Statistical Mechanics - Physics at Oregon State University

160 CHAPTER 8. MEAN FIELD THEORY: CRITICAL TEMPERATURE.
Even this form is hard to analyze. Fortunately, in most cases the interactions
in solids are screened and as a result they have a short range. We therefore take
interactions into account between nearest neighbors only on the lattice and
assume that the interactions are the same between all nearest neighbor pairs.
A sum over nearest neighbors only is indicated by < ij > under the summation
sign. We also include the factor of four and Planck’s constant in the interaction
strength J and find for our final model for the energy eigenvalues
E {σ1, · · · , σN } = −J
(8.9)
σiσj
The sum over nearest neighbors again counts each bond only once. For
example, for a simple mono-atomic structure
1
1 = 2Nq where q is the
number of nearest neighbors of each site. Although geometrical effects are no
longer included via the exchange interaction strength, they are important due
to the lattice summations. The state labels do not depend on geometry.
Including a magnetic field.
Very often we need the presence of a magnetic field. The energy term associated
with an external field is simplified by assuming that each total spin
couples to the external field H according to
Hint( Si) = − H • γG Si
(8.10)
This represents the fact that the magnetic moment is proportional to the orbital
moment, according to Mi = γG Si. The components of S parallel to H are the
only ones that play a role. The factors γG and 1
2 are now included in the
strength of the field h, leading to
Eint = − H • M = −h
(8.11)
Hence we define the total magnetic moment or total spin (equivalent in this
case, since we redefined the units of magnetic field by including the appropriate
factors)
M {σ1, · · · , σN } =
(8.12)
Thermodynamic limit.
So far we have not specified the actual positions of the atoms. The atomic
models are valid for all values of N, but a comparison with thermodynamics is
only possible in the thermodynamic limit. If we want to study bulk properties,
we assume that N → ∞ and calculate the energy and magnetic moment per
particle. In that limit all atomic positions become equivalent if the structure
i
i
σi
σi