Introduction to CRYSTAL PHYSICS

By

M. SILAMBARASAN

Centre for Photonics and Nanotechnology

Department of Physics

Sona College of Technology

Salem-636 005, Tamil Nadu, India

http://www.sonapan.com/

E-mail: silambu.physicist@gmail.com

INTRODUCTION

Crystals• Periodically arrangement of

atoms (or) molecules (or) ions in a three dimensional pattern

• The crystalline solids have well defined geometrical form (pattern)

• Further when crystals breaks, all the broken pieces will have a regular shape

• The phase change from liquid (or) gas to solid is called Crystallization

Single Crystal• In, Which the solid contains only one crystal• These crystals are produced artificially from vapour or

liquid state• Sharp melting point

Poly Crystal

• Which has an aggregate of many small crystalsseparated by well defined boundaries

Molecular Solids Covalent Solids

Ionic solidsMetallic solids

Na+

Cl-

Structures of crystalline solid types

Non-crystalline solids

• Amorphous solids or Non crystalline solids.

• Atoms (or) molecules are not in an orderly fashion

• In amorphous solids the same atomic groups arearranged randomly in all direction

• Super cooled liquids: glass. ( some of the propertiesof solid), but not crystals

Crystals Vs Amorphous S.No

Crystalline Material Amorphous materials

1. They have a definite and regular geometrical shapes which extend throughout the crystals.

They do not have definite geometrical shapes.

2. They are anisotropic substance. (directional properties)

They are isotropic substance. (no directional properties)

3. They are most stable. They are less stable.

4. They have sharp meting point. They do not have sharp meting point.

5. Ex: NaCl, Diamond, KCl, Iron, etc Ex: Glasses, Plastics, Rubber, etc

Crystallography definitions

Lattice: An array of points which are imaginarily kept torepresent the position of atoms in crystal such that everylattice points has got the same environment as that of theothers

Crystallography : The study of the Physical properties and

geometrical form of the crystalline solids, using X-ray (or)

electron beam (or) neutron beam etc., is termed as the science of

Crystallography.

Space lattice (or) crystal lattice: A 3D collection of points in space are called space lattice, the environment about any particular point is in everyway the same

• Lattice points: It denotes the position of atoms (or)molecules in the crystals. (or) Points in the spacelattice.

Basis (or) Motif: An unit assembly of atoms (or)molecules which are identified with respect to theposition of lattice points, identical in composition andorientation. The number of atoms in the basis may be1 or 2 or 3 etc. it may be go even above 1000.

Crystal Structure=

Space lattice + Basis

• Lattice Plane: A set of parallel and equally spaced planes in a space lattice

• Unit Cell: The fundamental elementary patternwith minimum number of atoms and, molecules (or)group of molecules which represents the totalcharacteristic of the crystals.

• Smallest geometric figure.

• Lattice parameters (or) Unit Cell parameters: Thelines drawn parallel to the lines of intersection of anythree faces of the unit cell which don’t lie in the sameplane are called crystallography axis.

a, b, c – intercepts or primitive or axial length.

α, β and γ – interfacial angles.

• Primitive cell

• Non Primitive cell

The Crystal System

The crystal systems are classified into 7 crystal

systems on the basis of lattice parameters. Viz ( axial

length a, b, c & axial angle α, β and γ

In this universe all the solids comes under

7 crystal systems

14 Bravais lattice

32 Point groups and

230 Space groups

Rotation

Translation

Reflection

Inversion

Point group

Space group

Symmetry Operation

A symmetry operation is atransformation (operation)performed on a body so that thebody can go to a new position inspace, which is similar to that theoriginal position

Crystal Systems

• Triclinic

• Monoclinic

• Orthorhombic

• Tetragonal

• Hexagonal

• Trigonal (Rhomohedral)

• Cubic

Triclinic• The Lattice parameters

a b c

α β γ 90º

Monoclinic

The Lattice parameters

a b c

α = β = 90º γ

Orthorhombic• The Lattice parameters

a b c

α = β = γ = 90º

Tetragonal

The Lattice parameters

a = b c

α = β = γ = 90º

Hexagonal The Lattice parameters

a = b c

α = β = 90º, γ = 120º

Trigonal (Rhombohedral)

The Lattice parameters

a = b = c

α = β = γ 90º

Cubic

The Lattice parameters

a = b = c

α = β = γ = 90º

Baravais Lattice

In, 1880 Bravais lattice, studied by Auguste Bravais. There

are 14 possible types of space lattices in 7 crystal system.

