MAGNETIC SCATTERING OF NEUTRONS - Rare Earth Magnetism: Structures and Excitations - page 167

neglecting the diamagnetic term of second order in µ N . A e denotes the
additional contribution to the total vector potential from the surround-
ing electrons, or an external magnetic field. The prime on p only plays
aroleif A e is non-zero, in which case p = p + e

A e .Wenotethat A n

c

commutes with p , because

∇ e · A n = ∇· A n and

( r ( r

∇· A n ( r )= ∇·{− µ n ×∇ ) } = µ n ·∇×∇ )=0 ,

( r

recalling that r /r 3 = ).

The Fourier transform of A n with respect to the neutron coordinate,
defining x = r n − r e ,is
A n ( r e − r n ) e −i κ · r n d r n = e −i κ · r e A n (

−∇

− x ) e −i κ · x d x
= −e −i κ · r e ( µ n × x ) x − 3 e −i κ · x d x = −e −i κ · r e 4 π

iκ µ n × κ ,

where κ (the integration is performed straight-

forwardly in spherical coordinates). Applying Green's theorem and as-
suming V to be a sphere of radius r ,

is a unit vector along κ

∇× e −i κ · x A n ( x ) d x ∝ ( κr ) − 1

→ 0 or r →∞ ,

from which we deduce
∇× A n ( x ) e −i κ · x d x =

∇

e −i κ · x × A n ( x ) d x

−

e −i κ · x A n ( x ) d x =4 π κ × µ n × κ

= i κ ×

(we note that ∇× A n ( r )= ∇ ( x ) × A n ( x )). From these results, we obtain

)= H int ( r e , r n ) e −i κ · r n d r n

=2 µ B e −i κ · r e 4 π i

H int ( κ

) ,

hκ µ n × κ · p + s ·

( κ × µ n × κ

or

i
hκ κ × p + κ × s × κ

e −i κ · r e .

H int ( κ )=8 πµ B µ n · (4 . 1 . 6)

κ × p commutes with

κ · r e and therefore also with exp( − i κ · r e ), and

we have made use of the identity κ × a × κ = a − ( κ · a ) κ .