ME2151-Chp4

4-1

C H A P T E R 4

C R Y S T A L S T R U C T U R E S

4.1 ATOMIC ARRANGEMENTS

4.2 LATTICES AND UNIT CELLS 4.2.1 Unit Cel l s in Space

4.2.2 Atomic Pack ing in Crysta l s

4.3 METALLIC CRYSTAL STRUCTURES 4.3.1 Face-Centred Cubic (FCC)

Crysta l St ructure

4.3.2 Body-Centred Cubic (BCC) Crysta l

St ructure

4.3.3 Hexagonal Close-Packed (HCP)

Structure

4.3.4 A l lot ropy/Polymorphism

4.4 CLOSE-PACKED CRYSTAL STRUCTURES

4.5 INTERSTITIAL POSITIONS AND SIZES

4-2

4.6 CRYSTALLOGRAPHIC DIRECTIONS AND PLANES 4.6.1 Coordinates of Points

4.6.2 Indices of Di rect ions

4.6.3 Fami l ies of Di rect ions

4.6.4 Hexagonal Crysta l s (Mi l ler -

Brava is Indices)

4.6.5 Indices of P lanes

4.6.6 Fami l ies of P lanes

4.6.7 P lanes in Hexagonal Crysta l s

4.7 CRYSTALLINE MATERIALS

4-3

4.1 ATOMIC ARRANGEMENTS

• The properties of a material depend not only on atomic

bonding and forces, but also, equally important, on how

atoms pack together. There are 3 levels in which atoms

may be arranged:

• No order: there is no special positional

relationship or interaction between the

atoms; e.g. inert gases (Fig. 4.1-1a).

• Short-range order: the specific

arrangement of atoms extends only to

an atom's nearest neighbours. Materials

exhibiting short-range order are

amorphous (glassy) (Fig. 4.1-1b).

• Long-range order: there is a special

arrangement of atoms that is repeated

throughout the entire material.

Crystalline materials exhibit both short-

range and long-range order. The

repetitive pattern formed by atoms in a

crystalline solid is known as a lattice (Fig.

4.1-1c).

(a) No order.

(b) Short-rangeorder.

(c) Long-rangeorder.

Fig. 4.1-1 The levelsof atomic

arrangement.

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4.2 LATTICES AND UNIT CELLS

• A lattice is a collection of points (positions in space)

arranged in a periodic pattern so that surroundings of each

point in the lattice are identical. The points that make up

the lattice are called lattice points (Figs. 4.2-1, 4.2-2). An

everyday example in 2-dimensions is wall-paper.

Fig. 4.2-1 Lattices and unit cells in 2D.

Fig. 4.2-2 Lattices and unit cells in 3D.

• A unit cell is the smallest subdivision of a lattice that

contains all the characteristics of the entire lattice. A

complete crystal can be formed by translating its unit cell

along each of its edges.

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• The properties of a unit cell are therefore the same as those

of the whole crystal.

• Each unit cell is described by lattice parameters, which

are the lengths of the cell edges and the angles between

the axes. (Figs. 4.2-1, 4.2-2)

4.2.1 Unit Cells in Space

• By geometry, there are only 7 unique unit cell shapes that

may be stacked together in space. And there are only a

total of 14 possible ways in which atoms may be arranged

inside these unit cells. These 14 different unit cells are

known as Bravais lattices and they fall into one of 7 crystal

systems (Fig. 4.2-3).

• One or more atoms may be associated with each lattice

point. The "group of atoms" located at each lattice point is

the basis. The actual crystal structure itself is defined by a

combination of crystal lattice and crystal basis (Fig. 4.2-4).

Crystal lattice

+

+

crystal basis

=

=

crystal structure

4-6

Fig. 4.2-3 14 Bravais lattices grouped into 7

crystal systems.

