Electrostatic potential determined from electron diffraction data

Anatoly Avilov

Shubnikov Institute of Crystallography of Russian academy of sciences, Moscow

Strong interaction with substances. In the age of nanoscience, there is an urgent need for a

method of rapidly solving new, inorganic nanostructured materials. Many are fine-grained, with light elements crystallites, thin films which cannot be solved by XRD.

Scattering on the electrostatic potential, possibility to reconstruct the potential from ED experiments

ED is very sensitive to ionicity, so study of bonding

More easy than XRD the localization of light atoms in the presence of heavy ones, solvable problems of hydrogen localization

Great intensity of the signal

Challenge of electron crystallography

GENERAL for modern SA:

Investigation of crystal structure and properties - one of the most important problems of physics and chemistry of solids.

Contents of this is changing as development of experimental techiques and theoretical presentations. Study of the features of distribution of ED and inner crystalline field and establishment of their connection with physical properties is one of the major question of modern SA.

The relation of ( r ) and ( r )with physical properties

Properties directly Properties indirectlydepending on depending on ( r ) and ( r ) ( r ) and ( r )_________________________________________________Diamagnetic Electron staticsusceptibility polarizability

Dipole, Nonlinearquadrupole and ( r ) opticalother momentum ( r ) characteristics

of nuclearCharacteristics of Intermolecularthe electrostatic interactions

fieldEnergy of

electrostatic interaction

Electron diffraction structure analysis (EDSA)

What is it?

EDSA of thin polycrystalline films - advantages

Wide beam (100-400 mkm), great amount of micro-crystals in irradiated area - special types of DP-s. Possibly to extract from one DP sufficiently full (3-dim.) set of structure amplitudes. Detailed SA: determination of structure parameters, reconstruction ESP and ED.

Small sizes of micro-crystallites. More often kinematical or quasi-kinematical scattering

Small effects of diffuse scattering, easy to subtract the background

Wide beam - small current density and small radiation damage (good for organic and metal-organic substances)

Prof. Boris Vainshtein

Founder of EDSA Vainshtein B.K. (1964) Structure analysis by electron

diffraction. Pergamon Press, Oxford

(translation of the revised Russian eddition (1956))

Vainshtein, B. K., Zvyagin, B. B. & Avilov, A. S. (1992).

Electron Diffraction Techniques, Vol. 1, edited by J. M. Cowley, p. 216.

Oxford University Press.

Theoretical fundamentals EDSA

Geometrical theory formation of DPTheory of reflexion intensitiesEstimations of the limits of validity of

kinematical theory of diffractionExperimental techiques and

preparation Fourier analysis

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 .

Electron diffraction camera EMR-110К

1. Electron gun2. Condensors3. Crystal holder4. Camera5. Optical microscope6. Tubus7. Photochamber

Transmission diffraction

Effect of mosaicity on the formation DP

Peculiarities of EDSA

Wide plane-parallel beam (100-400 mkm) Samples-polycrystalline thin films with

different degree of orientations ofmicrocrystallites (polycrystals, texteres,mosaic monocrystals)

mosaic single crystalline Au-H films with effects superperiodicity and twinning

Pd3H4 powder ED pattern

OTED pattern from clay mineral

Problems of development of the precise EDSA

Elaboration of the methods of many beam calculations or some type of corrections of dynamical effects

Development of the precise technique of measurements of electron DPs

Improvement of the means for the accounting for the inelastic scattering

Working out the methods of modelling ESP on the base of experimental information and the estimation of its real accuracy

Elaboration of the methods of treatment of uninterrupted ESP distribution in terms of conception of physics and chemistry of solids

Kinematical dynamical

How to avoid dynamic scattering or to account for it?

