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Solution of problem 3

Solution 3. Change of the coordinate system.
For the problem, see p. [*].

The new origin $O'$ has the coordinates $\mbox{\textit{\textbf{p}}}=0,\bar{\frac{1}{4}},
\frac{1}{8}$ referred to the present origin $O$. Therefore, the change of coordinates consists of subtracting 0, $\bar{\frac{1}{4}},
\frac{1}{8}$ from the old values, i.e. leave the $x$ coordinate unchanged, add $\frac{1}{4}=0.25$ to the $y$ coordinate, and subtract $\frac{1}{8}=0.125$ from the $z$ coordinate.

Answers

The new coordinates are

(i)
$Zr:\ (a)\ \ 0,\frac{1}{4},\bar{\frac{1}{8}}\sim
\frac{7}{8};\ 0,\frac{3}{4},\fr...
...};\ \frac{1}{2},\frac{1}{4},
\frac{5}{8};\ \frac{1}{2},\frac{3}{4},\frac{3}{8};$

(ii)
$Si:\ (b)\ \ 0,\frac{1}{4},\frac{3}{8};\
0,\frac{3}{4},\frac{5}{8};\ \frac{1}{2...
...{1}{4},\frac{1}{8};\
\frac{1}{2},\frac{3}{4},\bar{\frac{1}{8}}\sim\frac{7}{8};$

(iii)
$O:\ (h)\ \ 0,\,0.20+0.25,\,0.34-0.125=0,\,
0.45,\,0.215.$

This oxygen atom is obviously not the one (0,0.067,0.198) listed by the Structure Reports but must be a symmetrically equivalent one. Therefore, it is necessary to determine also the new coordinates of the other oxygen atoms.

(iv)
$O:\ (h)$ The coordinates of the other oxygen atoms are (normalized $0\leq x_i<1$):

$\begin{array}{llll} 0,\,0.05,\,0.215 & 0.20,\,0.25,\,0.535 &
0.80,\,0.25,\,0.53...
...l also with} \\
(\frac{1}{2},\,\frac{1}{2},\,\frac{1}{2})+. & & &
\end{array}$

The first one of these oxygen atoms corresponds to the one representing the results of the later refinement with higher accurancy. The $Si-O$ distance is reduced from 1.62 Å to 1.61 Å.

Answers

The change of basis to the primitive cell is described by the matrix

$\mbox{\textit{\textbf{P}}}=\left(
\begin{array}{rrr} 1&0&1/2\\ 0&1&1/2\\ 0&0&1/2
\end{array} \right) $.

One determines the inverse matrix $\mbox{\textit{\textbf{P}}}^{-1}= \left(
\begin{array}{rrr} 1&0&\bar{1}\\ 0&1&\bar{1}\\ 0&0&2
\end{array} \right),$

by which the coordinates are transformed using the formula (5.3.6):

$\mbox{\textit{\textbf{x}}}'=\mbox{\textit{\textbf{P}}}^{-1}
\mbox{\textit{\textbf{x}}}$. The coordinates x are those referred to the origin in $2/m$.

(v)
The new coordinates of the first $Zr$ atom are

$0-\frac{7}{8},\,\frac{1}{4}-\frac{7}{8},\,2\cdot\frac{7}{8}\ \sim\
\frac{1}{8},\,\frac{3}{8},\,\frac{3}{4}.$

(vi)
The new coordinates of the first $Si$ atom are

$0-\frac{3}{8},\,\frac{1}{4}-\frac{3}{8},\,2\cdot\frac{3}{8}\
\sim\ \frac{5}{8},\,\frac{7}{8},\,\frac{3}{4} .$

(vii)
The new coordinates of the first $O$ atom are

$0-0.215,\,0.45-0.215,\,2\cdot0.215 \sim 0.785,\,0.235,\,0.430$.


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