CRYSTAL STRUCTURE

Introduction A crystal is a solid composed of atoms or other microscopic particles arranged in an orderly repetitive array. Further Solids can be broadly classified into Crystalline and Non-crystalline or Amorphous. In crystalline solids the atoms are arranged in a periodic manner in all three directions, where as in non crystalline the arrangement is random. Non crystalline substances are isotropic and they have no directional properties. Crystalline solids are anisotropic and they exhibit varying physical properties with directions. Crystalline solids have sharp melting points where as amorphous solids melts over a range of temperature.

Space lattice A Space lattice is defined as an infinite array of points in three dimensions in which every point has surroundings identical to that of every other point in the array. X X X X X Where a and b are called the repeated translation vectors.

Three dimensional lattice Lattice planes Lattice lines Lattice points

UNIT CELL The unit cell is a smallest unit which is repeated in space indefinitely, that generates the space-lattice. BASIS A group of atoms or molecules identical in composition is called the Basis. Lattice + basis = Crystal structure

CRYSTALLOGRAPHIC AXES The lines drawn parallel to the lines of intersection of any three faces of the unit cell which do not lie in the same plane are called Crystallographic axes. PRIMITIVES: The a, b and c are the dimensions of an unit cell and are known as Primitives. INTERFACIAL ANGLES The angles between three crystallographic axes are known as Interfacial angles α ,β and γ.

1. Primitives decides the size of the unit cell. β α γ X Y Z a b c NOTE 1. Primitives decides the size of the unit cell. 2. Interfacial angles decides the shape of the unit cell.

The unit cell is formed by primitives is called primitive cell. A primitive cell will have only one lattice point. β α γ X Y Z a b c LATTICE PARAMETERS The primitives and interfacial angles together called as lattice parameters.

X X X X X X

BRAVIAS LATTICES There are only fourteen distinguishable ways of arranging the points independently in three dimensional space and these space lattices are known as Bravais lattices and they belong to seven crystal systems

SYMBOLS Simple Cubic P Base Centered C Body Centered I Face Centered F CRYSTAL TYPE BRAVAIS LATTICE 1. Cubic Simple Body centered Face centered 2. Tetragonal Simple 3. Orthorhombic Simple Base centered 4. Monoclinic Simple Base centered 5. Triclinic Simple 6. Trigonal Simple 7. Hexgonal Simple SYMBOLS Simple Cubic P Base Centered C Body Centered I Face Centered F

Crystal System Unit Vector Angles Cubic a = b = c α = β = γ = 90˚ Tetragonal a = b ≠ c α = β = γ = 90˚ Ortho rhombic a ≠ b ≠ c α = β = γ = 90˚ Mono clinic a ≠ b ≠ c α = β = 90 ≠ γ Triclinic a ≠ b ≠ c α ≠ β ≠ γ ≠ 90˚ Trigonal a = b = c α = β = γ ≠ 90˚ Hexagonal a = b ≠ c α = β = 90˚,γ =120˚

Cubic Crystal System 1 2 3 4 5 6 a = b = c & α = β = γ =90˚

Tetragonal Crystal System a = b ≠ c & α = β = γ =90˚

Ortho Rhombic Crystal System 1 2 3 4 5 6 a ≠ b ≠ c & α = β = γ =90˚

Monoclinic Crystal System a ≠ b ≠ c & α = β = 90 ≠ γ

Triclinic clinic Crystal System a ≠ b ≠ c & α ≠ β ≠ γ ≠90˚

Trigonal Crystal System a = b = c & α = β = γ ≠90˚

Hexagonal Crystal System a = b ≠ c & α = β =90˚,γ =120˚

NEAREST NEIGHBOUR DISTANCE The distance between the centers of two nearest neighboring atoms is called nearest neighbor distance. CO – ORDINATION NUMBER Co-ordination number is defined as the number of equidistance nearest neighbors that an atom has in a given structure.

ATOMIC PACKING FACTOR Atomic packing factor is the ratio of volume occupied by the atoms in an unit cell to the total volume of the unit cell. It is also called packing fraction.

Void Space Vacant space left or unutilized space in unit cell , and more commonly known as interstitial space. Void space = ( 1-APF ) X 100

SIMPLE CUBIC STRUCTURE - PACKING FACTOR Effective number of atoms per unit cell (8 x 1/8) =1 Atomic radius r = a / 2 Nearest neighbor distance 2r = a Co-ordination number = 6

5. Atomic packing factor 6.Void space = (1-APF) X 100 = (1-0.52)X 100 = 48% Example: Polonium.

BCC STRUCURE – PACKING FACTOR Effective number of atoms per unit cell (8 x 1/8) + 1 =2 2. Atomic radius r = √3a /4 3. Nearest neighbor distance 2r =√3a/2 4. Co-ordination number = 8

5.Atomic packing factor 6.Void space = (1-APF) x 100 = (1-0.68) x 100 = 32% Ex: Na, lithium and Chromium.

FCC Crystal Structure – APF Effective number of atoms per unit cell (8 x 1/8) + 1/2 X 6 = 4 2. Atomic radius r = a / 2√2 3. Nearest neighbor distance 2r = a /√2 4. Co-ordination Number = 12

5.Atomic packing factor 6.Void space = (1-APF) X 100 = (1-0.74) X 100 = 26% Ex: Cupper , Aluminum, Silver and Lead

Diamond Structure: Diamond is a combination of interpenetrating Fcc - Sub lattices along the body diagonal by 1/4th Cube edge.

