Wolfram Language Paclet Repository
Community-contributed installable additions to the Wolfram Language
The association structure of crystallographic data
An is a list structure with entries being a pair of elements, referred to as and . Each is associated with a certain using the notation . One can think of the structure as a one-way dictionary.
keys
values
key
value
keyvalue
Associations can be nested with both lists and associations. This feature, along with a fast and easy way of inquiring values, makes it useful for storing information about point groups and space group in an hierarchical form.
The package must be loaded:
In[1]:=
<<MaXrd`
returns information on the given point group | |
returns information on the given space group | |
Looking up point- and space group information.
Examples of queries
First a point group example:
Let us inquire information on the point group :
4/m
In[2]:=
 | $PointGroups |
Out[2]=
Symbol4/m,PointGroupNumber11,CrystalSystemTetragonal,LaueClass4/m,SchoenfliesSymbol,Setting,ClassNamesGrothdipyramidal,Friedelparahemihedry,PropertiesCentrosymmetricTrue,EnantiomorphicFalse,PolarFalse,SymmetryOperationsMatrixOperations{{{1,0,0},{0,1,0},{0,0,1}},{{-1,0,0},{0,-1,0},{0,0,1}},{{0,-1,0},{1,0,0},{0,0,1}},{{0,1,0},{-1,0,0},{0,0,1}},{{-1,0,0},{0,-1,0},{0,0,-1}},{{1,0,0},{0,1,0},{0,0,-1}},{{0,1,0},{-1,0,0},{0,0,-1}},{{0,-1,0},{1,0,0},{0,0,-1}}},SymmetryOperationsITA1,,,,,,,,SymmetryOperationsSeitz1,,,,,,,,SubgroupsSubgroupClassType1,SubgroupSymmetry1,SubgroupList{1},SubgroupClassType,SubgroupSymmetry,SubgroupList{1,5},SubgroupClassType2,SubgroupSymmetry,SubgroupList{1,2},SubgroupClassTypem,SubgroupSymmetry,SubgroupList{1,6},SubgroupClassType2/m,SubgroupSymmetry,SubgroupList{1,2,5,6},SubgroupClassType4,SubgroupSymmetry,SubgroupList{1,2,3,4},SubgroupClassType,SubgroupSymmetry,SubgroupList{1,2,7,8},SubgroupClassType4/m,SubgroupSymmetry,SubgroupList{1,2,3,4,5,6,7,8}
C
4h
2
[001]
+
4
[001]
-
4
[001]
1
m
[001]
+
4
[001]
-
4
[001]
2
z
4
z
3
4
z
1
m
z
4
z
3
4
z
1
1
2
z
m
z
2
z
m
z
4
z
4
4
z
4
z
m
z
Notice how the output is itself an association, with the values of and being "sub-associations". Read more about the output structure on the documentation page for .
"Properties"
"SymmetryOperations"
Now a little more advanced:
Selecting all centrosymmetric point groups:
In[4]:=
centr=Keys@Select
,#"Properties","Centrosymmetric"&
 | $PointGroups |
Out[4]=
{-1,2/m,mmm,4/m,4/mmm,-3,-3m,6/m,6/mmm,m-3,m-3m}
Verify that the set of centrosymmetric point groups is the same set as the Laue classes:
In[5]:=
centr===Keys@
 | $LaueClasses |
Out[5]=
True
Find the 13 alternative settings of point groups (and obtain the formatted symbols):
In[6]:=
pgWithAlt=Keys@DeleteMissing@
All,"AlternativeSettings"
| $PointGroups |
Out[6]=
{2,m,2/m,-42m,3,-3,32,3m,-3m,-62m}
In[7]:=
 | $PointGroups |
Out[7]=
{2,m,2/m,2m,3,,32,3m,m,2m}
4
3
3
6
In[8]:=
Dataset@
["3m"]
| $PointGroups |
Out[8]=
| |||||||||||||||||||||||||||||
Auxiliary tools
In this section we will learn more about the two functions and , which are built-in to the MaXrd package and help query data from point groups, space groups and crystal data.
SymmetryData
Associations only support a single key for each entry. The associations and only have one “main entry” for each group,
In[1]:=
Length/@
,
| $PointGroups |
| $SpaceGroups |
Out[1]=
{32,230}
All data concerned with the alternative settings are stored as sub-entries. For example, the orthorhombic space group has five alternative settings: one with an -permutation of the axes, and one with a -permutation, and the remaining versions with , and permutations.
