= Barnett response of a magnetic material
{c}
The two-state calculation is not a general formula for a <ferromagnet>. <Exchange interactions>, <magnetic domains>, <magnetic domain-wall pinning>, <magnetic anisotropy>, and <hysteresis> can determine which response is accessible. Both <spin angular momentum> and <orbital angular momentum> can contribute; a single constant <gyromagnetic ratio> may be only a restricted effective description. Rotation sets an internal bias, while the material and its preparation determine <magnetization>. The <Barkhausen effect> illustrates why a smooth bias can produce jumps instead of the toy smooth curve.
For a limited, reversible, isotropic linear model, suppose the same intrinsic <volume magnetic susceptibility> $\chi_v$ responds to the applied bias and to $B_\Omega/\mu_0$. For a uniformly magnetized <ellipsoid> along a principal axis, include its <demagnetizing factor> $N_d$:
$$
H_{\rm int}=H_{\rm app}-N_dM_z,\qquad
M_z=\chi_v\left(H_{\rm app}-N_dM_z+\frac{B_\Omega}{\mu_0}\right).
$$
Solving this <linear equation> yields
$$
M_z=\frac{\chi_v}{1+N_d\chi_v}
\left(H_{\rm app}+\frac{\Omega}{\mu_0\gamma}\right).
$$
Here $H_{\rm int}$ and $H_{\rm app}$ are actual <magnetic field intensities>; the added rotational bias is an equivalent drive in the constitutive response. In <SI>, dimensionless $\chi_v$ multiplies an intensity in $\mathrm{A/m}$, so $M_z=\chi_v\Omega/\gamma$ would omit $\mu_0$ and have the wrong units. A susceptibility defined instead as $\partial M/\partial B$ has different units. State the convention before comparing formulas.
At zero applied <magnetic field intensity>, a stipulated $\chi_v=3$ gives prefactor $3/2$ in a <sphere>, with $N_d=1/3$, and a prefactor approaching $3$ in a long axial <ellipsoid>. This is a geometry check of the assumed linear model. A real cylinder does not have an exactly uniform <demagnetizing field>, and an irreversible <ferromagnet> cannot in general be represented by this single linear equation. Inferring an intrinsic <gyromagnetic ratio> requires an independently calibrated material response and a geometry-dependent conversion of the measured stray <magnetic field>.
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