1 2 3 4 5 6

Diamond - APF Effective number of atoms per unit cell x p z y 2r Effective number of atoms per unit cell (8 x 1/8) + 1/2 X 6 + 4 = 8. 2. Atomic radius r = √3a / 8. 3. Nearest neighbor distance 2r = √3a / 4. 4. Co-ordination number = 4.

x p y z a/4 a/2 a

5. Atomic packing factor 6. Void space = (1-APF) x 100 = (1-0.34) x 100 = 66% Ge, Si and Carbon atoms are possess this structure

Hexagonal Close Packed Structure Effective number of atoms per unit cell 2 x (6x 1/6) + 2 x 1/2 + 3 = 6. 2. Atomic radius r = a / 2. 3. Nearest neighbor distance 2r = a 4. Co-ordination number = 12.

5.Volume of the HCP unit cell The volume of the unit cell determined by computing the area of the base of the unit cell and then by multiplying it by the unit cell height. Volume = (Area of the hexagon) x (height of the cell) Area of the hexagon A B C a 60° If c is the height of the unit cell

c/a ratio: The three body atoms lie in a horizontal plane at a height c/2 from the base or at top of the Hexagonal cell. A B p o a 30° c/2 2r x N q

5. Atomic packing factor 6. Void space = (1-APF) x 100 = (1-0.74) x 100 = 26% Ex: Mg, Cd and Zn.

Sodium Chloride Structure Nacl Crystal is an ionic crystal. It consists of two FCC sub lattices. One of the chlorine ion having its origin at the (0, 0, 0) point and other of the sodium ions having its origin at (a/2,0,0). Each ion in a NaCl lattice has six nearest neighbor ions at a distance a/2. i,e its Co-ordination number is 6.

Sodium Chloride structure Cl Na

Each unit cell of a sodium chloride as four sodium ions and four chlorine ions. Thus there are four molecules in each unit cell. Cl : (0,0,0) (1/2,1/2,0) (1/2,0,1/2) (0,1/2,1/2) Na : (1/2,1/2,1/2),(0,0,1/2) (0,1/2,0)(1/2,0,0)

Structure of Cesium chloride: Cscl is an ionic Compound. The lattice points of CsCl are two interpenetrating simple cubic lattices. One sub lattice occupied by cesium ions and another occupied by Cl ions. The co-ordinates of the ions are Cs : (000),(100),(010),(001),(110), (110),(011),(111). Cl : (1/2,1/2,1/2). Cs Cl

Some important directions in Cubic Crystal Square brackets [ ] are used to indicate the directions The digits in a square bracket indicate the indices of that direction. A negative index is indicated by a ‘bar’ over the digit . Ex: for positive x-axes→[ 100 ] for negative x-axes→[ 100 ]

Fundamental directions in crystals z x y [100] [000] [001] [010]

Crystal planes & Miller indices Reciprocals of intercepts made by the plane which are simplified into the smallest possible numbers or integers and represented by (h k l ) are known as Miller Indices. (or) The miller indices are the three smallest integers which have the same ratio as the reciprocals of the intercepts having on the three axes. These indices are used to indicate the different sets of parallel planes in a crystal.

Procedure for finding Miller indices Find the intercepts of desired plane on the three Co-ordinate axes. Let they be (pa, qb, rc). Express the intercepts as multiples of the unit cell dimensions i.e. p, q, r. (which are coefficients of primitives a, b and c) Take the ratio of reciprocals of these numbers i.e. a/pa : b/qb : c/rc. which is equal to 1/p:1/q:1/r. Convert these reciprocals into whole numbers by multiplying each with their L.C.M , to get the smallest whole number. These smallest whole numbers are Miller indices (h, k, l) of the crystal.

Important features of miller indices When a plane is parallel to any axis, the intercept of the plane on that axis is infinity. Hence its miller index for that axis is zero. When the intercept of a plane on any axis is negative a bar is put on the corresponding miller index. All equally spaced parallel planes have the same index number (h, k, l). If a plane passes through origin, it is defined in terms of a parallel plane having non-zero intercept.

The numerical parameters of the plane ABC are (2,2,1). The reciprocal of these values are given by (1/2,1/2,1). LCM is equal to 2. Multiplying the reciprocals with LCM we get Miller indices [1,1,2]. x y z 2 1

Construction of [100] plane z x [1 0 0] plane y

Set of [100] parallel planes z x y Set of [100] parallel planes

x y z [010] plane.

Set of ( 0 1 0 ) parallel planes x y z Set of ( 0 1 0 ) parallel planes

x y z [ 001 ] plane

Set of ( 0 0 1 ) parallel planes x y z [ 001 ]

Construction of [110] plane y x [110] z

Set of [110] parallel planes x y z [110] Set of [110] parallel planes

Construction of ( ī 0 0) Planes z x y

Intercepts of the planes are 1,1,1 Reciprocals of intercepts are 1/1,1/1,1/1 Miller indices:(111) x y z ( 1 1 1 ) plane

Inter planner spacing of orthogonal crystal system: Let ( h ,k, l ) be the miller indices of the plane ABC. Let ON=d be a normal to the plane passing through the origin ‘0’. Let this ON make angles α, β and γ with x, y and z axes respectively. Imagine the reference plane passing through the origin “o” and the next plane cutting the intercepts a/h, b/k and c/l on x, y and z axes.

OA = a/h, OB = b/k, OC = c/l A normal ON is drawn to the plane ABC from the origin “o”. the length “d” of this normal from the origin to the plane will be the inter planar separation. from∆ ONA from∆ ONB from∆ ONC Where cosα, cosβ ,cosγ are directional cosines of α,β,γ angles.

According to law of directional cosines

This is the general expression for inter planar separation for any set of planes. In cubic system as we know that a = b = c, so the expression becomes