INTERNATIONAL SCHOOL ON FUNDAMENTAL CRYSTALLOGRAPHY

SPACE-GROUP SYMMETRY (short overview) Mois I. Aroyo Universidad del Pais Vasco, Bilbao, Spain

SPACE GROUPS Crystal pattern: infinite, idealized crystal structure (without disorder, dislocations, impurities, etc.) Space group G: The set of all symmetry operations (isometries) of a crystal pattern Translation subgroup H G: The infinite set of all translations that are symmetry operations of the crystal pattern Point group of the space groups PG: The factor group of the space group G with respect to the translation subgroup T: PG G/H

SYMMETRY OPERATIONS AND THEIR MATRIX-COLUMN PRESENTATION

Mappings and symmetry operations Definition: A mapping of a set A into a set B is a relation such that for each element a A there is a unique element b B which is assigned to a. The element b is called the image of a. An isometry leaves all distances and angles invariant. An ʻisometry of the first kindʼ, preserving the counter clockwise sequence of the edges ʻshort middle longʼ of the triangle is displayed in the upper mapping. An ʻisometry of the second kindʼ, changing the counter clockwise sequence of the edges of the triangle to a clockwise one is seen in the lower mapping. A parallel shift of the triangle is called a translation. Translations are special isometries. They play a distinguished role in crystallography.

Example: Matrix presentation of symmetry operation Mirror symmetry operation Mirror line my at 0,y Matrix representation drawing: M.M. Julian Foundations of Crystallography ctaylor & Francis, 2008 my x = -x y y = -1 1 x y Fixed points my x f yf = x f yf det -1 1 =? tr -1 1 =?

Description of isometries coordinate system: {O, a, b, c} isometry: X X ~ (x,y,z) (x,y,z) ~ ~ ~ ~ x = F1(x,y,z)

Matrix notation for system of linear equations

Matrix-column presentation of isometries linear/matrix part translation column part matrix-column Seitz symbol pair

EXERCISES Problem Referred to an orthorhombic coordinated system (a b c; α=β=γ=90) two symmetry operations are represented by the following matrix-column pairs: ( ) ( ) -1 0-1 1/2 (W1,w1)= 1 0 (W2,w2)= 1 0-1 0-1 1/2 Determine the images Xi of a point X under the symmetry operations (Wi,wi) where Can you guess what is the geometric nature of (W1,w1)? And of (W2,w2)? X= 0,70 0,31 0,95 Hint: A drawing could be rather helpful

EXERCISES Problem Characterization of the symmetry operations: det( ) tr( ) -1-1 1 =? 1 =? -1-1 What are the fixed points of (W1,w1) and (W2,w2)? ( -1 1-1 1/2 0 1/2 )xf yf zf = xf yf zf

Short-hand notation for the description of isometries (W,w) isometry: X X ~ notation rules: -left-hand side: omitted -coefficients 0, +1, -1 -different rows in one line examples: -1 1-1 1/2 0 1/2 { -x+1/2, y, -z+1/2 x+1/2, y, z+1/2

EXERCISES Problem 2.14 Construct the matrix-column pair (W,w) of the following coordinate triplets: (1) x,y,z (2) -x,y+1/2,-z+1/2 (3) -x,-y,-z (4) x,-y+1/2, z+1/2

Combination of isometries (U,u) X X ~ (W,w) (V,v) X ~

EXERCISES Problem Consider the matrix-column pairs of the two symmetry operations: (W1,w1)=( ) ( ) 0-1 0-1 1/2 1 0 0 (W2,w2)= 1 0 1 0-1 1/2 Determine and compare the matrix-column pairs of the combined symmetry operations: (W,w)=(W1,w1)(W2,w2) (W,w) =(W2,w2)(W1,w1) combination of isometries:

Inverse isometries X ~ X (W,w) (C,c)=(W,w) -1 X ~ I = 3x3 identity matrix (C,c)(W,w) = (I,o) o = zero translation column (C,c)(W,w) = (CW, Cw+c) CW=I Cw+c=o C=W -1 c=-cw=-w -1 w

EXERCISES Problem Determine the inverse symmetry operations (W1,w1) -1 and (W2,w2) -1 where (W1,w1)=( ) ( ) 0-1 0-1 1/2 1 0 0 (W2,w2)= 1 0 1-1 1/2 0 Determine the inverse symmetry operation (W,w) -1 (W,w)=(W1,w1)(W2,w2) inverse of isometries:

