[Ifeffit] symmetry in Atoms

Bruce Ravel ravel at phys.washington.edu
Mon Aug 19 08:39:21 CDT 2002

Here is an interesting question that was waiting in my mailbox this
morning.  I thought folks on the ifeffit list might also find it
interesting and, in any case, it seems useful to have the question and
the answer in the ifeffit list's archive.


   The attached EXCEL file is the structure data for Fe17Y2 referred
   from Pearson's Handbook.
   Actually I get FEFF.inp file by Atoms using the input data as
   attached list.  I wondered, howevere, how Atoms recognize site
   symmetry thorough the Atoms.inp data.
   My question is that the input data for each symmetric site should
   fulfill the site symmetry like as following list.
   In case of Fe17Y2 (No. 194), when I input (0 0 1/4) then Atoms will
   automatically recognizes this is 2b site.
   But when I input the number (0.1672 0.3344 0.9874) for 12k-site,
   how Atoms recognizes this is 12k site.
   Or if input numerical numbers do not fulfill the symmetry
   condition, then how Atoms deal with it.
    Thank you for your attention.

   space group number  :   194
   setting             :   1
   Hermann-Mauguin     :   P 6_3/m 2/m 2/c
   Laue group          :   6/mmm
   point group         :   6/m mm


  24 l   x,y,z                 -y,x-y,z              -x+y,-x,z
         -x,-y,z+1/2           y,-x+y,z+1/2          x-y,x,z+1/2
         y,x,-z                x-y,-y,-z             -x,-x+y,-z
         -y,-x,-z+1/2          -x+y,y,-z+1/2         x,x-y,-z+1/2
         -x,-y,-z              y,-x+y,-z             x-y,x,-z
         x,y,-z+1/2            -y,x-y,-z+1/2         -x+y,-x,-z+1/2
         -y,-x,z               -x+y,y,z              x,x-y,z
         y,x,z+1/2             x-y,-y,z+1/2          -x,-x+y,z+1/2

  12 k   x,2x,z                -2x,-x,z              x,-x,z
         -x,-2x,z+1/2          2x,x,z+1/2            -x,x,z+1/2
         2x,x,-z               -x,-2x,-z             -x,x,-z
         -2x,-x,-z+1/2         x,2x,-z+1/2           x,-x,-z+1/2

  12 j   x,y,1/4               -y,x-y,1/4            -x+y,-x,1/4
         -x,-y,3/4             y,-x+y,3/4            x-y,x,3/4
         y,x,3/4               x-y,-y,3/4            -x,-x+y,3/4
         -y,-x,1/4             -x+y,y,1/4            x,x-y,1/4

  12 i   x,0,0                 0,x,0                 -x,-x,0
         -x,0,1/2              0,-x,1/2              x,x,1/2
         -x,0,0                0,-x,0                x,x,0
         x,0,1/2               0,x,1/2               -x,-x,1/2

   6 h    x,2x,1/4              -2x,-x,1/4            x,-x,1/4
          -x,-2x,3/4            2x,x,3/4              -x,x,3/4

   6 g    1/2,0,0               0,1/2,0               1/2,1/2,0
          1/2,0,1/2             0,1/2,1/2             1/2,1/2,1/2

   4 f    1/3,2/3,z             2/3,1/3,z+1/2         2/3,1/3,-z

   4 e    0,0,z                 0,0,z+1/2             0,0,-z

   2 d    1/3,2/3,3/4           2/3,1/3,1/4

   2 c    1/3,2/3,1/4           2/3,1/3,3/4

   2 b    0,0,1/4               0,0,3/4

   2 a    0,0,0                 0,0,1/2

The attachment contained this:

     Fe17Y2  "structure type" "Pearson Smmbol" "space group"  No
              Fe17Ho2          hP44             P6_3/mmc      194

     Wyckoff         x       y       z       occupncy        
     Y1      2b              0       0       0.25    0.41    
     Y2      2c              0.3333  0.6666  0.25    0.35    
     Y3      2d              0.3333  0.6666  0.75    1       
     Fe1     4e              0       0       0.1103  0.28    
     Fe2     4f              0.3333  0.6666  0.1049  0.71    
     Fe3     6g              0.5     0       0       1       
     Fe4     12j             0.3341  0.964   0.25    1       
     Fe5     12k             0.1672  0.3344  0.9871  1       

And my response:

   Well, you made it very easy for me to answer your question by
   including the symmetry table for your space group in your email.

   The answer is that Atoms reads a data base of the space group
   symmetries.  For your example, #194, the data base entry contains
   this information:

       x,y,z                 -y,x-y,z              -x+y,-x,z
       -x,-y,z+1/2           y,-x+y,z+1/2          x-y,x,z+1/2
       y,x,-z                x-y,-y,-z             -x,-x+y,-z
       -y,-x,-z+1/2          -x+y,y,-z+1/2         x,x-y,-z+1/2
       -x,-y,-z              y,-x+y,-z             x-y,x,-z
       x,y,-z+1/2            -y,x-y,-z+1/2         -x+y,-x,-z+1/2
       -y,-x,z               -x+y,y,z              x,x-y,z
       y,x,z+1/2             x-y,-y,z+1/2          -x,-x+y,z+1/2

   which you will recognize as Wyckoff's position "l".  Atoms doesn't
   exactly distinguish between Wyckoff's position l and position b.
   Instead it assumes that everything is position l and discards
   repeated positions.  To understand this, apply the symmetry
   operations in the list above to the position (0,0,1/4).  You will
   notice that 12 of them generate the position (0,0,1/4) and 12 of
   them generate (0,0,3/4).  Atoms notices that and discards 11 of

   So, you see, position b is not actually different from position l,
   it's just degenerate due to symmetry.  Atoms will notice these
   degeneracies a posteriori and does not need to know about the a

   So, where did Wyckoff get his notation scheme from?  Well, the
   Wyckoff positions are simply ordered in terms of decreasing
   degeneracy.  Position a is the most degenerate and the highest
   lettered position is the non-degenerate one.  This is not actually
   a useful notation scheme because there is no consistency between
   space groups.  That is, position "c", for example, in group 194 has
   no relationship to position "c" in any other space group.

   The last thing worth mentioning is that Atoms works with 5
   significant digits.  If you use, as you suggest, (0.1672 0.3344
   0.9874) then Atoms will find the correct amount of degeneracy.
   However, how you write (1/3, 2/3, z) is very important.  Atoms 3.0
   is able to read the string "1/3" as a coordinate.  However if you
   choose to write coordinates in decimal notation, you must be
   careful.  (0.33333, 0.66667, z) is fine and will generate 4
   positions.  (0.33, 0.67, z) will have problems related to numerical

   Hope that's clear,

 Bruce Ravel  ----------------------------------- ravel at phys.washington.edu
 Code 6134, Building 3, Room 222
 Naval Research Laboratory                          phone: (1) 202 767 5947
 Washington DC 20375, USA                             fax: (1) 202 767 1697

 NRL Synchrotron Radiation Consortium (NRL-SRC)
 Beamlines X11a, X11b, X23b, X24c, U4b
 National Synchrotron Light Source
 Brookhaven National Laboratory, Upton, NY 11973

 My homepage:    http://feff.phys.washington.edu/~ravel 
 EXAFS software: http://feff.phys.washington.edu/~ravel/software/exafs/

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