The gyration tensor is a tensor that describes the second moments of position of a collection of particles-1...

where is the m^th Cartesian component of the position of the i^th particle and which has been defined such that

In the continuum limit,

where represents the number density of particles at position .

The gyration tensor is related to the moment of inertia tensor. The chief difference is that the particle positions are weighted by mass in the inertia tensor. Moment of inertia, also called mass moment of inertia and, sometimes, the angular mass, (SI unit kilogram metre squared kg m2) quantifies the rotational inertia of an object, i. ... Mass is a property of a physical object that quantifies the amount of matter it contains. ...

Diagonalization

Since the gyration tensor is a symmetric 3x3 matrix, a Cartesian coordinate system can be found in which it is diagonal Look up matrix in Wiktionary, the free dictionary. ... Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ...

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 The radius of gyration describes the way in which the total cross-sectional area is distributed around its centroidal axis. ...

The asphericity b is defined by The gyration tensor is a tensor that describes the second moments of position of a collection of particles where is the Cartesian component of the position of the particle and which has been defined such that In the continuum limit, where represents the number density of particles at position . ...

which is always non-negative and zero only for a spherically symmetric distribution of particles. Similarly, the acylindricity c is defined by The gyration tensor is a tensor that describes the second moments of position of a collection of particles where is the Cartesian component of the position of the particle and which has been defined such that In the continuum limit, where represents the number density of particles at position . ...

which is always non-negative and zero only for a cylindrically symmetric distribution of particles. Finally, the relative shape anisotropy κ² is defined The gyration tensor is a tensor that describes the second moments of position of a collection of particles where is the Cartesian component of the position of the particle and which has been defined such that In the continuum limit, where represents the number density of particles at position . ...

which is bounded between zero and one.

Reference

Mattice WL and Suter UW. (1994) Conformational Theory of Large Molecules, Wiley Interscience. ISBN 0471843385

Theodorou DN and Suter UW. (1985) Macromolecules, 18, 1206-1214.