MAGNETIC SCATTERING OF NEUTRONS - Rare Earth Magnetism: Structures and Excitations - page 174

where J jα ( t ) is the angular-momentum operator in the Heisenberg pic-
ture, as in (3.2.1),

J jα ( t )= e iHt/h J jα e −iHt/h .

At thermal equilibrium, the differential cross-section can then be written

d 2 σ
dEd Ω =
k
k

hγe 2
mc 2

2

e − 2 W ( κ )
αβ

κ α κ β )

jj { 2

} j { 2

( δ αβ − gF ( κ ) gF ( − κ ) } j

∞

1
2 πh

dt e iωt e −i κ · ( R j − R j )

× J jα ( t ) J j β (0) . (4 . 2 . 1)

−∞

If the magnetic atoms are all identical, the form factor may be taken
outside the summation and the cross-section reduces to

hγe 2
mc 2

2

2

d 2 σ
dEd Ω

k
k

| 2

e − 2 W ( κ )

αβ (

= N gF ( κ ) | ( δ αβ − κ α κ β ) S κ ,ω ) ,

αβ

(4 . 2 . 2 a )
where we have introduced the Van Hove scattering function (Van Hove
1954)

∞

N
jj

1
2 πh

dt e iωt 1

αβ (

e −i κ · ( R j − R j ) S κ ,ω )= J jα ( t ) J j β (0) ,

−∞

(4 . 2 . 2 b )
which is (2 πh ) − 1 times the Fourier transform, in space and time, of
the pair-correlation function

is added
and subtracted, the scattering function may be written as the sum of a
static and a dynamic contribution:

J jα ( t ) J j β (0) . f J jα J j β

αβ
d

αβ ( αβ (

S κ ,ω )= S κ )+ S ( κ ,ω ) , (4 . 2 . 3 a )

where the static or elastic component is

N

)= δ ( hω ) 1

αβ (

e −i κ · ( R j − R j )

S κ

jj

J jα J j β (4 . 2 . 3 b )

and the inelastic contribution is

1
2 πh S αβ (

,ω )= 1
π

1

αβ
d

e −βhω χ αβ (

S ( κ ,ω )= κ κ ,ω ) . (4 . 2 . 3 c )

1 −