Environmental Engineering Reference
In-Depth Information
These are the sum rules relating the spectral frequency-moments with
the thermal expectation-values of operators obtainable from
B
,
A
,and
H
.If
B
=
A
=
A
†
, then (3.3.1) shows that
F
BA
(
ω
)isevenin
ω
,andall
the odd moments vanish. In this case, the even moments are
2
n−
1
A, A
]
χ
AA
(0)
(
hω
)
2
n
AA
=
−
[
L
.
(3
.
3
.
18
b
)
3.4 Linear response of the Heisenberg ferromagnet
In this section, we shall illustrate the use of linear response theory by
applying it to the case of the three-dimensional Heisenberg ferromagnet,
with the Hamiltonian
2
1
H
=
−
i
=
j
J
(
ij
)
S
i
·
S
j
,
(3
.
4
.
1)
where
S
i
isthespinonthe
i
th ion, placed in a Bravais lattice at the
position
R
i
. The spatial Fourier transform of the exchange coupling,
with the condition
J
(
ii
)
≡
0, is
N
(
ij
)
e
−i
q
·
(
R
i
−
R
j
)
=
j
(
q
)=
1
(
ij
)
e
−i
q
·
(
R
i
−
R
j
)
,
J
ij
J
J
(3
.
4
.
2
a
)
and conversely
N
V
N
(2
π
)
3
(
ij
)=
1
(
q
)
e
i
q
·
(
R
i
−
R
j
)
=
(
q
)
e
i
q
·
(
R
i
−
R
j
)
d
q
,
J
q
J
J
(3
.
4
.
2
b
)
depending on whether
q
, defined within the primitive Brillouin zone, is
considered to be a discrete or a continuous variable (we shall normally
assume it to be discrete).
N
is the total number of spins,
V
is the
volume, and the inversion symmetry of the Bravais lattice implies that
J
(
q
)=
J
(
−
q
)=
J
∗
(
q
). The maximum value of
J
(
q
) is assumed to
be
(
q
=
0
), in which case the equilibrium state at zero temperature,
i.e. the ground state, is the ferromagnet:
J
S
i
=
S
z
at
T
=0
,
(3
.
4
.
3)
where
z
is a unit vector along the
z
-axis, which is established as the
direction of magnetization by an infinitesimal magnetic field. This result
is exact, but as soon as the temperature is increased above zero, it is
necessary to make a number of approximations.
As a first step, we
Search WWH ::
Custom Search