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What is What is Bragg’sBragg’s Law and why is it Law and why is it

Important?Important?

•• Bragg’sBragg’s lawlaw refersrefers toto aa simplesimple equationequation

derivedderived byby EnglishEnglish physicistsphysicists SirSir WW.. HH..

BraggBragg andand hishis sonson SirSir WW.. LL.. BraggBragg inin

19131913..

•• ThisThis equationequation explainsexplains whywhy thethe facesfaces ofof

crystalscrystals appearappear toto reflectreflect (diffract)(diffract) XX--rayray

beamsbeams atat certaincertain anglesangles ofof incidenceincidence θθ..

•• ThisThis observationobservation isis anan exampleexample ofof XX--rayray

wavewave interference,interference, knownknown asas XX--rayray

diffractiondiffraction (XRD)(XRD)

•• Bragg’sBragg’s lawlaw cancan easilyeasily bebe derivedderived byby consideringconsidering thethe conditionsconditions

necessarynecessary toto makemake thethe phasesphases ofof thethe beamsbeams coincidecoincide whenwhen thethe

incidentincident angleangle == reflectingreflecting angleangle (Figure(Figure 11))

•• TheThe secondsecond incidentincident beambeam bb continuescontinues toto thethe nextnext layerlayer wherewhere itit isis

scatteredscattered byby atomatom CC

•• TheThe secondsecond beambeam mustmust traveltravel thethe extraextra distancedistance BCBC ++ CDCD ifif thethe twotwo

beamsbeams aa && bb areare toto continuecontinue travellingtravelling adjacentadjacent andand parallelparallel

•• ThisThis extraextra distancedistance mustmust bebe anan integralintegral (n)(n) multiplemultiple ofof thethe wavelengthwavelength

forfor thethe phasesphases ofof thethe twotwo beamsbeams toto bebe thethe samesame

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Figure 1: Derivation of Bragg’s Law

DDBB

CC

AA

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�� ,, thethe distancedistance

�� SinceSince BCBC == CD,CD, wewe havehave::

�� ThenThen afterafter substitutionsubstitution givesgives

CDBCn +=λ θsindBC =

BCn 2=λ

θλ sin2dn =

Bragg’s Law

d d : lattice interplanar spacing of the crystal: lattice interplanar spacing of the crystal

θθ : x: x--ray incidence angle (Bragg angle)ray incidence angle (Bragg angle)

λλ : wavelength of the characteristic x: wavelength of the characteristic x--raysrays

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•• TheThe processprocess ofof diffractiondiffraction isis describeddescribed inin termsterms ofof incidentincident

andand diffracteddiffracted (reflected)(reflected) rays,rays, eacheach makingmaking anan angleangle θθ withwith

aa fixedfixed crystalcrystal planeplane

•• ReflectionReflection occursoccurs fromfrom planesplanes setset atat anan angleangle θθ withwith respectrespect

toto thethe incidentincident beambeam andand generatesgenerates aa reflectedreflected beambeam atat anan

angleangle 22θθ fromfrom thethe incidentincident beambeam

•• TheThe possiblepossible ““d”d” spacingspacing defineddefined byby thethe millermiller indices,indices, h,h, k,k, ll

areare determineddetermined byby thethe shapeshape ofof thethe unitunit cellcell

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�� Rewriting Bragg’s law”Rewriting Bragg’s law”

�� Thus,Thus, thethe possiblepossible 22θθ valuesvalues wherewhere wewe cancan havehave reflectionsreflections areare

determineddetermined byby thethe unitunit cellcell dimensionsdimensions

�� TheThe intensitiesintensities ofof thethe reflectionsreflections dependdepend onon whatwhat kindkind ofof atomsatoms andand

theirtheir locationlocation inin thethe unitunit cellcell

d2sin λθ =

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•• DiffractionDiffraction onlyonly occursoccurs whenwhen thethe BraggBragg conditioncondition isis satisfiedsatisfied..

•• InIn orderorder toto bebe suresure ofof satisfyingsatisfying Bragg’Bragg’ law,law, eithereither λλ oror θθ mustmust bebe

continuouslycontinuously variedvaried duringduring thethe experimentexperiment..

