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

The structure factor is a fundamental concept in X-ray crystallography, yet it is often introduced via formal sum expressions that make it difficult to realize its connection to the atomic structure. Therefore, a one-dimensional Fourier analysis method is proposed as an alternative approach for teaching. The method is a graphical discrete Fourier analysis-based approach that is used to determine the cosine and sine coefficients, A and B, respectively, of the structure factor from projections. To allow seamless and easy integration into existing curriculums, worksheets with the corresponding function graphs for determining the coefficients are provided. The application of the method is demonstrated by several examples, which illustrate its use for enhancing the understanding between structure and systematic extinctions and existing constraints for structure factor phases. Although this approach has not been developed for crystal structure analysis, it does provide the user with an intuitive framework for developing a conceptual understanding between the structure factors and atomic arrangement in crystalline materials.