where the axes are chosen such that the diagonal elements are ordered .
These diagonal elements are called the principal moments of the gyration tensor.
Shape descriptors
The principal moments can be combined to give several parameters that describe the distribution of particles. The squared radius of gyration is the sum of the principal moments divided by the number of particles N:
The
asphericity
is defined by
which is always non-negative and zero only when the three principal moments are equal, λx = λy = λz. This zero condition is met when the distribution of particles is spherically symmetric (hence the name asphericity) but also whenever the particle distribution is symmetric with respect to the three coordinate axes, e.g., when the particles are distributed uniformly on a cube, tetrahedron or other Platonic solid.
Similarly, the
acylindricity
is defined by
which is always non-negative and zero only when the two principal moments are equal, λx = λy.
This zero condition is met when the distribution of particles is cylindrically symmetric (hence the name, acylindricity), but also whenever the particle distribution is symmetric with respect to the two coordinate axes, e.g., when the particles are distributed uniformly on a regular prism.
Finally, the relative shape anisotropy is defined
which is bounded between zero and one. = 0 only occurs if all points are spherically symmetric, and = 1 only occurs if all points lie on a line.
References
Mattice, WL; Suter, UW (1994). Conformational Theory of Large Molecules. Wiley Interscience.