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 approximation2. Atomic scattering3. Temperature factor4. Structure amplitude5. Reflexion intensitiesideal single crystal

mosaic crystalline film

secondary scattering

texture film

polycrystalline film

6. Dynamical corrections 7. Structure analysis methods

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

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

0.1 0.5 sin / x 10-8

fe-curves for atoms with Z=1-10.

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.

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>

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 + iВ hkl , |Ф| = (A2 + B2)1/2,

А=|Ф| cos , В = |Ф| sin , tg =B/A

reflection intensities ideal 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 crystal Laue 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

D2 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 / ai

2 = Ni2

To the calculation of integrated intensities from a single crystal

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 A3 2

• 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

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

Account for the crystal film mosaicity

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)

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

“right” texture

2π range of azimuthal orientations (angles 1) of micro- crystals around the ТА (texture axis) f(2) - disorientation function

R = LHhk0 , d = L dh3/ R = dh3/Hhk0 = dhkodh3

integral intensity of an arc :

Ihkl = J0 2|Фhkl / |2 V’ (dhk0 / 2 ) p

Ihkl / J0S = 2|Ф / |2 t (dhk0 / 2 ) p

relative values |Фhkl |2 ~ Ihkl / dhk0 p

p - the multiplicity factor.

“oblique” texture

dx1 = L dh1 / sin φ, the tilting angle φ

dx2 = L dh2 R/R’

integral intensity

Ihkl / J0S = 2|Ф / |2 t Lp / 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.

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 = LHhkl

I’hkl = Ihkl dhkl /2 L.

I’hkl = J0 2|Фhkl / |2 V’ d2hkl p / 4 L

relative values - Фhkl 2 d2hkl p ~ I’ hkl

Problems of practical EDSA

• Dynamical interactions

• Secondary scattering

• Background

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)

Secondary diffraction effects

Kinematical dynamical scattering

Kinematical case - Ihkl << J0 S

Limit case - tel 1

tel – “extinction lenght”

Kinematical case - hkl 2 ~ Ihkl

Two – beam case - hkl ~ Ihkl

Many-beam case – complex relations from dynamical theory

AgTlSe2 , textured film (Imamov, Pinsker)

a = 9,70 0,04 A , b = 8,25 0,04 A, sp.gr. D1

3d

AgTlSe2 , textured film, many beam calculations (Turner, Cowley)

All crystals are of equal thickness, and have agaussian distribution of width about the texture

axis; i.e.:

T ( ) = exp (- 2 / 2 ) ,

H (D) = (D),

J’hk0 = Ihk0 T ( ) H (D) d d D

= Ihk0 exp (- 2 / 2 ) dStructure 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.

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

Dynamical two-beams corrections

Iohkl / I

chkl = ()-1 0

Jo(2x) dx = Dhkl ()

Iohkl / I

chkl D

ohkl() thkl tav

Davhkl() Io,corr

hkl / Ichkl I

o,corrhkl

F o r p o l y c r y s t a l i t i s n e c e s s a r y t o i n t e g r a t eo n v a r i o u s a n g l e s o f i n c i d e n c e b e a m :

A

dxxJAdww

wA

0

02

2/122

)2(1

])1([sin

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

Atom x/a y/b z/c B11 B33 B12 B13

Mg 0 0 0 222(13) 81(9)

O 1/3 2/3 0.2205(3) 228(13) 82(9)

H 0.362(5) 0.724(7) 0.416(4) 664(165) 243(73) 537(190) -54(84)

a = 3.149(2), c = 4.769(2) A, P -3m1

Blackman corrections, tav = 440 A

ESP sections with subtracted oxygen atom. Isolines are drown at a step 10V. The numbers indicate thevalues of the ESP at the maxima: a) (110) section, b) (001) section passing through the H atom. One cansee the elongation of ESP isolines in the direction of three nearest oxygen atoms of the adjacent layer,which favors the three-positional model for hydrogen atoms

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)];

When the Bragg conditionsfor one reflection is satisfied,the other reflections of“systematic set” alwayshave the same“excitation errors”

vg/(2 - kg2) << 1 ; vh-g/(2 - kg

2) << 1

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

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 {iKhr)

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;

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)

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

ihkl

hkl 2exp(1

Hr ) = r )

Allowance of Fourier – expansion:

One-, two-, three-dimensional

synthesises;Patterson and Foureir- maps;Differencial Fourier- synthesises .

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)

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)

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

(a) potential map (b) LSQ ref. (c) X-ray diffr.

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)

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)

Electron crystallography investigations of polycrystals and single crystals

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...