The GAP package CrystKit

Description

CrystKit is a GAP package designed for the manipulation and analysis of crystallographic groups. It offers a variety of functionalities, including space group simplification on any basis, minimal generating set computation for solvable space groups, isomorphism detection between space groups, enantiomorphic pair existence checking, efficient orbit computation for the space group vector system (SNoT), and rapid identification of space groups (up to six-dimension) based on CARAT. It also encompasses the calculation of allowed orders of point groups in any dimensional space groups (an extension of the crystallographic restriction theorem) and conjugacy determination between two finite matrix groups.

Main Features

Simplification of space groups represented on any basis: simplification to more simple generator forms.

SGGenSet227me:=[ [ [ 0, -1, 0, 1/2 ], [ 0, 0, -1, 1/2 ], [ -1, 0, 0, 1/2 ], [ 0, 0, 0, 1 ] ], [ [ -15/4, 29/4, -15/4, -15/16 ], [ -33/8, 55/8, -25/8, -25/32 ], 
      [ -25/8, 55/8, -33/8, -41/32 ], [ 0, 0, 0, 1 ] ] ];
S:=AffineCrystGroupOnLeft(SGGenSet227me);
conj1:=ConjugatorSpaceGroupSimplification(S);
S1:=S^(conj1^-1);
GeneratorsOfGroup(S1);

Minimal generating set calculation for solvable space groups: e.g., identifying the minimal number of generators for 3D space group 227 as follows.

S2:=SpaceGroupOnLeftIT(3,227);
Length(GeneratorsOfGroup(S2));
minSgen:=MinimalGeneratingSetAffineCrystGroup(S2);
Length(minSgen.mgs);

Isomorphism determination between space groups represented on any basis: checking isomorphism between two space groups.

AffineIsomorphismSpaceGroups(S,S2);
S^(last^-1)=S2;

Existence check for enantiomorphic pairs in any basis represented space groups: finding the conjugator connecting its enantiomorphic partners for a given space group.

S4:=SpaceGroupOnLeftIT(3,212);
conj2:=ConjugatorSpaceGroupEnantiomorphicPartner(S4);
S5:=S4^(conj2^-1);

Efficient calculation of orbits of the space group vector system (SNoT): orbit calculation using the normalizer of the point group.

P4:=PointGroup(S4);
d:= DimensionOfMatrixGroup(S4) - 1;
norm := GeneratorsOfGroup(Normalizer(GL(d,Integers), P4)); 
orbnpg := OrbitSpaceGroupStdByNormalizerPointGroup(S4, GeneratorsOfGroup(P4), norm);

Rapid identification of space groups (up to six dimensions) based on any basis (using CARAT): quick identification method for space groups.

IdentifySpaceGroup(S4);
IdentifySpaceGroup(S5);

Allowed orders calculation for point groups in any dimensional space groups: the crystallographic restriction theorem in higher dimensions by calculating new finite orders of GLNZ.

NewFiniteOrdersOfGLNZ(2);
NewFiniteOrdersOfGLNZ(4);

Conjugacy determination between two finite matrix Groups (using RepnDecomp): e.g., determines if two point groups are conjugate.

P:=PointGroup(S);;
S6:=SpaceGroupOnLeftIT(3,222);;
P6:=PointGroup(S6);;
ConjugatorMatrixGroups(P, P6); 

Install

CrystKit is distributed with GAP, and does not require any installation. It is loaded with the GAP command

gap> LoadPackage( "CrystKit" ); 

Contact

For bug reports, suggestions and other comments please use the issue tracker on the GitHub page of the package:

https://github.com/hongyi-zhao/CrystKit/issues

For further information, contact us at: hongshengzhao@hevute.edu.cn or hongyi.zhao@gmail.com.