The relationship between the anomalous transmission and the negative extintion in the symmetrical Bragg case near the absorption edge is studied in detail. The negative extinction results from the anomalous transmission and takes the largest value when the diffraction is induced only by the imaginary part of the atomic scattering factor near the absorption edge. The appearance of the negative extinction also depends on temperature, which can be used for the identification of the theoretical model for calculating the Debye-Waller factor.Podrobno se razmatra odnos anomalne transmisije i negativnog gašenja za simetričan Braggov slucaj. Negativno gašenje je posljedica anomalne transmisije i najveće je kada je difrakcija uzrokovana samim imaginarnim dijelom atomskog faktora raspršenja u blizini apsorpcijskog ruba. Pojava negativnog gašenja ovisi također o temperaturi, što se može primijeniti za prepoznavanje teorijskog modela pri izračunavanju Debye-Wallerovog faktora

Changlin, Guo

Zhangcheng, Xu

Zongyan, Zhao

English

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ISSN1330–0008CODEN FIZAE4THE RELATIONSHIPBETWEENTHE ANOMALOUS TRANSMISSIONAND THENEGATIVE EXTINCTION IN THE BRAGGCASEXU ZHANGCHENGa   b, GUOCHANGLINb andZHAO ZONGYANca CCAST(World Laboratory),P.O.Box.8730,Beijing100080,People’s Republicof Chinab ShanghaiInstituteof Ceramics,ChineseAcademyof Sciences,Shanghai200050,People’s Republicof China(mailingaddress)c Departmentof Physics,AnhuiUniversity, Heifei 230039,People’s Republicof ChinaReceived9 June1997UDC 538.951PACSnumber:61.10.DpThe relationshipbetweenthe anomaloustransmissionandthe negative extinction in thesymmetricalBraggcaseneartheabsorptionedgeis studiedin detail.Thenegativeextinc-tion resultsfrom theanomaloustransmissionandtakesthelargestvaluewhenthediffrac-tion is inducedonly by theimaginarypartof theatomicscatteringfactorneartheabsorp-tion edge.Theappearanceof thenegative extinction alsodependson temperature,whichcanbeusedfor theidentificationof thetheoreticalmodelfor calculatingtheDebye-Wallerfactor.1. IntroductionX-ray diffraction theoriesarecalleddynamicalor kinetic, accordingto whetherthemulti-scatteringeffect is taken into accountor not [1-5]. The observation of BorrmanneffectandPendell̈osungfringesprovesthevalidity of thedynamicaltheory.With X-rays from a synchrotronradiationsource,it is possibleto makeFhrsmallerthanFhi, evenFhr  0, nearanabsorptionedgeof a constituteatomof a perfectcrystalFIZIKA A 6 (1997)3, 103–110 103XU ZHANGCHENG ET AL .: THE RELATIONSHIP BETWEEN THE ANOMALOUS . . .[6-7]. Here,Fhr andFhi arethecontributionsof therealandimaginaryparts,respectively,of theatomicscatteringfactorto thecrystalstructurefactor. In 1992,Kato presentedthegeneralformulaefor calculatingthereflectivity for a semi-infinitecrystal,which is validevenwhenFhr  0 [8]. Thecorrespondingformulaefor afinite parallel-planecrystalwerederivedby Xu Zhangchengetal. in 1995[9]. Morerecently, thepresentauthorsfoundthatthe famousKato-Ewald theoremis not valid when  Fhi  is comparative to Fhr  neartheabsorptionedge[10,11].It is well known, the extinction effect in the Braggcaseundernormalexperimentalconditionsis oneof themostimportantdynamicaleffects.This effect manifestsitself notonly in the integratedintensity, but also in the secondaryfluorescenceemission[3,12].However, theanomaloustransmissioncantakeplaceat theexactBragganglein theBraggcasewhen  Fhr  is smallerthan  Fhi  neartheabsorptionedge,aspredictedin Ref. 9. Thepurposeof thepresentwork is to show how suchaneffectaffectsthereflectivity, especiallytheintegratedintensityfor theperfectcrystal.2. TheoreticalbasisTheparametersusedin thepresentwork arephysicallythesamewith thosein Ref.1,avoiding