SPACE GROUPS. International Tables for Crystallography, Volume A: Space-group Symmetry. Mois I. Aroyo Universidad del Pais Vasco, Bilbao, Spain

Size: px

Start display at page:

Download "SPACE GROUPS. International Tables for Crystallography, Volume A: Space-group Symmetry. Mois I. Aroyo Universidad del Pais Vasco, Bilbao, Spain"

Rosaline McKinney
5 years ago
Views:

2 SPACE GROUPS International Tables for Crystallography, Volume A: Space-group Symmetry Mois I. Aroyo Universidad del Pais Vasco, Bilbao, Spain

3 SPACE GROUPS Crystal pattern: Space group G: A model of the ideal crystal (crystal structure) in point space consisting of a strictly 3-dimensional periodic set of points 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

4 INTERNATIONAL TABLES FOR CRYSTALLOGRAPHY VOLUME A: SPACE-GROUP SYMMETRY Extensive tabulations and illustrations of the 17 plane groups and of 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;

5 GENERAL LAYOUT: LEFT-HAND PAGE

6 General Layout: Right-hand page

7 {a1, a2}: Primitive and centred lattice basis in 2D two translation vectors, linearly independent, form a lattice basis Primitive basis: Centred basis: If all lattice vectors are expressed as integer linear combinations of the basis vectors If some lattice vectors are expressed as linear combinations of the basis vectors with rational, non-integer coefficients Conventional centred basis (2D): c Number of lattice points per primitive and centred cells

8 Crystal families, crystal systems, conventional coordinate system and Bravais lattices in 2D

9 3D-unit cell and lattice parameters lattice basis: {a, b, c} unit cell: the parallelepiped defined by the basis vectors C B A primitive P and centred unit cells: A,B,C,F, I, R number of lattice points per unit cell

10 Crystal families, crystal systems, lattice systems and Bravais lattices in 3D

11 HEADLINE BLOCK

12 Short Hermann- Mauguin symbol Schoenflies symbol Crystal class (point group) Crystal system Number of space group Full Hermann- Mauguin symbol Patterson symmetry

13 HERMANN-MAUGUIN SYMBOLISM FOR SPACE GROUPS

14 Hermann-Mauguin symbols for space groups The Hermann Mauguin symbol for a space group consists of a sequence of letters and numbers, here called the constituents of the HM symbol. (i) The first constituent is always a symbol for the conventional cell of the translation lattice of the space group (ii) The second part of the full HM symbol of a space group consists of one position for each of up to three representative symmetry directions. To each position belong the generating symmetry operations of their representative symmetry direction. The position is thus occupied either by a rotation, screw rotation or rotoinversion and/ or by a reflection or glide reflection. (iii) Simplest-operation rule: pure rotations > screw rotations; pure rotations > rotoinversions reflection m > a; b; c > n > means has priority

15 crystal family 14 Bravais Lattices

16 Symmetry directions A direction is called a symmetry direction of a crystal structure if it is parallel to an axis of rotation, screw rotation or rotoinversion or if it is parallel to the normal of a reflection or glide-reflection plane. A symmetry direction is thus the direction of the geometric element of a symmetry operation, when the normal of a symmetry plane is used for the description of its orientation.

17 Hermann-Mauguin symbols for space groups

18 Example: Hermann-Mauguin symbols for space groups primary direction secondary direction tertiary direction

19 PRESENTATION OF SPACE-GROUP SYMMETRY OPERATIONS IN INTERNATIONAL TABLES FOR CRYSTALLOGRAPHY, VOL. A

20 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

21 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

22 Matrix formalism linear/matrix part translation column part matrix-column Seitz symbol pair

23 Space group Cmm2 (No. 35) Diagram of symmetry elements 0 b How are the symmetry operations represented in ITA? a Diagram of general position points General Position

24 General position (i) ~ coordinate triplets of an image point X of the original point X= x y under (W,w) of G z ~ -presentation of infinite image points X under the action of (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

