Unexpectedly strong shape anisotropy of out-of-phase magnetic susceptibility due to eddy currents of internally isotropic copper cylinders

In order to better understand the anisotropy of out-of-phase magnetic susceptibility (opAMS) due to eddy currents of rocks and minerals, we investigated the ipAMS and opAMS of copper cylinders, because copper is diamagnetic, internally isotropic magnetically, and its opMS is no doubt due to eddy currents. Results of laboratory measurements were compared to theoretical models.  The degree of AMS (P=k₁/k₃, k₁>k₂>k₃) of internally isotropic substance is controlled by the internal susceptibility and demagnetizing factors solely controlled by the substance shape. If the internal susceptibility is very low and this is the case of magnetostatic susceptibility of most diamagnetic substances, there is virtually no shape anisotropy of such material. The magnetostatic susceptibility of copper is about -8 x 10^-6 [SI units], which gives rise to undetectably low degree of magnetostatic AMS. The ipMS is also negative, but several orders lower depending on grain size, grain shape, and operating frequency. The opMS is on the other hand positive, relatively high and also dependent on operating frequency. Calculated from signed (negative) k₁, k₂, k₃ values, the degree of ipAMS, P<1, and decreases with increasing elongation of copper cylinders. It also depends on grain size and operating frequency. The degree of opAMS, P>1 and increases with increasing cylinder elongation. It also depends on grain size and operating frequency. Parallel to the axes of elongated cylinders are the ipk₁ and opk₃ directions, while in flattened cylinders are parallel to the cylinder axes the ipk₃ and opk₁ directions. The above results are useful in the structural interpretation of the opAMS of rocks and ores whose opAMS is due eddy currents (in electrically conductive materials).