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 < s2 >0(1/2).*Source: *

Purple Book, 1^{st} ed., p. 48 (http://old.iupac.org/publications/books/author/metanomski.html)

Purple Book, 1