Using samples of small thickness t tel

or to estimate suitable situation according criteria :

hkl t 1

Using dynamical corrections:

a) Two-beam corrections by “ Blackman curve”

b) Using “Bethe potentials” - influence of weak beams

Direct many-beam calculations

Corresponding algorithms 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

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 conditions for one reflexion are satisfied, the other reflections of the«systematic set» always have the same «excitation errors»

Main problem in using direct many-beam calculation – to find the distribution functions on sizes and orientations of microcrystals…Additional EM studies of micro-structure are very useful

Inelastic scattering

The neglect the absorption in very thin polycrystalline films of substances with the small atomic numbers does not cause noticeable errors in the determination of structure amplitudes

Using system of energy filtration of electrons at the filter resolution within 2-3 eV improves the situation.

Construction of smooth background line provides partially to take into account for the thermal diffuse scattering

Electron diffractometry

Types of measuring detectors for DP

photographic registration – dynamic range (DR) ~ 102

scintillator + PM – DR ~ 104 (limitation - nonlinearity) CCD – camera – DR ~ 104 (measurements of 2D

patterns) Image plates – DR ~ 106 (high linearity)

Control program determines mode of measuring and its accuracy

Accuracy of measurements depends on the mode: «accumulation mode» or «constant time mode»

Scheme of electron diffractometer

Accumulation mode – statistical acc. ~ 1-2%

Statistical treatment, quasimonitoring – improvement of accuracy

New method for measurement-direct current measuringvery high linearity and wide DR

Ultramicroscopy. 107 (2007), 431-444.

1. Faraday cup 2. electronic amplifier 3. window comparator 4. pulse counter 5. quartz oscillator 6. PC

How to reconstruct the electrostatic potential for quantitative analysis?

Summing of the Fourier series with using experimental structure amplitudes hkl is not good (!)

Analytical reconstruction in the direct space on the parameters of the model, obtained from the experiment

φ (r) =

{σ (r’) / r – r’ } dr’,

full charge density σ (r’) = aZa ( r’ – Ra) - (r’)

Za and Ra - nuclear charge and nuclear position of atom “a” (r’) – electron density

Fourier method in EDSA

hkl = i fэлi exp (2 i (h xi + k yi + l zi ))

hkl and hkl are determined from the EP

hkl ~ Ihkl / d2 hkl («kinematical approximation)

Retrieval of the right model of structure is realized By the trial and error method or by «direct methods».

= | | expi

Influence of the break of Fourier series for reconstruction

potential’s maps (synchr.exp.data of U.Pietsch for GaAs) Fourier maps for the ESP: (100) - left,

and (110) - right.

Two upper rows are experimental

series up to (sin/)max 1,3 A-1.

Two lower rows present theoretical Hartree-Fock calculations and experimental amplitudes with adding theoretical ones ( 15 A-1) .

Appearence of false peaks (5-10 % from true peaks) and distortions of the forms of the ESP peaks GaAs examples is seen

The reconstruction of the ESP by analytical methods

Model’s parameters are found by adjustment to experimental structure amplitudes

The calculation of ESP is realized in direct space by using Hartree-Fock wave functions

Analytical methods are free from many errors: - the break of Fourier series; - inaccuracies of transition to structure amplitudes and noises with intensity measurements Static ESP is calculated for the following analysisThis approach allows quantitatively to establish: features of

ESP in inter-nuclear area, intensity of electric field (gradient ESP), to make a topological analysis ESP

So the analytical methods of

the reconstruction are needed.

Chemical bonding in EDSAMultipole model Hansen-Coppens:

( r ) = Pcore core ( r ) + Pval val ( r) + Rl ( r) Plm ylm (r/r)

( r ) – electron density of each pseudoatom, core ( r ) and val ( r )

– core and spherical densities of valence electron shells

Pval and Plm (multipoles) describe electron shell occupations - and describe spherical deformation - y (r/r) is geometrical functions

For ionic bonding – spherical approximation (kappa –model):

( r ) = Pcore core ( r ) + Pval val ( r) Electron structure amplitude, using Mott-formula:

(g) = ( g ) {Z – [ f core(g) + Pval fval (g/ )]}

Rl ( r) Plm ylm (r/r) - nonspherical part, describing space anisotropy of the electron density

