Non-systematic Three-beam Effects in Dynamical Electron Diffraction and Their Use in Determination of Amplitude and Phase of Structure Factors

The treatment of non-systematic multiple-beam effects in dynamical diffraction is extended. Expressions for Bloch wave degeneracies are given in the centrosymmetrical four-beam case and for some symmetrical directions. These degeneracies can be determined experimentally either as critical voltages or by locating the exact diffraction condition at a fixed voltage. The accuracy when applied to structure factor determination is comparable with the systematical critical voltage, namely 1% in UfT The three-beam case 0, g, h is treated as well in the non-centrosymmetrical case, where it can be used for determination of phases. It is shown that the contrast features can be represented .by an effective structure factor defined by the gap at the dispersion surface. From the variation in the gap with diffraction condition, a method to determine the three-phase structure invariant I\J = 9 + _ h + h _ 9 is given. The method is based upon the contrast asymmetry in the weaker diffracted beam and can be applied in Kikuchi, convergent beam or channelling patterns. Calculations relating to channelling in backscattering are also presented.

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Three-phase structure invariants and structure factors determined with the quantitative convergent-beam electron diffraction method

Acta Crystallographica Section A Foundations of Crystallography ◽

10.1107/s0108767398008903 ◽

1999 ◽

Vol 55 (2) ◽

pp. 188-196 ◽

Cited By ~ 1

Author(s):

R. Høier ◽

C. R. Birkeland ◽

R. Holmestad ◽

K Marthinsen

Keyword(s):

Electron Diffraction ◽

Phase Structure ◽

Diffraction Method ◽

Convergent Beam Electron Diffraction ◽

Beam Electron ◽

Structure Factors ◽

Refinement Method ◽

Three Phase ◽

Phase Sensitive ◽

Convergent Beam

Quantitative convergent-beam electron diffraction is used to determine structure factors and three-phase structure invariants. The refinements are based on centre-disc intensities only. An algorithm for parameter-sensitive pixel sampling of experimental intensities is implemented in the refinement procedure to increase sensitivity and computer speed. Typical three-beam effects are illustrated for the centrosymmetric case. The modified refinement method is applied to determine amplitudes and three-phase structure invariants in noncentrosymmetric InP. The accuracy of the results is shown to depend on the choice of the initial parameters in the refinement. Even unrealistic starting assumptions and incorrect temperature factor lead to stable results for the structure invariant. The examples show that the accuracy varies from 1 to 10° in the electron three-phase invariants determined and from 0.5 to 5% for the amplitudes. Individual phases could not be determined in the present case owing to spatial intensity correlations between phase-sensitive pixels. However, for the three-phase structure invariant, stable solutions were found.

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Structure factor measurement by 3-beam convergent beam electron diffraction

Acta Crystallographica Section A Foundations and Advances ◽

10.1107/s2053273314083764 ◽

2014 ◽

Vol 70 (a1) ◽

pp. C1623-C1623

Author(s):

Yueming Guo ◽

Philip Nakashima ◽

Joanne Etheridge

Keyword(s):

Electron Diffraction ◽

Structure Factor ◽

Convergent Beam Electron Diffraction ◽

Beam Electron ◽

Direct Observations ◽

Structure Factors ◽

Factor Measurement ◽

Relative Structure ◽

Convergent Beam ◽

Approximate Measurement

It has been shown mathematically that both the magnitudes and 3-phase invariants of the structure factors of a centrosymmetric crystal can be expressed explicitly in terms of the distances to specific features in the 3-beam convergent beam electron diffraction (CBED) pattern [1].This theoretical inversion can be implemented experimentally, enabling direct observations of 3-phase invariants and the approximate measurement of structure factor magnitudes. This method then enables a different approach to crystal structure determination, which is based on the observation of phases, rather than the measurement of amplitudes. It has been shown that by inspection of just a few phases using 3-beam CBED patterns, centrosymmetric crystal structures can be determined directly to picometre precision without the need to measure magnitudes [2]. Here, we will explore a different approach for measuring structure factor magnitudes from 3-beam CBED patterns. It has been demonstrated that the relative structure factor magnitudes can be determined directly from the ratio of the intensity distributions along specific lines within the CBED discs [3]. We will investigate the potential of using this approach for the relatively fast measurement of approximate structure factor magnitudes from nano-scale volumes of crystals.

