MAGNETIC SCATTERING OF NEUTRONS - Rare Earth Magnetism: Structures and Excitations - page 164

means, derive the magnetic cross-section for unpolarized neutrons in the
simple dipole approximation , which is normally adequate for scattering
by rare earth ions, and will therefore suce in our further discussion.
A neutron interacts with the nuclei in a solid through the nuclear force
and, through its magnetic moment, with the magnetic field due to the
electrons. In solids with unpaired electrons, the two kinds of scatter-
ing mechanism lead to cross-sections of the same order of magnitude.
The magnetic field of the electrons may be described by a multipole ex-
pansion, and the first term in this series, the dipole term, leads to the
dominating contribution to the cross-section at small scattering vectors.
We use this approximation in a derivation from first principles of a gen-
eral expression for the differential cross-section (Trammel 1953), which
we then separate into elastic and inelastic components. Using linear re-
sponse theory, we derive the different forms which the inelastic part may
exhibit, and illustrate some of the results by means of the Heisenberg
ferromagnet. A detailed treatment of both the nuclear and magnetic
scattering of neutrons may be found in Marshall and Lovesey (1971),
and Lovesey (1984), while a brief review of some of the salient features
of magnetic neutron scattering and its application to physical problems
has been given by Mackintosh (1983).

4.1 The differential cross-section in the dipole
approximation
A neutron-scattering experiment is performed by allowing a collimated
beam of monochromatic (monoenergetic) neutrons to impinge upon a
sample, and then measuring the energy distribution of neutrons scat-
tered in different directions. As illustrated in Fig. 4.1, a uniform en-
semble of neutrons in the initial state | ks n > is created, typically by

utilizing Bragg-reflection in a large single-crystal monochromator, plus
suitable shielding by collimators. We may write the state vector for this
initial plane-wave state

| ks n > = V − 1 / 2 exp( i k · r n ) | s n >,

representing free neutrons with an energy ( hk ) 2 / 2 M and a flux j ( ks n )=
V − 1 h k /M . When passing through the target, the probability per unit
time that a neutron makes a transition from its initial state to the state
| k s n > is determined by Fermi's Golden Rule :

h
if

W ( ks n , k s n )= 2 π

2 δ ( hω + E i −

|H int | k s n ; f> P i | < ks n ; i | E f ) .

(4 . 1 . 1)