hikari.symmetry package

Submodules

hikari.symmetry.group module

This file contains class definition and necessary tools for constructing and evaluating all symmetry groups.

class hikari.symmetry.group.Group(*generators)

Bases: object

Base immutable class containing information about symmetry groups. It stores information for point and space groups and, among others, allows for iteration over its elements from hikari.symmetry.SymmOp.

AXIS_PRIORITY_RULES = '6>61>62>63>64>65>-6>4>41>42>43>-4>-3>3>31>32>2>21'

BRAVAIS_PRIORITY_RULES = 'A+B+C=F>R>I>C>B>A>H>P'

PLANE_PRIORITY_RULES = 'm>a+b=e>a+c=e>b+c=e>a>b>c>n>d'

class System(value)

Bases: Enum

Enumerator class with information about associated crystal system

cubic = 5

property directions

hexagonal = 6

monoclinic = 1

orthorhombic = 2

tetragonal = 4

triclinic = 0

trigonal = 3

property auto_generated_name

Name of the group generated automatically. Use only as approx.

property centering_symbol

classmethod create_manually(generators, operations)

Generate group using already complete list of generators and operators. :param generators: A complete list of group generators :type generators: List[np.ndarray] :param operations: A complete list of group operations :type operations: List[np.ndarray] :return: :rtype:

property generators

property is_achiral

property is_centrosymmetric

Returns

True if group has centre of symmetry; False otherwise.

Return type

bool

property is_chiral

property is_enantiogenic

Returns

True if determinant of all operations in group are positive.

Return type

bool

property is_polar

property is_sohncke

Returns

True if determinant of all operations in group are positive.

Return type

bool

property is_symmorphic

lauefy()

Returns

New PointGroup with centre of symmetry added to generators.

Return type

Group

property operations

property order

reciprocate()

property system

Returns

Predicted crystal system associated with this group

Return type

self.CrystalSystem()

transform(m)

Transform the group using 4x4 matrix. For reference, see bilbao resources or IUCr pamphlet no. 22.

Example

>>> import numpy
>>> from hikari.symmetry import SG
>>> matrix = numpy.array([(1,0,1,0),(0,1,0,0),(-1,0,0,0),(0,0,0,1)])
>>> SG['P21/c'].transform(matrix).auto_generated_name
P 21/n

Parameters

m (np.ndarray) – A 4x4 array containing information about new base and origin.

Returns

Group with new, transformed basis and origin.

Return type

Group

hikari.symmetry.operations module

This file contains class and definitions of 3-dimensional symmetry operations.

class hikari.symmetry.operations.SymmOp(transformation, translation=array([0, 0, 0]))

Bases: object

Class storing information about symmetry operations, with clear string representation and intuitive syntax for combining / utilising operations: “=” to compare two symmetry operations for logical equivalence, “*” to combine two symmetry operations or transform a vector, “** n” to apply symmetry operation n times, “% n” to restrict symmetry operation to n unit cells. Some of the functions may work incorrectly for rhombohedral unit cells #TODO

class Type(value)

Bases: Enum

Enumerator class storing information about type of operation

identity = 3

inversion = 0

reflection = 2

rotation = 1

rotoinversion = 4

rototranslation = -1

transflection = -2

translation = -3

at(point)

Transform operation as little as possible so that its symmetry element contains “point”. To be used after “into” if used together.

Parameters

point (np.array) – Target coordinates of point which should lie in element

Returns

New symmetry operation which contains “point” in its element

Return type

SymmOp

property code

property det

Returns

determinant of 3x3 transformation part of operation’s matrix

Return type

int

extincts(hkl)

Return boolean array with truth whenever reflection should be extinct

Parameters

hkl (np.ndarray) – An array containing one hkl or multiple hkls in columns

Returns

array of booleans, where extinct reflections are marked as True

Return type

np.array

property fold: int

Number of times operation must be repeated to become identity, inversion or translation: n for n-fold axes, 2 for reflections, 1 for other (max 6)

classmethod from_code(code)

Create new symmetry operation using symmetry code “x’,y’,z’” containing transformation of each individual coordinate from x,y,z to x’,y’,z’.

Parameters

code (str) – string representing new coordinates after operation

Returns

Symmetry operation generated from given coordinate triplet code

Return type

SymmOp

classmethod from_matrix(matrix)

Create new symmetry operation using augmented 4x4 transformation matrix

Parameters

matrix (np.ndarray) – augmented 4x4 matrix

Returns

Symmetry operation generated based on augmented matrix

Return type

SymmOp

classmethod from_pair(matrix, vector)

Create new symmetry operation using point transformation 3x3 matrix and 3-length translation vector. Alias for standard creation method.

