Precession technique and electron diffractometry as new tools for crystal structure analysis and chemical bonding determination

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Ultramicroscopy 107 (2007) 431–444 www.elsevier.com/locate/ultramic

Precession technique and electron diffractometry as new tools for crystal structure analysis and chemical bonding determination A. Avilova,, K. Kuligina, S. Nicolopoulosb,c, M. Nickolskiya,c, K. Boulahyad, J. Portilloc,e, G. Lepeshova, B. Soboleva, J.P. Collettef, N. Martinf, A.C. Robinsg, P. Fischioneg a

Institute of Crystallography of Russian Academy of Sciences, Leninsky prosp. 59, Moscow 119333, Russian Federation b Universidad Politecnica de Valencia, Anenida de los Naranjos s/n 46022, Valencia, Spain c Nanomegas sprl, Blvd Edmond Machtens 79, B-1080 Brussels, Belgium1 d Universidad Complutense de Madrid, Dep. de Quimica Inorga´nica 28040, Madrid, Spain e Serveis Cientificotecnics, Universitat de Barcelona c/ Sole´ i Safaris s/n 08028, Spain f Advanced Technologies Smart Sensors, Center Spatiale de Lie`ge, B-4031 Angleur Liege, Belgium g E.A. Fischione Instruments, Inc.9003 Corporate Circle Export, PA 15632, USA Received 18 February 2006; accepted 7 September 2006

Dedicated to the memory of Prof. B. Zvyagin

Abstract We have developed a new fast electron diffractometer working with high dynamic range and linearity for crystal structure determinations. Electron diffraction (ED) patterns can be scanned serially in front of a Faraday cage detector; the total measurement time for several hundred ED reflections can be tens of seconds having high statistical accuracy for all measured intensities (1–2%). This new tool can be installed to any type of TEM without any column modification and is linked to a specially developed electron beam precession ‘‘Spinning Star’’ system. Precession of the electron beam (Vincent–Midgley technique) reduces dynamical effects allowing also use of accurate intensities for crystal structure analysis. We describe the technical characteristics of this new tool together with the first experimental results. Accurate measurement of electron diffraction intensities by electron diffractometer opens new possibilities not only for revealing unknown structures, but also for electrostatic potential determination and chemical bonding investigation. As an example, we present detailed atomic bonding information of CaF2 as revealed for the first time by precise electron diffractometry. r 2006 Elsevier B.V. All rights reserved. Keywords: Electron diffractometer; Electron beam precession; Chemical bonding; Electrostatic potential distribution

1. Introduction 1.1. Measuring accurately ED intensities Structure analysis (SA) by single crystal X-ray diffraction is a well-established technique for investigating structures of new compounds, provided that their crystal size has dimensions near to 0.1 mm. However, generally and mostly due to modern nanotechnology needs, lots of Corresponding author. Tel.: +7 495 1351020; fax: +7 495 1351011. 1

E-mail address: [email protected] (A. Avilov). [email protected]

0304-3991/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.ultramic.2006.09.006

newly synthesized materials are of nm scale and are therefore completely out of range for single crystal X-ray SA. As an alternative, powder X-ray diffraction with present refinement techniques, enables several known phases to be identified; however it is hard to refine ab initio unknown new phases, especially in cases where more than one phase coexists in the powder diffraction pattern or they are poorly crystallized. Electron microscopy is an alternative for studying nanocrystals by HREM and electron diffraction. Interpretation of HREM micrographs, however, is not straightforward as the small thicknesses involved require appropriate (e.g. Scherzer) defocus in order to interpret

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image contrast as a direct projection of the structure. In all other cases, specific software may be needed to directly interpret the projected crystal structure from an HREM image. Again, in a medium voltage TEM (120–200 kV) with conventional LaB6 sources, image resolution is usually worse than 1.9 A˚, and is not comparable with the resolution limit attainable in X-ray diffraction (o 0.5 A˚). Alternatively, using electron diffraction as a method to resolve structures of nanocrystals has several advantages: electron beams, even in a conventional TEM, can be as small as several nm, enabling the selection and study of specific individual crystallites; and electron diffraction is much more sensitive than X-ray to the redistribution of valence electrons and to the presence of light elements such as H, O and N in a compound [1]. Also the resolution in a modern electron microscope used for diffraction SA can be extended by more than 0.5 A˚, comparable to resolution achieved using X-ray SA. The location of light atoms and investigation of chemical bonding require kinematical structure factor determination with high precision. Errors of the structure amplitudes may include experimental measurement errors, errors related with inelastic background subtraction, many-beam dynamical effects, and errors related with the accuracy of formula of conversion between intensities and structure amplitudes. Inaccuracy of experimental measurement of reflection intensities and many-beam effects mostly contribute to these errors. The influence of such errors (related with structure amplitudes) on the precise determination of atomic positions can be estimated by means of the following formulas derived by Vainstein [1]. If b is the mean relative error of the structure amplitude (Fhkl/ equivalent to Fhkl in X-ray diffraction) and all experimental errors are reduced to error measuring intensities, then approximately b  DjFhkl j=jFhkl j for structure amplitudes and for intensities b  DjFhkl j2 =2jFhkl j2 . If the threedimensional (3D) structure consists of n identical atoms with Z-atomic number, the error in atomic coordinates of every atom can be expressed by [1]. pffiffiffiffiffiffiffiffiffiffiffiffi Dx(A˚)0.023BbZ0.07 ðn=OÞ, where B is the thermal factor and O is the cell volume. According to that formula, despite a small dependence of Dx on Z, heavy atoms can be determined more accurately in a structure containing identical atoms. A simple estimation shows that a maximum error Dx for organic compounds of 0.05 A˚ for crystal projections coupled with an error in interatomic distances of 0.01 A˚ this corresponds to an error b in measured intensities as high as 20%. The accuracy of the electrostatic potential determination when termination error Fourier series is absent can be written in simple relation through b according to [1] qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Dj error ¼ ð1=OÞ Sjb Fhkl j2 . It is clear that the accuracy of all structural parameters depends directly on the accuracy of intensity reflection determination in the diffraction patterns.

