Environmental Engineering Reference
In-Depth Information
The derivatives
F
θθ
and
F
φφ
are directly related to the static sus-
ceptibilities, as shown in Section 2.2.2. When
θ
0
=
π
2
,weobtainfrom
eqn (2.2.18)
2
/F
θθ
2
/F
φφ
.
χ
xx
(
0
,
0) =
N J
z
;
χ
yy
(
0
,
0) =
N J
z
(5
.
3
.
7)
These results are of general validity, but we shall proceed one step further
and use
F
(
θ, φ
) for estimating the frequency dependence of the bulk
susceptibilities. When considering the uniform behaviour of the system,
we may to a good approximation assume that the equations of motion
for all the different moments are the same:
h∂
J
/∂t
=
J
×
h
(eff)
.
(5
.
3
.
8)
By equating it to the average field, we may determine the effective field
from
F
=
F
(0)
−
N
J
·
h
(eff)
,
(5
.
3
.
9
a
)
corresponding to
N
isolated moments placed in the field
h
(eff). The free
energy is
F
=
F
(
θ
0
,φ
0
)+
2
F
θθ
(
δθ
)
2
+
2
F
φφ
(
δφ
)
2
− N
J
·
h
,
(5
.
3
.
9
b
)
and, to leading order,
δθ
=
−
J
x
/
J
z
and
δφ
=
−
J
y
/
J
z
. Hence
1
N
∂F
N
F
θθ
1
J
x
h
x
(eff) =
−
=
h
x
−
2
,
(5
.
3
.
10
a
)
∂
J
x
J
z
and similarly
N
F
φφ
1
J
y
h
y
(eff) =
h
y
−
2
.
(5
.
3
.
10
b
)
J
z
Introducing a harmonic field applied perpendicular to the
z
-axis into
eqn (5.3.8), we have
1
ihω
J
x
=
F
φφ
J
y
−
h
y
J
z
N
J
z
(5
.
3
.
11)
1
ihω
J
y
=
−
F
θθ
J
x
−
h
x
J
z
,
N
J
z
and
∂
/∂t
= 0, to leading order in
h
. Solving the two equations for
h
x
= 0, we find
J
z
1
N
F
θθ
E
0
(
T
)
χ
yy
(
0
,ω
)=
J
y
/h
y
=
(
hω
)
2
,
(5
.
3
.
12
a
)
−
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