Anisotropy analysis by IMT – 3D

In three dimensions, irreducible Minkowski tensors (IMT’s) are defined by an expansion of the density of surface normal directions, in complete analogy to the two-dimensional case. The distribution of surface normals uniquely determines a convex object. (This statement is known as Minkowski’s problem. In this context, the surface normal density is sometimes called the extended Gaussian image. [1])

It is useful to select the complete orthogonal set of Spherical harmonics as basis functions. For a polyhedron with face areas A_k and normals \vec n_k, the IMT’s are given by

\displaystyle \psi_{s,m} = \sum_k A_k Y_s^m(\vec n_k).

Here Y_s^m(\vec n) is the obvious notation for the spherical harmonic evaluated with the direction corresponding to the normal \vec n. The index s = 0, 1, 2, \dotsc labels increasing tensor ranks, and m=-s, \dotsc, s is the “magnetic quantum number”. The complex coefficients \psi_{s,m} have the usual transformation properties of spherical harmonics under rotations.

Often, it is useful to define rotational invariants which can serve as fingerprints of a body irrespective of its orientation in space.  The quadratic invariants in particular are

\displaystyle q_s = \frac{4\pi}{2s+1} \times \frac{1}{A} \times \sum_{m=-s}^s |\psi_{s,m}|^2.

For convenience, we have also divided by the total surface area, such that the q_s are invariant under translation, rotation, and scaling.

Three-dimensional irreducible Minkowski Tensors are discussed in greater detail in Ref. [2]. When computed for Voronoi cells, they generalize Paul Steinhardt’s bond-orientational order parameters [3].

[1] [doi] B. K. Horn, “Extended Gaussian images,” Proceedings of the IEEE, vol. 72, iss. 12, p. 1671–1686, 1984.
[2] [doi] W. Mickel, S. C. Kapfer, G. E. Schröder-Turk, and K. Mecke, “Shortcomings of the bond orientational order parameters for the analysis of disordered particulate matter,” Journal of Chemical Physics, vol. 138, iss. 4, p. 44501, 2013.
[3] [doi] P. J. Steinhardt, D. R. Nelson, and M. Ronchetti, “Bond-orientational order in liquids and glasses,” Phys. Rev. B, vol. 28, p. 784–805, 1983.