Fundamentals. Crystal patterns and crystal structures. Lattices, their symmetry and related basic concepts

Transcription

1 Fundamentals. Crystal patterns and crystal structures. Lattices, their symmetry and related basic concepts Didactic material for the MaThCryst schools, France

2 Ideal vs. real crystal, perfect vs. imperfect crystal Ideal crystal: Perfect periodicity, no static (vacancies, dislocations, chemical heterogeneities, even the surface!) or dynamic (phonons) defects. Real crystal: A crystal whose structure differs from that of an ideal crystal for the presence of static or dynamic defects. Perfect crystal: A real crystal whose structure contains only equilibrium defects. Imperfect crystal: A real crystal whose structure contains also non-equilibrium defects (dislocations, chemical heterogeneities...). What follows describes the structure of an ideal crystals (something that does not exist!), whereas a real crystal is rarely in thermodynamic equilibrium.

3 What do you get from a (conventional) diffraction experiment? Time and space averaged structure! Time-averaged because the time span of a diffraction experiment is much larger that the time of an atomic vibration. The instantaneous position of an atom is replaced by the envelop (most often an ellipsoid) that describes the volume spanned by the atom during its vibration. Space-averaged because a conventional diffraction experiment gives the average of the atomic position in the whole crystal volume, which corresponds to the position of the atom only if perfect periodicity is respected. Importance of studying the ideal crystal Non-conventional experiments (time-resolved crystallography, nanocrystallography etc.) allow to go beyond the ideal crystal model, but also some information that are often neglected in a conventional experiment (diffuse scattering) can give precious insights on the real structure of the crystal under investigation.

4 Space and time-averaged structure

5 Transformations Affine transformation Euclidean transformation or isometry Symmetry operation

6 Symmetry operations of a crystal pattern A crystal pattern is an idealized crystal structure which makes abstraction of the defects (including the surface!) and of the atomic nature of the structure. A crystal pattern is therefore infinite and perfect. A crystal pattern has both translational symmetry and point symmetry; these are described by its space group. We have to define the concepts of group and its declinations in crystallography.

7 Crystal structure/pattern vs. crystal lattice Crystal Molecule or coordination polyhedron Crystal pattern (crystal structure) Unit cell Lattice nodes Translations

8 The minimal unit you need to describe a crystal pattern Asymmetric unit* fi Unit cell *In mathematics, it is called fundamental region

9 The concepts of chirality and handedness (the left-right difference) Symmetry operations are classified into first kind (keep the handedness) and second kind (change the handedness). If the object on which the operation is applied is non-chiral, the effect of the operation on the handedness is not visible but the nature of the operation (first or second kind) is not affected! The determinant of the matrix representation of a symmetry operation is +1 (first kind) or -1 (second kind)

10 Global vs. local domain of action Real numbers Summation, substraction, multiplication: global operations (can be applied to any pair of numbers) Division: local operation (cannot applied to 0 as divisor). Crystallography Substructure 2 Substructure 1 i,00 i,10 Structure Substructure 0 Substructure 3 Substructure 4 pp local operations, qp partial operations Global (total) operations:those that can be applied to the whole structure

11 Binary operations acting on a set A binary operation An internal law of composition S S S (closure property) A set S We've got a magma Note: An external law of composition acts on two disjoint sets S S' S S S'

12 Possible properties associated to a magma Associativity: a (b c) = (a b) c Presence of identity (neutral element): left identity e a = a right identity a e = a Presence of singular element: left singular s a = s right singular a s = s Presence of the inverse (symmetric) element of each element: left inverse: a -1 a = e right inverse: a a -1 = e Commutativity: a b = b a Presence of a second binary operation: a S b S = c S

13 Algebraic structures globality associativity identity element inverse element Magma Quasigroup Semigroup Loop Monoid Group Semicategory Category Groupoid

14 The notion of symmetry group A symmetry group (G, ) is a set G whose elements are symmetry operations having the following features: the combination (successive application) of two symmetry operations g i and g j of the set G is still a symmetry operation g k of the set G (closure property): g i G g j G g k G (g i g j = g k ) the binary operation is associative: g i (g j g k ) = (g i g j )g k the set G includes the identity (left and right identical): eg i = g i e for each element of the set G (each symmetry operation) the inverse element (inverse symmetry operation) is in the set G: g i -1 g i = g i g i -1 = e In the following we will normally speak of a group G: it is a shortened expression for group (G, ) where is the successive application of symmetry operations g G. Rigorously speaking, G is not a group but just a set!