25 Space Groups: infinite order Coset decomposition G:TG General position (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) Factor group G/TG isomorphic to the point group PG of G Point group PG = {I, W2, W3,,Wi}

26 Example: P12/m1 Coset decomposition G:TG Point group PG = {1, 2, 1, m} General position TG TG 2 1 TG TG m 1 (I,0) (2,0) (,0) (m,0) 1 (I,t1) (2,t1) (, t1) (m, t1) 1 (I,t2) (2,t2) (, t2) (m,t2) (I,tj) (2,tj) (, tj) (m, tj) n1 n2 1 at n1/2 n2/2 inversion centres (1,t): -1 n3 n3/2

27 EXAMPLE Coset decomposition P121/c1:T Point group? General position (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)

28 Symmetry Operations Block TYPE of the symmetry operation SCREW/GLIDE component ORIENTATION of the geometric element LOCATION of the geometric element GEOMETRIC INTERPRETATION OF THE MATRIX- COLUMN PRESENTATION OF THE SYMMETRY OPERATIONS

29 Example: Cmm2 Diagram of symmetry elements 0 b General position TG TG 2 TG my TG mx (I,0) (2,0) (my,0) (mx,0) a Diagram of general position points (I,t1) (2,t1) (my,t1) (mx, t1) (I,t2) (2,t2) (my,t2) (mx,t2) (I,tj) (2,tj) (my,tj) (mx, tj)

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

32 Bilbao Crystallographic Server Problem: Matrix-column presentation Geometrical interpretation GENPOS space group 14

33 Example GENPOS: Space group P21/c (14) Space-group symmetry operations short-hand notation matrix-column presentation Geometric interpretation Seitz symbols General positions ITA data

34 SEITZ SYMBOLS FOR SYMMETRY OPERATIONS Seitz symbols { R t } short-hand description of the matrix-column presentations of the symmetry operations of the space groups rotation (or linear) part R - - specify the type and the order of the symmetry operation; orientation of the symmetry element by the direction of the axis for rotations and rotoinversions, or the direction of the normal to reflection planes. 1 and 1 m 2, 3, 4 and 6 3, 4 and 6 identity and inversion reflections rotations rotoinversions translation part t translation parts of the coordinate triplets of the General position blocks

35 Glazer et al. Acta Cryst A 70, 300 (2014) EXAMPLE Seitz symbols for symmetry operations of hexagonal and trigonal crystal systems

36 EXAMPLE Matrix-column presentation Geometric interpretation Seitz symbols (1) {1 0} (2) { /21/2 } (3) {1 0} (4) {m010 01/21/2}

37 EXERCISES Problem 2.16 (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

38 SPACE-GROUPS DIAGRAMS

39 Diagrams of symmetry elements 0 b c 0 three different settings a a permutations of a,b,c c 0 b Diagram of general position points

42 Space group Cmm2 (No. 35) Diagram of symmetry elements 0 b conventional setting a General Position How many general position points per unit cell are there? Diagram of general position points

43 EXAMPLE Space group Cmm2 (No. 35) Geometric interpretation 0 b glide plane, t=1/2a at y=1/4, b glide plane, t=1/2b at x=1/4, a a General Position Matrix-column presentation of symmetry operations x+1/2,-y+1/2,z -x+1/2,y+1/2,z

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

45 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.

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

47 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 A l l r o t a t i o n s a n d s c r e w 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

50 ORIGINS AND ASYMMETRIC UNITS

51 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.

52 Example: Different origins for Pnnn

53 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.

