Environmental Engineering Reference
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exchange and Zeeman energies,
ρ
MF
(
x
)=
1
(
0
)
J
2
σ
+
gµ
B
JH
Z
exp(
xJ
z
/J
)
;
x
=
β
{J
}
,
(2
.
2
.
1)
where
σ
=
M/M
0
is the relative magnetization, the direction of which
is assumed to be parallel to the field. In this case the
n
th moment of
J
z
is determined as
J
p
J
n
exp(
xp/J
)
.
1
Z
(
J
z
/J
)
n
σ
n
=
=
(2
.
2
.
2)
p
=
−J
This equation offers the possibility of relating the higher moments
σ
n
to
the first moment, which is the relative magnetization
σ
1
=
σ
, without
referring explicitly to the MF value of
x
in eqn (2.2.1). According to the
analysis of Callen and Shtrikman (1965), the functional dependence of
σ
n
on
σ
has a wider regime of validity than the MF approximation, be-
cause it only utilizes the exponential form of the density matrix, which
is still valid when correlation effects are included in the random-phase
approximation, where the excitations are collective spin-waves, as we
shall discuss in Section 3.5. Furthermore, they found that the functions
σ
n
=
σ
n
(
σ
);
n
2, derived from (2.2.2), only depend very weakly on the
actual value of
J
, and for increasing values these functions rapidly con-
verge towards the results obtained in the limit of infinite
J
(Callen and
Callen 1965). In this limit, the sums in (2.2.2) are replaced by integrals,
and the reduced diagonal matrix-elements of the Stevens operators are
(1
/J
(
l
)
)
<J
z
=
p
≥
J
z
=
p>
J→∞
O
l
|
|
=
δ
m
0
c
l
P
l
(
u
=
p/J
)
,
(2
.
2
.
3)
where the
J
(
l
)
are defined by eqn (1.5.25),
P
l
(
u
) are the Legendre poly-
nomials, and
c
l
are constants. Multiplying the terms in the sum in
(2.2.2) by ∆
p
=
J
∆
u
= 1, and then taking the limit
J
→∞
,weobtain
P
l
(
u
)
e
xu
du
1
−
1
=
1
−
1
(
x
)
I
2
1
c
l
J
(
l
)
(
x
)=
I
l
+
2
O
l
e
xu
du
=
I
l
+
2
(
x
)
.
(2
.
2
.
4)
I
l
+
2
(
x
) is the usual shorthand notation for the ratio of
I
l
+
2
(
x
)to
I
2
(
x
),
i
)
l
+
2
J
l
+
2
and the functions
I
l
+
2
(
ix
) are the modified spherical
(or hyperbolic) Bessel functions. The relative magnetization
(
x
)=(
−
1
x
σ
=
I
2
(
x
)=coth
x
−
is the familiar Langevin function
L
(
x
) and, eliminating
x
in (2.2.4) by
L
−
1
(
σ
), we finally arrive at
writing
x
=
[
σ
]=
I
l
+
2
L
−
1
(
σ
)
,
(2
.
2
.
5)
=
δ
m
0
c
l
J
(
l
)
I
l
+
2
ith
I
l
+
2
O
l
(
σ
)
[
σ
]
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