+-----------------------------------------------------------------------------+ | P R O G R A M S E X I E | +-----------------------------------------------------------------------------+ | Calculation of Coordination Shells and Interatomic Distances | | Bernhard Rupp | | Version 5.4, Revision 07/01/97 | | Proprietary code of Bernhard Rupp (C) 1988-97 | +-----------------------------------------------------------------------------+ | Title : Webjob on Tue Feb 25 1997 at 16:35:45 from 128.115.150.112 | | Control parameters : | | No atom card detected - space group decoding only | +-----------------------------------------------------------------------------+
Header. This program is the basis for several others I wrote, including geometry calculations, water analysis, structure factor calculations etc. etc. Sexie is a historic acronym for Shells for EXafs InteractivE. I have chosen this mostly to irritate the political correctness Mafia.
+-----------------------------------------------------------------------------+ | Space group P 21 21 21, space group number 19, Laue class mmm | +-----------------------------------------------------------------------------+
The symbol followed by the group number and the Laue class. The Laue class (or symmetry) is of interest for the data collection, we'll revisit this in the next chapter.
+-----------------------------------------------------------------------------+ | Non centrosymmetric space group | | Bravais translation primitive | +-----------------------------------------------------------------------------+
We know this is the most common enantiomorphic space group for proteins, so it will not have a center of symmetry due to the absence of inversions and mirrors. Note that a Bravais centering would be allowed, example I 2 2 2.
+-----------------------------------------------------------------------------+ | 4 equipoint transformations [X]'=[R]*[X]+[T] : | +-----------------------------------------------------------------------------+
This line gives us the number of equipoint transformations created by the application of the symmetry operations, not including Bravais translations and centering. Those are listed later. [X]'=[R]*[X]+[T] means that the new position vector (x') is obtained from the original position (x) upon application of a rotation matrix (R) originating from the point group part and a translation vector (T) from the internal elements (glide, screw). Here is the matrix notation :
Note that T is NOT the Bravais translation yet! You'll notice that for this space group there are translations by 1/2 in T caused by the 21 operation. Try a space group without screws and glides, and there will be zeros only in T.
+-----------------------------------------------------------------------------+ | R(11) R(12) R(13) R(21) R(22) R(23) R(31) R(32) R(33) T(1) T(2) T(3) | | 1.0 .0 .0 .0 1.0 .0 .0 .0 1.0 .000 .000 .000 | | 1.0 .0 .0 .0 -1.0 .0 .0 .0 -1.0 .500 .500 .000 | |-1.0 .0 .0 .0 1.0 .0 .0 .0 -1.0 .000 .500 .500 | |-1.0 .0 .0 .0 -1.0 .0 .0 .0 1.0 .500 .000 .500 | +-----------------------------------------------------------------------------+
As the matrix notation is not easy to read without practice, the transformations are now listed in a more intuitive format :
+-----------------------------------------------------------------------------+ | 4 equipoint transformations decoded into atom coordinate format : | +-----------------------------------------------------------------------------+ | X , Y , Z | |1/2 +X,1/2 -Y, -Z | | -X ,1/2 +Y,1/2 -Z | |1/2 -X, -Y ,1/2 +Z | +-----------------------------------------------------------------------------+ | 4 equipoints generated by point symmetry | | yielding the general position multiplicity of 4 | +-----------------------------------------------------------------------------+
In the box above you would find further operations like inversion and Bravais centering and the final number of equivalent positions generated in the unit cell. See exercises for more.
+-----------------------------------------------------------------------------+ | Conditions limiting reflections due to Bravais centering | +-----------------------------------------------------------------------------+ | HKL : none | +-----------------------------------------------------------------------------+ | Conditions limiting reflections due to space group symmetry | +-----------------------------------------------------------------------------+ | H00 : H=2N only | | 0K0 : K=2N only | | 00L : L=2N only | +-----------------------------------------------------------------------------+
In the box above you find a listing of conditions limiting possible reflections for this spacegroup. Extinctions are due to translational elements either from the Bravais centering or symmetry operations containing a translational element. Click here to find more about systematic absences.
+-----------------------------------------------------------------------------+ | Asymmetric unit of intensity data for Laue group mmm | +-----------------------------------------------------------------------------+ | h : 0 to inf | | k : 0 to inf | | l : 0 to inf | | No other exclusions | | Coverage extends over 1/8 of reciprocal space | +-----------------------------------------------------------------------------+
The listing above shows the extent of the corresponding asymmetric unit of reciprocal space, which is the range of reflections you need to allow for a complete Fourier reconstruction of the real space asymmetric unit. Click here to learn more about Laue groups, unique reflections and Friedel pairs.
The final listing below just makes it easier to copy the symbolic representations into another program.
Alternate listing of symmetry operators follows
X, Y, Z
1/2 +X,1/2 -Y, -Z
-X,1/2 +Y,1/2 -Z
1/2 -X, -Y,1/2 +Z
Go ahead and try other space groups. Have fun!
Back to space group tutorial.
Back to X-ray Facility Introduction
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This World Wide Web site conceived and maintained by
Bernhard Rupp (br@llnl.gov)
Last revised February 26, 2002 14:07
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