15 Abelian and cyclic groups Commutativity is NOT included in the group properties (closure, associativity, presence of identity and of the inverse of each element). A group which includes the commutativity property is called Abelian. A special case of Abelian groups: cyclic groups A cyclic group is a special Abelian group in which all elements of the group are generated from a single element (the generator). G = {g, g 2, g 3,., g n = e}

16 The notion of order Order of a group element: the smallest n such that g n = e. If n = 2, then g -1 = g and the element is known as an involution. Order of a group: the number of elements of the group (finite or infinite) If n = the group is infinite

17 A note about nomenclature The elements of the group (G, ) are operations The operations are performed about geometric elements A geometric element together with the set of symmetry operations performed about it is called a symmetry element

18 One-dimensional lattices Symmetry operations Identity Translations Reflections How many 1D lattices are there? Infinite! etc. etc. How many types of 1D lattice are there? One

19 The world in two dimensions E 2 : the two-dimensional Euclidean space

20 Coordinates of lattice nodes, direction indices [uv] [01] [13] [10] [11]

21 Choice of the unit cell t(1,0), t(0,1) t(1,0), t(0,1), t(½,½) t(1,0), t(0,1), t(⅔,⅓), t(⅓,⅔) t(1,0), t(0,1), t(¾,¼), t(½,½), t(¼,¾)

22 Choice of the unit cell (cont.) t(1,0), t(0,1) : primitive (p) unit cell t(1,0), t(0,1), t(½,½) : centred (c) unit cell Conventional unit cell 1. basis vectors form a right-handed reference (applies in 3D space); 2. edges of the cell are parallel to the symmetry directions of the lattice (if any); 3. if more than one unit cell satisfies the above condition, the smallest one is the conventional cell. Reduced cell 1. basis vectors form a right-handed reference (applies in 3D space); 2. basis vectores correspond to the shortest lattice translation vectors; 3. the angles between pairs of basis vectors are either all acute (type I reduced cell) or non-acute (type II reduced cell).

23 Coordinates of lattice nodes, direction indices [uv] [01] ½[13] [10] ½[11]

24 Symmetry in E 2 Operations that leave invariant all the space (2D): the identity Operations that leave invariant one direction of the space (1D): reflections Operations that leave invariant one point of the space (0D): rotations Operations that do not leave invariant any point of the space: translations The subspace left invariant (if any) by the symmetry operation has dimensions from 0 to N (= 2 here) Two independent directions in E 2 two axes (a, b) and one interaxial angle ()

25 Types of symmetry elements and operations in E 2 (Hermann-Mauguin symbols) Operations of the first kind (no change of handedness) Operations of the second kind (change of handedness) Element Operation Rotation point Rotation 1 2/1 2 2/2 3 2/3 4 2/4 6 2/6 Element Reflection line (mirror) m Operations obtained as combination with a translation are introduced later Operation The orientation in space of a reflection line is always expressed with respect to the lattice direction to which it is perpendicular m

26 Exercise

27 Exercise no symmetry direction for this lattice symmetry of the lattice: 2 no symmetry restriction on the cell parameters: a; b; O a b

28 Exercise

29 Exercise two symmetry directions for this lattice, that are taken as axes a and b symmetry of the lattice: 2 m m [10][01] symmetry restriction on the cell parameters: a; b; = 90º O a c b p b c a p Conventional unit cell Primitive unit cell

30 Exercise

31 Exercise two symmetry directions for this lattice, that are taken as axes a and b symmetry of the lattice: 2 m m [10][01] symmetry restriction on the cell parameters: a; b; = 90º O a c=p b c=p Conventional (primitive) unit cell

32 Exercise

33 Exercise This lattice has four symmetry directions. Those corresponding to the shortest period are taken as axes a 1 and a 2.

