The definition of the dipole tensor and of multipole tensors with Kanzaki forces assumes that they are applied to atoms in a perfect crystal and selected such that they produce a strain field that is identical to the actual strain field in a crystal with the defect present. In particular, the dipole tensor reproduces the long-range part of this real strain field, and it can be determined from the Huang scattering measure­ments of crystals containing the particular defects. The actual specification of the exact Kanzaki forces is therefore not necessary. However, if the crystal is
under the influence of external loads, the Kanzaki forces may be different in the deformed reference crystal. Consider for example the case of a crystal with a vacancy and under external pressure. In the absence of pressure, the vacancy relaxation volume has a certain value. However, under pressure, the volume of the vacancy may change by a different amount than the average volume per atom, and therefore, the Kanzaki forces necessary to reproduce this additional change will have to change from their values in the crystal under no pressure. The change of the Kanzaki forces induced by the extraneous strain field may then also be viewed as a change in the dipole tensor by dPj. Assuming that this change is to first-order linear in the strains,

dpij = aijki [46]

The tensor a. jjki has been named diaelastic polariz­ability by Kroner27 based on the analogy with dia­magnetic materials.

When the change of the dipole tensor is included in the derivation of the interaction energy per­formed in the previous section, an additional contri­bution arises, namely

W2 = -2 a4u ej e0 [47]

The factor of 1/2 appears here because when the extraneous strain field is switched on for the purpose of computing the work, the induced Kanzaki forces are also switched on. This additional contribution W2, the diaelastic interaction energy, is quadratic in the strains in contrast to the size interaction, eqn [44], which is linear in the strain field.

A crystalline sample that contains an atomic frac­tion n of well-separated defects and is subject to external deformation will have an enthalpy of

H(c) = 2(Cijki — O j4e0i + O(V — pij4) N

per unit volume.

It follows from this formula that the presence of defects changes the effective elastic constants of the sample by

n

DCijki о aijki [49]

Such changes have been measured in single crystal specimens of only a few metals that were irradiated at cryogenic temperatures by thermal neutrons or elec­trons. Significant reductions of the shear moduli C44 and C’ = (C11-C12)/2 are observed from which the corresponding diaelastic shear polarizabilities listed in Table 9 are derived.25,28 These values are per Frenkel pair, and hence each one is the sum of the shear polarizabilities of a self-interstitial and a vacancy. By annealing these samples and observing the recovery of the elastic constants to their original values, one can conclude that the shear polarizabil­ities of vacancies are small and that the overwhelming contribution to the values listed in Table 9 comes from isolated, single self-interstitials.

The softening of the elastic region around the self­interstitial to shear deformation is not intuitively obvious. However, the theoretical investigations by Dederichs and associates29 on the vibrational proper­ties of point defects have provided a rather convinc­ing series of results, both analytical and by computer simulations. According to these results, the self­interstitial dumbbell axis is highly compressed, up to 0.6 of the normal interatomic distance between neighboring atoms. Therefore, the dumbbell axis can be easily tilted by shear of the surrounding lattice and thereby release some of this axial compression. The weak restoring forces associated with this tilt introduce low-frequency vibrational modes that are also responsible for the low migration energy of self­interstitials in pure metals.

Computer simulations carried out by Dederichs et ai. with a Morse potential for Cu gave the results presented in Table 10.

While the shear polarizabilities compare favor­ably with the experimental results for Cu listed in Table 9, the bulk polarizability in the last column of Table 10 is too large, and most likely of the wrong sign for the self-interstitial. The experimental results for Cu indicate that the bulk polarizability for the Frenkel pair is close to zero. Atomistic simulations

Table 9 Diaelastic shear polarizabilities per Frenkel pair Metal Al Cu Mo $44 (eV) 269 ± 20 377 ± 44 162 ± 60 (an-ai2)/2 (eV) 125 ± 16 111 ± 14 298 ± 43

