Gyration Tensor - Shape Descriptors

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

$R_{g}^{2} = \lambda_{x}^{2} + \lambda_{y}^{2} + \lambda_{z}^{2}$

The asphericity is defined by

$b \ \stackrel{\mathrm{def}}{=}\ \lambda_{z}^{2} - \frac{1}{2} \left( \lambda_{x}^{2} + \lambda_{y}^{2} \right)$

which is always non-negative and zero only when the three principal moments are equal, λ_x = λ_y = λ_z. 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

$c \ \stackrel{\mathrm{def}}{=}\ \lambda_{y}^{2} - \lambda_{x}^{2}$

which is always non-negative and zero only when the two principal moments are equal, λ_x = λ_y. 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

$\kappa^{2} \ \stackrel{\mathrm{def}}{=}\ \frac{b^{2} + (3/4) c^{2}}{R_{g}^{4}}$

which is bounded between zero and one.

Read more about this topic: Gyration Tensor

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