The electrostatic potential in a crystal (revealed by electron diffraction) is the sum of nuclear and electronic potentials; there is less difference between heights of electrostatic potential peaks for different atoms in comparison with heights of electron density peaks (revealed by X-ray diffraction) [1]. An integral of electron density over the atomic volume is equal to Z, while an integral on the potential of the atom—‘‘full potential’’—is proportional to Z1/3. In conclusion, there is weak dependence of electron structure amplitudes on Z, therefore the light atoms contribution in a total scattering of a given structure in comparison with heavy atom contribution, is much more important in electron diffraction than in X-ray diffraction. Hence, the detectability of light atoms in the presence of heavy ones will be easier with any structure resolved with electron diffraction rather than X-ray diffraction. One can estimate the quantitative difference in the detectability of light atoms by comparing electron diffraction and X-ray diffraction using a formula by Vainstein [1] w=w0  ðZ h =Z l Þ0:5 , where Zh and Zl—the atomic numbers of heavy and light atoms, w and w0 —their corresponding ratios for electron and X-ray diffraction. This implies, for example, that assuming high accuracy structure amplitudes determination and with the absence of termination effects in Fourier synthesis, detectability of hydrogen atoms in organic structures is approximately in 2–2.5 times easier using electron diffraction compared to that of X-ray diffraction. Similar analysis indicates that the accuracy of determination of atom coordinates in a structure consisting of light atoms will be approximately 1.5 times better with electron diffraction than in X-ray diffraction.Therefore, it becomes clear that improvement of accuracy of crystal SA depends requires methods for precise determination of electron diffraction intensities. However, resolving atomic structures by precise electron diffraction measurements is not the only application of electron diffraction; in fact this technique can be applied for the precise investigation of chemical bonding and electrostatic potential distribution. Electrostatic potential and charge density are intrinsic crystal properties that determine most of the physical and chemical properties. To detect chemical bonding effects, which mainly contribute to low scattering angles, accuracy of measurement of low order structure factor measurement is very crucial. Indeed, electron scattering factors are much more sensitive to atomic bonding variations than equivalent X-ray scattering factors. For example, in LaMnO3 a change of 1.45% in electron structure factors amplitudes of the strongest reflections (detectable by quantitative convergent beam electron diffraction—QCBED) account for a change of 0.26% or less for equivalent X-ray structure factors, which is hardly detectable [2]. Considerable progress on resolving crystal bonding in atoms has been reported recently with QCBED using an energy filter and a CCD camera [3]. The

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advantage of filtered QCBED is the possibility to precisely measure values of a few low order structure factors. The filtered CBED profiles technique [3] uses electron diffraction data from nanocrystals to accurately determine modulus and phase of structure factors with an accuracy of 1% and 11, respectively. Measurements are made by comparing experimental intensity profiles across CBED discs (rocking curves) with many-beam dynamical calculations. Unfortunately, the QCBED technique is not practical for collecting a large number of reflections, therefore for structure refinements and in order to complete a data set, data from middle and high angle reflections are taken from precise X-ray amplitudes or from corresponding theoretical calculations. For the precise determination of electron diffraction intensities one of the present authors (Avilov [4–6]) recently reported about a scintillator/photomultiplier (PM) based electron diffractometry device (adapted to electron diffraction camera (EDC)) and was able to reveal electrostatic potential and bonding in a number of compounds (LiF, MgO, NaF, Ge) prepared as thin polycrystalline films. This technique has clear advantages over QCBED since all reflections (weak and strong) can be measured at once with the same statistical accuracy for all reflections.

1.2. Comparison of different recording ED techniques In view of the previous discussion relating to the required accuracy of experimental intensities for structure determination by electron diffraction, it is useful to review and compare the existing registration systems for electron diffraction intensities in a modern TEM. The electron diffraction pattern dynamic range can be as high as 105 or more (intensity difference between strongest central beam and weakest relection). In order to register all observed intensities a system having the same dynamic range is a minimum requirement. Photographic film is commonly used in TEM work, but the dynamic range is very limited up to 102 linear range; therefore several time exposures are needed to ensure that the whole dynamic range of electron diffraction intensities are measured. The number of gray levels obtainable by film is about 256 (8 bits). Slow scan charge-coupled device (CCD) cameras have also been used for registration of electron diffraction intensities; the CCD performance can be as high as 12–14 bit dynamic range/4096–16.384 gray levels (although the lowest 2–4 bits are obscured by noise), thus the effective range is 2  103. Imaging plates (IP) can also be used for quantification of ED intensities since their dynamic range is reported to be between 16 and 20 bits or roughly 105–106. They are similar to film, but must be read out using a special film reader [7]. Image plates (IP) have a better performance than a CCD camera at low dose levels due to low dark current and readout performance. At low dose range, a CCD camera is limited by readout noise and dark current; the CCD response can be limited by

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linear noise which can be reduced by averaging over several frames. As both systems (CCD and IP) are used in constant time mode (same time for every measured point of diffraction pattern) have poor statistical accuracy, 415–20% for medium and high-angle (40.8 A˚1) weak electron diffraction reflections. Nevertheless they have been successful for ab initio structure determinations [8–11] from precession electron diffraction data, detecting atomic positions of medium and high Z atoms in structures. Scintillator/PM systems—being much more sensitive than CCD cameras—have been incorporated into early electron diffractometers. They were mainly used in constant time mode and were successful in accurate determination of intensities with about 5% mean statistical accuracy for all reflections. Accurate measurement of different types (ring, texture, mosaic crystals) of electron diffraction patterns were performed by scanning and recording in serial mode individual electron diffraction reflections, bringing them in front of a fixed scintillator with PM [12,13]. These early electron diffractometers (in analogy with single X-ray diffractometers) have been very successful in the past (interfaced with EDCs) revealing detailed structures of nanocrystallites in minerals like 2 M1 muscovite, 1 M celadonite, 1 M phengite [14]. Hydrogen atoms in brucite, lepidocrocite and lizardite structures have also been precisely localized in mineral structures [15–17]. Some later electron diffractometers using scintillators, working in the so-called ‘‘accumulation’’ mode, measured all recorded electron diffraction intensities with high statistical accuracy (1–2%) [4] thus allowing the precise investigation of the chemical bonding and electrostatic potential in crystals [5,6]. The accuracy of these structure determination by electron diffraction resolved structures was comparable to single crystal X-ray refinements. It is now widely accepted [18–20] that the combination of precise measurements of electron diffraction intensities, at limited dynamical scattering affecting texture or polycrystalline patterns, was the reason for the success of structure determination by electron diffractometry measurements. Although electron diffractometry proved to be successful for nanostructure determination, serious limiting factors have prohibited its wider use in crystal determination: (a) Detector ‘‘dead time’’ effect: despite high sensitivity, an important disadvantage for PMs is the non-linear character of signal transition, even effecting results for intensities as low as 1013–1012 A. This is the result of the so-called ‘‘dead time’’ effect for such devices: (b) Poor measurement speed: it generally took 2–3 h to measure up to 1 0 0 reflections; (c) Use of those early electron diffractometers has been limited due to fact that they were adapted to EDC cameras uniquely and they were never interfaced to conventional TEMs. Besides that, the beam size of

