 Independent spin-one-half two-level system

ID: independent-spin-one-half-two-level-system

For

N

distinguishable noninteracting spins, let a fraction

α

occupy the level

+ ϵ

and the remainder occupy

- ϵ

S = - N k_{B} [α lo g α + (1 - α) lo g (1 - α)], E = N ϵ (2 α - 1) .

The relation

1/ T = \partial S / \partial E

yields

α = \frac{1}{1 + e ^{2 ϵ / (k_{B} T)}}, E = - N ϵ tanh (\frac{ϵ}{k _{B} T}) .

Positive temperature corresponds to

0 \leq α < 1/2

1/2 < α \leq 1

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