34 Exercise b,a 2 a,a 1 This lattice has four symmetry directions. Those corresponding to the shortest period are taken as axes a 1 and a 2.

35 Exercise b,a 2 a,a 1 This lattice has four symmetry directions. Those corresponding to the shortest period are taken as axes a 1 and a 2.

36 Exercise b,a 2 a,a 1 Lattice symmetry: 4 m m 101

37 Exercise b,a 2 a,a 1 Lattice symmetry: 4 m m p cell (conventional) 101

38 Exercise b,a 2 a,a 1 Restrictions on cell parameters: a = b; = 90

39 This lattice has six symmetry directions. Two among the three corresponding to the shortest period are taken as axes a 1 et a 2.

40 Restrictions on the cell parameters: a = b; = 120

41 Direction indices [uv] in the hexagonal lattice of E 2 a 1 a 2 [01] [11] [12] [10] [21] [11]

42 Conventional cell parameters and symmetry directions in the four crystal families of E 2 monoclinic No restriction on a, b, No symmetry direction orthorhombic No restriction on a, b ; = 90º [10] and [01] tetragonal a = b ; = 90º 10 ([10] and [01]) 11 ([11] and [11]) hexagonal a = b ; = 120º 10 ([10], [01] and [11]) 1 ([21], [12] and [11]) a b b a a 1 mp op a 2 a 2 a 1 a p b p tp hp a c b c Crystal families: monoclinic, orthorhombic, tetragonal, hexagonal Type of lattice*: primitive, centred *Lattice whose conventional unit cell is primitive or centred Kettle S.F.A, Norrby L.J., J. Chem. Ed. 70(12), 1993, oc

43 Why not 5, then?

44 Now let's put something IN our unit cell!

45 The concept of holohedry Lattice «Object» (content of the unit cell) Pattern Symmetry of the lattice : S L = 4mm Symmetry of the object : S o = 4mm Symmetry of the pattern : S p = 4mm S p = S L : holohedry

46 The concept of merohedry Symmetry of the lattice: S L = 4mm Symmetry of the object: S o = 3m Symmetry of the pattern : S p = m = 4mm 3m S p < S L : merohedry

47 The concept of merohedry Symmetry of the lattice: S L = 4mm Symmetry of the object: S o = 3m Symmetry of the pattern : S p = 1 = 4mm 3m S P < S L : merohedry

48 The concept of merohedry The crystallographic restriction (n = 1, 2, 3, 4, 6) applies to the lattice and to the pattern, but not to the content of the unit cell! Symmetry of the lattice : S L = 4mm Symmetry of the object : S o = 5m Symmetry of the pattern : S p = m = 4mm 5m S p < S L : merohedry

49 The notion of subgroup

50 Exercise: find the symmetry elements, the symmetry operation and the symmetry group of a square in E Symmetry elements: Symmetry operations: 4, m [10], m [01], m [11], m [11] 4 1, 4 2 =2, 4 3 =4-1, 4 4 =1, m [10], m [01], m [11], m [11] m [10] Check that these operations do form a grop 4 m [11] 1 m [11] The identity belong to the set (it corresponds to 4 4 and m 2 ) The inverse of each operation belongs to the set (1-1 =1; m - 1 =m; 2-1 =2, 4-1 =4 3 ) To check the closure and associativity, build the multiplication (Cayley) table. m [01] a 2 (b) a 1 (a)

