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5. Systematic Absences

If h $\cdot$ R = h and h $\cdot$ t $\neq$ 0 (modulo 1) then the reflection h is extinct, or absent, i.e. its amplitude is $\equiv$ 0. If we have a 2-fold symmetry, like in P21, the two contributions to the structure factor are equally large but have exactly opposite directions and they cancel each other. This is illustrated in Fig. 4a. In space groups with 3-fold symmetry, such as P31, there will be systematic absences for reflections where hR = h and h $\cdot$t = $\frac{1}{3}$ or $\frac{2}{3}$, as illustrated in Fig. 4. Because of the 3-fold symmetry the atoms are divided into 3 groups rather than 2 as was the case with a 2-fold symmetry. With 4- or 6-fold symmetry the situation is much like that of the 3-fold, only we now have 4 or 6 contributions, each unlike in size but differing by 90$^{\circ}$ and 60$^{\circ}$ respectively. See Figs. 4c and 4d.


 
Figure 4: The rise of systematic absences in space groups with (a) 2-fold, (b) 3-fold, (c) 4-fold and (d) 6-fold symmetry elements.
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\includegraphics {fig4.ps}
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Next: 6. Systematic Enhancement, Epsilon Up: Rotation Matrices and Translation Vectors in Previous: 4. Phase Restrictions

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