## Formalism II - Neutron Cross section

mcdiff uses the standard formalism to calculate the elastic neutron scattering cross section, which can be obtained from the elastic part of the double differential cross section:

 (24)

In equation (25) denotes the double differential cross section, the number of atoms, and the wave vectors of the incoming and scattered neutron, respectively. is the gyromagnetic ratio of the neutron and  fm is the classical electron radius, the scattering vector, the energy transfer and the are the nuclear and magnetic Van Hove scattering functions. Both are a sum of elastic and inelastic contributions:

 (25) (26)

The nuclear elastic scattering can be split into a coherent and an incoherent part:

 (27)

The nuclear elastic coherent scattering function is given by a product of lattice factor and nuclear structure factor NSF:

 (28) (29)

denotes the number of atoms in the basis and the sums run over all reciprocal lattice vectors and over all atoms in the unit cell (unit cell volume is ). , and are the coherent scattering length, the unit cell position vector and the Debye Waller factor of the atom ( ), respectively.

Similar, the magnetic elastic scattering function can be written as a product:

 (30) (31)

In dipole approximation the products are evaluated as follows (see also appendix J):

[rare earth ( in sipf file):] [transition metals ( in sipf file) given (only) magnetic moments in mcdiff.in:] (i.e. a quenched orbital moment is assumed) [transition metals ( in sipf file) given and in mcdiff.in:]

Here , , , and denote the Landé factor, the form factor, the expectation value of the spin-, orbital- and total angular momentum operator of the atom in the unit cell, respectively.

Program mcdiff calculates the scattering angle of a reflection according to

 (32)

Nuclear elastic coherent and magnetic intensities are calculated according to:

 (33)

 (34)

 (35) (36) (37) (38) (39) (40) (41) (42) (43) (44) (45)

Mind: no absorption and extinction corrections are calculated.

Martin Rotter 2017-01-10