Localized moments and Curie susceptibility

Use dimensionless spin operators $\mathbf s=\mathbf S_{\mathrm{phys}}/\hbar$, with $\mathbf s^2=S(S+1)$ and projection quantum numbers m from minus S to plus S. The positive number g is a spin-only or effective multiplet factor. For an electron-like localized moment, $\boldsymbol\mu=-g\mu_B\mathbf s$. Thus a field along positive z favors negative angular-momentum projection while the magnetic moment aligns with the field. The Bohr magneton has units of joules per tesla.

The Zeeman Hamiltonian is minus moment dotted into magnetic induction B, so its sign becomes positive when expressed through the electron's spin. A spin-half doublet therefore has energies plus and minus half of $g\mu_B B$. A crystal-field doublet may require an anisotropic g tensor instead of this scalar approximation. If orbital motion is not quenched, one must first identify the appropriate total-angular-momentum multiplet; simply inserting the bare electron spin can give the wrong Curie constant.

At zero field all projections in the multiplet have equal statistical weight, so their mean is zero and their mean square is $S(S+1)/3$. Expanding the Boltzmann weights to first order in B makes the mean spin projection negative. Multiplication by the negative moment factor then gives positive magnetization. Equivalently, the susceptibility measures the zero-field variance of the moment component; the factor one third reflects rotational isotropy rather than a classical approximation to the spin length.

For number density n, magnetization M is moment per volume, measured in amperes per metre. The dimensionless SI volume susceptibility is $\chi=M/H$, where H is the magnetic field strength. In a weakly magnetized sample with negligible demagnetizing correction, $B\simeq\mu_0H$, introducing the factor of vacuum permeability. A response defined instead as M divided by B has different units and lacks that factor. Molar susceptibilities introduce the number of moments per mole and must not be substituted silently.

The linear expansion requires $g\mu_BBS\ll k_BT$. At stronger fields or lower temperatures, populations become unequal enough that magnetization bends toward saturation rather than diverging as an unrestricted linear law. For spin one half the exact independent-moment result is a hyperbolic tangent. Its small-argument expansion recovers the Curie law, while its large-argument limit gives the saturated moment. This limit is an immediate check on signs and factors of two.

Exchange, crystal-field splittings, itinerant electrons, and temperature-independent orbital contributions can all modify the measured susceptibility. A nonzero Curie–Weiss intercept can suggest interactions, but a fit alone does not establish an ordered phase or identify a microscopic exchange path. Finite collections of independent moments do not undergo a thermal ordering transition. The observed field response in this lesson is population redistribution within a statistical ensemble, not coherent precession of a single pure spin state.