statistisk fysik

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This problem deals with a standard situation in statistical physics, a single system in contact
with a surrounding heat reservoir at temperature T. The system has several energy levels and
the probabilities for occupation of the different levels depend on T:
In a solid there is a certain kind of defect which is characterized by a polarisation vector p,
with constant length p, but with 8 possible directions given by normal vectors in an x,y,z
coordinate system,


p= (±1,±1,±1) p/√3
In an electric field along the (111) direction: E =E·(1,1,1)/ √3 , the energy of the defect is given
by a scalar product between the vectors p and E,
E(p) = - p ·E .
1. Show that the defect can have 4 different energies, and determine the degeneracy of
each energy level.
2. Show that the partition function for one defect can be written as
Z(T ) = 8 cosh 3 (x),
where x = pE/3kT .
3. Determine the average energy of one defect as a function of T. Give a sketch of this
function and comment on the limits T → 0 and T → ∞ .