 Solution

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

At weak field or high temperature,

μ_{B} B ≪ k_{B} T

and

tanh x ≃ x

, so

M ≃ \frac{n μ _{B}^{2} B}{k _{B} T} .

Using

B ≃ μ_{0} H

for the applied-field response and neglecting demagnetizing corrections, the SI volume magnetic susceptibility is

χ = \frac{\partial M}{\partial H}_{H = 0} = \frac{C}{T}, C = \frac{μ _{0} n μ _{B}^{2}}{k _{B}} .

This is Curie's law. If susceptibility is instead defined as

\partial M / \partial B

, its Curie constant is

n μ_{B}^{2} / k_{B}

without

μ_{0}

; stating which field is used prevents a units error.

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