lie at the corners of a rectangular plane as shown in
Figure 9. As the jumping atom crosses this plane,
they are displaced such as to open the channel. This
coordinated motion can be viewed as a particular
strain fluctuation and described in terms of phonon
excitations. In this manner, Flynn11 has derived the
following formula for the energy of vacancy migra-
tion in cubic crystals.

Em = ______ 15CH C44(C11 ~ C12 VX 14i

V 2[C11(C11 — C12) + C44(5C11 — 3 C12)]

Here, a is the lattice parameter, Ch1, C12, and C44 are
elastic moduli, and X is an empirical parameter that
characterizes the shape of the activation barrier and
can be determined by comparing experimental
vacancy migration energies with values predicted by
eqn [14]. Ehrhart eta/.7,12 recommend that X = 0.022
for fcc metals and X = 0.020 for bcc metals.

2.5

In the derivation of Flynn,11 only the four nearest neighbor atoms are supposed to move, while all other atoms are assumed to remain in their normal lattice positions. On the other hand, Kornblit eta/.13 treat the expansion of the diffusion channel as a quasistatic elastic deformation of the entire surrounding mate­rial. The extent of the expansion is such that the opened channel is equal to the cross-section of the jumping atom, and a linear anisotropic elasticity cal­culation is carried out by a variational method to determine the energy involved in the channel expan­sion. A vacancy migration energy is obtained for fcc metals of

[15]

Figure 9 Second nearest neighbor atom (blue) jumping
through the ring of four next-nearest atoms (green) into
adjacent vacancy in a fcc structure.

("£max _ e™1). As a result, Kornblit14 assumes that the vacancy migration energy for bcc metals is given by

To determine the preexponential factor for self­diffusion

if E^ _ £yin < 2kT if EVlax _ E™11 > 2kT

max EV ’ max min 2EV EV ;

DSd — VLV*2exp( (4 + sm) /k) [21]

requires the values for the entropy 4 + and for the attempt frequency nLV. Based on theoretical esti­mates, Seeger and Mehrer5 recommend a value of 2.5 k for the former. The atomic vibration of nearest neighbor atoms to the vacancy is treated within a sinusoidal potential energy profile that has a maxi­mum height of Ef. For small-amplitude vibrations, the attempt frequency is then given by

m

EV —

[18]

Using the formulae of Flynn and Kornblit, we compute the vacancy migration energies and compare them with experimental values in Figure 10.

With a few exceptions, both the Flynn and the Kornblit values are in good agreement with the exper­imental results.

The self-diffusion coefficient determines the transport of atoms through the crystal under condi­tions near the thermodynamic equilibrium, and it is defined as

Vlv

Dsd — D? CVq — VLV«2exp( (sV + Sm)/k) exp(_Qso/kT) — D0Dexp(_QsD/kT) [19]

where the activation energy for self-diffusion is

Qsd — 4 + Em [20]

The most accurate measurements of diffusion coef­ficients are done with a radioactive tracer isotope of the metal under investigation, and in this case one obtains values for the tracer self-diffusion coeffi­cient dTd — fDsD that involves the correlation factor f For pure elemental metals of cubic structure, f is a constant and can be determined exactly by com — putation.15 For fcc crystals, f— 0.78145, and for bcc crystals, f— 0.72149.

[22]

Vlv

crystals where M is the atomic mass.

In contrast, Flynn11 assumes that the atomic vibra­tions can be derived from the Debye model for which the average vibration frequency is

2

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