CRYSTALLOGRAPHIC SYMMETRY OPERATIONS Mois I. Aroyo Universidad del Pais Vasco, Bilbao, Spain miércoles, 9 de octubre de 13 SYMMETRY OPERATIONS AND THEIR MATRIX-COLUMN PRESENTATION miércoles, 9 de octubre de 13 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. !2 !1 and X The relation of the point X to the points X is not a mapping because the image point is not uniquely defined (there are two image points). The five regions of the set A (the triangle) are mapped onto the five separated regions of the set B. No point of A is mapped onto more than one image point. Region 2 is mapped on a line, the points of the line are the images of more than one point of A. Such a mapping is called a projection. miércoles, 9 de octubre de 13 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 ‘isometry of the first kind’, preserving the counter–clockwise sequence of the edges ‘short–middle–long’ of the triangle is displayed in the upper mapping. An ‘isometry of the second kind’, 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. miércoles, 9 de octubre de 13 Description of isometries coordinate system: isometry: {O, a, b, c} ~ X ~ ~~ (x,y,z) X (x,y,z) ~ x = F1(x,y,z) miércoles, 9 de octubre de 13 Matrix-column presentation of isometries linear/matrix part matrix-column pair miércoles, 9 de octubre de 13 translation column part Seitz symbol Short-hand notation for the description of isometries isometry: (W,w) X ~ X notation rules: -left-hand side: omitted -coefficients 0, +1, -1 -different rows in one line examples: 1/2 -1 0 1 -1 miércoles, 9 de octubre de 13 1/2 { -x+1/2, y, -z+1/2 x+1/2, y, z+1/2 EXERCISES Problem 2.19 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 miércoles, 9 de octubre de 13 Combination of isometries (U,u) X (W,w) ~ X (V,v) ~ ~ X miércoles, 9 de octubre de 13 Inverse isometries X ~ ~ X (W,w) ~ X (C,c)=(W,w)-1 I = 3x3 identity matrix (C,c)(W,w) = (I,o) o = zero translation column (C,c)(W,w) = (CW, Cw+c) CW=I -1 C=W miércoles, 9 de octubre de 13 Cw+c=o c=-Cw=-W-1w Matrix formalism: 4x4 matrices augmented matrices: point X −→ point X̃ : miércoles, 9 de octubre de 13 4x4 matrices: general formulae point X −→ point X̃ : combination and inverse of isometries: miércoles, 9 de octubre de 13 EXERCISES Problem 2.19 (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 miércoles, 9 de octubre de 13 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. miércoles, 9 de octubre de 13 Crystallographic symmetry operations characteristics: fixed points of isometries (W,w)Xf=Xf geometric elements Types of isometries preserve handedness identity: the whole space fixed translation t: no fixed point rotation: one line fixed rotation axis φ = k × 360 /N no fixed point screw axis screw vector screw rotation: miércoles, 9 de octubre de 13 x̃ = x + t ◦ Crystallographic symmetry operations miércoles, 9 de octubre de 13 Crystallographic symmetry operations Screw rotation n-fold rotation followed by a fractional p translation n t parallel to the rotation axis Its application n times results in a translation parallel to the rotation axis miércoles, 9 de octubre de 13 do not Types of isometries preserve handedness roto-inversion: inversion: reflection: glide reflection: miércoles, 9 de octubre de 13 centre of roto-inversion fixed roto-inversion axis centre of inversion fixed plane fixed reflection/mirror plane no fixed point glide plane glide vector Symmetry operations in 3D Rotoinvertions miércoles, 9 de octubre de 13 Symmetry operations in 3D Rotoinvertions miércoles, 9 de octubre de 13 Symmetry operations in 3D Rotoinvertions miércoles, 9 de octubre de 13 Crystallographic symmetry operations Glide