78
3 Two (or More) Magnetic Centers
3.8 Confirm that the Bleaney–Bowers expression for a dimer with S 1 = S 2 =
1
2 equals
χ =
2N A µ 2
B g 2
e
kT
3 + exp(−J/kT)
.
This is rather trivial as long as dimeric systems are concerned, because there is
actually only one parameter in the analytical expression of χ and no new information
is obtained by calculating χ(T ) from the theoretical estimates of the J -value. The
situation is different when polynuclear systems are considered. Most importantly,
there are not many systems for which an exact expression of χ has been derived. In
addition to the above stated expression for binuclear complexes, Boˇ ca derived expression for tri- and tetra-nuclear systems [3], which turn out to be rather lengthy. The
situation is even more complicated for extended systems, which have (in principle)
an infinite number of interacting magnetic centers.
In fact, the one-dimensional uniform Heisenberg chain is the only extended system for which an exact solution has been derived making use of the Bethe Ansatz [4].
Bonner and Fisher extended this T = 0 solution to finite temperatures by extrapolating the results obtained for small chains to chains of infinite length [5]. The
Bonner-Fisher expression is still widely used to fit magnetic susceptibility data to
determine the magnetic coupling strength in systems with a magnetic chain-like
topology.
χ(T ) =
N A µ 2
B g 2
e
kT
A + Bx + Cx 2
1 + Dx + Ex 2 + Fx 3
(3.51)
where the values of A–F are given in Appendix D and x =| J |/2kT. Similar
strategies were used to derive expressions for χ(T ) in magnetic chains in which the
magnetic centers alternately interact through J 1 and J 2 [6]. Defining the Hamiltonian
as
ˆ
H =−J
i=1
ˆ
S 2i · ˆ
S 2i+1 + α ˆ
S 2i · ˆ
S 2i−1
(3.52)
with the same quadratic/cubic equation as for the uniform Heisenberg chain for which
A-F are also listed in the Appendix. Note that the Bonner-Fisher expression is only
valid for 2kT/|J | > 0.5 and hence the low-temperature data should not be included
in the fitting procedure. Improvements upon the Bonner-Fisher expression for low
temperatures have been published [7] and many more expressions for the magnetic
susceptibility can be found in Ref. [8].
Magnetic susceptibility data in two-dimensional extended systems are often interpreted based on the work of Rushbrooke and Wood [9], who derived an expression
for χ(T ) valid for high temperatures. The discovery of the high T c superconductors
renewed the interest in the 2D Heisenberg lattices and the original work was extended
to lower temperatures. A workable expression for a uniform lattice—characterized
3 Two (or More) Magnetic Centers
3.8 Confirm that the Bleaney–Bowers expression for a dimer with S 1 = S 2 =
1
2 equals
χ =
2N A µ 2
B g 2
e
kT
3 + exp(−J/kT)
.
This is rather trivial as long as dimeric systems are concerned, because there is
actually only one parameter in the analytical expression of χ and no new information
is obtained by calculating χ(T ) from the theoretical estimates of the J -value. The
situation is different when polynuclear systems are considered. Most importantly,
there are not many systems for which an exact expression of χ has been derived. In
addition to the above stated expression for binuclear complexes, Boˇ ca derived expression for tri- and tetra-nuclear systems [3], which turn out to be rather lengthy. The
situation is even more complicated for extended systems, which have (in principle)
an infinite number of interacting magnetic centers.
In fact, the one-dimensional uniform Heisenberg chain is the only extended system for which an exact solution has been derived making use of the Bethe Ansatz [4].
Bonner and Fisher extended this T = 0 solution to finite temperatures by extrapolating the results obtained for small chains to chains of infinite length [5]. The
Bonner-Fisher expression is still widely used to fit magnetic susceptibility data to
determine the magnetic coupling strength in systems with a magnetic chain-like
topology.
χ(T ) =
N A µ 2
B g 2
e
kT
A + Bx + Cx 2
1 + Dx + Ex 2 + Fx 3
(3.51)
where the values of A–F are given in Appendix D and x =| J |/2kT. Similar
strategies were used to derive expressions for χ(T ) in magnetic chains in which the
magnetic centers alternately interact through J 1 and J 2 [6]. Defining the Hamiltonian
as
ˆ
H =−J
i=1
ˆ
S 2i · ˆ
S 2i+1 + α ˆ
S 2i · ˆ
S 2i−1
(3.52)
with the same quadratic/cubic equation as for the uniform Heisenberg chain for which
A-F are also listed in the Appendix. Note that the Bonner-Fisher expression is only
valid for 2kT/|J | > 0.5 and hence the low-temperature data should not be included
in the fitting procedure. Improvements upon the Bonner-Fisher expression for low
temperatures have been published [7] and many more expressions for the magnetic
susceptibility can be found in Ref. [8].
Magnetic susceptibility data in two-dimensional extended systems are often interpreted based on the work of Rushbrooke and Wood [9], who derived an expression
for χ(T ) valid for high temperatures. The discovery of the high T c superconductors
renewed the interest in the 2D Heisenberg lattices and the original work was extended
to lower temperatures. A workable expression for a uniform lattice—characterized