Dislocation line energy in documentation too simple for proper GND calculation on HCP?

[Tijmenvermeij]

Hi,

I think we already discussed this once at the Mtex workshop, but I forgot what the outcome was.
In any case, here it is again:

In the documentation for GND calculation (https://mtex-toolbox.github.io/GND.html), and in other places, the line energy dS.u is always calculated as:

dS = dislocationSystem.bcc(ebsd.CS);
dS(dS.isEdge).u = 1;
dS(dS.isScrew).u = 1 - 0.3;

This is a good approximation for fcc and bcc, however, for hcp this is incorrect (when considering all dislocation systems) because <a> and <c+a> dislocations have different Burgers vector lengths. This gives quite substantial different energies, which has a strong (detrimental) effect on GND calculations. The documentation does remark on this a bit (see below), by discussing the full equations for the energy, but I feel like this is an easy trap for casual users who want to use their code on hcp. I do also agree that this may be one's own responsibility.

% There is no single accepted way of setting these energies. Formulae in use
% include |U = 1 - nu| as above, and |U = c * G * |b|^2| with |G| the shear
% modulus, i.e. an energy per unit length squared. Which one is appropriate
% depends on the model you are comparing against, so |u| is left for you to
% set rather than being fixed by MTEX.

Besides this point, the initialization of dislocationSystem includes the following code, where the line energies get a "rough" value of:

% line energy
dS.u = 1 + dS.isEdge;

This is rather different from what is suggested in the documentation. Also, the documentation mentions that dS.u is by default 1, which is only true for screw dislocations.

Because of all this, I am curious if it would make sense to:

1. Extend the standard definition of dS.u to something like:

nu = 0.3;
dS(dS.isEdge).u = dS(dS.isEdge).b.norm.^2;
dS(dS.isScrew).u = dS(dS.isScrew).b.norm.^2 * (1 - nu);

This would assure that the documentation code would also run correctly for hcp, at least as correct as published literature (Although I guess dislocation core parameters also vary between and <c+a>…). Also, it would not mess with the fcc and bcc calculations.
2. Change the calculation of dS.u in the initialization of the class to something more reasonable, or give a warning?

I would be happy to make these changes. Also, we could think of including an hcp example (from our published research) in the documentation, if that would be of interest.

Cheers,
Tijmen