plane reflection followed by a fractional translation 1 t parallel to the plane 2 Its application 2 times results in a translation parallel to the plane miércoles, 9 de octubre de 13 Matrix-column presentation of some symmetry operations Rotation or roto-inversion around the origin: ( ) ( ) ( ) W11 W12 W13 0 0 W21 W22 W23 0 0 W31 W32 W33 0 0 0 w1 x x+w1 w2 y w3 z = 0 0 Translation: 1 1 1 = y+w2 z+w3 Inversion through the origin: -1 -1 -1 miércoles, 9 de octubre de 13 0 x 0 y 0 z = -x -y -z Geometric meaning of (W , w ) W information (a) type of isometry order: Wn=I rotation angle miércoles, 9 de octubre de 13 EXERCISES Problem 2.19 (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 (4) x,-y+1/2, z+1/2 (3) -x,-y,-z (a) type of isometry miércoles, 9 de octubre de 13 Geometric meaning of (W , w ) W information Texto (b) axis or normal direction (b1) rotations: = (b2) roto-inversions: reflections: miércoles, 9 de octubre de 13 : EXERCISES Problem 2.19 (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 rotations: reflections: miércoles, 9 de octubre de 13 (4) x,-y+1/2, z+1/2 Y(W) = Wk-1 + Wk-2 + ... + W + I 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 miércoles, 9 de octubre de 13 Fixed points of isometries solution: point, line, plane or space (W,w)Xf=Xf no solution Examples: solution: ( ) ( ) NO solution: miércoles, 9 de octubre de 13 -1 0 0 0 x 0 1 0 0 y 0 0 -1 1/2 z = What are the fixed points of this isometry? x y z -1 0 0 0 x 0 1 0 1/2 y 0 0 -1 1/2 z = x y z Geometric meaning of (W,w) (W,w)n=(I,t) (W,w)n=(Wn,(Wn-1+...+ W+I)w) =(I,t) (A) intrinsic translation part : glide or screw component (A1) screw rotations: = (A2) glide reflections: w miércoles, 9 de octubre de 13 t/k Glide or Screw component (intrinsic translation part) (W,w)k= (W,w). (W,w). ... .(W,w)=(I,t) (W,w)k=(Wk,(Wk-1+...+ W+I)w) =(I,t) screw rotations : glide reflections: miércoles, 9 de octubre de 13 t/k=1/k (Wk-1+...+ W+I)w w EXERCISES Problem 2.20 (cont.) Determine the intrinsic translation parts (if relevant) of the following symmetry operations (1) x,y,z (3) -x,-y,-z screw rotations: glide reflections: miércoles, 9 de octubre de 13 (2) -x,y+1/2,-z+1/2 (4) x,-y+1/2, z+1/2 t/k=1/k (Wk-1+...+ W+I)w w Fixed points of (W,w) Location (fixed points xF ): (B1) t/k = 0: (B2) t/k ≠ 0: miércoles, 9 de octubre de 13 EXERCISES Problem 2.19 (cont.) Determine the fixed points of the following symmetry operations: (1) x,y,z (3) -x,-y,-z fixed points: miércoles, 9 de octubre de 13 (2) -x,y+1/2,-z+1/2 (4) x,-y+1/2, z+1/2 Space group P21/c (No. 14) EXAMPLE Matrix-column presentation Geometric interpretation miércoles, 9 de octubre de 13 c . w w w miércoles, 9 de octubre de 13 y r e . t s u h s .e Crystallographic databases Group-subgroup relations Structural utilities Representations of point and space groups Solid-state applications miércoles, 9 de octubre de 13 Crystallographic Databases International Tables for Crystallography miércoles, 9 de octubre de 13 EXERCISES Problem 2.19 Construct the matrix-column pairs (W,w) of the following coordinate triplets: (1) x,y,z (3) -x,-y,-z (2) -x,y+1/2,-z+1/2 (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. miércoles, 9 de octubre de 13 Bilbao Crystallographic Server Problem: Geometric Interpretation of (W,w) miércoles, 9 de octubre de 13 SYMMETRY OPERATION Problem 2.19 SOLUTION (i) (W,w)(1)= 1 0 1 0 1 (W,w)(3)= -1 0 -1 (ii) miércoles, 9 de octubre de 13 0 1 0 1/2 -1 0 0 -1 (W,w)(2)= -1 (W,w)(4)= 1 1/2 0 -1 1/2 1 1/2 ITA description: under Symmetry operations CO-ORDINATE TRANSFORMATIONS IN CRYSTALLOGRAPHY miércoles, 9 de octubre de 13 & & & & & & & They are briefly reviewed in