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Electrostatic potential determined from electron diffraction data
Anatoly Avilov
Shubnikov Institute of Crystallography of Russian academy of sciences, Moscow
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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
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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.
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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
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Electron diffraction structure analysis (EDSA)
What is it?
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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)
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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.
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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
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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 .
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Electron diffraction camera EMR-110К
1. Electron gun2. Condensors3. Crystal holder4. Camera5. Optical microscope6. Tubus7. Photochamber
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Transmission diffraction
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Effect of mosaicity on the formation DP
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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
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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
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Kinematical dynamical
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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
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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
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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»
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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
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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
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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»
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Scheme of electron diffractometer
Accumulation mode – statistical acc. ~ 1-2%
Statistical treatment, quasimonitoring – improvement of accuracy
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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
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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
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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
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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
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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
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So the analytical methods of
the reconstruction are needed.
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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
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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
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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.
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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).
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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.
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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
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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
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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)
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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
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“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
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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
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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
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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
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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
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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 -
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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
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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
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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
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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…
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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
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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
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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
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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...
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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
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The developed methods can be
used for other ED methods
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Different schemes of electron diffraction in TEM
SAED MBED HRED HDED CBED SMBED
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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)
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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)
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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
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Why the precession method is good?
Three-dimensional set of experimental data
Intensities of reflexions are angle-integrated.
Precession data is "more kinematic"
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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
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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.
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Great thanks for the attention!