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EDC camera ranges from 0.1 to 0.5 mm making it impossible to study individual nanocrystallites. 1.3. Factors affecting electron diffraction intensities— dynamical scattering Crystal SA is based on a measurement of a complete set of experimental kinematical structure amplitudes. However, strong electron interaction with matter (103 times stronger than X-rays) leads to many beam dynamical scattering effects that may dramatically change electron diffraction intensities from their kinematical values. Until recently all electron diffractometry experiments performed with EDC cameras were performed on either polycrystalline or textured samples and were composed from very small microcrystallites. So strong dynamical scattering was absent and mainly kinematical scattering could be applied for structure determination; dynamical scattering has been taken into account by correcting a limited number of strong intensities affected by dynamical scattering using the Blackman correction curve or using Bethe-potentials technique [19]. However, in TEM, where small single nanocrystals are studied, the situation is different, especially when a thicker part of the sample may be examined. In that case dynamical diffraction may be important and must be taken into account. In many cases dynamical scattering can be revealed by appearance of forbidden reflections. However, it must be emphasized that dynamical effects are at the same time very sensitive to very fine structural details such as valence electron redistribution during chemical bond formation, anharmonic character of thermal motion, presence of inversion center, etc. In general, an electron diffraction pattern contains a lot of very useful information about the fine detail of crystal structure and chemical bonding. It now becomes clear that in order to adapt precise electron diffractometry to conventional TEM (not only EDC) it is necessary to reduce dynamical effects for solving traditional structural problems or elaborate methods for taking into account possible dynamical interactions in the investigations of fine structure details as well as atomic structure. As a solution for that problem, the electron precession technique, first developed by Vincent and Midgley [21], is a very promising technique to get intensities of electron diffraction reflections closer to their kinematical values. The real advantage of this technique is the possibility of applying standard direct method phasing techniques (as in X-ray crystallography) to the experimental electron diffraction intensities to reveal the structure of crystals. The Vincent–Midgley technique originally stemmed from studies of HOLZ reflections in CBED patterns, where precession was a logical extension.2 The 2 It should be noted that reflections of the first-, second- and higherorder Laue zones may appear due to the curvature of the Ewald sphere or due to the high divergence of the primary beam (like e.g. in CBED). In the precession technique used here there is a near parallel beam with small

geometry of precession can be seen at Fig. 1 [21] where the beam is rocked in a conical fashion with a wide symmetrical tilt above the specimen. After interaction with the sample, transmitted and diffracted beams are descanned to re-form a point pattern (Fig. 1); scanning and de-scanning of the beam must be exactly compensated in order to have a stationary pattern. Precession electron diffraction patterns have several advantages over conventional electron diffraction patterns: Precession patterns may be indexed like conventional electron diffraction patterns (Fig. 2a,b). With increasing precession angle additional reflections become visible from high angle crystallographic planes. Their intensities are closer to kinematical values due to their large extinction distances and due to electron scattering averaging within Bragg reflection width as a result of the precession integration effects. Even dynamical effects such as Kikuchi lines, having a complex diffuse and inelastic scattering nature, are greatly reduced with precession. Precession is much less sensitive to misorientation effects; in fact with precession the Ewald sphere sweeps through the reciprocal space and can effectively compensate (small) disorientation effects in crystals. With an increasing precession angle many more reflections become visible from both ZOLZ and FOLZ that have more kinematical character; the acquisition of an increased number of mostly kinematical spots permits structure solution by crystallographic direct methods techniques. Again, by comparing periodicities between visible ZOLZ and FOLZ reflections direct determination of the space/ point group might be possible [22]. Initial studies on AlFe, Ti2P [23,24] and more recent studies by several researchers [25–30] showed that using PED intensities to a large variety of samples, such as oxides, minerals and ceramics, electron diffraction intensities can be approximately treated as pseudo-kinematical or near to two-beam dynamical (in some cases Blackman type correction might be needed). The acquisition of an increased number of quasi-kinematical reflections permits structure solution by crystallographic direct method techniques. In this paper, we describe the development and first results of a new generation high-speed electron diffractometer linked with an electron beam precession device that can be interfaced to any TEM and can be useful not only to reveal unknown nanocrystal atomic structures, but also for determining fine atomic details such as electrostatic potential and chemical bonding in crystals.

(footnote continued) divergence, so for every particular beam direction the pattern contains mainly ZOLZ reflections and it is more correct to refer to the rest of observed reflections as 3d reflections according to their position to the Ewald sphere. Due to the full rotation of the beam in precession, the symmetry of the resulting pattern is analogous to a CBED pattern; therefore in the text the terms FOLZ, ZOLZ etc. will be used taking into account the precession geometry.

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Fig. 1. Precession electron diffraction pattern of a Si (1 1 0) crystal without (a) and with (b) applying descan in the electron beam. Precession angle can be calibrated by measuring radius of descan circle (a).

a

b

h

h 100

100

010

010

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Fig. 2. ED pattern of cubic LTA zeolite (Pm3m, a ¼ 12 A˚) at 200 kV before (a) and after applying precession (b).

2. Development and characteristics of new universal precession device ‘‘Spinning Star’’ In the past some beam precession instruments (fitted to Philips EM430 and CM30T microscopes) were described in the literature [21,24] where the first electron diffraction precession experiments were performed. However, we must stress here that in those early instruments although beam scanning was not a problem (at least for those instruments having STEM and working at low precession frequency so30 Hz) descanning of the beam was not correct, so as a result precession electron diffraction pattern was not completely stationary. The reason for this is the poor precision in the digitalization of scan and descan signals by the electron microscope CPU unit. Moreover, the CPU unit cannot control TEM coils simultaneously for precession and descanning. In all those early instruments probe size was approximately 100 nm, mainly due to optical distortions. Recently, Own [26,29] reported design of a precession device fitted in a Hitachi TEM working with parallel illumination mode at a frequency of 60 Hz. The same author has reported the fitting of a more recent version to a JEOL TEM resulting in high-quality precession patterns obtained with a probe of up to 25 nm for very

well aligned TEM experiments. For beam precession, we designed a special universal interface ‘‘Spinning Star’’ with some important built-in characteristics: (a) Precession device can be adapted independently to any TEM. (b) Precession device is designed with a proper interface to work in combination with an electron diffractometer. (c) Beam precession is possible with either parallel or convergent beam while keeping beam size as small as possible. (d) Precession angle can be varied continuously (without need of any further alignment) in order to achieve maximum usable electron diffraction area and best compromise to avoid FOLZ and ZOLZ reflections overlapping with increasing precession angle [21]. (e) During precession the tilted beam precesses, tracing out a circuit of theta (0, 2p); ZOLZ reflections can potentially be recovered as partial scans of (0, p) tilt avoiding problems of FOLZ-ZOLZ overlapping at high precession angles. In order to design the precession spinning star system major technical difficulties had to be addressed:

