INTRODUCTION TO CERAMIC MINERALS 1.7 BASIC CRYSTALLOGRAPHY

BASIC CRYSTALLOGRAPHY Crystallography is the experimental science of determining the arrangement of atoms in solids. In older usage, it is the scientific study of crystals. The word "crystallography" is derived from the Greek words crystallon = cold drop / frozen drop, with its meaning extending to all solids with some degree of transparency, and graphein = write.

BASIC CRYSTALLOGRAPHY UNIT CELL > a convient repeating unit of a space. >The axial length and axial angles are the lattice constants of the unit cell.

BASIC CRYSTALLOGRAPHY LATTICE: >an imaginative pattern of points in which every point has an environment that is identical to that of any other point in the pattern. >A lattice has no specific origin as it can be shifted parallel to itself.

BASIC CRYSTALLOGRAPHY PLANE LATTICE: >A plane lattice or net represents a regular arrangement of points in two-dimensions.

BASIC CRYSTALLOGRAPHY SPACE LATTICE: > a three dimensional array of points each of which has identical surrounding

BASIC CRYSTALLOGRAPHY BRAVAIS LATTICE: >Unique arrangement of lattice points >crystal systems are combined with the various possible lattice centering >describe the geometric arrangement of the lattice points, and thereby the translational symmetry of the crystal >also refer as SPACE LATTICES.

BASIC CRYSTALLOGRAPHY SEVEN STRUCTURE SYSTEM OF CRYSTAL

THE BRAVAIS LATTICE 14 BRAVAIS LATTICE TRICLINIC MONOCLINIC SIMPLE BASED-CENTERED ORTHORHOMBIC SIMPLE ORTHORHOMBIC BASED-CENTERED ORTHORHOMBIC BODY-CENTERED ORTHORHOMBIC FACE-CENTERED HEXAGONAL RHOMBOHEDRAL (TRIGONAL) TETRAGONAL SIMPLE TETRAGONAL BODY-CENTERED CUBIC (ISOMETRIC) SIMPLE CUBIC (ISOMETRIC) BODY-CENTERED CUBIC (ISOMETRIC) FACE-CENTERED 14 BRAVAIS LATTICE

SYMMETRY Symmetry in an object may be defined as the exact repetition, in size, form and arrangement, of parts on opposite sides of a plane, line or point. Symmetry element is a simple geometry operation such as: translation, inversion, rotation or combinations thereof.

Figure 34: Symmetry Elements For Cubic Form

SYMMETRY Symmetry operations can include: mirror planes, which reflect the structure across a central plane, rotation axes, which rotate the structure a specified number of degrees, center of symmetry or inversion point which inverts the structure through a central point  

INTERFACE ANGLE Interface angle ( α ) is an angle between two crystal faces that is measured in a plane perpendicular to both of the crystal faces concerned. This may be done with a contact goniometer   Figure 35: (a) Contact Goniometer and (b) Measurement 0o Reference

ZONES & ZONE AXIS Zones the arrangement of a group of faces in such a manner that their intersection edges are mutually parallel.  Zones axis An axis parallel to these edges is known In Figure 36 the faces m’, a, m and b are in one zone, and b, r, c, and r’ in another. The lines given [001] and [100] are the zones axis.

BASIC CRYSTALLOGRAPHY MILLER INDICES

BASIC CRYSTALLOGRAPHY Miller Indices ~a notation system in crystallography for planes and directions in crystal (Bravais) lattices ~a shorthand notation to describe certain crystallographic directions and planes in a material ~ (h,k,l) Miller-Bravais Indices A special shorthand notation to describe the crystallographic planes in hexagonal close packed unit cell

William Hallowes Miller (1801 – 1880) Miller Indices of directions and planes William Hallowes Miller (1801 – 1880)

ATOMIC COORDINATE z Coordinate of Points 0,0,1 x y 0,0,0 1,1,0 0,0,1 1,0,0 1,1,1 Coordinate of Points We can locate certain points, such as atom position in the lattice or unit cell by constructing the right-handed coordinate system Distance is measured in terms of the number of lattice parameter we must move in each of the x,y and z coordinates to get form the origin to the point Atom position in S.C : 000, 100, 110, 010, 001,101,111,011

Atomic coordinate FCC: Face Centered Cubic 8 Cu (corner) 6 Cu (face) (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)

