Roger Johnson Structure and Dynamics: The 230 space groups Lecture 3

Transcription

1 Roger Johnson Structure and Dnamics: The 23 space groups Lecture Summar In the first two lectures we considered the structure and dnamics of single molecules. In this lecture we turn our attention to 3D crstals materials with periodic lattices. To describe the structure of crstals we must introduce translational smmetr The crstal lattice and basis Crstal structures are periodic arras of atoms subdivided into a basis and a lattice. The basis is a single pattern of atoms, much like a single molecule. The (direct) lattice is a three dimensional framework of points that maps out the periodic repetition of the basis. All points of the lattice are equivalent b the translational smmetr of the crstal, such that the crstal looks the same from an point of the lattice. The 3D lattice is defined b three vectors, a, b, and c, known as the lattice vectors, from which all lattice points at position r n = n 1 a + n 2 b + n 3 c can be generated. The lattice is generall parametrised b si numbers; three describe the lengths of the lattice vectors, a = a, b = b, and c = c, and three describe the angles between them, α (angle between b and c), β (angle between a and c), and γ (angle between a and b). The si numbers together are known as the lattice parameters. The basis and the lattice must have compatible smmetries, which is categorised b the crstal sstem. The crstal sstem is dependent upon the point group smmetr of the basis, as summarised in Table 1, and it imposes constraints on the lattice parameters (Table 1). The convention for the orientation of lattice aes, a, b, and c, with respect to the smmetr operators of the basis is as follows (eaminable crstal sstems onl). Triclinic: arbitrar orientation. Monoclinic: 2 parallel to b or b normal to m. Orthorhombic: a, b, and c parallel to the three 2 or m elements. Tetragonal: c parallel to 4 or 4. The lattice can be divided into repeating unit cells whose edges are the lattice vectors a, b, and c. The primitive unit cell is the unit cell of the lattice with smallest volume (Figure 1). The primitive unit cell, therefore, has a lattice point at each verte, and n 1, n 2, and n 3 are integers. In most cases, the primitive unit cell preserves the point smmetr of the crstal sstem. If the primitive unit cell does not preserve the point smmetr of the crstal sstem, we define a centred unit cell that does (red cell in Figure 2). Such cells include additional translational smmetries with rational but non-integter n 1, n 2, and n 3. These additional smmetries are summarised in Table 2. Table 1: Point group and lattice parameter constraints that define the seven crstal sstems (trigonal, heagonal, and cubic crstal sstems are non-eaminable). Crstal sstem Point smmetr Lattice parameters Triclinic One 1 or 1 a b c, α β γ Monoclinic One 2 or m( 2) a b c, α = γ = 9, β 9 Orthorhombic Three orthogonal 2 or m( 2) a b c, α = β = γ = 9 Tetragonal One 4 or 4 a = b c, α = β = γ = 9 Trigonal One 3 or 3 a = b = c, α = β = γ 9 Heagonal One 6 or 6 a = b c, α = β = 9, γ = 12 Cubic Four 3 or 3 a = b = c, α = β = γ = 9 Page 1 of 8

2 Roger Johnson Structure and Dnamics: The 23 space groups Lecture 3 Figure 1: Eample of a primitive lattice. The primitive unit cell is drawn in red. Figure 2: Eample of a centred lattice. The centred unit cell is drawn in red, and the primitive unit cell in blue. Table 2: Centring translations of the 14 Bravais Lattices (rhombohedral centring is non-eaminable). Centring tpe Description Translation A Base-centred in bc-plane (, 1 2, 1 2 ) B Base-centred in ac-plane ( 1 2,, 1 2 ) C Base-centred in ab-plane ( 1 2, 1 2, ) F Face-centred ( 1 2, 1 2, ) and ( 1 2,, 1 2 ) and (, 1 2, 1 2 ) I Bod-centred ( 1 2, 1 2, 1 2 ) R Rhombohedral centring ( 2 3, 1 3, 1 3 ) and ( 1 3, 2 3, 2 3 ) Page 2 of 8