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Most of the modern TEM are digitally (computer) controlled. They all offer the possibility to write scripting in order to perform any type of task or to control the current of any coils. However, combining the signals for scan and descan in precession mode, scripting is not suitable because of the software reaction speed. In fact, in order to correctly perform precession (where scanning and descanning must be exactly compensated) it is required to control simultaneously 8 different TEM coils. If we were to use scripting for that, relevant commands would be executed line by line (in a sequence) and as a consequence the current of those 8 coils would also be updated sequentially one by one. And as a result, the final diffraction pattern would constantly move back and forth. In Figs. 3a and b, we can see the block diagram of the precession device interfaced to a TEM. In the ‘‘Spinning Star’’ system newly developed by Nanomegas for electron precession all those coils may be controlled by dedicated electronic hardware which provides all necessary signals for beam rotation and descan. The device can dynamically control the beam tilt (at an adjustable angle and frequency) and the TEM imaging coils to perform descan. Alignment of the pivot points is done in ‘‘Spinning Star’’ by controlling amplitude and phase of the current of 8 specific coils. When working in precession mode, fine tuning of Beam tilt and Descan alignment can be done interactively through the user interface. When fine tune alignment is done, the precession and descan process can be controlled at any precession angle via single potentiometer. During operation, full TEM functionality is preserved so routine TEM alignment is not affected in any way. During operation, the user can keep precession beam probe size to a minimum by adjusting the beam alignment knobs. By adjusting the descan alignment knobs, the user can achieve a completely stationary electron diffraction pattern when the beam is precessed. A user interface allows the user to vary the precession angle continuously in order to get maximum usable electron diffraction pattern area; the electron diffraction pattern can then be centered interactively by using the diffraction centering knobs in the front panel of our control unit. Ultimate precession performance is limited by TEM objective lens aberrations. The diameter d of the minimum probe size in conventional microdiffraction is given by the following relation: d ¼ 4C s f2 a, where f stands for the precession semi angle, 2a is the beam convergence angle and Cs is the spherical aberration [21]. Measured minimum probe size at eucentric height position depends upon the TEM lens configuration however values of 50 nm can be easily obtained for 25 mrad precession angle (values less than 25 nm has been achieved with high resolution lens Topcon 200 kV TEM for f ¼ 20 mrad). The maximum precession angle available

depends on the type of the TEM’s objective lens and can be estimated by using maximum dark field tilt angle available on the microscope. It can be as high as 751 depending on the TEM lenses. The precession angle can be calibrated accurately by the user switching off the descan signal. With descan off, the diffraction pattern will turn on a circle and the precession semi-angle can be accurately calibrated (Fig. 1a). Precession frequency can be adjusted from 0.33 to 2.340 Hz (user dependent). ‘‘Spinning star’’ has been interfaced successfully to several TEMs (such as Philips EM4xx, Philips CM, FEI Tecnai, JEOL, Topcon); its function being independent from the presence (or not) of an STEM unit on the TEM. ‘‘Spinning star’’ can be adapted to any type of TEM be it either a digitally controlled microscope or an older analog instrument. When the device is switched off, all connections to the microscope remain unaffected to avoid possible interference with the TEM. The unit is also designed to be connected to an electron diffractometer device, which is described next. 3. Development and characteristics of new electron diffractometry device ‘‘Pleiades’’ for TEM In order to enable for precise electron diffraction measurement for analyzing structure and chemical bonding of nanocrystals, a new electron diffractometer named ‘‘Pleiades’’ was developed with the following characteristics: (a) the device can be interfaced to ‘‘Spinning Star’’ to accurately measure ED intensities in precession or standard SAED mode; (b) the device can also be retrofitted to any TEM type or EDC and can measure any type of patterns (single nanocrystals or powder patterns); (c) high electron diffraction measurement accuracy (up to 1%) is combined with wide dynamic range (min 106) in order to accommodate the dynamic range of any diffraction pattern; (d) maximum accuracy is compatible with speed of measurement (seconds range) of the whole electron diffraction pattern (100–500 reflections) to minimize possible sample beam damage degradation; (e) the device can be interfaced to external CCD cameras, allowing the possibility of setting up experimental parameters through CCD. Is important to note that the diffraction pattern registration process in all previously mentioned devices is a two-stage process: for example, in CCD and scintillator/ PM devices electron pulses are transformed to light quanta and then converted to current. During such conversions energy losses are inevitable, as both processes have transformation coefficients less than unity. Again, this conversion process cannot be linear. In order facilitate a high accuracy solution in both dynamic range and linearity

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a

Rear panel

Front panel

Transmission Electron Microscope

Spinningstar

Beam Deflection coils (4x)

Amplitude and phase control Analog Power Regulator From +/-15 Volts To +/-12 Volts

Analog Power Suppl y

Beam Alignment

Descan Alignment

Precession Angle

Sin Oscillator from 0.33 Hertz to 2340 Hertz

Output signal To control Beam deflection coils and Image deflection coils

Cos

220 Volts

Image Deflectio Coils (4x)

Input signal From Pleiades

b TEM

Gun

Software: NanoDrive Record diffraction Image from FC

Scan generator setting

C1

Analysis of the diffraction image

C2 B D

Personal Computer Obj

Specimen I D

To Beam deflection coils (BD) for precession

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Electronic rack of microscope To image deflection coil (ID) for DESCAN and to scan diffraction pattern

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Precession mode+ Universal interface compatible to any Transmission Electron Microscope

Retractable Faraday Cage

X and Y Scan signal

NanoMax2 Controler:

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CPU controlled interface to manage high precision high dynamic high speed Preamplifier to measure intensity of reflection spots + Scan generator

FC

Fig. 3. (a) Front and back face of the precession interface and block schematics. (b) Schematics of the new electron diffractometer and precession interface for accurate measurement of ED intensities and (b) electron diffractometer control unit, precession unit interfaced to an EM 400 microscope.

