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CRYSTAL PHYSICS

General Objective

To develop the knowledge of crystal

structure and their properties.

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Specific Objectives

1. Differentiate crystalline and amorphous

solids.

2. To explain nine fundamental terms of

crystallography.

3. To discuss the fourteen Bravais lattice

of seven crystal system.

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‘Crystal Physics’ or ‘Crystallography’ is

a branch of physics that deals with the

study of all possible types of crystals and

the physical properties of crystalline

solids by the determination of their actual

structure by using X-rays, neutron beams

and electron beams.

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Solids can broadly be classified into two types

based on the arrangement of units of matter.

The units of matter may be atoms, molecules or

ions.

They are,

Crystalline solids

Non-crystalline (or) Amorphous solids

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Ex: Metallic and non-metallic

NaCl, Ag, Cu, AuEx: Plastics, Glass and Rubber

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Grains

Grain boundaries

A lattice is a regular and periodic

arrangement of points in three dimension.

It is defined as an infinite array of points in

three dimension in which every point has

surroundings identical to that of every other

point in the array.

The Space lattice is otherwise called the

Crystal lattice

SPACE LATTICE

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A crystal structure is formed by associating every

lattice point with an unit assembly of atoms or

molecules identical in composition, arrangement and

orientation.

This unit assembly is called the `basis’.

When the basis is repeated with correct periodicity in

all directions, it gives the actual crystal structure.

The crystal structure is real, while the lattice is

imaginary.

BASIS

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UNIT CELL

A unit cell is defined as a fundamental building block

of a crystal structure, which can generate the

complete crystal by repeating its own dimensions in

various directions.

XA

Y

B

Z

C

O

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• Inter axial lengths: OA = a , OB = b and OC = c

• Inter axial angles: α,β and γ

• Primitives: The intercepts OA, OB, OC are called Primitives

7 Crystal systems:

1. Cubic

2. Orthorhombic

3. Monoclinic

4. Triclinic

5. Hexagonal

6. Rhombohedral

7. Tetragonal

Fourteen Bravais Lattices in Three Dimensions

Fourteen Bravais Lattices …

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MILLER INDICES

Chapter 3 -19

MILLER INDICES

d

DIFFERENT LATTICE PLANES

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MILLER INDICES

The orientation of planes or faces in a crystal can be

described in terms of their intercepts on the three

axes.

Miller introduced a system to designate a plane in a

crystal.

He introduced a set of three numbers to specify a

plane in a crystal.

This set of three numbers is known as ‘Miller Indices’

of the concerned plane.

Chapter 3 -21

MILLER INDICES

Miller indices is defined as the reciprocals of

the intercepts made by the plane on the three

axes.

Chapter 3 -22

MILLER INDICES

Procedure for finding Miller Indices

Step 1: Determine the intercepts of the plane

along the axes X,Y and Z in terms of

the lattice constants a,b and c.

Step 2: Determine the reciprocals of these

numbers.

Chapter 3 -23

Step 3: Find the least common denominator (lcd)

and multiply each by this lcd.

Step 4:The result is written in paranthesis.This is

called the `Miller Indices’ of the plane in

the form (h k l).

This is called the `Miller Indices’ of the plane in the form

(h k l).

MILLER INDICES

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ILLUSTRATION

PLANES IN A CRYSTAL

Plane ABC has intercepts of 2 units along X-axis, 3

units along Y-axis and 2 units along Z-axis.

Chapter 3 -25

DETERMINATION OF ‘MILLER INDICES’

Step 1:The intercepts are 2,3 and 2 on the three axes.

Step 2:The reciprocals are 1/2, 1/3 and 1/2.

Step 3:The least common denominator is ‘6’.

Multiplying each reciprocal by lcd,

we get, 3,2 and 3.

Step 4:Hence Miller indices for the plane ABC is (3 2 3)

ILLUSTRATION

Chapter 3 -26

MILLER INDICES

IMPORTANT FEATURES OF MILLER INDICES

A plane passing through the origin is defined in terms of a

parallel plane having non zero intercepts.

