Environmental Engineering Reference
In-Depth Information
the
c
-direction, for which the double-zone representation may be used.
Introducing the Fourier transforms, we may write
H
1
=
q
A
q
a
q
a
q
+
B
2
−
q
+
a
q
a
−
q
)
,
(
a
q
a
+
(5
.
2
.
17)
with
A
q
=
A
+
J
{J
(
0
)
−J
(
q
)
}
.
(5
.
2
.
18)
H
1
is quadratic in the Bose operators, and it can be diagonalized by
performing a
Bogoliubov transformation
.AnewBoseoperator
α
q
is
introduced, such that
v
q
α
+
2
2
=1
,
a
q
=
u
q
α
q
−
;
|
u
q
|
−|
v
q
|
(5
.
2
.
19)
−
q
in terms of which
H
0
+
H
1
is transformed into
H
1
=
U
0
+
U
1
+
q
E
q
α
q
α
q
,
H
0
+
(5
.
2
.
20)
when
u
q
and
v
q
are adjusted appropriately. Here they can both be
chosen to be real quantities, and are determined by the equation
v
q
)
2
=(
A
q
±
(
u
q
±
B
)
/E
q
.
(5
.
2
.
21)
The energy parameters are
E
q
=
A
q
−
2
q
U
1
=
1
(
E
q
−
A
q
)
;
B
2
.
(5
.
2
.
22)
When
B
is different from zero, as occurs if either
B
2
or
B
6
is non-zero,
the product of the
|
J
iz
=
J>
=
|
0)
i
-states is no longer the (MF) ground
state.
Q
2
and
Q
6
give rise to couplings between the single-ion states
|J>
,
2
>
etc. as reflected in the term proportional to
B
in (5.2.17).
The new ground state established by the Bogoliubov transformation
has the energy
U
0
+
U
1
(=
U
0
−
q
B
2
/
4
E
q
to leading order in
B
),
which is always smaller than
U
0
. The admixture of (predominantly) the
|
|J −
2
>
-state into the ground state implies that the system is no longer
fully polarized at
T
= 0, as assumed in (5.2.5). Using (5.2.19) and the
conditions
J
−
α
q
α
q
α
q
α
q
=
=0,whereas
1
e
βE
q
−
α
q
α
q
=
n
q
=
(5
.
2
.
23)
1
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