Magnetic anisotropy in orientational trapping

Magnetic anisotropy can create a restoring torque that aligns a specimen in a magnetic field. For a small linearly responding object with uniaxial susceptibility difference $\Delta \chi$, volume $V$ and axis angle $\theta$ in an approximately uniform field, an orientation-dependent energy is $U=-V\Delta \chi B^2{\cos }^2\theta /(2{\mu }_0)$, up to an angle-independent term.

Positive $\Delta \chi$ favors alignment along the field; negative $\Delta \chi$ favors a perpendicular axis. Thermal angular fluctuations compete with this energy, so a preferred orientation becomes pronounced when the anisotropy scale exceeds $k_{B}T$. This induced-moment mechanism differs from the $-\mathbf{\mu }\cdot \mathbf{B}$ energy of a fixed permanent moment and is relevant to interpreting magnetic tweezers.