CRYSTALLOGRAPHIC SYMMETRY OPERATIONS. Mois I. Aroyo Universidad del Pais Vasco, Bilbao, Spain

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2 CRYSTALLOGRAPHIC SYMMETRY OPERATIONS Mois I. Aroyo Universidad del Pais Vasco, Bilbao, Spain

3 SYMMETRY OPERATIONS AND THEIR MATRIX-COLUMN PRESENTATION

4 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. The relation of the point X to the points and X2 is not a mapping because the image point is not uniquely defined (there are two image points). X 1 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.

5 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.!

Example: Matrix presentation of symmetry operation Mirror symmetry operation Mirror line my at,y Matrix representation drawing: M.M. Julian Foundations of Crystallography c Taylor & Francis, 28 my x = -x y y = -1 1 x y Fixed points det -1 1 =?

6 Example: Matrix presentation of symmetry operation Mirror symmetry operation Mirror line my at,y Matrix representation drawing: M.M. Julian Foundations of Crystallography c Taylor & Francis, 28 my x = -x y y = -1 1 x y Fixed points det -1 1 =? tr -1 1 =? my x y = x y

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

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

9 EXERCISES Problem 2.14 Referred to an orthorhombic coordinated system (a b c; α=β=γ=9) two symmetry operations are represented by the following matrix-column pairs: ( ) ( ) /2 (W1,w1)= 1 (W2,w2)= /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=,7,31,95 Hint: A drawing could be rather helpful

10 EXERCISES Problem 2.14 Characterization of the symmetry operations: det( ) tr( ) =? 1 =? -1-1 What are the fixed points of (W1,w1) and (W2,w2)? ( /2)x y 1/2 z = x y z

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

12 EXERCISES Problem 2.15 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

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

14 EXERCISES Problem 2.14(cont) Consider the matrix-column pairs of the two symmetry operations: (W1,w1)=( ) ( ) /2 1 (W2,w2)= /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:

15 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

16 EXERCISES Problem 2.14(cont) Determine the inverse symmetry operations (W1,w1) -1 and (W2,w2) -1 where (W1,w1)=( ) ( ) /2 1 (W2,w2)= /2 Determine the inverse symmetry operation (W,w) -1 (W,w)=(W1,w1)(W2,w2) inverse of isometries:

17 EXERCISES Problem 2.14(cont) Consider the matrix-column pairs

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

19 4x4 matrices: general formulae point X point X : combination and inverse of isometries:

20 EXERCISES Problem 2.15 (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

21 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 12 and 24 around its centre, reflections through the three thick lines intersecting the centre, and the identity operation.!

22 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 36 /N screw rotation: no fixed point screw axis screw vector

23 Crystallographic symmetry operations

$Crystallographic symmetry operations Screw rotation n-fold rotation followed by a fractional p translation t$

24 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

25 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

26 Symmetry operations in 3D Rotoinvertions

27 Symmetry operations in 3D Rotoinvertions

28 Symmetry operations in 3D Rotoinvertions

$Crystallographic symmetry operations Glide plane reflection followed by a 1 2 fractional$

29 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

30 Matrix-column presentation of some symmetry operations Rotation or roto-inversion around the origin: ( ) W W W W W W W W W = Translation: ( ) w1 w2 w3 x y z = x+w1 y+w2 z+w3 Inversion through the origin: ( ) -1 x -x = -1 y -y -1 z -z

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

32 EXERCISES Problem 2.15 (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

33 EXERCISES Problem 2.14(cont.) Consider the matrix-column pairs Determine the type and order of isometries that are represented by the matrices A, B, C and D:

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

35 Direction of rotation axis/normal Example: (W,w)=( /2 1/2 ) det W=? tr W=? What is the type and order of the isometry? Determine its rotation axis? Y(W) = W k-1 + W k W + I Y(W)= = W 3 W 2 W I

36 EXERCISES Problem 2.15 (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 W + I reflections: Y(-W) = - W + I

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

38 Sense of rotation Example: (W,w)=( /2 1/2 ) det W=1 tr W=1 W=41 What is its sense of rotation? det(z): u= x= Wx= -1 Z= det Z=?

39 Fixed points of isometries (W,w)Xf=Xf point, line, plane or space -1 1 solution: ( ) -1 1/2 x y z = x y z NO solution: ( ) /2 1/2 x y z = x y z Fixed points? w1 translation part w= w2 w3 intrinsic (screw, glide) location

40 Glide or Screw component (intrinsic translation part) (W,w) k = (W,w). (W,w).....(W,w)=(I,t) (W,w) k =(W k,(w k W+I)w) =(I,t) screw rotations : t/k=1/k (W k W+I)w glide reflections: w

41 EXERCISES Problem 2.15 (cont.) Determine the intrinsic translation parts (if relevant) 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: t/k=1/k (W k W+I)w glide reflections: w

42 Fixed points of (W,w) Location (fixed points ): xf (B1) t/k = : (B2) t/k :

43 EXERCISES Problem 2.15 (cont.) Determine 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 fixed points:

44 Space group P21/c (No. 14) EXAMPLE Matrix-column presentation Geometric interpretation

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

47 Crystallographic Databases International Tables for Crystallography

48 EXERCISES Problem 2.15 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.

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

50 CO-ORDINATE TRANSFORMATIONS IN CRYSTALLOGRAPHY

51 Co-ordinate transformation a, b, c ˆ a, b, c P 1 P 11 P 12 P 13 B C ˆ a, b, 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 ) 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

52 EXAMPLE

53 EXAMPLE

54 Transformation matrix-column pair (P,p) (P,p)=( ) 1/2 1/2 1/2 1-1/2 1/2 1/4 1 (P,p) -1 =( ) /4-3/4 a =1/2a-1/2b b =1/2a+1/2b a=a +b b=-a +b c =c c=c O =O+ 1/2 1/4 O=O + -1/4-3/4

55 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 = ( ) P P P P P P P P P p1 p p -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

56 Short-hand notation for the description of transformation matrices Transformation matrix: (a,b,c), origin O (P,p)= ( ) P P P P P P P P P p1 p p (a,b,c ), origin O notation rules: -written by columns -coefficients, +1, -1 -different columns in one line -origin shift example: /4-3/4 { a+b, -a+b, c;-1/4,-3/4, 1

57 EXERCISES Problem 2.16 The following matrix-column pairs (W,w) are referred with respect to a basis (a,b,c): (1) x,y,z (2) -x,y+1/2,-z+1/2 (3) -x,-y,-z (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 with respect to the new basis (a,b,c ). X=,7,31,95

MATRIX CALCULUS APPLIED TO CRYSTALLOGRAPHY. (short revision) Mois I. Aroyo Universidad del Pais Vasco, Bilbao, Spain

MATRIX CALCULUS APPLIED TO CRYSTALLOGRAPHY (short revision) Mois I. Aroyo Universidad del Pais Vasco, Bilbao, Spain INTRODUCTION TO MATRIX CALCULUS Some of the slides are taken from the presentation Introduction