Transcript
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
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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.
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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.
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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.
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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.
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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
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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).
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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.
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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.
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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.
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