All equally spaced parallel planes have same ‘Miller

indices’ i.e. The Miller indices do not only define a particular

plane but also a set of parallel planes. Thus the planes

whose intercepts are 1, 1,1; 2,2,2; -3,-3,-3 etc., are all

represented by the same set of Miller indices.

Chapter 3 -27

MILLER INDICES

IMPORTANT FEATURES OF MILLER INDICES

It is only the ratio of the indices which is important in this

notation. The (6 2 2) planes are the same as (3 1 1) planes.

If a plane cuts an axis on the negative side of the origin,

corresponding index is negative. It is represented by a bar,

like (1 0 0). i.e. Miller indices (1 0 0) indicates that the

plane has an intercept in the –ve X –axis.

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MILLER INDICES OF SOME IMPORTANT PLANES

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Simple Cubic Structure (SC)

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• Rare due to low packing density (only Po has this structure)

• Close-packed directions are cube edges.

• Coordination # = 6

(# nearest neighbors)

Simple Cubic Structure (SC)

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• APF for a simple cubic structure = 0.52

APF =

a3

4

3p (0.5a) 31

atoms

unit cellatom

volume

unit cell

volume

Atomic Packing Factor (APF):SC

APF = Volume of atoms in unit cell*

Volume of unit cell

*assume hard spheres

close-packed directions

a

R=0.5a

contains 8 x 1/8 = 1 atom/unit cell

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• Coordination # = 8

• Atoms touch each other along cube diagonals.--Note: All atoms are identical; the center atom is shaded

differently only for ease of viewing.

Body Centered Cubic Structure (BCC)

ex: Cr, W, Fe (), Tantalum, Molybdenum

2 atoms/unit cell: 1 center + 8 corners x 1/8

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Atomic Packing Factor: BCC

a

APF =

4

3p ( 3a/4)32

atoms

unit cell atom

volume

a3

unit cell

volume

length = 4R =

Close-packed directions:

3 a

• APF for a body-centered cubic structure = 0.68

aRAdapted from

Fig. 3.2(a), Callister &

Rethwisch 8e.

a2

a3

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• Coordination # = 12

Adapted from Fig. 3.1, Callister & Rethwisch 8e.

• Atoms touch each other along face diagonals.--Note: All atoms are identical; the face-centered atoms are shaded

differently only for ease of viewing.

Face Centered Cubic Structure (FCC)

ex: Al, Cu, Au, Pb, Ni, Pt, Ag

4 atoms/unit cell: 6 face x 1/2 + 8 corners x 1/8Click once on image to start animation

(Courtesy P.M. Anderson)

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• APF for a face-centered cubic structure = 0.74

Atomic Packing Factor: FCC

maximum achievable APF

APF =

4

3p ( 2a/4)34

atoms

unit cell atom

volume

a3

unit cell

volume

Close-packed directions:

length = 4R = 2 a

Unit cell contains:6 x 1/2 + 8 x 1/8

= 4 atoms/unit cella

2 a

Adapted from

Fig. 3.1(a),

Callister &

Rethwisch 8e.

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A sites

B B

B

BB

B B

C sites

C C

CA

B

B sites

• ABCABC... Stacking Sequence

• 2D Projection

• FCC Unit Cell

FCC Stacking Sequence

B B

B

BB

B B

B sitesC C

CA

C C

CA

AB

C

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Hexagonal Close-Packed Structure

(HCP)

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• Coordination # = 12

• ABAB... Stacking Sequence

• APF = 0.74

• 3D Projection • 2D Projection

Adapted from Fig. 3.3(a),

Callister & Rethwisch 8e.

6 atoms/unit cell

ex: Cd, Mg, Ti, Zn

• c/a = 1.633

c

a

A sites

B sites

A sites Bottom layer

Middle layer

Top layer

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Crystal structure coordination # packing factor close packed directions

Simple Cubic (SC) 6 0.52 cube edges

Body Centered Cubic (BCC) 8 0.68 body diagonal

Face Centered Cubic (FCC) 12 0.74 face diagonal

Hexagonal Close Pack (HCP) 12 0.74 hexagonal side

Thank You…

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