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the scintillator and related PM were replaced with specially designed Faraday cage (FC). This gives the advantage of being linear in all dynamic ranges and free from previously indicated shortcomings related with PM. FCs have already been used since 1960–1970s for ED patterns registration. It has been shown that an FC has high accuracy and an extremely wide dynamic range; experimental dynamic ranges may vary from 1018 to 1011–1012 A˚. However, FC current amplifiers that were used in the past were so slow that ED measurement time took several hours. In the present equipment a new approach was applied whereby measurement for strong intensities can be reduced to microseconds and as a result the whole diffraction pattern can be measured within tens of seconds. FCs are used in our new system as a current integrator; the electron diffractometer utilizes the integrant converter ‘‘current-time’’ principle. The device consists of an FC, a dual level electronic comparator, a timer and a scanning system. The principle of measurement is as follows: before measurement the FC is positively charged; during measurement the positive FC potential is decreased by the electron diffraction beam current at speed directly proportional to the beam current. It is important, before measurement takes place, that the operator establishes the difference between potential values at the beginning and end of the cycle. When the potential value is inside thresholds of the comparator a timer is switched on. A second comparator switches off the timer when the FC potential becomes smaller than the lower threshold (end of measurement). Operation speed of diffractometer is determined by the setting up of the intervals between thresholds of the electronic comparator. Measurement time is inversely proportional to the ED pattern intensity. A specially designed scanning system brings electron beam reflection spots to FC. The scanning system operates in an area of size 4096  4096 points. Maximum positioning time for a reflection spot is 5 ms, and a typical value is 1 ms. The system has the following advantages: 1. FC can act as a perfectly linear detector from extremely small currents (extremely weak intensities) up to very high currents (strong diffraction intensities). 2. FC has a huge dynamic range (more than 106); the lower level of the dynamic range is limited by the input operational amplifier of the electrometer, its current stability and its level is estimated in several fA. As far as high current is concerned, it is possible for FC to measure even a high intensity primary beam. 3. Since the diffractometer works in an accumulation mode, all statistical accuracy is constant for any intensity dynamic range and can be 1–2%. Mean square error can be estimated as s ¼ 1/ON , where N is the number of counts, which is true for Poisson distribution of counts with time. 4. The device speed is very fast, e.g. for a current of 1 pA measurement takes approximately 50 ms (with a com-

parator window set to 100 mV). Decreasing the comparator window can reduce measurement time further but measurement accuracy would be sacrificed. In order to increase our system flexibility the FC has been coupled it to a specially designed electrometer developed in collaboration with Spatial Research center Liege, Belgium (CSL). The electrometer is working in a fixed time counting mode; development of such specific devices is common for space applications and astronomy where extremely weak and strong currents must be accurately measured within a very high dynamic range. The electrometer has an input leakage current of 25 fA. The analog section of the electrometer is capable of amplifying the current of the FC from 0.05 to 5000 pA in one single scale. The digital section of the electrometer is able to convert this signal into 24 bits (or 16.000.000 gray levels) using an analog to digital converter (ADC) in one single step. The digital part of the electrometer can be connected to a PC via a USB port and controlled by software loaded into the CPU. A processor inside the electrometer controls the digital section. This electron diffractometer device can measure electron diffraction patterns over a vast dynamic range accommodating a wide range of intensities (from strong beams to the weakest reflection in FOLZ/SOLZ). The scanning unit brings all reflections sequentially in front of the FC to be measured with high precision in two different modes and speed: 1. Fast mode: This mode allows a fixed analog to digital conversion time of 1 ms (1000 conversions/s (maximum speed)). Fixed total scan time will depend strictly on the number of pixels of the electron diffraction pattern. As an example, the scan time for 1000 electron diffraction reflection spots (having same 50 pixel each including a number of pixels for background calculation) results in data acquisition time for 1000  50 pixels of 50 s. 2. High precision mode: This mode permits the same measurement accuracy for all individual reflections regardless of their intensity. To perform measurement, the necessary average number of repetitive and cumulative measurements is automatically calculated from the measured current of the FC. As a result, for high beam individual intensities, only a small number of repetitive/cumulative measurements will be applied, whereas for low beam intensities, a higher average number of measurements will be applied. Total scan time, therefore, varies according to the desired measurement precision on all intensities and the brightness of the diffraction pattern. It is important to stress that measurement time in both modes is combined with a number of complete precession periods, in order that measured integrated intensity can be sampled adequately. That is for example, for the fastest conversion rate of 103 s. and a precession frequency of

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2000 Hz, reflections scanned in constant time mode are sampled twice during two precession periods. In Fig. 3b, we can observe the whole TEM precession electron diffractometer interface. 4. Setting up scanning parameters/interface CCD camera In order to scan the electron diffraction pattern to measure all observed intensities, firstly all individual reflection positions must be accurately known in order that the scanning diffractometer unit may accurately scan reflections through the FC. An initial spiral scan can be started until two nearest to origin (0 0 0) reflections (equivalent to the smallest nonlinear vectors in reciprocal space) can be found and indexed by the user interactively through the user interface. The rest of the observed reflections can be indexed subsequently by simple geometrical considerations in the reciprocal space. Once accurate positions of all reflections have been calculated by the software, the user may fix the scanning step which determines the pixel size of the measured areas intensity. It is clear that better accuracy can be achieved when a finer grid of pixels is defined for intensity measurement, but at the expense of whole measurement time. It is also possible to scan the whole diffraction pattern area (e.g. for more accurate background calculation, see Fig. 4). We can observe in that figure that the intensities measured accurately (including the central spot) really reflect the huge dynamical range in a typical ED pattern. Another way for setting prescan is to use a CCD camera mounted on a TEM column. The electron diffraction pattern image can be read by the electron diffractometer software and indexing of reflections can be done automatically through the user interface. Again, the scanning step, which determines pixel size of measured areas intensity, can be determined by the user in the same way as before. However, it is important to stress that the diffractometer software allows the user to scan/measure 134.502

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either individual or a pre-selected number of reflections or specific areas of a reciprocal electron diffraction pattern, dependent on the symmetry of the diffraction pattern. Again, it is important to note that scanning of spots is realised in a spiral way, beginning from (0 0 0) central reflections. This is important, because working in accumulation mode, the scanning unit acts much more rapidly with intense structure determining reflections that usually are in a region of o0.5 A˚1; the rest of high resolution reflections are generally weaker, and consequently the unit spends more time to scan them with the required precision. Scanning is then progressed in spiral mode through circular areas of increased resolution. At the end of scanning revisiting reflections is always possible to check possible intensity variations due to specimen degradation or change orientation effects. 5. Mechanical TEM/EDC interface The sensitive part of the device can be incorporated into a special interface designed by the Fischione Corporation (see Fig. 5) which is a high vacuum outgas free design that is compatible with all major TEM columns through the 35 mm camera port. The device is compatible with other possible detectors using the same port such as an HAADF detector. A retraction mechanism has been designed that is operated by a pneumatic system and fitted to the TEM column opposite to the HAADF position. The design includes a safety switch that allows safe removal of either detector prior to insertion of other. 6. Using electron diffractometer for investigation of bonding and electrostatic potential in fluorite The first results with the new version of the electron diffractometer installed on a Philips EM 400 TEM working at 100 kV have already been published [30]. In this work, because of the high accuracy of the measured intensities, the device was able to detect Li atoms in spinel compounds. Furthermore, using an electron diffractometer to determine the nature of chemical bonding in many compounds, can give impressive results establishing this technique as an

7.877.388

356.086

146.026

Fig. 4. Whole ED pattern scanned by electron diffractometer for BaCoO3; hexagonal crystal along b-axis; measured intensities of selected reflections (arbitrary values ) at 24 bit resolution are shown. The central (0 0 0) beam is not saturated.

