MAGNETIC SCATTERING OF NEUTRONS - Rare Earth Magnetism: Structures and Excitations - page 176

It must be either zero, or positive or negative with ω (such functions are
called herglotz functions ), because a negative value of the cross-section
is clearly unphysical.
If the magnetic moments in a Bravais lattice are ordered in a static
structure, described by the wave-vector Q ,wemaywrite

J α ∗ e −i Q · R j ,

= 2 e i Q · R j + J jα J α (4 . 2 . 4)

allowing to be complex in order to account for the phase. The

static contribution to the cross-section is then proportional to

J α

)=
αβ

αβ (

J β ∗ }

( δ αβ − κ α κ β )

S κ

( δ αβ − κ α κ β )Re

{ J α

αβ

δ ( hω ) (2 π ) 3
υ

1
4

× (1 + δ Q 0 ) { δ ( τ + Q − κ )+ δ ( τ − Q − κ ) } ,

τ

(4 . 2 . 5)
where δ Q 0 is equal to 1 in the ferromagnetic case Q = 0 , and zero oth-
erwise, and υ is the volume of a unit cell. The magnetic ordering of
the system leads to δ -function singularities in momentum space, corre-
sponding to magnetic Bragg scattering , whenever the scattering vector
is equal to

± Q plus a reciprocal lattice vector τ . The static and dy-

αβ
d

namic contributions from ,ω ) to the total integrated

scattering intensity may be comparable, but the dynamic contributions,
including possibly a quasi-elastic diffusive term, are distributed more or
less uniformly throughout reciprocal space. Consequently, the elastic
component, determined by

S

αβ (

κ )and S ( κ

αβ (

), in which the scattering is condensed
into points in reciprocal space, is overwhelmingly the most intense con-
tribution to the cross-section dσ/d Ω, obtained from the differential cross-
section (4 . 2 . 2 a ) by an energy integration:

S κ

N hγe 2
mc 2

2

2

dσ
d Ω

e − 2 W ( κ ) | 2 J β ∗ } gF ( κ ) | ( δ αβ − κ α κ β )Re { J α

αβ

(2 π ) 3
υ

1
4

× (1 + δ Q 0 ) { δ ( τ + Q − κ )+ δ ( τ − Q − κ ) } .

τ

(4 . 2 . 6)
dσ/d Ω is the cross-section measured in neutron diffraction experiments,
in which all neutrons scattered in the direction of k are counted without
energy discrimination, i.e. without the analyser crystal in Fig. 4.1. This
kind of experiment is more straightforward to perform than one in which,
for instance, only elastically scattered neutrons are counted. In the
ordered phase, (4.2.6) is a good approximation, except close to a second-
order phase transition, where J α is small and where critical fluctuations