Hari Prasad

CRYSTAL STRUCTURES

UNIT-I

Hari PrasadAssistant Professor

MVJCE-Bangalore

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Learning objectives

• After the chapter is completed, you will be able to answer:

• Difference between crystalline and noncrystalline structures

• Different crystal systems and crystal structures

• Atomic packing factors of different cubic crystal systems

• Difference between unit cell and primitive cell• Difference between single crystals and poly

crystals

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What is space lattice?

• Space lattice is the distribution of points in 3D in such a way that every point has identical surroundings, i.e., it is an infinite array of points in three dimensions in which every point has surroundings identical to every other point in the array.

Common materials: with various ‘viewpoints’

Glass: amorphous

Ceramics

Crystal

Graphite

PolymersMetals

Metals and alloys Cu, Ni, Fe, NiAl (intermetallic compound), Brass (Cu-Zn alloys) Ceramics (usually oxides, nitrides, carbides) Alumina (Al2O3), Zirconia (Zr2O3)

Polymers (thermoplasts, thermosets) (Elastomers) Polythene, Polyvinyl chloride, Polypropylene

Common materials: examples

Based on Electrical Conduction Conductors Cu, Al, NiAl Semiconductors Ge, Si, GaAs Insulators Alumina, Polythene*

Based on Ductility Ductile Metals, Alloys Brittle Ceramics, Inorganic Glasses, Ge, Si

* some special polymers could be conducting

MATERIALS SCIENCE & ENGINEERING

PHYSICAL MECHANICAL ELECTRO-CHEMICAL

TECHNOLOGICAL

• Extractive• Casting• Metal Forming• Welding• Powder Metallurgy• Machining

• Structure• Physical Properties

Science of Metallurgy

• Deformation Behaviour

• Thermodynamics• Chemistry• Corrosion

The broad scientific and technological segments of Materials Science are shown in the diagram below.

To gain a comprehensive understanding of materials science, all these aspects have to be studied.

Lattice the underlying periodicity of the crystal

Basis Entity associated with each lattice points

Lattice how to repeatMotif what to repeat

Crystal = Lattice + Motif

Motif or Basis: typically an atom or a group of atoms associated with each lattice point

Definition 1

Translationally periodic arrangement

of motifs

CrystalTranslationally periodic

arrangement of points

Lattice

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An array of points such that every point has identical surroundings

In Euclidean space infinite array

We can have 1D, 2D or 3D arrays (lattices)

Space Lattice

Translationally periodic arrangement of points in space is called a lattice

or

A lattice is also called a Space Lattice

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Unit cell: A unit cell is the sub-division of the space lattice that still retains the overall characteristics of the space lattice. Primitive cell: the smallest possible unit cell of a lattice, having lattice points at each of its eight vertices only.A primitive cell is a minimum volume cell corresponding to a single lattice point of a structure with translational symmetry in 2 dimensions, 3 dimensions, or other dimensions. A lattice can be characterized by the geometry of its primitive cell.

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• atoms pack in periodic, 3D arraysCrystalline materials...

-metals-many ceramics-some polymers

• atoms have no periodic packingNon-crystalline materials...

-complex structures-rapid cooling

crystalline SiO2 (Quartz)

"Amorphous" = Noncrystalline

Materials and Packing

Si Oxygen

• typical of:

• occurs for:

noncrystalline SiO2 (Glass)

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Crystal Systems

7 crystal systems

14 crystal lattices

Unit cell: smallest repetitive volume which contains the complete lattice pattern of a crystal.

a, b, and c are the lattice constants

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The Unite Cell is the smallest group of atom showing the characteristic lattice structure of a particular metal. It is the building block of a single crystal. A single crystal can have many unit cells.