Parameters

• matrix (np.ndarray) – 3x3 point transformation matrix

• vector (np.ndarray) – 3-length translation vector

Returns

Symmetry operation generated based on matrix - vector pair

Return type

SymmOp

property glide: ndarray

Part of the translation vector stemming from operations’ glide

property glide_fold: int

Number of types glide component of the operation must be repeated in order to contain only integer values, eg.: 3 for “6_2”, 4 for “d”

into(direction, hexagonal=False)

Rotate operation so that its orientation changes to “direction”, while preserving fractional glide. To be used before respective “at” method. Will most likely not work for unimplemented rhombohedral unit cells.

Parameters

• direction (np.ndarray) – Target orientation for element of symmetry operation

• hexagonal (bool) – True if operation is defined in hexagonal coordinates

Returns

New symmetry operation whose orientation is “direction”

Return type

SymmOp

property invariants

Returns

List of directions not affected by this symmetry operation

Return type

list[np.ndarray]

property matrix

Returns

Augmented 4 x 4 transformation matrix with float-type values

Return type

np.ndarray

property name

Returns

short name of symmetry operation, eg.: “m”, “3” or “2_1”

Return type

str

property order: int

Number of times operation has to be repeated to become a translation, eg.: n for all n-fold axes, 2 for other (max 6)

property orientation

Returns

Direction of symmetry element (if can be defined) else None

Return type

Union[np.ndarray, None]

property origin: ndarray

Selected point invariant to the symmetry operation

property reciprocal

Returns

relevant symmetry operation in reciprocal hkl space

Return type

SymmOp

property sense

Returns

“+” or “-”, the “sense” of rotation, as given in ITC A, 11.1.2

Return type

str

property tf

property tl

property trace

Returns

trace of 3x3 transformation part of operation’s matrix

Return type

transform(other)

Transform a column containing rows of coordinate points

Parameters

other (np.ndarray) – A vertical numpy array of coordinate triplets kept in rows

Returns

Same-shaped array of coordinate triplets transformed by self

Return type

np.ndarray

property translational

Returns

True if operation has any glide component, False otherwise

Return type

bool

property typ

Returns

crystallographic type of this symmetry operation

Return type

SymmOp.Type

hikari.symmetry.point_groups module

Dictionary containing all known point groups written as Group along with alternative axis settings. The point groups in this dictionary can be accessed using their short Hermann-Maugin notation, as presented below.

No.	CRYSTAL SYSTEM	Hermann-Maugin notation	Schoenflies notation	Can be accessed using
1	triclinic	1	C1	PG[‘1’]
2		-1	Ci	PG[‘-1’]
3	monoclinic	2	C2	PG[‘2’]
4		m	Cs	PG[‘m’]
5		2/m	C2h	PG[‘2/m’]
6	orthorhombic	222	D2	PG[‘222’]
7		mm2	C2v	PG[‘mm2’]
8		mmm	D2h	PG[‘mmm’]
9	tetragonal	4	C4	PG[‘4’]
10		-4	S4	PG[‘-4’]
11		4/m	C4h	PG[‘4/m’]
12		422	D4	PG[‘422’]
13		4mm	C4v	PG[‘4mm’]
14		-42m	D2d	PG[‘-42m’]
		-4m2	D2d	PG[‘-4m2’]
15		4/mmm	D4h	PG[‘4/mmm’]
16	trigonal	3	C3	PG[‘3’]
17		-3	C3i	PG[‘-3’]
18		321	D3	PG[‘321’]
		312	D3	PG[‘312’]
19		3m1	C3v	PG[‘3m1’]
		31m	C3v	PG[‘31m’]
20		-3m1	D3d	PG[‘-3m1’]
		-31m	D3d	PG[‘-31m’]
21	hexagonal	6	C6	PG[‘6’]
22		-6	C3h	PG[‘-6’]
23		6/m	C6h	PG[‘6/m’]
24		622	D6	PG[‘622’]
25		6mm	C6v	PG[‘6mm’]
26		-6m2	D3h	PG[‘-6m2’]
		-62m	D3h	PG[‘-62m’]
27		6/mmm	D6h	PG[‘6/mmm’]
28	cubic	23	T	PG[‘23’]
29		m-3	Th	PG[‘m-3’]
30		432	O	PG[‘432’]
31		-43m	Td	PG[‘-43m’]
32		m-3m	Oh	PG[‘m-3m’]

hikari.symmetry.space_groups module

Dictionary containing all known space groups written as Group along with alternative axis settings. The point groups in this dictionary can be accessed using their short Hermann-Maugin notation, as presented below. To access origin choice 1 or 2, append key with “#1” or “2”, eg.: SG[“Fd-3c#2”].

