MILLER INDICES

PREPARED BY:- Akashdeep Singh B120020024 Alisha Bansal B120020029 Pratibha Kochar B120020191 Puneet Deewan B120020198

CONTENTS Miller Indices History Crystal Structure Cubic Bravais Lattices Lattice Structure Case of Cubic Structures Case of hexagonal and rhombohedral

structures Representation of Miller indices Planes with different Miller Indices Sample Diamond Structure Cesium Chloride Structure

MILLER INDICES

The orientation of a surface or a crystal plane may be defined by considering how the plane intersects the main crystallographic axes of the solid.

Miller Indices form a notation system in crystallography for planes in crystal lattices.

In particular, a family of lattice planes is determined by three integers h, k, and ℓ.

The notation {hkℓ} denotes the set of all planes that are equivalent to (hkℓ) by the symmetry of the lattice.

The notation ⟨hkℓ⟩ denotes the set of all directions that are equivalent to [hkℓ] by symmetry.

HISTORY Miller indices were introduced in 1839 by the British

mineralogist William Hallowes Miller. The method was also historically known as the

Millerian system, and the indices as Millerian. The precise meaning depends upon a choice of

lattice vectors for the crystal. Usually, three primitive lattice vectors are used.

However, for cubic crystal systems, the cubic lattice vectors are used even when they are not primitive.

CRYSTAL STRUCTURE

It is a unique arrangement of atoms/molecules in a crystalline liquid/solid.

Composed of a pattern, a set of atoms arranged in a particular way of & a lattice exhibiting long range-order & symmetry.

Patterns are located upon the points of a lattice, which is an array of points repeating periodically in 3D.

CUBIC LATTICES

Primitive cubic

Body centered Cubic

Face Centered

LATTICE SYSTEMS

They are grouping of crystal according to axial system used to describe their lattices.

Each lattice Consists of a set three axes in a particular geometrical arrangement.

They are similar to but not quite the same as seven crystals.

CASE OF CUBIC STRUCTURES For the special case of simple cubic crystals, the

lattice vectors are orthogonal and of equal length similar to the reciprocal lattice.

The Miller indices (hkℓ) and [hkℓ] both simply denote normals /directions in Cartesian coordinates.

For cubic crystals with lattice constant a, the interplanar spacing d between adjacent (hkℓ) lattice planes is:-

CASE OF HEXAGONAL & RHOMBOHEDRAL STRUCTURES

With hexagonal and rhombohedral lattice systems, it is possible to use the Bravais-Miller index which has 4 numbers (h k i ℓ).

i = −(h + k). Here h, k and ℓ are identical to the Miller index,

and i is a redundant index

← Miller-Bravais indices

REPSENTATION OF MILLER INDICES

Let us consider a simple cubic system (α=β=γ=90º) & (a=b=c).

For this system the miller indices are showed corresponding to-

(1`,0,0), (1,0,0),(0,0,1),(1,1,0),(1,0,1),(1,1,1) planes in a cubic crystal respectively.

← Planes with different Miller indices in cubic crystals

SAMPLE A plane cuts intercepts 2a, 3b, c along the

crystallographic axes in a crystal. Determine the miller indices of plane?

Ans. Intercepts are 2a, 3b, c From law of rational indices- 2a:3b:c=a/h:b/k:c/l 1/h:1/k:1/l=2:3:1 h:k:l=1/2:1/3:1 =3:2:6 →Miller indices of plane are=(3 2 6)

DIAMOND STRUCTURE The space lattice of diamond is FCC. In diamond

structure we have two FCC lattices placed at (0,0,0) & (1/4,1/4,1/4) which superimpose each other.

In diamond structure we have two carbon atoms placed at (0,0,0) & (1/4,1/4,1/4).the diamond crystal structure is shown below.

CESIUM CHLORIDE STRUCTURE

In CsCl crystal, we have a BBC structure lattice in which Cs atom is placed at the origin(0,0,0) i.e. at the body centre point & Cl atoms are placed at the corners of the BBC lattice whose coordinates are given by (1/2,1/2,1/2).