•• TheThe waysways inin whichwhich thesethese parametersparameters areare variedvaried distinguishdistinguish thethe twotwo

mainmain diffractiondiffraction methodsmethods::

� Laue Method : λλ isis variedvaried andand θθ isis fixedfixed

� Powder Method : λλ isis fixedfixed andand θθ isis variedvaried

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LAUE METHODLAUE METHOD

•• InIn thisthis method,method, continuouscontinuous radiationradiation isis usedused

•• ThisThis radiationradiation fallsfalls onon aa stationarystationary crystalcrystal.. TheThe crystalcrystal diffractsdiffracts thethe XX--

rayray beambeam andand producesproduces aa patternpattern ofof spotsspots whichwhich conformconform exactlyexactly withwith

thethe internalinternal symmetrysymmetry ofof thethe crystalcrystal

•• TheThe LaueLaue methodmethod cancan bebe usedused inin twotwo waysways::

�� TransmissionTransmission methodmethod (Figure(Figure 22a)a)

�� BackBack--reflectionreflection methodmethod (Figure(Figure 22b)b)

•• DependingDepending onon thethe relativerelative positionposition ofof thethe XX--rayray source,source, thethe crystalcrystal andand

thethe photographicphotographic filmfilm (to(to detectdetect thethe diffracteddiffracted XX--rays)rays)

LaueLaue

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(a) Transmission method (b) Back-reflection method

Figure 2: Laue methods

Crystal

X-ray beam

Photographic film

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•• InIn thethe LaueLaue method,method, thethe crystalcrystal isis fixedfixed inin aa positionposition relativerelative toto thethe XX--

rayray beam,beam,

•• Thus,Thus, notnot onlyonly isis thethe valuevalue forfor dd fixed,fixed, butbut thethe valuevalue ofof θθ isis alsoalso fixedfixed

•• TheThe diffracteddiffracted beambeam isis producedproduced byby diffractiondiffraction fromfrom thethe planesplanes whichwhich

belongbelong toto aa particularparticular zonezone axisaxis (ZA)(ZA) ofof thethe crystalcrystal

•• TheThe beambeam inin eacheach setset allall lielie onon thethe surfacesurface ofof anan imaginaryimaginary conecone:: thethe

axisaxis ofof thisthis conecone isis thethe zonezone axisaxis (Figure(Figure 33))

•• WhenWhen thisthis beambeam intersectsintersects withwith thethe planeplane ofof thethe photographicphotographic filmfilm itit

producesproduces spotsspots (Figure(Figure 44))

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Figure 3: Location of Laue spots (a) onellipses in transmission method and(b) on hyperbolas in back-reflection

method. (C: crystal, F: film, Z.A: zoneaxis)

a

b

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Figure 4: Laue diffraction patterns (a)

Transmission method and (b) Back-

reflection method.

(a)(b)

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•• ForFor transmissiontransmission patternspatterns thethe curvescurves areare generallygenerally ellipsesellipses oror

hyperbolashyperbolas (Figure(Figure 33a)a).. ForFor backback reflectionreflection patternspatterns theythey areare usuallyusually

hyperbolashyperbolas (Figure(Figure 33b)b)

•• TheThe spotsspots whichwhich lielie onon anyany oneone curvecurve areare reflectionsreflections fromfrom planesplanes whichwhich

belongbelong toto oneone zonezone

•• EachEach diffracteddiffracted beambeam inin thethe LaueLaue methodmethod hashas aa differentdifferent wavelengthwavelength

•• TheThe LaueLaue methodmethod isis usedused mainlymainly forfor thethe determinationdetermination ofof crystalcrystal

orientationorientation andand assessmentassessment ofof crystalcrystal qualityquality becausebecause thethe positionspositions ofof

thethe spotsspots onon thethe filmfilm dependdepend onon thethe orientationorientation ofof thethe crystalcrystal withwith

respectrespect toto thethe incidentincident beambeam..

