In contrast to the formation energy of vacancies, there exists no direct measurement for the formation energy of self-interstitials. We have mentioned in Section 1.01.2 that the displacement energy required to create a Frenkel pair is much larger than the combined formation energies of the vacancy and the self-interstitial. As pointed out, there exist a large energy barrier to create the Frenkel pair, namely the displacement energy Td, and this barrier is mainly associated with the insertion of the intersti­tial into the crystal lattice. However, although this barrier should be part of the energy to form a self­interstitial, it is by convention not included. Rather, the formation energy of a self-interstitial is consid­ered to be the increase of the internal energy of a crystal with this defect in comparison to the energy of the perfect crystal. In contrast, since vacancies can be created by thermal fluctuations at surfaces, grain boundaries, and dislocation cores by accepting an atom from an adjacent lattice site and leaving it vacant, no similar barrier exists. The activation energy for this process is simply the sum of the actual formation energy EV and the migration energy Em, that is, the energy for self-diffusion, QsD.

When Frenkel pairs are created by irradiation at cryogenic temperatures, self-interstitials and va­cancies can be retained in the irradiated sample. Subsequent annealing of the sample and measuring the heat released as the defects migrate and then disappear provide an indirect method to measure the energies of Frenkel pairs. Subtracting from these calorimetric values, the vacancy formation energy should give the formation energy of self-interstitials. The values so obtained for Cu7 vary from 2.8 to

4.2 eV, demonstrating just how inaccurate calorimet­ric measurements are. Besides, measurements have only been attempted on two other metals, Al and Pt, with similar doubtful results. As a result, theoreti­cal calculations or atomistic simulations provide per­haps more trustworthy results.

For a theoretical evaluation of the formation energy, we can consider the self-interstitial as an inclusion (INC) as described in Appendix A. Accord­ingly, a volume O of one atom is enlarged by the amount A V or in other words, is subject to the trans­formation strain

ej = rb{j where 3r = AV/Q [30]

The energy associated with the formation of this inclusion is given in Table A2, and it can be written as

rr _ 2KmQ fAv2

U = 3K + 4m Q

This expression for the so-called dilatational strain energy provides a rough approximation to the forma­tion energy of a self-interstitial in fcc metals when the above volume expansion results are used.

However, as the nearest neighbor cells depicted in Figures 13 and 14 show, their distorted shapes can­not be adequately described with a radial expansion of the original cell in the ideal crystal as implied by eqn [31]. As the detailed analysis by Wolfer19 indicates, the [001] dumbbell interstitial in the fcc lattice does not change the cell dimension in the [001] direction. In fact, it shortens it slightly, imply­ing that e33 = —0.01005. The volume change is there­fore due to the nearest neighbor atoms moving on average away from the dumbbell axis. This can be represented by the transformation strain components en = e22 being determined by

AV

2en + Є33 = — = 1.10164

The transformation strain tensor for the [001] self­interstitials in fcc crystals is then

and it can be divided into an dilatational part, and a shear part, ~y, as shown.

To find the transformation strain tensor for the [011] self-interstitial in bcc crystal, it is convenient to use a new coordinate system with the x3-axis as the dumbbell axis and the x1- and x2-axes emanating from the midpoint of the dumbbell axis and pointing toward the corner atoms. While the distance between these corner atoms and the central atom in the original bcc unit cell is the interatomic distance r0, in the distorted cell containing the self-interstitial, their distance is reduced to v/3r0/2 as shown by Wolfer.19 As a result,

P3

єн = Є22 = — y — 1 ~ -0.134

the remaining strain component is then determined by DV

2eu + Є33 = о = 0.6418

The strain energy associated with the shear part can be shown (see Mura20) to be

rr 2(9K+8m)mOr

U1 12

1 5(3K + 4m) 2

where /2 = —e11 ?22 — ?22?33 — ?33?П [34]

and is equal to /2fcc = 0.1068 and /2bcc = 0.3633 for self­interstitials in fcc and bcc metals, respectively.

We consider now the total strain energy as a reasonable approximation for the formation energy of self-interstitials, namely

Ef « U = U0 + U1 [35]

The dilatational and the shear strain energies for some elements are listed in Table 7 in the fourth and fifth column, respectively. For the fcc elements, the ratio of Ux/U0 is about 0.2. In contrast, for the bcc elements (in italics) this ratio is about two.