Introduction to Crystallography - Flaney · PDF fileMaterials Science & Engineering MSE 271 Unit 1– 2 1 Introduction to Crystallography Instructor: Joshua U.Otaigbe Iowa State University

1M S E 2 7 1 Uni t 1– 2M a t e r i a l s S c i e n c e & E n g i n e e r i n g

Introduction to CrystallographyIntroduction to Crystallography

Instructor: Joshua U.Instructor: Joshua U. OtaigbeOtaigbeIowa State UniversityIowa State University


Goals for this unitGoals for this unit

§§ Define basic terms in crystallographyDefine basic terms in crystallography§§ Be able to identify 7 crystal systems and 14Be able to identify 7 crystal systems and 14

BravaisBravais latticeslattices§§ Name and draw specific atomic positions, Name and draw specific atomic positions,

crystallographic directions and planescrystallographic directions and planes


Crystal Structure and PeriodicityCrystal Structure and Periodicity§§ Crystalline Materials Crystalline Materials -- atoms are in an ordered atoms are in an ordered

33--D periodic array extending over many atomic D periodic array extending over many atomic distancesdistances§§ metals,ceramics, semiconductors, some metals,ceramics, semiconductors, some

polymerspolymers§§ single crystals; polycrystalline massessingle crystals; polycrystalline masses

§§ Amorphous Materials Amorphous Materials -- short range but no long short range but no long range orderrange order§§ glasses, many polymersglasses, many polymers

§§ Intermediates Intermediates


Crystal Structure and PeriodicityCrystal Structure and Periodicity

§§ Lattice Lattice -- a 3a 3--D array of points in space with which D array of points in space with which atom(s) are associated in describing a crystalatom(s) are associated in describing a crystal§§ Unit Cells Unit Cells -- the smallest repeat unit which defines the smallest repeat unit which defines

the crystal structurethe crystal structure

§§ lattice constants (lattice parameters) lattice constants (lattice parameters) -- unit cell unit cell edges lengths ( a,b,c) and angles (edges lengths ( a,b,c) and angles (α, β, γα, β, γ))

§§ Crystal Systems Crystal Systems -- unique cell shapes which can fill unique cell shapes which can fill 33--D space (there are 7)D space (there are 7)


Generalized cell axes and anglesGeneralized cell axes and angles

§§ a, b and c are lengths of cell edgesa, b and c are lengths of cell edges§§ α, β, γ α, β, γ are angles (are angles (αα is across from a, etc.)is across from a, etc.)

ab

c

αβ

γ


Unit Cells (2Unit Cells (2-- D examples)D examples)

Shackelford 3.1


Crystal SystemsCrystal Systems

Every faceis a square

One faceis square

Every face isa rectangle

Cubic Tetragonala=b=c a=b°c

α=β=γ=90α=β=γ=90Orthorhombic

a°b°cα=β=γ=90


Crystal SystemsCrystal Systems -- (cont.)(cont.)

“Squished”tetragonal

“Pushed over”cube

“Pushed over”orthorhombic

(in one direction)

“Pushed over”orthorhombic

(in two directions)

Triclinic

a�b�cα�β�γ �9 0

Monoclinic

a�b�cα = γ = 90 , β�90

Rhombohedrala=b=c

α = β = γ 9 0�

Hexagonal

a=b�cα = β = 9 0 γ = 1 2 0


Alternative representation of Alternative representation of hexagonal unit cellhexagonal unit cell

1 0M S E 2 7 1 Uni t 1– 2M a t e r i a l s S c i e n c e & E n g i n e e r i n g

Drawing Crystal SystemsDrawing Crystal Systems§§ Crystal Systems are Crystal Systems are

drawn with just drawn with just lattice points lattice points

§§ Actual structures can Actual structures can be considered to be be considered to be space filling (with space filling (with one or more atoms one or more atoms per lattice point)per lattice point)


The 14The 14 BravaisBravais LatticesLattices

System Bravais LatticeCubic Simple, body-centered,

face-centeredTetragonal Simple, body-centered

Ortho-rhombic

Simple, body-centered,base-centered, face-centered

Monoclinic Simple, base-centered

Rhombohedral, Hexagonal, and Triclinic have only simple forms


The 14The 14 BravaisBravais LatticesLattices


Lattice Positions, Directions and PlanesLattice Positions, Directions and Planes

§§ Many material properties vary with Many material properties vary with direction in the crystaldirection in the crystal

§§ It is often necessary to be able to specify It is often necessary to be able to specify certain atom positions, as well as directions certain atom positions, as well as directions and planes in crystals.and planes in crystals.

