LDA Calculations of Dislocation Mobility in Fe & Mo Page: 4 of 8
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conditional convergence.
Our approach is explained almost completely by stating that at its equi-
librium, the distortion minimizes the elastic energy subject to constraints im-
posed by the defect. We impose periodic boundary conditions by noting that
the distortion field is a periodic function of position and so can be expressed
in a series of plane waves (a generalization of a Fourier Series). The elastic
energy, which is quadratic in the distortion, is then represented in terms of
an infinite sum over reciprocal lattice vectors. Thus, to form a complete
solution, the distortion must match any topological conditions imposed by
the defects, and any freedom left in the distortion must minimize the elastic
energy in the cell.
This is enough to completely specify the solution. The solution is ex-
pressed as an infinite sum, but the sum can be shown to be absolutely con-
vergent, in contrast to other recent solutions which involve carefully balancing
opposing terms.
W(d) for s=O and s=1/2
X0.6~"
0.4
0.2
0 0.2 0.4 0.6 0.8 1
d
Figure 2: Elastic energy as a function of dipole separation in the periodic
array. The curves are calculated for two different shaped cells, as described
by a parameter s.
In Fig. 2, we show the elastic energy (per unit cell per length) as a function
of dipole separation d for two values of cell offset s. Clearly for this geom-
etry the dislocations are in unstable equilibrium for d = 1/2. If the dipole
is allowed to collapse, the energy vanishes; this is possible because we have
smeared the dislocation cores and so the annihilation of oppositely signed
dislocations can occur without singularity. Likewise, expanding the dipoles
beyond the equilibrium causes the members to annihilate with oppositely-
signed partners from other periodic cells. One can show numerically that the
energy depends logarithmically on rs.
The resulting displacement fields are shown in Fig. 3.4
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Daw, Murray S. & Chrzan, Daryl. LDA Calculations of Dislocation Mobility in Fe & Mo, report, July 13, 2007; United States. (https://digital.library.unt.edu/ark:/67531/metadc888164/m1/4/: accessed April 23, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.