Optical X-ray & Neutron Techniques

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1、1.DIFFRACTIONANDIMAGING1.1.OnedimensionClassicalopticstreatsscatteringofawaveintermsofadistributionofHuygens’sourceswhichthescatteredradiationequallyinalldirection.Itistheinterferencebetweenthewavefrontsfromthesourceswhichgivesrisetodiffraction.Diffractionfrom1-Dobject•Fraunhofergeometr

2、y–L>>λ,L>>a•Huygens'sourceatx,strengthρ(x)δx.ρ(x)δxsScatteredbeam

3、s0

4、=

5、s

6、=1S=s-s0x.sx0x.ss02θIncidentbeamScatteringOvectorS2θθAmplitudediffractedscatteredfromelementatxisproportionaltoA0ρ(x)δxwhere:A0istheincidentamplitude,ρ(x)isthescatteringdensity.Pathdifferencebetweenraysscatteredat0

7、andxthroughanangle2θ=−=xsxs..x.SwhereSss=−;S=2sinθooPhasedifference=2π/λx.SIgnoringfactorsrelatedtothemechanismofinteractionbetweentheincidentradiationandthescatteringmaterial,thecontributiontodiffractedamplitudeatangle2θisgivenbyx.SδρδA=Axxo()exp2πiλTotalamplitudescattered∞x.SA

8、Ax()S=o∫ρπ()exp2idxλ−∞(1.1)ThescatteredamplitudeA()Sandthescatteringdensity,ρ()xarerelatedbyaFourierTransform.mgb110/04ExamplesofapplyingFourierTransforms:1.SingleslitxsθSxso2θ2θxS.s=xin2θρρ()xa=−≤≤122,;xax()=0otherwwisea222πθixsinAA()S=o∫expdxλ−a2a222πθixsinexxpλ=A

9、o22πθisinλ−a2πθasin2sinλsinφAA()S=o=A′πθsin2φwhereA'=aA0λsin2θ/λperiodicinaStandardFraunhoferdiffractionpatternEnvelopedecreaseswithangle2.Pairofslits
spacingbbbρδ()xx=−++δx22b2bbi22πθxsinAA()S=−o∫δx++δxexpdx22λ−b2π

10、θbsin2AA()S=2ocosλ"Young'sSlits"EnvelopeconstantSimplestdiffractiongrating"Scale"ofdiffractingobject~1/"Scale"ofdiffractionpatternmgb210/04ReverseTransformationFouriertransformscanbereversed.imageofdiffractingobjectisrecoveredbyFouriertransformingthediffractionpattern:∞x.Sρπ()xA

11、A=o∫()Sexp−2idSλ−∞LimitstoS(Smax&Smin)sotheimagecannotbeidenticaltothediffractingobject:Smaxx.Sρπ()xAA=o∫()Sexp−2idS(1.2)λSminThisisrepresentedveryclearlyintheAbbetheoryoftheresolutionoftheopticalmicroscope.ThephaseproblemByrecordingmagnitude(intensity)(asinX-rayandne