# Conventions and Notations used in **byxtal**

1. A fixed **global** orthogonal Cartesian coordinate frame is denoted by a row of basis-vectors as:

\begin{equation}
\mathcal{B}_{GO} = \left[ \hat{e}_x, \hat{e}_y, \hat{e}_z \right]
\end{equation}

The subscript **GO** refers to the fact that the reference is global (**G**) and orthogonal (**O**).

2. A vector is expressed as the product of a row of basis vectors and a column comprising the components of this vector along the basis vectors. For example, the vector $\vec{a}$ in the Cartesian frame is expressed as:

  \begin{equation}
    \label{eq:vec}
  \vec{a} = \left[ \hat{e}_x, \hat{e}_y, \hat{e}_z \right] 
 \begin{bmatrix}
 a_x \\ a_y \\ a_z
 \end{bmatrix}
  = \mathcal{B}_{GO} 
 \begin{bmatrix}
 a_x \\ a_y \\ a_z
 \end{bmatrix}
 = \mathcal{B}_{GO} a_{GO}    
  \end{equation}

Here, $ a_{GO}$ refers to the column vector of components of $\vec{a}$ expressed in the global orthogonal Cartesian coordinate reference frame ($GO$).


3. A Bravais lattice, with Pearson symbol $P$, is denoted as $\mathcal{B}_{P}$ and is defined by the basis vectors spanning the **primitive unit cell**, i.e.,

\begin{equation}
\mathcal{B}_{P}= [\vec{a} ,\, \vec{b} ,\, \vec{c}]
\end{equation}

4. If one were to use the components of the vectors in the $GO$ reference frame, then:

\begin{equation}
\mathcal{B}_{P}= \mathcal{B}_{GO} [ a_{GO}, \, b_{GO}, \, c_{GO}] = \mathcal{B}_{GO} \Lambda_{P}^{GO}
\end{equation}

where $\Lambda_{P}^{GO}$ is a $3 \times 3$ matrix of components of the basis vectors in the $GO$ reference frame.

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ipynb
Jupyter Notebook
docs/source/Tutorials/byxtal/Conventions-Notations.ipynb
spatala/gbpy
71ed719ad795176508250b2d8b1c18ffbbd537c0
BSD-3-Clause
docs/source/Tutorials/byxtal/Conventions-Notations.ipynb
spatala/gbpy
71ed719ad795176508250b2d8b1c18ffbbd537c0
BSD-3-Clause
2020-12-23T19:29:06.000Z
2021-04-20T17:30:10.000Z
docs/source/Tutorials/byxtal/Conventions-Notations.ipynb
spatala/gbpy
71ed719ad795176508250b2d8b1c18ffbbd537c0
BSD-3-Clause
2021-08-04T17:20:46.000Z
2021-08-04T17:20:46.000Z
Qwen/Qwen-72B
1. YES
2. YES