EXERCISES Problem Consider the matrix-column pairs

Matrix formalism: 4x4 matrices augmented matrices: point X point X :

4x4 matrices: general formulae point X point X : combination and inverse of isometries:

EXERCISES Problem 2.14 (cont) Construct the (4x4) matrix-presentation of the following coordinate triplets: (1) x,y,z (2) -x,y+1/2,-z+1/2 (3) -x,-y,-z (4) x,-y+1/2, z+1/2

Crystallographic symmetry operations Symmetry operations of an object The isometries which map the object onto itself are called symmetry operations of this object. The symmetry of the object is the set of all its symmetry operations. Crystallographic symmetry operations If the object is a crystal pattern, representing a real crystal, its symmetry operations are called crystallographic symmetry operations. The equilateral triangle allows six symmetry operations: rotations by 120 and 240 around its centre, reflections through the three thick lines intersecting the centre, and the identity operation.

Crystallographic symmetry operations characteristics: fixed points of isometries geometric elements (W,w)Xf=Xf Types of isometries preserve handedness identity: the whole space fixed translation t: no fixed point x = x + t rotation: one line fixed rotation axis φ = k 360 /N screw rotation: no fixed point screw axis screw vector

Crystallographic symmetry operations Screw rotation n-fold rotation followed by a fractional p translation t parallel n to the rotation axis Its application n times results in a translation parallel to the rotation axis

Types of isometries do not preserve handedness roto-inversion: centre of roto-inversion fixed roto-inversion axis inversion: centre of inversion fixed reflection: plane fixed reflection/mirror plane glide reflection: no fixed point glide plane glide vector

Crystallographic symmetry operations Glide plane reflection followed by a 1 2 fractional translation t parallel to the plane Its application 2 times results in a translation parallel to the plane

Matrix-column presentation of some symmetry operations Rotation or roto-inversion around the origin: ( ) W11 W12 W13 W21 W22 W23 W31 W32 W33 0 0 0 0 0 0 = 0 0 0 Translation: ( ) 1 1 1 w1 w2 w3 x y z = x+w1 y+w2 z+w3 Inversion through the origin: ( ) -1 0 x -x = -1 0 y -y -1 0 z -z

Geometric meaning of (W, w) W information (a) type of isometry order: W n =I rotation angle

EXERCISES Problem 2.14 (cont) Determine the type and order of isometries that are represented by the following matrix-column pairs: (1) x,y,z (2) -x,y+1/2,-z+1/2 (3) -x,-y,-z (4) x,-y+1/2, z+1/2 (a) type of isometry

EXERCISES Problem (cont.) Consider the matrix-column pairs Determine the type and order of isometries that are represented by the matrices A, B, C and D:

Geometric meaning of (W, w) W information Texto (b) axis or normal direction : (b1) rotations: = (b2) roto-inversions: reflections:

EXERCISES Problem 2.14 (cont) Determine the rotation or rotoinversion axes (or normals in case of reflections) of the following symmetry operations (2) -x,y+1/2,-z+1/2 (4) x,-y+1/2, z+1/2 rotations: Y(W) = W k-1 + W k-2 +... + W + I reflections: Y(-W) = - W + I

Geometric meaning of (W, w) W information (c) sense of rotation: for rotations or rotoinversions with k>2 det(z): x non-parallel to

Geometric meaning of (W, w) w -information (A) intrinsic translation part : glide or screw component t/k (A1) screw rotations: (A2) glide reflections: w

EXAMPLE Space group P21/c (No. 14) Geometric interpretation of (W,w) -screw or glide components? -fixed points? symmetry-elements diagram general-position diagram

Fixed points of isometries (W,w)Xf=Xf solution: point, line, plane or space no solution Examples: solution: ( ) 0 1 0-1 0 0 0 0 1 0 1/2 0 x y z = x y z What are the fixed points of this isometry? no solution: ( ) 0 1 0-1 0 0 0 0 1 0 0 1/2 x y z = x y z

Geometric meaning of (W,w) (W,w) n =(I,t) (W,w) n =(W n,(w n-1 +...+ W+I)w) =(I,t) (A) intrinsic translation part : glide or screw component t/k (A1) screw rotations: = (A2) glide reflections: w

Fixed points of (W,w) x F (B) location (fixed points ): (B1) t/k = 0: (B2) t/k 0:

