Commit 18dfad12 authored by Tobias WEBER's avatar Tobias WEBER
Browse files

continued with writing lecture: derived B matrix

parent da5cd8b3
......@@ -17,9 +17,92 @@
Useful formulas, T. Weber,, July 13, 2018.
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\section{Fractional Coordinates}
\subsection*{Basic Properties}
From the cosine theorem we get:
\begin{equation} \left< a | b \right > = ab \cos \gamma, \label{ab} \end{equation}
\begin{equation} \left< a | c \right > = ac \cos \beta, \label{ac} \end{equation}
\begin{equation} \left< b | c \right > = bc \cos \alpha. \label{bc} \end{equation}
\subsection*{Basis Vectors}
We first choose $\left| a \right>$ along $x$,
\begin{equation} \boxed{ \left| a \right> = \left( \begin{array}{c} a_1 = a \\ 0 \\ 0 \end{array} \right), } \label{avec} \end{equation}
$\left| b \right>$ in the $xy$ plane,
\begin{equation} \left| b \right> = \left( \begin{array}{c} b_1 \\ b_2 \\ 0 \end{array} \right), \end{equation}
and $\left| c \right>$ in general:
\begin{equation} \left| c \right> = \left( \begin{array}{c} c_1 \\ c_2 \\ c_3 \end{array} \right). \end{equation}
Inserting $\left| a \right>$ and $\left| b \right>$ into Eq. \ref{ab} gives:
\begin{equation} \left< a | b \right > = a_1 b_1 = ab \cos \gamma, \end{equation}
\begin{equation} b_1 = b \cos \gamma. \end{equation}
Using the cross product between $\left| a \right>$ and $\left| b \right>$, we get:
\begin{equation} \left\Vert \left| a \right> \times \left| b \right> \right\Vert =
\left\Vert \left( \begin{array}{c} 0 \\ 0 \\ a_1 b_2 \end{array} \right) \right\Vert =
ab \sin \gamma, \label{crossab}
\begin{equation} b_2 = b \sin \gamma, \end{equation}
\begin{equation} \boxed{ \left| b \right> = \left( \begin{array}{c} b \cos \gamma \\ b \sin \gamma \\ 0 \end{array} \right). } \label{bvec} \end{equation}
Inserting $\left| a \right>$ and $\left| c \right>$ into Eq. \ref{ac} gives:
\begin{equation} \left< a | c \right > = a_1 c_1 = ac \cos \beta, \end{equation}
\begin{equation} c_1 = c \cos \beta. \end{equation}
Inserting $\left| b \right>$ and $\left| c \right>$ into Eq. \ref{bc} gives:
\begin{equation} \left< b | c \right > = b_1 c_1 + b_2 c_2 = bc \cos \alpha, \end{equation}
\begin{equation} b \cos \gamma \cdot c \cos \beta + b \sin \gamma \cdot c_2 = bc \cos \alpha, \end{equation}
\begin{equation} c_2 = \frac{c \cos \alpha - c \cos \gamma \cos \beta}{\sin \gamma}. \end{equation}
The last component, $c_3$, can be obtained from the vector length normalisation, $ \left< c | c \right> = c^2 $:
\begin{equation} \left< c | c \right > = c_1^2 + c_2^2 + c_3^2 = c^2, \end{equation}
\begin{equation} c_3^2 = c^2 - c_1^2 - c_2^2, \end{equation}
\begin{equation} c_3^2 = c^2 \left[1 - \cos^2 \beta - \left(\frac{\cos \alpha - \cos \gamma \cos \beta}{\sin \gamma} \right)^2 \right], \end{equation}
\begin{equation} \boxed{ \left| c \right> = \left( \begin{array}{c}
c \cdot \cos \beta \\
c \cdot \frac{\cos \alpha - \cos \gamma \cos \beta}{\sin \gamma} \\
c \cdot \sqrt{ 1 - \cos^2 \beta - \left(\frac{\cos \alpha - \cos \gamma \cos \beta}{\sin \gamma} \right)^2 }
\end{array} \right). } \label{avec} \end{equation}
The crystallographic $A$ matrix, which transforms real-space fractional to lab coordinates (A), is formed with the basis vectors in its columns:
A = \left(
\left| a \right> & \left| b \right> & \left| c \right>
The $B$ matrix, which transforms reciprocal-space relative lattice units (rlu) to lab coordinates (1/A), is:
\begin{equation} B = 2 \pi A^{-t}, \end{equation}
where $-t$ denotes the transposed inverse.
% ------------------------------------------------------------------------------------------------------------------------------------
\section{Scattering Triangle}
\includegraphics[width = 0.2 \textwidth]{triangle}
......@@ -131,6 +131,10 @@ def get_A(lattice, angles):
(cs[0]-cs[1]*cs[2]) / s2, \
(np.sqrt(1. -,cs) + 2.*cs[0]*cs[1]*cs[2])) / s2])
# testing equality with own derivation
#print((np.sqrt(1. -,cs) + 2.*cs[0]*cs[1]*cs[2])) / s2)
#print(np.sqrt(1. - cs[1]*cs[1] - ((cs[0] - cs[2]*cs[1])/s2)**2.))
# the real-space basis vectors form the columns of the A matrix
return np.transpose(np.array([a, b, c]))
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