X-Rays The basic understanding of XRD analysis

When the energy of the accelerated electrons is higher than a certain threshold value (which depends on the metal anode), a second type of spectrum is obtained superimposed on top of the white radiation. It is called the characteristic radiation and is composed of discrete peaks.
The energy (and wavelength) of the peaks depends solely on the metal used for the target and is due to the ejection of an electron from one of the inner electron shells of the metal atom.
This results in an electron from a higher atomic level dropping to the vacant level with the emission of an X-ray photon characterised by the difference in energy between the two levels.

1. X-Rays
The basic understanding of XRD analysis
ChandraPrakash Singh
Electromagnetic Spectrum

2. LaboratoryX-raysourcescanbeclassifiedintotwotypes:sealed-tubeandrotatinganode.
BothmaybeusedtogeneratemonochromaticX-rayradiationandtheybasicallydifferonlyintheintensityoftheradiation
produced.
A Typical X-ray Spectrum from a Copper Target
White Radiation
X-raysaregeneratedwhenmatterisirradiatedbyabeam
ofhigh-energychargedparticlessuchaselectrons.
Inthelaboratory,afilamentisheatedtoproduceelectrons
whicharethenacceleratedinvacuumbyahighelectric
fieldintherange20-60kVtowardsametaltarget,which
beingpositiveiscalledtheanode.
Thecorrespondingelectriccurrentisintherange5-
100mA.
Theprocessisextremelyinefficientwith99%ofthe
energyofthebeambeingdissipatedasheatinthetarget.
Thelossofenergyoftheelectronsbycollisionwiththe
atomsusuallytakesplaceviamultipleevents.Theresultis
theproductionofacontinuousspectrumofX-raysknown
aswhiteradiation.
Generation of X-Rays

3. The maximum energy lost,E (max), determines the shortest wavelength, λ(min), that can be obtained according to the
equation
E=e V=h c/ λ
Where:
eis the charge on the electron,
Vis the accelerating voltage,
his Planck's constant, and
cis the speed of light.
A more practical form of this equation is given by
λ = 12.398 /V
Where:
Vis in kilovolts and
λ is in Angstroms (1Å=0.1nm).
Thus, the higher the accelerating voltage of the X-ray generator, the shorter the minimum wavelength that can be obtained.
The maximum in the intensity of the white radiation occurs at a wavelength that is roughly 1.5×λ(min).
Longer wavelengths are obtained by multiple-collision processes.
The total intensity,I(w) of the white radiation is approximately proportional to the filament current,i, the atomic number of
the anode target,Z, and the square of the accelerating voltage,V.

4. Whentheenergyoftheacceleratedelectronsishigher
thanacertainthresholdvalue(whichdependsonthe
metalanode),asecondtypeofspectrumisobtained
superimposedontopofthewhiteradiation.Itiscalled
thecharacteristicradiationandiscomposedofdiscrete
peaks.
Theenergy(andwavelength)ofthepeaksdepends
solelyonthemetalusedforthetargetandisduetothe
ejectionofanelectronfromoneoftheinnerelectron
shellsofthemetalatom.
Thisresultsinanelectronfromahigheratomiclevel
droppingtothevacantlevelwiththeemissionofanX-
rayphotoncharacterisedbythedifferenceinenergy
betweenthetwolevels.
Characteristic Radiation