51 Multiplication (Cayley) table of a square in E m [10] m [11] m [01] m [11] m [10] m [10] m [11] m [01] m [11] m [10] m [11] m [01] m [11] m [10] m [11] m [01] m [11] m [01] m [11] m [10] m [11] m [11] m [01] m [11] m [10] m [11] m [10] m [11] m [01] m [01] m [11] m [11] m [11] m [10] m [01] m [10] m [11] m [11] m [11] m [01] m [10] m [11] m [11] m [01] Show that the portion of the table within the red square corresponds to a subgroup a 2 (b) a 1 (a)

52 Order and index of a subgroup group G, order G group H, order H H G H is a divisor of G The ratio i = G / H is called the index of H in G

53 Desymmetrisation of a square Order Index 8 1 4mm = {1, 4 1, 2, 4 3, m [10], m [01], m [11], m [11] } 4 2 {1, 4 1, 2, 4 3 } = 4 {1, 2, m [10], m [01], } = 2mm. {1, 2, m [11], m [11] } = 2.mm 2 4 {1, 2} = 2 {1, m [10] } =.m. {m [01] } =.m. {1, m [11] } =..m {1, m [11] } =..m 1 8 {1} = 1

54 Desymmetrisation of a square Order Index 8 4mm 1 4 2mm. 4 2.mm 2 2.m [10]..m [01]. 2..m [11]..m [11]

55 Holohedry 2 : {1,2} Subgroup 1 : {1} Holohedry 2mm : {1,2,m [10],m [01] } Subgroups 2 : {1,2}, m : {1,m [10] }, {1,m [01] }, 1 : {1} Holohedry 4mm : {1, 4 1, 4 2 = 2, 4 3 = 4-1, m [10], m [01], m [11], m [11] } Subgroups 4 : {1, 4 1, 4 2, 4 3 }, 2mm : {1, 2, m [10], m [01] }, {1, 2, m [11], m [11] }, 2 : {1, 2}, m : {1,m [10] }, {1,m [01] }, {1,m [11] }, {1,m [11] }, 1 : {1} Holohedry 6mm : {1, 6 1, 6 2 = 3, 6 3 = 2, 6 4 = 3-1, 6 5 = 6-1, m [10], m [01], m [11], m [11], m [12], m [21] } Subgroups 6 : {1, 6 1, 6 2, 6 3, 6 4, 6 5 }, 3m : {1, 3 1, 3 2, m [10], m [01], m [11] }, {1, 3 1, 3 2, m [11], m [12], m [21] }, 3 : {1, 3, 3-1 }, 2mm : {1, 2, m [10], m [12] }, {1, 2, m [01], m [21] }, {1, 2, m [11], m [11] }, 2 : {1,2}, m : {1,m [10] }, {1,m [01] }, {1,m [11] }, {1,m [11] }, {1,m [12] }, {1,m [21] }, 1 : {1}

56 Holohedries and merohedries 4mm 4mm 6mm 6mm 6 3m 2.mm 2mm. 4 2mm 3m1 31m 2mm 2 m 3 m [010] m [100] 1 Tree of maximal group-subgroups (in bold the holohedries) Normal subgroup Conjugate subgroups

57 Crystal families Point-group types that satisfy the following criteria belong to the same crystal family: 1.the correspondent to the same holohedry; 2.they are in group-subgroup relation; 3.the types of Bravais lattice on which they act have the same number of free parameters. Teétragonal family: 1 free parameter (a) 4mm 6mm Hexagonal family:1 free parameter (a) 6 3m 4 2mm 2 Monoclinic family: 3 free parameters (a,b,) m 1 Orthorhombic family:2 free parameters (a,b) 3

58 Crystal system Point-group types that act on the same types of Bravais lattices belong to the same crystal system. Type of group mp op oc tp hp Crystal system 2, 1 monoclinic 2mm, m orthorhombic 4mm, 4 tetragonal 6mm, 6, 3m, 3 hexagonal No. of free parameters

59 Lattice system Point-group types that correspondent to the same lattice symmetry belong to the same lattice system. 6mm, 6, 3m, 3 : hexagonal lattice system 4mm, 4 : tetragonal lattice system 2mm, m : orthorhombic lattice system 2, 1 : monoclinic lattice system