Fig. 5. Mechanical interface for electron diffractometer adaptable to TEM 35 mm port, designed by Fischione Inc. and compatible with HAADF imaging.

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alternative to filtered QCBED. On the other hand, as an electron diffraction refinement study combined with ‘‘Spinning Star’’ precession system has already been performed to determine bonding effects of Si single crystal in a 100 kV TEM (results and corresponding article are to be published in the same special issue of Ultramicroscopy) we decided to carry out precise diffraction experiments in polycrystalline samples (CaF2) without using precession to further evaluate electron diffractometry technique for bonding studies. We have tested our electron diffractometry system investigating chemical bonds of powder CaF2 sample (sp.gr. Fm3m, a ¼ 5.463 A˚) [31]. In our experiments we used a previously described electron diffractometer coupled to an EDC working at 75 kV; thin polycrystalline films were prepared by ultrasound disintegration of industrial grown single CaF2 crystals. Mean crystal sizes were estimated to be about 150–200 A˚. To avoid a possible preferential orientation of crystallites (textured patterns), which can bring systematic errors to the transition from the intensities to the structure amplitudes, a special test of samples was devised. For this test, intensity measurements were produced from the sample inclined at 30–501 in three different directions (approximately at 50–601 angular interval). From the test results, samples selected for investigations did not give a deviations of the intensities for identical reflections for all directions more than 0.5–1%. Intensities of the diffraction rings were azimuthally integrated. Intensities of 23 rings (64 independent reflections) were measured to the value of sin y/lE1.06 A˚1 with statistical accuracy less then 1%. Scaling of intensities and refinement of isotropic thermal parameter B were performed for independent reflections having sin y/l40.75 A˚1 with ASTRA software [32]. High angle reflections were used for thermal factor B refinement as they are much more sensitive to atomic positions and thermal motion. Relativistic atomic scattering factors from [33] were used. Final refined values for isotropic thermal parameters were B(Ca) ¼ 0.615(23) A˚2, B(F) ¼ 0.788 (48) A˚2, at this a value of the reliability factor R on the structure amplitudes was 2.47%. Values of B are in good agreement with results of precise synchrotron experiments [34]. Analysis shows that intensities of most reflections are well described within kinematical theory. At the same time, several reflections (1 1 1, 4 0 0, 4 2 2, 4 4 0, 6 2 0 and 4 4 4) show deviations from the kinematical values of intensities, caused by the dynamical scattering effects, mainly of two-beam nature. Using the technique described in [19], Blackman (two beam) correction was introduced into reflections 1 1 1, 4 0 0, 4 2 2 and 6 2 0 at calculated mean crystal thickness 180 A˚. Intensities of 4 4 0 and 4 4 4 reflections, even after this twobeam correction, shows lower intensity values in comparison with kinematical theory; it is our opinion that this was connected to dynamical effects of a more complex nature; therefore these two reflections have been excluded from the following refinement of the structural model.

Information about chemical bonding is contained in all reflections; however ,small angle reflections are much more sensitive to electron occupancies, medium angle reflections are more sensitive to compressibility kappa parameter and high angle reflections are more sensitive to atomic positions. CaF2 is a typical ionic crystal, so the structural kappa-model of Hansen and Coppens [35] has been used for refinement of ion states in this crystal. The electron structure amplitude Fa of each atom ‘‘a’’ was expressed as the sum of the fixed component fcore,a, connected with the atomic core and the variable valence component Pval,a fval,a, which took into account the charge transfer between the atoms and the expansion–compression of an ion in the crystal. Thus, a full structural amplitude for the electron diffraction is written as [5] FðsÞ ¼ ðp O jsj2 Þ1 Sa fZ a  ½f core;a ðsÞ þ Pval;a f val;a ðs=ka Þ expð2pisra  Ba s2 g. Here s is the reciprocal—lattice vector, fcore,a (s) and fval,a (s/ka) are the X-ray atomic scattering factors for the spherically averaged atomic (ionic) core and densities of valence electrons per electron, ka is the atomic parameter responsible for the compression–extension, and Pval,a are the occupancies of the ions. The Hartree–Fock wave functions of the valence and the core electrons of neutral atoms have been used [34]. The results of the refinement of the k-model with the final reliability factors R are presented in Table 1. The calculated value of charge transfer (0.95e per F atom) corresponds to the preferentially ionic character of the bond. Charge transfer is accompanied with a 6% radius increase for F atom and a 10% decrease of Ca atom radius. Therefore, values for k(Ca) is interpreted as 10% cation compression and k(F) is interpreted as 6% anion decompression. Results of our electron diffraction refinement are confirmed by using fluorite static structure factors which were ab initio calculated [36] by linear combination of atomic orbital Hartree–Fock method as implemented in the CRYSTAL software package (bottom line of Table 1). Further multipole refinement has been done by using software package MOLDOS96 [38]. Refined parameters of the multipole model were used for the electrostatic potential calculation and Laplacian of electron density. Laplacian of electron density calculation has been refined with software package MOLPROP [37,39] using electron wave functions in direct space assuming that inner electron atomic shells stay without changes and all deformation is taken in account for outer electronic shells (4 s for Ca and

Table 1 Refinement results of the ionic crystal structural k model for CaF2 B (Ca) (A˚2)

B(F) (A˚2)

k (Ca)

k (F)

Pv (Ca)

R (%)

0.615 (23) —

0.788 (48) —

1.106 (15) 1.106

0.937 (15) 0.939

0.104 (22) 0.092

3.2 0.51

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2 s, 2p for F). Such approach has the advantage in comparison with Fourier type refinement to avoid errors related with termination error Fourier series effect during refinement using experimental intensities [5]. A topological analysis of the latter has also been made. In terms of Bader topological analysis [40] critical points (CPs) are the points where the first derivative on all three potential coordinates is equal to 0: rj(rcp) ¼ 0. The value of the second derivative determines the type of CP: corresponding to one-dimensional (1D) and twodimensional (2D) minima which are denoted as (3,1) and (3,+1), 3D maximum and minimum are denoted as (3,3) and (3,+3). Here, 3 is the number of a non-singular, nonzero eigenvalue li of the Hessian matrix for the ESP, and the second number in brackets is the sum of algebraic signs of l1, l2 and l3. It is interesting to describe some peculiarities of electrostatic potential distribution in the FCC lattice plane (Fig. 6). As can be seen, electrostatic potential along cation–cation line is quite smooth with axial 1D minimum. This minimum indicates transfer of electron charge in the crystal, in particular acquisition of redundant electron charge by anions. Similar minima can be observed in the center of anion–anion line along [0 0 1] direction. In both cases, the shape of the minima are characterized by negative curvature in two directions, normal to the bond, and a positive curvature in one direction along the bond. This minimum turns into two 1D maxima (or 2D minima) nearer to anions (Fig. 6). Note that analogous electrostatic potential distribution along anion–anion lines was observed in two-component ionic crystals LiF, NaF, MgO [5]. These maxima reflect the acquisition of excess electronic

Fig. 6. Fragment of electrostatic potential map distribution along (1 1 0) plane in e/A˚. CPs are denoted as: circle—(3,1), triangle—(3,+1), square—(3,+3). Negative lines—drawn as dashed lines, positive lines— as solid lines, zero potential line—shown as dot–dashed line. Positive interval contour values are: (2, 4, 8)  10n e/A˚, 1pnp2. Negative line correspond to the potential value 0.65 e/A˚.

