Past exam of the materials science course of the University of Cambridge / 2015 / Paper 1 / 7 / d / Solution

A single-domain precipitate has no internal domain wall available to move. Magnetization changes therefore require rotation, curling or reversal of a particle's magnetization, rather than ordinary wall displacement through that particle. Relative to a larger multidomain material this can increase coercivity, but the actual coercivity depends on anisotropy, shape and particle interactions. If thermal activation over the anisotropy barrier is rapid on the observation timescale, superparamagnetic behavior can instead suppress remanence. Ferromagnetic composition plus small size alone does not prove a permanent hard magnet.

For an elementary wall-energy comparison, imagine one planar wall through a sphere of radius $r$. Its area is $A_w=\pi r^2$. If that trial two-domain state removes essentially all of the specified stray-field energy while adding wall energy ${\gamma }_w{A}_w$, stability of the uniform state requiresWith $r\simeq 0.33\mu \mathrm{m}$ and the supplied $U_{\mathrm{stray}}=5\times 10^{-18}\mathrm{J}$,This is the requested scale under a full stray-energy-release and one-planar-wall approximation; it is a lower-bound estimate within that model.

A more exact comparison uses the actual energy reduction $\Delta U_{\mathrm{stray}}$ of the competing domain pattern, not necessarily the whole original stray energy:No domain geometry, anisotropy energy or barrier is supplied to determine $\zeta$ or establish that the observed state is the thermodynamic ground state rather than kinetically trapped. Thus the given information alone does not establish an unconditional positive numerical lower bound independent of the competing pattern; the elementary estimate must retain its stated assumptions.