No.	CRYSTAL SYSTEM	Hermann-Maugin short notation	Additional notes	Can be accessed using
1	triclinic	P1		SG[‘P1’]
2		P-1		SG[‘P-1’]
3	monoclinic	P2	unique axis b	SG[‘P2’]
4		P21	unique axis b	SG[‘P21’]
5		C2	unique axis b	SG[‘C2’]
6		Pm	unique axis b	SG[‘Pm’]
7		Pc	unique axis b	SG[‘Pc’]
8		Cm	unique axis b	SG[‘Cm’]
9		Cc	unique axis b	SG[‘Cc’]
10		P2/m	unique axis b	SG[‘P2/m’]
11		P21/m	unique axis b	SG[‘P21/m’]
12		C2/m	unique axis b	SG[‘C2/m’]
13		P2/c	unique axis b	SG[‘P2/c’]
14		P21/c	unique axis b	SG[‘P21/c’]
		P1121/a	z,x,y: c-unique	SG[‘P1121/a’]
15		C2/c	unique axis b	SG[‘C2/c’]
16	orthorhombic	P222		SG[‘P222’]
17		P2221		SG[‘P2221’]
18		P21212		SG[‘P21212’]
19		P212121		SG[‘P212121’]
20		C2221		SG[‘C2221’]
21		C222		SG[‘C222’]
22		F222		SG[‘F222’]
23		I222		SG[‘I222’]
24		I212121		SG[‘I212121’]
25		Pmm2		SG[‘Pmm2’]
26		Pmc21		SG[‘Pmc21’]
27		Pcc2		SG[‘Pcc2’]
28		Pma2		SG[‘Pma2’]
29		Pca21		SG[‘Pca21’]
30		Pnc2		SG[‘Pnc2’]
31		Pmn21		SG[‘Pmn21’]
32		Pba2		SG[‘Pba2’]
33		Pna21		SG[‘Pna21’]
		Pn21a	x,z,-y orientation	SG[‘Pn21a’]
34		Pnn2		SG[‘Pnn2’]
35		Cmm2		SG[‘Cmm2’]
36		Cmc21		SG[‘Cmc21’]
37		Ccc2		SG[‘Ccc2’]
38		Amm2		SG[‘Amm2’]
39		Aem2		SG[‘Aem2’]
40		Ama2		SG[‘Ama2’]
41		Aea2		SG[‘Aea2’]
42		Fmm2		SG[‘Fmm2’]
43		Fdd2		SG[‘Fdd2’]
44		Imm2		SG[‘Imm2’]
45		Iba2		SG[‘Iba2’]
46		Ima2		SG[‘Ima2’]
47		Pmmm		SG[‘Pmmm’]
48		Pnnn	origin at 222	SG[‘Pnnn’]
			origin at -1	SG[‘Pnnn#2’]
49		Pccm		SG[‘Pccm’]
50		Pban	origin at 222/n	SG[‘Pban’]
			origin at -1	SG[‘Pban#2’]
51		Pmma		SG[‘Pmma’]
52		Pnna		SG[‘Pnna’]
53		Pmna		SG[‘Pmna’]
54		Pcca		SG[‘Pcca’]
55		Pbam		SG[‘Pbam’]
56		Pccn		SG[‘Pccn’]
57		Pbcm		SG[‘Pbcm’]
58		Pnnm		SG[‘Pnnm’]
59		Pmmn	origin at mm2/n	SG[‘Pmmn’]
			origin at -1	SG[‘Pmmn#2’]
60		Pbcn		SG[‘Pbcn’]
61		Pbca		SG[‘Pbca’]
62		Pnma		SG[‘Pnma’]
63		Cmcm		SG[‘Cmcm’]
64		Cmca		SG[‘Cmca’]
65		Cmmm		SG[‘Cmmm’]
66		Cccm		SG[‘Cccm’]
67		Cmme		SG[‘Cmme’]
68		Ccce	origin at 222	SG[‘Ccce’]
			origin at -1	SG[‘Ccce#2’]
69		Fmmm		SG[‘Fmmm’]
70		Fddd	origin at 222	SG[‘Fddd’]
			origin at -1	SG[‘Fddd#2’]
71		Immm		SG[‘Immm’]
72		Ibam		SG[‘Ibam’]
73		Ibca		SG[‘Ibca’]
74	Imma		SG[‘Imma’]
75	tetragonal	P4		SG[‘P4’]
76		P41		SG[‘P41’]
77		P42		SG[‘P42’]
78		P43		SG[‘P43’]
79		I4		SG[‘I4’]
80		I41		SG[‘I41’]
81		P-4		SG[‘P-4’]
