= Barnett equivalent magnetic field
{c}
{title2=$B_\Omega=\Omega/\gamma$}
= Barnett field
{c}
{synonym}
Let the specimen rotate uniformly with <angular velocity> $\boldsymbol\Omega=\Omega\widehat{\mathbf z}$. In its <rotating reference frame>, the internal <Hamiltonian> contains
$$
\mathcal H_{\rm rot}=\mathcal H_0-\Omega J_z.
$$
Here $J_z$ is the internal <angular momentum operator> about the rotation axis, and $\mathcal H_0$ contains the internal interactions and any imposed <magnetic field> expressed in that frame. It does not include the separately treated rigid body's <rotational kinetic energy>. For a sector with <magnetic moment> $m_z=\gamma J_z$, the rotational term has exactly the form of a <magnetic dipole potential energy>:
$$
-\Omega J_z=-m_zB_\Omega,\qquad
B_\Omega=\frac{\Omega}{\gamma}.
$$
This <Barnett equivalent magnetic field> has <tesla (unit)> units, with the angular convention for the signed <gyromagnetic ratio>: $\Omega$ is in radians per second, not revolutions per second. Its sign matters. For an electronic contribution with $\gamma<0$, positive $\Omega$ favors positive $J_z$ and negative $m_z$. A sector with $\gamma>0$ has the opposite magnetic sign. Equal $\Omega$ need not correspond to equal <Barnett fields> for sectors having different <gyromagnetic ratios>.
A short derivation fixes the sign independently of any material model. Put $U(t)=\exp(-\mathrm i\Omega tJ_z/\hbar)$ and transform a laboratory <quantum state> by $|\psi_{\rm rot}\rangle=U^\dagger|\psi_{\rm lab}\rangle$. The <Schrödinger equation> gives
$$
\mathcal H_{\rm rot}
=U^\dagger\mathcal H_{\rm lab}U-\mathrm i\hbar U^\dagger\dot U
=\mathcal H_0-\Omega J_z.
$$
The transformed laboratory term is denoted by $\mathcal H_0$; a laboratory perturbation not stationary in the rotating frame can make it time dependent. Establishing a steady thermal response also requires exchange of internal <angular momentum> with the rotating body, for example through <spin-lattice relaxation>. A mere relabeling of coordinates cannot polarize isolated, unrelaxed <spins>.
The <Barnett field> is a useful description of <spin-rotation coupling>, not a claim that empty laboratory space acquires a uniform <magnetic flux density> $\Omega/\gamma$. In particular, inserting it into the <Lorentz force> on an unrelated test <electric charge> is unjustified. A nearby <magnetometer> detects the specimen's actual stray <magnetic field> and backgrounds, rather than directly detecting this internal <Hamiltonian> parameter.
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