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

55 GENERAL AND SPECIAL WYCKOFF POSITIONS SITE-SYMMETRY

56 Group Actions Group Actions Orbit and Stabilizer Equivalence classes

57 General and special Wyckoff positions Orbit of a point Xo under G: G(Xo)={(W,w)Xo,(W,w) G} Site-symmetry group So={(W,w)} of a point Xo (W,w)Xo = Xo Multiplicity ( ) a b c d e f g h i w 1 w 2 w 3 x0 y0 z0 = x0 y0 z0 Multiplicity: P / So General position Xo S={(1,o)} 1 Multiplicity: P Special position Xo S> 1 ={(1,o),...,} Multiplicity: P / So Site-symmetry groups: oriented symbols

58 General position (i) ~ coordinate triplets of an image point X of the original point X= x y under (W,w) of G z ~ -presentation of infinite image points X under the action of (W,w) of G: 0 xi<1 (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

59 General Position of Space groups ~ As coordinate triplets of an image point X of the original point X= x y z under (W,w) of G General position (I,0)X (W2,w2)X... (Wm,wm)X... (Wi,wi)X (I,t1)X (W2,w2+t1)X... (Wm,wm+t1)X... (Wi,wi+t1)X (I,t2)X (W2,w2+t2)X... (Wm,wm+t2)X... (Wi,wi+t2)X (I,tj)X (W2,w2+tj)X... (Wm,wm+tj)X... (Wi,wi+tj)X ~ -presentation of infinite image points X of X under the action of (W,w) of G: 0 xi<1

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

61 Example Space group P4mm

62 EXAMPLE Space group P4mm General and special Wyckoff positions of P4mm

63 Bilbao Crystallographic Server Problem: Wyckoff positions Site-symmetry groups Coordinate transformations WYCKPOS space group Standard basis ITA settings Transformation of the basis

64 Bilbao Crystallographic Server

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

66 EXERCISES Problem 2.17 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.

67 EXERCISES Problem 2.18 Consider the Wyckoff-positions data of the space group I41/amd (No. 141), origin choice 2. Determine the site-symmetry groups of Wyckoff positions 4a, 4c, 8d and 8e. Compare the results with the listed ITA data. Compare your results with the results of the program WYCKPOS. Characterize geometrically the isometries (3), (7), (12), (13) and (16) as listed under General Position. Compare the results with the corresponding geometric descriptions listed under Symmetry operations block in ITA. Comment on the differences between the corresponding symmetry operations listed under the sub-blocks (0, 0, 0) and (1/2, 1/2, 1/2). Compare your results with the results of the program SYMMETRY OPERATIONS. How do the above results change if origin choice 1 setting of I4 1 /amd is considered?

68 CO-ORDINATE TRANSFORMATIONS IN CRYSTALLOGRAPHY

69 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, 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 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

70 EXAMPLE

71 EXAMPLE

72 EXAMPLE

73 EXAMPLE

74 Transformation matrix-column pair (P,p) (P,p)=( ) 1/2 1/2 0 1/2 1-1/2 1/2 0 1/ (P,p) -1 =( ) /4-3/4 0 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 0 O=O + -1/4-3/4 0

75 Short-hand notation for the description of transformation matrices Transformation matrix: (a,b,c), origin O (P,p)= ( ) P11 P12 P13 P21 P22 P23 P31 P32 P33 p1 p2 p3 (a,b,c ), origin O notation rules: -written by columns -coefficients 0, +1, -1 -different columns in one line -origin shift example: /4-3/4 { a+b, -a+b, c;-1/4,-3/4,0 1 0

76 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) : EXAMPLE X =(P,p) -1 X=( /4-3/4 0 )3/4 1/4 0 = 1/4 1/4 0

77 Covariant and contravariant crystallographic quantities direct or crystal basis c)( ) P11 P12 P13 (a,b,c )=(a, b, c)p =(a, b, P21 P22 P23 reciprocal or dual basis P31 P32 P33 a* b* c* a* ( a* P11 P12 P13 = P -1 b* = b* P21 P22 P23 c* P31 P32 P33 )-1 c* covariant to crystal basis: Miller indices (h,k,l )=(h, k, l)p contravariant to crystal basis: indices of a direction [u] ( )-1 u P11 P12 P13 u v = P21 P22 P23 v w P31 P32 P33 w

