Environmental Engineering Reference
In-Depth Information
The approximate expression is obtained by using
hω
±
∆inthetem-
perature denominator.
The above results are only valid as long as the excitation energies
remain positive for all
q
. The mode of lowest energy is found at the wave-
vector
Q
at which
J
αα
(
q
) has its maximum. Introducing the critical
parameter
χ
αα
(0)
χ
αα
(
Q
,
0)
,
R
(
T
)=1
−
(7
.
1
.
6
a
)
which, in the present approximation, depends on
T
through
n
01
:
E
Q
/
∆
2
=
n
01
R
0
2
M
α
J
αα
(
Q
)
∆
R
(
T
)=1
−
;
R
0
=
,
(7
.
1
.
6
b
)
we find that the excitation energies are all positive as long as
R
(
T
)
<
1. This parameter increases monotonically when the temperature is
lowered and, if the zero-temperature value
R
0
is greater than one, the
energy
E
Q
of the
soft mode
vanishes at a temperature
T
=
T
N
(or
T
C
if
Q
=
0
) determined by
R
(
T
N
) = 1. Correspondingly, the susceptibility
χ
αα
(
Q
,
0) becomes infinite at this temperature. This indicates that the
system undergoes a second-order phase transition, from a paramagnetic
phase to one which has the same symmetry as the soft mode. In this case,
this means that
J
αi
=
J
α
cos (
Q
·
R
i
+
ϕ
), where the MF equations
have a non-zero solution for
below, but not above,
T
N
.
We shall assume ferromagnetic ordering with
Q
=
0
.FortheIsing
model in a transverse field, the development of a ferromagnetic moment
below
T
C
corresponds to a rotation of the moments away from the di-
rection of the 'transverse field'. The MF Hamiltonian in the (
J
α
|
0
>
|
1
>
)-
basis is
H
MF
(
i
)=
E
0
−
δ
;
δ
=
M
α
J
αα
(
0
)
J
α
.
(7
.
1
.
7)
−
δE
1
Introducing the new eigenstates
0
>
=cos
θ
|
|
0
>
+sin
θ
|
1
>
(7
.
1
.
8
a
)
1
>
=cos
θ
|
|
1
>
−
sin
θ
|
0
>,
we find that the coupling parameter
δ
, due to the molecular field, gives
rise to a non-zero moment
<
0
|
0
>
=
M
α
sin 2
θ
in the ground state.
Because it is a singlet, the ground state
J
α
|
0
>
in the paramagnetic phase is
necessarily 'non-magnetic', in zero field. This condition does not apply
in the ordered phase, so the nomenclature
induced-moment
system is
frequently used. In the ordered phase, the splitting between the two
singlets is ∆
/
cos 2
θ
,and
|
J
α
=
n
01
M
α
sin 2
θ
(where
n
0
and
n
1
are now
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