Consider a lattice embedded in a particle reservoir. A lattice site can be empty or occupied by a particle. When it is o

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Consider a lattice embedded in a particle reservoir. A latticesite can be empty or occupied by a particle. When it is occupied,the site carries energy ε and magnetic moment µ. An empty site hasneither energy not magnetic moment . When a uniform externalmagnetic field B is applied, each occupied site gains additionalenergy of σμB. The sign of the magnetic energy is determined by thestate of the spin σ = ± 1.
The Hamiltonian for a single site is then given by
H1 = n (ε – μBσ)
Where n = 1 if the site is occupied and n = 0 if not. Each sitecan be considered as a system with four degrees of freedom (n,σ).
Z = 1/(1-zZ1)
Where z and Z1 are fugacityand partition functions for a single size, respectively. [Thenumber of particles N is in the current context is the number ofoccupied sites, which varies from 0 to infinity.]
Z = 1/[1-2z(1+ e-βε cosh(βμB))].
Recall that Z1 = ∑e-βH
All possible states
N = 2z[1+ e-βε cosh(βμB)]/ [1-2z(1+e-βε cosh(βμB))]
U/N = [εcosh(βμB) - µBsinh(βµB)]/[ eβε +cosh(βμB)]
M/N = [µsinh(βµB)]/[ eβε + cosh(βμB)]
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