# Chapter 3: The Structure of Crystalline Solids

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1 Sapphire: cryst. Al 2 O 3 Insulin : The Structure of Crystalline Solids Crystal: a solid composed of atoms, ions, or molecules arranged in a pattern that is repeated in three dimensions A material in which atoms are situated in a repeating or periodic array over large atomic distances 3.1 Classification Crystalline materials - atoms(ions or molecules) in repeating 3D pattern (called a lattice) - long-range order; ex.: NaCl, Amorphous (noncrystalline) materials - Short range order, not periodic; ex.: liquid water, glass Fractals - Long-range order, symmetry, but not repeating Liquid crystals - long range order of one type; disorder of another - Nematic and smectic 1

2 Translational Symmetry Consider perfect ideal solid The translational group represented by space lattice or Bravais net l Lattice sites defined by: a2 O a1 l = l1 a1 + l2 a2 + l3 a3 The actual definition of a unit cell is to some extent arbitrary NB: atoms do not necessarily coincide with space lattice Space lattice Positions of atoms in crystals can be defined by referring the atoms to the point of intersection with 3D coordinate system -Y +Z 000 -X +X Such network is called space or crystal lattice +Y -Z In ideal crystal the grouping of lattice points about any given point are identical with the grouping about any other lattice point in the crystal lattice 2

3 Unit cell a = a α (0, 0, 0) b c γ β a The size and shape of the unit cell can be described by three lattice vectors: a, b, c or the axial lengths a, b, and c and interaxial angles α, β, γ NB!!!: atoms do not necessarily coincide with lattice points How to choose unit cell? Unit cell: a convenient repeating unit of a crystal lattice; the axial lengths and axial angles are the lattice constants of the unit cell Wigner Seitz cell Larger than needed Wigner Seitz Cell : place the symmetry centre in the centre of the cell; draw the perpendicular bisector planes of the translation vectors from the chosen centre to the nearest equivalent lattice site 3

4 Crystal systems (7) symmetry crystallographic point group set of symmetry operations, (ex.: rotations or reflections), that leave a point fixed while moving each atom of the crystal to the position of an atom of the same kind 14 Bravais Lattices 4

5 230 lattices or spacegroups Lowest in symmetry #1: a b c; α β γ 90o Even higher symmetry structure 5

6 Metallic crystal structures (will talk about metal oxides later) >90% of elemental metals crystallize upon solidification into 3 densely packed crystal structures: Body-centered cubic (bcc) Face-centered cubic (fcc) Hexagonal closepacked (hcp) ex.: Fe, W, Cr ex.: Cu, Ag, Au ex.: Zr, Ti, Zn Important to know: Distance between atoms (d) - in terms of a Number of atoms in the unit cell - each corner atoms shared by 8 cells: 1/8 - each face atom shared by 2 cells: ½ - each edge atom shared by 4 cells: 1/4 Coordination number - Number of nearest neighbours (n.n.); for metals all equivalent Atomic Packing Factor (APF) APF = Volume of atoms in unit cell / Volume of unit cell (a 3 ) 6

7 Coordination number The number of atoms that surround a central atom in a solid is called coordination number C.N. = 3 Corners of triangle C.N. = 4 Corners of tetrahedron C.N. = 6 Corners of octahedron C.N. = 8 Corners of cube Body-Centered Cubic (bcc) Distance between atoms: Atoms/cell: Coordination number: a Atomic Packing Factor (APF): 7

8 Face-Centered Cubic (fcc) Distance between atoms: Atoms/cell: Coordination number: Atomic Packing Factor (APF): Hexagonal close-packed (hcp) Distance between atoms: a and c Atoms/cell: Coordination number: Atomic Packing Factor (APF): 8

9 Density calculations Calculate theoretical density of gold (Au), if the molar density is given: w = kg/mol; and a(au)=0.408nm 3.4 Atom coordinates in the cubic unit cell bcc + Z (0, 0, 0) - X (1, 0, 0) (0, 1, 0) (0, 0, 1) - Y a Y (1, 1, 1) (1, 1, 0) (1, 0, 1) (0, 1, 1) (½, ½, ½) + X - Z 9

10 Face-Centered Cubic (fcc) 8 Cu 6 Cu (0, 0, 0) (½, ½, 0) (1, 0, 0) (0, ½, ½) (0, 1, 0) (½, 0, ½) (0, 0, 1) (½, ½, 1) (1, 1, 1) (1, ½, ½) (1, 1, 0) (½, 0, ½) (1, 0, 1) (0, 1, 1) Sodium Chloride (NaCl) 10

11 Strontium titanate (Sr x Ti y O z ) Sr Ti O 3.5 Directions in cubic unit cell The crystallographic direction indices [u, v, w] - the vectors of the direction resolved along each of the coordinate axes and reduced to the smallest integers + Z - X [2, 0, 0] -Y Y = [1, 0, 0] + X 2 - Z 11

12 Examples: Crystallographically equivalent Parallel direction vectors have the same direction indices: ex.: determine the direction indices of the cubic direction shown Directions are crystallographically equivalent if the atom spacing along each directions are the same ex.: [100], [010], [001], [-100], [0-10], [00-1] = <100> 12

13 3.6 Miller indices for crystallographic planes in cubic unit cells Miller indices of a crystal plane are the reciprocals of the fractional intercepts that the plane makes with the crystallographic x, y, and z axes Most important crystallographic planes: (100), (110), (111) (100) intercepts: 1,, Atomic Packing in Different Planes bcc (100) (110) (111) fcc (100) (110) (111) 13

