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In-Depth Information
antiferromagnetic order. On the contrary,
J
0 favours the spins being oriented
ferromagnetically. The case of a 1D antiferromagnetic spin-1/2 lattice was solved
exactly by H. Bethe with coupling only between nearest neighbours (Bethe, 1931).
In the presence of an external magnetic field
B
, Eq. (1.47) becomes:
>
N
ion
N
ion
H
spin
=−
J
S
n
S
n
+
1
−
µ
m
B
S
n
,
(1.48)
n
=
1
n
=
1
where
µ
=
g
µ
B
(
µ
=
e ¯h
/
2
m
e
). The total magnetization
M
(
=
χ
B
) for this 1D
m
B
chain is given by the expression (Ising, 1925):
sinh
µ
m
B
k
B
T
=
N
ion
µ
sinh
µ
m
B
.
(1.49)
M
m
2
e
−
4
J
+
k
B
T
k
B
T
One of the striking consequences of the Isingmodel, pointed out by E. Ising himself,
is the absence of permanent magnetization since the total magnetization
M
vanishes
for
B
0 (Ising, 1925). This behaviour is a characteristic property of 1D magnetic
systems and will be discussed in Section 6.2 with some examples.
Spin frustration in a linear antiferromagnetic chain can be achieved by adding
a second-neighbour antiferromagnetic exchange, since nearest-neighbour antifer-
romagnetic exchange favours second-neighbour ferromagnetic ordering. On the
other hand, if
J
is modulated the system exhibits spin waves (magnons) and re-
sults in spin-Peierls effects in analogy to the phonon-mediated Peierls instabilities
discussed before.
Equation (1.47) can be generalized to the 2D case of an (
n
×
m
) lattice. The ex-
pression then becomes, again only valid for near-neighbour interactions but without
cyclic boundary conditions:
=
N
ion
−
1
N
ion
−
1
H
spin
=−
1
{
J
n
,
n
+
1
,
m
S
n
,
m
S
n
+
1
,
m
+
J
n
,
m
,
m
+
1
S
n
,
m
S
n
,
m
+
1
}
.
(1.50)
m
=
1
n
=
Equation (1.50) can be further simplified assuming that
J
n
,
n
+
1
,
m
≡
J
and
J
n
,
m
,
m
+
1
≡
J
⊥
.If
J
J
⊥
or
J
J
⊥
we fall into the 1D case of independent 1D
chains. However, if
J
<
J
⊥
, we have parallel chains with some degree of interchain
interaction. In this case we have arbitrarily set that
J
and
J
⊥
represent the inter- and
intrachain interactions, respectively. If
n
max
=
2 we have a two-leg ladder with
N
ion
rungs. More complex spin ladders can be built for
n
2. Sr
2
CuO
3
, whose crystal
structure consists of linear chains of corner-sharing CuO
4
squares, is an inorganic
example of a spin-1/2 antiferromagnetic Heisenberg chain. (DTTTF)
2
[Au(mnt)
2
], a
mixed-valence salt, formed by segregated DTTTF and [Au(mnt)
2
] stacks with a her-
ringbone arrangement, is an example of a two-leg spin-ladder (Rovira
et al.
, 1997).
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