82		I-4		SG[‘I-4’]
83		P4/m		SG[‘P4/m’]
84		P42/m		SG[‘P42/m’]
85		P4/n	origin at -4 on n	SG[‘P4/n’]
			origin at -1 on n	SG[‘P4/n#2’]
86		P42/n	origin at -4	SG[‘P42/n’]
			origin at -1 on n	SG[‘P42/n#2’]
87		I4/m		SG[‘I4/m’]
88		I41/a	origin at -4	SG[‘I41/a’]
			origin at -1 on b	SG[‘I41/a#2’]
89		P422		SG[‘P422’]
90		P4212		SG[‘P4212’]
91		P4122		SG[‘P4122’]
92		P41212		SG[‘P41212’]
93		P4222		SG[‘P4222’]
94		P42212		SG[‘P42212’]
95		P4322		SG[‘P4322’]
96		P43212		SG[‘P43212’]
97		I422		SG[‘I422’]
98		I4122		SG[‘I4122’]
99		P4mm		SG[‘P4mm’]
100		P4bm		SG[‘P4bm’]
101		P42cm		SG[‘P42cm’]
102		P42nm		SG[‘P42nm’]
103		P4cc		SG[‘P4cc’]
104		P4nc		SG[‘P4nc’]
105		P42mc		SG[‘P42mc’]
106		P42bc		SG[‘P42bc’]
107		I4mm		SG[‘I4mm’]
108		I4cm		SG[‘I4cm’]
109		I41md		SG[‘I41md’]
110		I41cd		SG[‘I41cd’]
111		P-42m		SG[‘P-42m’]
112		P-42c		SG[‘P-42c’]
113		P-421m		SG[‘P-421m’]
114		P-421c		SG[‘P-421c’]
115		P-4m2		SG[‘P-4m2’]
116		P-4c2		SG[‘P-4c2’]
117		P-4b2		SG[‘P-4b2’]
118		P-4n2		SG[‘P-4n2’]
119		I-4m2		SG[‘I-4m2’]
120		I-4c2		SG[‘I-4c2’]
121		I-42m		SG[‘I-42m’]
122		I-42d		SG[‘I-42d’]
123		P4/mmm		SG[‘P4/mmm’]
124		P4/mcc		SG[‘P4/mcc’]
125		P4/nbm	origin at 4/22	SG[‘P4/nbm’]
			origin at 2/m	SG[‘P4/nbm#2’]
126		P4/nnc	origin at 422/n	SG[‘P4/nnc’]
			origin at -1	SG[‘P4/nnc#2’]
127		P4/mbm		SG[‘P4/mbm’]
128		P4/mnc		SG[‘P4/mnc’]
129		P4/nmm	origin at -4m2	SG[‘P4/nmm’]
			origin at 2/m	SG[‘P4/nmm#2’]
130		P4/ncc	origin at -4/ncn	SG[‘P4/ncc’]
			origin at -1	SG[‘P4/ncc#2’]
131		P42/mmc		SG[‘P42/mmc’]
132	P42/mcm		SG[‘P42/mcm’]
133	P42/nbc	origin at -4121/c	SG[‘P42/nbc’]
		origin at -1	SG[‘P42/nbc#2’]
134	P42/nnm	origin at -42m	SG[‘P42/nnm’]
		origin at 2/m	SG[‘P42/nnm#2’]
135	P42/mbc		SG[‘P42/mbc’]
136	P42/mnm		SG[‘P42/mnm’]
137	P42/nmc	origin at -4m2/n	SG[‘P42/nmc’]
		origin at -1	SG[‘P42/nmc#2’]
138	P42/ncm	origin at -4cg	SG[‘P42/ncm’]
		origin at 2/m	SG[‘P42/ncm#2’]
139	I4/mmm		SG[‘I4/mmm’]
140	I4/mcm		SG[‘I4/mcm’]
141	I41/amd	origin at -4m2	SG[‘I41/amd’]
		origin at 2/m	SG[‘I41/amd#2’]
142	I41/acd	origin at -4c21	SG[‘I41/acd’]
		origin at -1	SG[‘I41/acd#2’]
143	trigonal	P3		SG[‘P3’]
144		P31		SG[‘P31’]
145		P32		SG[‘P32’]
146		R3	hexagonal axes	SG[‘R3’]
147		P-3		SG[‘P-3’]
148		R-3	hexagonal axes	SG[‘R-3’]
149		P312		SG[‘P312’]
150		P321		SG[‘P321’]
