 Solution

ID: past-exam-of-the-general-relativity-and-cosmology-and-condensed-matter-physics-paper-of-part-b-of-the-physics-course-of-the-university-of-oxford/2011/section-vi/8/unlettered-1/solution

For positive

λ

, a fluctuation of magnetization produces a Weiss molecular field in the same direction, reinforcing alignment. The mean-field self-consistency equation is

M = M_{s} B_{J} [\frac{g μ _{B} J μ _{0}}{k _{B} T} (H_{applied} + λ M)] .

Linearizing in the paramagnetic phase gives

M = \frac{C}{T} (H_{applied} + λ M), χ = \frac{C}{T - λ C} = \frac{C}{T - T _{C}}, T_{C} = λ C = \frac{μ _{0} λn ( g μ _{B} ) ^{2} J ( J + 1 )}{3 k _{B}} .

At zero applied field, the right-hand side's slope at

M = 0

T_{C} / T

. Below

T_{C}

this exceeds one, and the saturation of the Brillouin function allows stable nonzero solutions of opposite sign: spontaneous ferromagnetism. Above

T_{C}

, only the zero-field solution

M = 0

remains stable. The stated linear susceptibility is the paramagnetic branch for

T > T_{C}

; it must not be continued as a negative susceptibility below the transition.

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