Strain DoF#

Simple cubic with strain DoF#

This example constructs the prim for simple cubic crystal system with strain degrees of freedom (DoF), using the Green-Lagrange strain metric.

To construct this prim, the following must be specified:

the lattice vectors
basis site coordinates
occupant DoF
strain DoF

This example uses a fixed “A” sublattice for occ_dof, which by default are created as isotropic atoms.

import numpy as np
import libcasm.xtal as xtal

# Lattice vectors
lattice_column_vector_matrix = np.array([
    [1., 0., 0.],  # a
    [0., 1., 0.],  # a
    [0., 0., 1.],  # a
]).transpose()  # <--- note transpose
lattice = xtal.Lattice(lattice_column_vector_matrix)

# Basis sites positions, as columns of a matrix,
# in fractional coordinates with respect to the lattice vectors
coordinate_frac = np.array([
    [0., 0., 0.],  # coordinates of basis site, b=0
]).transpose()

# Occupation degrees of freedom (DoF)
occ_dof = [
    ["A"],  # occupants allowed on basis site, b=0
]

# Global continuous degrees of freedom (DoF)
Hstrain_dof = xtal.DoFSetBasis("Hstrain")  # Hencky strain metric
global_dof = [Hstrain_dof]

# Construct the prim
prim = xtal.Prim(lattice=lattice,
                 coordinate_frac=coordinate_frac,
                 occ_dof=occ_dof,
                 global_dof=global_dof,
                 title="simple_cubic_Hstrain")

This prim as JSON: simple_cubic_Hstrain.json

Simple cubic with strain DoF, symmetry-adapted basis#

This example constructs the prim for simple cubic crystal system with strain DoF, using the Green-Lagrange strain metric, with the symmetry-adapted basis $\vec{e}$ , as described by Thomas and Van der Ven [TVandVen17]:

\begin{array}{r} \vec{e} = (\begin{array}{ccc} e_{1} \\ e_{2} \\ e_{3} \\ e_{4} \\ e_{5} \\ e_{6} \end{array}) = (\begin{array}{ccc} (E_{x x} + E_{y y} + E_{z z}) / \sqrt{3} \\ (E_{x x} - E_{y y}) / \sqrt{2} \\ (2 E_{z z} - E_{x x} - E_{y y} +) / \sqrt{6} \\ \sqrt{2} E_{y z} \\ \sqrt{2} E_{x z} \\ \sqrt{2} E_{x y} \end{array}), \end{array}

The symmetry-adapted basis $\vec{e}$ decomposes strain space into irreducible subspaces which do not mix under application of symmetry.

To construct this prim, the following must be specified:

the lattice vectors
basis site coordinates
occupant DoF
strain DoF

This example uses a fixed “A” sublattice for occ_dof, which by default are created as isotropic atoms.

import numpy as np
import libcasm.xtal as xtal
from math import sqrt

# Lattice vectors
lattice_column_vector_matrix = np.array([
    [1., 0., 0.],  # a
    [0., 1., 0.],  # a
    [0., 0., 1.],  # a
]).transpose()  # <--- note transpose
lattice = xtal.Lattice(lattice_column_vector_matrix)

# Basis sites positions, as columns of a matrix,
# in fractional coordinates with respect to the lattice vectors
coordinate_frac = np.array([
    [0., 0., 0.],
]).transpose()  # coordinates of basis site, b=0

# Occupation degrees of freedom (DoF)
occ_dof = [
    ["A"],  # occupants allowed on basis site, b=0
]

# Global continuous degrees of freedom (DoF)
Hstrain_dof = xtal.DoFSetBasis(
    dofname="Hstrain",
    axis_names=["e_{1}", "e_{2}", "e_{3}", "e_{4}", "e_{5}", "e_{6}"],
    basis=np.array([
        [1. / sqrt(3), 1. / sqrt(3), 1. / sqrt(3), 0.0, 0.0, 0.0],
        [1. / sqrt(2), -1. / sqrt(2), 0.0, 0.0, 0.0, 0.0],
        [-1. / sqrt(6), -1. / sqrt(6), 2. / sqrt(6), 0.0, 0.0, 0.0],
        [0.0, 0.0, 0.0, 1.0, 0.0, 0.0],
        [0.0, 0.0, 0.0, 0.0, 1.0, 0.0],
        [0.0, 0.0, 0.0, 0.0, 0.0, 1.0],
    ]).transpose())
global_dof = [Hstrain_dof]

# Construct the prim
prim = xtal.Prim(lattice=lattice,
                 coordinate_frac=coordinate_frac,
                 occ_dof=occ_dof,
                 global_dof=global_dof,
                 title="simple_cubic_Hstrain_symadapted")

This prim as JSON: simple_cubic_Hstrain_symadapted.json