CASTEP > Tasks in CASTEP > Analyzing CASTEP results > Calculating elastic constants

# Calculating elastic constants

CASTEP results for the Elastic Constants task are returned as a set of `.castep` output files. Each of them represents a geometry optimization run with a fixed cell, for a given strain pattern and strain amplitude. The naming convention for these files is:

`seedname_cij__m__n`

Where m is the current strain pattern and n is the current strain amplitude for the given pattern.

CASTEP can then be used to analyze the calculated stress tensors for each of these runs and generate a file with information about elastic properties. The information in this file includes a summary of the input strains and calculated stresses; results of linear fitting for each strain pattern, including quality of the fit; the correspondence between calculated stresses and elastic constants for a given symmetry; a table of elastic constants, Cij, and elastic compliances, Sij; and finally, the derived properties such as bulk modulus and its inverse, compressibility, Young modulus and Poisson ratios for three directions and the Lame constants that are needed for modeling the material as an isotropic medium.

To calculate elastic constants

1. Choose Modules | CASTEP | Analysis from the Materials Studio menu bar.
2. Select Elastic constants from the list of properties.
3. Use the Results file selector to pick the appropriate results file.
4. Click the Calculate button.
5. A new text document, `seedname Elastic Constants.txt`, is created in the results folder.

## Description of the elastic constants file

Below, the example of a hexagonal crystal, BeO, is used to explain the contents of the `Elastic Constants.txt` output file produced by CASTEP.

Two strain patterns are required for this lattice type. For each strain pattern there is a summary of calculated stresses as extracted from the respective `.castep` files:

```         Summary of the calculated stresses
**********************************

Strain pattern:      1
======================

Current amplitude: 1
Transformed stress tensor (GPa) :
-1.249661        0.000000        0.000000
0.000000       -1.227407        0.001706
0.000000        0.001706        0.990234

Current amplitude: 2
Transformed stress tensor (GPa) :
-1.423318        0.000000        0.000000
0.000000       -1.400907        0.001555
0.000000        0.001555        0.089948
....
```

Any information about the connection between components of the stress, strain and elastic constants tensors is provided. At this stage each elastic constant is represented by a single compact index rather than by a pair of ij indices. The correspondence between the compact notation and the conventional indexing is provided later in the file:

``` Stress corresponds to elastic coefficients (compact notation):
8  8  3  0  0  0

as induced by the strain components:
3  3  3  0  0  0
```

A linear fit of the stress-strain relationship for each component of the stress is given in the following format:

```     Stress    Cij       value of          value of
index    index       stress            strain
1        8        -1.249661         -0.003000
1        8        -1.423318         -0.001000
1        8        -1.592620          0.001000
1        8        -1.757343          0.003000
Error on C       :       0.706339
Correlation coeff:       0.999930
Stress intercept :      -1.505736
```

The gradient provides the value of the elastic constant (or a linear combination of elastic constants), the quality of the fit, indicated by the correlation coefficient, provides the statistical uncertainty of that value. The stress intercept value is not used in further analysis, it is simply an indication of how far the converged ground state was from the initial structure.

The results for all the strain patterns are then summarized:

```  ============================
Summary of elastic constants
============================

id  i  j       Cij (GPa)
1   1  1     408.16095 +/-   0.601
3   3  3     447.05940 +/-   1.117
4   4  4     129.43210 +/-   0.094
7   1  2     114.84665 +/-   0.889
8   1  3      84.91535 +/-   0.392
```

The errors are only provided when more than two values for the strain amplitude were used, since there is no statistical uncertainty associated with fitting a straight line to only two points.

Elastic constants are then presented in a conventional 6×6 tensor form, followed by a similar 6×6 representation of the compliances:

```      =====================================
Elastic Stiffness Constants Cij (GPa)
=====================================

408.16095   114.84665    84.91535     0.00000     0.00000     0.00000
114.84665   408.16095    84.91535     0.00000     0.00000     0.00000
84.91535    84.91535   447.05940     0.00000     0.00000     0.00000
0.00000     0.00000     0.00000   129.43210     0.00000     0.00000
0.00000     0.00000     0.00000     0.00000   129.43210     0.00000
0.00000     0.00000     0.00000     0.00000     0.00000   146.65715

========================================
Elastic Compliance Constants Sij (1/GPa)
========================================

0.0027235  -0.0006858  -0.0003870   0.0000000   0.0000000   0.0000000
-0.0006858   0.0027235  -0.0003870   0.0000000   0.0000000   0.0000000
-0.0003870  -0.0003870   0.0023839   0.0000000   0.0000000   0.0000000
0.0000000   0.0000000   0.0000000   0.0077261   0.0000000   0.0000000
0.0000000   0.0000000   0.0000000   0.0000000   0.0077261   0.0000000
0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0068186

```

The final part of the file contains the derived properties:

```Bulk modulus    =   203.62075 +/-  0.321 (GPa)

Compressibility =     0.00491 (1/GPa)

Axis   Young Modulus        Poisson Ratios
(GPa)
X      367.17384       Exy=  0.2518 Exz=  0.1421
Y      367.17384       Eyx=  0.2518 Eyz=  0.1421
Z      419.48574       Ezx=  0.1624 Ezy=  0.1624
```

The last section of the file contains average properties that describe the elastic response of a polycrystal, for example:

```====================================================
Elastic constants for polycrystalline material (GPa)
====================================================
Voigt       Reuss        Hill
Bulk modulus              :     235.70320   235.70320   235.70320
Shear modulus (Lame Mu)   :      75.75417    64.59893    70.17655
Lame lambda               :     185.20042   192.63725   188.91883

Universal anisotropy index:       0.86342```

This output contains the bulk modulus and shear modulus averaged according to Voigt, Reuss, and Hill schemes (Nye 1957).

In addition an universal anisotropy index suggested by Ranganathan and Ostoja-Starzewski (2008) is evaluated.