S.No

Crystal Systems Unit Cell Parameters Baravais Lattice

1 Triclinic a b c,α β γ 90º

Simple

2 Monoclinic a b c,α = β = 90º γ

Simple, Base-centred.

3 Orthorhombic a b c,α = β = γ= 90º

Simple, Base-centred,Body-centred, Face-centred.

4 Tetragonal a = b c,α = β = γ= 90º

Simple, Body-centred

5 Hexagonal a = b c,α = β = 90º, γ=120º

Simple

6 Trigonal(Rhomohedral)

a = b = c,α = β = γ 90º

Simple

7 Cubic a = b = c,α = β = γ = 90º

Simple, Body-centred, Face-centred.

Triclinic

Monoclinic

Simple

Simple Base- Centre

OrthorhombicSimple

Body Centered

Base- Centred

Face Centered

Simple

Body- Centre

Tetragonal Hexagonal

Simple

Cubic

Trigonal (Rhombohedral)

Simple

Simple Body Centered Face Centered

Relation Between Lattice Constant and Density

• Density = Mass/Volume

• Mass = number of atoms per unit cell * mass of each atom

• mass of each atom = atomic weight/avogadro’s number

Miller Indices

Introduction:

Crystals are made up of largenumber of parallel and equalspaced planes, passingthrough the lattice points iscalled lattice planes.

A set of three numbers todesignate a plane in a crystal.That set of three numbersare called miller indices ofthe concern plan.

Miller Index (hkl)

Definition:

Miller indices is defined as the reciprocal of the intercepts

made by plane on the crystallography axis which are reducedto smallest numbers.

(or)

Miller indices are the three smallest possible integers, whichhave the same ratio as the reciprocals of the intercepts ofthe plane concerned along the three axes.

(2,0,0)

(0,3,0)

(0,0,1) Select an origin not on the

plane: O

select a crystallographic

coordinate system:

XYZ

Find intercepts along axes:

2 :3 :1

Take reciprocal

1/2: 1/3 :1/1x

y

z

Steps for finding Miller Index

Convert to smallest integers in the same ratio i.e, multiplying each and every

reciprocal with their least common Multiplier (LCM) :

Here LCM =6. and get 3:2:6

Enclose in parenthesis: (326)

Generally Miller Indices are denoted by (hkl)

h:k:l= a/p:b/q:c/r for any system.

h:k:l= a/p:a/q:a/r for cubic system.

origin

intercepts

reciprocals

Miller Indices

A B

C

D

O

x

y

z

E

x

y

zABCD

O

1 ∞ ∞

1 0 0

(1 0 0)

O*

O*

Plane OCBE

1 -1 ∞

1 -1 0

(1 1 0)

_

Miller Indices for planes

Miller Indices should be enclosed only in this parenthesis: i.e., ( )

There should be no comma’s in-between the numbers.

(2 6 3) means it should be read as only two six three.

The Negative Miller indices can be represented by (2 6 3)

If a plane is parallel to any one of the coordinate axis, then its intercepts will be at infinity. i.e., Miller index for that particular axis is zero. Ex: (1 0 0)

All equally spaced parallel plans have the same Miller Index

i.e., Same ratio: (8 4 4), (4 2 2) and (2 1 1)

The Indices (h k l) do not define a particular plane, but a set of parallel plans.

Important Feature of Miller Index

Sketching the plane from the given miller Indices

If the miller index are say

(a b c )

h=a; k=b; l=c.

Take the reciprocals of the given miller indices.

1/a: 1/b:1/c

Multiplying the reciprocals by the LCM

We get intercepts of the plane.

