CHAPTER 3 THE STRUCTURE OF CRYSTALLINE SOLIDS CRYSTALLINE SOLIDS.

CHAPTER 3CHAPTER 3

THE STRUCTURE THE STRUCTURE OF OF

CRYSTALLINE CRYSTALLINE SOLIDSSOLIDS

SOLIDS

AMORPHOUS CRYSTALLINEAtoms in an amorphous

solid are arranged randomly- No Order

Atoms in a crystalline solid are arranged in a repetitive three dimensional pattern

Long Range Order All metals are crystalline solids

Many ceramics are crystalline solids

Some polymers are crystalline solids

3.2 FUNDAMENTAL CONCEPTS

LATTICE

Lattice -- points arranged in a pattern that repeats itself in three dimensions.

The points in a crystal lattice coincides with atom centers

3-D view of a lattice

3.3 3.3 UNIT CELLUNIT CELL

►Unit cellUnit cell -- smallest grouping -- smallest grouping which can be arranged in three which can be arranged in three dimensions to create the lattice. dimensions to create the lattice. Thus the Thus the Unit CellUnit Cell is basic is basic structural unit or building block of structural unit or building block of the crystal structurethe crystal structure

Unit cell & LatticeUnit cell & Lattice

Lattice Unit Cell

4

• Tend to be densely packed.

• Have several reasons for dense packing:

-Typically, only one element is present, so all atomic radii are the same.-Metallic bonding is not directional.-Nearest neighbor distances tend to be small in order to have lower bonding energy.

• Have the simplest crystal structures.

Let us look at three such structures...

3.4 METALLIC CRYSTALS3.4 METALLIC CRYSTALS

FACE CENTERED CUBIC STRUCTURE (FCC)

FACE CENTERED CUBIC STRUCTURE (FCC)

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

BODY CENTERED CUBIC STRUCTURE (BCC)

BODY CENTERED CUBIC STRUCTURE (BCC)

Cr, Fe, W, Nb, Ba, V

HEXAGONAL CLOSE-PACKED STRUCTURE HCP

Mg, Zn, Cd, Zr, Ti, Be

SIMPLE CUBIC STRUCTURE (SC)

Only Pd has SC structure

Number of atoms per unit Number of atoms per unit cellcell

BCCBCC 1/8 corner atom x 8 corners + 1 body center 1/8 corner atom x 8 corners + 1 body center atomatom

==2 atoms/uc2 atoms/uc

FCCFCC 1/8 corner atom x 8 corners + ½ face atom 1/8 corner atom x 8 corners + ½ face atom x 6 facesx 6 faces


HCPHCP 3 inside atoms + ½ basal atoms x 2 bases + 3 inside atoms + ½ basal atoms x 2 bases + 1/ 6 1/ 6 corner atoms x 12 cornerscorner atoms x 12 corners


Relationship between atomic radius Relationship between atomic radius and edge lengths and edge lengths

For FCC:For FCC:a = 2R√2a = 2R√2

For BCC:For BCC:a = 4R a = 4R //√3√3

For HCPFor HCPa = 2Ra = 2Rc/a = 1.633 (for ideal case)c/a = 1.633 (for ideal case)

Note: c/a ratio could be less or more than the ideal Note: c/a ratio could be less or more than the ideal value of 1.633value of 1.633

Face Centered Cubic (FCC)Face Centered Cubic (FCC)

ra 42 0

a0

a0

r

r

2r

Body Centered Cubic (BCC)Body Centered Cubic (BCC)

02a

03aa0

ra 43 0

Coordination NumberCoordination Number

►The number of touching or nearest The number of touching or nearest neighbor atomsneighbor atoms

►SC is 6SC is 6►BCC is 8BCC is 8►FCC is 12FCC is 12►HCP is 12HCP is 12

6

APF = Volume of atoms in unit cell*

Volume of unit cell

*assume hard spheres

• APF for a simple cubic structure = 0.52

APF = a3

4

3(0.5a)31

atoms

unit cellatom

volume

unit cellvolumeclose-packed directions

a

R=0.5a

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

ATOMIC PACKING FACTORATOMIC PACKING FACTOR

aR

• APF for a body-centered cubic structure = 0.68

Unit cell contains: 1 + 8 x 1/8 = 2 atoms/unit cell

ATOMIC PACKING FACTOR: BCCATOMIC PACKING FACTOR: BCC

APF = a3

4

3( 3a/4)32

atoms

unit cell atomvolume

unit cell

volume

a = 4R /√3

• Coordination # = 12

• Close packed directions are face diagonals.--Note: All atoms are identical; the face-centered atoms are shaded differently only for ease of viewing.