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charge by F atoms in a crystal, and their existence is a consequence of the specific r-dependence of the electrostatic potential of negatively charged atoms. Analysis shows that the more the negative charge of any single ion increases, the more the corresponding negative potential minimum value is lowered; the position of this maximum is slightly shifted to the nucleus. Global minima in the same (1 1 0) plane in the cube center (and in the center of cube edge) are observed along ‘‘non-connecting’’ cation–cation lines. All three CPs are in fields of negative values of electrostatic potential field. Existence of those areas originates from special anion potential dependence from distance to the nucleus. Analysis shows that the more positive is the charge of any isolated cation and the greater is the deviation parameter of spherical deformation k from 1, the less is the negative minimum. The electron density map r(r) (Fig. 7) reveals charge redistribution during crystal formation. As can be observed, there is a big area with low electron density value and global minimum (3,+3) in the center of octahedron between F and Ca atoms. In particular this can be explained by the possibility of additional F atoms during formation of non-stoichiometric fluorides with rare-earth elements. Comparison of electrostatic potential (Fig. 6) and electron density (Fig. 7) reveals that CP positions do not match. Nucleus potential distribution also gives a CPs map different from electron diffraction map. This illustrates the well-known fact that electron density (and energy) of many electron system is determined by factors other than just electrostatic fields inside the crystal. The distribution of the Laplacian of the electron density r2r(r) along (1 1 0) plane of CaF2 is shown in Fig. 8. One can see the distribution of inner cation and anion electron shells. The specific form of ion polarization can also be observed; the outer electron F shell is strongly polarized to the nearest cations. This seems to be common feature for all compounds with ionic bonds.

Fig. 7. Electron density map r(r) in fluorite (1 1 0) plane. Main interval contour value: (2, 4, 8)  10n e/A˚3, 2pnp2.

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Fig. 8. Laplacian electron density distribution r2r (r) of fluorite along (1 1 0) plane. Electron shells of Ca and F atoms can be clearly seen.

Electrostatic potential determines electrostatic field E(r) ¼ rj(r) inside the crystal, which originates from the sum of all nuclei and electrons fields. Electrostatic potential also determines the value of classic electrostatic (single electron) Coulomb force in the point r. Since j(rcp) ¼ 0 CPs of electrostatic potential are the points where electric field and electrostatic force, acting on the charge at point rcp, are equal to 0. In the nuclear positions this agrees with the demands of Hellman–Feynman theorem for a system being in equilibrium. There are some points inside internucleus space where the Coulomb force acting on the electron density is equal to 0 as well. Using the equation of electrostatic energy w(r) ¼ (1/8p)|E(r)|2 we can say that w(r) at the CPs of the electrostatic potential is also equal to 0. Lines of pairs of electric field gradients terminated at CP (3,1) in the electrostatic potential correspond to the shortest cation– cation and anion–anion distances in the crystals with structure of CaF2. Since the electric field and interacting charges have oneto-one correspondence, these lines can be treated as images of electrostatic atomic interactions. These observations reveal a defect of the model for pair Coulomb interaction among all point ions, a model which is widely used in the calculations of electrostatic energy of crystals: the lines of gradient connecting anions in rj(r) field are not observed. Thus, taking into account gradient field observations we

can conclude that the finite size long-distance Coulomb atomic interactions take place in crystals in the form of atom–atom interaction, which is unique for each structure type.

7. Discussion Ten years after the development of beam precession and initial experiments on a limited number of compounds [21] preliminary theoretical calculations showed that PED intensities could be treated as pseudo-kinematical, at least for reasonable medium thickness (o500 A˚). Inspired by those promising results for electron crystallography, very recent work on a number of known compounds based on PED intensities has been undertaken by several research groups to check further the validity of those initial conclusions (most of this recent work is published in this issue). From early times of electron diffraction SA Russian school investigated atomic structure of thin polycrystalline film where randomly oriented nanosized crystals in powder, texture or mosaic crystals were examined in the EDC. It is evident that in such samples many-beam dynamical effects, which are always present to a greater or lesser extent, are averaged in different directions where different crystals have a wide range of thickness.

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Since electron beam size in EDC can be very large (up to half mm size) lots of crystals take part in a scattering and formation of diffraction pattern. A large part of microcrystallites found in arbitrary orientations can give both near and far angle reflections with large extinction distances. The resulting pattern (of polycrystalline and textured type) contains a large number of reflections which is enough for quantitative SA. It is important to note that, in such a case the intensity of reflections could be treated as pseudo-kinematical and in the most general case it is enough to apply two-beam Blackman correction or manybeam Bethe correction (Bethe potentials). In the case of strong dynamic diffraction, averaging of intensities in polycrystalline samples leads to loss of detailed structure information. An elaboration of the strict dynamical theory for polycrystalline films is a very complex problem. For weak dynamical effects (small and thin crystals, low Z numbers, large cells) it is possible to study not only the structure but chemical bonds and distribution of electrostatic potential by means of very precise experiments [5,6]. Precession method is similar to diffraction of a polycrystalline sample which is performed with large beams in EDC. The primary beam falls on a crystal with an angle in relation to the crystallographic axis resulting in excitation of high angle reflections. Since in precession the beam runs along a hollow-cone, reflection intensities are summed without account for their phases, and as a result reflection intensity in precession diffraction pattern represents integral intensities. The main difference between TEM precession diffraction and EDC diffraction is that precession diffraction is applied to single individual nanocrystals. That means that strict many beam theory (Bloch wave method) can be used directly for the description of the precession intensities [41]. In this work we have presented, the recently developed high precision electron diffractometer which allows not only to study atomic structures but also to refine fine structural details like light atoms position, thermal motion anharmonicity, chemical bonding and to reveal fine details of electrostatic potential distribution. The quality of results is comparable with filtered QCBED and EELS techniques [42]. It is important to underline that the electron diffractometer is not like a 2D detector such as the CCD or IP where parallel detection is applied. In the case of the electron diffractometer, the diffraction pattern is scanned serially; however, although 2D flat detectors are faster as they use parallel detection they are less precise in the estimation of middle and high angle reflections intensities which contain very important information on the atomic structure and structural bonding effects. The electron diffractometer can work in accumulation mode and reveals the same statistical accuracy for the total dynamic range of observed electron diffraction reflection intensities in a fraction of minute. The main point is that only with accuracy of measurement can meaningful information be obtained. Therefore, accurate electron diffractometry

