CRYSTALLOGRAPHIC POINT GROUPS I (basic facts) Mois I. Aroyo Universidad del Pais Vasco, Bilbao, Spain Bilbao Crystallographic Server http://www.cryst.ehu.es C´esar Capillas, UPV/EHU 1 GROUP THEORY (brief introduction) Crystallographic symmetry operations Symmetry operations of an object The symmetry operations are isometries, i.e. they are special kind of mappings between an object and its image that leave all distances and angles invariant. 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. Symmetry operations in the plane Matrix representations Mirror symmetry operation Mirror line my at 0,y drawing: M.M. Julian Foundations of Crystallography c Taylor & Francis, 2008 Matrix representation Fixed points x -x -1 x my y = y = 1 y xf xf my = yf yf det -1 = ? tr -1 = ? 1 1 Geometric element and symmetry element 2. Group axioms Group properties 1. Order of a group ∣ G∣ : number of elements crystallographic point groups: 1≤∣ G∣ ≤48 space groups: ∣ G∣=∞ 2. Abelian group G: gi.gj = gj.gi ⋁gi, g j ∈ G 3. Cyclic group G: G={g, g2, g 3, ..., g n} finite: ∣ G∣ =n, gn=e infinite: G= <g, g-1> order of a group element: gn=e Group Properties 4. How to define a group Multiplication table Group generators a set of elements such that each element of the group can be obtained as a product of the generators Crystallographic Point Groups in 2D Point group 2 = {1,2} Motif with symmetry of 2 Where is the two-fold point? x -x -1 x 2z y = -x = -1 y det -1 = ? -1 -1 drawing: M.M. Julian tr = ? Foundations of Crystallography -1 c Taylor & Francis, 2008 Crystallographic Point Groups in 2D Point group 2 = {1,2} -group axioms? Motif with -1 -1 1 symmetry of 2 2 x 2 = -1 x -1 = 1 -order of 2? -multiplication table drawing: M.M. Julian Foundations of Crystallography -generators of 2? c Taylor & Francis, 2008 Crystallographic symmetry operations in the plane Mirror symmetry operation Mirror line my at 0,y Where is the mirror line? Matrix representation x -x -1 x my y = y = 1 y -1 -1 drawing: M.M. Julian det = ? tr = ? Foundations of Crystallography 1 1 c Taylor & Francis, 2008 Crystallographic Point Groups in 2D Point group m = {1,m} Motif with symmetry of m -group axioms? -1 -1 1 m x m = 1 x 1 = 1 -order of m? -multiplication table drawing: M.M. Julian Foundations of Crystallography -generators of m? c Taylor & Francis, 2008 Isomorphic groups . Φ(g)=g’ G’ G . Φ-1(g’)=g Φ(g1)Φ(g2)= Φ(g1 g2) Point group 2 = {1,2} Point group m = {1,m} -groups with the same multiplication table Isomorphic groups. Φ(g1) g1. g. 2 Φ G (g2) . G’ . Φ(g1g2) g1g2 Φ(g)=g’ G={g} G’={g’} Φ-1(g’)=g Φ: G G’ Φ-1: G’ G homomorphic Φ Φ Φ condition (g1) (g2)= (g1g2) -groups with the same multiplication table Crystallographic Point Groups in 2D Point group 1 = {1} Motif with -group axioms? symmetry of 1 1 1 1 1 x 1 = 1 x 1 = 1 -order of 1? -multiplication table drawing: M.M. Julian Foundations of Crystallography c Taylor & Francis, 2008 -generators of 1? SEITZ SYMBOLS FOR SYMMETRY OPERATIONS point-group - specify the type and the order of the symmetry symmetry operation operation 1 and 1 identity and inversion m reflections 2, 3, 4 and 6 rotations 3, 4 and 6 rotoinversions - orientation of the symmetry element by the direction of the axis for rotations and rotoinversions, or the direction of the normal to reflection planes. SHORT-HAND NOTATION OF SYMMETRY OPERATIONS -left-hand side: omitted x’ x R11 R12 x =R = notation: -coefficients 0, +1, -1 y’ y R21 R22 y -different rows in one line, 1 separated by commas x’=R11x+R12y 0 -1 -y, -x+y y’=R21x+R22y -1 1 { y, x+y Problem 2.1 Consider the model of the molecule of the organic semiconductor pentacene (C22H14): m10 y m01 x Determine: -symmetry operations: matrix and (x,y) presentation -generators -multiplication table Problem 2.3 Consider the symmetry group of the equilateral triangle. Determine: (-1,-1) -symmetry operations: matrix and (x,y) presentation (0,1)(0,1) y -generators -multiplication table x (1,0) Visualization of Crystallographic Point Groups - general position diagram - symmetry elements diagram Stereographic Projections P’’ Points P in the projection plane Rotation axes Symmetry-elements diagrams filled polygons with the same number of sides as the foldness of the axes Mirror planes Combinations of symmetry elements • line of intersection of any two mirror planes must be a rotation axis. Stereographic Projections of EXAMPLE mm2 Point group mm2 = {1,2,m10,m01} Molecule of m10 pentacene y m01 x Stereographic projections diagrams general position symmetry elements EXAMPLE Stereographic