§§ Atom positions, directions and planes are Atom positions, directions and planes are described by sets of three integers described by sets of three integers --“indices”“indices”


Specifying atom positionsSpecifying atom positions

§§ Select corner of unit cell as origin (this Select corner of unit cell as origin (this position is considered to be the 000 position is considered to be the 000 position)position)

§§ Determine the number (or fractions) of Determine the number (or fractions) of translations along each unit cell axis from translations along each unit cell axis from 000 needed to arrive at the position of 000 needed to arrive at the position of interest (x units along a, y units along b, etc)interest (x units along a, y units along b, etc)

§§ Express position indices as xyz or x,y,zExpress position indices as xyz or x,y,z


Atom position indicesAtom position indices


Characteristic TranslationsCharacteristic Translations

§§ In each crystal structure, theIn each crystal structure, the BravaisBravais lattice has a lattice has a characteristic translation that will take you to an characteristic translation that will take you to an identical featureidentical feature§§ simplesimple 1,1,11,1,1§§ bccbcc 1,1,1 (trivial) 1,1,1 (trivial) a n da n d 1/2,1/2,1/21/2,1/2,1/2§§ fccfcc 1,1,1 (trivial) 1,1,1 (trivial) a n da n d 1/2,1/2,0 1/2,1/2,0 ( a l l 3 )( a l l 3 )

§§ base centered 1,1,1 (trivial) base centered 1,1,1 (trivial) a n da n d 1/2,1/2,0 1/2,1/2,0 ( o n l y )( o n l y )

§§ This will be helpful when we look at actual crystal This will be helpful when we look at actual crystal structurestructure


Direction indicesDirection indices

§§ Select origin in the cell Select origin in the cell §§ Draw vector from origin in the desired Draw vector from origin in the desired

direction (or parallel to it)direction (or parallel to it)§§ Determine the three components of this Determine the three components of this

vector in terms of the three unit cell axes vector in terms of the three unit cell axes lengthslengths

§§ Write three component lengths in orderWrite three component lengths in order abcabc§§ Clear fractions and enclose in [ ] bracketsClear fractions and enclose in [ ] brackets


Direction indices (cont.)Direction indices (cont.)

§§ Negative components are indicated with a Negative components are indicated with a bar over them (rather than with a negative bar over them (rather than with a negative sign) sign) §§ e.g., [113] not [11e.g., [113] not [11--3]3]

§§ Commas are never used to separate Commas are never used to separate direction indices within the [ ]direction indices within the [ ]§§ e.g., [212] but not [2,1,2]e.g., [212] but not [2,1,2]


DirectionDirection indiciesindicies (cont.)(cont.)

||lel directn

same notation

“cos only origin

shifted



§§ Families of “Families of “crystallographicallycrystallographically equivalent” equivalent” directions are indicated by < > bracketdirections are indicated by < > bracket

§§ E.g., [100], [010] and [001] directions in cubic E.g., [100], [010] and [001] directions in cubic crystals arecrystals are crystallographicallycrystallographically equivalent and equivalent and belong to the <100> family of directionsbelong to the <100> family of directions

§§ [100] and [010] but not [001] belong to the same [100] and [010] but not [001] belong to the same family (<100>) in tetragonal crystalsfamily (<100>) in tetragonal crystals



§§ The <111> family of directions inThe <111> family of directions in cubicscubics


Crystallographic DirectionsCrystallographic Directions

x

y

z

x

y

z

x

y

z

x

y

z

x

y

z

x

y

z


Crystallographic DirectionsCrystallographic Directions

x

y

z

x

y

z

x

y

z

x

y

z

x

y

z

x

y

z

[111] [110] [111]

[210] [010] [111]


Crystal Plane IndicesCrystal Plane Indices

§§ This special category of crystal indices is called This special category of crystal indices is called “Miller Indices”“Miller Indices”

§§ Draw the plane within a unit cell (or several Draw the plane within a unit cell (or several adjacent cells if needed)adjacent cells if needed)

§§ Select a cell corner as the origin (not part of Select a cell corner as the origin (not part of plane)plane)----This is an arbitrary choice.This is an arbitrary choice.

§§ Determine the intercepts of the plane along the Determine the intercepts of the plane along the three cell axes in axis lengthsthree cell axes in axis lengths§§ x lengths along a, y along b, z along cx lengths along a, y along b, z along c§§ If a plane is parallel to an axis, the intercept on that axis If a plane is parallel to an axis, the intercept on that axis

is taken as infinityis taken as infinity


Miller indices for planesMiller indices for planes

§§ Write these intercepts in the order xyz Write these intercepts in the order xyz (without separating commas)(without separating commas)§§ Take reciprocals of x y and z and reduce to Take reciprocals of x y and z and reduce to

lowest integer values lowest integer values §§ These integers are designated h k and lThese integers are designated h k and l

§§ Enclose in ( ) brackets, e.g., (Enclose in ( ) brackets, e.g., (hklhkl))


Miller Indices (cont.)Miller Indices (cont.)

