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Electron diffraction structure analysis EDSA of thin polycrystalline films Part 2 Reflexion intensit

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Reflexion intensities in ED ... scattered by a crystal for a definite reflexion: ... 1 - reflexion 200, 2 - 220, 3 - 600, 4 - 10.0.0; Textured PbSe films ... – PowerPoint PPT presentation

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Title: Electron diffraction structure analysis EDSA of thin polycrystalline films Part 2 Reflexion intensit


1
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

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
  • F (S) ? f (r) exp (2 pi Sr) dvr
  • ? f (r) exp 2 pi (xx yy zz) dx dy dz
    ? f
  • (? is Fourier
    operator)
  • absolute value
  • Fabs K F (S), K 2 p 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 lt r2 gt,
  • lt r2 gt - the mean square radius of the
    electronic shell of the atom

5
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6
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
    fx electrons are scattered in more narrow range
    of sin ?/?
  • fe (0) Z1/3 and fx(0) Z, while for large s,
    fe Z and fx Z3/2, i.e. fe is less dependent
    on Z than fx
  • electrons are scattered by light atoms in the
    presence of heavy ones relatively more strongly
    than X-rays.

7
Temperature factor
  • thermal motion of an atom - distribution function
    w(r) ?T (r) ? (r r) w(r
    ) dvr ? (r ) w(r)
  • ? fat (r ) fe , ? w(r ) fT ,
    fe,T fe (s) fT (s)
  • for Gaussian law the vibrations are spherically
    symmetrical
  • w(r) (2 ? ltu2gt) -3/2 exp (- r2 / 2 lt u2
    gt)
  • ? w(r) dvr 1, ltu2gt -1/2 - mean square
    displacement of

  • an atom from the equilibrium

  • position
  • fT (s) exp (ltu2gts2/2) exp-B(sin ?/?)2
    ,
  • ?8? ltu2gt

8
Structure amplitude
  • FH ? f (r) exp (2 pi Hr) dvr
  • ?
  • ? f (r) exp 2 pi (hx ky lz) dx dy
    dz
  • ?
  • f (r) ? fi (r ri ), Fhkl ? fei exp(2 pi
    Hri )
  • In general Fhkl is a complex quantity
  • Fhkl ? hkl i? hkl , ? (A2 B2)1/2,
  • ?? cos ?, ? ? sin ?, tg? B/A

9
reflection intensities ideal single crystal
(kinematical approximation)
  • for spherical wave scattered by a crystal for a
    definite reflexion
  • Ihkl (h1h2h3) (J0/r2) ? hkl 2
    D(h1h2h3))2
  • for a parallel piped-shaped crystal Laue
    interference function
  • D(h1h2h3)2 ? sin2 p Ai hi / ( p Ai hi)2
  • i 1,2,3
  • Ai - linear dimensions of crystal, ai - unit
    cell edges
  • ??D?2 dhi sin2 p Ai hi/(p ai hi)2 dhi Ai
    / ai2
  • at the maximum (i.e. for hi 0) is Di (0)2 Ai2
    / ai2 Ni2

10
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11
ideal single crystal (continued)
  • Ihkl (h3) ? Ihkl (h1h2h3) dx1 dx2
  • (J0 ? hkl 2 / L2) ? D(h1h2h3)2
    dx1 dx2 SA1A2, VA1A2A3 , ? a1a2a3 , RL L?
  • Ihkl(h3) / J0S ? 2? / ?2 sin2 ?A3h3 / ?2
    h32
  • in reflections maximum (at h3 0)
  • Ihkl / J0S ? 2? / ?2 A3 2
  • Scattering is kinematic if Ihkl(h3) ltlt J0S
  • Ihkl J0S - transition to the region of
    dynamic scattering
  • ? ?hkl / ? A 3 ? 1
  • the block thickness - A3

12
Effect of mosaicity on the formation DP
13
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?hkl / ?2 t dhkl / ?
  • t - mean film thickness, S -
    illuminated film area
  • perfect crystal in a reflecting position
  • ?hkl 2 Ihkl
  • mosaic film ?hkl 2 Ihkl /dhkl

14
Account for the crystal film mosaicity
15

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)
16
DP-s for right (needle) and oblique textures
17
right texture
  • 2p range of azimuthal orientations (angles ?1) of
    micro- crystals around the ?? (texture axis)
    f(?2) - disorientation
    function
  • R L?Hhk0 , d ? L? dh3/ R dh3/Hhk0
    dhkodh3
  • integral intensity of an arc
  • Ihkl J0 ? 2?hkl / ?2 V (dhk0 / 2? ) p
  • Ihkl / J0S ? 2? / ?2 t (dhk0 / 2? ) p
  • relative values ?hkl 2 Ihkl / dhk0 p
  • p - the multiplicity factor.

