Dynamics of a Magnetic Needle Magnetometer: Sensitivity to Landau-Lifshitz-Gilbert Damping

An analysis of a single-domain magnetic needle (MN) in the presence of an external magnetic field $\mathbf{B}$ is carried out with the aim of achieving a high-precision magnetometer. We determine the uncertainty $\mathrm{\Delta }B$ of such a device due to Gilbert dissipation and the associated internal magnetic field fluctuations that give rise to diffusion of the MN axis direction $\mathbf{n}$ and the needle orbital angular momentum. The levitation of the MN in a magnetic trap and its stability are also analyzed.

Figure 1

(a) The normalized spin vector $\mathbf{m}$ versus time for the low-field case at short times (5 orders of magnitude shorter than in Fig. 2) when the initial spin is not along the fast axis. (b) The reduced orbital angular momentum vector $\mathbf{\ell }(t)$. (c) The inner product $\mathbf{m}(t)·\mathbf{n}(t)$ (the projection of the spin on the fast magnetic axis of the needle).

Figure 2

Dynamics for the low-field case (${\omega }_B=1$ ${\mathrm{s}}^{-1}$), over relatively long timescales relative to those in Fig. 1. (a) $\mathbf{m}$ versus time in units of seconds (note that $\mathbf{n}$ is indistinguishable from $\mathbf{m}$ on the scale of the figure). (b) $\mathbf{\ell }(t)$ (note that it stays small compared to $S$). (c) Parametric plot of the needle spin vector $\mathbf{m}(t)$ showing that nutation is almost imperceptible for small fields [contrast this with the large-field result in Fig. 3]; only precession is important.

Figure 3

High-field case (${\omega }_B={10}^5{\mathrm{s}}^{-1}$). (a) $\mathbf{m}(t)$ [which is almost numerically equal to $\mathbf{n}(t)$]. (b) $\mathbf{\ell }(t)$ (note the ordinate axis scale is $[-40,40]$). (c) Parametric plot of the needle spin vector $\mathbf{m}(t)$ showing that strong nutation occurs for large fields in addition to precession.