FPTE 1.2.1

The FPTE package is a collection of tools for finite pressure temperature elastic constants calculation. Features include, but are not limited to stress-strain method for getting second order elastic tensors using DFT package VASP as well as, ab initio molecular dynamic method for temperaturedependent elastic constants.

Finite Pressure Temperature Elasticity (FPTE) package

Installation

Dependencies

FPTE requires:

• Python (>= 3.7)
• NumPy (>= 1.16.5)
• Pandas (>= 0.25.3)
• Matplotlib (>= 2.2.4)
• joblib (>= 0.11)

FPTE 1.2.0 and later require Python 3.7 or newer. FPTE 1.1.0 and later require Python 3.4 or newer.

FPTE plotting capabilities (i.e., functions start with plot_ and classes end with "Display") require Matplotlib (>= 2.2.4).

User installation

If you already have a working installation of numpy and scipy, the easiest way to install FPTE is using pip:

pip install -U FPTE

or install from source:

git clone https://github.com/MahdiDavari/FPTE
cd FPTE
python setup.py install

In order to check your installation you can use:

python -m pip show FPTE  # to see which version and where FPTE is installed
python -m pip freeze  # to see all packages installed in the active virtualenv
python -c "import FPTE; print(FPTE.__version__)"

Note that in order to avoid potential conflicts with other packages it is strongly recommended to use a virtual environment (venv).

Theory

Elastic Stifness Coefficients from Stress-Strain Relations:

According to Hooke's law, the second-rank stress and strain tensors for a slightly deformed crystal are related by

$$ $$

where the fourth rank tensors c_ijkl and s_ijkl are called the elastic stiffness coefficients and elastic compliance constants respectively. Here we deal with elastic stiffness coefficients c_ijkl, which govern the proper stress-strain relations at nite strain. In general, we can write

$$ $$

where X and x are the coordinates before and after the deformation. There are 81 independent stiffness coefficients in general; however, this number is reduced to 21 by the requirement of the complete Voigt symmetry. In Voigt notation (c_ij), the elastic constants form a symmetric 6x6 matrix

$$ $$

In single suffix notation (running from 1 to 6), we can also use the matrix representations for stress and strain

$$ $$
and

$$ $$

where the stress components are σ₁ = σ_xx ; σ₂ = σ_yy ; σ₃ = σ_zz ; σ₄ = σ_yz ; σ₅ = σ_zx ; σ₆ = σ_xy, and the strain components are ε₁ = ε _xx ; ε₂ = ε_yy ; ε₃ = ε_zz ; ε₄ = ε_yz ; ε₅ = ε_zx ; ε₆ = ε_xy. When a crystal lattice is deformed with strain (ε), new lattice vectors a are related to old vectors ** a**₀ by a = (I + ε) a₀, where I is identity matrix. The stress-strain relations are then simply given by

$$ $$

The presence of the symmetry in the crystal reduces further the number of independent c _ij . A cubic crystal having highest symmetry is characterized by the lowest number (only three) of independent elastic constants, c₁₁, c₁₂ and c₄₄, which in matrix notation is

$$ $$

Crystal System	Space Group Number	No. of Elastic Constants
Cubic	195-230	3
Hexagonal	168-194	5
Trigonal	143-167	6-7
Tetragonal	75-142	6-7
Orthorhombic	16-74	9
Monoclinic	3-15	13
Triclinic	1 and 2	21

Note: For more information regarding the second-order elastic constant see reference:

1. Golesorkhtabar, Rostam, et al., “ElaStic: A Tool for Calculating Second-Order Elastic Constants from First Principles.” Computer Physics Communications 184, no. 8 (2013): 1861–73.

2. Karki, Bijaya B. “High-Pressure Structure and Elasticity of the Major Silicate and Oxide Minerals of the Earth’s Lower Mantle,” 1997.

3. Barron, THK, and ML Klein. “Second-Order Elastic Constants of a Solid under Stress.” Proceedings of the Physical Society 85, no. 3 (1965): 523.