SOLID
Consist atoms or molecules that can take the form as

Lattice + basis
• Lattice : set of periodic geometrical point in space
• Basis : collection of atoms at a lattice points

• Crystal : collection of atoms at a set of lattice points in space

Wigner-Seitz cell
(4 steps to follow)

1.Selected a lattice point.
2.Draw line A to connect the given lattice point to all nearest lattice point
3.Draw new line at the midpoint and normal to line A.
4.The WS cell is the smallest volume enclosed in the region.

Non Primitive
• More than one lattice point per cell
• Integral multiples of the area of primitive cell

UNIT CELL
• Smallest portion of crystal lattice that represent a three dimensional pattern of the entire crystal.
• A unit cell is a parallelepiped formed by and (2-D case), or (3-D case)
• Three common unit cell in 3D is simple cubic, body-centered cubic and face-centered cubic

BL in 2D

Non Bravais Lattice (NBL)
• Atoms can be different kind
• Some lattice points are not equivalent

Monoclinic

Trigonal

Tetragonal

Triclinic

CUBIC

Monoclinic

Orthorombic

No of atoms per unit cell
• Product of the number of atoms per lattice point and the number of lattice point per unit cell
Lattice constant
• Physical dimension of unit cells in a crystal lattice.
Coordination number
• Number of atoms touching particular atoms
Packing Factor
• Fraction of the volume of a unit cell that is occupied by "hard sphere" atoms or ions.
• Packing factor = (number of atoms/cell) x (volume of each atom)/(volume of unit cell)
Atomic Packing Factor
• APF = (volume of atoms in unit cell/volume of unit cell)
Lattice Points
• Points in a crystal with specific arrangements of atoms, reproduced many times in crystal
Lattice Vectors
• Represent the edges of a unit cell of a lattice
Theoretical Density

Simple Cubic

• Lattice constant (a) = 2r
• Volume of unit cell (v) = a^3
• Number of atom per unit cell =1
• Volume of primitive cell =a^3
• Coordinate number =6
• nearest neighbors distance =a
• Packing fraction (PF) =0.52
• Example = Po

Body Centered Cubic

• Lattice constant (a) =(4/√ 3) r
• Volume of unit cell (v) =a^3
• Number of atom per unit cell =2
• Volume of primitive cell =a^3/2
• Coordinate number =8
• nearest neighbors distance =√ 3/2
• Packing fraction (PF) =0.68
• Example =Li,Na,K

Face Centered Cubic

• Lattice constant (a) = (4/√ 2) r
• Volume of unit cell (v) =a^3
• Number of atom per unit cell =4
• Volume of primitive cell =(a^3)/4
• Coordinate number =12
• nearest neighbors distance =(1/√ 2)a
• Packing fraction (PF) =0.74
• Example =Al,Cu

Miller Indices(Indices of Plane)
• A method used to determine the orientation of lattice plane in (hkl).
• There are 4 steps involved
1.Identify the plane intercept on the x, y and z axis
2.Take the reciprocals of the fractional intercept.
3.Reduce the numbers to three smallest integers by multiplying the number with the smallest denominator.
4.• The (hkl) pattern are known as Miller indices.

Indices Notation
[ h, k, l] : direction
< h, k , l>: Family of direction
( h, k, l): Plane
{h, k , l} : Family of plane
Negative sign of plane is written as

Miller Indices for (Directions)
• Draw vector and set tail as origin
• Determine the length of vector projection
• Remove fraction by multiplying by the smallest factor
• Enclose in square bracket

Miller Indices (Head and Tail Procedure for Crystallographic direction)
1.Find coordinate points of head and tails
2.Subtract the coordinate points(Head - Tail)
3.Remove fractions
4.Enclose in [ ]

Lattice Sites

Hexagonal Close Packed (HCP)

Symmetry operation
• An operation that change the position of a lattice point however, there is another lattice point at the same position as before the operation
• Types of symmetry operation:
1.Translation-operation that satisfies eqn:
T = n1a+n2b+n3c
2.Point Operation-operation that consist of rotation an axis,reflection on plane,and inverse of point
3.Combination Operation-combination of translation and point operation

3.Reflection
• A reflection on a plane or mirror plane
• Reflection on a horizontal plane,
• Reflection on vertical plane,
• Plane of reflection is in diagonal,
ex:Reflection Plane

2.n -Fold Rotation
• n=360/θ

4.Inversion
• Inversion operation on a point via an origin (inversion center)
i(x, y,z) -> (-x,-y,-z)

1.Identity operation,E = No change in the object

9 Planes of Symmetry of the cube

Angle=180

Angle=120