Dynamical Theory of Electron Diffraction. Dr. Hongzhou Zhang SNIAM

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1 Dynacal Teoy of lecton Dffacton D. Honou Zan SNIAM

2 Lectue 4 Indexn Dffacton Patten To etod l, caea contant, ato of te dtance ZOLZ opute Poae ttp://ce.epfl.c/people/tadelann/es/ esv3_68.t

3 Outlne Fenel Dffacton olun Appoxaton Lt of te neatcal Teoy Te Dynacal Teoy Dan-Hoe-Welan quaton envalue Dpeve uface

4 Huyen Pncple vey Pont on a popaatn avefont eve a te ouce of pecal econday avelet, uc tat te avefont at oe late te te envelop of tee avelet Specal Wavelet: d P A ds / pae ft elatve to te ncdent ave Te avefont at oe late te: d S S ds A A co nty n te popaatn decton Zeo n te evee decton ds

5 Fenel Dffacton Te adu of t FZ: P d S S ds A co d n d P A Q ax A d ds d A Q Te apltude-pae daa: addn d t ncean pae: Fo cle : Te ft Fenel-one: - = / adu of te ccle deceae: A Te cle convee to te cente: Half of te ft Fenel-Zone: ax / A d d P A Q

6 Te Wavefont ut belo te Specen Te ave functon at te ncdent plane: Total ave functon at te ext plane: To-bea P Queton: Te contbuton of te dffacted ave at a catten anle B to te total ave font at P

7 olun Appoxaton Incdent ave: Plane ave, Fenel Dffacton: Nea feld - Te adu of te t FZ: = n, =e Specen A =.6 n - A colun t a daete of - n contbutn to te apltude at P B Te eoluton of dffacton contat >.5 n A nao colun paallel to te dffacted ave P =t

8 Te ntenty of te Dffacted bea ontbuton of d at a dept of - # of unt cell F pe unt aea: - Apltude at P: d d e S F d F e d F e ds d - xtncton dtance: d - ontbuton of te colun t t t d d d e Incdent plane ave: ds d t F e d d d # of catten cente/avelet t Specen B - Te dffacton ntenty at P I * A n t / / P = d d =t

9 Ltaton of te neatcal Appoxaton neatcal Appoxaton : Te Intenty of te ncdent ave - Te Ba condton: = I I - I I t I 4 t / / A caactetc lent fo a patcula dffacted bea e F t ~ Te tuctue of te cytal: e, F Te atoc nube of te pecen: F Te avelent of te electon bea: Te dffacton anle, dffacton plane: F t 4 4 n I,ax 4, ax I Only one dffacted bea excted n-bea etct te valdty of neatcal teoy to even alle tcne I * n t / /

10 Te Dynacal Teoy: Dan-Hoe-Welan quaton oupln beteen te dffacted ave and te dect ave and vay t To enfoceent decton - Foad: d d d d d d d Inoe te pae facto - Ba Anle: d - Foad: - Ba Anle: d d d d d d d d onevaton of te total ntenty of te ave d d * * d d d d BF and DF ae ae utually copleentay : te devaton fo te exact Ba anle : te extncton dtance Pefect cytal xp: pae facto due to te catten poce : pae cane of / due to te catten poce

11 Solvn te quaton d d d d d d d d d d d d A oluton: A Te ubttuton doen t cane te eauable quantte, e.. ntenty Inet t nto te above equaton 4 aactee te tlt out of te Ba condton = 4 To ndependent oluton: Te coponent of te ave vecto odfed We nae te ne ave vecto:

12 Te ave Functon To poble ndependent oluton total ave, dect and dffacted: b, b, Noale te coeffcent co :Te ae x, y coponent a, A Te ave functon n te cytal: n cot b, b, co n co nequal odfcaton to te ave vecto fo te to oluton: qute all tou Detened by bounday condton n At Ba poton: =; =/ < B : <; >/ : te ave vecto n te cytal - Te ae x, y coponent a, te ave vecto n te vacuu : decton pependcula to e F

13 Bounday ondton co n co n At =: Dect bea apltude = co n Dffacted bea apltude = - Dect bea co n co n co n onde coponent Pependcula to te and paallel to aple uface: co co n - Dffacted bea At te botto of te aple: Te ntenty t t n n co n n 4 co n t n n eff t / / eff eff Wen eff 4 4 neatcal appoxaton

14 Te Intenty of te Dect and Dffacted Bea: Tcne Fne t t n t n n eff t / / eff eff 4 4 ede Bt Feld Iae Tcne Fne u+7%al Alloy F e Pendellöun of te dynacal teoy

15 Te Intenty of te Dect and Dffacted Bea: Bend ontou t n eff t / 4 t t n n 4 / eff eff t 4 Syetca l about = : all and potve anoalou tanon Lo tanon <, lo tanon tanon tanon Bt Feld Iae Bend ontou

16 Te Intenty of te Dect and Dffacted Bea: Bend ontou

17 Te Dynacal Teoy: an envalue Poble Te one-body equaton fo te ncdent electon: n n n n e Z d e d d ;... ; * - : upepoton of all te atoc potental, te ae peodcty a te lattce - e: attbuted to eac pont of te ecpocal lattce d q d F 3 3 Stuctue Facto: Foue See d e 3 e F 4 e F xtncton dtance: F e, Bloc ave: Te oluton to te one-body equaton b Supecpt: ac Bloc ave a a nle value of : : c not te ave vecto of te ncdent o dffacted ave Plane ave apltude, -ub- Su ove all te pont n ecpocal pace Te total ave functon Bloc-ave feld b xctaton apltude/coeffcent A patcula eflected ave

18 Te Fundaental quaton of Dynacal Teoy - nde te cytal: c e Mean value, nne potental Subttute te Bloc ave and te peodc potental nto te equaton: ollectn up te contann Te Fundaental quaton ecula eqn =: n - n vacuu: lae tan Dpeon elaton: ave vecto ~ eney - Te potental of te cytal xe te Bloc ave - Ln Ba bea and Bloc ave * Wave vecto of te Bloc ave

19 To-Bea ondton ae non-eo fo to bea: dect bea and te dffacted bea Fo =, Fo = ; = = ; = Solve te equaton: 4 Inne potental: < e -bea: > e, >> Wen te electon ae n te vacuu, = Suface of contant eney: alloed Tey ave te ae antude Wen G bea tonly excted, t act le a dect bea

20 To-Bea ondton 4 4 Wen Bounday condton: ave vecto tanental to te uface Wave vecto n te vacuu To Bloc ave exted Te lne To dffacted B-ave

21 Suay eve of Dynacal Teoy Dan-Hoe-Welan quaton envalue Dpeve uface To Bea cae No lectue fo te next to ee Pleae o on te eay and pepae fo you nteve Lectue 6: eve of TM ontat Mecan

22 Dynacal ffect t t n t n n eff t / / eff eff 4 4 Ocllaton ll occu f o t vae.

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