Last Modified 17 December 1999
In the Equation of State (EOS) section we found E(V) for Titanium by minimizing the total energy at each volume with respect to the c/a ratio. This gives us the true equation of state, but we can ask the question ``what happens if we keep a fixed value of c/a?'' The answer, of course, depends upon the precise value of c/a chosen. For our purposes here, we'll set c/a equal to the equilibrium value of c/a found in the EOS,
Let e(V) = E(V,(c/a)0), the energy at any volume using the equilibrium c/a ratio, and
E(V) = E(V,(c/a)(V)), the energy calculated in the EOS section and the minimum energy at a give volume.
Then we know that
It follows from this that the equilibrium bulk modulus derived from e(V) is greater than the true bulk modulus:
If we go through the derivation of b(V) from e(V), we find that in terms of the elastic constants,
In this section we'll determine e(V) = E(V,(c/a)0) as
a function of volume, and then use the Birch fit to calculate
b(V0). This will give us our second relationship between
elastic constants.
The setup is reasonably simple. Since we are not changing the
symmetry of the crystal we can use the same space group file, work/spcgp1.hcp and k-point mesh,
work/kmesh.194. Of course we use the same
tight-binding parameters, which are stored in work/ti_par. The only thing new
is the SKIN file, work/fixca/SKIN. You'll
notice that this SKIN file contains every volume used in the final
EOS determination, but in every
case the c/a ratio is 1.61326030.
Running the calculation in the
work/fixca directory, we get the following results:
Since we know e(V) and E(V) are identical at V0, we
can use the same fitting procedure here as in the EOS calculation, with only
a very slightly modified
work/fixca/hcpfit2.gnu
gnuplot fitting file.
We see that the bulk modulus is just a bit larger than in the true EOS calculation (122.02
vs 121.74 GPa). The total energy plot is very similar,
too, so I'll leave that for you to plot out.
Finally, the result of this section is that
1/9 [ 2 (C11 + C12) +
C33 + 4 C13 ] = 122.0 GPa .
Go back to the Example X home page.
Look at other examples.
Get other parameters from the Tight-binding periodic
table. Return to the static Reference
Manual.
$ static > output
$ ls -l
-rw-r----- 1 mehl users 1134 Jun 4 14:41 SKENG
-rw-r----- 1 mehl users 13124 Jun 4 14:38 SKIN
-rw-r----- 1 mehl users 51120 Jun 4 14:41 SKOUT
-rw-r----- 1 mehl users 39727 Jun 4 14:41 output
$ gnuplot hcpfit2.gnu
Fit set for vo = 220.088 Bohr^3 and
Eo = -0.009018141 Ry. Change hcpfit2.gnu if necessary.
Press Enter to continue
Equilibrium Volume = 220.088 Bohr^3 = 32.613683827428 Angstroms^3
Equilibrium Energy = -0.009018141 Ry
Equilibrium Bulk Modulus = 0.00829452139631914 Ry/Bohr^3 = 122.016693030704 GPa
dB/dP (P=0) = 3.81463566444934
Press Enter to end program
$