Last Modified 17 December 1999

The ``Static'' Tight-Binding Program: Example X

``Best'' Elastic Constants for hcp Titanium

Previous parts of this Example have shown you how to calculate the elastic constants of a hexagonal close-packed (hcp) crystal, specifically, Titanium, using our tight-binding parameters. In then end, we calculated seven different combinations of elastic constants. However, a hexagonal crystal only has five independent elastic constants. (A sixth, C66, is equal to ½(C11-C12) by symmetry.) Obviously this means that several elastic constants are overdetermined. In a perfect world, this would not matter. Even an exact determination of the Cij from our tight-binding parameters would give precise agreement between all of the calculations. However, in our case two numerical problems pop up:

  1. We determine total energies by an approximate integration of the eigenvalues over the Brillouin zone, using a fixed k-point mesh . Other choices of k-point mesh will give slightly different answers. (You should repeat this example with another k-point mesh to check the accuracy of our results.) In addition, our method of calculating total energies means that we slightly change the lattice for each calculation. This changes the eigenvalue spectra our k-point mesh samples, and means that the space sampled by two different elastic constants is slightly different. Thus we can't really expect perfect agreement between elastic constants.
  2. We obtain the elastic constants by fitting energy versus strain calculations to polynomials. Choosing lower or higher order polynomials will change the results somewhat (try it and see). Using a different form of fitting function, e.g. exponentials, would also change the results.

So the question arises, how consistent are our elastic constants? Let's review them and see:

Section Elastic Constant Value (GPa)
i.
(C11 + C12) C33 - 2 C132
C11 + C12 + 2 C33 - 4 C13
121.7
ii. 1/9 [2 (C11 + C12) + C33 + 4 C13] 122.0
iii. C11 + C12 + 2 C33 - 4 C13 567.0
iv. C11 + C12 295.5
v. C33 261.2
vi. C11 - C12 133.5
vii. C44 70.3

Let's do a little consistency checking. First, by arguments given in the respective sections, we should have the elastic constant (ii.) larger than (i.). It is, slightly. Second, if we assume (iv.) and (v.) give the correct results for C11+C12 and C33, respectively, then

Given the numerical errors involved, this is probably as close as we can hope to achieve. Therefore we'll tentatively assign
C13 = 62.0 GPa   .

If we apply this to (1), we find that our predicted value for (i) would be

121.9 GPa    
compared to our actual value of
121.7 GPa &nbps; .
Again, this is about as close as we can hope to get.

As a result, we'll list the following values for the elastic constants of our Tight-Binding parameters for Titanium at the tight-binding equilibrium as

Elastic Constant Value (GPa)
C11 214.5
C12 81.0
C33 261.2
C13 62.0
C44 70.3
C66 66.75

We don't expect these to be exactly equal to the experimental elastic constants of Titanium. For one thing, we fitted our results to the first-principles local density approximation calculations for titanium. For another thing, we've evaluated the elastic constants at the TB equilibrium lattice constants rather than at experiment. We have shown that evaluating the elastic constants at the experimental volume gives results close to experiment. You may want to try evaluating the elastic constants at other volumes and c/a ratios and see what you find.

Return to the Example 10 home page.

Look at other examples.

Get other parameters from the Tight-binding periodic table.

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