r " x a ( y b ( z c" The same point X is given with respect to a new coordinate system, & & & & i.e. the new basis vectors a , b , c and the new origin O (Fig. 5.1. TRANSFORMATIONS OF THEthe COORDINATE SYSTEM In this section, relations between the primed and unprimed 5.1.3.1), by the position vector quantities are treated. P ! Q"1 : notation & & transformation !3-dimensional (affine transformation) of the space r& " x& a& ( yThe b ( general z& c& " $ ! $ ' '# coordinate system consists of two parts, a linear part and a shift a a are written in the following '& '# & # borigin In this section, the relations between primed andmatrix of origin.theThe #3 ! 3$ of the linear #3 ! y, 1$ z) bpart ! P #O: !and the (a,unprimed b,Pc), point X(x, c' c'# the shift vector p, column matrix p, containing the components of quantities are treated. the transformation by the symbol The general transformationdefine (affine transformation)uniquely. of the It is represented (P,p) These part transformation rules apply also to the quantities covariant (P, p). coordinate system consists of two parts, a linear and a shift ' ' ' , b , c and contravariant! with! ! with respect to the basis vectors a ! ! ! of Thespace #3 ! 3$ matrix P of(i)theThe linear part and the #3 ! 1$ orsorigin. of direct (a , b , c ), origin O’: point X(xorare, ythe, z ) linear partto implies a change of orientation or length respect a, b, c, which are written as column matrices. They column components of the p,direct space, [uvw], which are transformed dices of amatrix planep,(orcontaining a set of theboth indices ofshift a direction of the basis vectors a,vector b, c, ini.e. neral affine consisting of a shiftItofisorigin define thetransformation, transformation uniquely. represented by& & &by the symbol ect space or the coordinates by a shift vector pTransformation with components p1 and p2 and amatrix-column change #a , b , c $ " #a, b, pair c$P! (P,p) (P, p). # # space $ ! $ reciprocal m a, b to a , b . This implies a change in the coordinates of # # & u u # # y. rom(i) x, yThe to x , linear #11 part implies a change of orientation or length or #P & !PQ12# vP&13! v rices: $ # ' (i) linear change of orientation orb, length: both of the basis vectorspart: a, b, c, i.e. w21 P22 wP23 ( " #a, c$% P es of matrices a pointofinP& direct space verse and & p& are needed. They are #a , bspace , c $ " #a, b, c$P P31 P32 above, P33 the components of a ors of reciprocal In contrast to all quantities mentioned # & Q ! P"1 thePcoordinates of a point X in direct space P11 P12 position P13 vector " #Pr11ora ( f a direction in direct space 21 b ( P31 c, x, y, z ' depend also on the shift of the origin in direct space. The $ nts of a shift vector from P32 c,by P12 a ( P22 b is(given " #a, b, c$% P21 P22 general P23 ((affine) transformation & "1 he new origin q!O "P p! ! #P$13 a (!P23$b ( P33 c$" P P P 31 32 33 x x nts of of anthe inverse origin onsists components of the negative shift vector & " #% " % gintheOcoordinate to origin O, with P b ( P c, y ! Q q # & # yshift & $ vector b# , #P c# , 11 i.e.a ( to system a# , " 31 For 21 a pure linear transformation, the p is zero and the (ii) shift by a shift vector p(p #1,p2,p3): # # origin # symbol is (P, o). z z q ! q1 a $ q2 b $ q3 c ! P12 a ( P22 b ( P32 c, $ n part of a symmetry The determinant of P, det#P$,!should be positive. If det#P$ is x $ Q y $ Q z $ q Q 11 12 13 1 nsformation (Q, q) is the inverse P transformation of the origin O’ has a ( P b ( P c$" 13 23 33 in direct space negative, a right-handed coordinate system is transformed " % into a # O # #+ p O’ = x ,p $ 3Q)22in y $ Q23 z $ q2 &! !