4-7

4.2.2 Atomic Packing in Crystals

• When describing crystal structures, atoms are assumed to

be solid spheres that touch one another. This is known as

the hard sphere model. The centres of the solid spheres

coincide with the lattice points in unit cells (Fig. 4.2-5).

Fig. 4.2-5 Hard sphere model of

a unit cell.

• The coordination number is the number of nearest

neighbours (touching atoms) to any atom. For materials

with non-directional bonding (i.e. metals and ionic solids),

the lowest energy (most stable) configuration is obtained

when atoms pack as closely as possible, separated only by

their equilibrium bond lengths. In such crystals, the

number of nearest neighbours (i.e. coordination number)

would be as high as possible.

• The atomic packing factor (APF) is the fraction of space

occupied by atoms in a unit cell.

APF =

Volume of atoms in unit cellVolume of unit cell

=

(Number of atoms in cell)(Volume of one atom)Volume of unit cell

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• The properties of an entire crystal, such as the theoretical

density, may be calculated from just one of its unit cells.

Density =

Mass of unit cellVolume of unit cell

=

(Number of atoms in cell)(Mass of one atom)Volume of unit cell

=

(Number of atoms in cell)(Atomic mass)(Volume of unit cell)(Avogadro's number)

4.3 METALLIC CRYSTAL STRUCTURES In pure metals, only one metal ion occupies each lattice

point. Many common metals may be defined by one of 3

crystal structures: face-centred cubic (FCC), body-centred

cubic (BCC) and hexagonal close-packed (HCP) crystal

structures (Table 4.3-1).

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4.3.1 Face-Centred Cubic (FCC) Crystal Structure

• Cubic geometry with one atom at each corner and one at

the centre of each face. (Fig. 4.3-1)

Fig. 4.3-1 FCC structure.

• Corner atoms touch face-centred atoms (along the face

diagonal) but corner atoms do not touch one another.

Face-centred atoms also touch adjacent (but not opposite)

face-centred atoms in the midplanes of the cube.

• Unit cell length: a = 2R 2 (where R is the atomic radius)

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• Coordination number: consider a corner atom, which

touches 4 face-centred atoms on each of the 3 mutually

perpendicular planes passing through the corner atom

itself (Fig. 4.3-2), giving CN = 12.

Fig. 4.3-2 Finding the

coordination number in FCC structures.

• Atomic packing factor: each atom at the centres of the

cube faces is shared by 2 cells; each corner atom is shared

by 8 cells (Fig. 4.3-3), such that: • Atoms per cell = 4; ! APF = 0.74

Fig 4.3-3 Sharing of face and corner atoms in FCC structures.

• Examples of FCC metals: Al, Au, Ag, Cu, Ni, Pb, Pt.

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4.3.2 Body-Centred Cubic (BCC) Crystal Structure

• Cubic geometry with one atom at each corner and one at

the centre of the cube. (Fig. 4.3-4)

Fig. 4.3-4 BCC structure.

• Atoms touch along the body diagonal.

• a =

4R3

• CN = 8.

• Atoms per cell = 2; APF = 0.68

• Examples of BCC metals: Cr, Fe, W, Mo, V.

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4.3.3 Hexagonal Close-Packed (HCP) Structure

• Hexagonal faces at the top and bottom are linked by 6

rectangular side faces. There is one atom at each corner of

the top and bottom hexagons surrounding one atom at

the centre of each hexagon. 3 other atoms are located on

a plane midway between the hexagons (Fig. 4.3-5).

Fig. 4.3-5 HCP structure and its smaller primitive unit cell.

• Corner atoms at the top and bottom hexagons are shared

between 6 cells; the central atom in each hexagon is

shared between 2 cells; the 3 atoms in the midplane

belong to only one cell.

• Lattice parameters: a = 2R; c =

46

a

• By considering the central atom in basal plane, CN =12.

• Atoms per cell = 6 (or 2 per primitive unit cell); APF = 0.74

• Examples of HCP metals: Co, Mg, Ti, Zn.

