radius of gyration, \(s\)

https://doi.org/10.1351/goldbook.R05121
A parameter characterizing the size of a particle of any shape. For a rigid particle consisting of mass elements of mass \(m_{i}\), each located at a distance \(r_{i}\) from the centre of mass, the radius of gyration, \(s\), is defined as the square root of the mass-average of \(r_{i}^{2}\) for all the mass elements, i.e. \[s = \sqrt{\frac{\sum _{i}m_{i}\ r_{i}^{2}}{\sum _{i}m_{i}}}\] For a non-rigid particle, an average over all conformations is considered, i.e. \[\sqrt{ < s^{2} > } = \frac{\sqrt{< \sum _{\begin{array}{c} i \end{array}}m_{i}\ r_{i}^{2} > }}{\sqrt{\sum _{\begin{array}{c} i \end{array}}m_{i}}}\] The subscript zero is used to indicate unperturbed dimensions, as in \(\text{<}s^{2}\text{>}_{0}^{1/2}\).
Source:
Purple Book, 1st ed., p. 48 [Terms]