60 The world in three dimensions E 3 : the three-dimensional Euclidean space

61 The inversion an operation that exists only in odd-dimensional spaces Why? Inversion centre (point)

62 x y z w... x y z w 1 = -I n det(1) = (-1) n odd-dimension spaces: -1 even-dimension spaces: +1 1 operation Example in E 2 Two-fold rotation! Matrix representation of the inversion in E n

63 Symmetry in three dimensions (E 3 : the three-dimensional Euclidean space) Operations that leave invariant all the space (3D): the identity Operations that leave invariant a plane (2D): the reflections Operations that leave invariant one direction of the space (1D): the rotations Operations that leave invariant one point of the space (0D): the roto-inversions Operations that do not leave invariant any point of the space : the translations The subspace left invariant (if any) by the symmetry operation has dimensions from 0 to N (= 3 here) Three independent directions in E 3 three axes (a, b, c) and three interaxial angles (,, )

64 Types of symmetry elements and operations in E 3 (Hermann-Mauguin symbols) Operations of the first kind (no change of handedness) Operations of the second kind (change of handedness) Element Operation Direct axis Rotation 1 2/1 2 2/2 3 2/3 4 2/4 6 2/6 Element Operation Inverse axis Rotoinversion 1 (centre) inversion 2 (m) reflection 3 2/3 + inversion 4 2/4 + inversion 6 2/6 + inversion Operations resulting from a combination with a translation are introduced later The orientation in space of a rotation axis or mirror plane is always expressed with respect to the lattice direction to which these are parallel or perpendicular respectively

65 Labelling of axes and angles in E 3 c b a

66 Graphic symbols for symmetry elements n-fold rotation axis n-fold rotoinversion axis n-fold rotation axis and mirror plane perpendicular to it 2 2-fold rotation axis 1 one-fold rotoinversion axis (inversion centre) 2/m 3 3-fold rotation axis 4 4-fold rotation axis 6 6-fold rotation axis 3 three-fold rotoinversion axis 4/m 4 four-fold rotoinversion axis 6/m 6 (3/m) six-fold rotoinversion axis two-fold rotoinversion axis is a mirror plane First-kind operations Second-kind operations Operations including translations are introduced later The orientation of a mirror plane is indicated by the vector normal to it

67 The orientation of a mirror plane is indicated by the vector normal to it Only one Infinite!

68 Equivalence of 2 and m b I d II II q

69 Combination of 2 and 1 gives m b I d II II II p q I Applies to even-fold rotations as well, because they all contain a twofold rotation 4 2 = 2; 6 3 = 2

70 Types of unit cells that bring a letter in E 3 a b c P a b c a b c C B a b c A S a b c I a b c F The unit cell of type R is introduced later

71 Crystal families and types of lattices in E 3 : the triclinic family a mp b + a third direction inclined on the plane 2D 3D = 1 triclinic crystal family (anorthic) ap One symmetry element: the inversion centre No symmetry direction The conventional unit cell is not defined no reason to choose à priori a centred cell Symmetry of the lattice: 1 No restriction on a, b, c,,,

72 Crystal families and types of lattices in E 3 : the monoclinic family a b + a third direction perpendicular to the plane mp 2D 3D = 2/m monoclinic crystal family One symmetry direction (usually taken as the b axis). The conventional unit cell has two right angles ( and ) Symmetry of the lattice: 2/m Two independent types of unit cell respect these conditions mp ms

73 A lattice of type mp is equivalent to mb c B c P c B b B a B a P = (a B -c B )/2 ; c P = (a B +c B )/2 a B b B b P a P a P = (a B -c B )/2 c B c P a B b B b P a P

74 A lattice of type mp is NOT equivalent to mc c C b P b C a C b C a P a C c C b P b C a C b C a P a C

75 Lattices of type mc, ma, mi and mf are all equivalent (ms) c C c A ma mc b A b C a A a A = -c C c A = a C a C b C mc mf mc mi a I = a C -c C c F = -a C +2c C c I c F b I b A a F a C a I b F b C

76 Three monoclinic settings b-unique c-unique a-unique unrestricted by symmetry unrestricted by symmetry unrestricted by symmetry mb = mp ma = mc = mi = mf mc = mp ma = mb = mi = mf ma = mp mb = mc = mi = mf The symbol of a monoclinic space group will change depending on which setting and what type of unit cell you choose, leading to up to 21 possible symbols for a monoclinic space group (the monoclinic monster ).

