Titanium alpha (hcp) to omega phase transition

Basic ideas

Energy barrier of pathways

Computational method

To describe the atomic shuffle, we use a single parameter  x that varies from 0 to 1.  Within our supercell calculation, we know where each atom starts in the hcp lattice (r0), and where it ends in the omega lattice (r1).  If we use supercell coordinates, then each atom is located at  r(x) = r0 + (r1-r0) * x.  This produces a homogenous simultaneous shuffle of all of the atoms in the supercell.

To describe the shear, we use a single parameter  y that varies from 0 to 1.  If our total unit cell shear is e0, then the shear  e(y) = 1 + (e0-1) * y.

Then, with the potential energy surface E(x, y), we can find a saddle point--the transition state.  The energy barrier is then the difference of the energy at the alpha structure and the energy at the transition state.  We can determine a pathway between the alpha and omega states going through the transition state by using a transition state following algorithm (steepest descent from the transition state).

The energy at each shuffle and shear value on a 10 x 10 grid is calculated using a new tight-binding parameterization.  For each mechanism, we determine a k-point set which is equivilant to a 18x18x12 grid in the original hcp lattice.  Convergence of this k-point set was determined by evaluating the energy at the four corners and the center of the shuffle/shear space, and is converged to within 0.1 meV/atom for all mechanisms.  A fermi broadening of 5 mRyd was used to aid in convergence.

Results

Silcock mechanism

Burgers  [111] mechanism

Burgers  [111] mechanism