78 Transformation of symmetry operations (W,w) original coordinates new coordinates (W,w )=(P,p) -1 (W,w)(P,p)

79 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

80 EXERCISES Problem 2.19 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 (i) 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. (ii) Determine the coordinates X of a point with respect to the new basis (a,b,c ). X= 0,70 0,31 0,95 Hints (W,w )=(P,p) -1 (W,w)(P,p) (X )=(P,p) -1 (X)

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

82 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

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

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

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

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

87 EXERCISES Problem 2.23 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.

88 EXERCISES Problem 2.24 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)

89 METRIC TENSOR

the basis vectors C B A primitive P and centred unit

90 3D-unit cell and lattice parameters lattice basis: {a, b, c} unit cell: the parallelepiped defined by the basis vectors C B A primitive P and centred unit cells: A,B,C,F, I, R number of lattice points per unit cell

91 Lattice parameters (3D) An alternative way to define the metric properties of a lattice L Given a lattice L of V 3 with a lattice basis: {a1, a2, a3} Remark: the lengths of basis vectors are measured in nm (1nm=10-9 m) Å (1Å=10-10 m) pm (1pm=10-12 m) Metric tensor G in terms of lattice parameters

92 METRIC TENSOR (FUNDAMENTAL MATRIX) Given a lattice L of V 3 with a lattice basis: Metric tensor G of L {a1, a2, a3} The lattice L inherits the metric properties of the Euclidean space and they are conveniently expressed with respect to a lattice basis (right-handed coordinate system) G ={a1, a2, a3} T. {a1, a2, a3}= a1 a2 a3 Metric tensor G is symmetric:.{a1, a2, a3}= Gik=Gki Scalar product of arbitrary vectors: (r,t)=r T Gt G11 G12 G13 G21 G22 G23 G31 G32 G33 Gik=(ai,ak)=aiakcosαj, Transformation properties of G under basis transformation {a 1, a 2, a 3}= {a1, a2, a3} P G ={a 1, a 2, a 3} T. {a 1, a 2, a 3}= P T {a1, a2, a3} T. {a1, a2, a3} P G =P T G P

93 Crystallographic calculations: Volume of the unit cell

94 Volume of the unit cell in terms of lattice parameters (Buerger,1941) Basis vectors with respect to Cartesian basis det (A)=det(A T ) = det (G)

95 EXERCISE (Problem 2.20) Write down the metric tensors of the seven crystal systems in parametric form using the general expressions for their lattice parameters. For each of the cases, express the volume of the unit cell as a function of the lattice parameters. For example: tetragonal crystal system: a=b, c, α=β=γ=90 a G = 0 a 2 0 V=? 0 0 c 2

96 EXERCISES Problem 2.21 A body-centred cubic lattice (ci) has as its conventional basis the conventional basis (ap,bp,cp) of a primitive cubic lattice, but the lattice also contains the centring vector 1/2aP+1/2bP+1/2cP which points to the centre of the conventional cell. Calculate the coefficients of the metric tensor for the body-centred cubic lattice: (i) for the conventional basis (ap,bp,cp); (ii) for the primitive basis: ai=1/2(-ap+bp+cp), bi=1/2(ap-bp+cp), ci=1/2(ap+bp-cp (iii) determine the lattice parameters of the primitive cell if ap=4 Å Hint metric tensor transformation G =P t G P

97 EXERCISES Problem 2.22 A face-centred cubic lattice (cf) has as its conventional basis the conventional basis (ap,bp,cp) of a primitive cubic lattice, but the lattice also contains the centring vectors 1/2bP+1/2cP, 1/2aP+1/2cP, 1/2aP+1/2bP, which point to the centres of the faces of the conventional cell. Calculate the coefficients of the metric tensor for the face-centred cubic lattice: (i) for the conventional basis (a P,bP,cP); (ii) for the primitive basis: af=1/2(bp+cp), bf=1/2(ap+cp), cf=1/2(ap+bp) (iii) determine the lattice parameters of the primitive cell if ap=4 Å

INTERNATIONAL SCHOOL ON FUNDAMENTAL CRYSTALLOGRAPHY

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