Files Chapters ME215 Ch3 R

7/31/2019 Files Chapters ME215 Ch3 R

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CHAPTER 3CHAPTER 3


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Crystal StructuresCrystal Structures

Why study t he st ructure of crystall ine st ructure?Why study t he st ructure of crystall ine st ructure?

TheThe propertiesproperties of some materials are directly related to their crystalof some materials are directly related to their crystalstructure.structure.

For example, pure and undeformed magnesium and beryllium,For example, pure and undeformed magnesium and beryllium,having one crystal structure, are much more brittle than pure anhaving one crystal structure, are much more brittle than pure anddundeformed metals such as gold and silver that have yet anotherundeformed metals such as gold and silver that have yet another

crystal structure.crystal structure.

SignificantSignificant propert y dif ferencespropert y dif ferences exist between crystalline andexist between crystalline andnoncrystallinenoncrystalline materials having same composition.materials having same composition.

NoncrystallineNoncrystalline ceramics and polymers normally are opticallyceramics and polymers normally are opticallytransparent; the same material in crystalline form tend to betransparent; the same material in crystalline form tend to beopaque or, at best, translucent.opaque or, at best, translucent.


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3.2 Fundamental Concepts3.2 Fundamental Concepts

Chapter 2 was concerned primarily with the various types of atomChapter 2 was concerned primarily with the various types of atomicic

bonding, which are determined by the electron structure of thebonding, which are determined by the electron structure of the

individual atoms.individual atoms.

Next level of the structure of the materials deals with some ofNext level of the structure of the materials deals with some ofthethe

arrangement that may be assumed by the atoms in the solid state.arrangement that may be assumed by the atoms in the solid state.

Within this framework, concepts of crystallinity and noncrystallWithin this framework, concepts of crystallinity and noncrystallinity areinity are

introduced.introduced.

Solid materials may be classified according to the regularity wiSolid materials may be classified according to the regularity with whichth which

atoms or ions are arranged with respect to one another.atoms or ions are arranged with respect to one another.


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Crystalline mater ialCrystalline mater ial atoms are situatedatoms are situated

in a repeating or periodic array over largein a repeating or periodic array over large

atomic distancesatomic distances

Upon solidification, the atoms will position themselves inUpon solidification, the atoms will position themselves in aarepet it ive t hreerepet it ive t hree--dimensional pat t erndimensional pat t ern , in which each, in which each

atom is bonded to itsatom is bonded to its nearestnearest -- neighbor atomsneighbor atoms..

All metals, many ceramic materials, and certain polymersAll metals, many ceramic materials, and certain polymersform crystalline structure under normal solidificationform crystalline structure under normal solidification

conditions.conditions.

Noncrystall ine or amorphous materialsNoncrystall ine or amorphous materials do notdo not

crystallize.crystallize.


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3.2 FUNDAMENTAL CONCEPTS

SOLIDS

AMORPHOUS CRYSTALLINEAtoms in a crystalline solid

are arranged in a repetitive

three dimensional pattern

Long Range Order

Atoms in an amorphous

solid are arranged

randomly- No Order

All metals are crystalline solids

Many ceramics are crystalline solids

Some polymers are crystalline solids


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3.2 Fundamental Concepts (Contd.)3.2 Fundamental Concepts (Contd.)

When describing crystallineWhen describing crystallinestructures, atoms ( or ions ) arestructures, atoms ( or ions ) arethought of as beingthought of as being solid spheressolid sphereshaving wellhaving well--defined diameters.defined diameters.

This is termed theThis is termed the atomic hardatomic hardsphere modelsphere model in which spheresin which spheresrepresenting nearestrepresenting nearest--neighborneighbor

atoms touch one another.atoms touch one another.

An example of the hard sphereAn example of the hard spheremodel for the atomic arrangementmodel for the atomic arrangement

found in some common elementalfound in some common elementalmetals is displayed in Figure 3.1cmetals is displayed in Figure 3.1c

LatticeLattice a 3a 3--D array of pointsD array of points

coinciding with atom positions orcoinciding with atom positions orsphere centers.sphere centers.


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LATTICE

Lattice -- points arranged in a pattern thatrepeats itself in three dimensions.

The points in a crystal lattice coincideswith atom centers


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3.3 Unit Cells3.3 Unit Cells

The atomic order in crystalline solids indicates that small grouThe atomic order in crystalline solids indicates that small groupspsof atoms form a repetitive pattern.of atoms form a repetitive pattern.

in describing crystal structures, it is often convenient toin describing crystal structures, it is often convenient tosubdivide the structure into small repeat entities calledsubdivide the structure into small repeat entities called unitunit

cellscells.. Unit cellUnit cell :: The basic structural unit or building block of theThe basic structural unit or building block of the

crystal structure and defines the crystal structure by virtue ofcrystal structure and defines the crystal structure by virtue ofitsitsgeometry and the atom positions within.geometry and the atom positions within.