FACE CENTERED CUBIC STRUCTURE FACE CENTERED CUBIC STRUCTURE (FCC)(FCC)

APF = a3

4

3( 2a/4)34

atoms

unit cell atomvolume

unit cell

volume

Unit cell contains: 6 x 1/2 + 8 x 1/8 = 4 atoms/unit cella

• APF for a face-centered cubic structure = 0.74

ATOMIC PACKING FACTOR: FCCATOMIC PACKING FACTOR: FCC

a = 2R√2

3.5 Density Computations3.5 Density Computations►► Density of a material can be determined Density of a material can be determined

theoretically from the knowledge of its crystal theoretically from the knowledge of its crystal structure (from its Unit cell information)structure (from its Unit cell information)

► DensityDensitymass/Volumemass/Volume► Mass is the mass of the unit cell and volume is Mass is the mass of the unit cell and volume is

the unit cell volume.the unit cell volume.► mass = ( number of atoms/unit cell) “mass = ( number of atoms/unit cell) “nn” x ” x

mass/atommass/atom ► mass/atom = atomic weight “mass/atom = atomic weight “AA”/Avogadro’s ”/Avogadro’s

Number “Number “NNAA””► Volume = Volume of the unit cell “Volume = Volume of the unit cell “VVcc””

THEORETICAL DENSITYTHEORETICAL DENSITY

n AVcNA

# atoms/unit cell Atomic weight (g/mol)

Volume/unit cell (cm3/unit cell)

Avogadro's number (6.023 x 1023 atoms/mol)

Example problem on Density ComputationExample problem on Density Computation

Problem:Problem: Compute the density of CopperCompute the density of CopperGiven: Given: Atomic radius of Cu = 0.128 nm (1.28 x 10Atomic radius of Cu = 0.128 nm (1.28 x 10-8 -8

cm)cm) Atomic Weight of Cu = 63.5 g/molAtomic Weight of Cu = 63.5 g/mol Crystal structure of Cu is FCCCrystal structure of Cu is FCC

Solution:Solution: = n A / V = n A / Vc c NNAA

nn = 4= 4

VVcc= a= a33 = (2R√2) = (2R√2)33 = 16 R = 16 R3 3 √2√2

NNAA = 6.023 x 10 = 6.023 x 102323 atoms/mol atoms/mol

= 4 x 63.5 g/mol / 16 √2(1.28 x 10= 4 x 63.5 g/mol / 16 √2(1.28 x 10-8 -8 cm)cm)3 3 x 6.023 x x 6.023 x 10102323 atoms/mol atoms/mol

Ans = 8.98 g/cmAns = 8.98 g/cm33

Experimentally determined value of density of Cu = 8.94 Experimentally determined value of density of Cu = 8.94 g/cmg/cm33

3.6 Polymorphism and Allotropy3.6 Polymorphism and Allotropy► Polymorphism Polymorphism The phenomenon in some metals, as well The phenomenon in some metals, as well

as nonmetals, having more than one crystal structures.as nonmetals, having more than one crystal structures.

► When found in elemental solids, the condition is often called When found in elemental solids, the condition is often called allotropyallotropy..

► Examples:Examples: Graphite is the stable polymorph at ambient conditions, Graphite is the stable polymorph at ambient conditions,

whereas diamond is formed at extremely high pressures.whereas diamond is formed at extremely high pressures. Pure iron is BCC crystal structure at room temperature, Pure iron is BCC crystal structure at room temperature,

which changes to FCC iron at 912which changes to FCC iron at 912ooC.C.

POLYMORPHISM AND ALLOTROPYPOLYMORPHISM AND ALLOTROPY

BCC (From room temperature to 912 BCC (From room temperature to 912 ooC)C)

FeFe FCC (at Temperature above 912 FCC (at Temperature above 912

ooC)C)

912 912 ooCC

Fe (BCC)Fe (BCC) Fe (FCC)Fe (FCC)

3.7 Crystal Systems3.7 Crystal Systems

► Since there are many different possible crystal Since there are many different possible crystal structures, it is sometimes convenient to divide structures, it is sometimes convenient to divide them into groups according to unit cell them into groups according to unit cell configurations and/or atomic arrangements.configurations and/or atomic arrangements.

