Wolfram Language Paclet Repository
Community-contributed installable additions to the Wolfram Language
Computations on reflections
In this tutorial, we will see how to make use of the package when working with reflections on the form , symbolising Miller indices .
{h,k,l}
hkl
The package must be loaded:
In[1]:=
<<MaXrd`
MergeSymmetryEquivalent | delete duplicate reflections by symmetry equivalence |
generates lists of reflections | |
checks whether a reflection is extinct | |
gives the «standard representative» reflection |
Useful functions concerning reflections and systematic absence (extinct- or forbidden reflections).
hkl
converts formatted string hkl {h,k,l} | |
converts list {h,k,l} hkl |
Functions for formatting and clearing formatting of reflections.
Generating lists of reflections
In this demonstration we wish to generate an appropriate list of reflections for sodium chloride.
Reflections are characterised by their Miller indices, , which are represented by lists of three elements in Mathematica, .
hkl
{h,k,l}
Let us start by making a generic list of reflections; every possible whole number combinations, up to index (except the reflection):
2
000
In[2]:=
hkl=
[2]
 | ReflectionList |
Out[2]=
{{0,0,1},{0,1,0},{1,0,0},{0,0,-1},{0,-1,0},{-1,0,0},{0,0,2},{0,1,1},{0,2,0},{1,0,1},{1,1,0},{2,0,0},{0,0,-2},{0,1,-1},{1,0,-1},{0,-2,0},{0,-1,1},{1,-1,0},{0,-1,-1},{-2,0,0},{-1,0,1},{-1,1,0},{-1,0,-1},{-1,-1,0},{0,1,2},{0,2,1},{1,0,2},{1,1,1},{1,2,0},{2,0,1},{2,1,0},{0,1,-2},{0,2,-1},{1,0,-2},{1,1,-1},{2,0,-1},{0,-2,1},{0,-1,2},{1,-2,0},{1,-1,1},{2,-1,0},{0,-2,-1},{0,-1,-2},{1,-1,-1},{-2,0,1},{-2,1,0},{-1,0,2},{-1,1,1},{-1,2,0},{-2,0,-1},{-1,0,-2},{-1,1,-1},{-2,-1,0},{-1,-2,0},{-1,-1,1},{-1,-1,-1},{0,2,2},{1,1,2},{1,2,1},{2,0,2},{2,1,1},{2,2,0},{0,2,-2},{1,1,-2},{1,2,-1},{2,0,-2},{2,1,-1},{0,-2,2},{1,-2,1},{1,-1,2},{2,-2,0},{2,-1,1},{0,-2,-2},{1,-2,-1},{1,-1,-2},{2,-1,-1},{-2,0,2},{-2,1,1},{-2,2,0},{-1,1,2},{-1,2,1},{-2,0,-2},{-2,1,-1},{-1,1,-2},{-1,2,-1},{-2,-2,0},{-2,-1,1},{-1,-2,1},{-1,-1,2},{-2,-1,-1},{-1,-2,-1},{-1,-1,-2},{1,2,2},{2,1,2},{2,2,1},{1,2,-2},{2,1,-2},{2,2,-1},{1,-2,2},{2,-2,1},{2,-1,2},{1,-2,-2},{2,-2,-1},{2,-1,-2},{-2,1,2},{-2,2,1},{-1,2,2},{-2,1,-2},{-2,2,-1},{-1,2,-2},{-2,-2,1},{-2,-1,2},{-1,-2,2},{-2,-2,-1},{-2,-1,-2},{-1,-2,-2},{2,2,2},{2,2,-2},{2,-2,2},{2,-2,-2},{-2,2,2},{-2,2,-2},{-2,-2,2},{-2,-2,-2}}
In[3]:=
 | $CrystalData |
Out[3]=
Fmm
3
In[4]:=
extinct=
["Fmm",hkl]
| SystematicAbsentQ |
3
Out[4]=
{True,True,True,True,True,True,False,True,False,True,True,False,False,True,True,False,True,True,True,False,True,True,True,True,True,True,True,False,True,True,True,True,True,True,False,True,True,True,True,False,True,True,True,False,True,True,True,False,True,True,True,False,True,True,False,False,False,True,True,False,True,False,False,True,True,False,True,False,True,True,False,True,False,True,True,True,False,True,False,True,True,False,True,True,True,False,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,False,False,False,False,False,False,False,False}
In[5]:=
hkl=Pick[hkl,extinct,False]
Out[5]=
{{0,0,2},{0,2,0},{2,0,0},{0,0,-2},{0,-2,0},{-2,0,0},{1,1,1},{1,1,-1},{1,-1,1},{1,-1,-1},{-1,1,1},{-1,1,-1},{-1,-1,1},{-1,-1,-1},{0,2,2},{2,0,2},{2,2,0},{0,2,-2},{2,0,-2},{0,-2,2},{2,-2,0},{0,-2,-2},{-2,0,2},{-2,2,0},{-2,0,-2},{-2,-2,0},{2,2,2},{2,2,-2},{2,-2,2},{2,-2,-2},{-2,2,2},{-2,2,-2},{-2,-2,2},{-2,-2,-2}}
In[6]:=
| MergeSymmetryEquivalentReflections |
3
Out[6]=
{{0,0,2},{1,1,1},{0,2,2},{2,2,2}}
Two comments before we proceed further: We chose to remove the extinct reflections before we merged the symmetry equivalents, but this could have been done in the opposite order. The other thing is that we could also had used the name of the crystal directly instead of its space group. In the next lines we do this, and see that we end up with the same four reflections , , and :
002
111
022
222
In[7]:=
hkl=
[2];hkl=
["SodiumChloride",hkl]
 | ReflectionList |
| MergeSymmetryEquivalentReflections |
Out[8]=
{{0,0,1},{0,0,2},{0,1,1},{0,1,2},{1,1,1},{0,2,2},{1,1,2},{1,2,2},{2,2,2}}
In[9]:=
extinct=
["SodiumChloride",hkl]
| SystematicAbsentQ |
Out[9]=
{True,False,True,True,False,False,True,True,False}
In[10]:=
Pick[hkl,extinct,False]
Out[10]=
{{0,0,2},{1,1,1},{0,2,2},{2,2,2}}
Another thing to consider is the Bragg angles of the reflections. The wavelength will limit how far out in reciprocal space we will see reflections. For example:
Given the wavelength (), we see that the reflection is OK, but the higher indices of the reflection yields an undefined angle:
λ=1.54Å
Cu
K
α1
335
557
In[11]:=
| BraggAngle |
Out[11]=
{,Undefined}
63.541
°
Thus, we also need to run our reflections through this check. Since we only generated reflections with a maximum index , we won't have this problem with the four reflections we got. This begs another question: Which maximum index is appropriate for our crystal, given a certain wavelength? Luckily, the function does this and above steps automatically.