Table 10 Diaelastic simulations of Cu	polarizabilities from computer
Diaelastic	a.44 (eV)	(a.11-a.12)/2	(o.11^2a.12)/3
polarizability		(eV)	(eV)
Frenkel pair	481.6	109.5	117.7
Self-interstitial	443.9	77.7	90.3
Vacancy	37.7	31.8	27.4

have also been reported by Ackland31 using an effec­tive many-body potential. The predicted diaelastic polarizabilies all turn out to be of the opposite sign than those reported by Dederichs and those obtained from the experimental measurements. Furthermore, Ackland also reports that the simulation results are dependent on the size of the simulation cell, that is, on the number of atoms. Evidently, the predictions depend very sensitively on the type and the particu­lar features of the interatomic potential as well as on the boundary conditions imposed by the periodicity of the simulation cell.

The model of the inhomogeneous inclusion pio­neered by Eshelby may be instructive to explain the diaelastic polarizabilities of vacancies and self­interstitials. A defect is viewed as a region with elastic constants different from the surrounding elastic con­tinuum. We suppose that this region occupies a spherical volume of NO, has isotropic elastic con­stants K and G, and is embedded in a medium with elastic constants K and G. Here, O is the volume per atom, N the number of atoms in the defect region, and Kand G the bulk and shear modulus, respectively. As Eshelby has shown, when external loads are applied to this medium and they produce a strain field Є0 in the absence of the spherical inhomogene­ity, then an interaction is induced upon forming it that is given by

W = —N Q (KA e0 4 + 2 GB [50]

31

aG = NQGB к a44 + (a11 — a12) [55]

The approximation in the last equation is based on Voigt’s averaging of the shear moduli of cubic mate­rials to obtain an isotropic value.

Let us first apply the formulae [52] to [55] to a vacancy. It seems plausible to select N = 1 and assume that K* G* 0. Then 15(1 — v) 7 — 5v

With these expressions, it is easy to compute the bulk and shear polarizabilities for vacancies, and their values are listed in the second and third columns of Table 11.

Next, we consider the bulk polarizability of self­interstitials. The two atoms that form the dumbbell are under compression, and the local bulk modulus that controls their separation distance may be esti­mated as follows:

K*» K + dK d. DU

dp dV

Here, Au’ is the volume compression of the dumbbell exerted on it by the surrounding material, and as shown in Section 1.01.4, it is given by

Au’ = Au — AV =(1/gE — 1)A V [59]

surrounds the dumbbell becomes significantly dilated due to nonlinear elastic effects. This additional dila­tation, 8V/O, has to be added to A V/O to obtain relaxation volumes that agree with experimental values. We repeat the values for 8V/O in the last column of Table 11 as a reminder. As a result of this additional dilatation, the atomic structure adja­cent to the dumbbell is more like that in the liquid phase, as it lost its rigidity with regard to shear. For this dilated region, consisting of Np atoms that include the two dumbbell atoms, we assume that its shear modulus G = 0. Then

15(1 — v)

7 — 5v and

ap = NpQGBI

If the dilated region extends out to the first, sec­ond, or third nearest neighbors, then Np = 14, 20, or 44, respectively, for fcc crystals, and Np = 10, 16, or 28 for bcc crystals. From these numbers we shall select those that enable us to predict a value for ap that comes closest to the experimental value. Matching it for fcc Cu indicates that the dilated region reaches out to third nearest neighbors, and hence Np = 44. However, the best match for bcc Mo is obtained with Np = 10, a region that only includes the dumbbell and its first nearest neighbors. These respective values for Np are also adopted for the other fcc and bcc elements in Table 11 , and the shear polarizabilities so obtained are listed in the fifth column.

To compare these estimates with experimental results, the approximation given in eqn [55] is used with the data in Table 9 for the shear polarizabilities of Frenkel pairs. These isotropic averages are listed in the sixth column of Table 11, and they are to be compared with (ap + ap). It is seen that the
inhomogeneity model is quite successful in explain­ing the experimental results, in spite of its simplicity and lack of atomistic details.