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instrumentation is a key not only for routine structure determinations but also for future investigations related to the nature of the crystal bonding and electrostatic potential which determines most physical properties of crystals. Acknowledgments Financial support through RFBR project No. 04-0216241 and No. 05-0565242 are acknowledged. We would like to thank Prof. DelPlancque and Dr. Jean Dille for support in our measurements in ULB and Mr. Roger D’Haens for technical assistance during our experiments. References [1] B. Vainstein, Structural Analysis by Electron Diffraction, Pergamon, Oxford, 1964. [2] B. Jiang, J.M. Zuo, Q. Chen, J.C.H. Spence, Acta Crystallogr. A 58 (2002) 4. [3] J.C.H. Spence, J.M. Zuo, Electron Microdiffraction, Plenum Press, New York, 1992. [4] A.S. Avilov, A.K. Kuligin, U. Pietch, J.C.H. Spence, V.G. Tsirelson, J.M. Zuo, J. Appl. Crystallogr. 32 (1999) 1033. [5] V.G. Tsirelson, A.S. Avilov, G.G. Lepeshov, A.K. Kuligin, J. Stahn, U. Pietsch, J.C.H. Spence, J. Phys. Chem. B 105 (2001) 5068. [6] A. Avilov, G. Lepeshov, U. Pietsch, V. Tsirelson, J. Phys. Chem. Solids 62 (2001) 2135. [7] Ditabis application notes (www.ditabis.de). [8] J. Gjonnes, V. Hansen, A. Krerneland, Microscopy Microanalysis 10 (2004) 16. [9] T.E. Weirich, J. Portillo, G. Cox, H. Hibst, S. Nicolopoulos, Ultramicroscopy 106 (2006) 164. [10] M. Gemmi, S. Nicolopoulos, Ultramicroscopy, to be published in this issue. [11] M.S. Nickolskiy, Ph.D. Dissertation, Institute of Mineralogy and Ore Deposits IGEM RAS, Moscow, 2006. [12] A.S. Avilov, Sov. Phys. Crystallogr. 21 (1976) 646. [13] A.S. Avilov, Sov. Phys. Crystallogr. 24 (1979) 103. [14] V. Drits, Electron Diffraction and High Resolution Electron Microscopy of Mineral Structures, Springer, Berlin, 1987. [15] M.S. Nickolsky, S. Nicolopoulos, A.P. Zhuklistov, B.B. Zvyagin, R. Ochs, Acta Crystallogr. A 58 (2002) C173. [16] A.P. Zhuklistov, B.B. Zvyagin, Crystallogr. Rep. 43 (6) (1998) 950. [17] A.P. Zhuklistov, Crystallogr. Rep. 42 (5) (1997) 841. [18] A.S. Avilov, Z. Kristallogr. 218 (2003) 247. [19] B.K. Vainshtein, B. Zvyagin, A.S. Avilov, Electron Diffraction Techniques, Oxford University Press, Oxford, vol. 1, 1992, pp. 216–312 (Chapter 6). [20] D.L. Dorset, Structural Electron Crystallography, Plenum Press, New York, 1995. [21] R. Vincent, P.A. Midgley, Ultramicroscopy 53 (1994) 271. [22] J.P. Morniroli, A. Redjaimia, S. Nicolopoulos, Ultramicroscopy, to be published in this issue. [23] J. Gjonnes, V. Hansen, A. Krerneland, Microscopy Microanalysis 10 (2004) 16. [24] M. Gemmi, X. Zou, S. Hovmoller, A. Migliori, M. Vennstrom, Y. Andersson, Acta Crystallogr. A 59 (2003) 117. [25] C.S. Own, A.K. Subramanian, L.D. Marks, Microscopy Microanalysis 10 (2004) 96. [26] C. Own, Ph.D. Dissertation, Northwestern University Evanston Illinois, 2005 /http://www.numis.northwestern.edu/Research/Current/ precessionS. [27] M. Gemmi, S. Nicolopoulos, Ultramicroscopy, to be published in this issue.

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[28] D. Dorset, C. Gilmore, J.L. Jorda, S. Nicolopoulos, Ultramicroscopy, to be published in this issue. [29] C.S. Own, L.D. Marks, W. Sinklair, Rev. Sci. Instrum. 76 (2005) 033703. [30] S. Nicolopoulos, A.S. Kuligin, K. Kuligin, K. Boulahya, G. Lepeshov, J.L. DelPlancke, A.S. Avilov, M. Nickolskiy, A. Ponce Electron Crystallography NATO Science Series II, vol. 211, Springer, Berlin, 2006. [31] G.G. Lepeshov, A.O. Larina, A.P. Dudka, A.S. Avilov, B.P. Sobolev, Fifth National Conference on Applying X-rays, Synchrotron Irradiations, Neutrons and Electrons for the Investigation of Nanomaterials and Nanosystems, Abstracts RSNE-NANO 2005 Moscow, 2005, p. 229. [32] A.P. Dudka, Crystallogr. Rep. 47 (2002) 152. [33] Z. Su, P. Coppens, Acta Crystallogr. A 53 (1997) 749.

[34] R. Bachmann, H. Kohler, H. Schulz, H.-P. Weber, Acta Crystallogr. A 41 (1985) 35. [35] N.K. Hansen, P. Coppens, Acta Crystallogr. A 34 (1978) 909. [36] A. Lichanot, M. Rarat, Acta Crystallogr. A 51 (1995) 323. [37] Z. Su, P. Coppens, Acta Crystallogr. A 48 (1992) 188. [38] Protas J., MOLDOS96/MOLLY PC-DOS, 1997, private communication. [39] Z. Su, P. Coppens, J. Appl. Crystallogr. 27 (1994) 89. [40] R.F.W. Bader, Atoms in Molecules A Quantum Theory, Oxford University Press, Oxford, 1990. [41] A. Dudka, A. Avilov, S. Nicolopoulos, Ultramicroscopy, to be published in this issue. [42] Y. Zhu, Measurements of valence electron distribution using QCBED Elcryst2005 electron crystallography School Brussels, September 2005.