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Electron diffraction structure analysis (EDSA) of thin polycrystalline films – Part 2 Reflexion intensities in ED patter

Electron diffraction structure analysis (EDSA) of thin polycrystalline films – Part 2 Reflexion intensities in ED patterns. Anatoly Avilov Institute of Crystallography of Russian Academy of sciences. 1. Kinematical approximation 2. Atomic scattering 3. Temperature factor

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Electron diffraction structure analysis (EDSA) of thin polycrystalline films – Part 2 Reflexion intensities in ED patter

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  1. Electron diffraction structure analysis (EDSA) of thin polycrystalline films– Part 2Reflexion intensities in ED patterns Anatoly Avilov Institute of Crystallography of Russian Academy of sciences

  2. 1. Kinematical approximation • 2. Atomic scattering • 3. Temperature factor • 4. Structure amplitude • 5. Reflexion intensities • ideal single crystal • mosaic crystalline film • secondary scattering • texture film • polycrystalline film • 6. Dynamical corrections • 7. Structure analysis methods

  3. Kinematical approximation • kinematical approximation is derived from the first Born approximation • Φ (S) =  φ (r) exp (2 πi Sr) dvr = =  φ (r) exp [2 πi (xx* + yy* + zz*)] dx dy dz = Ғ [φ] (Ғ is Fourier operator) • absolute value : Φabs = K Φ (S), K = 2 π me/h2 S - vector of Fourier space S 2 sin 

  4. Atomic scattering • atomic amplitude in Born approximation: fe(s) = 4 K  (r) r2(sin sr/sr) dr (*) • Poisson equation Mott formula : fe (s) = me2/(2h2) {[Z – fx (s)] / s2 Z - nuclei charge • if s = sin / 0 : interpolation of (*) or using : f (0) = [4 2 me2 / (3h2)] Z < r2 >, < r2 > - the mean square radius of the electronic shell of the atom

  5. Comparison of fe and fx • According Thomas-Fermi statistic theoryat ~ at2/3- the atomic potential function is more smeared than that of the electronic density • fe – curves slope more sharply with sin / than fxelectrons are scattered in more narrow range of sin / • fe (0) ~ Z1/3and fx(0) = Z, while for larges, fe ~ Z and fx ~ Z3/2,i.e. feis less dependent on Z than fx • electrons are scattered by light atoms in the presence of heavy ones relatively more strongly than X-rays.

  6. Temperature factor • thermal motion of an atom - distribution function w(r) T (r) =  (r – r’) w(r’ ) dvr =  (r ) * w(r) Ғ [φat (r ) ] = fe , Ғ [w(r )] = fT , fe,T = fe (s) fT (s) • for Gaussian law the vibrations are spherically symmetrical w(r) = (2  <u2>) -3/2 exp (- r2 / 2 < u2 >)  w(r’) dvr = 1, <u2> -1/2 - mean square displacement of an atom from the equilibrium position fT (s) = exp (<u2>s2/2) = exp{-B(sin /)2 }, В=8 <u2>

  7. Structure amplitude • ΦH =  φ (r) exp (2 πi Hr) dvr =  =  φ (r) exp 2 πi (hx + ky + lz) dx dy dz  φ (r) =  φi (r – ri ), Φhkl =  fei exp(2 πi Hri ) • In general Φhkl is a complex quantity: Φhkl = А hkl + iВhkl , |Ф| = (A2 + B2)1/2, А=|Ф| cos , В = |Ф| sin , tg =B/A

  8. reflection intensitiesideal single crystal (kinematical approximation) for spherical wave scattered by a crystal for a definite reflexion: Ihkl (h1h2h3) = (J0/r2)| Ф hkl| 2 D(h1h2h3))2 for a parallel piped-shaped crystalLaue interference function : |D(h1h2h3)|2 = П sin2 π Ai hi / ( π Ai hi)2 i = 1,2,3 Ai - linear dimensions of crystal, ai - unit cell edges D2 dhi = [ sin2 π Ai hi/(π ai hi)2 ] dhi = Ai / ai2 at the maximum (i.e. for hi = 0) is Di (0)2= Ai2 / ai2 = Ni2

