MAGNETIC SCATTERING OF NEUTRONS - Rare Earth Magnetism: Structures and Excitations - page 172

Experimental studies of the form factor and the associated moment
densities have been reviewed by Sinha (1978). For an accurate interpre-
tation of the data, it is generally necessary to proceed beyond the dipole
approximation. In the heavy rare earths, the deduced 4 f densities are
in good agreement with atomic calculations, provided that relativistic
effects are included, but the conduction-electron distributions are much
less certain. In the light elements, crystal-field effects become impor-
tant, as observed for example in Pr and Nd by Lebech et al. (1979). Of
especial interest is Sm, where the opposition of spin and orbital moments
leads to a form factor which has its maximum at a non-zero κ ,andthe
conduction-electron polarization seems to be very strong (Koehler and
Moon 1972).
Labelling quantities pertaining to the j th atom with the index j ,
and summing over all the atoms in the sample, we find that the total
H int ( κ ) (4.1.7), in the dipole approximation, is given by

)=8 πµ B
j

} j e −i κ · R j

{ 2 H int ( κ gF ( κ ) µ n · ( κ × J j × κ ) .

The squared matrix element in (4.1.4) may furthermore be written
< s n ; i | s n ; f>< s n ; f | s n ; i> .

We shall only consider the cross-section for unpolarized neutrons, so
that we sum over all the spin states

|H int ( κ ) |H int ( − κ )

| s n > of the scattered neutrons,

and average over the spin-states | s n > , with the distribution P s ,ofthe

incoming neutrons. With an equal distribution of up and down spins,
P s = 2 , and introducing Q j = κ × J j × κ , we find that the cross-section

is proportional to

P s < s n | µ n · Q j | s n >< s n | µ n · Q j | s n >

s n s n

| s n > = 2 g n µ N 2

=
s

P s < s n | µ n · Q j )( µ n · Q j )

Q j · Q j , (

as may readily be shown by using the Pauli-matrix representation, in
which Tr { σ α σ β } =2 δ αβ . We have further that Q j · Q j may be written

( κ × J j × κ ) · ( κ × J j × κ )=( J j − κ ( J j · κ )) · ( J j − κ ( J j · κ ))

)=
αβ

δ αβ −

κ α κ β J jα J j β , = J j · J j − ( J j · κ )( J j · κ

in terms of the Cartesian components. Defining ( J ⊥ ) j to be the projec-

tion of J j on the plane perpendicular to κ ,wehave

δ αβ −

κ α κ β J jα J j β =( J ⊥ ) j ·

( J ⊥ ) j .

αβ