77 Crystal families and types of lattices in E 3 : the orthorhombic family a b a b op oc + a third direction perpendicular to the plane orthorhombic crystal family 2D 3D 2mm 2mm 1 = 2/m2/m2/m op oi os of Three symmetry directions (axes a, b, c) The conventional unit cell has three right angles (,, ) Symmetry of the lattice: 2/m 2/m 2/m Four types of cell respect these conditions The unit cell with one pair of faces centred can be equivalently described as A, B or C by a permutation of the (collective symbol : S)

78 Three possible setting for the os type of lattice in E 3 c b a a b c a b oc ob oa os c

79 Crystal families and types of lattices in E 3 : the tetragonal family a b tp + a third direction perpendicular to the plane tp ti tetragonal crystal family Five symmetry directions (c, a&b, the two diagonals in the a-b plane) Symmetry of the lattice: 4/m 2/m 2/m The conventional cell has three right angles and two identical edges Two independent types of unit cell respect these conditions: tp (equivalent to tc) and ti (equivalent to tf) ½ 2D ½ 3D 4mm 4mm 1 = 4/m2/m2/m tc ½ ½ ti ½ tp ½ tf ½

80 Why unit cells of type ta et tb cannot exist? fold axis! a 1 a

81 Why unit cells of type ta et tb cannot exist? fold axis! a 1 a

82 Why unit cells of type ta et tb cannot exist? fold axis! a 1 a

83 Why unit cells of type ta et tb cannot exist? fold axis! a 1 a tf!

84 Why unit cells of type ta et tb cannot exist? fold axis! a 1 a tf! ti!

85 Crystal families and types of lattices in E 3 : the cubic family a 2 + a third direction perpendicular to the plane AND c = a = b a 1 cubic crystal family tp cp ci cf

86 Three-fold rotoinversion along the 111 direction

87 Crystal families and types of lattices in E 3 : the cubic family a 2 + a third direction perpendicular to the plane AND c = a = b a 1 tp cp ci cf cubic crystal family 2D 3D 4mm 4mm 3 = 4/m32/m Thirteen symmetry directions (the 3 axes; the 4 body diagonals ; the six face diagonals) Symmetry of the lattice: 4/m 3 2/m The conventional unit cell has three right angles and three identical edges Three types of unit cell respect these conditions : cp, ci et cf

88 Crystal families and types of lattices in E 3 : the hexagonal family a 1 a 2 + a third direction perpendicular to the plane 2D hp 3D 6mm 6mm 1 = 6/m2/m2/m C hexagonal crystal family + a new type of unit cell! A 1 A 2

89 Uniqueness of the hexagonal crystal family in E 3 Symmetry directions C a 3 a 2 a 1 A 2 a 3 A 1 a 1 a 2 A 1, A 2, C hexagonal axes parallel to the symmetry directions A a 1 A 2 1, a 2, a 3 rhombohedral axes NOT parallel to the symmetry directions Two types of lattice with different symmetry in the hexagonal crystal family

90 Symmetry difference between hp and hr types of lattice 6/m2/m2/m [110] H [210] H [100] H 32/m [101] R A 3 [110] R [110] H [120] H a 3 a 2 [010] H [011] R A 2 [010] H [011] R a 1 [120] H [110] H A 1 z = 0 [100] H [210] H [110] H z = 1/3 [110] R [101] R z = 2/3

