the structure of cry stall in solid

8/9/2019 The Structure of Cry Stall In Solid

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The Structure ofThe Structure of

crystalline solidscrystalline solids


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Why Study The Structure OfWhy Study The Structure Of

Crystalline Solid ?Crystalline Solid ?Because the properties of some materials are directly related to theirBecause the properties of some materials are directly related to theircrystal structure.crystal structure.

Solid MaterialSolid Material

NonNon--Crystalline MaterialCrystalline Material(Amorphous)(Amorphous)

In which there is no longIn which there is no long--range atomic order.range atomic order.

Crystalline MaterialCrystalline Material

"In which the atoms or"In which the atoms orions are arrangedions are arranged

with respect to onewith respect to one

another " or "There isanother " or "There isa repetition ora repetition or

periodic array overperiodic array overlarge atomiclarge atomic

distances."distances."


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Lattice:Lattice: Lattice means a threeLattice means a three

dimensional array of pointsdimensional array of pointscoinciding with atom positions (orcoinciding with atom positions (or

sphere centers)sphere centers)

Unit Cell:Unit Cell: It is the smallest groupIt is the smallest groupof atoms or ions of material byof atoms or ions of material by

repeating it in space we obtain therepeating it in space we obtain thewhole lattice.whole lattice.

Metallic Crystal StructureMetallic Crystal Structure


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The FaceThe Face--Centered Cubic CrystalCentered Cubic Crystal

Structure (FCC)Structure (FCC)

The crystal structure found for many metals has a unit cellThe crystal structure found for many metals has a unit cell

of cubic geometry, with atoms located at each of cornersof cubic geometry, with atoms located at each of corners

and the centers of all the cube faces. Some of theand the centers of all the cube faces. Some of the

familiar metals having RR this crystal structure arefamiliar metals having RR this crystal structure arecopper , aluminum , silver and gold.copper , aluminum , silver and gold.


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Characteristics Of (FCC) CrystalCharacteristics Of (FCC) Crystal

Structure :Structure :The cube edge length (a) and the atomic radius (R) are related throughThe cube edge length (a) and the atomic radius (R) are related through

a=a=22RR

The corner atom is shared among eight unit cells , but the faceThe corner atom is shared among eight unit cells , but the face--centered atomcentered atombelongs to only two unit cells.belongs to only two unit cells.

No. of atoms/unit cell =No. of atoms/unit cell = 88 ++ 66 = = 44 atoms/unit cell.atoms/unit cell.

Corner and face position are really equivalent because of the translation of theCorner and face position are really equivalent because of the translation of the

cube corner from an original corner atom to the center of a face atom will notcube corner from an original corner atom to the center of a face atom will notalter the cell structure.alter the cell structure.

The coordination number of each atom is the no. of nearestThe coordination number of each atom is the no. of nearest--neighbor or touchingneighbor or touchingatoms .The coordination no for FCC isatoms .The coordination no for FCC is 1212..

The atomic packing factor (APF) is the fraction of solid sphere Volume in a unitThe atomic packing factor (APF) is the fraction of solid sphere Volume in a unitcell. The APF of FCC will be :cell. The APF of FCC will be :

Which is the maximum packing possible for spheres all having the same diameter.Which is the maximum packing possible for spheres all having the same diameter.


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The BodyThe Body--Center Cubic CrystalCenter Cubic Crystal

Structure (BCC)Structure (BCC)The crystal structure has a cubic unit cellThe crystal structure has a cubic unit cell

with atoms located at all eight corners an awith atoms located at all eight corners an a

single atom at the cub center.single atom at the cub center.


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Characteristics Of (BCC) CrystalCharacteristics Of (BCC) Crystal

StructureStructure The cube edge length (a) and the atomic radius (R) areThe cube edge length (a) and the atomic radius (R) are

related throughrelated through

The corner atom is shared among eight unit cells and aThe corner atom is shared among eight unit cells and asingle center atom.single center atom.

No. of atoms/unit cell = 8No. of atoms/unit cell = 8 + 1 = 2 atoms/unit + 1 = 2 atoms/unit

cell.cell. The coordination no. for BCC is 8.The coordination no. for BCC is 8.

The APF for BCC is 0.68.The APF for BCC is 0.68.