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Determination of structure factors of copper by convergent beam electron diffraction

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100149271 ◽

1993 ◽

Vol 51 ◽

pp. 688-689

Author(s):

John Mansfield ◽

Martin Saunders ◽

George Burgess ◽

David Bird ◽

Nestor Zaluzec

Keyword(s):

Electron Diffraction ◽

National Laboratory ◽

Pure Copper ◽

Argonne National Laboratory ◽

Convergent Beam Electron Diffraction ◽

Beam Electron ◽

Structure Factors ◽

Convergent Beam ◽

Ray Methods

There has been considerable recent interest in the determination of structure factors from convergent-beam electron diffraction (CBED) patterns and the ultimate goal is the ability to retrieve the crystal structure of an unknown crystal by inversion of a CBED pattern. There are a number of different methods that have been used to extract structure factor information. The zone-axis pattern fitting technique of Bird and Saunders has recently been used to obtain structure factors for silicon that compare well with those obtained by X-ray methods. This work extends the techniques to f.c.c. metals, specifically copper.CBED patterns were recorded from [110] zone axes of electropolished foils of pure copper (99.999% purity) in the Philips EM420T at Argonne National Laboratory. The patterns were energy-filtered by scanning the whole pattern across the entrance aperture of a Gatan #607 serial energy loss spectrometer and collecting the zero loss intensity only (energy window ∼5eV).

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On the experimental determination of low-order structure factors in TiAl by energy-filtered convergent-beam electron diffraction

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100149143 ◽

1993 ◽

Vol 51 ◽

pp. 662-663

Author(s):

S. Swaminathan ◽

I. P. Jones ◽

N. J. Zaluzec ◽

D. M. Maher ◽

H. L. Fraser

Keyword(s):

Electron Diffraction ◽

Charge Density ◽

Structure Factor ◽

Convergent Beam Electron Diffraction ◽

Order Structure ◽

Electron Charge Density ◽

Beam Electron ◽

Structure Factors ◽

Convergent Beam ◽

Low Order

It has been claimed that the effective Peierls stresses and mobilities of certain dislocations in TiAl are influenced by the anisotropy of bonding charge densities. This claim is based on the angular variation of electron charge density calculated by theory. It is important to verify the results of these calculations experimentally, and the present paper describes a series of such experiments. A description of the bonding charge density distribution in materials can be obtained by utilizing the charge deformation density (Δρ (r)) defined by(1) where V is the volume of the unit cell, Fobs is the experimentally determined low order structure factor and Fcalc is the structure factor calculated using the Hartree-Fock neutral atom model. To determine the experimental low order structure factors, a technique involving a combination of convergent beam electron diffraction (CBED) and electron energy loss spectroscopy (EELS) has been used.

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Critical Voltage for Non-Centro. Crystals and its Application to the Accurate Measurement of Structure Factor Phases by CBED

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100136258 ◽

1990 ◽

Vol 48 (2) ◽

pp. 530-531

Author(s):

J.C.H. Spence ◽

J.M. Zuo

Keyword(s):

Structure Factor ◽

Dynamical Theory ◽

Critical Voltage ◽

Bragg Condition ◽

Two Phase ◽

Transmission Electron ◽

Structure Factors ◽

Three Phase ◽

Highly Sensitive ◽

Image Position

The application of the three-beam dynamical theory of transmission electron diffraction to centrosymmetric and non-centrosymmetric crystals shows that certain regions of three-beam CBED patterns are highly sensitive to the three-phase structure invariant By applying this method to the systematics geometry, a method has been described which allows structure factor phases to be measured with an accuracy of 0.07°. Using the Bethe potentials, we find a degeneracy in eigenvalues at the critical voltage VA for acentric crystals in the systematics orientation whereHere VA is now seen to depend both on the structure factors Vg (in volts) and on the systematics “two=phase” invariant In this paper we consider the application of this method to BeO (wurtzite structure), with g = (004) and h = (002). Then the CBED intensity is most sensitive to Y near the (004) Bragg condition at 46 kV. But since the theory contains only the product KSg, other voltages and orientations are also sensitive.

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On the determination of enanttomorphs from non-systematic many-beam effects in CBED patterns

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100154391 ◽

1989 ◽

Vol 47 ◽

pp. 484-485

Author(s):

K. Marthinsen ◽

R. Høier

Keyword(s):

Structure Factor ◽

High Accuracy ◽

Convergent Beam Electron Diffraction ◽

Bragg Condition ◽

Sign Problem ◽

Three Phase ◽

Convergent Beam ◽

Diffraction Effects ◽

Beam Diffraction

A convergent beam electron diffraction (CBED) method which makes it possible to determine structure factor magnitudes and phases with high accuracy has recently been suggested. It is based on detailed simulations of non-systematic many-beam diffraction effects in the disks. Basis for the phase determination is an asymmetry which may appear in a line h with respect to the Bragg condition of the coupled reflection g near a three-beam condition. Approximate analytical three-beam solutions show that the sign and size of this asymmetry depends on the structure factor phases Θh of the reflections h involved through a term cos(Φ) where Φ is the three phase structure invariant, Φ = Θh + Θg + Θh-g. The magnitude of the phase invariant is thus in principle available, but not the sign. The aim of the present work has been to discuss the origin of the sign problem and the possibilities of distinguishing +/−Φ.

