3.3 From Micro to Macro: The Bottom-Up Approach
79
Fig. 3.7 Some examples of two-dimensional magnetic lattices for which analytical expressions
have been derived to fit J from the temperature dependence of the magnetic susceptibility
by one single J , see Fig. 3.7 left—was given by Woodward and co-workers and reads
χ(T ) =
N A µ 2
B g 2
e
kT
5
n=1
a n J/kT
b n J/kT
(3.53)
More general expressions were derived by Curély for S = 1/2, for 2D lattices with
different magnetic interaction paths (Fig. 3.7 middle) and to hexagonal (or honeycomb) lattices (Fig. 3.7 right) [10, 11].
When no analytical expression can be used to fit χ(T ), the experimental data
are interpreted by defining a magnetic model with the magnetic interactions that are
considered ap r i o r ito be the most important ones. The corresponding Heisenberg
Hamiltonian is then diagonalized and the resulting eigenvalues are substituted in
the van Vleck equation. The J -values of the magnetic model are adjusted to give
an optimal fit of the experimental data. However, one has to be aware that a multiparameter fit can have several solutions of equal quality and that this way of deriving
experimental J -values can be subject to uncertainties. Actually, this is where computations can be helpful to discern the important interactions from less important
ones and determine the sign and order of magnitude of the interactions. This would
in principle lead to a well-founded magnetic model that will lead to reliable J -values
from the fitting procedure.
A closely related procedure allows theoreticians to take the full journey from
microscopic to macroscopic in a three-step strategy [12]. In the first stage, one calculates as exhaustive as possible the interactions among the different magnetic centers.
This should not be restricted to nearest neighbors and preferably also include three- or
four-body interactions, see Sect. 3.4. Secondly, a magnetic model is defined by writing down the Heisenberg Hamiltonian with the most important interactions. When
dealing with an extended system, periodic boundary conditions can be applied. This is
best illustrated taking the 1D Heisenberg chain as example. As illustrated in Fig. 3.8,
the first center in the chain not only interacts with center 2 on the right, but also with
the last center in the chain. In this way, there is no open end in the chain, exactly as in
an infinite 1D chain. The topology of the model is actually a ring, but this turns out
79
Fig. 3.7 Some examples of two-dimensional magnetic lattices for which analytical expressions
have been derived to fit J from the temperature dependence of the magnetic susceptibility
by one single J , see Fig. 3.7 left—was given by Woodward and co-workers and reads
χ(T ) =
N A µ 2
B g 2
e
kT
5
n=1
a n J/kT
b n J/kT
(3.53)
More general expressions were derived by Curély for S = 1/2, for 2D lattices with
different magnetic interaction paths (Fig. 3.7 middle) and to hexagonal (or honeycomb) lattices (Fig. 3.7 right) [10, 11].
When no analytical expression can be used to fit χ(T ), the experimental data
are interpreted by defining a magnetic model with the magnetic interactions that are
considered ap r i o r ito be the most important ones. The corresponding Heisenberg
Hamiltonian is then diagonalized and the resulting eigenvalues are substituted in
the van Vleck equation. The J -values of the magnetic model are adjusted to give
an optimal fit of the experimental data. However, one has to be aware that a multiparameter fit can have several solutions of equal quality and that this way of deriving
experimental J -values can be subject to uncertainties. Actually, this is where computations can be helpful to discern the important interactions from less important
ones and determine the sign and order of magnitude of the interactions. This would
in principle lead to a well-founded magnetic model that will lead to reliable J -values
from the fitting procedure.
A closely related procedure allows theoreticians to take the full journey from
microscopic to macroscopic in a three-step strategy [12]. In the first stage, one calculates as exhaustive as possible the interactions among the different magnetic centers.
This should not be restricted to nearest neighbors and preferably also include three- or
four-body interactions, see Sect. 3.4. Secondly, a magnetic model is defined by writing down the Heisenberg Hamiltonian with the most important interactions. When
dealing with an extended system, periodic boundary conditions can be applied. This is
best illustrated taking the 1D Heisenberg chain as example. As illustrated in Fig. 3.8,
the first center in the chain not only interacts with center 2 on the right, but also with
the last center in the chain. In this way, there is no open end in the chain, exactly as in
an infinite 1D chain. The topology of the model is actually a ring, but this turns out