Common planes in a simple cubic structure

• Three important crystal planes

( 1 0 0) (1 1 1)(1 1 0)

(100) plane(-100) plane

(100) planes

[100] vector

Family of Symmetry Related Planes

Direction of the plan can be represented by [1 0 0]

To represent the family of planes we can use this bracket:

{ }

The {1 0 0} family includes 6 planes:

(1 0 0)_

( 1 0 0 )

( 0 1 0 )

( 0 0 1 )_

( 0 0 1 )

( 0 1 0 )_ { 1 0 0 }

(1 1 0)

_

( 1 1 0 )

( 1 0 1 )

( 0 1 1 )

_

( 0 1 1 )

( 1 0 1 )

_

{ 1 1 0 }

{ 1 1 0 } = Plane ( 1 1 0 ) and all other planes related

by symmetry to ( 1 1 0 )

The {1 1 1} family includes 8 planes:

The {1 1 0} family includes 12 planes:

(1 1 0)

_

( 1 1 0 )

( 1 0 1 )

( 0 1 1 )

_

( 0 1 1 )

( 1 0 1 )

_

_

_

_

_

_

_

Interplanar Distance

222222 ///

1

clbkahd crystals

hkl

222 lkh

ad cubic

hkl

The distance between any two successive planes.

Parameters Determining the Crystal Structure of Materials

• Number of Atoms per Unit Cell (or) Effective Number:

The total number of atoms present in (or) Shared by an unit cell is known as number of atoms per unit cell

• Co-ordination Number:

Co-ordination number is the number of nearest neighboring atoms to a particular atom

(or)

Co-ordination number is the number of nearest neighbors directly surrounding a given atom

• Atomic Radius:

Atomic radius is defined as half of the distance between any

two nearest neighbor atoms which have with each other, in a crystal of pure element.

• Atomic Packing Factor (or) Packing Density

Simple Cubic Structure (SC)• Number of Atoms per Unit Cell (n):

n = Number of corner atoms/8

= 1

• Co-ordination Number in SC:

= (No. of nearest neighbors)

= 6

• Atomic Radius (r) in SC:

a=2r

r = a/2

• Atomic Packing Factor (APF) in SC:

Lattice constant

close-packed directions

a

R=0.5a

• APF for a simple cubic structure = 0.52

APF = (No. of Atoms per unit cell * Volume of

one atom) / Total volume of the unit cell

Body Centered Cubic structure (BCC)• Number of Atoms per Unit Cell (n):

Unit cell c ontains:

1 + 8 x 1/8

= 2 atoms/unit cell

Co-ordination Number = 8

• Close packed directions are cube diagonals.

Atomic Radius (r) in BCC:

aR

r = √3 a

4

• Atomic Packing Factor (APF) in BCC:

• APF for a body-centered cubic structure = p (3/8)= 0.68

Face Centered Cubic Structure (FCC)

• Number of Atoms per Unit Cell (n):

Unit cell c ontains:

6 x 1/2 + 8 x 1/8

= 4 atoms/unit cell

Co-ordination Number = 12

Atomic Radius (r) in FCC:

a

r = √2 a

4

• Atomic Packing Factor (APF) in FCC:

• APF for a body-centered cubic structure = p/(32)

= 0.74(best possible packing of identical spheres)

Hexagonal Close-Packed Structure (HCP)

• Number of Atoms per Unit Cell (n):n1 = (No. of corner atoms/6) =12/6 =2

n2 = (No. of base atoms/2) = 2/2 =1n3 = No. of middle atoms = 3

n = 2 + 1 +3 = 6

• 3D Projection of HCP

A sites

B sites

A sites

• 2D Projection of HCP

• Co-ordination Number in HCP:

= 12

• Atomic Radius (r) in HCP:

a = 2r

r = a/2

• The relation between (c/a):

– c/a = (√8/3)

– for ideal c/a ratio of 1.633

• Atomic Packing Factor (APF) in HCP:

Close packed crystals

A plane

B plane

C plane

A plane

…ABCABCABC… packing

[Face Centered Cubic (FCC)]

…ABABAB… packing

[Hexagonal Close Packing (HCP)]

Close Packing

Diamond Cubic Structure

Diamond Cubic Structure is a FCC sub lattice with the basis of two carbon atoms

This structure has two sub lattice A and B.

