|
International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. B. ch. 2.3, p. 238
Section 2.3.1.5. The Patterson synthesis of the second kind
aDepartment of Biological Sciences, Purdue University, West Lafayette, Indiana 47907, USA, and bCABM & Rutgers University, 679 Hoes Lane, Piscataway, New Jersey 08854-5638, USA |
Patterson also defined a second, less well known, function (Patterson, 1949![[link]](/graphics/greenarr.gif)
) as ![[\eqalign{ P_{\pm} ({\bf u}) &= {\int} \rho ({\bf u + x}) \cdot \rho ({\bf u - x})\;\hbox{d}{\bf x}\cr &= {2 \over V^{2}} {\sum\limits_{{\bf h}}^{\rm hemisphere}} F_{{\bf h}}^{2} \cos (2 \pi 2{\bf h \cdot u} - 2 \alpha_{{\bf h}}).}]](/teximages/bach2o3/bach2o3fd16.svg)
This function can be computed directly only for centrosymmetric structures. It can be calculated for noncentrosymmetric structures when the phase angles are known or assumed. It will represent the degree to which the known or assumed structure has a centre of symmetry at u. That is, the product of the density at ![[{\bf u} + {\bf x}]](/teximages/bach2o3/bach2o3fi42.svg)
and ![[{\bf u} - {\bf x}]](/teximages/bach2o3/bach2o3fi43.svg)
is large when integrated over all values x within the unit cell. Since atoms themselves have a centre of symmetry, the function will contain peaks at each atomic site roughly proportional in height to the square of the number of electrons in each atom plus peaks at the midpoint between atoms proportional in height to the product of the electrons in each atom. Although this function has not been found very useful in practice, it is useful for demonstrating the presence of weak enantiomorphic images in a given tentative structure determination.
References
Patterson, A. L. (1949). An alternative interpretation for vector maps. Acta Cryst. 2, 339–340.Google Scholar