Projections of 3m (-1,-1) (0,1)(0,1) y Point group 3m = - {1,3+,3 ,m10, m01, m11} x (1,0) Stereographic projections diagrams general position ? ? symmetry elements EXAMPLE Stereographic Projections of 3m (-1,-1) (0,1)(0,1) y Point group 3m = - {1,3+,3 ,m10, m01, m11} x (1,0) Stereographic projections diagrams general position symmetry elements (-1,-1) Conjugate elements + (0,1)(0,1) 3 y m10 ~ m01 m01 x (1,0) 3m m10 m01 m10 Conjugate elements Conjugate elements gi ~ gk if ∃ g: g-1gig = gk, where g, gi, g k, ∈ G Classes of conjugate L(gi)={gj| g-1gig = gj, g∈G} elements Conjugation-properties (i) L(gi) ∩ L(gj) = {∅}, if gi ∉ L(gj) (ii) |L(gi)| is a divisor of |G| (iii) L(e)={e} (iv) if gi, g j ∈ L, then (gi)k=(gj)k= e Problem 2.3 (cont.) Classes of conjugate elements Distribute the symmetry operations of the group of the equilateral triangle 3m into classes of conjugate elements (-1,-1) Point group 3m = - {1,3+,3 ,m10, m01, m11} (0,1)(0,1) 3m y Multiplication table of x (1,0) EXERCISES Problem 2.1 (cont) Distribute the symmetry elements of the group mm2 = {1,2,m10,m01} in classes of conjugate elements. multiplication table stereographic projection Problem 2.2 Consider the symmetry group of the square. Determine: 11 01 11 -symmetry operations: matrix and (x,y) presentation 10 10 -general-position and symmetry- elements stereographic projection diagrams; 11 01 11 -generators -multiplication table -classes of conjugate elements GROUP-SUBGROUP RELATIONS I. Subgroups: index, coset decomposition and normal subgroups II. Conjugate subgroups III. Group-subgroup graphs Subgroups: Some basic results (summary) Subgroup H < G 1. H={e,h1,h2,...,hk} ⊂ G 2. H satisfies the group axioms of G Proper subgroups H < G, and trivial subgroup: {e}, G Index of the subgroup H in G: [i]=|G|/|H| (order of G)/(order of H) Maximal subgroup H of G NO subgroup Z exists such that: H < Z < G Example Subgroups of point groups Molecule of pentacene y x Subgroups of mm2 {1,2,m10,m01} {1, m10} {1, 2} {2, m10} Subgroup graph Index 4 mm2 = {1,2,m10,m01} 1 2 {1, 2} {1,m10} {1,m01} 2 1 {1} 4 Problem 2.5 (i) Consider the group of the equilateral triangle and determine its subgroups; (ii) Construct the maximal-subgroup graph of 3m (-1,-1) (0,1)(0,1) y x (1,0) Multiplication table of 3m Coset decomposition G:H Group-subgroup pair H < G left coset G=H+g2H+...+gmH, gi∉H, decomposition m=index of H in G right coset G=H+Hg2+...+Hgm, g i∉H decomposition m=index of H in G Coset decomposition-properties (i) giH ∩ gjH = {∅}, if gi ∉ gjH (ii) |giH| = |H| (iii) giH = gjH, gi ∈ gjH Coset decomposition G:H Normal Hgj= gjH, for all gj=1, ..., [i] subgroups Theorem of Lagrange group G of order |G| |H| is a divisor of |G| then subgroup H<G of order |H| and [i]=|G:H| The order k of any Corollary element of G, gk=e, is a divisor of |G| Example: Coset decompositions of 3m (-1,-1) (0,1)(0,1) y (1,0) x Multiplication table of 3m Consider the subgroup {1, m10} of 3m of index 3. Write down and compare the right and left coset decompositions of 3m with respect to {1, m10}. Problem 2.7 Demonstrate that H is always a normal subgroup if |G:H|=2. Conjugate subgroups Conjugate subgroups Let H1<G, H2<G then, H1 ~ H2, if ∃ g∈G: g-1H1g = H2 (i) Classes of conjugate subgroups: L(H) (ii) If H1 ~ H2, then H1 ≅ H2 (iii) |L(H)| is a divisor of |G|/|H| Normal subgroup H G, if g-1H g = H, for ∀g∈G Problem 2.5 (cont.) Consider the subgroups of 3m and distribute them into classes of conjugate subgroups Multiplication table of 3m Complete and contracted group-subgroup graphs Complete graph of Contracted graph of maximal subgroups maximal subgroups International Tables for Crystallography, Vol. A, Chapter 3.2 Group-subgroup relations of point groups Hahn and Klapper EXERCISES Problem 2.4 (i) Consider the group of the square and determine its subgroups (ii) Distribute the subgroups into classes of conjugate subgroups; (iii) Construct the maximal-subgroup graph of 4mm 11 01 11 10 10 11 01 11 EXERCISES Problem 2.6 Consider the subgroup {e,2} of 4mm, of index 4: -Write down and compare the right and left coset decompositions of 4mm with respect to {e,2}; -Are the right and left coset decompositions of 4mm with respect to {e,2} equal or different? Can you comment why? FACTOR GROUP Factor group Kj={gj1,gj2,...,gjn} product of sets: G={e, g2, ...,g p} { Kk={gk1,gk2,...,gkm} Kj Kk={ gjpgkq=gr | gjp ∈ Kj, g kq ∈Kk} Each element gr is taken only once in the product Kj Kk factor group G/H: H G G=H+g2H+...+gmH, gi∉H, G/H={H, g2H, ..., gmH} group axioms: (i) (giH)(gjH) = gijH