The sub lattice A has it origin of (0,0,0) and sub lattice B has its origin at (a/4, a/4, a/4) along the body diagonal

Number of Atoms per Unit Cell (n) in Diamond Cubic

No. of corner atom per unit cell = (1/8)*8 = 1 atom

No. of face centered atoms per unit cell = (1/2)*6 = 3

No. atoms inside the unit cell = 4

Total number of atoms per unit cell = 1+3+4 = 8

• Co-ordination Number in Diamond Cubic:

= 4

• Atomic radius (r):

• Atomic Packing Factor (APF) in Diamond Cubic :

r = (a* 3)/8

(APF) =

Comparison Chart for Crystal Structures

S.No

Systems SC BCC FCC HCP Diamond

1 No. Atoms per unit cell

1 2 4 6 8

2 Coordination number

6 8 12 12 4

3 Atomic radius a/2 a3/4 a2/4 a/2 a3/8

4 Atomic PackingFactor (APF)

p/6

0.52

p3/8= 0.68

p2/6=0.74

p/32=0.74

p3/16 = 0.34

5 Packing density 52% 68% 74% 74% 34%

6 Example Polonium Na, Fe Pb, Ag Mg, Ti Ge, Si

Sodium Chloride Structure

• No. atoms per unit cell: Na+ = 4; Cl- = 4 atoms

• Coordination Number: Na+ = 6; Cl- = 6

• Atomic radius (r) = a/2

Zinc Blende (ZnS) Structure • Sphalerite ( FCC ZnS)

Wurtzite ( HCP ZnS)

Fluorite and anti fluorite

• Expanded FCC lattice

• Fluorite: Cations forming the lattice with the anions occupying both types of tetrahedral hole.

• Ex: CaF, UO2, PbO2

• Anti Fluorite: inverse of Fluorite structure• Ex: Na2S, Li2O, K2O

Polymorphism and Allotropy• The ability of a material to have more

than one structure

• If the change in structure is reversible, then the polymorphic change is know as allotropy– i.e., only physical properties change without

any change in the chemical properties.

Metal Temperature (°C)

Crystal Structure

Lattice constant& Property

Iron Up to768° C768 - 910° C ,910 -1400° C,1400 -1593° C

BCCBCCFCCBCC

-- , Highly Magnetic0.290nm, Non Magnetic0.363nm, Para Magnetic0.293nm, --

Cobalt Room TempAbove 477° C

HCPFCC

Graphite Structure

Crystal Defects

• An ideal crystal can be described in terms a three-dimensionally periodic arrangement of points called lattice and an atom or group of atoms associated with each lattice point called motif:

• However, there can be deviations from this ideality.

• These deviations are known as crystal defects.

Crystal = Lattice + Motif

Vacancy: A point defect

Defects Dimensionality Examples

Point 0 Impurity, Vacancy

Line 1 Dislocation

(Edge, Screw)

Surface 2 Free surface,

Grain boundary

Volume 3 Cavities or voids,

Cracks and Holes

vacancy Interstitial

impurity

Substitutional

impurity

Point Defects

Frenkel defect

Schottky

defect

Defects in ionic solids

Cation vacancy

+

cation interstitial

Cation vacancy

+

anion vacancy

Missing half planeA Defect

Line Defects Dislocations

An extra half plane…

…or a missing half plane

What kind of

defect is this?

A line defect?

Or a planar defect?

Extra half plane No extra plane!

Missing plane No missing plane!!!

An extra half plane…

…or a missing half plane

Edge

Dislocation

This is a line defect called an

EDGE DISLOCATION

The atom positions around an edge dislocation; extra

half-plane of atoms shown in perspective.

1 2 3 4 5 6 7 8 9

1 2 3 4 5 6 7 8 9

1 2 3 4 5 6 7 8 9

1 2 3 4 5 6 7 8 9

1 2 3 4 5 6 7 8 9

1 2 3 4 5 6 7 8 9

slip no slip

boundary = edge dislocation

Slip planeb

Burgers vector

Slip plane

slip no slip

dis

loca

tion

b

t

Dislocation: slip/no

slip boundary

b: Burgers vector

magnitude and

direction of the slip

t: unit vector tangent to

the dislocation line

• Dislocation Line:A dislocation line is the boundary between slip and no slip regions of a crystal

• Burgers vector:The magnitude and the direction of the slip is represented by a vector b called the Burgers vector,

• Line vectorA unit vector t tangent to the dislocation line is called a tangent vector or the line vector.