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POWDER METHODPOWDER METHOD

•• InIn thisthis method,method, characteristiccharacteristic XX--rayray radiationradiation ofof fixedfixed wavelengthwavelength(monochromatic)(monochromatic) isis usedused

•• TheThe materialmaterial toto bebe studiedstudied isis inin thethe formform ofof aa veryvery finefine powder,powder, eacheachparticleparticle ofof thethe powderpowder isis aa veryvery smallsmall crystalcrystal

•• InIn thisthis method,method, wewe taketake aa monochromaticmonochromatic XX--radiationradiation ofof oneone fixedfixedwavelengthwavelength andand placeplace thethe crystalcrystal (powder(powder materialmaterial toto bebe studied)studied) ininfrontfront ofof thethe beambeam (Figure(Figure 55a)a)..

•• IfIf oneone planeplane isis setset atat exactlyexactly thethe correctcorrect valuevalue ofof θθ forfor diffraction,diffraction, thenthenwewe observeobserve oneone andand onlyonly oneone reflectedreflected (diffracted)(diffracted) beambeam fromfrom thatthatcrystalcrystal..

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22θθθθθθθθ

θθθθθθθθ

Zone Axis

Diffracted beamDiffracted beam

Incident beamIncident beam

XX--ray beamray beam

Figure 5: Formation of a diffracted cone of radiation in the powder method

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�� ImagineImagine now,now, stillstill holdingholding thethe crystalcrystal fixedfixed atat thethe angleangle θθ,, wewe rotaterotate

thethe crystalcrystal aroundaround thethe directiondirection ofof thethe incidentincident XX--rayray beambeam soso thatthat thethe

planeplane causingcausing aa reflectionreflection isis stillstill setset atat thethe angleangle θθ relativerelative toto thethe XX--

rayray beambeam..

�� TheThe reflectedreflected beambeam willwill describedescribe aa conecone asas shownshown inin FigureFigure 55bb.. TheThe

axisaxis ofof thisthis conecone coincidescoincides withwith thethe axisaxis ofof thethe incidentincident beambeam

�� InIn thethe powderpowder materialmaterial thethe crystalscrystals areare notnot rotated. However,However, therethere

areare soso manymany randomlyrandomly orientedoriented crystalscrystals thatthat therethere willwill bebe somesome withwith

(hkl)(hkl) planesplanes whichwhich makemake thethe rightright BraggBragg angleangle withwith thethe incidentincident beambeam

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•• WeWe willwill havehave manymany reflectedreflected beamsbeams eacheach givinggiving oneone observableobservable pointpoint..

•• ImagineImagine whenwhen thesethese manymany crystalscrystals areare rotatedrotated aboutabout thethe axisaxis ofof thethe

incidentincident XX--rayray beam,beam, wewe willwill havehave manymany conescones tracedtraced outout byby thesethese

reflectedreflected beamsbeams..

•• SinceSince therethere areare millions of crystals in the powder material, therethere willwill bebe

manymany crystalscrystals inin thatthat powderpowder whichwhich willwill bebe inin aa positionposition toto diffractdiffract thethe

incidentincident beambeam andand therethere willwill bebe enoughenough ofof themthem toto getget thethe effecteffect ofof aa

continuouscontinuous pointpoint reflectionsreflections whichwhich willwill bebe lyinglying alongalong thethe arcarc ofof thethe conecone

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•• AA separateseparate conecone isis formedformed forfor eacheach setset ofof differentlydifferently

spacedspaced latticelattice planesplanes..

•• ThisThis isis thethe basisbasis ofof thethe powderpowder oror DebyeDebye--SherrerSherrer

methodmethod whichwhich isis thethe mostmost commoncommon techniquetechnique usedused

inin XX--rayray crystallographycrystallography

•• TheThe incidentincident beambeam isis inin thethe planeplane ofof thethe circlecircle

•• TheThe conescones ofof diffracteddiffracted radiationradiation interactinteract withwith thethe filmfilm

inin lineslines whichwhich areare generallygenerally curvedcurved exceptexcept atat 22θθ ==

9090oo inin whichwhich casecase theythey areare straightstraight

DebyeDebye

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•• FigureFigure 66aa showsshows schematicallyschematically threethree conescones andand FigureFigure 66bb showsshows whatwhat

thethe filmfilm lookslooks likelike whenwhen itit isis unrolledunrolled andand laidlaid outout flatflat..

