 Solution

ID: past-exam-of-the-quantum-condensed-matter-course-of-the-university-of-cambridge/2019/section-b/6/a/solution

J = 1/2

doublet has moment projections

\pm g_{J} μ_{B} /2

. The requested normalization assumes

g_{J} = 2

, so the projections are

\pm μ_{B}

and the two Zeeman energies are

\mp μ_{B} B

. For independent localized ions, the one-ion partition function and thermal moment are

Z_{1} = e^{β μ_{B} B} + e^{- β μ_{B} B} = 2 cosh (β μ_{B} B),

⟨ μ_{z} ⟩ = \frac{μ _{B} e ^{β μ_{B} B} - μ _{B} e ^{- β μ_{B} B}}{Z _{1}} = μ_{B} tanh (β μ_{B} B) .

M = n μ_{B} tanh (\frac{μ _{B} B}{k _{B} T}) .

It saturates at

n μ_{B}

for a field strong compared with

k_{B} T / μ_{B}

. For a different Landé g-factor, replace every moment

μ_{B}

g_{J} μ_{B} /2

;

J = 1/2

alone does not fix that factor.

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