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An Introduction to X-Ray Diffraction by Single Crystals and Powders

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An Introduction to X-Ray Diffraction by Single Crystals and Powders An Introduction to X-Ray Diffraction by Single Crystals and Powders Patrick McArdle NUI, Galway, Ireland 1 pma 2019 The Nature of Crystalline Materials • Crystalline materials differ from amorphous materials in that they have long range order. They also exhibit X-ray powder diffraction patterns. • Amorphous materials may have very short range order (e.g. molecular dimers) but do not have long range order and do not exhibit X-ray powder diffraction patterns. • The packing of atoms, molecules or ions within a crystal occurs in a symmetrical manner and furthermore this symmetrical arrangement is repetitive throughout a piece of crystalline material. • This repetitive arrangement forms a crystal lattice. A crystal lattice can be constructed as follows: 2 pma 2019 A 2-dimensional Lattice Pick any position within the 2 dimensional lattice in Fig. 1(a) and note the arrangement about this point. Place a dot at this position and then place dots at all other identical positions as in Fig. 1(b). Join these lattice points using lines to give a lattice grid. The basic building block of this lattice (unit cell) is indicated in Fig. 1(c). 3 pma 2019 Unit Cell Types and The Seven Crystal Systems Cubic a = b = c. = = = 90º. c Tetragonal a = b c. = = = 90º. b Orthorhombic a b c. = = = 90 º. a Monoclinic a b c. = = 90º, 90º. Orthorhombic Triclinic a b c.. 90º. Rhombohedral a = b = c. = = 90 º. (or Trigonal) Hexagonal a = b c. = = 90º, = 120º. In general, six parameters are required to define the shape and size of a unit cell, these being three cell edge lengths (conventionally, defined as a, b, and c), and three angles (conventionally, defined as , , and ). In the strict mathematical sense, a, b, and c are vectors since they specify both length and direction. is the angle between b and c, is the angle between a and c, is the angle between a and b. The unit cell should be right handed. Check the cell above with your right hand Unit Cells may centred on one or more faces or at the centre of the cell. When the unit cells listed above are combined with centring there are 14 different Bravais lattices. 4 pma 2019 The 14 Bravais Lattices Cubic Tetragonal Orthorhombic Body Face Monoclinic Centred Centred Cell Cell I F Triclinic End Face Centred Cell C Trigonal Hexagonal Primitive Cell P 5 pma 2019 Primitive and Centered cells On the previous slide you can see that that a Monoclinic unit cell can be primitive or centred (by convention on the c face). These are referred to as Monoclinic P or Monoclinic C. Choice of Unit Cell A unit cell can be any unit of a lattice array which when repeated in all directions, and always maintaining the same orientation in space, generates the lattice array. There is no unique way of choosing A B a unit cell. For example, each of the cells (A to D) in Fig. 2 are OK. However, the cell favoured by A crystallographers is smallest most orthogonal cell that displays all of the symmetry of the lattice. Thus, cells C and A are the C D B preferred unit cells for the lattices of Figs. 2 and 3 respectively. Fig.Fig. 2 1 Fig.Fig. 32 6 pma 2019 Four important points on crystal lattices: 1. Every crystal system has a primitive Bravais lattice or P type. 2. The distribution of lattice points in a cell must be such as to maintain the total symmetry of the crystal system. Thus, the cubic system cannot have a C-type cell. 3. The fact that a unit cell meets the symmetry requirements of a crystal system does not guarantee its inclusion within the crystal system. This could result if the lattice it generated could be equally well represented by a unit cell type which is already included within the crystal system. The C-type cell for the tetragonal system (see Fig. 4) provides a good example. P - Cell 4. If you apply point 3 to the orthorhombic system you will find that the primitive cell you generate will not have 90º angles. This would not C - cell be orthorhombic and thus orthorhombic C is included in the orthorhombic system. This is shown on the next slide. Fig. 4 7 pma 2019 Tetragonal C and Orthorhombic C centred cells A simplified view down c-axis can be