Crystal systemsCubic Three equal axes, mutually perpendicular

a=b=c ===90˚

Tetragonal Three perpendicular axes, only two equala=b≠c ===90˚

Hexagonal Three equal coplanar axes at 120˚ and a fourth unequal axis perpendicular to their planea=b≠c == 90˚ =120˚

Rhombohedral Three equal axes, not at right anglesa=b=c ==≠90˚

Orthorhombic Three unequal axes, all perpendiculara≠b≠c ===90˚

Monoclinic Three unequal axes, one of which is perpendicular to the other twoa≠b≠c ==90˚≠

Triclinic Three unequal axes, no two of which are perpendiculara≠b≠c ≠ ≠≠90˚

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Some engineering applications require single crystals:--diamond single crystals for abrasives

--turbine blades

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What is coordination number?• The coordination number of a central atom in a

crystal is the number of its nearest neighbours.What is lattice parameter?• The lattice constant, or lattice parameter, refers

to the physical dimension of unit cells in a crystal lattice.

• Lattices in three dimensions generally have three lattice constants, referred to as a, b, and c.

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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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• 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 ( 3 a/4 ) 32

atoms

unit cell atom

volume

a 3

unit cell

volume

length = 4R =

Close-packed directions:

3 a

• APF for a body-centered cubic structure = 0.68

aR

a 2

a 3

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• Coordination # = 12

• 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 1/8

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• APF for a face-centered cubic structure = 0.74

Atomic Packing Factor: FCC

maximum achievable APF

APF =

4

3 p ( 2 a/4 )34

atoms

unit cell atom

volume

a3unit 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

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

A B

+ +

FCC

=

Putting atoms in the B position in the II layer and in C positions in the III layer we get a stacking sequence ABC ABC ABC…. The CCP (FCC) crystal

A

BC

A

BC

C

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• Coordination # = 12

• ABAB... Stacking Sequence

• APF = 0.74

• 3D Projection • 2D Projection

Hexagonal Close-Packed Structure (HCP)

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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APF for HCP

c

a

A sites

B sites

A sites

C=1.633a

Number of atoms in HCP unit cell=(12*1/6)+(2*1/2)+3=6atoms

Vol.of HCP unit cell=area of the hexagonal face X height of the hexagonalArea of the hexagonal face=area of each triangle X6

a

ha

Area of triangle = Area of hexagon =

Volume of HCP= APF= 6

a=2r

APF =0.74

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SC-coordination number

6

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• Coordination # = 6 (# nearest neighbors)

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BCC-coordination number

8

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FCC-coordination number

4+4+4=12

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HCP-coordination number

3+6+3=12

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Theoretical Density, r

where n = number of atoms/unit cell A = atomic weight VC = Volume of unit cell = a3 for cubic NA = Avogadro’s number = 6.023 x 1023 atoms/mol

Density = =

VC NA

n A =

Cell Unit of VolumeTotalCell Unit in Atomsof Mass

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• Ex: Cr (BCC) A = 52.00 g/mol R = 0.125 nm n = 2

theoretical

a = 4R/ 3 = 0.2887 nm

ractual

aR

= a 3

52.002

atoms

unit cellmol

g

unit cell

volume atoms

mol

6.023 x 1023

Theoretical Density, r

= 7.18 g/cm3

= 7.19 g/cm3

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Polymorphism • Two or more distinct crystal structures for the same

material (allotropy/polymorphism) titanium , -Ti

carbondiamond, graphite

BCC

FCC

BCC

1538ºC

1394ºC

912ºC

-Fe

-Fe

-Fe

liquid

iron system

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Miller indices

Miller indices: defined as the reciprocals of the intercepts made by the plane on the three axes.

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Procedure for finding Miller indices

Determine the intercepts of the plane along the axes X,Y and Z in terms of the lattice constants a, b and c.

Step 1

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Determine the reciprocals of these numbers.

Step 2

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Find the least common denominator (lcd) and multiply each by this lcd

Step 3

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The result is written in parenthesis.This is called the `Miller Indices’ of the plane in the form (h k l).

Step 4

Find intercepts along axes → 2 3 1 Take reciprocal → 1/2 1/3 1 Convert to smallest integers in the same ratio

→ 3 2 6 Enclose in parenthesis → (326)

(2,0,0)

(0,3,0)

(0,0,1)

Miller Indices for planes

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X

Z

Y

Plane ABC has intercepts of 2 units along X-axis, 3 units along Y-axis and 2 units along Z-axis.