Quantitative data for the ionic crystals LiF, NaF, and MgOa

Structural - model Electron diffraction Hartree-Fock

Structure amplitudes structure amplitudes

Comp-d atom Pv R% Rw % Pv R%

LiF Li 0.06(4) 1b 0.99 1.36 0.06(2) 1b 0.52 F 7.94(4) 1b 7.94(2) 1.01(1) NaF Na 0.08(4) 1b 1.65 2.92 0.10(2) 1b 0.20 F 7.92(4) 1.02(4) 7.90(2) 1.01(1) MgO Mg 0.41(7) 1b 1.40 1.66 0.16(6) 1b 0.16 O 7.59(7) 0.960(5) 7.84(6) 0.969(3)a Structural - models were as followed- LiF: cation = 1s (r ) +

+ Pval 3 2s ( r ),

anion = 1s (r ) + Pval 3 2s,2p ( r ); NaF and MgO: cation = 1s,2s,2p (r ) +

+ Pval 3 3s ( r ), anion = 1s (r ) + Pval 3 2s,2p ( r )b Parameters were not refined

Theoretical calculation for the estimation of accuracy of experimental results.

Calculation for 3-dim. periodical crystals by non-empirical Hartree-Fock method with using CRYSTAL 95.

Broadened atomic basis 6-11G+, 8-511G, 7-311G*, 8-511G* и 8-411G* for Li+, Na+, F-, Mg2+, and O2- corr. were taken as initial ones and were optimized for achievement of minimum of crystal energy. An accuracy of such calculations for the infinite three-dim. crystal is about 1%.

From the theoretical ED X-ray structure amplitudes have been calculated, which then were recalculated in electron amplitudes and were used as experimental for the refinement of the model’s parameters.

ESP, TOPOLOGICAL ANALYSIS (1)

Classical electrostatic field is characterized by the gradient field ( r ) and curvature 2 ( r ) (these characteristics do not depend on the mean inner potential 0) :

E ( r ) = - ( r ) ESP exhibits maxima, saddle points, and minima

(nuclear positions, internuclear lines, atomic rings, and cages).

TOPOLOGICAL ANALYSIS ESP (2)Theory Bader (analog for the electron density) was used for the

ESPIn “critical points”: ( r ) = 0 Hessian matrix - H is composed from the second derivative ( r ) For ESP in critical points 1 + 2 + 3 0, because ( r ) 0 1 + 2 3 CPs are denoted as (3, i), i – algebraic sum of signs of : (3,-3), (3,-1), (3,+1), (3,+3)Nuclear of neighboring atoms and molecules in crystals are

separated in the E ( r ) by “zero-flux” surfaces S ( r ) E ( r ) n ( r ) = - ( r ) n ( r ) = 0 , r S ( r ) These surfaces define the electrically neutral bonded

pseudoatoms in statistic equilibrium at the accounting for Coulomb interaction. Inside surfaces nuclear charge is fully screened by the electronic charge.

ESP for binary compounds (analytical reconstruction), (110) - plane

circle - (3,-1) - bonding lines - one-dim. minimum

treangle - (3,+1) - two-dim. minimum

square - (3,+3) - absolut minimum

ESP (left) and ED for (100) plane of LiF

The location of CPs does not coincide, ESP does not fully determine the ED

In ESP the main input belongs to cations

ESP along bonding lines in binary crystals LiF, NaF, MgO

Distribution of ESP in binary compounds is along cation-cation (dotted),

anion-anion (solid) -left; The same one is along cation-anion - right side (ESP-values are in log of Volts)

ESP for bonded atoms in LiF and NaF

Electrostatic potential as a function of the distance from the point of observation to the center of an atom for remoted ions in LiF and NaF crystals with the parameters of the -model obtained from the electron diffraction data

“Bonded radii” derived from the electrostatic potential and electron densitya

“bonded radii” (A)

compound atom electrostatic potential electron density

LiF Li 1.084 0.779

F 0.928 1.233

NaF Na 1.355 1.064

F 0.964 1.255

MgO Mg 1.207 0.918

O 0.899 1.188a “Bonded ionic radii” is defined as a distance from a nuclear

position to the one-dimensional maximum in the electrostatic potential or electron density along the bond direction

