direct methods
Direct methods are a clever way to guess the missing phases directly from the measured amplitudes, with no need for heavy atoms or extra experiments. The trick is to lean on two facts that any real electron density must obey: it is never negative anywhere (you cannot have less than no electrons), and it is concentrated into sharp, atom-shaped lumps rather than smeared out. Those physical facts, it turns out, secretly constrain the phases far more than you might expect.
Here is the mechanism in plain steps. Because the density is positive and atom-like, the phases of the strongest reflections are not independent — they are tied together statistically. The key relationship is the triplet: for three strong reflections whose indices add to zero, phi(H) + phi(K) + phi(-H-K) is probably close to zero. Starting from a small set of assumed phases, you propagate these probabilistic links (through Sayre's equation and the tangent formula) to phase reflection after reflection, try many starting guesses, and keep the trial whose resulting electron-density map looks like sensible, well-separated atoms.
This is what made routine, automated small-molecule structure solution possible — the achievement recognised by the Nobel Prize to Hauptman and Karle. It works best when the atoms are roughly equal and light and when the data reach atomic resolution (spacings down to about 1 angstrom). Be honest about where it struggles: for very large structures such as proteins, and for lower-resolution data, the statistics weaken, and crystallographers turn instead to the Patterson function, heavy-atom methods, or density modification.
Given three strong reflections (2 3 1), (1 -1 0) and (-3 -2 -1), whose indices sum to zero, direct methods bet that their phases add to nearly zero — and stitching thousands of such triplets together phases the whole small-molecule data set automatically.
Positivity and atomicity secretly link the phases of strong reflections — direct methods exploit that.
Direct methods need data to near-atomic resolution and work best for small, roughly equal-atom structures; they largely fail for large proteins from X-ray data alone, where other phasing routes take over.