► One such scheme is based on the unit cell One such scheme is based on the unit cell geometry, i.e. the shape of the appropriate unit cell geometry, i.e. the shape of the appropriate unit cell parallelepiped without regard to the atomic parallelepiped without regard to the atomic positions in the cell.positions in the cell.

► Within this framework, an x, y, and z coordinate Within this framework, an x, y, and z coordinate system is established with its origin at one of the system is established with its origin at one of the unit cell corners; each x, y, and z-axes coincides unit cell corners; each x, y, and z-axes coincides with one of the three parallelepiped edges that with one of the three parallelepiped edges that extend from this corner, as illustrated in Figure.extend from this corner, as illustrated in Figure.

The Lattice ParametersThe Lattice Parameters

Lattice parametersLattice parameters

a, b, c, a, b, c, , , , , are called are called thethe latticelattice

Parameters.Parameters.

► Seven different Seven different possible combinations possible combinations of edge lengths and of edge lengths and angles give seven angles give seven crystal systems.crystal systems.

► Shown in Table 3.2Shown in Table 3.2

► Cubic system has the Cubic system has the greatest degree of greatest degree of symmetry.symmetry.

► Triclinic system has Triclinic system has the least symmetry.the least symmetry.

The Lattice ParametersThe Lattice Parameters

Lattice parameters areLattice parameters are

a, b, c, a, b, c, , , , , are called are called thethe latticelattice

Parameters.Parameters.

3.7 CRYSTAL SYSTEMS3.7 CRYSTAL SYSTEMS

3.8 Point Coordinates in an 3.8 Point Coordinates in an Orthogonal Coordinate System Orthogonal Coordinate System

Simple CubicSimple Cubic

3.9 Crystallographic Directions in Cubic 3.9 Crystallographic Directions in Cubic SystemSystem

► Determination of the directional indices in Determination of the directional indices in cubic system:cubic system:

Four Step Procedure (Text Book Method)Four Step Procedure (Text Book Method)1.1. Draw a vector representing the direction within the Draw a vector representing the direction within the

unit cell such that it passes through the origin of the unit cell such that it passes through the origin of the xyzxyz coordinate axes.coordinate axes.

2.2. Determine the projections of the vector on Determine the projections of the vector on xyzxyz axes. axes.3.3. Multiply or divide by Multiply or divide by common factorcommon factor to obtain the to obtain the

three smallest integer values.three smallest integer values.4.4. Enclose the three integers in Enclose the three integers in square bracketssquare brackets [ ]. [ ].

e.g.e.g. [ [uvwuvw] ]

uu, , vv, and , and ww are the integers are the integers

Crystallographic DirectionsCrystallographic Directions1. Vector repositioned (if necessary) to pass through origin.2. Read off projections in terms of unit cell dimensions a, b, and c3. Adjust to smallest integer values4. Enclose in square brackets, no commas

[uvw]

ex: 1, 0, ½ => 2, 0, 1 => [ 201 ]

-1, 1, 1

families of directions <uvw>

z

x

Algorithm

where overbar represents a negative index

[ 111 ]=>

y

Crystallographic Directions in Cubic SystemCrystallographic Directions in Cubic System

[120]

[110]

[111]

Head and Tail Procedure for determining Head and Tail Procedure for determining Miller Indices for Crystallographic Miller Indices for Crystallographic

DirectionsDirections

1.1. Find the coordinate points of head Find the coordinate points of head and tail points.and tail points.

2.2. Subtract the coordinate points of the Subtract the coordinate points of the tail from the coordinate points of the tail from the coordinate points of the head.head.

3.3. Remove fractions.Remove fractions.

4.4. Enclose in [ ]Enclose in [ ]

Indecies of Crystallographic Directions in Cubic Indecies of Crystallographic Directions in Cubic SystemSystem

Direction A

Head point – tail point

(1, 1, 1/3) – (0,0,2/3)

1, 1, -1/3

Multiply by 3 to get smallest integers

3, 3, -1

A = [33Ī]

_ _ B = [403]

C = [???] D = [???]