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4.3.4 Allotropy/Polymorphism

• Allotropy or polymorphism is the ability of an element or

compound to assume more than one crystal structure in

the solid state, depending on external conditions such as

temperature, pressure, magnetic and electric fields (Table 4.3-2)

• The change in crystal structure is usually accompanied by

changes in properties.

• Such property changes can be very useful; e.g. hardening/

softening of steel through controlled heating/cooling (Chps

9&10), piezoelectric transducers (Figs. 4.3-6 & 7).

• Or detrimental; e.g. distortion and cracking due to sudden

changes in volume, especially in brittle ceramics, but also

in metals (Fig. 4.3-8).

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Fig. 4.3-6 PZT (lead zirconate titanate)

ceramic changes its structure from

(a) cubic, to (b) tetragonal,

in response to an electric field.

Fig. 4.3-7 Use of piezoelectric effect

of PZT crystals in inkjet printer head.

Fig. 4.3-8 Tin changes from tetragonal to diamond structure below 13.2°C.

The volume expansion accompanying this transformation from

soft, ductile white tin to hard, brittle grey tin causes it to disintegrate.

4-15

4.4 CLOSE-PACKED CRYSTAL STRUCTURES

• Atoms in metals pack as closely as

possible, separated only by their

equilibrium bond lengths, r0, as this

gives the lowest energy (most

stable) configuration (Fig 4.4-1).

• Close-packed planes or directions

refer to the planes or directions in a

crystal, where the atoms are in

direct contact, assuming a hard

sphere model of atomic packing.

• Plastic deformation in metals

occurs most readily on close-

packed planes along close-packed directions in those

planes (Sec. 6.1). The number and relative positions of these

planes and directions influence properties such as ductility.

• Packing same-sized atoms in FCC or HCP structures gives

the smallest volume (i.e. highest density). FCC and HCP

are known as close-packed structures.

• FCC and HCP differ only in the arrangements of their

close-packed planes (Fig. 4.4-2, 4.4-3); this difference affects

plastic deformation and ductility (Sec. 6.1).

Fig. 4.4-1 Bonding energy is higher when atoms are away from equilibrium separation r0.

4-16

Fig. 4.4-2 Illustration of close-packed stacking sequence.

HCP FCC

Fig. 4.4-3 Close-packed stacking sequence and close-packed planes for HCP and FCC.

4-17

4.5 INTERSTITIAL POSITIONS AND SIZES

• Lattices not completely filled with atoms. Interstices are

the ‘holes’ between lattice atoms.

• Interstitial sites are classified by geometry, based on the

shape of the polyhedron formed by existing lattice atoms

surrounding a particular site. (Fig. 4.5-1 & Table 4.5-1)

• The size of an interstitial site is defined by the radius

ratio,

!

rR , where r is the radius of the largest sphere that can

completely fill the site without straining the adjacent lattice

atoms, and R is the radius of the lattice atoms.

Tetrahedral interstitial Octahedral interstitial Cubic interstitial

!

rR

= 0.225

!

rR

= 0.414

!

rR

= 0.732

Fig. 4.5-1 Interstitial sites and sizes in close-packed crystal structures.

• Atoms (alloy or impurity) occupying interstitial sites (Sec.

6.3.2) must be larger than the size of the holes; smaller

atoms are not allowed to “rattle” around loose in the sites.

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Table 4.5-1 The size and number of interstitial sites in FCC, BCC and HCP. Size of interstitial sites,

!

rR No. of sites per unit cell Crystal

structure Octahedral Tetrahedral Octahedral Tetrahedral

FCC 0.414 0.225 4 8 BCC 0.155 0.291 6 12 HCP 0.414 0.225 6 12

Fig. 4.5-2 Locations of the interstitial sites in FCC, BCC and HCP structures.

