for the following analysis, since already the presence of third order
terms prohibits to solve the equations of moveurent in the general case .
In mort solids
anharmonic contributions to the potential are, however, small justifying the assumption .
If the anharmonic effects
have to be taken into consideration (for example close to the
melting of the crystal lattice), perturbation theory is the usual technique .
Using the definition
D,, s (l, l') = d2 ,D(l ' l')
(13 .4)
du, (1)du,s (1-')
one obtains the equations of movement (a,,3 = x, y, z) to
T'he displacement of the atour l' in ,3-direction yields a force on the atour l in a-direction
of strength D,,s(1,l')uß(l'), D,, s (1,l') are therefore called the force-constants . 'The real
problem in treating lattice dynamics consists in the large number of there equations, there
are 3N equations to be solved with N being of the order of 10 23 .
In order to avoid this
complexity one makes the Ansatz of plane waves :
the movements of all atoms are given
by the displacement in one unit Gell at time zero propagating in time and space as a plane
wave
exp(i u~(l) = e~M -i / z
(1 l - w ot) (13 .6) .
Figure 3 shows the displacement pattern of a plane wave characterized by the wave
vector q, reflecting the propagation of the wave (planes perpendicular to (j are always
identically displaced), the vibration frequency wo
of each atour and the direction of the
oscillation given by the polarization vector c .
In the single atom lattice there are only
acoustic modes, they are called longitudinal (LA) if q is parallel to e and transversal (TA)
for q perpendicular to e .
The plane wave Ansatz (13 .6) for the equations of movement yields a system of three
equations for each q-value
13-4