151		P3112		SG[‘P3112’]
152		P3121		SG[‘P3121’]
153		P3212		SG[‘P3212’]
154		P3221		SG[‘P3221’]
155		R32	hexagonal axes	SG[‘R32’]
156		P3m1		SG[‘P3m1’]
157		P31m		SG[‘P31m’]
158		P3c1		SG[‘P3c1’]
159		P31c		SG[‘P31c’]
160		R3m	hexagonal axes	SG[‘R3m’]
161		R3c	hexagonal axes	SG[‘R3c’]
162		P-31m		SG[‘P-31m’]
163		P-31c		SG[‘P-31c’]
164		P-3m1		SG[‘P-3m1’]
165		P-3c1		SG[‘P-3c1’]
166		R-3m	hexagonal axes	SG[‘R-3m’]
167		R-3c	hexagonal axes	SG[‘R-3c’]
168	hexagonal	P6		SG[‘P6’]
169		P61		SG[‘P61’]
170		P65		SG[‘P65’]
171		P62		SG[‘P62’]
172		P64		SG[‘P64’]
173		P63		SG[‘P63’]
174		P-6		SG[‘P-6’]
175		P6/m		SG[‘P6/m’]
176		P63/m		SG[‘P63/m’]
177		P622		SG[‘P622’]
178		P6122		SG[‘P6122’]
179		P6522		SG[‘P6522’]
180		P6222		SG[‘P6222’]
181		P6422		SG[‘P6422’]
182		P6322		SG[‘P6322’]
183		P6mm		SG[‘P6mm’]
184		P6cc		SG[‘P6cc’]
185		P63cm		SG[‘P63cm’]
186		P63mc		SG[‘P63mc’]
187		P-6m2		SG[‘P-6m2’]
188		P-6c2		SG[‘P-6c2’]
189		P-62m		SG[‘P-62m’]
190		P-62c		SG[‘P-62c’]
191		P6/mmm		SG[‘P6/mmm’]
192		P6/mcc		SG[‘P6/mcc’]
193		P63/mcm		SG[‘P63/mcm’]
194		P63/mmc		SG[‘P63/mmc’]
195	cubic	P23		SG[‘P23’]
196		F23		SG[‘F23’]
197		I23		SG[‘I23’]
198		P213		SG[‘P213’]
199		I213		SG[‘I213’]
200		Pm-3		SG[‘Pm-3’]
201		Pn-3	origin at 23	SG[‘Pn-3’]
			origin at -3	SG[‘Pn-3#2’]
202		Fm-3		SG[‘Fm-3’]
203		Fd-3	origin at 23	SG[‘Fd-3’]
			origin at -3	SG[‘Fd-3#2’]
204		Im-3		SG[‘Im-3’]
205		Pa-3		SG[‘Pa-3’]
206		Ia-3		SG[‘Ia-3’]
207		P432		SG[‘P432’]
208		P4232		SG[‘P4232’]
209		F432		SG[‘F432’]
210		F4132		SG[‘F4132’]
211		I432		SG[‘I432’]
212		P4332		SG[‘P4332’]
213		P4132		SG[‘P4132’]
214		I4132		SG[‘I4132’]
215		P-43m		SG[‘P-43m’]
216		F-43m		SG[‘F-43m’]
217		I-43m		SG[‘I-43m’]
218		P-43n		SG[‘P-43n’]
219		F-43c		SG[‘F-43c’]
220		I-43d		SG[‘I-43d’]
221		Pm-3m		SG[‘Pm-3m’]
222		Pn-3n	origin at 432	SG[‘Pn-3n’]
			origin at -3	SG[‘Pn-3n#2’]
223		Pm-3n		SG[‘Pm-3n’]
224		Pn-3m	origin at -43m	SG[‘Pn-3m’]
			origin at -3m	SG[‘Pn-3m#2’]
225		Fm-3m		SG[‘Fm-3m’]
226		Fm-3c		SG[‘Fm-3c’]
227		Fd-3m	origin at -43m	SG[‘Fd-3m’]
			origin at -3m	SG[‘Fd-3m#2’]
228		Fd-3c	origin at 23	SG[‘Fd-3c’]
			origin at -3	SG[‘Fd-3c#2’]
229		Im-3m		SG[‘Im-3m’]
230		Ia-3d		SG[‘Ia-3d’]

Module contents

Module containing symmetry operations and point groups to be used by other objects.