Environmental Engineering Reference
In-Depth Information
where
F
G
(
)isthe
geometric structure factor
. The elastic cross-section
then becomes
κ
d
Ω
=
N
0
hγe
2
2
2
dσ
|
2
e
−
2
W
(
κ
)
gF
(
κ
)
|
(
δ
αβ
− κ
α
κ
β
)
|J
α
J
β
| ×
mc
2
αβ
(1 +
δ
Q
0
)Re
F
α
(
)
{
(2
π
)
3
υ
1
4
)
F
β
(
τ
τ
δ
(
τ
+
Q
−
κ
)+
δ
(
τ
+
Q
+
κ
)
}
τ
(4
.
2
.
9
a
)
where
N
0
is the number of unit cells, and the
structure factor
is
s
=1
r
J
α
|
−
1
e
−i
τ
·
d
s
.
F
α
(
τ
)=
|
J
sα
(4
.
2
.
9
b
)
As an example, we return to the Heisenberg ferromagnet discussed
in Chapter 3. The magnitude of the ordered moments and their direction
relative to the crystal lattice, defined to be the
z
-axis, may be determined
by neutron diffraction, since
d
Ω
=
N
hγe
2
2
2
(2
π
)
3
υ
dσ
|
2
e
−
2
W
(
κ
)
2
(1
κ
z
)
gF
(
κ
)
|
−
S
z
δ
(
τ
−
κ
)
.
mc
2
τ
(4
.
2
.
10)
The Bragg-peak intensity is thus proportional to the square of the or-
dered moment and to sin
2
θ
,where
θ
is the angle between the magne-
tization and the scattering vector. The elastic scattering is therefore
strongest when
is perpendicular to the magnetization. On the
other hand, the inelastic scattering is strongest when the scattering vec-
tor
κ
=
τ
κ
=
q
+
τ
is along the magnetization, in which case, from (3.4.11),
− e
−βhω
χ
xx
(
q
,ω
)+
χ
yy
(
q
,ω
)
,ω
)=
1
π
1
αβ
(
(
δ
αβ
−
κ
α
κ
β
)
S
κ
1
αβ
1
=
S
z
{
δ
(
hω
−
E
q
)
−
δ
(
hω
+
E
q
)
}
1
−
e
−βhω
=
S
z
{
(
n
q
+1)
δ
(
hω
−
E
q
)+
n
q
δ
(
hω
+
E
q
)
}
,
(4
.
2
.
11)
where
n
q
=(
e
βE
q
−
1)
−
1
is the Bose population factor. The magnon-
scattering intensity is thus proportional to the ordered moment, and
the
stimulated emission and absorption
of the boson excitations, i.e. the
magnons, due to the neutron beam, are proportional respectively to
(
n
q
+1)and
n
q
, which may be compared with the equivalent result for
light scattering from a gas of atoms.
Search WWH ::
Custom Search