For most crystal structures are parallelepipeds or prisms havingFor most crystal structures are parallelepipeds or prisms havingthree sets of parallel faces ( In case of Figure 3.1c, it is cubthree sets of parallel faces ( In case of Figure 3.1c, it is cube )e )

Parallelepiped corners coincides with centers of the hard sphereParallelepiped corners coincides with centers of the hard sphereatoms.atoms.

Generally use the unit cell having the highest level ofGenerally use the unit cell having the highest level ofgeomet rical symmet rygeometrical symmet ry . More than single unit cell may be. More than single unit cell may bechosen for a particular crystal structure.chosen for a particular crystal structure.


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Unit cell & LatticeUnit cell & Lattice


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LatticeUnit Cell


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3.4 Metallic Crystal Structures3.4 Metallic Crystal Structures

The atomic bonding in this group of materials is metallic, andThe atomic bonding in this group of materials is metallic, andthus nondirectional.thus nondirectional.

there arethere are no restrictionsno restrictions as to the number and position ofas to the number and position of

nearestnearest--neighbor atoms; this leads toneighbor atoms; this leads to relatively large numbersrelatively large numbersof nearest neighbors andof nearest neighbors and dense atomic packingsdense atomic packings for mostfor mostmetallic crystal structures.metallic crystal structures.

Have several reasons for dense packing:

Typically, only one element is present, so all atomic radii are thesame.

Metallic bonding is not directional. Nearest neighbor distances tend to be small in order to have lower bonding energy.


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Using the hard sphere model for the crystal structure of metals,Using the hard sphere model for the crystal structure of metals, each sphereeach sphere

represents anrepresents an ion coreion core..

Table 3.1 presents the atomic radii and crystal structure type.Table 3.1 presents the atomic radii and crystal structure type.

4


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Three relatively simple structures found are:Three relatively simple structures found are:

FaceFace--Centered Cubic (Centered Cubic (FCCFCC);); BodyBody--Centered Cubic (Centered Cubic (BCCBCC););

Hexagonal CloseHexagonal Close--Packed (Packed (HCPHCP).).


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FACE CENTERED CUBIC STRUCTURE (FCC)


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3.4 Metallic Crystal Structures (Contd.)3.4 Metallic Crystal Structures (Contd.)

The FaceThe Face--Centered Cubic Crystal StructureCentered Cubic Crystal Structure

Unit cells of cubic geometry, with atoms located at each of theUnit cells of cubic geometry, with atoms located at each of the cornerscornersand the centers of all the cube faces.and the centers of all the cube faces.

FCCFCC FaceFace--Centered CubicCentered Cubic

Found in Copper, Aluminum, Silver, and Gold.Found in Copper, Aluminum, Silver, and Gold.

The spheres or ion cores in FCC touch one another across a faceThe spheres or ion cores in FCC touch one another across a facediagonal; the cube edge length a and the the atomic radius R arediagonal; the cube edge length a and the the atomic radius R arerelated throughrelated through

a = 2Ra = 2R22 Each corner atom is shared among eight unit cells, whereas a facEach corner atom is shared among eight unit cells, whereas a facee--

centered atom belong to only two.centered atom belong to only two.

1/8 of each of the eight corner atoms and1/8 of each of the eight corner atoms and

of each of the six faceof each of the six face

atoms, or aatoms, or a total of four whole atomstotal of four whole atoms, may be assigned to a given unit, may be assigned to a given unitcell.cell.


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FACE CENTERED CUBIC STRUCTURE (FCC)

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


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The FaceThe Face--Centered Cubic Crystal StructureCentered Cubic Crystal Structure

Two other important characteristics of a crystal structure are:Two other important characteristics of a crystal structure are:

Coordination number and the atomic packing factor (APF).Coordination number and the atomic packing factor (APF).

Coordination numberCoordination number

The number of nearestThe number of nearest--neighbor or touching atoms for an atom.neighbor or touching atoms for an atom.

For metals, the number is same. For FCC, the coordination numberFor metals, the number is same. For FCC, the coordination numberis 12.is 12.

