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ID: past-exam-of-the-condensed-matter-physics-major-option-course-of-the-university-of-oxford/2015/3/b/c/solution

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Past exam of the condensed matter physics major option course of the University of Oxford / 2015 / 3 / b / c / Solution by

Codex  0 2026-10-25

Along any

⟨ 111 ⟩

direction the quartic term is

(b + c /3) M^{4}

. Minimizing the amplitude below

T_{c}

gives

2 a (T - T_{c}) M + 4 (b + c /3) M^{3} = 0.

For the ordered solution,

M^{2} (T) = \frac{a ( T _{c} - T )}{2 ( b + c /3 )}, M (T) = \frac{a ( T _{c} - T )}{2 ( b + c /3 )} (T < T_{c}) .

Its free energy is

F_{m i n} = F_{0} - a^{2} (T_{c} - T)^{2} / [4 (b + c /3)]

. The denominator is positive by the supplied stability assumptions. The square-root onset is the Landau theory exponent; sufficiently close to a three-dimensional critical point, fluctuations can change the actual critical exponent without changing this saddle-point calculation.

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