1 2 3 4 5 6 7 8 9

1 2 3 4 5 6 7 8 9

slip no slip

boundary = edge dislocation

Slip planeb

Burgers vector

t

• In general, there can be any angle between the Burgers vector b (magnitude and the direction of slip) and the line vector t (unit

vector tangent to the dislocation line)

• b t Edge dislocation

• b t Screw dislocation

• b t , b t Mixed dislocation

b

t

b || t

12

3

• If b || t

• Then parallel planes to the dislocation line lose their distinct identity and become one

continuous spiral ramp

• Hence the name SCREW DISLOCATION

Edge

Dislocation

Screw

Dislocation

Positive Negative

Extra half plane

above the slip

plane

Extra half plane

below the slip

plane

Left-handed

spiral ramp

Right-handed

spiral ramp

b parallel to t b antiparallel to t

Edge Dislocation

432 atoms

55 x 38 x 15 cm3

Screw Dislocation 525 atoms

45 x 20 x 15 cm3

Screw Dislocation (another view)

Johannes Martinus

BURGERS

Burgers vector Burger’s vector

Burgers vector

1

2

7

6

5

4

3

8

9

1 82 3 4 5 6 7 9 10 11 12 13

1

2

3

4

5

6

7

8

9

18 2345679101

1

1213

A closed Burgers

Circuit in an ideal

crystal

SF

14 15 16

141516

1

2

7

6

5

4

3

8

9

1 82 3 4 5 6 7 9 10 11 12 13 14 15

1

2

3

4

5

6

7

9

1234568 79101112131415

8

16

S

b 16

RHFS convention

F

Map the same

Burgers circuit on a

real crystal

b is a lattice translation

b

If b is not a complete lattice translation then a surface defect

will be created along with the line defect.

Surface defect

N+1 planes

N planes

Compression

Above the slip plane

Tension

Below the slip plane

Elastic strain field associated with an

edge dislocation

A dislocation line cannot end abruptly

inside a crystal

Slip plane

slip no slip

slip no slip

dis

loca

tio

n

b

Dislocation:

slip/no slip

boundary

Slip plane

F

B

A dislocation line cannot end abruptly

inside a crystal

F

M

T

A dislocation line cannot end abruptly

inside a crystal

It can end on a free surface

Grain 1 Grain 2

Grain

Boundary

Dislocation can end on a grain boundary

A dislocation line cannot end abruptly inside a crystal

• It can end on

– Free surfaces

– Grain boundaries

– On other dislocations at a point called a node

– On itself forming a loop

Slip plane

The plane containing both b and t is called the

slip plane of a dislocation line.

An edge or a mixed dislocation has a unique

slip plane

A screw dislocation does not have a unique

slip plane.

Any plane passing through a screw

dislocation is a possible slip plane

Glide of an Edge

Dislocation

crss

crss

crss is

critical

resolved

shear stress

on the slip

plane in the

direction of

b.

Glide of an Edge

Dislocation

A dislocation cannot end

abruptly inside a crystal

Burgers vector of a

dislocation is constant

Surface Defects

External Internal

Free surface Grain boundary

Stacking fault

Twin boundary

Interphase boundary

Same

phase

Different

phases

External surface: Free surface

If bond are broken over an area A then two

free surfaces of a total area 2A is created

Area A

Area A

Broken

bonds

External surface: Free surface

If bond are broken over an area A

then two free surfaces of a total

area 2A is created

Area A

Area A

Broken

bonds

nA=no. of surface atoms per

unit area

nB=no. of broken bonds per

surface atom

=bond energy per atom

BA nn2

1

Surface energy per

unit area

Grain 1Grain 2

Grain

Boundary

Internal surface: grain boundary

A grain boundary is a boundary between two regions of

identical crystal structure but different orientation

Photomicrograph an iron

chromium alloy. 100X.

Optical Microscopy

Edge dislocation model of

a small angle tilt

boundary

Grain 1

Grain 2

Tilt boundary

A

BC

2

2h

b

A

B

C

2sin

2

h

b

tanh

b

Or approximately

Stacking fault

C

B

A

C

B

A

C

B

A

A

C

B

A

B

A

C

B

A

Stacking

fault

FCC FCC

HCP

Twin PlaneC

B

A

C

B

A

C

B

A

C

B

A

C

A

B

C

A

B

C

B

A

C

B

A

Twin plane