Ambiguities in the X-ray analysis of crystal structures

Abstract

A mathematical construction is given for arbitrarily many distinct crystal structures all of which would give the same diffraction pattern. A. L. Patterson's concept of homometric sets is analyzed, and examples are given in one, two and three dimensions.Let al, a2,a3 be linearly independent vectors in threedimensional space. Let A be the three-by-three matrix whose columns are the aj. The vectors aj determine a lattice of points An = nlat + n2a2 + n3a3(1)where the nj are integers. The basic cell of the lattice is the set of points x = ~al + ~za2 + ~3a3 with 0 < #j < 1 .(2)The reciprocal lattice has the matrixIts columns bj satisfyThe reciprocal lattice consists of the points Bh, where the hj are integers (called Miller indices).Let the atoms in a crystal be located at rt,...,rN in the basic cell (2) and at all congruent points rj + An. By X-ray analysis, one tries to find the positions rj.

The use of higher-order invariants in the determination of generalized Patterson cyclotomic sets

“…This justifies the name 'homometric sets' for those with the same value of dkd_~, k = 0, 1 .... , N. This information (second-order invariants) is directly accessible from intensity measurements. Pairs of noncongruent homometric sets have been constructed starting with the work of Patterson (see Buerger, 1976;Chieh, 1979;Franklin, 1974;Patterson, 1944). Here is a general method for constructing homometric pairs.…”

“…A different line of attack on the determination of (strictly) homometric sets was undertaken by the mathematician Calder6n and the crystallographer Pepinsky (Caldertn & Pepinsky, 1952). Their work was later expanded by Franklin (1974) as well as by and Bloom & Golomb (1977). The examples that they produce are, however, of a different nature than those studied by Patterson.…”

“…The examples that they produce are, however, of a different nature than those studied by Patterson. They correspond to 'strictly homometric structures', in the nomenclature of Franklin (1974).…”

Resolution of ambiguities in Zhdanov notation: actual examples of homometric structures

“…Subsequently, Patterson (1944) has cited many theoretical examples of one-dimensional homometric sets and has extended them to two and three dimensions. Recently, Franklin (1974) has given a mathematical construction for many distinct arbitrary crystal structures all of which would give the same diffraction pattern. However, no practical example of homometric structures is known to date and it has been a common belief (e.g.…”

Homometrische kubische Punktkonfigurationen

“…Allgemeine Abhandlung: Buerger (1959); Eindeutigkeit der Kristallstrukturbestimmung: Engel (1979Engel ( , 1980; zyklotomische Komplexe: Buerger (1977), Chieh (1979Chieh ( , 1982, Iglesias (1981); Faltprodukte: Buerger (1961), Bullough (1961Bullough ( , 1964, Franklin (1974); Polytype Strukturen: Farkas-Jahnke und Dornberger-Schiff (1969), Dornberger-Schiff und Farkas-Jahnke (1970), Jain and Trigunayat (1977), Chadha (1981) und Fichtner (1986; Homometrische Quasikristalle: Zobetz (1992). Allgemeine Abhandlung: Buerger (1959); Eindeutigkeit der Kristallstrukturbestimmung: Engel (1979Engel ( , 1980; zyklotomische Komplexe: Buerger (1977), Chieh (1979Chieh ( , 1982, Iglesias (1981); Faltprodukte: Buerger (1961), Bullough (1961Bullough ( , 1964, Franklin (1974); Polytype Strukturen: Farkas-Jahnke und Dornberger-Schiff (1969), Dornberger-Schiff und Farkas-Jahnke (1970), Jain and Trigunayat (1977), Chadha (1981) und Fichtner (1986; Homometrische Quasikristalle: Zobetz (1992).…”