(A) Fixed points of isometries (W,w)Xf=Xf (W,w)=( ) 0 1 0 0 Example: -1 0 0 1/2 0 0 1 1/2 ( (W,w) 4 = )t/k= 1 0 0 0 1 0 0 0 1 0 0 solution: point, line, plane or space no solution (W,w-t/k)= 0 0 2 1/2 What are the fixed points of this isometry? ( ) 0 1 0-1 0 0 0 0 1 0 1/2 0 (B) Y(W) = W k-1 + W k-2 +... + W + I 0-1 0-1 0 0 0 1 0 1 0 0 0 0 0 Y(W)= 1 0 0 + 0-1 0 + -1 0 0 + 0 1 0 = 0 0 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 4 W 3 W 2 W I

EXERCISES Problem 2.14 (cont.) Determine the intrinsic translation parts (if relevant) and the fixed points of the following symmetry operations (1) x,y,z (2) -x,y+1/2,-z+1/2 (3) -x,-y,-z (4) x,-y+1/2, z+1/2 screw rotations: Y(W) = W k-1 + W k-2 +... + W + I glide reflections: w fixed points:

EXAMPLE Space group P21/c (No. 14) Geometric interpretation of (W,w) -screw or glide components? -fixed points? symmetry-elements diagram general-position diagram

PRESENTATION OF SPACE-GROUP SYMMETRY IN INTERNATIONAL TABLES VOL. A

INTERNATIONAL TABLES FOR CRYSTALLOGRAPHY VOLUME A: SPACE-GROUP SYMMETRY Extensive tabulations and illustrations of the 17 plane groups and the 230 space groups headline with the relevant group symbols; diagrams of the symmetry elements and of the general position; specification of the origin and the asymmetric unit; list of symmetry operations; generators; general and special positions with multiplicities, site symmetries, coordinates and reflection conditions; symmetries of special projections; extensive subgroup and supergroup data

GENERAL LAYOUT: LEFT-HAND PAGE

General Layout: Right-hand page

HERMANN-MAUGUIN SYMBOLISM FOR SPACE GROUPS

Hermann-Mauguin symbols for space groups primary direction secondary direction tertiary direction

14 Bravais Lattices

Hermann-Mauguin symbols for space groups

SPACE-GROUP SYMMETRY OPERATIONS

EXAMPLE Space group Cmm2 (No. 35) Geometric interpretation glide plane, t=1/2a at y=1/4, b glide plane, t=1/2b at x=1/4, a General Position Matrix-column presentation of symmetry operations x+1/2,-y+1/2,z -x+1/2,y+1/2,z

Problem 2.15 (a) General Position Space group Cmm2 (No. 35) Matrix-column presentation of symmetry operations A) Characterize geometrically the matrix-column pairs listed under General position of the space group Cmm2 in ITA. B) Try to determine the matrix-column pairs of the symmetry operations whose symmetry elements are indicated on the unit-cell diagram Geometric interpretation

www.cryst.ehu.es

Crystallographic databases Group-subgroup relations Structural utilities Representations of point and space groups Solid-state applications

Crystallographic Databases International Tables for Crystallography

Bilbao Crystallographic Server Problem: Matrix-column presentation Geometrical interpretation GENPOS space group 35

Example GENPOS: Space group Cmm2(35) Matrix-column presentation of symmetry operations Geometric interpretation ITA data

Bilbao Crystallographic Server Problem: Geometric Interpretation of (W,w) SYMMETRY OPERATION

EXERCISES Problem 2.14 Construct the matrix-column pairs (W,w) of the following coordinate triplets: (1) x,y,z (2) -x,y+1/2,-z+1/2 (3) -x,-y,-z (4) x,-y+1/2, z+1/2 Characterize geometrically these matrix-column pairs taking into account that they refer to a monoclinic basis with unique axis b, Use the program SYMMETRY OPERATIONS for the geometric interpretation of the matrix-column pairs of the symmetry operations.

EXERCISES Problem 2.15 (b) 1. Characterize geometrically the matrix-column pairs listed under General position of the space group P4mm in ITA. 2. Consider the diagram of the symmetry elements of P4mm. Try to determine the matrix-column pairs of the symmetry operations whose symmetry elements are indicated on the unit-cell diagram. 3. Compare your results with the results of the program SYMMETRY OPERATIONS

SPACE GROUPS DIAGRAMS

Diagrams of symmetry elements three different settings permutations of a,b,c Diagram of general position points

Space group Cmm2 (No. 35) Diagram of symmetry elements conventional setting Diagram of general position points How many general position points per unit cell are there?