  9. ideal single crystal (continued) • Ihkl (h3) =  Ihkl (h1h2h3) dx1 dx2 = = (J0 | Ф hkl| 2 / L2)  D(h1h2h3)2 dx1 dx2 S=A1A2, V=A1A2A3 ,  =a1a2a3 , RL= L Ihkl(h3) / J0S = 2|Ф / |2 sin2A3h3 / 2 h32 • in reflection’s maximum (at h3 = 0): Ihkl / J0S = 2|Ф / |2 A32 • Scattering is kinematic if Ihkl(h3) << J0S Ihkl = J0S - transition to the region of dynamic scattering :  |Фhkl / | A’ 3 1 the block thickness - A’3

  10. Intensity diffracted by a mosaic single crystal film • In a real mosaic specimen a certain angular distribution function : f()= f(1)f(2)f(3), d = dh3 /Hhkl Ihkl / J0S = 2|Фhkl / |2 t dhkl /  t - mean film thickness, S - illuminated film area • perfect crystal in a reflecting position: |Фhkl |2~ Ihkl • mosaic film: |Фhkl |2~ Ihkl /dhkl

  11. Account for the crystal film mosaicity

  12. To the calculation the reflection intensities for texture films for two cases:(a) needle (fiber) texture patterns and (b) oblique texture patterns (on the right)

  13. DP-s for “right” (needle) and “oblique” textures

  14. “right” texture 2π range of azimuthal orientations (angles 1) of micro- crystals around the ТА (texture axis) f(2) - disorientation function R = LHhk0 , d  = L dh3/ R = dh3/Hhk0 = dhkodh3 integral intensity of an arc : Ihkl = J02|Фhkl / |2V’ (dhk0 / 2 ) p Ihkl / J0S = 2|Ф / |2 t (dhk0 / 2 ) p relative values |Фhkl |2 ~ Ihkl / dhk0 p p - the multiplicity factor.

  15. “oblique” texture dx1 = L dh1 / sin φ, the tilting angle φ dx2 = L dh2 R/R’ integral intensity Ihkl / J0S = 2|Ф / |2 t Lp / 2 R’ sin φ |Фhkl |2 ~ Ihkl R’ . local intensity Ihkl = I’hkl / r as r = L / dhkl , Ihkl = Ihkl dhkl / L I’hkl / J0S = 2|Фhkl / |2 t  dhkl p dhk0 / 2 L.

  16. Reflexion intensities for polycrystalline films distribution over the whole solid angular interval 4л. DP - concentric rings as plane sections of spheres It is the local intensity which is of interest in this case reflexion radius r = LHhkl I’hkl = Ihkl dhkl /2 L. I’hkl = J02|Фhkl / |2 V’ d2hklp / 4 L relative values - Фhkl2 d2hkl p ~ I’ hkl

  17. Problems of practical EDSA • Dynamical interactions • Secondary scattering • Background

  18. Secondary scattering • strong diffraction beams may act as primary ones in their propagation through subsequent mosaic blocks forming additional DP's • identically oriented blocks - secondary reflexions coincide with those in the initial pattern • nonidentically oriented ones- do not coincide DP’s are not suitable for structural determinations (geometrical analysis is possible)

  19. Secondary diffraction effects

  20. Kinematical dynamical scattering

  21. AgTlSe2, textured film (Imamov, Pinsker) a = 9,70  0,04 A , b = 8,25  0,04 A, sp.gr. D13d

  22. AgTlSe2, textured film, many beam calculations (Turner, Cowley) All crystals are of equal thickness, and have a gaussian distribution of width  about the texture axis; i.e.: T ( ) = exp (- 2 / 2 ) , H (D) =  (D), J’hk0 =  Ihk0T ( ) H (D) ddD =  Ihk0 exp (- 2 / 2 ) d Structure parameters were approx. the same. General conclusion: many beam effects can distort structure parameters. But it does not exist alternative metod to EDSA for the finding of the draft model of structure. The refinement of the model should be made with the accounting for the dynamic scattering of electrons.

  23. How to avoid dynamic scatteringor to account for it? • Using samples of small thicknesst tel and to estimate suitable situation according criteria : A = hkl t  1 • using of the modern electron diffraction technics, e.g. “hollow cone” precession • Using dynamical corrections: a) Two-beam corrections by “ Blackman curve” b) Using “Bethe potentials” - influence of weak beams • Direct many-beam calculations Corresponding algorithmus have been developed for partly oriented polycrystalline films

  24. Dynamical two-beams corrections

  25. Brucite-Mg(OH)2-(Textured film) Zhukhlistov,Avilov etc. Cryst.Rep.(1997) 774

  26. Dynamical corrections by «Bethe potentials» • Two-beam scattering with accounting for weak reflexions. «Bethe potentials» - modified potentials in many beam theory:U0,h = vh - g’’[vg vh-g/(2 – kg2)];

