Explore Symmetries of Crystal Systems

Get symmetry, edge and angle properties of crystal systems using the Entity framework

Use

"CrystalSystem"

with EntityList to get a list of the seven named crystal systems in three dimensions:

In[1]:=

EntityList["CrystalSystem"]

Out[1]=



cubic

hexagonal

monoclinic

orthorhombic

tetragonal

triclinic

trigonal



In a crystal system, a set of point groups and their corresponding space groups are assigned to a lattice system. In three dimensions, the crystal families and systems are identical except that the hexagonal and trigonal crystal systems are combined into the single hexagonal crystal family:

In[2]:=

EntityList["CrystalFamily"]

Out[2]=



cubic

hexagonal

monoclinic

orthorhombic

tetragonal

triclinic



Obtain a list of properties available for each crystal system and family:

In[3]:=

EntityValue["CrystalSystem","Properties"]

Out[3]=



alternate names

angle relations

Bravais lattice

crystal family

list of point group standard names in each crystal system

list of space group standard names in each crystal system

edge relations

entity classes

entity type list

lattice systems

name

required point group symmetry

unit cell volume



Get a list of symmetry properties:

In[4]:=

symmetryprops={"Entity","RequiredPointGroupSymmetries","CrystallographicPointGroups","CrystallographicSpaceGroups","BravaisLattices","LatticeSystems"};

Create a table summarizing the symmetry properties of crystal systems using the built-in data:

In[5]:=

symmetryheader=StringReplace[Prepend[CommonName[EntityProperty["CrystalFamily",#]&/@Rest[symmetryprops]],"crystam system"],"crystallographic """];

In[6]:=

TextGridPrepend[MapAt[Length,EntityValue["CrystalSystem",symmetryprops],{All,3;;5}],symmetryheader],



Out[6]=

crystam system	required point group symmetry	point groups	space groups	Bravais lattice	lattice systems
cubic	4 3-fold rotation axes	5	36	3	 cubic 
hexagonal	1 6-fold rotation axis	7	27	1	 hexagonal 
monoclinic	1 2-fold rotation axis or 1 mirror plane	3	13	2	 monoclinic 
orthorhombic	3 2-fold rotation axes or 1 2-fold rotation axis and 2 mirror planes	3	59	4	 orthorhombic 
tetragonal	1 4-fold rotation axis	7	68	2	 tetragonal 
triclinic	none	2	2	1	 triclinic 
trigonal	1 3-fold rotation axis	5	25	2	 rhombohedral , hexagonal 

Similarly, create a concise table of crystal system edge and angle relations:

In[7]:=

edgeangleprops={"Entity","EdgeRelations","AngleRelations"};images=ImageResize[#,Small]&/@EntityValue[Select[EntityList["Lattice"],StringMatchQ[CanonicalName[#],"Simple*"]&],"Image"];

In[8]:=

edgeangledata=ReplaceMapThreadPrepend,EntityValue["CrystalSystem",edgeangleprops]/.

...

,images,

;

In[9]:=

TextGridPrependedgeangledata,

headers

,

style



Out[9]=

images

crystal system

edge relations

angle relations

cubic

a.b.c.

q.p.r.

hexagonal

a.≠c.

q.

&&r.

120

monoclinic

a.≠b.≠c.

q.≠

&&p.r.

orthorhombic

a.≠b.≠c.

q.p.r.

tetragonal

a.b.&&b.≠c.

q.p.r.

triclinic

a.≠b.≠c.

q.≠p.≠r.

trigonal

a.b.c.

a.≠c.

q.p.r.&&r.≠

q.

&&r.

120

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Contributed by: Wolfram Staff

Explore Symmetries of Crystal Systems

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