5. The diagram show the electronic energy levels
for a copper atom

6. ThecharacteristiclinesinthistypeofspectrumarecalledK,L,M,...andtheycorrespondtotransitionstoorbitalswith
principalquantumnumbers1,2,3,...
Whenthetwoorbitalsinvolvedinthetransitionareadjacent(e.g.2→1),thelineiscalledα.
Whenthetwoorbitalsareseparatedbyanothershell(e.g.3→1),thelineiscalledβ.
Sincethetransitionforβisbiggerthanforα,i.e.ΔE
β
>ΔE
α
,thenλ
β
<λ
α
.
ThisisdemonstratedbythevaluesoftheKαandKαwavelengthsinthetablebelowfortwocommonanodematerials:
Anode Kα Kβ
Cu 1.54184 Å 1.39222 Å
Mo 0.71073 Å 0.63229 Å
InthecopperX-rayspectrum,only2characteristiclinesareseenatlow-energyresolution.
However,athigherresolutiontheKα
1
lineisreadilyseentobeadoublet,whichislabelledasKα
1
andKα
2
where
ΔEα
1
>ΔEα
2
.
Thesplittingofthe2porbitalsincopper,i.e.thesplittingoftheenergylevelsL
II
andL
III
,isverysmall(0.020keV)and
sothetwowavelengthsKα
1
(=1.54056Å)andKα
2
(=1.54439Å)areverysimilar.

7. Spectral Line Shape
Thepictureisactuallyasimplifiedversionofreality
sinceahigh-resolutionanalysisofthespectrallines
of,say,CuKαshowsthatboththeα
1
andα
2
peaks
aredistinctlyasymmetric.
Anexplanationoftheoriginofthisasymmetryis
importantinunderstandingtheso-
calledfundamentalparameterapproachtothe
profilefittingofpowderdiffractiondatapeaks.
Thede-excitationprocessinwhichanouter
2pelectronfillstheinner1selectronshellisfast
(≈10
-12
s),butnotfastenoughtostopdouble
ionizationevents.
Inparticular,theejectionoftheinitial1selectron
canbefollowedbythelossofoneofthe2sor2p
electronsfromtheenergylevelsL
I
,L
II
,orL
III
.
Theeffectoftheincreasedionizationontheatomis
tochangeslightlytheenergygapbetweentheKand
Llevelsresultinginslightlydifferentwavelengths
fortheemittedX-rayphoton.
Theresultingpeakasymmetryinthespectral
distributionoftheKαlinesofcopperisshown
inredinthediagram.
The dotted colouredlines represent individual spectral
contributions to the total.

8. Spectral Intensity
TheintensityoftheKα
1
peakisalmostexactlydoubletheintensityoftheKα
2
peak.
TheintensityofaKlineisgivenapproximatelybytheformula
I
K
=ci(V-V
K
)
n
Where
iistheelectronbeamcurrent,
cisaconstant,and
V
K
istheexcitationpotentialoftheKline(asgivenearlierbyV
K
=12.398[kV/Å]/λ).
Theexponentnisapproximately1.5,butdropstowards1.0whenV>2V
K
.
TheratioI
K
:I
white
isamaximumwhentheacceleratingvoltageVisapproximately4×theexcitationpotentialV
K
.
ForaCuKαanode,whereV
K
is8.0kV,runwithatypicaloperatingvoltageof40kV,theKαlineisapproximately
90×moreintensethanthewhiteradiationofasimilarwavelength.
Thusthewhiteradiationfromacopperanodeistooweaktobeofanypracticaluseforpowderdiffractioninthe
laboratory.
WhatabouttheintensityoftheKβradiation?
Againconsideringacopperanode,theintensityoftheKαlinesisapproximately5timesthatofKβ.Hence,all
instrumentalsetupsareoptimizedaroundtheKαradiation,andpreferablyaroundKα
1
whenhighresolution
monochromatorsareusedaspartoftheX-rayoptics.

9. Choice of X-ray Target
Thewavelength,λ,ofthecharacteristiclinegivingrisetoaparticulartransitionisgivenbyMoseley'sLaw:
1 / λ =c(Z-σ)
2
Where
candσareconstants,and
Zistheatomicnumberofthemetalusedfortheanode.
Fromthisequationitcanseenthatastheatomicnumberofthetargetincreases,thenthewavelengthofthecharacteristic
radiationdecreases.
Sincethetargethastobemetallic(sothatitconductselectrons)andhastohaveareasonablyhighmeltingpoint(40kV
at30mAgenerates1.2kWofheat),thislimitsthechoiceofanodematerialtochromium(Cr),iron(Fe),cobalt(Co),
copper(Cu),molybdenum(Mo),andafewotherlesscommonlyusedmaterialsforX-raypowderdiffraction.
ThetablebelowshowstheKαradiationforeachelement:AnodeCr Fe Co Cu Mo Ag
Kα (Å)2.29 1.94 1.79 1.54 0.71 0.56
Copper anodes are by far the most common since copper gives the shortest wavelength above 1Å.
The wavelengths provided bymolybdenum and silver are normally too short for most powder diffraction work in the
laboratory.
Short wavelengths both scatter weakly and contract the diffraction pattern towards low Bragg angles with consequent loss
ofdspacing accuracy and resolution.

