Environmental Engineering Reference
In-Depth Information
may lead to strong inelastic or quasi-elastic scattering in the vicinity of
the magnetic Bragg peaks.
Independently of whether the magnetic system is ordered or not,
the total integrated scattering intensity in the Brillouin zone has a def-
inite magnitude, determined by the size of the local moments and the
following sum rule:
∞
−∞
S
N
q
1
αα
(
q
,ω
)
d
(
hω
)
α
N
j
N
q
1
1
2
+
=
J
jα
S
αα
(
q
,t
=0)
α
α
N
j
1
=
J
jα
J
jα
=
J
(
J
+1)
,
(4
.
2
.
7)
α
and taking into account the relatively slow variation of the other param-
eters specifying the cross-section. This implies, for instance, that
dσ/d
Ω
is non-zero in the paramagnetic phase, when
= 0, but the distri-
bution of the available scattered intensity over all solid angles makes it
hard to separate from the background. In this case, much more useful in-
formation may be obtained from the differential cross-section measured
in an inelastic neutron-scattering experiment.
For a crystal with a basis of
p
magnetic atoms per unit cell, the
ordering of the moments corresponding to (4.2.4) is
J
α
J
sα
∗
e
−i
Q
·
R
j
s
,
=
2
e
i
Q
·
R
j
s
+
J
j
s
α
J
sα
(4
.
2
.
8
a
)
where
R
j
s
=
R
j
0
+
d
s
,
with
s
=1
,
2
,
···
,p.
(4
.
2
.
8
b
)
Here
R
j
0
specifies the position of the unit cell, and
d
s
is the vector
determining the equilibrium position of the
s
th atom in the unit cell.
The summation over the atoms in (4.2.2) may be factorized as follows:
e
−i
κ
·
(
R
i
−
R
j
)
ij
=
i
0
j
0
p
p
e
−i
κ
·
(
R
i
0
−
R
j
0
)
e
−i
κ
·
(
R
i
s
−
R
i
0
)
e
i
κ
·
(
R
j
r
−
R
j
0
)
s
=1
r
=1
p
=
i
0
j
0
e
−i
κ
·
(
R
i
0
−
R
j
0
)
2
e
−i
κ
·
d
s
,
|
F
G
(
κ
)
|
;
F
G
(
κ
)=
s
=1
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