Values of the Electrostatic potentials (V) at the nuclear positions in crystals and free

atoms and mean inner potentials (0 )

Comp. atom electron Hartree-Fock (crystal)

diffraction free 0

- model direct reciprocal atoms

space space

LiF Li -158(2) -159.6 -158.1 -155.6 7.07

F -725(2) -726.1 -727.2 -721.6

NaF Na -968(3) -967.5 -967.4 -964.3 8.01

F -731(2) -726.8 -727.0 -721.6

MgO Mg -1089(3) -1090.5 -1088.7 -1086.7 11.47

O -609(2) -612.2 -615.9 -605.7

Laplacian of the ED for LiF and MgO, plane (110)

Laplacian (-2 ( r )) allows one to analyse the overflow of the electronic charge at the bonding formation

Inner electronic shells are

seen

ESP in GaAs, plane (110)

-5.00 -4.00 -3.00 -2.00 -1.00 0.00 1.00 2.00 3.00 4.00 5.00-5.00

-4.00

-3.00

-2.00

-1.00

0.00

1.00

2.00

3.00

4.00

5.00

Distribution of ESP in (110),intervals: (2, 4, 8) 10n eÅ-1, -2 n 2 .

The distribution of electrical field Е = - grad

Ge - covalent bondingMultipole model Hansen-Coppens:

( r ) = Pcore core ( r ) + Pval val ( r) +

+ Rl ( r) Plm ylm (r/r)

( r ) – electron density of each pseudoatom, core ( r ) and val ( r )

– core and spherical densities of valence electron shells

Pval and Plm (multipoles) describe electron shell occupations

- and describe spherical and complex deformation in anysotropic cases

- y (r/r) is geometrical functions

for ionic bonding – spherical approximation (kappa –model):

( r ) = Pcore core ( r ) + Pval val ( r)

it should be taken into account for nonspherical part :

Rl ( r) Plm ylm (r/r) , describing space anisotropy of

the electron density

radial functions Rl ( r) = r exp (- r) and =2.1 a.u. are

calculated theoretically

Results of the multipole model refinement of Ge crystal on the electron diffraction data

Electron Refinement with LAPW

Diffraction structure factors [Lu et all,1993]

' 0.922(47) 0.957

P32 0.353(221) 0.307

P40 - 0.333(302) - 0.161

R(%) 1.60 0.28

Rw(%) 1.35 0.29

GOF 1.98 -

ED and ESP for (110) plane in Ge

Location of

critical points

is equivalent

circle - (3,-1) - bonding

square - (3,+3) - absolut minimum

treangle - (3,+1) - two-dim. minimum

Distribution ED in plane (110) ESP along (110) for Ge

Laplacian of electron density for Ge

-2 ( r )fragment of structure of

Ge along plane (110), reconstructed from the ED-data

The formation of Ge crystal is accompanied by the shift electron density to the Ge-Ge bonding line

The inner electron shells are seen

Topological characteristics of the electron density in Ge at the bond, cage and ring critical points First row presents the ED results, second row presents the our calculations based on model parameters, obtained by LAPW dataCharacteristics of the CP (3,-1) for the procrystal: = 0.357, 1 = 2 = - 0.65, 3 = 1.85

critical point type and (eÅ-3) 1(eÅ-5) 2 (eÅ-5) 3 (eÅ-5) Wyckoff positionBond critical point, 0.575(8) -1.87 -1.87 2.04 16c 0.504 - 1.43 - 1.43 1.68

Ring critical point, 0.027(5) - 0.02 0.013 0.013 16d 0.030 - 0.02 0.014 0.014

Cage critical point, 0.024(5) 0.05 0.05 0.05 8 b 0.022 0.05 0.05 0.05

Quantitative analysis of ESP is important for:

Comparison atomic potential of identical atoms in different structure - analysis of composition, chemical bonding

Crystal-chemistry analysis for the decision more general questions on the crystal formation

Solving of the problems with quantitative investigations of the chemical bonding and electrostatic potential