10. Metalfoilfiltersareonewayofachievingthis.
ThephotographshowsthetypicalmetalsusedtofilterX-
raysproducedbyasealedX-raytube,i.e.Ni,Fe,Mn,V,
orZr.
FilterspreferentiallyreducetheintensityoftheKβlinein
theX-rayspectrumcomparedtoKαasexplainedbelow.
Notethatabsorptionfilterscannotbeusedtoremovethe
unwantedKα
2
componentfromKαradiation.
FiltersexploittheX-rayabsorptionedgeoftheparticular
element.
Atwavelengthslongerthantheabsorptionedge(i.e.just
abovetheedge),theabsorptionoftheX-raysis
considerablylessthanforwavelengthsshorterthanthe
absorptionedge(i.e.justbelowtheedge)asshownbelow
fornickelmetal:
X-Ray Filters
The spectrum from a sealed X-ray tube is composed of several X-ray lines.
Laboratory powder diffraction requires an X-ray source that is essentially monochromatic and so the Kβ line in the X-ray
spectrum needs to be removed.

11. Notethatthefilteralsoremovesmuchofthehighenergy
backgroundradiation.
Thechoiceoffiltermaterialdependsuponthechoiceofanode
materialintheX-raytubeasshowninthefollowingtable:
AnodeCuCoFeCrMo
FilterNiFeMnVZr
Fromthetableitcanbeseenthattheidealchoice
ofmaterialforanX-rayfilterisametalwhose
atomicnumber,Z,isonelessthanthatofthe
anodetargetmetalforfirstrowtransitionmetals
(ortwolessforsecondrowtransitionmetals).
Theabsorptionedgeofnickelmetalat1.488ÅliesbetweentheKα(λ=1.542Å)andKβ(λ=1.392Å)X-rayspectrallines
ofcopper.HencenickelfoilofanappropriatethicknesscanbeusedtoreducetheintensityoftheCuKβX-raysasshown:

12. Theoptimumthickness,xofthefiltercanbedeterminedfromthemass-absorptionlaw:
I(λ) /I
o
(λ) = exp{− (μ / ρ)
λ
ρx}
Where:
(μ/ρ)isthemassabsorptioncoefficientatthewavelengthλ,
ρisthedensityofthematerial,whichfornickelmetalis8.92g/cm
3
,
I(λ)andI
o
(λ)arethetransmittedandincidentX-rayintensities,respectively.
The mass absorption coefficients of nickel for Cu Kα and Cu Kβ are 49.2 and 286cm
2
/g, respectively.
The table below shows the percentage transmission for various thicknesses of nickel foil:
Thickness (cm)I/I
o
(%) for Cu KαI/I
o
(%) for Cu KβReduction Ratio
0.0010 64.5 7.8 8
0.0015 51.8 2.2 24
0.0020 41.6 0.6 68
0.0025 33.4 0.2 197
Itcanbeseenfromthetablethattheoptimumthicknesshastobeacompromisebetweenreducingtheintensityofthe
unwantedCuKβandreducingtheintensityofthedesiredCuKα.
Mostcommercialsystemsemployinganickelfilterwithacopperanodetargetwillchoosethethicknessofthefoilsoas
togiveareductionratiointherange25:1to50:1,i.e.foilsbetween15and20µmthick.
Fromthetable,itcanbeseenthatthisrangeoffoilthicknesswilldiminishthedesiredradiationbyapproximatelya
factorof2

13. Thanks
ChandraPrakash Singh