Graphical Fourier-coefficient analysis as a paper-based method for teaching structure factors

Weirich, T.E.

doi:10.1107/S2056989026000745

Figure 3
Fourier analysis plots for the projection of plane group pg along the b axis (see Fig. 1

) with atom positions at x′₁ = 0.2 and x′₂ = 0.8 that allow investigation of the reflections along the a axis. The positions of the atoms were marked initially on the upper lines on each side and then (using a ruler) projected downwards into the graphs. In order to derive the contribution of each atom, it is only necessary to read the function value at the position of the atoms (positive and negative contributions are indicated by lines of different colours). After all atoms have been processed in each a plot, the individual contributions are summed to yield the values for $[A\left(h \right)]$ and $[B\left(h \right)]$ , respectively. In agreement with plane group pg, no reflections exist with a structure-factor amplitude $[\left| {F\left(h \right)} \right| = 0]$ (use eq. 5

). However, as all sine terms vanish due to the presence of the inversion centre (mirror point) in the one-dimensional unit cell, the structure-factor phases for the h0 reflections in plane group pg are restricted to $[0^\circ]$ [the positive amplitudes $[A\left(h \right)]$ ] and $[180^\circ]$ for the amplitudes with negative signs (use eq. 4

CRYSTALLOGRAPHIC
COMMUNICATIONS

ISSN: 2056-9890

Volume 82| Part 2| February 2026| Pages 235-243

https://doi.org/10.1107/S2056989026000745

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