Study of relation with properties…

Programs used in the workEDSA - measurements and treatment of intensity,

refinement of kappa-model, Fourier

reconstruction of ESP- mapsAREN - refinement of structure parameters

(scaling, B)CRYSTAL-95 - theoretical calculations on

Hartree-Fock methodMOLDOS - refinement of multipole’s parametersMOLPROP - analytical calculations of maps ESP,

ED, CPs, Laplacian

Direct calculation of some physical properties

Diamagnetic susceptibility - d

spherical symmetry, ionic bonding classical Langevin equation, with accounting for

symmetry:

d = - (0 e2 NA a2 / 4m) [ N/96 + 1/(22 ) (-1)h/2 F (h00) / h2 ]

N – number of electrons in elem.cube, NA – Avogadro constant,

a – parameter of cell, 0 – permeability of vacuum. F (h00) – structure amplitude for h00.Static electron polarizability - Kirkwood relation between number of electrons in

molecules and mean-square radius-vector of electrons in atom

= 16 a4 /(a0 Ne) [ Ne /96 1/(2 2) (-1) h/2 F(h00) /(2 2 h2)]2

Ne – number of electrons in the molecular unit,

a0 – Bohr radius

Values of diamagnetic susceptibility d and static electron polarizability (0)

compound d (x10-10 м3/mole) (0) (х10-30 м3)

EDSA Magnetic EDSA Optical

measur-ts measur-ts

LiF 1,37 1,31 12,3 11,66

NaF 2,02 1,93 15,6 15,10

MgO 2,10 2,31 23,2 18,61

How the EDSA is developed? (perspectives)

development of the precise methods of EDSA :

- technique of measurements of diffraction pattern

- using new methods of measuring techniques with

applying precession mode in EM

- applying of energy filtering for quasielastic scattering

- improvement of the methods of accounting for many

beam scattering in the process of structure refinement

(integrat. on angle) far investigations of ESP distribution and chemical bonding, relation of

the atomic structure with properties; modification of the methods of structure analysis (automatic indexing,

direct methods, low and high temperatures) and its using for solving more complex structure : metallo-organic and organic films, polymers, catalysts, nano-materials etc...

Main conclusions:

The achieved level of the EDSA in the combination of the topological analysis of ESP and ED allow one to get a reliable quantitative information about chemical bonding and properties depending on it

The precise EDSA data about the distribution of ESP sufficiently add to a physical picture of interaction of atoms and molecules, obtained from the ED distributions

The developed methods can be

used for other ED methods

Different schemes of electron diffraction in TEM

SAED MBED HRED HDED CBED SMBED

Convergent beam electron diffraction

Specimen

Back Focal Plan

Condenser II

Upper Objective

Lower Objective

• diffraction on single crystals• dynamical diffraction• high accuracy of structure determinations

Goodman (1960-70), Spence (1980-90), Steeds et al)

Direct observation of d- holes in Cu2O

Experimental difference

map between ED of crystal

and superpositional map of

ED from spherical ions,

blue – negative charge;

red – positive charge

J.M. Zuo, M. Kim, M. O'Keeffe and J.C. Spence, Nature 401, 49-52 (1999)

Projection of

Incident beam

Precession

Thin electron beam is rotated along the surface of cone. Its top is placed on investigated microcrystal.

PED – precession electron diffraction

Vincent and Midgely 1994

Why the precession method is good?

Three-dimensional set of experimental data

Intensities of reflexions are angle-integrated.

Precession data is "more kinematic"

Separate nanocrystals

Properties of nanocrystals distinguish from

properties of macroobjects

The precession diffraction with the method of

the quantitative reconstruction and analysis by

precise EDSA can give answer on the question:

what properties (structural and physical) are on

the nanoparticles

Program «ASTRA» for the structure calculations on the PED data

Dudka A.P., Avilov A.S., Nicolopoulos S. «Hollow-cone-

program crystal structure refinement using Bloch-

wave method for precession electron diffraction».

Ultramicroscopy. 107 (2007), 474.

Great thanks for the attention!