MAGNETIC SCATTERING OF NEUTRONS - Rare Earth Magnetism: Structures and Excitations - page 175

We have introduced the dynamic correlation function S αβ ( κ ,ω ), defined

by eqn (3.2.13), with
α = N − 2
j

and β = N − 2
j

J jα e −i κ · R j J j β e i κ · R j ,

and the corresponding susceptibility function χ αβ ( ,ω ), utilizing the

relation between the two functions given by the fluctuation-dissipation
theorem (3.2.18).
An important consequence of (4.2.2-3) is that the inelastic scat-
tering of neutrons is proportional to the correlation function S αβ (

κ

,ω ),
which is essentially the Fourier transform of the probability that, if the
moment at site j has some specified vector value at time zero, then the
moment at site j has some other specified value at time t .Aninelas-
tic neutron-scattering experiment is thus extremely informative about
the dynamics of the magnetic system. Poles in the correlation function,
or in χ αβ (

κ

,ω ), are reflected as peaks in the intensity of the scattered
neutrons. According to (4.1.2) and (4.1.3), each neutron in such a scat-
tering peak has imparted energy hω and momentum h

κ

to the sample,
so the peak is interpreted, depending on whether hω is positive or neg-
ative, as being due to the creation or annihilation of quasi-particles or
elementary excitations in the system, with energy

κ

| hω | and crystal mo-

mentum h q = h ( κ − τ ), where τ is a reciprocal lattice vector. A part

of the momentum h may be transferred to the crystal as a whole. If

the sample is a single crystal, with only one magnetic atom per unit
cell, S αβ (

τ

is normally chosen so
that q lies within the primitive Brillouin zone. The form factor in the
scattering amplitude is not however invariant with respect to the addi-
tion of a reciprocal lattice vector. This interpretation of the poles in
S αβ ( q ,ω ) governs the choice of sign in the exponential arguments in
both the temporal and the spatial Fourier transforms.
The relation (4 . 2 . 3 c ) between the scattering function and the gen-
eralized susceptibility implies that the neutron may be considered as
a magnetic probe which effectively establishes a frequency- and wave-
vector-dependent magnetic field in the scattering sample, and detects
its response to this field. This is a particularly fruitful way of look-
ing at a neutron scattering experiment because, as shown in Chapter
3, the susceptibility may be calculated from linear response theory, and
thus provides a natural bridge between theory and experiment. Using
the symmetry relation (3.2.15), which may here be written χ αβ ( q ,z )=
χ αβ (

κ ,ω )= S αβ ( q = κ − τ ,ω ), where τ

z ∗ ), it is straightforward to show that χ αβ ( q ,ω )+ χ βα ( q ,ω )is − q , −

real and equal to Im χ αβ ( q ,ω )+ χ βα ( q ,ω ) . In addition, the form of the
inelastic cross-section, and also the result (3.3.2) for the dissipation rate,
impose another analytic condition on the function χ αβ ( q ,ω )+ χ βα ( q ,ω ).