Atom coordinate (0, 0, 0) (1, 0, 0) (0, 1, 0) (0, 0, 1) (1, 1, 1) BCC : Based centered cubic +z (0, 0, 0) (1, 0, 0) (0, 1, 0) (0, 0, 1) (1, 1, 1) (1, 1, 0) (1, 0, 1) (0, 1, 1) (½, ½, ½) -x -y +y +x -z

Anisotropy of crystals 191.1 GPa Young’s modulus of FCC Cu 130.3 GPa 66.7 GPa

Anisotropy of crystals (contd.) Different crystallographic planes have different atomic density And hence different properties Si Wafer for computers

MILLER INDICES OF DIRECTIONS

Miller Indices of Directions z 1. Choose a lattice point on the direction as the origin 2. Choose a crystal coordinate system with axes parallel to the unit cell edges y 3. Find the coordinates, in terms of the respective lattice parameters a, b and c, of another lattice point on the direction. 4. Reduce the coordinates to smallest integers. 1a+1b+0c x [110] 5. Put in square brackets […]

Miller Indices of Directions (contd.) x y z OA=1/2 a + 1/2 b + 1 c Q x y z 1/2, 1/2, 1 A [1 1 2] PQ = -1 a -1 b + 1 c O -1, -1, 1 [ 1 1 1 ] _ P

MILLER INDICES FOR PLANES

Miller Indices for planes x y z (0,0,1) 1. Select an origin not on the plane 2. select a crystallographic coordinate system (0,3,0) 3. Find intercepts along axes : 2 3 1 (2,0,0) 4. Take reciprocal : 1/2 1/3 1 5. Convert to smallest integers in the same ratio : 3 2 6 (1/2 1/3 1) X 6 6. Enclose in parenthesis : (326)

_ Miller Indices for planes (contd.) x y z x y z E Plane ABCD OCBE A B origin O O* 1 -1 ∞ 1 ∞ ∞ intercepts 1 -1 0 reciprocals 1 0 0 (1 1 0) _ Miller Indices (1 0 0)

INTRODUCTION TO CERAMIC MINERALS z x y A B C Planes in the Unit Cell Procedure Identify the points at which the plane intercepts the x,y and z coordinates in terms of the number of lattice parameters. If the plane passes through thr origin, the origin of the coordinate system must be moved Take reciprocals of these intercepts Clear fractions but do not reduce to lowest integers Enclose the resulting umbers in parenthess ( ). Again, negative numbers should be written with a bar over the number

INTRODUCTION TO CERAMIC MINERALS Planes in the Unit Cell z x y A B C Plane A x=1, y=1, z=1 1/x=1, 1/y=1, 1/z=1 No fractions to clear Miller Indices =(111) Plane B The plane never intercepts the z-axis, so x=1, y=2 and z=∞ 1/x=1, 1/y=1/2, 1/z=0 Clear fractions: 1/x=2, 1/y=1, 1/z=0 Miller Indices =(210) Plane C We must move the origin, since the plane passes through 0,0,0.Let’s move the origin one lattice parameter in the y-direction. Then, x=∞, y=-1, and z=∞ 1/x=0, 1/y=-1, 1/z=0 No fraction to clear Miller Indices = (010)

Family of Symmetry Related Directions [ 0 0 1 ] Identical atomic density Identical properties [ 1 0 0 ] _  1 0 0  [ 0 1 0 ] _ [ 0 1 0 ] [ 1 0 0 ] [ 0 0 1 ] _ 1 0 0= [ 1 0 0 ] and all other directions related to [ 1 0 0 ] by symmetry x y z

Family of Symmetry Related Directions type: <100> Equivalent directions: [100],[010],[001] type: <110> Equivalent directions: [110], [011], [101], [-1-10], [0-1-1], [-10-1], [-110], [0-11], [-101], [1-10], [01-1], [10-1] type: <111> Equivalent directions: [111], [-111], [1-11], [11-1]

Family of Symmetry Related Planes z 4. (1 1 0) _ 1. ( 1 01 ) 2. ( 1 0 1 ) 6. ( 0 1 1 ) 3. ( 1 1 0) 5. ( 0 1 1 ) y { 1 1 0 } x { 1 1 0 } = Plane ( 1 1 0 ) and all other planes related by symmetry to ( 1 1 0 )

Summary [uvw] = components of a vector in the direction reduced to smallest integers (hkl)= reciprocal of intercepts of a plane reduced to smallest integers <uvw>= family of symmetry related directions {hkl}= family of symmetry related planes