4-19

4.6 CRYSTALLOGRAPHIC DIRECTIONS AND PLANES

Some physical and mechanical material properties vary

with the direction or plane within a crystal in which they

are measured. Miller Indices provide a convenient system

of describing points, directions and planes in crystals.

4.6.1 Coordinates of Points

• It is customary to use the right-hand coordinate system.

• The most common orientation is to align the coordinate

axes with the edges of the unit cell, putting the origin at

one corner of the cell.

• Each lattice point is defined by u, v, w, which correspond

to fractions of the lattice parameters, a, b, c.

Fig. 4.6-1 Coordinates of some points

in a unit cell. Note that coordinate axes are not necessarily perpendicular to one another and lattice parameters

may not be the same length. The choice of origin is arbitrary.

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4.6.2 Indices of Directions Procedure for finding Miller indices of directions: 1. Determine the coordinates of 2 points that lie in the

direction of interest. 2. Subtract coordinates of ‘start’ point from ‘end’ point. 3. Clear fractions and/or reduce to lowest integers. 4. Write the indices in square brackets [ ] without commas.

Negative integer values are indicated by placing a bar over the integer. (Fig. 4.6-3)

Fig. 4.6-2 Some common directions in a cubic unit cell.

Worked Example (Fig. 4.6-3) 1. Coordinates of ‘end’ and ‘start’

points: 0,0,1; 12, 1, 0.

2. ‘End’ - ‘start’ coordinates:

(0 - 12), (0 - 1), (1 - 0) = -

12

, -1, 1

3. Clear fractions:

2(- 12

, -1, 1) = -1, -2, 2

4. Miller index: [ 1 2 2]

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4.6.3 Families of Directions

• Certain directions in a unit cell, e.g. the cube edges, have

identical atomic arrangements. Their Miller indices are

different because of the choice of origin and axes (Fig. 4.6-4).

These directions are said to be symmetrically equivalent.

Fig. 4.6-4 Symmetrically equivalent directions in the <100> family.

• Equivalent directions that are related by symmetry may be

grouped as a family of directions. 'Members' in the same

family share the same set of Miller indices, but in different

permutations, including negative indices (Fig. 4.6-4).

• The family is represented by the Miller indices of one

family ‘member’, but enclosed in angle brackets <u v w>. It

is customary to choose one of the positive indices.

• In cubic systems, the directions along the cube edges

belong to the same family of <100> directions. Face

diagonals form a different family, <110>, and body

diagonals yet another family, <111>.

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4.6.4 Hexagonal Crystals (Miller-Bravais Indices)

• Due to the hexagonal geometry, equivalent directions will

not have the same set of Miller indices. To reflect the

symmetry of hexagonal systems, a 4-axis Miller-Bravais

coordinate system may be used instead (Fig. 4.6-5).

Fig. 4.6-5 3- and 4-axis coordinate systems for hexagonal crystals. Since only 3 axes are

required to define any 3D geometry, the extra axis, a3, in the basal plane is actually redundant; the 3rd index, t , in the Miller-Bravais system [uvtw],

is thus a function of u and v, such that t = -(u + v).

• For directions in hexagonal system, first find Miller indices

(Sec. 4.6-2), then convert to Miller-Bravais indices as follows:

• Miller to Miller-Bravais:

[u’ v’ w’] ! [uvtw]

u =

n3 (2u’ - v’)

v =

n3 (2v’ - u’)

t = -

n3 (u’ + v’)

w = w’

• Miller-Bravais to Miller:

[uvtw] ! [u’ v’ w’]

u’ = u - t

v’ = v - t

w’ = w

where n is the integer required to clear fractions and/or reduce to lowest integers

4-23

4.6.5 Indices of Planes Procedure for finding Miller indices of planes:

1. Find the coordinates of the points where the plane

intercepts the 3 axes. If the plane passes through the

origin, shift the origin of the coordinate system.

