Calculation of the equilibrium separation of dislocations

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Mag[x_] := Sqrt[x . x]

Place a 90 degree dislocation at the origin with dislocation line running in the -z direction and burgers vector running in +x direction.  Within isotropic elasticity theory the stress field around this dislocation is given by equation 3-43 of Hirth and Lothe.  

σxx = (μ b1)/(2 π (1 - ν)) (y (3 x^2 + y^2))/(x^2 + y^2)^2 ;

σxy = (μ b1)/(2 π (1 - ν)) (x (x^2 - y^2))/(x^2 + y^2)^2 ;

σxz = 0 ;

σyx = σxy ;

σyy = (μ b1)/(2 π (1 - ν)) (y (x^2 - y^2))/(x^2 + y^2)^2 ;

σyz = 0 ;

σzx = σxz ;

σzy = σyz ;

σzz = ν (σxx + σyy) ;

σ[r_] := {{σxx, σxy, σxz}, {σyx, σyy, σyz}, {σzx, σzy, σzz}}/.{xr[[1]], yr[[2]], zr[[3]]} ;

Here we set the material dependent parameters.  

a = 5.653*10^-10 ;

Lattice parameter taken from:
Data in Science and Technology : Semiconductors, edited by O. Madelung (Springer-Verlag, New York, 1992).

b1 = 6^(1/2)/6a ;

Elastic parameters taken from
H. J. McSkimin, A. Jayaraman, and P. A. Jr., Journal of Applied Physics  38, 2362 (1967).
Below are the Voigt averages of these numbers (units of Pa for μ)

μ = 4.867*10^10 ;

ν = 0.2344 ;

Surface Energy taken at 77K from:
Moon, W-J and Umeda, T. and Saka, H. Philosophical Magazine Letters Vol 83 No 4 P 233-240 (2003)
(units of J/m^2)

γ = 0.015 ;

For a 30 degree partial located x_oaway from the 90 the force per length felt is given by equation 3-90.  

b2 = {Sin[π/6], 0, Cos[π/6]} * 6^(1/2)/6a ; ξ = {0, 0, -1} ;

F[x_] := Cross[(b2 . σ[{x, 0, 0}]), ξ]

The attractive force is from the stacking fault.  The stacking fault energy is given above and the energy per line length is γ.x.  The force in the x direction will be the gradient of the energy taken with respect to the x direction so the force per line length is just γ.  

So the equilibrium length is just the point at which the force equals zero

Roots[F[x][[1]] - γ0, x]

x == -1.79624*10^-8

So the equilibrium separation is approximately 17.9 nm

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