Example: P4mm Diagram of symmetry elements Diagram of general position points

Symmetry elements Symmetry elements } Geometric element + Element set } Fixed points Symmetry operations that share the same geometric element Examples Rotation axis } line 1 st,..., (n-1) th powers + all coaxial equivalents All rotations and screw rotations with the same axis, the same angle and sense of rotation and the same screw vector (zero for rotation) up to a lattice translation vector. Glide plane } plane defining operation+ all coplanar equivalents All glide reflections with the same reflection plane, with glide of d.o. (taken to be zero for reflections) by a lattice translation vector.

Symmetry operations and symmetry elements Geometric elements and Element sets P. M. de Wolff et al. Acta Cryst (1992) A48 727

Example: P4mm Diagram of symmetry elements Element set of (00z) line Symmetry operations that share (0,0,z) as geometric element } all Element set of (0,0,z) line 1 st, 2 nd, 3 rd powers + coaxial equivalents All rotations and screw rotations with the same axis, the same angle and sense of rotation and the same screw vector (zero for rotation) up to a lattice translation vector. 2 -x,-y,z 4+ -y,x,z 4- y,-x,z 2(0,0,1) -x,-y,z+1......

ORIGINS AND ASYMMETRIC UNITS

Space group Cmm2 (No. 35): left-hand page ITA Origin statement The site symmetry of the origin is stated, if different from the identity. A further symbol indicates all symmetry elements (including glide planes and screw axes) that pass through the origin, if any. Space groups with two origins For each of the two origins the location relative to the other origin is also given.

Example: Different origins for Pnnn

Example: Asymmetric unit Cmm2 (No. 35) ITA: (output cctbx: Ralf Grosse-Kustelve) ITA: An asymmetric unit of a space group is a (simply connected) smallest closed part of space from which, by application of all symmetry operations of the space group, the whole of space is filled.

Example: Asymmetric units for the space group P121 b a c (output cctbx: Ralf Grosse-Kustelve)

SITE-SYMMETRY GENERAL POSITION SPECIAL WYCKOFF POSITIONS

General and special Wyckoff positions Site-symmetry group So={(W,w)} of a point Xo (W,w)Xo = Xo ( ) a b c d e f g h i w1 w2 w3 x0 y0 z0 = x0 y0 z0 General position Xo S={(1,o)} 1 Special position Xo S> 1 ={(1,o),...,} Site-symmetry groups: oriented symbols

General position (i) ~ coordinate triplets of an image point X of the original point X under (W,w) of G (ii) short-hand notation of the matrix-column pairs (W,w) of the symmetry operations of G -presentation of infinite symmetry operations of G (W,w) = (I,tn)(W,w0), 0 wi0<1

General Position of Space groups Coset decomposition G:TG (I,0) (W2,w2)... (Wm,wm)... (Wi,wi) (I,t1) (W2,w2+t1)... (Wm,wm+t1)... (Wi,wi+t1) (I,t2) (W2,w2+t2)... (Wm,wm+t2)... (Wi,wi+t2).................. (I,tj) (W2,w2+tj)... (Wm,wm+tj)... (Wi,wi+tj) General position.................. Factor group G/TG isomorphic to the point group PG of G Point group PG = {I, W1, W2,...,Wi}

Coset Decomposition Coset decomposition P21/c:T (I,0) (2,0 ½ ½) (1,0) (m,0 ½ ½) (I,t1) (2,0 ½ ½+t1) ( 1,t1) (m,0 ½ ½ +t1) (I,t2) (2,0 ½ ½ +t2) ( 1,t2) (m,0 ½ ½ +t2).................. (I,tj) (2,0 ½ ½ +tj) ( 1,tj) (m,0 ½ ½ +tj).................. inversion centers 21screw axes ( 1,p q r): 1 at p/2,q/2,r/2 (2,u ½+v ½ +w) (2,0 ½+v ½) (2,u ½ ½ +w)

Example: Calculation of the Site-symmetry groups Group P-1 S={(W,w), (W,w)Xo = Xo} ( ) -1-1 -1 0 0 0 1/2 0 1/2 = -1/2 0-1/2 Sf={(1,0), (-1,101)Xf = Xf} Sf {1, -1} isomorphic

Bilbao Crystallographic Server Problem: Wyckoff positions Site-symmetry groups WYCKPOS space group

Example WYCKPOS: Wyckoff Positions Ccce (68) 2 1/2,y,1/4 2 x,1/4,1/4

EXERCISES Problem 2.16 Consider the special Wyckoff positions of the the space group P4mm. Determine the site-symmetry groups of Wyckoff positions 1a and 1b. Compare the results with the listed ITA data The coordinate triplets (x,1/2,z) and (1/2,x,z), belong to Wyckoff position 4f. Compare their site-symmetry groups. Compare your results with the results of the program WYCKPOS.