Coordination no. for FCC (8) > Coordination no. for BCCCoordination no. for FCC (8) > Coordination no. for BCC(12).(12).

3

4Ra!

3

4Ra!


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The Hexagonal CloseThe Hexagonal Close--PackedPacked

Crystal Structure (HCP)Crystal Structure (HCP) The crystal structure has a hexagonal unit cell. The top andThe crystal structure has a hexagonal unit cell. The top and

bottom faces of the unit cell consist of six atoms that form regularbottom faces of the unit cell consist of six atoms that form regularhexagons and surround a single atom in the center.hexagons and surround a single atom in the center.

Another three atoms situated between the top and the bottomAnother three atoms situated between the top and the bottomplane , these atoms are adjacent two planes. From the fig. the c/aplane , these atoms are adjacent two planes. From the fig. the c/aratio should be (ratio should be (11..633633).).

Some of the familiar metals having this crystal structure areSome of the familiar metals having this crystal structure arecadmium , magnesium , titanium, zinc and Ca.cadmium , magnesium , titanium, zinc and Ca.


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Characteristics Of (HCP) Crystal StructureCharacteristics Of (HCP) Crystal Structure

XX The characteristics of HCP crystal structure is the same of theThe characteristics of HCP crystal structure is the same of the

FCC crystal structure.FCC crystal structure.

YY No. of atoms/unit cell =No. of atoms/unit cell = 1212 11//66 ++ 22 + + 33 == 66 atoms/unit cellatoms/unit cell

ZZ The coordination no for HCP isThe coordination no for HCP is 1212..

[[ The atomic packing factor (APF) isThe atomic packing factor (APF) is 00..7474 ..

Density ComputationsDensity ComputationsComputation of metallic solid theoretical densityComputation of metallic solid theoretical density


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Polymorphism and AllotropyPolymorphism and Allotropy

It is the phenomenon in which the solidIt is the phenomenon in which the solidmaterial have more than one crystal structure.material have more than one crystal structure.This material calledThis material called allotropyalallotropyal material . Thematerial . Theprevailing crystal structure depends on bothprevailing crystal structure depends on both

the temperature and external pressure.the temperature and external pressure. Ex :Ex :

CarbonCarbon:: Graphite is the stable polymorph atGraphite is the stable polymorph atambient conditions whereas diamond is formedambient conditions whereas diamond is formed

at extremely high pressures.at extremely high pressures.Pure ironPure iron:: BCC crystal structure at roomBCC crystal structure at room

temperature which changes to FCC iron attemperature which changes to FCC iron at 912912ooC (C (16741674o F).o F).


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Crystal SystemCrystal SystemMM CubicCubic

a = b = ca = b = c

EE == FF == KK == 9090oo

NNHexagonalHexagonal

a = ba = b {{ cc

EE == FF = 90 &= 90 & KK = 120= 120oo


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TetragonalTetragonal

a = ba = b {{ cc

EE == FF == KK = 90= 90

OrthorhombicOrthorhombic

aa {{ bb {{ cc

EE == FF == KK == 9090aa {{ bb {{

c

c

EE == FF == 9090 KK == FF

RhombohedralRhombohedral

MonoclinicMonoclinica = b = ca = b = c

EE == FF == KK {{ 9090

TriclinicTriclinica = b = ca = b = c

EE {{ FF {{ KK {{ 9090


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Crystallographic Points,Crystallographic Points,

Direction, PlanesDirection, Planes::Crystallographic PointsCrystallographic PointsThe position of any points located within a unit cell may beThe position of any points located within a unit cell may be

specified in terms ofspecified in terms of(q , r , s)(q , r , s) wherewhere qq is some fractionalis some fractional

length oflength ofaa along xalong x--axis and similarly foraxis and similarly for rr andand ss..


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Ex :Ex : Specify point coordinates for allSpecify point coordinates for allatom points for BCC unit cell.atom points for BCC unit cell.

Point No. Coordinates

1 0 0 0

2 1 0 0

3 1 1 0

4 0 1 0

5

6 0 0 1

7 1 0 1

8 1 1 1

9 0 1 1


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Ex:Ex: (a) locate the point having(a) locate the point having

coordination coordination 11 ..

