Similar, the magnetic inelastic scattering is given by

The total magnetic cross section is barn. In (230) the first term in the bracket corresponds to the total and the second term to the elastic scattering. If we split the index into basis and lattice part and compare equation (203), we see that the scattering function depends on the correlation function between the magnetisation operator , which is the observable in the case of magnetic neutron scattering ( ).

is the Debye-Waller
factor of the atom number in the magnetic unit cell^{28}. denotes the number of magnetic
atoms in the magnetic unit cell.
Therefore, if the generalised eigenvalue problem (200) for the dynamical matrix
has been solved, the magnetic neutron scattering function can
be evaluated with the help of equations (204) and (212):

Once the eigenvectors
of the system have been determined, this expression can be
evaluated.
*mcdisp* evaluates for every mode the expression (232) with exception
of the -function and multiplies it by in order to get the nuclear Intensity in
barns/meV formula unit. In addition, the components of the magnetic scattering function
can be output (to be used to interpret
polarised magnetic neutron scattering). then refer to either the xyz coordinate system
(,
and perpendicular to and )
or the uvw coordinate system
(
, perpendicular to the scattering plane (as determined by the cross product of
subsequent vectors in the input q-vector list of *mcdisp*)
and perpendicular to and , such that uvw for a righthanded system).

Form factor effects on the scattering intensity are
due to the -dependence of the magnetisation operator, which means that
the transformation matrices
and the eigenvalues are also
-dependent. These quantities have
to be calculated by evaluating the transition
matrix elements^{29}
of
for every
scattering vector
and diagonalising the matrix (208)
with
.
For small this procedure can be simplified by using the dipole approximation,
which is described below.

Martin Rotter 2017-01-10