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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