Last Modified 17 December 1999
The Bain path traces the energy of a bcc crystal as it is strained along the <001> axis until it becomes an fcc crystal. The behavior of the energy along this path determines, for example, the formation of martensitic phases in intermetallic systems.
In this example we use our tight-binding parametrization to study the behavior of tungsten along the Bain path. We should do this by finding the minimum energy volume at each strain, but we'll simplify the calculation by working only at the experimental volume.
As any solid state physics textbook will tell you, both the bcc and fcc structures can be viewed as high-symmetry special cases of the body-centered tetragonal (bct) lattice. When c/a = 1 we have a bcc lattice. When c/a = 2½ the lattice becomes fcc. At other values we have the structures known as Aa (alpha Pa) and A6 (In).
Once again, obtain the tight-binding parameters for tungsten from the tight-binding periodic table under ``w'', and save these parameters in a working directory as w_par.
The space group file can be obtained from any of the lattices having the I4/mmm symmetry, for example, A6 (In). Click on ``Cartesian'' and save the file in your work directory as spcgrp.
Once again, k-points can be generated by hand. Since we are going to cover a wide range of lattices, I would recommend a large mesh. On the other hand, I happen to have one here, called bcct.r24. Save it in your work directory, too.
Finally, take this copy of the SKIN (Slater-Koster INput) file, and save it in your working directory under the name SKIN. Let's examine this file.
The first few lines of the file should be familiar from previous examples:
Mode=3 Calculate E Only
0.002 0.500 (T_{Fermi}, Eigenvalue cutoff for P calculation)
w_par
The Bain path is essentially a stretching or compression of the bcc lattice by changing the length of the Z axis relative to the X and Y lattice. Looking at the lattice types table, we see that lattice 13 (or -13) is a body-centered tetragonal lattice which is written very much like the bcc. (I wonder how that got there.) Since we want to keep the volume fixed and change the c/a ratio we'll use -13 for the index. The first structure of our SKIN file thus looks like this:
BCC Tungsten -- Bain Path c/a= 0.792893218813452 (20 character label for SKENG) 0.00 (Electrons in addition to nominal W charge (=6/atom)) -13 (bct lattice, read in volume and c/a next:) 106.495045 0.792893218813452 0 (No additional strains applied) 1 (Atoms in the unit cell) 4 4 4 (Neighbor search cutoff indices) F (Logical variable -- no internal displacements) 1 0.000 0.000 0.000 0 0 0 (Position of atom in Lattice coordinates) NEWSYM=T (Generate new set of k-points) LATTIC=1 (Lattice type / Next is spacegroup file name:) spcgrp ILAT=F (Space group file in Cartesian Coordinates) -1313 (Read k-points from a file/ Next is file name:) bcct.r24
where the c/a ratio is about 0.793. The remaining structures look similar, except that we don't need the k-point information any more:
BCC Tungsten -- Bain Path c/a= 0.834314575050762 0.00 (Electrons in addition to nominal W charge (=6/atom)) -13 (bct lattice, read in volume and c/a next:) 106.495045 0.834314575050762 0 (No additional strains applied) 1 (Atoms in the unit cell) 4 4 4 (Neighbor search cutoff indices) F (Logical variable -- no internal displacements) 1 0.000 0.000 0.000 0 0 0 (Position of atom in Lattice coordinates) NEWSYM=F (Generate new set of k-points)
und so weiter.
Now go to the output discussion to see what will happen.
Look at other examples.
Get other parameters from the Tight-binding periodic table.
Return to the static Reference Manual.