Environmental Engineering Reference
In-Depth Information
The off-diagonal matrix elements involving the ground state are
J>
=
6(
J
h
(
J/
2)
1
/
2
sin
θ
−
2
)
B
2
cos
θ
<J
−
|H|
−
1
J>
=
2
{
2
J
(2)
1
/
2
B
2
sin
2
θ.
<J
−
2
|H|
}
We shall only be concerned with terms up to second order in
B
2
and
h
,so
that we may use second-order perturbation theory, and it is suciently
accurate to approximate the energy differences between the ground state
and the first and second excited-states by respectively ∆
1
=
J
J
(
0
)and
∆
2
=2
J
J
(
0
).
Because of the mixing of the states,
σ
=
J
z
/J
=
1
m
becomes slightly smaller than 1, but this only affects the exchange
contribution quadratic in
m
,as(1
−
1
2
σ
)
σ
=
2
(1
m
2
). To second order,
−
−
the ground-state energy is found to be
−
2
J
2
hJ
cos
θ
+
B
2
J
(2)
(3 cos
2
θ
E
0
(
h
)=
J
−
−
(
0
)
1)
6(
J
h
2
sin
2
θ/
−
2
−
2
−
4
−
2
)
B
2
cos
θ
)(
B
2
)
2
sin
4
θ/
−
J
(
0
)
(
J
(
0
)
.
(2
.
2
.
21)
J
The minimum condition
∂E
0
/∂θ
=0leadsto
)
B
2
1+3
B
2
sin
2
θ/
}
cos
θ
−
2
h
=
h
0
=6(
J
{
2
J
J
(
0
)
or
sin
θ
=0
,
to second order in
B
2
. The free energy
F
(
θ, φ
) at zero temperature is
then,inbothcases,
F
(
θ, φ
)
/N
=
E
0
(
h
0
)+
h
0
Jσ
cos
θ
=
−
2
(
0
)+
2
1) +
4
J
2
κ
2
(3 cos
2
θ
bκ
2
sin
4
θ,
J
−
with
κ
2
=2
B
2
J
(2)
3
B
2
/
;
b
=
−
{
2
J
J
(
0
)
}
,
(2
.
2
.
22
a
)
2
)
b
2
sin
4
θ
.The
b
-
parameter introduced here is the leading order contribution to
b
,de-
fined in (2.2.11), when
θ
=
π/
2. One important feature illustrated by
this calculation is that the
O
2
-term in
Q
2
,with
m
odd, is cancelled
by the Zeeman contribution, to second order in
B
2
. This is a conse-
quence of the freedom to replace the equilibrium condition
∂F/∂θ
=0
by the requirement that
1
and the relative magnetization is
σ
=1
−
(
J
−
) should vanish, by definition,
with the implication that the matrix-element
<J
J
x
(and
J
y
J>
must be
zero within the present approximation. Bowden (1977) did not take the
Zeeman effect into account, and therefore obtained an erroneously strong
renormalization of the anisotropy. The second derivatives of
F
(
θ, φ
)are
F
φφ
=0,and
−
1
|H|
b
sin
2
θ
)cos2
θ
+
2
3
κ
2
(1
κ
2
b
sin
2
2
θ.
F
θθ
/N
=
−
−
(2
.
2
.
22
b
)
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