(p ng (Q, q) to the basis vectors a , b , c and the origin #1Q,p 212 coordinates left-handed one (or vice versa). If det#P$ " 0, the new basis vectors d #4 ! 1$ column matrix of sis a, b, c withtransformation, origin O are obtained. Forvectors a pure linear the shift vector p is zero and the Q31 x $system Qa32 ycomplete $ Q33 z $ qcoordinate 3 the old coordinate # # are linearly dependent and do not form es of a point in direct space -dimensional transformation of a and b , some symbol is (P, o). Q are set de 13as follows: Qsystem. and miércoles, 9 de octubre 33 ! 1 Co-ordinate transformation EXAMPLE miércoles, 9 de octubre de 13 EXAMPLE miércoles, 9 de octubre de 13 Transformation matrix-column pair (P,p) ( ( (P,p)= (P,p)-1= miércoles, 9 de octubre de 13 1/2 1/2 -1/2 1/2 0 0 1 1 0 -1 1 0 ) ) 0 0 1 1/2 1/4 0 0 0 1 -1/4 -3/4 0 Transformation of the coordinates of a point X(x,y,z): (X’)=(P,p)-1(X) =(P-1, -P-1p)(X) x’ y’ z’ = ( P11 P12 P13 p1 P21 P22 P23 p2 P31 P32 P33 p3 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: miércoles, 9 de octubre de 13 G´=Pt G P ) -1 x y z Short-hand notation for the description of transformation matrices Transformation matrix: (a,b,c), origin O 5.1. TRANSFORMATIONS OF THE COORDINATE SYSTEM P ! Q"1 : ( P11 P12 P13 (P,p)= P21 P22 P23 P31 covariant P32 P33 These transformation rules apply also to the quantities 3.1. General affine transformation, consisting of a shift of origin O to O# by a shift vector p with components p1 and p2 and a change is from a, b to a# , b# . This implies a change in the coordinates of int X from x, y to x# , y# . ! $ ! '# $ a' a ' # b & ! P # b'# &! c' c'# with respect to the basis vectors a' , b' , c' and contravariant with respect to a, b, c, which are written as column matrices. They are the indices of a direction in direct space, [uvw], which are transformed by ! #$ ! $ u u # v# & ! Q# v &! w# w p1 p2 p3 ) (a’,b’,c’), origin O’ -written by columns In contrast to all quantities0, mentioned above, the components of a -coefficients +1, -1 notation Q ! P rules: position vector r or the coordinates of a point X in direct space x, y, z depend also on the shift of the origin in direct space. The -different columns general (affine) transformation is given byin one line q ! "P p! ! $ ! $ x x -origin shift rix q consists of the components of the negative shift vector " % " % # y & ! Q# y & $ q refer to the coordinate system a , b , c , i.e. the inverse matrices of P and p are needed. They are "1 "1 # # # # # z# q ! q1 a# $ q2 b# $ q3 c# ! he transformation (Q, q) is the inverse transformation of Applying (Q, q) to the basis vectors a# , b# , c# and the origin 1 -1 old basis vectors a, b, c with origin O are obtained. # # a two-dimensional transformation of a and b , some 1 s of Q are set as follows: Q33 !11 and Q23 ! Q31 ! Q32 ! 0. quantities which transform in the same way as the basis a, b, c are called covariant quantities and are written as row . miércoles, They are: 9 de octubre de 13 example: -1/4 Example -3/4 ! z Q11 x $ Q12 y $ Q13 z $ q1 $ " % ! # Q21 x $ Q22 y $ Q23 z $ q2 &! { a+b, -a+b, c;-1/4,-3/4,0 Q31 x $ Q32 y $ Q33 z $ q3 1If no shift 0 of origin is applied, i.e. p ! q ! o, the position vector r of point X is transformed by ! $ ! $ EXERCISES Problem 2.20 The following matrix-column pairs (W,w) are referred with respect to a basis (a,b,c): (1) x,y,z (3) -x,-y,-z (2) -x,y+1/2,-z+1/2 (4) x,-y+1/2, z+1/2 Determine the corresponding matrix-column pairs (W’,w’) with respect to the basis (a’,b’,c’)= (a,b,c)P, with P=c,a,b. Determine the coordinates X’ of a point X= with respect to the new basis (a’,b’,c’). 0,70 0,31 0,95 miércoles, 9 de octubre de 13