3 Roger Johnson Structure and Dnamics: The 23 space groups Lecture 3 Each crstal sstem has a primitive unit cell plus an centred cells that are compatible with the smmetr of the crstal sstem. Taken together, one finds 14 unique 3D lattices these are the 14 Bravais lattices (see Supplementar Material 1). 3.3.Fractional coordinates The position of atoms within a crstal are defined within the basis of the lattice, r = a + b + c where,, and are fractional coordinates tpicall given for the first unit cell onl < < 1, < < 1, < < 1. All smmetr equivalent atoms in the crstal can be found b operating on the fractional coordinates Smmetr operators of a 3D crstal The translational smmetr of the lattice is represented b an infinite set of translational smmetr operators, written as column vectors in the basis of the lattice. The lattice is described b an infinite set of translation operations, therefore, unlike point groups, the smmetr group of a crstal is infinite. We represented the smmetr operators of the molecular point groups as orthogonal matrices. As is the case for translation operators, we will now represent the rotation and rotation-inversion operators as matrices in the basis of the lattice. For eample, the smmetr operation 3 in a Cartesian basis is: which, in a heagonal crstal basis, is simplified to: Not onl does the crstal smmetr group contain an infinite set of translation operators, it will also contain an infinite set of rotations and rotation-inversion operators (should the eist) related b translational smmetr. As ever unit cell is identical, we need onl consider the rotation and rotation-inversion operators of a single unit cell. Unlike molecular point groups, smmetr elements of a crstal do not intersect at a single point. We must therefore define the position of the rotation or rotation-inversion smmetr elements within the unit cell b including a set of fractional coordinates in the notation. For eample, 2 [,,] is a twofold rotation about, and through the origin. An smmetr operation can be written as a rotational component plus a translational component. Consider, for eample, 2 [,, ] operating on an atom with fractional coordinates [,, ]. Step 1: Translate [,, ] b a vector that brings the smmetr operation to the origin: Page 3 of 8

4 Roger Johnson Structure and Dnamics: The 23 space groups Lecture 3 Step 2: Perform rotation about the origin (R 2 is in the basis of the lattice): R 2 Step 3: Translate back: R 2 + In general, if a smmetr operator, R, is located at [,, ], the smmetr operation can be written: = R + R where the first term is a rotation, and the second term is a translation. In addition to the grid of smmetr elements mapped out b the direct lattice vectors, an additional set of smmetr elements will be generated, also b translational smmetr. Take, for eample, an atom at [,, ] and an inversion centre at [,,]. The inversion at the origin generates an atom at [,, ], which has an equivalent position at [ + 1,, ] b translational smmetr [1,,], i.e., + 1 = R = R 1 + R 1 B comparison with the general epression derived above, the inversion at the origin plus a lattice translation is equivalent to inversion about the point [,,]. Etending to 3D, if a unit cell has an inversion centre at the origin, it will also have inversion centres at [,,], [,,], [,,], [,,], [,,], [,,], and [,,] Screw aes and glide planes If the translational part of the smmetr operator has a component orthogonal to the rotation ais, (i.e. it is not invariant under the rotation), then the smmetr operation is simpl a rotation centred at a position determined b the translational part. In special cases the translational component is parallel to the rotational component, and in this case new smmetr elements are generated, known as roto-translations, so long as: a) The rotational operator is either 2, 3, 4, 6 or m, and is alwas applied first. b) The translational part is equal to a lattice vector divided b the order of the rotational part, such that the translational smmetr of the lattice is preserved. Roto-translation operators composed of a rotation operator (2, 3, 4, or 6) are known as screw aes (see Supplementar Material 2). Roto-translation operators composed of a mirror, m, are known as glide planes (see Supplementar Material 2). Page 4 of 8