A

C

B

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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)

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For the cubic crystal especially, the important features of Miller indices are, A plane which is parallel to any one of the co-

ordinate axes has an intercept of infinity (). Therefore the Miller index for that axis is

zero; i.e. for an intercept at infinity, the corresponding index is zero.

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.

IMPORTANT FEATURES OF MILLER INDICES

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Worked Example: Calculate the miller indices for the plane with

intercepts 2a, - 3b and 4c the along the crystallographic axes.

The intercepts are 2, - 3 and 4

Step 1: The intercepts are 2, -3 and 4 along the 3 axes

Step 2: The reciprocals are

Step 3: The least common denominator is 12.

Multiplying each reciprocal by lcd, we get 6 -4 and 3

Step 4: Hence the Miller indices for the plane is

6 4 3

Intercepts → 1 Plane → (100)Family → {100} → 3

Intercepts → 1 1 Plane → (110)Family → {110} → 6

Intercepts → 1 1 1Plane → (111)Family → {111} → 8(Octahedral plane)

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Miller Indices :   (100)

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Intercepts :   a , a , ∞ Fractional intercepts :   1 , 1 , ∞ Miller Indices :   (110)

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Intercepts :   a , a , a Fractional intercepts :   1 , 1 , 1 Miller Indices :   (111)

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Intercepts :   ½ a , a , ∞ Fractional intercepts :   ½ , 1 , ∞ Miller Indices :   (210)

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(101)

Z

Y

X

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(122)

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(211)

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Crystallographic Directions

The crystallographic directions are fictitious lines linking nodes (atoms, ions or molecules) of a crystal.

Similarly, the crystallographic planes are fictitious planes linking nodes.

The length of the vector projection on each of the three axes is determined; these are measured in terms of the unit cell dimensions a, b, and c.

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To find the Miller indices of a direction, Choose a perpendicular plane to that direction.

Find the Miller indices of that perpendicular plane.

The perpendicular plane and the direction have the same Miller indices value.

Therefore, the Miller indices of the perpendicular plane is written within a square bracket to represent the Miller indices of the direction like [ ].

Summary of notations

Symbol

Alternate

symbols

Direction

[ ] [uvw] →Particular direction

< > <uvw> [[ ]] →Family of directions

Plane( ) (hkl) → Particular plane

{ } {hkl} (( )) → Family of planes

Point. . .xyz. [[ ]] → Particular point

: : :xyz: → Family of point*A family is also referred to as a symmetrical set

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For each of the three axes, there will exist both positive and negative coordinates.Thus negative indices are also possible, which are represented by a bar overthe appropriate index. For example, the 1

The above image shows [100], [110], and [111] directions within aunit cell

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The vector, as drawn, passes through the origin of the coordinate system, and therefore no translation is necessary. Projections of this vector onto the x, y, and z axes are, respectively,1/2, b, and 0c, which become 1/2, 1, and 0 in terms of the unit cell parameters (i.e., when the a, b, and c are dropped). Reduction of these numbers to the lowest set of integers is accompanied by multiplication of each by the factor 2.This yields the integers 1, 2, and 0, which are then enclosed in brackets as [120].

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Worked Example Find the angle between the directions [2 1 1]

and [1 1 2] in a cubic crystal.

The two directions are [2 1 1] and [1 1 2]

We know that the angle between the two directions,

1 2 1 2 1 2

2 2 2 2 2 21 1 1 2 2 2

½ ½

u u v v w wcos

(u v w ) (u v w )

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In this case, u1 = 2, v1 = 1, w1 = 1, u2 = 1, v2 = 1, w2 = 2

(or) cos = 0.833

= 35° 3530.

2 2 2 2 2 2

(2 1) (1 1) (1 2) 5cos

62 1 l 1 1 2

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http://core.materials.ac.uk/repository/doitpoms/tlp/miller_indices/indexing_a_plane_embed.swf

http://core.materials.ac.uk/repository/doitpoms/tlp/miller_indices/drawing_lattice_planes.swf

http://core.materials.ac.uk/repository/doitpoms/tlp/miller_indices/miller.swf

Reference