Atomic Packing Factor (APF)Atomic Packing Factor (APF)

APF is the fraction of solid sphere volume in a unit cell, assumAPF is the fraction of solid sphere volume in a unit cell, assuminging

the atomic hard sphere model, orthe atomic hard sphere model, or

APH = (Volume of atoms in a unit cell ) / (Total unit cell volumAPH = (Volume of atoms in a unit cell ) / (Total unit cell volume )e )

For FCC, APF=0.74For FCC, APF=0.74


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BODY CENTERED CUBIC STRUCTURE (BCC)


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The BodyThe Body--Centered Cubic Crystal StructureCentered Cubic Crystal Structure

BCCBCC BodyBody--Centered CubicCentered Cubic

This metallic crystal structure has a cubic unit cell with atomsThis metallic crystal structure has a cubic unit cell with atoms locatedlocatedat all eight corners and a single atom at the cube center.at all eight corners and a single atom at the cube center.

Center and corner atoms tough one another along cube diagonals.Center and corner atoms tough one another along cube diagonals.

Unit cell length (a) and atomic radius (R) are related throughUnit cell length (a) and atomic radius (R) are related through

a = (4R) /a = (4R) / 33

Examples: Chromium, iron, tungsten, and others exhibit BCC strucExamples: Chromium, iron, tungsten, and others exhibit BCC structure.ture.


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BODY CENTERED CUBIC STRUCTURE (BCC)

Cr, Fe, W, Nb, Ba, V


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The BodyThe Body--Centered Cubic Crystal Structure (Contd.)Centered Cubic Crystal Structure (Contd.)

Two atomsTwo atoms are associated with each BCC unit cell.are associated with each BCC unit cell.

The coordination number forThe coordination number for BCC is 8BCC is 8..

Since the coordination number is less for BCC thanSince the coordination number is less for BCC than

FCC, so also is the atomic packing factor for BCCFCC, so also is the atomic packing factor for BCC

lower _____ 0.68 versus 0.74.lower _____ 0.68 versus 0.74.


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22 September 2003 ME215: Chapter 3 23


The Hexagonal CloseThe Hexagonal Close--Packed Crystal StructurePacked Crystal Structure

HCPHCP Hexagonal Close Packed crystal structureHexagonal Close Packed crystal structure..

Not all metals have unit cells with cubic symmetry; in HCP it isNot all metals have unit cells with cubic symmetry; in HCP it ishexagonal.hexagonal.

TheThe top and bottom facestop and bottom faces of the unit cell consists of six atoms thatof the unit cell consists of six atoms that

form regular hexagons and surround a single atom in the center.form regular hexagons and surround a single atom in the center.

Another planeAnother plane that provides three additional atoms is situated betweenthat provides three additional atoms is situated betweenthe top and the bottom planes.the top and the bottom planes.

The equivalent ofThe equivalent ofsix atomssix atoms is contained in each unit cell; oneis contained in each unit cell; one--sixth ofsixth ofeach of the 12 top and bottom face corner atoms, oneeach of the 12 top and bottom face corner atoms, one--half of each ofhalf of each ofthe 2 center face atoms, and all the 3 midplane interior atoms.the 2 center face atoms, and all the 3 midplane interior atoms.


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HEXAGONAL CLOSE-PACKED STRUCTURE HCP

Mg, Zn, Cd, Zr, Ti, Be


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22 September 2003 ME215: Chapter 3 25

The Hexagonal CloseThe Hexagonal Close--Packed Crystal StructurePacked Crystal Structure

(Contd.)(Contd.)

If a and c represent, respectively, the short andIf a and c represent, respectively, the short and

long unit cell dimensions, c/a ratio should belong unit cell dimensions, c/a ratio should be1.633.1.633.

The coordination number and APF for HCP crystalThe coordination number and APF for HCP crystalstructure are same as for FCC:structure are same as for FCC: 12 and 0.7412 and 0.74,,

respectively.respectively.

HCP metal includes: cadmium, magnesium,HCP metal includes: cadmium, magnesium,

titanium, and zinc.titanium, and zinc.


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SIMPLE CUBIC STRUCTURE (SC)

Rare due to low packing density (only Pd has this structure) Close-packed directions are cube edges.

Coordination # = 6(# nearest neighbors)


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Number of atoms per unit cellNumber of atoms per unit cellBCCBCC 1/8 corner atom x 8 corners + 1 body center atom1/8 corner atom x 8 corners + 1 body center atom

==2 atoms/ uc2 atoms/ uc

FCCFCC 1/8 corner atom x 8 corners +1/8 corner atom x 8 corners +

face atom x 6face atom x 6

facesfaces


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


R l ti hi b t t i di dRelationship between atomic radius and


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Relationship between atomic radius andRelationship between atomic radius and

edge lengthsedge lengths

For FCC:For FCC:a = 2Ra = 2R22

For BCC:For BCC:

a = 4Ra = 4R

//33

For HCPFor HCPa = 2Ra = 2R

c/a = 1.633 (for ideal case)c/a = 1.633 (for ideal case)Note: c/ a rat io could be less or more than the ideal value ofNote: c/ a ratio could be less or more than the ideal value of