18
oblique texture
  • dx1 L? dh1 / sin f, the tilting angle f
  • dx2 L? dh2 R/R
  • integral intensity
  • Ihkl / J0S ? 2? / ?2
    t L?p / 2? R sin f
  • ?hkl 2 Ihkl R
    .
  • local intensity
  • Ihkl Ihkl ? / r?

    as r L? / dhkl ,
    Ihkl Ihkl ? dhkl / L??
  • Ihkl / J0S ? 2?hkl /
    ?2 t ? dhkl p dhk0 / 2? L??.

19
Reflexion intensities for polycrystalline films
  • distribution over the whole solid angular
    interval 4?.
  • 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
  • Ihkl Ihkl dhkl ? /2? L?.
  • Ihkl J0? 2?hkl / ?2 V d2hkl ?p / 4? L?

  • relative values - ?hkl 2 d2hkl p I hkl

20
Problems of practical EDSA
  • Dynamical interactions
  • Secondary scattering
  • Background

21
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
  • DPs are not suitable for structural
  • determinations
  • (geometrical analysis is possible)

22
Secondary diffraction effects
23
Kinematical dynamical scattering
24
AgTlSe2 , textured film (Imamov, Pinsker)
  • a 9,70 ? 0,04 A , b 8,25 ? 0,04 A, sp.gr.
    D13d

25
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),
  • Jhk0 ?? Ihk0 T (? ) H (D) d ? d D
  • ? 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.

26
How to avoid dynamic scattering or to account
for it?
  • Using samples of small thickness t ? 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

27
Dynamical two-beams corrections
28
Brucite-Mg(OH)2-(Textured film)
Zhukhlistov,Avilov etc. Cryst.Rep.(1997) 774
29
Dynamical corrections by Bethe potentials
  • Two-beam scattering with accounting for weak
    reflexions. Bethe potentials - modified
    potentials in many beam theory U0,h vh -
    ?gvg vh-g/(?2 kg2)

30
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

31
Partly oriented films (many beam calculations)
  • M?? x?, M - Dynamical matrix
  • vgh (4???? ?gh , ph 2 K ?h ?h2, K (?2
    v0)-1/2
  • ?hv(r) ? ?i ?0i ?hi exp izxi/ 2? exp iKh?r)

32
mosaic films (LiF)
  • I av 1/(t1 - t2) ? I (?,?,?,t) f1 (?) f2 (?)x
  • x f3 (?) f4 (t) d? d? d? dt
  • 1 - reflexion 200, 2 - 220, 3 - 600, 4 - 10.0.0

33
Textured PbSe films
  • MANY BEAM CALCULATIONS
  • 200 (curve-1), 220 (2) and 400 (3) ?- ? 300, U
    25?V ?- ? 300, U 50?V ?- ? 300, U
    75?V?- ? 50, U 75?V ?- ? 450, U 75?V
    ?- ? 600, U 75?? (RIGHT UPPER)
  • 111 (curve- 1), 311 (2), 331 (3) and
  • 600 (4) for ? 600, U 75?? (RIGHT DOWN)
  • electron diffraction pattern for PbSe (DOWN)

34
Fourier - method in EDSA
  • Integral chafacteristics - first attempt of
    quantitative estimation of ESP
  • 1. Estimation of errors
  • 2. Atom potential in
  • structures
  • 3. Analysis of the Fourier-
  • syntesises

35
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
    ? ?hkl? 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)

36
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)

37
Phase determination with the tangent formula and
LSQ refinement. D.Dorset, M.McCourtActa Cryst.
(1994) A50, 287
  • Diketopiperazine - C4H6N2O A5.20 , B11.45,
    C3.97?, ?81.90

38
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)

39
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)

40
Electron crystallography investigations of
polycrystals and single crystals
41
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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