EXERCISES Problem 2.17 Consider the special Wyckoff positions of the the space group P42/mbc (No. 135). Determine the site-symmetry groups of Wyckoff positions 4a, 4c, 4d and 8g. Compare the results with the listed ITA data. Compare your results with the results of the program WYCKPOS.

CO-ORDINATE TRANSFORMATIONS IN CRYSTALLOGRAPHY

Co-ordinate transformation a 0, b 0, c 0 ˆ a, b, c P 0 1 P 11 P 12 P 13 B C ˆ a, b, c @ P 21 P 22 P 23 A P 31 P 32 P 33 ˆ P a P b P c, 3-dimensional space (a, b, c), origin O: point X(x, y, z) (P,p) Transformation matrix-column pair (P,p) (i) linear part: change of orientation or length: (a, b, c ), origin O : point X(x,y,z ) 0 B @ ˆ P 11 a P 21 b P 31 c, (ii) origin shift by a shift vector p(p1,p2,p3): O = O + p P 12 a P 22 b P 32 c, P 13 a P 23 b P 33 c : the origin O has coordinates (p1,p2,p3) in the old coordinate system

EXAMPLE

Transformation of the coordinates of a point X(x,y,z): (X )=(P,p) -1 (X) =(P -1, -P -1 p)(x) x y z = ( ) P11 P12 P13 P21 P22 P23 P31 P32 P33 p1 p2 p3-1 x y z special cases -origin shift (P=I): -change of basis (p=o) : Transformation of symmetry operations (W,w): (W,w )=(P,p) -1 (W,w)(P,p) Transformation by (P,p) of the unit cell parameters: metric tensor G: G =P t G P

Problem: SYMMETRY DATA ITA SETTINGS 530 ITA settings of orthorhombic and monoclinic groups

SYMMETRY DATA: ITA SETTINGS Monoclinic descriptions abc c ba Monoclinic axis b Transf. abc ba c Monoclinic axis c abc ā c b Monoclinic axis a C 12/c1 A12/a1 A112/a B112/b B2/b11 C 2/c11 Cell type 1 HM C 2/c A12/n1 C 12/n1 B112/n A112/n C 2/n11 B2/n11 Cell type 2 I 12/a1 I 12/c1 I 112/b I 112/a I 2/c11 I 2/b11 Cell type 3 Orthorhombic descriptions No. HM abc ba c cab c b a bca a cb 33 Pna2 1 Pna2 1 Pbn2 1 P2 1 nb P2 1 cn Pc2 1 n Pn2 1 a

Bilbao Crystallographic Server Problem: Coordinate transformations Generators General positions GENPOS space group Transformation of the basis ITA-settings symmetry data

Example GENPOS: default setting C12/c1 (W,w)A112/a= (P,p) -1 (W,w)C12/c1(P,p) final setting A112/a

Example GENPOS: ITA settings of C2/c(15) default setting A112/a setting

Bilbao Crystallographic Server Problem: Coordinate transformations Wyckoff positions WYCKPOS space group ITA settings Transformation of the basis

EXERCISES Problem 2.18 Consider the space group P21/c (No. 14). Show that the relation between the General and Special position data of P1121/a (setting unique axis c ) can be obtained from the data P121/c1(setting unique axis b ) applying the transformation (a,b,c )c = (a,b,c)bp, with P= c,a,b. Use the retrieval tools GENPOS (generators and general positions) and WYCKPOS (Wyckoff positions) for accessing the space-group data. Get the data on general and special positions in different settings either by specifying transformation matrices to new bases, or by selecting one of the 530 settings of the monoclinic and orthorhombic groups listed in ITA.

EXERCISES Problem 2.19 Use the retrieval tools GENPOS or Generators and General positions, WYCKPOS (or Wyckoff positions) for accessing the space-group data on the Bilbao Crystallographic Server or Symmetry Database server. Get the data on general and special positions in different settings either by specifying transformation matrices to new bases, or by selecting one of the 530 settings of the monoclinic and orthorhombic groups listed in ITA. Consider the General position data of the space group Im-3m (No. 229). Using the option Non-conventional setting obtain the matrix-column pairs of the symmetry operations with respect to a primitive basis, applying the transformation (a,b,c ) = 1/2(-a+b+c,a-b+c,a+b-c)