1.6331.633


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

ra 42 0 =

a0

a0

r

r

2r


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

ra43 0 =

02a

03aa0


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Coordination NumberCoordination NumberThe number of touching or nearestThe 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


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ATOMIC PACKING FACTORATOMIC PACKING FACTOR


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ATOMIC PACKING FACTORATOMIC PACKING FACTOR

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 cell

volumeclose-packed directions

a

R=0.5a

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

ATOMIC PACKING FACTOR: BCCATOMIC PACKING FACTOR: BCC


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ATOMIC PACKING FACTOR: BCCATOMIC PACKING FACTOR: BCC

aR

APF for a body-centered cubic structure = 0.68

Unit cell contains:1 + 8 x 1/8

= 2 atoms/unit cell

a = 4R /3

APF =

a3

4

3 ( 3a/4)32

atomsunit cell atom

volume

unit cell

volume

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


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

Close packed directions are face diagonals.--Note: All atoms are identical; the face-centered atoms are shaded

differently only for ease of viewing.

Coordination # = 12

ATOMIC PACKING FACTOR: FCCATOMIC PACKING FACTOR: FCC


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ATOMIC PACKING FACTOR: FCCATOMIC PACKING FACTOR: FCC

APF =

a3

43 ( 2a/4)34

atoms

unit cell atom

volume

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

a = 2R2

3 5 Density Computations3 5 Density Computations


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3.5 Density Computations3.5 Density Computations

Density of a material can be determined theoreticallyDensity of a material can be determined theoreticallyfrom the knowledge of its crystal structure (from itsfrom the knowledge of its crystal structure (from itsUnit cell information)Unit cell information)

DensityDensity== mass/Volumemass/Volume Mass is the mass of the unit cell and volume is theMass is the mass of the unit cell and volume is the

unit cell volume.unit cell volume.

mass = ( number of atoms/unit cell)mass = ( number of atoms/unit cell)nnx mass/atomx mass/atom mass/atom = atomic weightmass/atom = atomic weightAA/Avogadro/Avogadros Numbers Number

NNAA

Volume = Volume of the unit cellVolume = Volume of the unit cellVVcc

O C STHEORETICAL DENSITY


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THEORETICAL DENSITYTHEORETICAL DENSITY

= n A

VcNA

# 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


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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--88 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 FCCSolution:Solution: = n A / V= n A / Vcc NNAAnn = 4= 4

VVcc= a= a33

= (2R= (2R2)2)33

= 16 R= 16 R33

22NNAA = 6.023 x 10= 6.023 x 102323 atoms/molatoms/mol

= 4 x 63.5 g/mol / 16= 4 x 63.5 g/mol / 16 2(1.28 x 102(1.28 x 10--88 cm)cm)33 x 6.023 xx 6.023 x

10102323 atoms/molatoms/mol

Ans = 8.98 g/cmAns = 8.98 g/cm33

Experimentally determined value of density of Cu = 8.94 g/Experimentally determined value of density of Cu = 8.94 g/cmcm33

D i i f M i l ClD iti f M t i l Cl


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Densities of Material ClassesDensities of Material Classes

metals > ceramics >polymers

Why?

Data from Table B1, Callister 7e.

(g/cm

)3

Graphite/

Ceramics/Semicond

Metals/Alloys Composites/fibersPolymers

1

2

20

30Based on data in Table B1, Callister

*GFRE, CFRE, & AFRE are Glass,Carbon, & Aramid Fiber-Reinforced

Epoxy composites (values based on60% volume fraction of aligned fibersin an epoxy matrix).10

34

5

0.3

0.4

0.5

Magnesium

Aluminum

Steels

Titanium

Cu,Ni

Tin, Zinc

Silver, Mo

TantalumGold, WPlatinum

Ggraphite

Silicon

Glass -sodaConcrete

Si nitrideDiamondAl oxide

Zirconia

HDPE, PSPP, LDPE

PC

PTFE

PETPVCSilicone

Wood

AFRE*

CFRE*

GFRE*

Glass fibers

Carbon fibers

Aramid fibers

Metals have...

close-packing(metallic bonding)

often large atomic masses

Ceramics have... less dense packing

often lighter elements

Polymers have...

low packing density(often amorphous)

lighter elements (C,H,O)

Composites have...

intermediate values

In general

3.6 Polymorphism and Allotropy3.6 Polymorphism and Allotropy


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3.6 Polymorphism and Allotropy3.6 Polymorphism and Allotropy

PolymorphismPolymorphism

The phenomenon in some metals, asThe phenomenon in some metals, as

well as nonmetals, having more than one crystalwell as nonmetals, having more than one crystal

structures.structures.

