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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 $N_d$:Solving this linear equation yieldsHere $H_{\mathrm{int}}$ and $H_{\mathrm{app}}$ 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 $A/m$, 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.

Spatial variations in energy, including crystal defects, can trap a magnetic domain wall. This is magnetic domain-wall pinning. Increasing the applied magnetic field can release the wall into a collective jump.

A full $180^{\circ }$ reversal leaves saturated isotropic magnetostrictive strain unchanged. A $90^{\circ }$ rotation generally changes it. In the simplest unconstrained orientation average over magnetic domains, replace $m_i{m}_j$ by $⟨m_i{m}_j⟩$, not by $⟨m_i⟩⟨m_j⟩$. Equal populations pointing along $+z$ and $-z$ have zero net magnetization but the same preferred strain as a single domain along $z$. An isotropic orientation distribution instead has $⟨m_i{m}_j⟩={\delta }_{ij}/3$ and zero average preferred strain. Elastic compatibility between real magnetic domains can modify the macroscopic response.

Reversing an applied magnetic field does not guarantee that all microscopic orientations retrace the same path. Hysteresis, magnetic domain-wall pinning, and changes in domain populations can alter the measured curve. The reversal identity concerns the same pair of magnetic states with opposite local magnetization; it is not a universal prediction that an arbitrary experimental sweep is a single-valued even function of applied magnetic field.