used to illustrate points 3 and 4 on the previous slide. Orthorhombic a ≠ b ≠ c, a = b = = 90º Tetragonal a = b ≠ c, a = b = = 90º a P - Cell a b b C - cell Fig. 4 Smaller cell is Tetragonal P and is preferred Angle not 90° thus the smaller cell is not orthorhombic and must be rejected 8 pma 2019 Symmetry - Point Groups and Space Groups C2 • Point Groups describe symmetry about z a point. • Formaldehyde has the symmetry of the O C2v point group. y • The symmetry operations of C2v are C2 (rotation about z of 360/2), two planes x C of symmetry sv and sv’ (vertical planes) H H sv’ and the identity operation. s • C2v is the Schoenflies symbol for the v point group • mm2 is the Hermann-Mauguin symbol for this point group. + + • Stereographic projections can be used + + to represent point groups • There are 32 crystallographic point groups (also called classes) Stereographic projection • Point groups cannot describe a crystal of C2v or mm2 (pick any lattice – Space Groups are required. + and apply C2 and s to get the others) 9 pma 2019 Crystal System 32 Crystallographic Point Groups Triclinic 1 1 Monoclinic 2 m 2/m Orthorhombic 222 mm2 mmm Tetragonal 4 4 4/m 422 4mm 4 2푚 4/mmm Trigonal 3 3 32 3m 3 m Hexagonal 6 6 6/m 622 6mm 6 2푚 6/mmm Cubic 23 푚3 432 4 3푚 푚3 푚 The reduction of a space group to a point group is described on slide 16 10 pma 2019 Space Groups from Point Groups. Point Group symmetry operations (sym.ops.) • Identity x,y,z → x,y,z • Inversion x,y,z → -x,-y,-z • Mirror e.g. xy plane mirror x,y,z → x,y,-z • Rotation axis rotation by 360/n n = 1,2,3,4 or 6. Space Group sym.ops. also have translational symmetry – screw axes and glide planes A screw axis is represented by nm where n is the rotation (360/n) and m/n is the fraction of the unit cell length of the translation e.g. a 21 along b • 21 along b x,y,z → -x,1/2+y, -z A glide plane has translation (often ½) and a reflection • b glide with a yz mirror x,y,z → -x,½+y,z • Combining these symmetry operations with the 32 point groups leads to the 230 possible 3d Space Groups. 11 pma 2019 The 230 Space Groups The distribution of Space Groups among the Bravais lattice type is shown in Fig. 9. CRYSTAL SYSTEMS (7) BRAVAIS LATTICES (14) SPACE GROUPS (230) P 15 Cubic F 11 36 The International Tables for I 10 Crystallography list the symmetry P 49 properties for all 230 Space Groups. Tetragonal 68 I 19 The 2nd edition was in one volume and P 30 edited by Kathleen Lonsdale. The F 5 59 current edition runs to 7 volumes. Orthorhombic I 9 The CSD or Cambridge Data Base is a C and A 15 repository for the structures of organic P 8 Monoclinic 13 and organometallic compounds which C 5 in 2019 exceeded 1000000 entries. Triclinic P 2 2 Rhombohedral P and R 25 25 Space Group determination is an important step in crystal Hexagonal P 27 27 structure determination. Fig. 9 12 pma 2019 The ABSEN program within Oscail can provide Bar Charts of the contents of the Cambridge Data Base (CSD) No 14 most entries The number of entries by crystal system Entries in the first 25 space groups 13 pma 2019 Space Group No. 14 P21/c • This monoclinic space group has the most entries on the CSD • Read its name as “p21 upon c” • The full name is P 1 21/c 1 (there is no symmetry on a or c) • There are 4 general positions 1 x,y,z 2 -x,1/2+y,1/2-z 3 -x,-y,-z 4 x,1/2-y,1/2+z (21 at 0,y,1/4) Inversion at (0,0,0) Glide at (x,1/4,z) 2 4 glide normal to screen at 1/4b 1 3 Stereographic view down b View down a 21 screw axis glide plane at 1/4b inversion centre 14 pma 2019 Space Group No. 14 P21/c, Example Benzoic acid CSD BENZAC 0 a 1 • The molecules in the unit cell of 2 1 benzoic acid illustrate the 2 positions in the stereographic projection of the space group • 1 and 2 are related by a 21 screw 4 4 • 1 and 3 by an inversion centre 3 • 1 and 4 by a c glide 3 c 15 pma 2019 Converting a Space Group to a Point Group • When the translational parts of the symmetry operations are removed the Space Group is reduced to a point group • The P21/c symm ops are; 1, x,y,z 2, -x,½+y,½-z 3 -x,-y,-z 4 x,½-y,½+z • Removing the ½ s will remove the translations leaving • 1, x,y,z 2 -x,y,-z, 3 -x,-y,-z 4 x,-y,z Stereographic projection of 2/m • With respect to 1 symm ops 2,3 and 4 now are a 2-fold axis along b, inversion and a mirror normal to b.
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