ESO 214: Nature and Properties of Materials Ashish Garg, Dept. of MME, IITK

ESO 214: Nature and Properties of Materials Points in Space Illustrate the concept of point lattice Lack of regular arrangement No periodicity Non-identical neighbourhood Regular arrangement Periodicity Identical neighbourhood of each point Ashish Garg, Dept. of MME, IITK

ESO 214: Nature and Properties of Materials Point Lattice a b  Periodic array of points in space with each point having identical neighborhood Ashish Garg, Dept. of MME, IITK

Example: Point Lattice ESO 214: Nature and Properties of Materials Example: Point Lattice A B A B Not A Point Lattice! Is A Point lattice A and B dont have identical surroundings A and B have identical surroundings Ashish Garg, Dept. of MME, IITK

Motif + Point Lattice = Crystal Structure ESO 214: Nature and Properties of Materials Motif + Point Lattice = Crystal Structure Motif can be defined as a unit of pattern. For a crystal, it is an atom, an ion or a group of atoms or ions or a formula unit or formula units. When motif replaces points in a periodic point lattice, it gives rise to what is called as a crystal with a defined structure. Ashish Garg, Dept. of MME, IITK

Primitive/Non-primitive lattices ESO 214: Nature and Properties of Materials Primitive/Non-primitive lattices Primitive unit-cell: Consists of one lattice point Non-primitive cell: More than one lattice points / unit cells Volume of NP Cell = No of Motifs x Volume of Primitive Unit Cell Ashish Garg, Dept. of MME, IITK

ESO 214: Nature and Properties of Materials Unit Cells Smallest repeatable unit Many unit cells can be formed by As shown by cell vectors Choice of cell is not unique Lattice vector R can be represented as Lattice parameters are usually defined as Lattice Translations a, b, and c Angle ,, and  Ashish Garg, Dept. of MME, IITK

ESO 214: Nature and Properties of Materials Summary Delete this Line Ashish Garg, Dept. of MME, IITK

ESO 214: Nature and Properties of Materials Symmetry in Crystals Symmetry An operation which brings the object back to its original confiscation. Symmetry elements underlying a point lattice (see the figures) Reflection: reflection across a mirror plane Rotation: rotation around a crystallographic axis by an angle such as 360˚/ is an integer of value 1, 2, 3, 4 and 6. Inversion: a point at x,y,z becomes its equivalent at (–x,-y,-z) Rotation-Inversion: Rotation followed by inversion OR Rotation-Reflection: Rotation followed by reflection. Ashish Garg, Dept. of MME, IITK

ESO 214: Nature and Properties of Materials (b) Rotation: for four fold axis, A1 goes to A2, A3 and then A4; for three fold axis A1 goes to A3; two fold axis, A1 goes to A4 (a) Reflection about a plane: point A1 reflects to A2 (d) Rotation-Inversion center: A1 becomes A’1 due to four fold rotation and then inversion takes A’1 to A2 (c) Inversion center Ashish Garg, Dept. of MME, IITK

ESO 214: Nature and Properties of Materials Crystal Systems Based on the symmetry considerations, for various primitive lattice shapes, unit-cells can be divided into seven crystal systems. Crystal system and lattice parameters Minimum symmetry elements: Cubic; a=b=c, ===90 1. Four 3-fold rotation axes Tetragonal; a=bc ===90 2. One 4-fold rotation (or rotation-inversion) axis 3. Orthorhombic; abc ===90 Three perpendicular 2-fold rotation (or rotation-inversion) axis Ashish Garg, Dept. of MME, IITK

ESO 214: Nature and Properties of Materials Crystal system and lattice parameters Minimum symmetry elements: 4. Rhombohedral; a=b=c, ==90 One 3-fold rotation (or rotation-inversion) axis Hexagonal; a=bc ==90 =120 One 6-fold rotation (or rotation-inversion) axis 5. Ashish Garg, Dept. of MME, IITK

ESO 214: Nature and Properties of Materials Crystal system and lattice parameters Minimum symmetry elements: Monoclinic; abc ==90 6. One 2-fold rotation (or rotation-inversion) axis Triclinic abc 90 7. None Ashish Garg, Dept. of MME, IITK

ESO 214: Nature and Properties of Materials Further, seven crystal systems can be categorized in 14 Bravais lattices including both primitive and non-primitive .types. 1. Cubic; a=b=c, ===90 5. Hexagonal; a=bc, ==90, =120 Crystal Systems & Bravais Lattices (Plz put it page title) 4. Rhombohedral; a=b=c, ==90 2. Tetragonal; a=bc, ===90 6. Monoclinic; abc, ==90 7. Triclinic abc; 90 3. Orthorhombic; abc, ===90 Ashish Garg, Dept. of MME, IITK

ESO 214: Nature and Properties of Materials Planes and Directions Essential for the completeness of crystal structures Millers Indices (in the names of William Hallowes Miller) Crystallographic Planes Identification of various faces seen on the crystal (h,k,l) for a plane or {h,k,l} for identical set of planes h, k, l are integers Directions Atomic directions in the crystal [u,v,w] for a direction or <u,v,w> for identical set of directions u, v, w are integers Ashish Garg, Dept. of MME, IITK

Determination of a Crystal Plane ESO 214: Nature and Properties of Materials Determination of a Crystal Plane A crystallographic plane in a crystal satisfies following equation h/a, k/b, and c/l are the intercepts of the plane on x, y, and z axes. a,b,c are the unit cell lengths h, k, l are the integers called as Miller indices and the plane is represented as (h, k, l) Ashish Garg, Dept. of MME, IITK

ESO 214: Nature and Properties of Materials Unit Cell Parameters 4A, 8A and 3A Fractional intercepts: 2A/4A, 6A/8A, 3A/3A Reciprocal of fractional intercepts: 2, 4/3, 1 Convert to smallest set of integers: (6, 4, 3) Ashish Garg, Dept. of MME, IITK

ESO 214: Nature and Properties of Materials Crystal Planes Interplanar angle is given by (cubic only) Interplanar spacing is given by (Cubic) Ashish Garg, Dept. of MME, IITK

ESO 214: Nature and Properties of Materials Ashish Garg, Dept. of MME, IITK

ESO 214: Nature and Properties of Materials Directions Denoted as [u,v,w] Vector components of the direction resolved along each of the crystal axis reduced to smallest set of integers Ashish Garg, Dept. of MME, IITK

ESO 214: Nature and Properties of Materials Crystal Directions How to locate a direction: Example: [231] direction would be 1/3 intercept on cell a-length 1/2 intercept on cell b-length and 1/6 intercept on cell c-length Directions are always denoted with [uvw] with square brackets and family of directions in the form <uvw> Ashish Garg, Dept. of MME, IITK

ESO 214: Nature and Properties of Materials Weiss Zone Law For a direction [u v w] lying in a plane (h k l) h.u + k.v + l.w = 0 Ashish Garg, Dept. of MME, IITK

ESO 214: Nature and Properties of Materials Summary Point lattice is regular arrangement of points in space with identical neighbourhood. Motif (Unit or basis) replaces the point to create crystal structure Unit cell: smallest repeatable unit. Unit-cell containing one lattice point is called as primitive whereas unit-cell containing more than one lattice points is called as non-primitive. There are a total of seven crystal systems and fourteen Bravais lattices. Unit-cell can be defined by crystal planes and directions in terms of their Miller indices. Ashish Garg, Dept. of MME, IITK