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Determination of structure factors in electron diffraction: A bloch wave picture

Ultramicroscopy ◽

10.1016/0304-3991(85)90036-1 ◽

1985 ◽

Vol 17 (2) ◽

pp. 171

Author(s):

Jon Gj∅nnes

Keyword(s):

Electron Diffraction ◽

Bloch Wave ◽

Structure Factors

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On the accuracy of structure-factor measurements in GaAs by CBED

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100106181 ◽

1988 ◽

Vol 46 ◽

pp. 824-825

Author(s):

J.M. Zuo ◽

M. O'Keeffe ◽

J.C.H. Spence

Keyword(s):

Structure Factor ◽

Phonon Scattering ◽

Three Dimensional ◽

Bloch Wave ◽

Bragg Angle ◽

Order Structure ◽

Structure Factors ◽

Convergent Beam ◽

Electron Energy Loss ◽

Low Order

By comparing the experimental intensity in convergent-beam electron diffraction (CBED) patterns along the [h,0,0], [h,h,0] and [h,h,h] systematics directions with three-dimensional Bloch-wave calculations, we have refined the low-order structure factor amplitudes of GaAs. (For Si, see) The experimental data were collected using a Philips EM400 electron microscope and a Gatan model 607 electron energy loss spectrometer (EELS) tuned to the elastic peak. By placing the scan coils of the microscope under the control of a PDP11 computer, the CBED patterns could be scanned over the EELS entrance slit. Data were collected at 120kV and -183°C to reduce phonon scattering and contamination. The angular resolution was 0.6% of the (200) Bragg angle. The refinement parameters in the calculations were high voltage (obtained from HOLZ lines), thickness (obtained from outer CBED fringes), absorption potentials (from the asymmetry of the (000) disk) and the low-order structure factors Vg (from inner peaks).

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Measurement of small structure factors by the critical-voltage effect in HEED: The superlattice reflections in β'nial, β'CoAl and γtial

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s042482010013050x ◽

1992 ◽

Vol 50 (2) ◽

pp. 1174-1175

Author(s):

Alan G. Fox ◽

Mark A. Tabbernor

Keyword(s):

Structure Factor ◽

High Energy ◽

Critical Voltage ◽

Intermetallic Alloys ◽

Rocking Curve ◽

X Ray ◽

Structure Factors ◽

Factor Measurement ◽

Small Structure ◽

Convergent Beam

The systematic critical voltage effect, Vc, in high energy electron diffraction has been used for some time to accurately measure low-angle x-ray structure factor structure factor amplitudes (see e.g. 1). It has a significant advantage over other methods for accurate structure factor measurement, such as systematic convergent beam rocking curve or x-ray Pendellösung techniques, in that it is capable of measuring very small structure factors such as the 222 ‘forbidden’ reflections in Si and Ge (see e.g. 2). In the present work the potential of the systematic Vc method for measuring small structure factor amplitudes and average Debye-Waller factors in the intermetallic alloys NiAl, CoAl and TiAl will be demonstrated.The structure factors, F, for ordered stoichiometric B2 alloys comprising A and B atoms such as Ni (Co)Al are given by

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Kinematical structure factors from dynamical diffraction?

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100149210 ◽

1993 ◽

Vol 51 ◽

pp. 676-677

Author(s):

Alwyn Eades

Keyword(s):

Electron Diffraction ◽

Structure Factor ◽

Multiple Diffraction ◽

Full Account ◽

Structure Factors ◽

Simplified Method ◽

Common Observation ◽

Symmetry Relations ◽

Convergent Beam ◽

Beam Diffraction

In convergent-beam diffraction it is a common observation that kinematically forbidden reflections show "dynamical extinctions", also known as Gjonnes-Moodie lines, G-M lines, black crosses or dark bars. These zeros of intensity can be understood as resulting from the pairing of multiple diffraction routes so that each pair cancels. If the multiple diffraction routes for a reflection that is not kinematically forbidden could be paired in the same way, we could locate a position in the convergent-beam disc where the intensity would depend only on the structure factor for that one reflection. This would be extremely valuable because it would provide electron diffraction with a greatly simplified method of solving crystal structures.It turns out that no such condition can be found. Here, an outline of the argument is given. A full account will be given elsewhere.The pairing of the multiple diffraction routes depends on the existence of symmetry relations between different reflections.

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