Ex : Specify point coordinates for all atompoints for BCC unit cell

Point No. Coordinates

1 0 0 0

2 1 0 0

3 1 1 0

4 0 1 0

5

6 0 0 1

7 1 0 1

8 1 1 1

9 0 1 1


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Crystallographic DirectionsCrystallographic Directions

It is defined as a line between two points , or a vector [It is defined as a line between two points , or a vector [uu

RRww]] uu ,, RR,, ww are integers corresponding to the reducedare integers corresponding to the reducedprojections alongprojections alongxx,,yy ,,zz--axes.axes.


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]]111111] and [] and [110110] , [] , [100100The [The [Ex :Ex :directions are common ones, theydirections are common ones, they

are drawn in the unit cell.are drawn in the unit cell.

xx yy zz

ProjectionsProjections 1 01 0 22

ReductionReduction 1 2 01 2 0

EnclosureEnclosure [[120120]]


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Ex :Ex :Draw [Draw [11 --1 01 0] , [] , [1 2 31 2 3] and [] and [ --2 12 1 --33]]

directions within a cubic unit cell.directions within a cubic unit cell.


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Family Of Directions < > :Family Of Directions < > :


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Crystallographic PlanesCrystallographic PlanesThe miller indices of the plane (h k l) are h , k , lThe miller indices of the plane (h k l) are h , k , l

X

Y

Z


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Ex :Ex :Constrict (Constrict (00 --11 --11) , () , (22 --1111) and) and

((102102) Planes within a cubic unit cell) Planes within a cubic unit cell


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Family Of Planes { }:Family Of Planes { }:

Linear And Planer Densities

Directional or planer equivalency are related to linear

or planer density , for a particular material equivalent

directions have the same linear densities. Also the

planes which have the same planer densities are equivalent.


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Li ear e sity (LD)Li ear De sity (LD)

Is defi ed as the o. of atoms per u it le gthIs defi ed as the o. of atoms per u it le gth

Ex:

Ex:Determi e the li ear de sity of the [Determi e the li ear de sity of the [110110] directio for the] directio for theFCC crystal structure.FCC crystal structure.


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Planer Density (PD)Planer Density (PD)Is the no. of atoms per unit area that centered on a particularIs the no. of atoms per unit area that centered on a particular

crystallographic planecrystallographic plane

Ex :Ex : Find the PD for (Find the PD for (110110) plane within FCC unit cell.) plane within FCC unit cell.


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Only oneOnly one--quarter of each of atoms (A , C , Dquarter of each of atoms (A , C , D

and F) and oneand F) and one--half of atoms (B and E) for ahalf of atoms (B and E) for atotal equivalence of justtotal equivalence of just 22 atoms are one onatoms are one on

the plane.the plane.


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CrystallineCrystalline--Crystalline and NonCrystalline and Non

MaterialMaterialSolidSolid

Single CrystalSingle Crystal InIn whichwhich thethe periodicperiodic andand

repeatedrepeated arrangementarrangement ofof thethe

atomsatoms isis perfectperfect oror extendsextends

throughoutthroughout thethe entirelyentirely of ofthethe specimenspecimen withoutwithout

interruptioninterruption

PolycrystallinePolycrystallineIt is composed of a collection ofIt is composed of a collection of

many crystals or grainsmany crystals or grains


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AnisotropyAnisotropy

It is directionality of properties is anisotropy.It is directionality of properties is anisotropy. (Anisotropic)(Anisotropic)

The materials in which the measured properties areThe materials in which the measured properties are

independent of the direction areindependent of the direction are isotropicisotropic


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Ray Diffraction and Bragg'sRay Diffraction and Bragg's--XXLawLaw

XX--raysrays areare aa formform ofof electromagneticelectromagnetic radiationradiation thatthat

havehave highhigh energiesenergies andand shortshort wavelengthswavelengths.. WhenWhen aa

beambeam ofofXX--raysrays impingesimpinges onon aa solidsolid materialmaterial ,, aa portionportion

ofof thisthis beambeam willwill bebe scatteredscattered inin allall directionsdirections byby thethe

electronselectrons associatedassociated withwith eacheach atomatom oror ionion lieslies withinwithin

thethe beam'sbeam's pathpath..


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TheThe iffr t eteriffr t eter


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the structure of cry stall in solid

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