Statistical Mechanics - Physics at Oregon State University

4.1. INTRODUCTION. 69
If we add one particle to the system, we assume that the orbital energies for this
particle are the same as for the other particles. For most practical applications
it is not necessary that this is true for an arbitrary number of particles in the
system. Fluctuations in the number of particles are never very large and if the
two formulas only hold for a certain range of values of N around the equilibrium
value < N >, they are already useful and the subsequent discussion is valuable.
This is the case in metals. The energy levels of the conduction electrons in
a metal are in first approximation independent of the state of the electronic
system for many changes in N which are of importance, but certainly not for
changes starting with zero particles! Even for small changes in N one has to be
aware of processes where the orbital levels do change because the correlation
between the particles changes.
Inclusion of correlation.
For atomic systems, as mentioned before, these correlation effects are always
important and a typical formula for the energy of a rare earth atom with nf
electrons in the 4f shell is
E(nf ) = E(0) + nf ɛf + 1
2 n2 f U (4.4)
which introduces a Coulomb interaction U, which is of the same order of magnitude
as ɛf (a few eV). This also shows why a starting point of independent
particles can be very useful. The previous formula can be generalized to
E(state s) = E(0) +
o
n s oɛo + 1
2
o,o ′
n s on s o ′Uo,o ′ (4.5)
If the Coulomb interactions are small, we can try to find solutions using perturbation
theory, based on the starting point of the independent particle approximation.
Of course, it is also possible to extend the equation above to include
third and higher order terms!
Inclusion of quantum statistics.
How many particles can there be in one orbital? That depends on the nature
of the particles! If the particles are Fermions, we can only have zero or one
( no = 0, 1). If the particles are Bosons, any number of particles in a given
orbital is allowed. Hence in the independent particle formalism the effects of
quantum statistics are easy to include! This is another big advantage.
Calculation for independent subsystems.
Suppose a gas of non-interacting particles is in thermal and diffusive contact
with a large reservoir. Hence both the temperature T and the chemical poten-