Crystallography & Symmetry
patterns derived from their groups
Concept

Heavy atom method — where it appears

Phasing a structure from the position of one much heavier atom, found in a Patterson map where its vectors stand above every other peak. The added symmetry of the map is what makes the search feasible, by concentrating those vectors onto sections rather than spreading them.

Named by 4 essays across one field — each of them below, with the objects they name alongside it.

A structure, and the vectors between its atoms. On the left, 4 atoms in a cell. On the right, every one of the 16 vectors between them, each drawn from a common origin: 13 distinct positions, with the 4-fold peak at the origin being each atom paired with itself. That right-hand picture is what a Patterson map shows, and it is the thing a diffraction experiment gives without phases. It has more peaks than the structure has atoms — n² against n — which is why interpreting one is hard, and why it is always symmetric about its centre.

The map that needs no phases

A diffraction experiment measures intensities and loses phases, so the electron density cannot be computed from it. One map can be: the transform of the intensities, whose peaks are not atoms but the vectors between them — every ordered pair, brought to a common origin.

diffraction · The Patterson function P2₁2₁2₁: the sections its symmetry forces. The Patterson cell of P2₁2₁2₁ with the sections marked. Each operation (M, t) sends an atom at x to Mx + t, so the vector between them is (I − M)x − t; where I − M is singular that vector cannot leave a plane, and the plane's equation comes from the left null space in integers. This group has 3 such operations, giving the sections w = 0.5, u = 0.5, v = 0.5. A heavy atom's vector to its own image is somewhere on one of them, which is what made structure solution possible before computers: a plane can be searched by eye and a volume cannot.

Where symmetry stacks the vectors

A Patterson map of a real structure is a blur with thousands of overlapping peaks. A screw axis rescues it: the vectors between symmetry-related atoms cannot leave a plane, so the search for a heavy atom is a search of a section rather than of a volume — and which plane it is falls out of the operation's matrix in integers.

diffraction · The Patterson function Every vector between every pair of atoms. The Patterson map of a four-atom structure: the transform of the intensities with every phase set to zero, so it is computable from a measurement and nothing else. Its peaks are not atoms but the vectors between them, and the ringed one is the strongest that is not the origin — taken here as an interatomic vector exactly as a crystallographer takes the vector between two heavy atoms. That it really is one of the structure's own vectors is checked rather than assumed: it lands within 0.0031 of a cell of a difference of two positions, and it sits 0.67 of a cell from the origin, well outside the skirt of the tall peak there. A candidate taken too close to the origin is the same peak seen again and the whole method fails quietly.

Solving from the vector set

A Patterson map contains a copy of the structure laid over every atom in turn. Shift it by one interatomic vector, take the pointwise minimum with itself, and the copies that fail to coincide are cut away — leaving the structure, together with its inverse, from a measurement that carries no phases at all.

diffraction · The Patterson function Two kinds of atom, and more ambiguity rather than less. Exhaustive searches on rings of four sizes. The third column counts homometric groups when every atom is identical; the fourth counts them when each atom may be one of two kinds. The fourth is larger at every size, and at nine and ten sites the third is nothing at all — there is no pair of arrangements of four identical atoms that a diffraction experiment cannot separate, and there are six and four once the atoms may differ. Distinguishing the atoms adds information to the structure and adds ambiguity to the measurement.

When the atoms are not all the same

Every homometric pair found so far is a pair of point sets, where an atom is a point and counts once. Give the atoms different scattering powers and the ambiguity does not go away — it grows. On a ring of nine there is no pair of four identical atoms that diffraction cannot separate, and there are six once two kinds of atom are allowed.

diffraction · Homometry

Named alongside it

The objects these essays reach for when they reach for this one.

Interatomic vectorThe Patterson functionPhase problemAutocorrelationConvolutionHarker sectionExhaustive searchFriedel lawHomometryMeasurementMinimum functionScrew axis

All concepts