Environmental Engineering Reference
In-Depth Information
neutron energy, is determined by again making use of Bragg-reflection
in a single-crystal analyser, so that only neutrons with energies in a
small interval
dE
around (
hk
)
2
/
2
M
strike the counter. The number of
neutrons in this range, corresponding to a state vector
|
k
s
n
>
for the
scattered neutrons, is
δN
=
V
(2
π
)
−
3
(
k
)
2
dk
d
Ω=
V
(2
π
)
−
3
(
Mk
/h
2
)
dEd
Ω
.
The number of neutrons arriving at the counter per unit time and per
incident neutron is proportional to the scattering area
dσ
=
|
−
1
|
j
(
ks
n
)
×
W
(
ks
n
,
k
s
n
)
δN
,ortothe
differential scattering cross-section
M
2
πh
2
2
d
2
σ
dEd
Ω
=
k
k
2
δ
(
hω
+
E
i
−
|
s
n
;
f>
P
i
|
<
s
n
;
i
|H
int
(
κ
)
|
E
f
)
,
if
(4
.
1
.
4
a
)
where
)=
H
int
e
−i
κ
·
r
n
d
r
n
.
H
int
(
κ
(4
.
1
.
4
b
)
This result of time-dependent perturbation theory, in the first Born ap-
proximation, is accurate because of the very weak interaction between
the neutrons and the constituents of the sample.
In order to proceed further, it is necessary to specify the interaction
Hamiltonian
H
int
. The magnetic moment of the neutron is
m
M
µ
B
=
eh
2
Mc
,
µ
n
=
−
g
n
µ
N
s
n
;
g
n
=3
.
827
;
µ
N
=
1
with
s
n
=
2
. In this chapter, in the interest of conformity with the rest
of the literature, we do not reverse the signs of the electronic angular-
momentum vectors, which are therefore antiparallel to the corresponding
magnetic moments, as is also the case for the neutron.
This magnetic dipole moment at
r
n
gives rise to a vector potential,
at the position
r
e
,
µ
n
×
r
/r
3
,
A
n
(
r
e
,
r
n
)=
A
n
(
r
=
r
e
−
r
n
)=
with
r
=
. The magnetic-interaction Hamiltonian for a neutron at
r
n
with a single electron of charge
|
r
|
−
e
, with coordinate
r
e
,momentum
p
,
and spin
s
is
p
+
e
c
(
A
n
+
A
e
)
2
p
+
e
c
A
e
2
+2
µ
B
s
·
B
n
1
2
m
1
2
m
H
int
(
r
e
,
r
n
)=
−
=2
µ
B
1
∇×
A
n
)
,
h
A
n
·
p
+
s
·
(
(4
.
1
.
5)
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