N Independent Spins
المؤلف:
Sidney B. Cahn, Gerald D. Mahan And Boris E. Nadgorny
المصدر:
A GUIDE TO PHYSICS PROBLEMS
الجزء والصفحة:
part 2 , p 41
30-8-2016
1553
N Independent Spins
Consider a system of N independent spin-1/2 particles. In a magnetic field H, in the ẑ direction, they can point either up or down with energy ±μH, where μ is the magnetic moment. Derive expressions for the
a) Partition function
b) Internal energy
c) Entropy
SOLUTION
a) The partition function is given by
(1)
where ξ = βμH. Each spin is independent, so one has the same result as for one spin, but raised to the Nth power
(2)
b) The internal energy is the derivative of ln Z with respect to β:
(3)
c) The entropy is derivative of ln Z with respect to τ:
(4)
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