•• FromFrom thethe measuredmeasured positionposition ofof aa givengiven diffractiondiffraction lineline onon thethe film,film, θθ cancan

bebe calculatedcalculated;; sincesince λλ isis fixedfixed andand known,known, thethe interplanarinterplanar spacingspacing dd ofof

thethe reflectingreflecting planesplanes whichwhich producedproduced thethe lineline cancan bebe calculatedcalculated..

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Figure 6: Debye-Sherrer powder method: (a) relation of film to specimen and incident beam, (b) appearance of film when laid out flat

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What is a powder Camera-(Debye Sherrer Camera)?

•• AA powderpowder cameracamera (Figure(Figure 77)) consistsconsists ofof aa metalmetal cylindercylinder atat thethe centrecentre

ofof whichwhich isis thethe samplesample..

•• TheThe powderedpowdered materialmaterial (which(which hashas aa diameterdiameter ofof aboutabout 00..33 mm)mm) isis

oftenoften gluedglued ontoonto aa glassglass rod,orrod,or placedplaced intointo aa thinthin glassglass tubetube..

•• TheThe samplesample mustmust bebe placedplaced accuratelyaccurately onon thethe axisaxis ofof thethe cylindercylinder andand

mustmust bebe rotatedrotated aboutabout itsits axisaxis soso thatthat thethe randomnessrandomness ofof thethe particlesparticles

ofof powderpowder shallshall bebe asas greatgreat asas possiblepossible..

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Figure 7: Schematic of Debye-

Sherrer camera, with cover

plate removed

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•• AA stripstrip ofof XX--rayray filmfilm isis placedplaced accuratelyaccurately insideinside thethe cylindercylinder (on(on itsits

perimeter)perimeter)..

•• PunchedPunched intointo oneone sideside ofof thethe filmfilm isis aa holehole forfor thethe beambeam collimatorcollimator andand

punchedpunched

•• TheThe appearanceappearance ofof thethe diffractiondiffraction patternpattern onon thethe filmfilm stripstrip afterafter

developmentdevelopment dependsdepends onon thethe wayway thethe filmfilm isis placedplaced inin thethe cameracamera..

•• ThereThere areare threethree mountingmounting inin commoncommon use,use, whichwhich differdiffer inin thethe positionposition

ofof thethe freefree endsends ofof thethe filmfilm relativerelative toto thethe incidentincident beambeam (Figure(Figure 88))

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Figure 8

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Determination of Accurate Lattice Parameters From Powder Photographs

•• TheThe patternpattern ofof lineslines onon aa photographphotograph representsrepresents possiblepossible valuesvalues ofof thethe

BraggBragg anglesangles θθ whichwhich satisfysatisfy thethe BraggBragg lawlaw forfor diffractiondiffraction..

•• WeWe needneed toto derivederive thethe valuesvalues ofof θθ fromfrom thethe powderpowder photographphotograph..

•• TheThe latticelattice parameter,parameter, ““a”a” cancan bebe calculatedcalculated usingusing anan appropriateappropriate

formulaformula (which(which dependsdepends onon thethe typetype ofof unitunit cell)cell)

•• TheThe wayway inin whichwhich θθ isis measuredmeasured dependsdepends onon thethe methodmethod ofof filmfilm

mountingmounting asas illustratedillustrated inin FigureFigure 88

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•• InIn thethe methodmethod shownshown inin FigureFigure 88c,c, wewe findfind thethe BraggBragg angleangle θθ forfor anyany

pairpair ofof lineslines byby measuringmeasuring thethe distancedistance SS betweenbetween thethe centrecentre ofof thethe exitexit

holehole andand thethe diffractiondiffraction lineline

•• TheThe angleangle betweenbetween aa pairpair ofof lineslines (two(two lines)lines) originatingoriginating fromfrom thethe samesame

conecone isis == 44θθ.. ThusThus::

•• RR isis thethe cameracamera radiusradius..

•• TypicalTypical camerascameras havehave RR == 2828..6565 mmmm ((55..7373 cm)cm);; thisthis givesgives 22ππRR == 180180..

•• TheThe measuredmeasured valuevalue ofof SS isis thereforetherefore givengiven byby::

360

4

2

θ

π=

R

S

θ2=S

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