Direction BHead point – tail point(0, 1, 1/2) – (2/3,1,1)-2/3, 0, -1/2Multiply by 6 to get smallest integers

Indices of Crystallographic Directions in Cubic Indices of Crystallographic Directions in Cubic SystemSystem

A = [???]

C = [0I2]

B= [???] D = [I2I]

Direction C

Head Point – Tail Point

(1, 0, 0) – (1, ½, 1)

0, -1/2, -1

Multiply by 2 to get the smallest integers

Direction DHead Point – Tail Point(1, 0, 1/2) – (1/2, 1, 0)1/2, -1, 1/2Multiply by 2 to get the smallest integers


[210]

CRYSTALLOGRAPHIC DIRECTIONS IN CRYSTALLOGRAPHIC DIRECTIONS IN HEXAGONAL UNIT CELLSHEXAGONAL UNIT CELLS

Miller-Bravais indices -- same as Miller-Bravais indices -- same as Miller indices for cubic crystals Miller indices for cubic crystals

except that there are 3 basal plane except that there are 3 basal plane axes and 1 vertical axis.axes and 1 vertical axis.

Basal plane -- close packed plane Basal plane -- close packed plane similar to the (1 1 1) FCC plane.similar to the (1 1 1) FCC plane.

contains 3 axes 120contains 3 axes 120oo apart. apart.

Direction Indices in HCP Unit CellsDirection Indices in HCP Unit Cells – –

[uvtw] where t=-(u+v)[uvtw] where t=-(u+v) Conversion from 3-index system to 4-index system:Conversion from 3-index system to 4-index system:

'

''

''

)(

)2(3

)2(3

wnw

vut

uvn

v

vun

u

][][ ''' uvtwwvu

HCP Crystallographic HCP Crystallographic DirectionsDirections► Hexagonal CrystalsHexagonal Crystals

4 parameter 4 parameter Miller-Bravais lattice Miller-Bravais lattice coordinates are related to the coordinates are related to the direction direction indices (i.e., indices (i.e., uu''vv''ww'') as follows.) as follows.

'ww

t

v

u

)vu( +-

)'u'v2(3

1-

)'v'u2(3

1-

]uvtw[]'w'v'u[

Fig. 3.8(a), Callister 7e.

-a3

a1

a2

z

Indices of a Family or FormIndices of a Family or Form

00]1[],1[000],1[0[001],[010],[100], >100 <

01]1[1],1[010],1[],101[],11[00],11[

]1[10],1[010],1[1[101],[011],[110],110 ><

]11[1],111[1],11[],111[

11],1[1],1[1],1[11[111],111 ><

3.10 MILLER INDICES FOR 3.10 MILLER INDICES FOR CRYSTALLOGRAPHIC PLANESCRYSTALLOGRAPHIC PLANES

► Miller Indices for crystallographic planes are Miller Indices for crystallographic planes are the reciprocals of the fractional intercepts the reciprocals of the fractional intercepts (with fractions cleared) which the plane (with fractions cleared) which the plane makes with the crystallographic x,y,z axes makes with the crystallographic x,y,z axes of the three nonparallel edges of the cubic of the three nonparallel edges of the cubic unit cell.unit cell.

► 4-Step Procedure:4-Step Procedure:

1.1. Find the Find the interceptsintercepts that the plane makes with the that the plane makes with the three axes three axes x,y,zx,y,z. If the plane passes through . If the plane passes through origin change the origin or draw a parallel plane origin change the origin or draw a parallel plane elsewhere (e.g. in adjacent unit cell)elsewhere (e.g. in adjacent unit cell)

2.2. Take the Take the reciprocalreciprocal of the of the interceptsintercepts

3.3. Remove fractionsRemove fractions

4.4. Enclose in ( )Enclose in ( )

Crystallographic PlanesCrystallographic Planesz

x

ya b

c

4. Miller Indices (110)

example a b cz

x

ya b

c


1. Intercepts 1 1 2. Reciprocals 1/1 1/1 1/

1 1 03. Reduction 1 1 0

1. Intercepts 1/2 2. Reciprocals 1/½ 1/ 1/

2 0 03. Reduction 2 0 0

example a b c

Crystallographic PlanesCrystallographic Planesz

x

ya b

c


example1. Intercepts 1/2 1 3/4

a b c

2. Reciprocals 1/½ 1/1 1/¾2 1 4/3

3. Reduction 6 3 4

(001)(010),

Family of Planes {hkl}

(100), (010),(001),Ex: {100} = (100),

Miller Indecies of Planes in Miller Indecies of Planes in Crystallogarphic Planes in Cubic SystemCrystallogarphic Planes in Cubic System