When found in elemental solids, the condition is oftenWhen found in elemental solids, the condition is often

calledcalled 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 highwhereas diamond is formed at extremely highpressures.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 912oo

C.C.


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Two or more distinct crystal structures for the sameTwo or more distinct crystal structures for the same

material (allotropy/polymorphism)material (allotropy/polymorphism)

titaniumtitanium,, --TiTi

carboncarbon

diamond, graphitediamond, graphite

BCC

FCC

BCC

1538C

1394C

912C

-Fe

-Fe

-Fe

liquid

iron system

POLYMORPHISM AND ALLOTROPYPOLYMORPHISM AND ALLOTROPY


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

FeFe

FCC (at Temperature above 912FCC (at Temperature above 912 ooC)C)

912912

oo

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

3 7 Crystal Systems3 7 Crystal Systems


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3.7 Crystal Systems3.7 Crystal Systems

Since there are many different possible crystal structures,Since there are many different possible crystal structures,it is sometimes convenient to divide them into groupsit is sometimes convenient to divide them into groups

according to unit cell configurations and/or atomicaccording to unit cell configurations and/or atomic

arrangements.arrangements.

One such scheme is based on the unit cell geometry, i.e.One such scheme is based on the unit cell geometry, i.e.

the shape of the appropriate unit cell parallelepipedthe shape of the appropriate unit cell parallelepipedwithout regard to the atomic positions in the cell.without regard to the atomic positions in the cell.

Within this framework, an x, y, and z coordinate system isWithin this framework, an x, y, and z coordinate system isestablished with its origin at one of the unit cell corners;established with its origin at one of the unit cell corners;

each x, y, and zeach x, y, and z--axes coincides with one of the threeaxes coincides with one of the three

parallelepiped edges that extend from this corner, asparallelepiped edges that extend from this corner, asillustrated in Figure.illustrated in Figure.

The Lattice ParametersThe Lattice Parameters


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The Lattice ParametersThe Lattice Parameters

Lattice parametersLattice parameters

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

Parameters.Parameters.

Seven different possibleSeven different possible

bi ti f dbi ti f d


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combinations of edgecombinations of edge

lengths and angles givelengths and angles giveseven crystal systems.seven crystal systems.

Shown in Table 3.2Shown in Table 3.2

Cubic system has theCubic system has the

greatest degree ofgreatest degree ofsymmetry.symmetry.

Triclinic system has theTriclinic system has theleast symmetry.least symmetry.

3.7 CRYSTAL SYSTEMS3.7 CRYSTAL SYSTEMS


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3.8 Point Coordinates in an Orthogonal3.8 Point Coordinates in an Orthogonal


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Coordinate System Simple CubicCoordinate System Simple Cubic

3.9 Crystallographic Direct ions in Cubic System3.9 Crystallographic Direct ions in Cubic System


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Determinat ion of t he direct ional indices in cubicDeterminat ion of t he direct ional indices in cubicsystem:system:

Four Step Procedure (Text Book Method)Four Step Procedure (Text Book Method)1.1. Draw a vector represent ing t he direct ion w it hin the unitDraw a vector represent ing the direct ion w it hin the unit

cell such t hat it passes t hrough the origin of t hecell such t hat it passes through t he origin of t he xyzxyzcoordinate axes.coordinate axes.

2.2. Determ ine the proj ect ions of t he vector onDeterm ine the proj ect ions of t he vector on xyzxyz axes.axes.

3.3. Mult iply or divide byMult iply or divide by comm on factorcomm on factor t o obtain the t hreeto obtain the t hree

smallest int eger values.smallest int eger values.4.4. Enclose the three int egers inEnclose t he three integers in square bracketssquare brackets [ ] .[ ] .

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

uu ,, vv , and, and ww are the int egersare the int egers

Crystallographic Directions in Cubic SystemCrystallographic Directions in Cubic System


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[120]

[110]

[111]



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Head and Tail Procedure for determiningHead and Tail Procedure for determining


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Miller Indices for Crystallographic DirectionsMiller Indices for Crystallographic Directions

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

points.points.2.2. Subtract the coordinate points of the tailSubtract the coordinate points of the tail

from the coordinate points of the head.from the coordinate points of the head.

3.3. Remove fractions.Remove fractions.

4.4. Enclose in [ ]Enclose in [ ]

Indecies of Crystallographic Directions in Cubic SystemIndecies of Crystallographic Directions in Cubic System


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Direction AHead 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 = [33]Direction B

Head point tail point

(0, 1, 1/2) (2/3,1,1)

-2/3, 0, -1/2Multiply by 6 to get smallest

integers

_ _B = [403] C = [???] D = [???]

