The excess volume is another important property in the characterization of grain boundaries. Excess volume was first proposed by Bishop in a private communication to Aaron and Bolling in 1972. It describes how much expansion is induced by the presence of a GB and is thought that the degree and susceptibility of segregation is directly proportional to this. Despite the name the excess volume is actually a change in length, this is because of the 2D nature of GBs the length of interest is the expansion normal to the GB plane. The excess volume (formula_12) is defined in the following way,