§§ Commas are never used in expressing Commas are never used in expressing Miller indicesMiller indices

§§ Families of “Families of “crystallographicallycrystallographicallyequivalent” planes are indicated by {equivalent” planes are indicated by {hklhkl}}

§§ e.g., ine.g., in cubicscubics (100) (010) (001) belong to (100) (010) (001) belong to the {100} family of planesthe {100} family of planes


Family of PlanesFamily of Planes


{110} Family of Planes{110} Family of Planes


Crystallographic PlanesCrystallographic Planes


Crystallographic PlanesCrystallographic PlanesName the Planes

x

y

z

x

y

z

x

y

z

x

y

z

x

y

z

x

y

z


Crystallographic PlanesCrystallographic Planes

x

y

z

x

y

z

x

y

z

x

y

z

x

y

z

x

y

z

(111) (011)

(201) (212) (100)

(001)


MillerMiller --BravaisBravais IndicesIndices

§§ For hexagonal crystals, if the unit cell is For hexagonal crystals, if the unit cell is drawn as a hexagonal prism (4 axes) a drawn as a hexagonal prism (4 axes) a special 4special 4--index system for planes is usedindex system for planes is used

§§ Called MillerCalled Miller--BravaisBravais system (system (hkilhkil))§§ Conversion between MillerConversion between Miller--BravaisBravais and and

ordinary Miller indices for a given plane ordinary Miller indices for a given plane uses (h + k) = uses (h + k) = --ii


MillerMiller --BravaisBravais indicesindices


Atomic densitiesAtomic densities

§§ Linear DensitiesLinear Densities§§ Fraction of line length along a particular Fraction of line length along a particular

direction that passes through atoms centered on direction that passes through atoms centered on that linethat line

§§ Planar DensitiesPlanar Densities§§ Fraction of total crystallographic plane area that Fraction of total crystallographic plane area that

is occupied by atoms centered on that planeis occupied by atoms centered on that plane


Planar densityPlanar density

§§ Why do we care?Why do we care?§§ Slip (plastic deformation in metals) depends Slip (plastic deformation in metals) depends

on density of atomic packingon density of atomic packing§§ Slip occurs in Slip occurs in planesplanes that have the greatest that have the greatest

density of atomsdensity of atoms§§ In such planes, slip occurs in the In such planes, slip occurs in the directionsdirections

having the greatest atomic packinghaving the greatest atomic packing


Planar Density and SlipPlanar Density and Slip


Calculate the planar densityCalculate the planar density

§§ Calculate the planar density of the (110) Calculate the planar density of the (110) plane for anplane for an fccfcc crystal (assume one atom crystal (assume one atom per lattice point and that atoms touch along per lattice point and that atoms touch along face diagonals)face diagonals)§§ Note: thisNote: this fccfcc structure would be called a cubic structure would be called a cubic

close packed (close packed (ccpccp) structure) structure§§ A direction along which atoms touch is called a A direction along which atoms touch is called a

close packed direction (e.g., the <110> close packed direction (e.g., the <110> directions in ourdirections in our ccpccp crystal)crystal)


Planar density Planar density -- exampleexampleA B C

D E F

AB

C

D

EF

• Compute planar area• Compute total “circle” area



A B C

D E F

Unit cell plane area A P :AC = 4RAD = 2R 2AP = (AC)(AD)= (4R)(2R 2) = 8R2 2



Total circle area - 1/ 4 of atoms A ,C,D and F1/ 2 of atoms B and E = 2 "whole " circlesAC = (2)π R2

Planar density

PD = AC

AP

=(2)π R2

8R2 20.555


Volume (true) densityVolume (true) density

§§ The true density of a crystal is determined The true density of a crystal is determined by dividing the mass of all of the atoms in a by dividing the mass of all of the atoms in a unit cell by the volume of the unit cell unit cell by the volume of the unit cell §§ We will make such calculations in a later unitWe will make such calculations in a later unit


End of Lecture, Unit 1End of Lecture, Unit 1––22

§§ READREAD§§ Lecture notes & Shackelford, pp. 54Lecture notes & Shackelford, pp. 54––6565

§§ Optional ReadingOptional Reading§§ Visualizations in Materials Science 2Visualizations in Materials Science 2

Documents

Introduction to Crystallography - Flaney · PDF fileMaterials Science & Engineering MSE 271 Unit 1– 2 1 Introduction to Crystallography Instructor: Joshua U.Otaigbe Iowa State University