terencezl

try:
    from urllib.parse import quote  # Py 3
except ImportError:
    from urllib2 import quote  # Py 2
import os
import sys

try:
    BLOG_DIR = os.environ['BLOG_DIR']
except KeyError:

terencezl

import warnings
import numpy as np
import pandas as pd
from pymatgen.analysis.elasticity.elastic import ElasticTensor
from pymatgen.symmetry.analyzer import SpacegroupAnalyzer


def get_symmetrized_elastic_tensor(C, structure, align=False, tol=1e-4):
    """
    Parameters:

terencezl

Questions

The definition of complex $R$ and point group $\mathrm{\textbf{R}}$

In the first half of Lecture II, the complex $R$, not necessarily a group $\mathrm{\textbf{R}}$, is introduced as such.

Consider only the purely translational symmetry operations of an infinite crystal. These operations form a group denoted $\mathrm{\textbf{T}}$, where $\mathrm{\textbf{T}}$ is a subgroup of the full space group $\mathrm{\textbf{G}}$ of the crystal. Denote by $R$, the set of all symmetry operations of $\mathrm{\textbf{G}}$ that involve pure rotations only (proper or improper) as well as rotations accompanied by a translation not in $\mathrm{\textbf{T}}$. $R$ is a subset of $\mathrm{\textbf{G}}$ called a "complex" whose elements are denoted by $\alpha$, $\beta$, $\gamma$, ... since in general $R$ is not a group.

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Problem 8

The ${200}$ set of wavefunctions are $\phi = \exp(i 2\pi/a (\pm 2x))$, $\exp(i 2\pi/a (\pm 2y))$, and $\exp(i 2\pi/a (\pm 2z))$. They can be represented as

$$ \begin{pmatrix} 1 \ 0 \ 0 \end{pmatrix} \begin{pmatrix} -1 \ 0 \ 0 \end{pmatrix} \begin{pmatrix} 0 \ 1 \ 0 \end{pmatrix} \begin{pmatrix} 0 \ -1 \ 0 \end{pmatrix} \begin{pmatrix} 0 \ 0 \ 1 \end{pmatrix} \begin{pmatrix} 0 \ 0 \ -1 \end{pmatrix}, $$ treating the magnitude of each as 1.

The classes are ${E}, {3C_2}, {6S_4}, {6\sigma}, {8C_3}$.

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Introduction

Density functional theory (DFT) has proven to be an effective way to obtain the physical properties of material systems. Using the linear stress-strain relationship, one is able to obtain the elastic constants from a set of DFT calculations. One particular challenge is that the elastic tensor thus obtained is usually not numerically symmetric with respect to the crystal symmetry. Neumann's principle, or principle of symmetry, states that [^1]:

If a crystal is invariant with respect to certain symmetry operations, any of its physical properties must also be invariant with respect to the same symmetry operations, or otherwise stated, the symmetry operations of any physical property of a crystal must include the symmetry operations of the point group of the crystal.

Therefore, it is worthwhile working out a routine to symmetrize the elastic tensor using the symmetry operations of the crystal.

Theoretical formulation

A $3\times 3 \times 3\times 3$ elastic tensor ${c_{ijkl}}$ has the in

terencezl

from pdfminer.pdfinterp import PDFResourceManager, PDFPageInterpreter
from pdfminer.converter import TextConverter, XMLConverter, HTMLConverter
from pdfminer.layout import LAParams
from pdfminer.pdfpage import PDFPage
from io import BytesIO

def convert_pdf(path, format='text', codec='utf-8', password=''):
    rsrcmgr = PDFResourceManager()
    retstr = BytesIO()
    laparams = LAParams()