Last Modified 17 December 1999

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

Equation of State and Elastic Constants of hcp Titanium

Cubic crystals are rather easy to deal with, especially those containing only one atom or atoms which remain on high symmetry sites under strain. Thus we could easily determine the equation of state of cubic materials such as palladium and Cu3Au in the A15 phase. We were also easily able to obtain the elastic constants C11-C12 of copper and C44 of niobium.

In a more complicated crystal, though, the atoms are allowed to move individually under strain, and they may not stay anywhere near their equilibrium positions. For example, suppose we want to find the elastic constant C44 of diamond. Since the lattice is cubic, we can use the strain described in Example 5. However, this strain breaks the symmetry of the crystal and the carbon atoms are free to move away from their equilibrium positions, even if we consider only the lattice coordinates of the basis vectors. Finding the correct positions of the atom under a given strain lowers the energy associated with that strain, so unless we minimize the energy we can only find an upper bound on C44.

The hcp lattice is even more difficult to deal with than diamond. Not only are there two atoms in the unit cell, it takes two parameters, (c,a) or (V,c/a), just to specify the lattice. This page is a brief tutorial on how to find the equilibrium structure of an hcp lattice; its equation of state (E(V) curve), which depends upon the behavior of c/a as a function of volume; and all of the elastic constants of the hcp lattice.

This example will be based upon our tight-binding parameters for Titanium. Since we'll need these parameters for all of the calculations, obtain them from the database and save the file under the name ti_par in a working directory. Since we'll be creating some subdirectories under this directory, we'll refer to the current directory as ``work'', and the file will be work/ti_par.

For the purposes of this tutorial, we'll use a fixed k-point mesh, generated by: 

    0
lspec=f,linfok=f
   18   18   12

Save this file fragment as work/kmesh. You may want to experiment with other k-point meshes, including those in the hexagonal k-point database, or construct your own mesh.

The tutorial has several parts. In the first five parts, the lattice maintains its full hcp (P63/mmc) symmetry so we can use the space group file spcgp1.hcp from Example 2. Save this file as work/spcgp1.hcp so that we can refer to it easily.

  1. The equation of state (E(V)) of Titanium, including determination of the equilibrium lattice structure. This will be in the subdirectory work/eos. The EOS can be used to obtain the bulk modulus of the crystal. In a hexagonal crystal this has the form:
    B = (C11 + C12) C33 - 2 C132   .
    C11 + C12 + 2 C33 - 4 C13


  2. The ``equation of state'' obtained by fixing the c/a ratio at the equilibrium value determined in (i) and varying the volume. Since the energy is not minimized with respect to c/a this produces an upper bound on E(V), and an upper bound on the bulk modulus,
    B < Bf = 1/9 [ 2 (C11 + C12) + C33 + 4 C13 ]   .

    This will be in the directory work/fixca.

  3. Finding E''(c/a) at fixed volume, the analog to the determination of C11-C12 in a cubic material. This will be in the directory work/czz and will allow us to determine
    Czz = C11 + C12 + 2 C33 - 4 C13   .


  4. Finding E''(a) at fixed c. This will be in the directory work/c11+c12 and will allow us to determine
    C11 + C12   .


  5. Finding E''(c) at fixed a. This will be in the directory work/c33 and will allow us to determine
    C33   .

We can only determine the remaining two lattice constants by breaking the symmetry of the crystal.

  1. Using an orthorhombic strain to determine
    C11 - C12   .

    This will be in the directory work/c11-c12.

  2. Using a monoclinic strain to determine
    C44   .

    Since this structure may have a complicated relaxation, I've divided it into two parts, the upper bound (no relaxation) section, including the theory, with working directory work/c44; and the relaxed calculation, with working directory work/c44r.

You'll notice that we have seven elastic constant relationships and only five elastic constants. We'll use these relationships to determine the best elastic constants and make consistency checks.

Look at other examples.

Get other parameters from the Tight-binding periodic table.

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