2
Generating reflections for sodium chloride, given the wavelength :
λ=1.54Å
In[12]:=
| ReflectionList |
Out[12]=
{{0,0,2},{1,1,1},{0,0,4},{0,2,2},{1,1,3},{0,2,4},{2,2,2},{1,1,5},{1,3,3},{0,4,4},{2,2,4},{1,3,5},{3,3,3},{2,4,4}}
You may have notices a progress bar during the computations. If we have a crystal with low symmetry or a very small wavelength, the process could take a little while.
A crystal with low symmetry (space group #14):
In[13]:=
| ReflectionList |
Out[13]//Shallow=
{{0,0,2},{0,1,1},{0,2,0},{1,0,1},{1,1,0},{2,0,0},{-1,0,1},{0,1,2},{0,2,1},{1,1,1},127}
Relatively short wavelength (could take a little time):
In[14]:=
| ReflectionList |
Out[14]//Shallow=
{{0,0,2},{1,1,1},{0,0,4},{0,2,2},{1,1,3},{0,0,6},{0,2,4},{2,2,2},{1,1,5},{1,3,3},1039}
Formatting the reflections
Once we have generated reflections, we may wish to present them in for instance a table.
Twelve random reflections with maximum index :
7
In[15]:=
hkl=RandomChoice
[5],12
 | ReflectionList |
Out[15]=
{{3,0,0},{-5,1,-2},{2,4,-5},{2,-1,0},{4,0,5},{3,1,-4},{2,2,3},{4,5,1},{-3,-4,-5},{-3,-1,2},{5,5,2},{-1,3,0}}
We can make formatted versions of the reflections (normal crystallographic notation) with . Here the function is mapped over each reflection:
ToMiller
In[16]:=
miller=
/@hkl
| MillerNotationToString |
Out[16]=
{(300),(1),(24),(20),(405),(31),(223),(451),(),(2),(552),(30)}
5
2
5
1
4
3
4
5
3
1
1
Let us make a table over the reflections and their corresponding Bragg angles (note: we do not care about extinct reflections at the moment):
In[17]:=
angle=
["SodiumChloride",1.54,hkl]
| BraggAngle |
Out[17]=
{,,,,,,,,,,Undefined,}
24.1781
°
48.3982
°
66.3248
°
17.7749
°
60.9487
°
44.1183
°
34.257
°
62.2247
°
74.8788
°
30.7191
°
25.5774
°
Let us gather the data into one list, and also sort the data by increasing Bragg angle (while pushing the to the bottom):
Undefined
In[18]:=
data=Transpose[{miller,hkl,angle}];data=SortBy[data,{!QuantityQ@#〚3〛,Abs@#〚3〛}&];Grid[Prepend[data,{"(hkl)","{h,k,l}",""}],Dividers{{False,True},{False,True}}]
θ
B
Out[20]=
(hkl) | {h,k,l} | θ B |
(2 1 | {2,-1,0} | 17.7749 ° |
(300) | {3,0,0} | 24.1781 ° |
( 1 | {-1,3,0} | 25.5774 ° |
( 3 1 | {-3,-1,2} | 30.7191 ° |
(223) | {2,2,3} | 34.257 ° |
(31 4 | {3,1,-4} | 44.1183 ° |
( 5 2 | {-5,1,-2} | 48.3982 ° |
(405) | {4,0,5} | 60.9487 ° |
(451) | {4,5,1} | 62.2247 ° |
(24 5 | {2,4,-5} | 66.3248 ° |
( 3 4 5 | {-3,-4,-5} | 74.8788 ° |
(552) | {5,5,2} | Undefined |
If we want to clear the formatting of the Miller indices, we use :
FromMiller
In[21]:=
| MillerNotationToList |
Out[21]=
{{3,0,0},{-5,1,-2},{2,4,-5},{2,-1,0},{4,0,5},{3,1,-4},{2,2,3},{4,5,1},{-3,-4,-5},{-3,-1,2},{5,5,2},{-1,3,0}}
Working with formatted strings
Subscripts are made with the key combination +hyphen-minus and overbars are made by highlighting the digit character and pressing +7 followed by and underscore.