 Solution

ID: past-exam-of-the-condensed-matter-physics-major-option-course-of-the-university-of-oxford/2016/2/d/solution

The two nonzero stationary points merge when their discriminant vanishes, giving the upper spinodal

T_{1} = T_{c} + \frac{b ^{2}}{4 a c}, κ_{1} = \frac{1}{4} .

a (T - T_{c}) = b P - c P^{2}

, so its free energy reduces to

F = P^{3} (b /6 - c P /4)

. Equality with

F (0) = 0

therefore requires

P = 2 b / (3 c)

. Substitution gives

T_{2} = T_{c} + \frac{2 b ^{2}}{9 a c}, κ_{2} = \frac{2}{9}, P = 0 or \frac{2 b}{3 c} .

Both have positive curvature

2 b^{2} / (9 c)

at coexistence. The intervening maximum is at

P = b / (3 c)

. For

T_{2} < T < T_{1}

the positive-polarization minimum is metastable; for

T_{c} < T < T_{2}

the zero-polarization minimum is metastable.

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