  27. Partly oriented films - textures and mosaic films (many beam calculations) Model for calculation : • Thin film consist of only slab of microcrystallites, so the effects of secondary extinction are absent • Crystallites are ideal and scatter incoherently , so the intensities of individual crystallites can be added without accounting for their phases • The distribution functions on the angles and dimensions are known

  28. Partly oriented films (many beam calculations) • M x, M - Dynamical matrix • vgh = (4 Фgh , ph= 2 K h + h2, K = (2 + v0)-1/2 • hv(r)  [i 0*i hi exp {izxi/ 2}] exp {iKhr)

  29. mosaic films (LiF) • I av= 1/(t1 - t2) I (,,,t) f1 ()f2 ()x xf3 ()f4 (t)d d d dt 1 - reflexion 200, 2 - 220, 3 - 600, 4 - 10.0.0;

  30. Textured PbSe films • MANY BEAM CALCULATIONS • 200 (curve-1), 220 (2) and 400 (3) а-  = 300, U = 25кV; б-  = 300, U = 50кV; в-  = 300, U = 75кV;г-  = 50, U = 75кV; д-  = 450, U = 75кV; е-  = 600, U = 75кВ (RIGHT UPPER) • 111 (curve- 1), 311 (2), 331 (3) and 600 (4) for  = 600, U = 75кВ (RIGHT DOWN) • electron diffraction pattern for PbSe (DOWN)

  31. Integral chafacteristics - first attempt of quantitative estimation of ESP 1. Estimation of errors 2. Atom potential in structures 3. Analysis of the Fourier- syntesises Fourier - method in EDSA

  32. Methods of structure analysis • The Patterson interatomic vector function • Superposition methods, introduced by Buerger in 1959, allow the vector sets of the Patterson function to be analyzed as being composed of the vector sets of the structure • Trial and error methods, in which intensities or Фhkl values calculated from a postulated structures are compared with those derived from experiment, may serve for relatively simple structures but are rarely used on present day SA • Direct methods, based on the use of equality or inequality relationships between sets of structure factors or their signs Some initial applications in EDSA have been reported, for example, by Dorset and Hauptmann (1976)

  33. The example of using of the Patterson interatomic vector function • (a) Projection of the Patterson function for BaCl2H2O. The strongest maximum (29) corresponds to the Ba-Ba distances, the next maximum (18) corresponds to the Ba-CI distance • (b) The Fourier map for the same structure (both maps are given in arbitrary units)

  34. Phase determination with the tangent formula and LSQ refinement. D.Dorset, M.McCourtActa Cryst. (1994) A50, 287 • Diketopiperazine - C4H6N2O A=5.20 , B=11.45, C=3.97Ǻ, =81.90

  35. Structure investigations by EDSA(1) • Ionic compounds (Pinsker, Vainshtein) • Ionization of atoms in crystals (Vainshtein, Dvoriankin) • Semiconductors (ternary halcogenides of metals I, III, Vb) (Semiletov, Imamov, Avilov) • Hydrogen position and hydrogen bonds. Long chains molecules (paraffines, polymeres),small-molecules (phospholipides etc.) (Vainshtein, Pinsker, Dorset, Moss); hydrides of metals (Ni, Pd, Sc) (Khodirev, Baranova) • Biological objects: a) polypeptide, poly--methil-L-glutamate, purple membrane (Vainshtein, Tatarinova, Dorset, Unwin, Henderson) b) mixed complexes of Cu with amino-acids (Diakon , co-workers)

  36. Structure investigations by EDSA(2) • Organic films (Klechkovskaya) • Oxides, carbides, nitrides (Khitrova, Klechkovskaya) • Minerals on OTED patterns and SAED (Zvyagin, Drits, Zhukhlistov) • Structure of molecules- gas EDSA (Vilkov, co-workers) • Chemical bonding, quantitative analysis of electrostatic potential, relation with physical properties (Avilov, co-workers, Tsirelson)

  37. Electron crystallography investigations of polycrystals and single crystals

  38. How the EDSA is developed? (perspectives) • development of the precise methods of EDSA : - technique of measurements of diffraction pattern - applying of energy filtering - improvement of the methods of accounting for many beam scattering in the process of structure refinement • investigations of ESP distribution and chemical bonding, relation of the atomic structure with properties; • modification of the methods of structure analysis and its using for solving more complex structure : metallo-organic and organic films, polymers, catalysts, nano-materials etc...

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