If the elastic distortions associated with self­interstitials could be adequately treated with linear elasticity theory, and if the repulsive interactions between the dumbbell atoms with their nearest neigh­bors were like that between hard spheres, then the volume change of a solid upon insertion of an intersti­tial atom would be equal to the volume change AVas derived above. This follows from the analysis of the inclusion in the center of a sphere given in Appendix A. From the results listed in Table A2, under column INC, we see that the volume change of the solid with a concentration S of inclusions is simply given by where 3m is the volume dilatation per inclusion as if it were not confined by the surrounding matrix. This remarkable result has been proven by Eshelby21 to be valid for any shape of the solid and any location of the inclusion within it, provided the inclusion and the solid can be treated as one linear elastic material. In other words, the elastic strains within the inclusion and within the matrix must be small.

However, this is not the case for the elastic strains produced by self-interstitials. Here, the elas­tic strains are quite large. For example, the volume of the confined inclusion, also listed in Table A2 under column INC, is given by

Au 3m 3m 1 + v

u 1 + o gE 3^ 3(1 — v)

and so it is reduced to about 62% of the uncon­strained volume for a Poisson’s ratio of v = 0.3.

This amounts to an elastic compression of 42% of the ‘volume’ of the self-interstitial in fcc materials, and 25% for the self-interstitial in bcc materials. Clearly, nonlinear elastic effects must be taken into account.

Zener22 has found an elegant way to include the effects of nonlinear elasticity on volume changes pro­duced by crystal defects such as self-interstitials and dislocations. If Urepresents the elastic strain energy of such defects evaluated within linear elasticity theory, and if one then considers the elastic constants in the formula for U to be in fact dependent on the pressure, then the additional volume change dV produced by the defects can be derived from the simple expression
found by Schoeck.23 Its application to the strain energy of self-interstitials leads to the following result.

dv = rBV/m+MTK — i 1 3K + 4m к

12 (K m — K m) 1 U [37]

(3K + 4m)(9K + 8m) m к 1

Here, m and K’ are the pressure derivatives of the shear and bulk modulus, respectively. The first term arising from the dilatational part of the strain energy was derived and evaluated earlier by Wolfer.19 It is the dominant term for the additional volume change for self-interstitials in fcc metals. Here, we evaluate both terms using the compilation of Guinan and Steinberg24 for the pressure derivatives of the elastic constants, and as listed in Table 7.

The calculated relaxation volumes for self­interstitials,

Vrel = AV + dv [38]

are given in the eighth column of Table 7, and they can be compared with the available experimental values also listed.

We shall see that the relaxation volume of self­interstitials is of fundamental importance to explain and quantify void swelling in metals exposed to fast neutron and charged particle irradiations.