Magnetotropic susceptibility

The magnetotropic susceptibility is the thermodynamic coefficient associated with the rotational anisotropy of the free energy in an external magnetic field and is closely related to the magnetic susceptibility. It emerges naturally in frequency-shift measurements of oscillating mechanical cantilevers, which are becoming an increasingly important tool in the quantitative study of the thermodynamics of modern condensed-matter systems. Here we discuss the basic properties of the magnetotropic susceptibility as they relate to the experimental aspects of frequency-shift measurements, as well as to the interpretation of those experiments in terms of the intrinsic properties of the system under study.

Figure 1

The sample is rotated in an external magnetic field $\mathit{B}$ around axis $\mathit{n}$. The magnetotropic susceptibility $k_{\mathit{n}}$ is the curvature—the second derivative—of the angular dependence of free energy in the applied magnetic field. $\mathit{n}$ indicates a slice of rotations by angle ${\theta }_{\mathit{n}}$ around axis $\mathit{n}$.

Figure 2

The magnetotropic susceptibility calculated for an isolated spin $1/2$ with an anisotropic $g$ factor, $g_{a,b}=2,g_c=1$. The rotation axis is along the $b$ axis of the crystal lattice. (a) Field scans up to 100 T. Magnetic field is along the $a$ axis. (b) Angular scans at 20 K. Easy axis is along the $ab$ plane. The hard axis is along the $c$ axis.

Figure 3

Resonating cantilever. (a) Schematic of an oscillating cantilever. The angle of $\mathrm{\Delta }\theta$ at the tip of the cantilever ($z=L$) is equal to the gradient of the displacement, $\mathrm{\Delta }\theta =\nabla \zeta \left(z\right)$. (b) Cross section of the cantilever. The blue dashed line indicates a neutral surface. The purple arrow indicates the magnitude of ${\sigma }_{zz}$ that changes signs across the neutral surface. The orange line indicates the radius of curvature $1/r\left(z\right)={\nabla }^2\zeta \left(z\right)$. The green arrow indicates the cross-sectional torque in the cantilever, Eq. (80). (c) Plot of the shape of the cantilever in fundamental (blue) and the next two modes of the cantilever, Eqs. (93) and (94).