One-Dimensional Ising Model
المؤلف:
Sidney B. Cahn, Gerald D. Mahan And Boris E. Nadgorny
المصدر:
A GUIDE TO PHYSICS PROBLEMS
الجزء والصفحة:
part 2 , p 40
30-8-2016
1520
One-Dimensional Ising Model
Consider N spins in a chain which can be modeled using the one dimensional Ising model
(i)
where the spin has the values s = ±1.
a) Find the partition function.
b) Find the heat capacity per spin.
SOLUTION
a) The partition function is defined as
(1)
where the product is taken over the n sites. Define K = βJ where β = 1/τ.
Start by evaluating the sum at one end, say for n = 1. The answer is independent of the value of s2 = ± 1:
(2)
Next we evaluate the sum over s2, which is also independent of the value of s3:
(3)
(4)
So each summation over sn gives the identical factor eK – e-K, and Z is the product of N such factors.
b) The heat capacity per spin is obtained using thermodynamic identities. The partition function is related to the free energy F:
(5)
The entropy is given by
(6)
Now, the heat capacity C is given by
(7)
The heat capacity per spin is
(8)
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