Drawing Plane of known Miller Indices in a Drawing Plane of known Miller Indices in a cubic unit cellcubic unit cell

Draw ( ) plane110

Miller Indecies of Planes in Crystallogarphic Miller Indecies of Planes in Crystallogarphic Planes in Cubic SystemPlanes in Cubic System

A = (IĪ0) B = (I22)

Origin for A

Origin for AOrigin for B

A = (2IĪ) B = (02Ī)

CRYSTALLOGRAPHIC PLANES AND CRYSTALLOGRAPHIC PLANES AND DIRECTIONS IN HEXAGONAL UNIT CELLSDIRECTIONS IN HEXAGONAL UNIT CELLS

Miller-Bravais indices -- same as Miller-Bravais indices -- same as Miller indices for cubic crystals Miller indices for cubic crystals

except that there are 3 basal plane except that there are 3 basal plane axes and 1 vertical axis.axes and 1 vertical axis.

Basal plane -- close packed plane Basal plane -- close packed plane similar to the (1 1 1) FCC plane.similar to the (1 1 1) FCC plane.

contains 3 axes 120contains 3 axes 120oo apart. apart.

Crystallographic Planes Crystallographic Planes (HCP)(HCP)► In hexagonal unit cells the same idea is In hexagonal unit cells the same idea is

used used

example a1 a2 a3 c

4. Miller-Bravais Indices (1011)

1. Intercepts 1 -1 12. Reciprocals 1 1/

1 0 -1-1

11

3. Reduction 1 0 -1 1

a2

a3

a1

z

Adapted from Fig. 3.8(a), Callister 7e.

Miller-Bravais Indices for crystallographic Miller-Bravais Indices for crystallographic planes in HCPplanes in HCP

_ (1211)

Miller-Bravais Indices for Miller-Bravais Indices for crystallographic directions and planes in crystallographic directions and planes in

HCPHCP

Atomic Arrangement on (110) plane in Atomic Arrangement on (110) plane in

FCCFCC

Atomic Arrangement on (110) plane in Atomic Arrangement on (110) plane in BCCBCC

Atomic arrangement on [110] Atomic arrangement on [110] direction in FCCdirection in FCC

3.11 Linear and Planar Atomic Densities3.11 Linear and Planar Atomic Densities

► Linear Density “LD”Linear Density “LD”

is defined as the number of atoms per is defined as the number of atoms per unit length whose centers lie on the unit length whose centers lie on the direction vector of a given direction vector of a given crystallographic direction. crystallographic direction.

vectordirection ofLength

directionon centered atoms of No.LD

Linear DensityLinear Density

LD for [110] in BCC.LD for [110] in BCC.

# of atom centered on the direction # of atom centered on the direction vector [110] vector [110]

= 1/2 +1/2 = 1= 1/2 +1/2 = 1

Length of direction vector [110] = Length of direction vector [110] = 2 2 aa

a = 4Ra = 4R/ / 3 3

[110]

2a

RRaLD

42

3

)3/4(2

1

2

1

Linear DensityLinear Density► LD of [110] in FCCLD of [110] in FCC

# # of atom centered on the direction of atom centered on the direction vector [110] = 2 atomsvector [110] = 2 atoms

Length of direction vector [110] = 4RLength of direction vector [110] = 4R

LD = 2 LD = 2 //4R4R

LD = 1/2RLD = 1/2R Linear density can be defined Linear density can be defined as reciprocal of the as reciprocal of the repeat repeat distance ‘r’distance ‘r’

LD = 1/rLD = 1/r

Planar DensityPlanar Density

►Planar Density “PD”Planar Density “PD” is defined as the number of atoms per unit is defined as the number of atoms per unit

area that are centered on a given area that are centered on a given crystallographic plane.crystallographic plane.

