  Micromagnetic tomography aims at reconstructing large numbers of individual
magnetizations of magnetic particles from combining high-resolution magnetic
scanning techniques with micro X-ray computed tomography (microCT). Previous
work demonstrated that dipole moments can be robustly inferred, and
mathematical analysis showed that the potential field of each particle is
uniquely determined. Here, we describe a mathematical procedure to recover
higher orders of the magnetic potential of the individual magnetic particles in
terms of their spherical harmonic expansions (SHE). We test this approach on
data from scanning superconducting quantum interference device microscopy and
microCT of a reference sample. For particles with high signal-to-noise ratio of
the magnetic scan we demonstrate that SHE up to order $n=3$ can be robustly
recovered. This additional level of detail restricts the possible internal
magnetization structures of the particles and provides valuable rock magnetic
information with respect to their stability and reliability as paleomagnetic
remanence carriers. Micromagnetic tomography therefore enables a new approach
for detailed rock magnetic studies on large ensembles of individual particles.
