Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2006). Vol. B. ch. 2.3, p. 239   | 1 | 2 |


M. G. Rossmanna* and E. Arnoldb

aDepartment of Biological Sciences, Purdue University, West Lafayette, Indiana 47907, USA, and  bCABM & Rutgers University, 679 Hoes Lane, Piscataway, New Jersey 08854-5638, USA
Correspondence e-mail:

Table| top | pdf |
Square matrix representation of vector interactions in a Patterson of a crystal with M crystallographic asymmetric units each containing N atoms

Peak positions [{\bf u}_{m1n1, \,  m2n2}] correspond to vectors between the atoms [{\bf x}_{m1n1}] and [{\bf x}_{m2n2}] where [{\bf x}_{mn}] is the nth atom in the mth crystallographic asymmetric unit. The corresponding weights are [w_{n1} w_{n2}]. The outlined blocks I1 and IM represent vector interactions between atoms in the same crystallographic asymmetric units (there are M such blocks). The off-diagonal blocks IIM1 and II1M represent vector interactions between atoms in crystal asymmetric units 1 and M; there are [M(M - 1)] blocks of this type. The significance of diagonal elements of block IIM1 is that they represent Harker-type interactions between symmetry-equivalent atoms (see Section[link]).

  [{\bf x}_{11}, w_{1}] [{\bf x}_{12}, w_{2}] [{\bf x}_{1N}, w_{N}] [{\bf x}_{M1}, w_{1}] [{\bf x}_{M2}, w_{2}] [{\bf x}_{MN}, w_{N}]
[{\bf x}_{11}, w_{1}] 0, [w_{1}^{2}] [{\bf u}_{11, \,  12}, w_{1} w_{2}] [{\bf u}_{11, \,  1N}, w_{1} w_{N}]          
[{\bf x}_{12}, w_{2}]   0, [w_{2}^{2}] [{\bf u}_{12, \,  1N}, w_{2} w_{N}]          
[\vdots]   [\ddots]   [\vdots]        
[{\bf x}_{1N}, w_{N}]       0, [w_{N}^{2}]          
  Block I1         Block II1M    
[\vdots]   [\vdots]     [\ddots]   [\vdots]    
[{\bf x}_{M1}, w_{1}] [{\bf u}_{M1, \,  11}, w_{1}^{2}] [{\bf u}_{M1, \,  12}, w_{1} w_{2}]              
[{\bf x}_{M2}, w_{2}] [{\bf u}_{M2, \,  11}, w_{2} w_{1}] [{\bf u}_{M2, \,  12}, w_{2}^{2}]              
[\vdots]   [\ddots]            
[{\bf x}_{MN}, w_{N}]       [{\bf u}_{MN, \,  1N}, w_{N}^{2}]          
  Block IIM1         Block IM