91

92 Lattice systems: classification based on the symmetry of the lattices ap mp ms tp ti 1 triclinic 2/m monoclinic 4/m 2/m 2/m (4/mmm) tetragonal op oi os of 2/m 2/m 2/m (mmm) orthorhombic hp 6/m 2/m 2/m (6/mmm) hexagonal hr 3 2/m (3m) rhombohedral cp ci cf 4/m 3 2/m (m3m) cubic

93 Crystal systems: morphological (macroscopic) and physical symmetry

94 Crystal systems: morphological (macroscopic) and physical symmetry A 1 1 or 1 A 2 2 or m or 2/m 3A 2 Three twofold elements triclinic monoclinic orthorhombic A 4 A 3 A 6 tetragonal trigonal hexagonal 4A 3 cubic

95 Crystal families, crystal systems, lattice systems and types of Bravais lattices in E 3 6 crystal families conventional unit cell 7 crystal systems (morphological symmetry) 7 lattice systems (lattice symmetry) 14 types of Bravais lattices (**) a = anortic* (triclinic, asynmetric, tetartoprismac...) no restriction on a ; b ; c,,, triclinic triclinic ap m = monoclinic (clinorhombic, monosymmetric, binary, hemiprismatic, monoclinohedral,...) no restriction on a ; b ; c ; = = 90º monoclinic monoclinic mp ( mb ) ms ( mc, ma, mi, mf ) o = orthorhombic (rhombic, trimetric, terbinary, prismatic, anisometric,...) t = tetragonal (quadratic, dimetric, monodimetric, quaternary...) no restriction on a ; b ; c. = = = 90º a = b ; = = = 90º no restriction on c orthorhombic tetragonal orthorhombic tetragonal op os ( oc, oa, ob ) oi of tp ( tc ) ti ( tf ) h = hexagonal (senaiey, monotrimetric...) a = b ; = = 90º, = 120º no restriction on c trigonal (ternary...) (***) hexagonal rhombohedral hexagonal hr hp c = cubic (isometric, monometric, triquaternary, regular, tesseral, tessural...) a = b = c = = = 90º cubic cubic cp ci cf (*) Synonyms within parentheses. (**) S = one pair of faces centred. Within parentheses the types of lattices that are equivalent (axial setting change see the monoclinic example). (***) Crystals of the trigonal crystal system may have a rhombohedral or hexagonal lattice

96 Symmetry directions of the lattices in the three-dimensional space (directions in the same box are equivalents by symmetry ) Lattice system Parameters First symmetry direction Second symmetry direction Third symmetry direction triclinic a ; b ; c any monoclinic (b-unique) a ; b ; c = 90º, any [010] orthorhombic a ; b ; c = 90º [100] [010] [001] tetragonal a = b ; c = 90º [001] [100] [010] 100 [110] [110] 110 rhombohedral rhombohedral axes hexagonal axes a = b = c a = b ; c = 90º, = 120º [111] [001] [110] [011] [101] 0 [100] [010] [110] 00 hexagonal a = b ; c = 90º, = 120º [001] [100] [010] [110] 00 [110] [120] [210] 0 cubic a = b = c = 90º [001] [100] [010] 001 [111] [111] [111] [111] 1 [110] [110] [011] [011] [101] [101] 0

97 Symmetry restrictions on cell parameters (conventional cell) Triclinic family: none Monoclinic family = = 90º Orthorhombic family = = = 90º Tetragonal family a = b = = = 90º Hexagonal family Hexagonal axes: a = b; = = 90º ; = 120º Rhombohedral axes: a = b = c; = = Cubic family a = b = c = = = 90º Restrictions incorrectly given in many textbooks a b c 90 a b c; = = 90 ; 90 a b c; = = = 90 a = b c; = = = 90 a = b c; = = 90 ; = 120 a = b = c; = = 90 a = b = c; = = = 90 For the crystal structure, too restrictive For the crystal lattice (geometric concept) insufficient

98 Why insufficient for the crystal lattice? A simple example mp if = 90 the lattice is op (obviously) Let us suppose cos= -a/2c c ob! a