theproblemof infinity whenFhr  0 [8,9].2.1. Thereflectionandtransmissioncoefficientsfor a finiteperfectcrystal[9]Thereflectivity in theBraggcasefor a finite perfectcrystalis givenbyPhP0 1  2psinδ  cosh 2AI1 	 cos 2AR1 P1   1whereP1  y2  g2  R21  I21  cosh 2AI1 	 2  yR1  gI1  sinh 2AI1    2 y2  g2  R21  I21  cos 2AR1   2  yI1  gR1 sin 2AR1  R1   L1  B  2  3I1    L1  B  2  4L1   B2  C2   5B  y2  g2  1  a2   6C  2  gy  pcosδ    7104 FIZIKA A 6 (1997)3, 103–110XU ZHANGCHENG ET AL .: THE RELATIONSHIP BETWEEN THE ANOMALOUS . . .g  1  b   χ0i2 b K  χh    8y   b   θ  θB  sin2θB  12  1  b   χ0r b K  χh    9b  sin ΘB  α   sin Θ  α    10A  πK  χh sin Θb  α  sin ΘB  α    11p   χhr χhi  χh  2   12a   2 χhi χh  2   13 χh  2   χhr  2  χhi  2   14χhr  χhr  eiαhr  e2Vπmν2 Fhr   15χhi  χhi  eiαhi  e2Vπmν2 Fhi   16δ  αhi  αhr   17χh  χhr  i χhi   18In (15)and(16),V is thevolumeof acrystalcell, ν theX-ray frequency, andmandethemassandthechargeof anelectron.In (11),ΘB is theBraggangle,α theanglebetweenthediffractionplaneandthecrystalsurface,andK thewavenumberin vacuum.In (3) thenegativesignis takenonly whenB  0 andC  0, but in (4) only whenB ! 0 andC  0.Thetransmissioncoefficient isPdP0 L1 exp" 2Ag 1  b   1  b #P1  19FIZIKA A 6 (1997)3, 103–110 105XU ZHANGCHENG ET AL .: THE RELATIONSHIP BETWEEN THE ANOMALOUS . . .2.2. IntegratedintensityTheintegratedintensityfor a finite perfectcrystalis givenbyRdyn $% & b & & χh &sin2θ Rydyn ' A0 ( g0 ( k) ( b* $% & b & & χh &sin2θ∞+, ∞PhP0dy- ' 20*Theintegratedintensityfor anideally imperfectcrystalis givenbyRkin $% & b & & χh &sin2θ Rykin ' A0 ( g0 ( k) ( b* ( ' 21*whereRykin ' A0 ( g0 ( k) ( b* $ ' 1 . e4Ag* π4 ' 1 . 2psinδ *& g & - ' 22*Fig. 1. Therockingcurvesof the transmittedbeamof a finite parallel-planecrystal.Theparametersare: A $ π ( δ $ 0, (1) χhi $ 0 ( g $ . 0 - 1 ( (2) & χhr & $ 10 & χhi & ( g $ . 1 - 0, (3)& χhr & $ 6 & χhi & ( g $ . 1 - 0, (4) & χhr & $ 2 & χhi & ( g $ . 1 - 0, (5) & χhr & $ & χhi & ( g $ . 1 - 0 and(6)χhr $ 0 ( g $ . 1 - 0.106 FIZIKA A 6 (1997)3, 103–110XU ZHANGCHENG ET AL .: THE RELATIONSHIP BETWEEN THE ANOMALOUS . . .WhenA / ∞ 0 if g 12 03 4 e4Ag / 0 0 233k5 26 χhr 66 χhi 6 4 0 243p 2 k5 1 0 1 7 k5 23 4 0 253a 28 21 0 1 7 k5 23 9 0 2633. Therockingcurvesof thetransmittedbeamsAccordingto Eq. (19), therockingcurvesof thetransmittedbeamof a finite parallel-planecrystalin thesymmetricalBraggcasearecalculatedfor all kindsof absorptioncases.Whenχhi 2 0 (Fig. 1, curve 1), thecurve shows a symmetricaldip with respectto y 2 0;whenχhrχhi 12 0 (curves2 to 5), thecurve is not symmetricalto y 2 0, which is dueto theasymmetryof theabsorptioncoefficient.Theanomaloustransmissionin theregion y : 0andtheanomalousabsorptionin the region y ; 0 areclearlyseen.When 6 χhr 6 ;; 6 χhi 6 ,theanomaloustransmissionisFig. 2. The dependenceof Rdyn 1 Rkin on the ratio of 6 χhr 6 and 6 χhi 6 . The parametersare:g0 2< 1, A50 2 2A1 0 A 7 π 3 , δ 2 0, (1) 6 χhr 6 2 10 6 χhi 6 , (2) 6 χhr 6 2 8 6 χhi 6 , (3) 6 χhr 6 2 6 6 χhi 6 ,(4) 6 χhr 6 2 4 6 χhi 6 , (5) 6 χhr 6 2 6 χhi 6 , (6) 6 χhr 6 2 0.FIZIKA A 6 (1997)3, 103–110 107XU ZHANGCHENG ET AL .: THE RELATIONSHIP BETWEEN THE ANOMALOUS . . .verysmall.Notethattheanomaloustransmissionoccursin theso-called”total reflection”region,when = χhr = >= χhi = . Whenχhr ? 0 (curve6), therockingcurveshowsasymmetricalpeakwith respecto y ? 