Indices of Crystallographic Directions in Cubic SystemIndices of Crystallographic Directions in Cubic System


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Direction C

Head Point Tail Point

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

0, -1/2, -1

Multiply by 2 to get the smallest integers

_ _

C = [0I2]Direction D

Head Point Tail Point

(1, 0, 1/2) (1/2, 1, 0)

1/2, -1, 1/2

Multiply by 2 to get the smallest

integers

_

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



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[210]



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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 ><

3.10 MILLER INDICES FOR3.10 MILLER INDICES FOR

CRYSTALLOGRAPHIC PLANESCRYSTALLOGRAPHIC PLANES


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CRYSTALLOGRAPHIC PLANESCRYSTALLOGRAPHIC PLANES

Miller I ndices for crystallographic planes are t heMiller I ndices for crystallographic planes are t he

reciprocals of t he f ract ional int ercepts (w it hreciprocals of t he f ract ional int ercepts (w it h

fract ions cleared) w hich t he plane makes w it hfract ions cleared) w hich t he plane makes w it h

t he crystallographic x,y,z axes of t he threethe crystallographic x,y,z axes of t he threenonparallel edges of t he cubic unit cell.nonparallel edges of t he cubic unit cell.

44--Step Procedure:Step Procedure:

1.1. Find theFind the interceptsintercepts that the plane makes with the threethat the plane makes with the threeaxesaxes x,y,zx,y,z. If the plane passes through origin change. If the plane passes through origin change

the origin or draw a parallel plane elsewhere (e.g. inthe origin or draw a parallel plane elsewhere (e.g. in

adjacent unit cell)adjacent unit cell)2.2. Take theTake the reciprocalreciprocal of theof the interceptsintercepts

3.3. Remove fractionsRemove fractions

4.4. Enclose in ( )Enclose in ( )

Miller Indecies of Planes in CrystallogarphicMiller Indecies of Planes in Crystallogarphic

Planes in Cubic SystemPlanes in Cubic System


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Planes in Cubic SystemPlanes in Cubic System

Drawing Plane of known Miller Indices in a cubic unitDrawing Plane of known Miller Indices in a cubic unit

cellcell


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cellcell

Draw ( ) plane110

Miller Indecies of Planes in Crystallogarphic Planes inMiller Indecies of Planes in Crystallogarphic Planes in

Cubic SystemCubic System


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Origin for AOrigin for B

Origin

for A

A = (I0) B = (I22) A = (2I) B = (02)

CRYSTALLOGRAPHIC PLANES ANDCRYSTALLOGRAPHIC PLANES AND


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C S OG C S

DIRECTIONS IN HEXAGONAL UNIT CELLSDIRECTIONS IN HEXAGONAL UNIT CELLS

MillerMiller

--Bravais indicesBravais indices

----

same as Millersame as Miller

indices for cubic crystals except that thereindices for cubic crystals except that there

are 3 basal plane axes and 1 vertical axis.are 3 basal plane axes and 1 vertical axis.

Basal planeBasal plane ---- close packed plane similarclose packed plane similar

to the (1 1 1) FCC plane.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

[ t ] h t[ t ] h t ( )( )


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[uvtw] where t=[uvtw] where t=--(u+v)(u+v)

Conversion from 3Conversion from 3--index system to 4index system to 4--index system:index system:

Miller Bravais indices are h,k,i,lMiller Bravais indices are h,k,i,lwith i =with i = --(h+k).(h+k).

Basal plane indices (0 0 0 1)

'

''

''

)(

)2(3

)2(3

wnw

vut

uvnv

vun

u

= +=

=

=][][ ''' uvtwwvu

Basal plane indices (0 0 0 1)

HCP Crystallographic DirectionsHCP Crystallographic Directions


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1. Vector repositioned (if necessary) to pass

through origin.

2. Read off projections in terms of unitcell dimensions a1, a2, a3, orc

3. Adjust to smallest integer values

4. Enclose in square brackets, no commas

[uvtw]

[1120]ex: , , -1, 0 =>

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

dashed red lines indicate

projections onto a1

and a2

axes

a1

a2

a3

-a32

a2

2

a1

-a3

a1

a2

z Algorithm

HCP Crystallographic DirectionsHCP Crystallographic Directions


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Hexagonal CrystalsHexagonal Crystals 4 parameter4 parameter MillerMiller--Bravais lattice coordinates areBravais lattice coordinates are

related to therelated to the direction indices (i.e.,direction indices (i.e., uu''vv''ww'') as) as

follows.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

Crystallographic PlanesCrystallographic Planes (HCP)(HCP)


