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Community-contributed installable additions to the Wolfram Language
Symmetry calculations
In this tutorial, we will learn about the correspondence between the information given in the International Tables for Crystallography, volume A and data stored in .
The package must be loaded:
In[1]:=
<<MaXrd`
returns equivalent postions of given coordinates | |
returns equivalent reflections of given Miller indices | |
check whether two or more reflections are equivalent | |
returns the symmetry operations of a given group |
Functions concerning symmetry equivalence and symmetry operations.
Working with symmetry equivalent positions
Shown below is an extract from the International Tables for Crystallography, volume A of the symmetry equivalent positions of space group (#114).
Pc
4
2
1
The same information may be obtained by the following:
In[2]:=
 | $SpaceGroups |
Out[2]=
| |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Let us generate these from the first coordinates from each site symmetry. We use the function called :
SymmetryEquivalentPostions
In[3]:=
?
 | SymmetryEquivalentPositions |
Out[3]=
Symbol | |
SymmetryEquivalentPositions[group,{x,y,z}] returns a list of coordinates that are equivalent to {x,y,z} group x 1 y 1 z 1 x 2 y 2 z 2 {x,i,z} i group | |
Wyckoff letter (general position):
e
In[4]:=
| SymmetryEquivalentPositions |
Out[4]=
{x,y,z},{-x,-y,z},{y,-x,-z},{-y,x,-z},-x,+y,-z,+x,-y,-z,-y,-x,+z,+y,+x,+z
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
Wyckoff letter :
d
In[5]:=
| SymmetryEquivalentPositions |
Out[5]=
0,,z,,0,-z,,0,-z,0,,+z
1
2
1
2
1
2
1
2
1
2
1
2
Wyckoff letter :
c
In[6]:=
| SymmetryEquivalentPositions |
Out[6]=
{0,0,z},{0,0,-z},,,-z,,,+z
1
2
1
2
1
2
1
2
1
2
1
2
Wyckoff letter :
b
In[7]:=
| SymmetryEquivalentPositions |
Out[7]=
0,0,,,,0
1
2
1
2
1
2
Wyckoff letter :
a
In[8]:=
| SymmetryEquivalentPositions |
Out[8]=
{0,0,0},,,
1
2
1
2
1
2
Users that are interested in how this function works should see the Mathematica code section in the documentation pages on this function.
Note that the multiplicity (the first column in the extract) is the number of symmetry equivalent positions times the number of centring vectors (which for is one).
P
Working with symmetry equivalent reflections
Analogous to is the function. Let us first see an example:
Reflections that are symmetry equivalent to reflection in :
hk0
P31m
In[1]:=
| SymmetryEquivalentReflections |
Out[1]=
{{h,k,0},{k,-h-k,0},{-h-k,h,0},{k,h,0},{h,-h-k,0},{-h-k,k,0}}
Let us now look more into the details of the function, and break the procedure into steps.
The functions starts by finding the Laue class of the space group:
In[2]:=
| GetLaueClass |
Out[2]=
3
Next we find the symmetry operations associated with that group:
In[3]:=
S=
["-3m"]
| GetSymmetryOperations |
Out[3]=
TransformationFunction
,TransformationFunction
,TransformationFunction
,TransformationFunction
,TransformationFunction
,TransformationFunction
,TransformationFunction
,TransformationFunction
,TransformationFunction
,TransformationFunction
,TransformationFunction
,TransformationFunction
1 | 0 | 0 | 0 |
0 | 1 | 0 | 0 |
0 | 0 | 1 | 0 |
0 | 0 | 0 | 1 |
0 | -1 | 0 | 0 |
1 | -1 | 0 | 0 |
0 | 0 | 1 | 0 |
0 | 0 | 0 | 1 |
-1 | 1 | 0 | 0 |
-1 | 0 | 0 | 0 |
0 | 0 | 1 | 0 |
0 | 0 | 0 | 1 |
0 | 1 | 0 | 0 |
1 | 0 | 0 | 0 |
0 | 0 | -1 | 0 |
0 | 0 | 0 | 1 |
1 | -1 | 0 | 0 |
0 | -1 | 0 | 0 |
0 | 0 | -1 | 0 |
0 | 0 | 0 | 1 |
-1 | 0 | 0 | 0 |
-1 | 1 | 0 | 0 |
0 | 0 | -1 | 0 |
0 | 0 | 0 | 1 |
-1 | 0 | 0 | 0 |
0 | -1 | 0 | 0 |
0 | 0 | -1 | 0 |
0 | 0 | 0 | 1 |
0 | 1 | 0 | 0 |
-1 | 1 | 0 | 0 |
0 | 0 | -1 | 0 |
0 | 0 | 0 | 1 |
1 | -1 | 0 | 0 |
1 | 0 | 0 | 0 |
0 | 0 | -1 | 0 |
0 | 0 | 0 | 1 |
0 | -1 | 0 | 0 |
-1 | 0 | 0 | 0 |
0 | 0 | 1 | 0 |
0 | 0 | 0 | 1 |
-1 | 1 | 0 | 0 |
0 | 1 | 0 | 0 |
0 | 0 | 1 | 0 |
0 | 0 | 0 | 1 |
1 | 0 | 0 | 0 |
1 | -1 | 0 | 0 |
0 | 0 | 1 | 0 |
0 | 0 | 0 | 1 |
Each symmetry operation (matrix) must be transposed and multiplied by the reflection. For example:
In[4]:=
s=RandomChoice[S]
Two reflections from the list above, so we know they are symmetry equivalent:
Example of two reflections that are not symmetry equivalent: