Three Ising Spins

Assume three spins are arranged in an equilateral triangle with each spin interacting with its two neighbors (see Figure 1.1). The energy expression for the Ising model in a magnetic field (F = μ . H) is

(1)

Derive expressions for the

a) Partition function

b) Average spin ⟨s⟩.

c) Internal energy ε.

Figure 1.1

SOLUTION

a) Define K = βJ and f = βF where β = 1/τ. The definition of the partition function is

(1)

A direct calculation gives

(2)

b) The average value of spin is

(3)

c) The internal energy is

(4)