Past exam of the condensed matter physics course of the University of Oxford / 2013 / 4

In the SI linear-response convention $\mathbf{M}={\chi }_H\mathbf{H}$, a diamagnetic contribution has negative susceptibility: the induced moment opposes the applied field. Orbital motion supplies a familiar microscopic mechanism, and closed-shell atoms can be diamagnetic even without permanent moments. Paramagnetism has positive susceptibility, with induced magnetization along the field; thermal alignment of permanent localized moments is one mechanism, while conduction-electron spin response is another.

Neither sign alone is a definition of ferromagnetic ordering. An ordinary paramagnetic phase has no spontaneous zero-field magnetization. A diamagnetic background can coexist with a larger paramagnetic contribution in the same specimen. Curie's inverse-temperature law describes an appropriate localized-moment model, not every possible paramagnet.

Interpret the source's $\mathbf{J}$ as the dimensionless angular-momentum operator, with $J_z|J,m_J⟩=m_J|J,m_J⟩$. If angular momentum is written with physical units, the magnetic-moment formula instead contains ${\mathbf{J}}_{\mathrm{phys}}/\hslash$; this is a convention, not an extra factor in the requested energy.

The field fixes the quantization axis, soThere are $2J+1$ Zeeman levels equally spaced in energy by $g{\mu }_{B}B$. For positive $gB$, the state with largest positive moment has lowest energy. The given sign defines $\mathbf{J}$ through the specified moment relation; inserting an additional electron-charge minus sign into it would change the source convention.

Let $x=g{\mu }_{B}B/(k_{B}T)$. The single-ion partition function and mean projected magnetic moment areFor $J>0$, the geometric sum also givesIn the high-temperature, weak-polarization regime $|x|J\ll 1$, expand the Boltzmann factors. The numerator's zero-order term vanishes because $\sum m_J=0$, and the first-order term uses $\sum m_J^2=J(J+1)(2J+1)/3$. The denominator is $2J+1+O(x^2)$. ThereforeThis fixed-field per-ion Curie coefficient describes mean moment per ion at fixed field, so it includes $B$. It is not the field-independent volume-susceptibility Curie constant. For number density $n$ and $B\simeq {\mu }_{0}H$, the latter is $C_{\chi }=n{\mu }_0(g{\mu }_B{)}^{2}J(J+1)/(3k_B)$, giving ${\chi }_H=C_{\chi }/T$. At $J=0$, the specified moment is zero and both expressions vanish.

Linear Curie behavior fails gradually when the Zeeman energy of a substantial moment is comparable to thermal energy. Its appropriate scale isAt $T=300\mathrm{K}$, $k_{B}T/{\mu }_B=446.6\mathrm{T}$. Thus $B_{\mathrm{nl}}\sim 446.6/(gJ)\mathrm{T}$: about $223\mathrm{T}$ for $g=2,J=1$, or $447\mathrm{T}$ for a spin-one-half moment with $g=2$. Larger moments saturate at smaller fields. The source does not specify numerical $g,J$, so it determines this parametric scale rather than one universal room-temperature threshold. At sufficiently large field the moment tends toward $g{\mu }_{B}J$ and cannot keep increasing linearly with $B$.

Room-temperature iron is ferromagnetic: interactions among its electronic moments produce spontaneous local order. The independent-ion Curie law therefore needs an interaction and an ordered-state description. In a Weiss mean-field approximation, replace the applied field by $H_{\mathrm{eff}}=H+\Lambda M$. Above the ordering temperature, linearizing the single-ion response yieldsThis is the Curie–Weiss approximation on the paramagnetic side. Merely substituting room temperature below $T_C$ into that high-temperature formula would produce a negative response and is not a valid model of stable iron.

Below $T_C$, use the full nonlinear moment response to the effective field; it has nonzero zero-field solutions. On a chosen ordered branch the local small-field response has the form $M\simeq M_s(T)+{\chi }_{\mathrm{diff}}(T)H$, rather than $M\propto H/T$ through the origin. Macroscopic magnetic domains and hysteresis further determine the measured net moment. Exchange interactions, arising from electronic quantum statistics and Coulomb correlations, supply the alignment energy; bare dipolar fields alone do not explain ordinary iron's strong ferromagnetism. A localized-ion Weiss model describes the qualitative ordering, while iron's itinerant electronic structure limits a literal independent-fixed-ion interpretation.