0, whichmeanstheanomaloustransmissionis thelargestatexactBraggangle.4. PrimaryextinctionandnegativeprimaryextinctionDarwinfirst predictedtheprimaryextinctionphenomena,whenheconstructedhisdy-namicaldiffractiontheory. Thatis to say, whenreflectionoccurs,thetransmissiondepthofX-raysis muchshallower thanwith normalabsorption.Only several thousandsof atomiclayersnearthecrystalsurfacecontributeto thediffractionandthecontributionof thelowerpartis nearlyzero.However, theanomaloustransmissionoccursin theso-called”total re-flection” region,whichmeansmoreatomiclayerscontributeto thediffractionin theperfectcrystalthanin thecorrespondingideally imperfectcrystal.This will resultin theappear-anceof negative primary extinction - the integratedintensity for a perfectcrystalbeinglargerthanthatfor a correspondingideally imperfectcrystal.As shown in Fig. 2, Rdyn @ Rkin is calculatedfor typical absorptioncasesby directinte-grationof the reflectedrockingcurves.Theparameterg0 ? χ0i @ = χhi = is setto – 1, whichmeanstheeffectof temperatureis not takeninto account.It canFig. 3a.Thedependenceof Rdyn @ Rkin on temperature.Theparametersare:AA0 ? 2A@ B A Cπ D , δ ? 0, = χhr = ? 10 = χhi = , (1) g0 ?FE 1 G 0, (2) g0 ?FE 1 G 05, (3) g0 ?E 1 G 1, (4) g0 ?E 1 G 2,and(5) g0 ?E 1 G 5.108 FIZIKA A 6 (1997)3, 103–110XU ZHANGCHENG ET AL .: THE RELATIONSHIP BETWEEN THE ANOMALOUS . . .be seenthat RdynH Rkin increaseswith the decreasingof kI . When J χhr J KLJ χhi J (curves1to 4), RdynH Rkin M 1, which is calledthe primary extinction. When J χhr J M J χhi J , (curves5,6), RdynH Rkin N 1, which is calledthe negative primaryextinction. Whendiffraction isinducedonly by the imaginarypart the atomicscatteringfactor (χhr O 0), the negativeprimaryextinction takesthe largestvalue.As shown in Fig. 3, the effect of temperatureon RdynH Rkin is studied.When J χhr J NPN J χhi J , (Fig. 3a), RdynH Rkin increasesand tendsto 1 whenthe temperatureincreases(g0 increasing).However, whenχhr O 0 (Fig. 3b),Rdyn H Rkin decreasesandtendsto 1 whenthe temperatureincreases.If the temperatureishigh enough,the negative primary extinction will disappear, which may be usedfor theidentificationof thetheoreticalmodelfor calculatingtheDebye-Waller factor.Fig. 3b. Thedependenceof RdynH Rkin on temperature.TheparametersarethesameasofFig. 3a,exceptfor χhr O 0.AcknowledgementsThefirst authorthanksProf.T. Fukamachin SaiamaInstituteof Technologyof Japanfor hissincerehelp.References1) W. H. Zachariasen,Theoryof X-raydiffractionin Crystals, Wiley, New York, 1945;2) R. W. James,SolidStatePhysics, Academic,New York, 1963;3) B. W. BattermanandH. Cole,Rev. Mod. Phys.36 (1964)681;FIZIKA A 6 (1997)3, 103–110 109XU ZHANGCHENG ET AL .: THE RELATIONSHIP BETWEEN THE ANOMALOUS . . .4) N. Kato,X-rayDiffraction, McGraw-Hill, New York, 1974;5) Z. G. Pinsker, DynamicalScatteringof X-raysin Crystals, Springer, Berlin, 1978;6) Z. Y. Zhao,J.H. Han,T. Fukamchi,M. Yoshizawa, T. Ehara,T. KawamuraandT. Nakajima,J.Phys.:Condens.Matter6 (1994)1619;7) T. Fukamachi,R.Negishi,M. Yoshizawa,T. Ehara,T. Kawamura,T. NakajimaandZ. Zhao,ActaCryst.A 49 (1993)573;8) N. Kato,ActaCryst.48 (1992)829;9) Z. C.Xu, Z. Y. Zhao,C.L. Guo,S.M. Zhou,T. Fukamachi,R.NegishiandT. Nakajima,J.Phys.:Condens.Matter7 (1995)8089;10) Z. C. Xu, Z. Y. ZhaoandC. L. Guo,J.Phys.:Condens.Matter8 (1996)5977;11) Z. C. Xu, C. L. GuoandZ. Y. Zhao,J.Phys.:Condens.Matter9 (1997)L275;12) B. W. Batterman,Phys.Rev. 133 (1964)A759.ODNOSANOMALNE TRANSMISIJEI NEGATIVNOG GAŠENJA U BRAGGOVOMSLUČAJUPodrobnose razmatraodnosanomalnetransmisijei negativnog gǎsenjaza simetrǐcanBraggov slučaj.Negativnogǎsenjeje posljedicaanomalnetransmisijei najvećeje kadajedifrakcija uzrokovanasamimimaginarnimdijelom atomskog faktoraraspřsenjau bliziniapsorpcijskog ruba. Pojava negativnog gǎsenjaovisi takodero temperaturi,̌sto semožeprimijeniti zaprepoznavanjeteorijskog modelapri izračunavanjuDebye-Wallerovog fak-tora.110 FIZIKA A 6 (1997)3, 103–110

Odnos anomalne transmisije i negativnog gašenja u Braggovom slučaju

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