Last Modified 23 February 2001

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

Finding C11-C12

The basic theory used to find the shear elastic constant C11-C12 can be found in Example 4. Although this formally covers only the fcc structure, it is easy to adapt to the bcc and simple cubic structures. It is actually fairly easy to adapt to the diamond structure, as well, be we won't cover that here.

In the following we use option 2 in the strain table to generate an orthorhombic strain along the <100>, <010>, and <001> directions:

e1 = [(1+x)/(1-x)]½ - 1

e2 = [(1-x)/(1+x)]½ - 1

e3,4,5,6 = 0

Then the energy behaves as

E(x) = E0 + V (C11 - C12) x2 + O[x4]  ,

so we can find the elastic constant simply by looking for the curvature around x = 0.

Notes:

  1. We calculate C11-C12 at the equilibrium lattice constant found in the equilibrium/bulk modulus calculation.
  2. Because of the distortion, the symmetry of these structures is lower than in the cubic case. This is reflected in the space group, and also in the choice of the k-point mesh. Note that the k-points used here are equivalent to the k-points used in calculating the bulk modulus. Thus, if you use the ``regular.08'' k-point mesh and e1 = 0 here you should get exactly the same energy using as the corresponding cubic lattice constant in the calculations above.
System Pt (fcc) W (bcc) Ru (s.c.)
Space Group Fmmm (#69) Immm (#71) Pmmm (#47)
Input Files
Space group file (Cartesian) spcgrp spcgrp spcgrp
SKIN file SKIN SKIN SKIN
K-points used: regular.08 regular.08 regular.04
More Regular K-point meshes
Mesh 04 regular.04 regular.04 regular.04
Mesh 06 regular.06 regular.06 regular.06
Mesh 08 regular.08 regular.08 regular.08
Mesh 10 regular.10 regular.10 regular.10
Mesh 12 regular.12 regular.12 regular.12
Mesh 14 regular.14 regular.14  
Mesh 16   regular.16  
Output Files (Text)
SKENG file SKENG SKENG SKENG
Graphical Output Files
GNUPLOT ``fitting'' file ptfit.gnu wfit.gnu rufit.gnu
GNUPLOT ``ploting'' file ptplot.gnu wplot.gnu ruplot.gnu
GNUPLOT output (PNG file) pt.png w.png ru.png
Polynomial Fit Results
C11-C12 (GPa) 156 348 396
Work Directory pt/c11-c12 w/c11-c12 ru/c11-c12

Find the Bulk Modulus

Find C44

Cubic Elastic Constant Home Page

Look at other examples.

Get other parameters from the Tight-binding periodic table.

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