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In hexagonal unit cells the same idea is usedIn hexagonal unit cells the same idea is used

a2

a3

a1

z

example a1 a2 a3 c1. Intercepts 1 -1 12. Reciprocals 1 1/

1 0

-1

-1

1

1

3. Reduction 1 0 -1 1

4. Miller-Bravais Indices (1011)

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

MillerMiller--Bravais Indices for crystallographic planesBravais Indices for crystallographic planes

in HCPin HCP


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in HCPin HCP

_

(1211)

MillerMiller--Bravais Indices for crystallographicBravais Indices for crystallographic

directions and planes in HCPdirections and planes in HCP


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directions and planes in HCPdirections and planes in HCP

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


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


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Atomic arrangement on [110] directionAtomic arrangement on [110] direction

in FCCin FCC


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in FCCin FCC

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


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Linear DensityLinear DensityLDLD

is defined as the number of atoms per unitis defined as the number of atoms per unit

length whose centers lie on the directionlength whose centers lie on the direction

vector of a given crystallographic direction.vector of a given crystallographic direction.

Linear DensityLinear DensityNumber of atoms


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ex: linear density of Al in [110]direction

a= 0.405 nm

Linear Density of AtomsLinear Density of Atoms LD =LD =

a

[110]

Unit length of direction vector

Number of atoms

# atoms

length

1

3.5 nma2

2LD

==

Linear DensityLinear Density


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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 a2 a

a = 4Ra = 4R// 33

RRaLD 42

3

)3/4(2

1

2

1

===[110]

2a

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


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## of atom centered on the directionof atom centered on the directionvector [110] = 2 atomsvector [110] = 2 atoms

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

LD = 2LD = 2//4R4R

LD = 1/2RLD = 1/2RLinear density can be defined asLinear density can be defined as

reciprocal of thereciprocal of the repeat distancerepeat distance

rrLD = 1/rLD = 1/r

Planar DensityPlanar Density


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Planar DensityPlanar DensityPDPDis defined as the number of atoms per unitis defined as the number of atoms per unitarea that are centered on a givenarea that are centered on a givencrystallographic 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 the


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# of atoms centered on the# of atoms centered on the

plane (110)plane (110)

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

Area of the planeArea of the plane

= (4R)(2R= (4R)(2R 2) = 8R2) = 8R2222

(111) Plane in FCC

24

1

28

222110

RR

atoms

PD == a = 2R 2

4R

Planar Density of (111) IronPlanar Density of (111) IronSolution (cont):Solution (cont): (111) plane(111) plane 1 atom in plane/ unit surface cell


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( ) ( ) p p

333

2

2

R3

16R

3

42a3ah2area =

===

atoms in plane

atoms above plane

atoms below plane

ah2

3=

a2

2D

repeatu

nit

1

= =nm2

atoms7.0

m2atoms

0.70 x 1019

3 2R

3

16Planar Density =

atoms

2D repeat unit

area

2D repeat unit

Planar Density of (100) IronPlanar Density of (100) Iron


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Solution:Solution: At T < 912At T < 912C iron has the BCC structure.C iron has the BCC structure.

(100)

Radius of iron R= 0.1241 nm

R3

34

a=

Adapted from Fig. 3.2(c), Callister 7e.

2D repeat unit

=Planar Density =a2

1

atoms

2D repeat unit

=nm2

atoms12.1

m2atoms

= 1.2 x 10191

2

R

3

34area

2D repeat unit

Closed Packed Crystal StructuresClosed Packed Crystal Structures


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FCC and HCP both have:FCC and HCP both have:CN = 12CN = 12 andand 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

FCCFCC --------(111) plane is the Closed Packed Plane(111) plane is the Closed Packed PlaneHCPHCP --------(0001) plane is the Closed Packed Plane(0001) plane is the Closed Packed Plane

The atomic staking sequence in the above twoThe atomic staking sequence in the above twostructures is different from each otherstructures is different from each other

Closed Packed StructuresClosed Packed Structures


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Closed Packed Plane Stacking in HCPClosed Packed Plane Stacking in HCP


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Closed Packed Plane Stacking in FCCClosed Packed Plane Stacking in FCC


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


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For a crystalline solid, when the periodic and repeatedFor a crystalline solid, when the periodic and repeatedarrangement of atoms is perfect or extends throughoutarrangement of atoms is perfect or extends throughoutthe entirety of the specimen without interruption, thethe entirety of the specimen without interruption, theresult is a single crystal.result is a single crystal.

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

Single crystals exist in nature, but may also be producedSingle crystals exist in nature, but may also be producedartificially.artificially.

They are ordinarily difficult to grow, because theThey are ordinarily difficult to grow, because theenvironment must be carefully controlled.environment must be carefully controlled.