14 fcc crystallographic planes Cu (100) fcc crystallographic planes Cu (110) Anisotropy of properties in two directions 14

15 fcc crystallographic planes Cu (111) 3 fold symmetry STM (Scanning tunneling microscopy) Cu (100) Cu (111) Atomically resolved STM of Cu(100) Custom Low Temperature UHV STM image of atomic resolution pattern on Cu(111). Image taken at 4.2K. 15

16 Crystallographic planes Single plane (h k l) Notation: planes of a family {h k l} (100); (010); (001); {100} Only for cubic systems: the direction indices of a direction perpendicular to a crystal plane have the same Miller indices as a plane Interplanar spacing d hkl : a d hkl = 2 2 h + k + l Planes in hexagonal crystals 4 coordinate axes (a 1, a 2, a 3, and c) of the HCP structure (instead of 3) Miller-Bravais indices - (h k i l) based on 4 axes coordinate system a 1, a 2, and a 3 are 120 o apart: h k i c axis is 90 o : l 3 indices (rarely used): h + k = - I (h k i l) (h k l) 16

17 Basal and Prizm Planes Basal planes; a 1 = ; a 2 = ; a 3 = ; c = 1 ( ) Prizm planes: ABCD a 1 = +1; a 2 = ; a 3 = -1; c = ( ) Directions in hexagonal crystals - a 3 + c - a 1 [u, v, t, w] or [u, v, w] u + v = -t - a a a 1 - c + a 3 17

18 Principle directions in hcp 3.8 Comparison of crystal structures FCC and HCP metal crystal structures void a void b A B Bb (111) planes of fcc have the same arrangement as (0001) plane of hcp crystal 3D structures are not identical: stacking has to be considered 18

19 FCC and HCP crystal structures void a void b A B C fcc B plane placed in a voids of plane A Next plane placed in a voids of plane B, making a new C plane Stacking: ABCABC A B hcp B plane placed in a voids of plane A Next plane placed in a voids of plane B, making a new A plane Stacking: ABAB 3.9 Volume, planar, and linear density unit cell calculations Volume density of a metal can be obtained by using the hard-sphere atomic model for the crystal structure unit cell mass / unit _ cell Volume _ density _ of _ metal = ρv = volume / unit _ cell Q.: Copper has an fcc crystal structure and an atomic radius of nm. Assuming the Cu atoms to be hard spheres that touch each other along the face diagonal of the fcc unit cell, calculate a theoretical value for the density of Cu in kg/m 3. 19

20 3.10 Allotropy and polymorphism (=..?) Allotropy the ability of element to exist in two or more crystalline structures Fe: bcc fcc bcc In case of compound it is called polymorphism Carbon allotropic forms:? 1. diamond 2. graphite 3. fullerene or buckyballs 4. nanotubes or buckysheets Carbon and its allotropes Diamond Cubic structure (0, 0, 0) (1/2, 1/2, 0) (1/4, 1/4, 1/4) Covalently bonded sp 3 orbitals Isotropic Stiffest, hardest (?) and least compressible High thermal conductivity But no electrical conductivity 20

21 Graphite Graphite hexagonal structure covalently bonded sp 2 orbitals and weak secondary bonds between planes anisotropic High thermal and electrical conductivity (in plane) Bad thermal and electrical conductivity (between plane, along c axis) Fullerenes (Buckyballs) Discovered in 1985 by Harry Kroto and Richard Smalley while blasting graphite with a laser (C 30 to C 100 ) 12 pentagons and 20 hexagons crystallize in fcc structure isotropic bonding: covalent and van der Waals forces 21

22 Carbon nanotubes hexagonal properties depend on the structure more later Carbon Nanotubes The structure can be specified by vector (n, m) which defines how the graphene sheet is rolled up A nanotube with the indices (6,3): the sheet is rolled up so that the atom (0,0) is superimposed on the one labeled (6,3) m = 0 for all zig-zag tubes, while n = m for all armchair tubes 22

23 Summary Unit cells, crystal systems and lattices Atomic coordinates and number of atoms in the unit cell Crystallographic directions and crystal planes Coordination numbers Finding distances between atoms and planes, and characteristic angles In order to find all this parameters: X-ray diffraction, electron microscopy (later) Problems 3.1. Molybdenum is bcc and has an atomic radius of 0.14nm. Calculate a value for its lattice constant a in nanometers Figure below illustrates unit cell of diamond crystal structure. (a) How many carbon atoms are there per unit cell? (b) What is the coordination number for each carbon atoms? (C.N is the number of equidistant nearest neighbors to an atom in a crystal structure) 3.3. Draw direction vectors in unit cubes for the following directions: (a) [111], (b) [1-1-1], (c) [-12-1]; (d) [- 1-13] 3.4. Define the Miller indices of the cubic crystal plane that intersects the following position coordinates: (a)1/2, 0, 1/2); (0, 0, 1); (1, 1, 1); (b) (0, 0, 1/2); (1, 0, 0); (1/2, 1/4, 0) 3.5. Draw in unit cubes the crystal planes that have the following Miller indices: (a) (1-1-1); (b) (10-2), (c) (1-21), (d) (21-3) 3.6 Allotropic phase change. Iron is observed to undergo a transformation from bcc to fcc cubic structure at C. Assuming that in each case the atoms behave as hard spheres and that the size of these spheres is not affected by the transformation, determine the percentage change in volume that occurs. 23

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