Past exam of the theoretical physics major option course of the University of Oxford / 2015 / 3 / b / Solution

Vary each field component independently and integrate the gradient term by parts, taking fixed boundary values or vanishing surface variations. With

ρ = m_{1}^{2} + m_{2}^{2} + m_{3}^{2}

, the Euler-Lagrange equations for the free-energy saddle are

- \nabla^{2} m_{1} + (t + r) m_{1} + 4 u ρ m_{1} - \nabla^{2} m_{2} + (t - r) m_{2} + 4 u ρ m_{2} - \nabla^{2} m_{3} + t m_{3} + 4 u ρ m_{3} = 0, = 0, = 0.

The anisotropy shifts the first two inverse susceptibilities in opposite directions. The derivative of

u ρ^{2}

is

4 u ρ m_{j}

, which fixes the factor needed for both the magnetization and the free energies in c. Under natural free-boundary conditions the integration by parts instead supplies

\partial_{n} m_{j} = 0

at the surface.

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