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Distances and angles

When considering crystal structures, idealized as crystal patterns, frequently the values of distances between the atoms (bond lengths) and of the angles between atomic bonds (bonding angles) are wanted. These quantities can not be calculated from the coordinates of the points (centers of the atoms) directly. Distances and angles are independent of the choice of the origin but the point coordinates depend on the origin choice, see Section 1.4. Therefore, bond distances and angles can only be calculated using the vectors (distance vectors) between the points participating in the bonding. In this section the necessary formulae for such calculations will be derived.

We assume the crystal structure to be given by the coordinates of the atoms (better: of their centers) in a conventional coordinate system. Then the vectors between the points can be calculated by the differences of the point coordinates.

Let $\stackrel{\longrightarrow}{XY}=\mathbf{r}=r_1\,
\mathbf{a}_{1}+
r_2\,\mathbf{a}_{2}+r_3\,\mathbf{a}_{3}$ be the vector from point $X$ to point $Y$, $r_i=y_i-x_i$, see equation 1.4.1. The scalar product $(\mathbf{r}\,,\,\mathbf{r})$ of r with itself is the square of the length $r$ of r. Thus

\begin{eqnarray*}r^2=(\mathbf{r}\,,\,\mathbf{r})& = & ((r_1\mathbf{a}_{1}+
r_2\...
...})
\,,\,(r_1\mathbf{a}_{1}+r_2\mathbf{a}_{2}+r_3\mathbf{a}_{3})).\end{eqnarray*}

Because of the rules for scalar products in equation (1.5.1), this can be written
$\displaystyle r^2$ $\textstyle =$ $\displaystyle (r_1\mathbf{a}_1\,,\,r_1\mathbf{a}_1)+
(r_2\mathbf{a}_2\,,\,r_2\mathbf{a}_2)+(r_3\mathbf{a}_3\,,\,r_3\mathbf{a}_3)+$  
    $\displaystyle 2\,(r_2\mathbf{a}_2\,,\,r_3\mathbf{a}_3)+2\,(r_3\mathbf{a}_3\,,\,r_1
\mathbf{a}_1)+2\,(r_1\mathbf{a}_1\,,\,r_2\mathbf{a}_2).$  

It follows for the distance between the points $X$ and $Y$
$\displaystyle r^2$ $\textstyle =$ $\displaystyle r_1^2\,a_1^2 + r_2^2\,a_2^2 + r_3^2\,a_3^2 +
2\,r_2\,r_3\,a_2\,a_3\,\cos\alpha_1 +$  
  $\textstyle +$ $\displaystyle 2\,r_3\,r_1\,a_3\,a_1\,\cos\alpha_2 +
2\,r_1\,r_2\,a_1\,a_2\,\cos\alpha_3.$ (1.6.1)

Using this equation, bond distances can be calculated if the coefficients of the bond vector and the lattice constants of the crystal are known.

The general formula (1.6.1) becomes much simpler for the higher symmetric crystal systems. For example, referred to an orthonormal basis, equation (1.6.1) is reduced to

\begin{displaymath}r^2=r_1^2+r_2^2+r_3^2.
\end{displaymath} (1.6.2)

Using the $\Sigma$ sign and abbreviating $(\mathbf{a}_i\,,\,\mathbf{a}_k)=G_{ik}=a_i\,a_k\,
\cos\alpha_j$ ($j$ is defined for $i\ne k$: then $k\ne j\ne i$), this formula can be written

\begin{displaymath}
% latex2html id marker 2003r^2=\sum_{i,k=1}^{3}G_{ik}\,r_i\,r_k,
\hspace{2em} \mbox{see also Subsection \ref{daa}}. \end{displaymath} (1.6.3)

\psfig{figure=angl.eps}
Fig. 1.6.1 The bonding angle $\Phi$ between the bond vectors $\stackrel
{\longrightarrow}{SX}\ =\mathbf{r}$ and $\stackrel{\longrightarrow}{SY}\
=\mathbf{t}$.

The (bonding) angle $\Phi$ between the (bond) vectors $\stackrel
{\longrightarrow}{SX}\ =\mathbf{r}$ and $\stackrel{\longrightarrow}{SY}\
=\mathbf{t}$ is calculated using the equation

\begin{displaymath}
% latex2html id marker 2416
(\mathbf{r}\,,\,\mathbf{t})=\ve...
...e{1em}\mbox{see Fig.}\,\ref{angl}.
\index{angle!calculation of}\end{displaymath}

One obtains

$\displaystyle {r\,t\cos\Phi = r_1\,t_1\,a_1^2 + r_2\,t_2\,a_2^2 +
r_3\,t_3\,a_3^2 + (r_2\,t_3+r_3\,t_2)\,a_2\,a_3\cos\alpha_1 + }$
    $\displaystyle + \,(r_3\,t_1+r_1\,t_3)\,a_1\,a_3\cos\alpha_2 +
(r_1\,t_2+r_2\,t_1)\,a_1\,a_2\cos\alpha_3.$ (1.6.4)

Again one can use the coefficients $G_{ik}$ to obtain, see also Subsection 2.6.2,

\begin{displaymath}\cos\Phi=\bigl( \sum_{i,k=1}^{3}G_{ik}\,r_i\,r_k\bigr)
^{-1/2...
...\sum_{i,k=1}^{3}G_{ik}\,r_i\,t_k.
\index{angle!calculation of}
\end{displaymath} (1.6.5)

For orthonormal bases, equation (1.6.4) is reduced to
\begin{displaymath}r\,t\cos\Phi=r_1\,t_1+r_2\,t_2+r_3\,t_3,
\end{displaymath} (1.6.6)

and equation 1.6.5 is replaced by

\begin{displaymath}
\cos\Phi=\frac{r_1t_1+r_2t_2+r_3t_3}{r\ t}.
\end{displaymath} (1.6.7)


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Next: Matrices and determinants Up: Points and vectors Previous: The scalar product and special bases

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