5 Roger Johnson Structure and Dnamics: The 23 space groups Lecture The 23 crstallographic space groups A space group is a group of smmetr operators that satisf the aioms of a group, and leave a 3D crstal structure (lattice + basis) invariant. There are 23 crstallographic space groups, which are tabulated in the International Tables for Crstallograph, published b IUCr (Wile). The space group notation is similar to the point group notation defined in Lecture 1. The space group smbol begins with a capital letter that denotes the lattice centring. This letter is followed b the smbols of the rotation, rotation-inversion, or roto-translation operators. The order of the smbols follows the same convention as for point group notation. Take orthorhombic P mc2 1 for eample: If a space group does not include roto-translations, there will be a point in the unit cell that is invariant under all rotation or rotation-inversion smmetr operations. In this case the group is known as smmorphic, and the point group smbol is found eplicitl in the space group smbol. For eample, space group P 2/m has point group 2/m. If a space group does include roto-translations then there is no invariant point in the unit cell. However, we still consider a respective point group that can be found b removing the translation from the roto-translation. For eample, space group P mc2 1 has point group mm2. The origin choice of the lattice is arbitrar, so we must decide ourselves where to place it with respect to the smmetr elements of the space group. Tpicall, if inversion centres eists, we choose the origin to coincide with an inversion centre. There is an advantage in doing so as it simplifies structure factor calculations (Lectures 4 and 6). We illustrate the space group using the same smmetr element smbols defined in Lecture 1, located on planar projections of the unit cell (see Figure 3) Wckoff positions General positions are smmetr equivalent positions generated b individuall appling all rotation, rotation-inversion, roto-translation, and lattice centring smmetr operators of the space group to the general fractional coordinates [,, ]. There are therefore as man smmetr equivalent general positions as there are smmetr elements in the point group, times the number of centring translations. Special positions eist within the unit cell when an atomic position intersects a rotation or rotationinversion smmetr element of the space group. Atoms are often found on these high smmetr positions known as Wckoff sites. The number of smmetr equivalent positions of a given Wckoff site is known as the site multiplicit. We can assign a site point group to each Wckoff site (not to be confused with the point group of the crstal structure), and the site multiplicit can be calculated b dividing the number of smmetr elements in the crstal point group b the number of smmetr elements in the site point group. Note that the general position is also a Wckoff site - that with highest multiplicit. Page 5 of 8

6 Roger Johnson Structure and Dnamics: The 23 space groups Lecture 3 Figure 3: Space group diagram for P 2/m Page 6 of 8

7 Supplementar material 1: The 14 Bravais lattices: (N.b. Trigonal, heagonal, and cubic are non-eaminable) Triclinic Monoclinic P P C Orthorhombic P C F I Tetragonal Trigonal P I R Heagonal Cubic P P F I

8 Supplementar Material 2: Roto-translations Screw aes (enantiomorphs are noted in colour) 1 st rotate about ais R, then translate along screw vector T = [,,t] R-ais Label R t Sketch Label R t Sketch (3 2 ) 3+ 1/3 6 1 (6 5 ) 6+ 1/6 6 2 (6 4 ) 6+ 1/3 3 2 (3 1 ) 3+ 2/ (4 3 ) 4+ 1/ (6 2 ) 6+ 2/3 6 5 (6 1 ) 6+ 5/6 4 3 (4 1 ) 4+ 3/4 Glide planes 1 st reflect in the mirror plane, m, then 'glide' along T. T is alwas of fractional length ½ as m m = E Label m T Notes a [,,] or [,,] [½ ] b [,,] or [,,] [ ½ ] c [,,] or [,,] [ ½] n [,,] or [,,] and equivalent [½ ½ ] and equivalent e [,,] and [,,] [½ ] and [ ½ ] When there eists two smmetr related parallel planes (rare!) d Man... [½ ½ ] or [½ ½ ½] F and I centred lattices onl