During diffraction experiments, X-ray radiation is scattered by the atoms present in the crystal and the intensity of the scattered X-rays are measured as reflections. However, the phase differences for the scattered X-rays are not known. Recovering this phasing information has been the focus of many efforts by many researchers.

In the early structure determinations, where there were only a few atoms in the unit cell, the atoms usually sat on symmetry elements and it was possible, with some guesswork, to deduce the positions of the atoms, given the space group was known. Indeed, for many years, researchers put pencil to paper, using trial and error as they tested their assumptions about the reflections' phase to determine the structure of their molecule of interest.

In 1934, Lindo Patterson published his keen insight on the use of Fourier theory to narrow the phasing search. His equation, called the Patterson function, used diffraction intensities to determine the interatomic distances within a crystal, setting limits to the possible phase values. Shortly thereafter, David Harker found that symmetry-related atoms produced peaks in the Patterson function at certain crystal planes. These findings cut down on manual computation time and allowed researchers to examine structures of even greater complexity, creating a boon for organometallic crystallography, which had heavy atoms that provided stronger diffraction intensities as guideposts. Organic compounds as well as molecules with more than ∼50 atoms remained a challenge.

In the early 1950s, David Sayre suggested that the phase problem could be more easily solved if you had at least one more intensity measurement beyond those of the Bragg peaks (Milestone 3) in each dimension. This idea was inspired by Claude Shannon's work on communication theory and is a concept understood today as oversampling. William Lawrence Bragg and Max Perutz, building on earlier work examining the dehydration of haemoglobin crystals, came to similar, but perhaps less precise, conclusions using Fourier analysis.

Meanwhile, Jerome Karle and Herbert Hauptman inferred that relationships must exist between the diffracted waves, as there were usually more measured reflections than atoms in the crystallized molecule. In Nobel prize-winning work, they were able to deduce such relationships based on the idea that a molecule's electron density can never have a negative value. Using probability theory, they developed extremely useful formulae for phase determination, known as direct methods. Sayre later developed an equation that gave rise to dominant relationships in triplets of strong reflections that produced similar results.

A map section presented in Patterson's 1934 publication. Figure reprinted with permission from A. L. Patterson Phys. Rev. 46, 372–376 (1934). Credit: © 1934 AMERICAN PHYSICAL SOCIETY

It took some time for the statistical methods proposed by Karle and Hauptman to gain traction, but the rapid increase in the strength of computing power helped the methods and their derivatives obtain widespread acceptance. Computationally intensive iterative techniques use a very simple mathematical framework developed from the above-mentioned oversampling methods.

Direct methods and the Patterson function are most effectively applied to the determination of small-molecule structures. Macromolecular crystallographers had to await the development of isomorphous replacement (Milestone 12), molecular replacement (Milestone 13) and anomalous diffraction (Milestone 19) techniques to address their phasing problems. However, direct methods and the Patterson function are routinely used as part of these structure solution efforts.