No of atoms centered on the planeNo of atoms centered on the plane

PD = PD = —————————————————————————— Area of the planeArea of the plane

Planar Density of (110) plane in FCCPlanar Density of (110) plane in FCC

# of atoms centered on # of atoms centered on the plane (110) the plane (110)

= 4(1/4) + 2(1/2) = = 4(1/4) + 2(1/2) = 22 atomsatoms

Area of the planeArea of the plane= (4R)(2R = (4R)(2R 2) = 8R 2) = 8R2222

4R

a = 2R 2

(111) Plane in FCC

24

1

28

222110

RR

atomsPD

Closed Packed Crystal StructuresClosed Packed Crystal Structures

► FCC and HCP both have:FCC and HCP both have:CN = 12 CN = 12 and and APF = 0.74APF = 0.74

APF= 0.74 is the most efficient packing.APF= 0.74 is the most efficient packing.

Both FCC and HCP have Closed Packed PlanesBoth FCC and HCP have Closed Packed Planes FCC ----(111) plane is the Closed Packed FCC ----(111) plane is the Closed Packed

PlanePlane HCP ----(0001) plane is the Closed Packed HCP ----(0001) plane is the Closed Packed

PlanePlaneThe atomic staking sequence in the above The atomic staking sequence in the above two structures is different from each othertwo structures is different from each other

Closed Packed StructuresClosed Packed Structures

Closed Packed Plane Stacking in HCPClosed Packed Plane Stacking in HCP

• Coordination # = 12

• ABAB... Stacking Sequence

• APF = 0.74

• 3D Projection • 2D Projection

Adapted from Fig. 3.3(a), Callister 7e.

Hexagonal Close-Packed Hexagonal Close-Packed Structure (HCP)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

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 SequenceFCC Stacking Sequence

B B

B

BB

B B

B sitesC C

CA

C C

CA

AB

C

Closed Packed Plane Stacking in FCCClosed Packed Plane Stacking in FCC

Crystalline and Noncrystalline MaterialsCrystalline and Noncrystalline Materials3.13 Single Crystals3.13 Single Crystals

► For a crystalline solid, when the periodic and For a crystalline solid, when the periodic and repeated arrangement of atoms is perfect or repeated arrangement of atoms is perfect or extends throughout the entirety of the specimen extends throughout the entirety of the specimen without interruption, the result is a single crystal.without interruption, the result is a single crystal.

► All unit cells interlock in the same way and have All unit cells interlock in the same way and have the same orientation.the same orientation.

► Single crystals exist in nature, but may also be Single crystals exist in nature, but may also be produced artificially.produced artificially.

► They are ordinarily difficult to grow, because the They are ordinarily difficult to grow, because the environment must be carefully controlled.environment must be carefully controlled.

► Example: Electronic microcircuits, which employ Example: Electronic microcircuits, which employ single crystals of silicon and other semiconductors.single crystals of silicon and other semiconductors.

Polycrystalline MaterialsPolycrystalline Materials3.13 Polycrytalline 3.13 Polycrytalline

MaterialsMaterialsPolycrystallinePolycrystalline crystalline crystalline

solids composed of many solids composed of many small crystals or grains.small crystals or grains.

Various stages in the Various stages in the solidification :solidification :

a)a) Small crystallite nuclei Small crystallite nuclei Growth of the crystallites.Growth of the crystallites.

b)b) Obstruction of some grains Obstruction of some grains that are adjacent to one that are adjacent to one another is also shown.another is also shown.

c)c) Upon completion of Upon completion of solidification, grains that are solidification, grains that are adjacent to one another is adjacent to one another is also shown.also shown.

d)d) Grain structure as it would Grain structure as it would appear under the appear under the microscope.microscope.

3.15 Anisotropy3.15 Anisotropy► The physical properties of single crystals of some The physical properties of single crystals of some

substances depend on the crystallographic direction in substances depend on the crystallographic direction in which the measurements are taken.which the measurements are taken.

► For example, modulus of elasticity, electrical conductivity, For example, modulus of elasticity, electrical conductivity, and the index of refraction may have different values in the and the index of refraction may have different values in the [100] and [111] directions.[100] and [111] directions.

► This directionality of properties is termed This directionality of properties is termed anisotropyanisotropy. .

► Substances in which measured properties are independent Substances in which measured properties are independent of the direction of measurement are of the direction of measurement are isotropicisotropic..

CHAPTER 3 THE STRUCTURE OF CRYSTALLINE SOLIDS CRYSTALLINE SOLIDS.

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