Open AccessExact direct‐space asymmetric units for the 230 crystallographic space groups
Author(s) -
GrosseKunstleve Ralf W.,
Wong Buddy,
Mustyakimov Marat,
Adams Paul D.
Publication year - 2011
Publication title -
acta crystallographica section a
Language(s) - English
Resource type - Journals
eISSN - 1600-5724
pISSN - 0108-7673
DOI - 10.1107/s0108767311007008
Subject(s) - notation , group (periodic table) , space (punctuation) , unit (ring theory) , table (database) , mathematics , basis (linear algebra) , transformation (genetics) , combinatorics , computer science , geometry , arithmetic , physics , data mining , chemistry , quantum mechanics , gene , operating system , biochemistry , mathematics education
It is well known that the direct‐space asymmetric unit definitions found in the International Tables for Crystallography , Volume A, are inexact at the borders. Face‐ and edge‐specific sub‐conditions have to be added to remove parts redundant under symmetry. This paper introduces a concise geometric notation for asymmetric unit conditions. The notation is the foundation for a reference table of exact direct‐space asymmetric unit definitions for the 230 crystallographic space‐group types. The change‐of‐basis transformation law for the conditions is derived, which allows the information from the reference table to be used for any space‐group setting. We also show how the vertices of an asymmetric unit can easily be computed from the information in the reference table.