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When X-rays interact with a crystal, the regularly spaced atoms scatter the radiation, and the scattered waves interfere constructively only in specific directions determined by Bragg's law (n-lambda = 2d sin-theta), producing a discrete pattern of diffraction spots. Each spot corresponds to a Fourier component of the crystal's electron density — its intensity gives the amplitude, but the phase is lost during measurement. The electron density is the inverse Fourier transform of the structure factors (complex numbers with amplitude and phase for each reflection). Understanding this Fourier relationship between real space (electron density) and reciprocal space (diffraction pattern) is essential for every step of crystallographic structure determination, from data processing to model building.
The physical basis of X-ray crystallography is the interaction between electromagnetic radiation and matter, and the mathematical framework that connects the experiment to the structure is the Fourier transform. Understanding this relationship — between what you measure (the diffraction pattern) and what you want to know (the electron density) — is the intellectual core of crystallography.
When an X-ray beam hits a crystal, each atom scatters the radiation in all directions. In a crystal, atoms are arranged in a regular, repeating lattice. The scattered waves from different atoms interfere with each other — constructively in some directions (producing bright diffraction spots) and destructively in others (producing darkness between spots). Bragg's law (n-lambda = 2d-sin-theta) specifies the directions of constructive interference: for a given set of parallel lattice planes with spacing d, constructive interference occurs at an angle theta where the path difference between waves scattered from adjacent planes is an integer multiple of the wavelength. Each family of lattice planes in the crystal produces a diffraction spot at a specific angle.
The key insight is that the diffraction pattern and the electron density are related by a Fourier transform — a mathematical operation that decomposes any function into its component frequencies. The electron density in a crystal is periodic (it repeats with the unit cell), so it can be expressed as a Fourier series — a sum of waves with specific frequencies, amplitudes, and phases. Each diffraction spot corresponds to one term in this series: its position tells you the frequency (which lattice planes it corresponds to), its measured intensity tells you the amplitude (specifically, intensity = amplitude squared), and its phase (which is NOT measured) tells you the relative timing of that wave component.
The phase problem arises because X-ray detectors record only photon counts (intensities), not the phase of the electromagnetic wave at each detector position. Both amplitude and phase are needed to perform the inverse Fourier transform that yields the electron density map. Remarkably, the phases carry more structural information than the amplitudes — computations using correct phases but wrong amplitudes produce better maps than correct amplitudes with wrong phases. This is why phase determination (by molecular replacement, isomorphous replacement, or anomalous dispersion) is the critical step. Once approximate phases are obtained, the electron density map reveals the molecular structure, and iterative refinement improves both the atomic model and the phases until the calculated diffraction pattern matches the observed pattern to within experimental error.