Pbcn
"ba"
c
"cab"
"ba"
c
"bca"
"ab"
c
In[2]:=
Keys@
["Pbcn","AlternativeSettings"]
| $SpaceGroups |
Out[2]=
{AxisPermutationBA-C,AxisPermutationCAB,AxisPermutation-CBA,AxisPermutationBCA,AxisPermutationA-CB}
Let us look at three different ways to access the data of, say the -version:
"bca"
In[3]:=
| $SpaceGroups |
Out[3]=
NameSymbolPbna,HermannMauguinShortP b n a,HermannMauguinFullP 2/b 21/n 21/a,HallString-P 2ac 2b,SettingAxisPermutationbca,SymmetryOperations{{{1,0,0},{0,1,0},{0,0,1}},{0,0,0}},{{-1,0,0},{0,-1,0},{0,0,1}},,0,,{{1,0,0},{0,-1,0},{0,0,-1}},0,,0,{{-1,0,0},{0,1,0},{0,0,-1}},,,,{{{-1,0,0},{0,-1,0},{0,0,-1}},{0,0,0}},{{1,0,0},{0,1,0},{0,0,-1}},-,0,-,{{-1,0,0},{0,1,0},{0,0,1}},0,-,0,{{1,0,0},{0,-1,0},{0,0,1}},-,-,-
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
2. Let look up the entry automatically (it works almost like a “switchboard” for the point- and space groups). You would need to know the space group symbol in advance:
In[4]:=
| $GroupSymbolRedirect |
Out[4]=
NameSymbolPbna,HermannMauguinShortP b n a,HermannMauguinFullP 2/b 21/n 21/a,HallString-P 2ac 2b,SettingAxisPermutationbca,SymmetryOperations{{{1,0,0},{0,1,0},{0,0,1}},{0,0,0}},{{-1,0,0},{0,-1,0},{0,0,1}},,0,,{{1,0,0},{0,-1,0},{0,0,-1}},0,,0,{{-1,0,0},{0,1,0},{0,0,-1}},,,,{{{-1,0,0},{0,-1,0},{0,0,-1}},{0,0,0}},{{1,0,0},{0,1,0},{0,0,-1}},-,0,-,{{-1,0,0},{0,1,0},{0,0,1}},0,-,0,{{1,0,0},{0,-1,0},{0,0,1}},-,-,-
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
3. Use to extract the data you want (see valid commands in the documentation). Say we want to know the space group number or whether the space group is a “main entry” or not. A similar function () can also extract the symmetry operations:
SymmetryData
SymmetryOperations
In[5]:=
| GetSymmetryData |
Out[5]=
60
In[6]:=
| GetSymmetryData |
Out[6]=
False
In[7]:=
| GetSymmetryOperations |
Out[7]=
TransformationFunction
,TransformationFunction
,TransformationFunction
,TransformationFunction
,TransformationFunction
,TransformationFunction
,TransformationFunction
,TransformationFunction
1 | 0 | 0 | 0 |
0 | 1 | 0 | 0 |
0 | 0 | 1 | 0 |
0 | 0 | 0 | 1 |
-1 | 0 | 0 | 1 2 |
0 | -1 | 0 | 0 |
0 | 0 | 1 | 1 2 |
0 | 0 | 0 | 1 |
1 | 0 | 0 | 0 |
0 | -1 | 0 | 1 2 |
0 | 0 | -1 | 0 |
0 | 0 | 0 | 1 |
-1 | 0 | 0 | 1 2 |
0 | 1 | 0 | 1 2 |
0 | 0 | -1 | 1 2 |
0 | 0 | 0 | 1 |
-1 | 0 | 0 | 0 |
0 | -1 | 0 | 0 |
0 | 0 | -1 | 0 |
0 | 0 | 0 | 1 |
1 | 0 | 0 | - 1 2 |
0 | 1 | 0 | 0 |
0 | 0 | -1 | - 1 2 |
0 | 0 | 0 | 1 |
-1 | 0 | 0 | 0 |
0 | 1 | 0 | - 1 2 |
0 | 0 | 1 | 0 |
0 | 0 | 0 | 1 |
1 | 0 | 0 | - 1 2 |
0 | -1 | 0 | - 1 2 |
0 | 0 | 1 | - 1 2 |
0 | 0 | 0 | 1 |