Example: Electronic microcircuits, which employ singleExample: Electronic microcircuits, which employ single

crystals of silicon and other semiconductors.crystals of silicon and other semiconductors.

Polycrystalline MaterialsPolycrystalline Materials

3.13 Polycryt alline Materials3.13 Polycryt alline Materials


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PolycrystallinePolycrystalline

crystalline solidscrystalline solidscomposed of many smallcomposed of many small

crystals or grains.crystals or grains.

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

a)a) Small crystallite nuclei GrowthSmall crystallite nuclei Growthof the crystallites.of the crystallites.

b)b) Obstruction of some grains thatObstruction of some grains that

are adjacent to one another isare adjacent to one another isalso shown.also shown.

c)c) Upon completion ofUpon completion of

solidification, grains that aresolidification, grains that are

adjacent to one another is alsoadjacent to one another is also

shown.shown.

d)d) Grain structure as it wouldGrain structure as it would

appear under the microscope.appear under the microscope.

Crystals as Building BlocksCrystals as Building Blocks

S i i li ti i i l t l


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Someengineering applications require single crystals:

Properties of crystalline materials

often related to crystal structure.

--Ex: Quartz fractures more easily

along some crystal planes than

others.

--diamond single

crystals for abrasives

--turbine blades

Fig. 8.33(c), Callister 7e.(Fig. 8.33(c) courtesy

of Pratt and Whitney).(Courtesy Martin Deakins,

GE Superabrasives,Worthington, OH. Used with

permission.)

(Courtesy P.M. Anderson)

Mostengineering materials are polycrystals.PolycrystalsPolycrystals Anisotropic


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Nb-Hf-W plate with an electron beam weld.

Each "grain" is a single crystal. If grains are randomly oriented,overall component properties are not directional.

Grain sizes typ. range from 1 nm to 2 cm

(i.e., from a few to millions of atomic layers).

Adapted from Fig. K,

color inset pages of

Callister 5e.

(Fig. K is courtesy ofPaul E. Danielson,

Teledyne Wah Chang

Albany)

1 mm

Isotropic

Single Crystals

P ti ith Data from Table 3 3

Single vs PolycrystalsSingle vs PolycrystalsE (diagonal) = 273 GPa


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-Properties vary with

direction: anisotropic.

-Example: the modulus

of elasticity (E) in BCC iron:

Polycrystals

-Properties may/may notvary with direction.

-If grains are randomly

oriented: isotropic.

(Epoly iron = 210 GPa)

-If grains are textured,

anisotropic.

200 m

Data from Table 3.3,Callister 7e.(Source of data is R.W.

Hertzberg, Deformationand Fracture Mechanicsof EngineeringMaterials, 3rd ed., JohnWiley and Sons, 1989.)

Adapted from Fig.4.14(b), Callister 7e.(Fig. 4.14(b) is courtesy

of L.C. Smith and C.

Brady, the National

Bureau of Standards,

Washington, DC [now

the National Institute ofStandards and

Technology,

Gaithersburg, MD].)

E (edge) = 125 GPa

3.15 Anisotropy3.15 Anisotropy

The physical properties of single crystals of someThe physical properties of single crystals of some

b t d d th t ll hi di ti ib t d d th t ll hi di ti i


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substances depend on the crystallographic direction insubstances depend on the crystallographic direction inwhich the measurements are taken.which the measurements are taken.

For example, modulus of elasticity, electricalFor example, modulus of elasticity, electricalconductivity, and the index of refraction may haveconductivity, and the index of refraction may have

different values in the [100] and [111] directions.different values in the [100] and [111] directions.

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

Substances in which measured properties areSubstances in which measured properties are

independent of the direction of measurement areindependent of the direction of measurement are

isotropicisotropic..


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SUMMARYSUMMARY Atoms may assemble into crystalline or


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Atoms may assemble into crystalline oramorphous structures.

Common metallic crystal structures are FCC, BCC, and

HCP. Coordination number and atomic packing factorare the same for both FCC and HCP crystal structures.

We can predict the density of a material, provided we

know the atomic weight, atomic radius, and crystalgeometry (e.g., FCC, BCC, HCP).

Crystallographic points, directions and planes are

specified in terms of indexing schemes.Crystallographic directions and planes are related

to atomic linear densities and planar densities.

SUMMARYSUMMARY


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Materials can be single crystals or polycrystalline.

Material properties generally vary with single crystal

orientation (i.e., they are anisotropic), but are generally

non-directional (i.e., they are isotropic) in polycrystalswith randomly oriented grains.

Some materials can have more than one crystal

structure. This is referred to as polymorphism (orallotropy).

X-ray diffraction is used for crystal structure and

interplanar spacing determinations.

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