[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.
B
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
positions
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
1/3,2/3,-z+1/2
4 e 0,0,z 0,0,z+1/2 0,0,-z
0,0,-z+1/2
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
each.
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
priori.
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
precision.
Hope that's clear,
Bruce
--
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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