2. Take reciprocals of these intercepts.

3. Clear fractions but do not reduce to lowest integers.

4. Write the indices in round brackets ( ) without commas.

Negative integer values are indicated by placing a bar

over the integer. (Fig. 4.6-6)

Worked example (Fig. 4.6-6)

1. Plane passes through origin, so

shift axes to x’, y’, z’.

Intercepts: x = -1, y = 1, z = "

2. Reciprocals:

1x = -1,

1y

= 1, 1z

= 0

3. No fractions to clear.

4. Miller indices: ( 1 10).

4-24

4.6.6 Families of Planes

• Planes that contain identical atomic arrangements in a unit

cell and are related by symmetry are equivalent.

• A family of equivalent planes share the same set of Miller

indices. The family is expressed by enclosing the Miller

indices of one ‘member’ in braces, {h k l}.

Fig. 4.6-7 The family of {110} planes contains (110) (101) (011) ( 11 0) ( 101 ) ( 01 1 ).

Fig. 4.6-8 Some planes in a cubic unit cell.

4-25

4.6.7 Planes in Hexagonal Crystals

• In order that equivalent planes have the same indices, the

4-axis Miller-Bravais coordinate system may be used.

• Miller-Bravais indices for planes are determined using the

same procedure as in 3-index Miller systems (Sec. 4.6-1) (no

formulae required), except that 4 intercepts are found,

giving indices of the form (h k i l).

Worked Example (Fig. 4.6-9)

Reciprocals of intercepts for 4-axis Miller-Bravais system Miller Indices

of planes

a1 a2 a3 z

Miller-Bravais Indices of planes

!

100( ) 1 0 -1 0 " ( 101 0) (

!

1 1 0) 1 -1 0 0 " ( 11 00) ( 01 0) 0 -1 1 0 " ( 01 10)

• Note that with only 3 axes, Miller indices are different even

though the planes are equivalent. The 4-index Miller-

Bravais notation yields similar indices for equivalent planes.

4-26

4.7 CRYSTALLINE MATERIALS

• In a single crystal, the same orientation and alignment of

unit cells is maintained throughout the entire crystal.

• Most materials are polycrystalline: their structures are

composed of many small crystals (grains) with identical

structures but different orientations (Fig. 4.7-1).

Fig. 4.7-1 The solidification of a polycrystalline material.

• Grain boundaries are the

interfaces where grains of

different orientations meet

(Fig. 4.7-2). Single crystals do

not contain grain boundaries.

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• The absence of grain boundaries in single crystals impart

unique properties but such crystals are extremely difficult

to grow, requiring carefully controlled conditions, which

are expensive. Single crystals are essential to some

applications; e.g. semiconductors and jet turbine blades,

piezoelectric transducers.

• Besides the absence of grain boundaries, single crystals

exhibit directionality in properties, such as magnetism, electrical conductivity and elastic modulus and creep resistance, which depend on the crystallographic direction of measurement. This directionality is called anisotropy.

• The random orientation of individual grains in a

polycrystalline material means that measured properties are independent of crystallographic direction. Such materials are said to be isotropic.

• Grain boundaries are disordered regions of atomic

mismatch where atoms are displaced from their

equilibrium positions (Fig. 4.4-1) and there are improper

coordination numbers (Sec. 4.2.2) across the boundaries.

Atoms at grain boundaries hence possess higher energy.

• This interfacial energy makes grain boundaries

preferential sites for chemical reactions and other chemical

changes.

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• Therefore, grain boundaries are attacked more aggressively

by chemical etchants. Under a microscope, the more

deeply etched grain boundaries scatter more light, and

appear darker (Figs. 4.7-3/4), thus revealing the microstructure.

This is the principle behind metallography.

Fig. 4.7-3 Observation of grains and grain boundaries in stainless steel sample. Note that different orientations of the grains result in differences in reflection.

Fig. 4.7-4 Observed microstructure in 2-D and the underlying 3-D structure.