research papers
Threebeam convergentbeam electron diffraction for measuring crystallographic phases
^{a}Department of Materials Science and Engineering, Monash University, Victoria 3800, Australia, and ^{b}Monash Centre for Electron Microscopy, Monash University, Victoria 3800, Australia
^{*}Correspondence email: joanne.etheridge@monash.edu
In honour of Professor Alexander F. Moodie, 1923–2018.
Under almost all circumstances, electron diffraction patterns contain information about the phases of structure factors, a consequence of the short wavelength of an electron and its strong Coulombic interaction with matter. However, extracting this information remains a challenge and no generic method exists. In this work, a set of simple analytical expressions is derived for the intensity distribution in convergentbeam electron diffraction (CBED) patterns recorded under threebeam conditions. It is shown that these expressions can be used to identify features in threebeam CBED patterns from which threephase invariants can be extracted directly, without any iterative i.e. the uncertainty of the phase measurement is ±22.5°). This approach is demonstrated with the experimental measurement of threephase invariants in two simple test cases: centrosymmetric Si and noncentrosymmetric GaAs. This method may complement existing methods by providing direct measurements of threephase invariants to replace `guessed' invariants in ab initio phasing methods and hence provide more stringent constraints to the structure solution.
processes. The octant, in which the threephase invariant lies, can be determined simply by inspection of the indexed CBED patterns (Keywords: crystallographic phase problem; threephase invariants; convergentbeam electron diffraction; structure determination; enantiomorph ambiguity; nanocrystals; dynamical studies; multiple scattering.
1. Introduction
1.1. Tackling the with electron diffraction
Solving a ); however, only magnitudes can be measured directly from the intensities of diffracted Xrays or neutrons (except under special experimental conditions, for example, Weckert & Hümmer, 1997). Specifically, under kinematic scattering conditions, the intensities are proportional to the modulus squared of the structure factors, so the phase cannot be measured experimentally [unless ptychography is used (Hoppe, 1969)]. This is known as the `phase problem' in crystallography. The loss of phase information can often be circumvented by the application of the (Patterson, 1934) or (Woolfson, 1971; Hauptman, 1991), provided that a sufficient number of structurefactor magnitudes are measured.
requires knowledge of both the magnitude and phase of the structure factors. The phases are stronger constraints to a structure solution than the magnitudes (Ramachandran & Srinivasan, 1970In the case of electron diffraction, the strength of the Coulombic interaction between the incident electrons and the crystal potential, means that the scattering
of electrons is four to five orders of magnitude larger than that of Xrays or neutrons, making dynamical scattering inevitable. Furthermore, the wavelength of the electron is small (the order of a picometre), so that the radius of the is large, resulting in a high probability that the Bragg condition will be satisfied simultaneously for more than one vector. As a consequence, the intensities of diffracted electron waves depend on the phases as well as the magnitudes of the structure factors; however, the relationship is extremely complicated.One approach to ab initio phasing techniques available for kinematic diffraction data, such as (Woolfson, 1971; Hauptman, 1991) and algorithms (Oszlányi & Sütő, 2004), can be applied with some degree of validity, although the structure solutions are usually less robust than those given by Xray diffraction. An example of such a technique is precession electron diffraction (PED) (Vincent & Midgley, 1994; Gjønnes, 1997; Midgley & Eggeman, 2015). As for Xray diffraction, the success of these methods improves with the number of structurefactor magnitudes that are measured and special experimental configurations, such as automated diffraction tomography (ADT)PED (Mugnaioli et al., 2009), have been devised to facilitate this. Another approach to using electrons is to embrace dynamical scattering and utilize the additional structural information it generates. If the dynamically diffracted intensities could be inverted, the phase information could be obtained through direct measurement, which can strongly confine structure solutions (Mo et al., 1996; Weeks et al., 2000). However, owing to the complexity of nbeam dynamic diffraction, no general analytical inversion (i.e. a mathematical description of structurefactor phases in terms of scattered intensities) under arbitrary conditions has been derived. As a commentary article has pointed out, `a general method of solving an unknown with dynamic electron diffraction is yet to be developed' (Zuo & Rouviere, 2015).
using electron diffraction is to develop techniques for mitigating the effects of dynamical scattering, so thatFor decades, there have been ongoing efforts to develop methods that would enable the extraction of all of the structural information that is present in dynamic electron diffraction intensities. In one line of approach, structure solutions from manybeam dynamic diffraction have been pursued through numerical methods, where structure factors (both the magnitudes and the phases) can be found through optimization procedures that impose few restrictions on the initial values of structure factors (Allen et al., 1998; Spence, 1998; Koch, 2008). However, the experimental implementation of these methods is challenging. So far, the retrieval of structure factors from experimental data using this class of methods has been limited to extremely thin specimens using the largeangle rockingbeam electron diffraction (LARBED) technique (Wang et al., 2016).
In another line of approach, analytical inversion of threebeam dynamic electron diffraction equations has been investigated, which has resulted in the determination of threephase invariants in centrosymmetric crystals (Moodie, 1979; Moodie et al., 1996; Nakashima et al., 2007, 2008, 2013). For noncentrosymmetric crystals, there have been some early attempts at analytical inversion, but these are limited to some special cases within threebeam electron diffraction, such as (i) the weak scattering case (Bird et al., 1987; Bird & James, 1988), (ii) the strong coupling case (Kambe, 1957a,b; Zuo et al., 1989) and (iii) the case where Bethe's approximation is valid (Bethe, 1928; Zuo et al., 1989). Analytical inversions, unlimited by special conditions, have so far proved impractical to implement experimentally, as they require the identification of features in the intensity distribution which cannot be easily isolated experimentally (Moodie et al., 1998).
In the present work, we consider an analytical description of threebeam dynamic electron diffraction, without recourse to special conditions, which reveals a general method for the qualitative determination of threephase invariants in both centrosymmetric and noncentrosymmetric crystals through simple inspection of convergentbeam electron diffraction (CBED) patterns.
1.2. A brief overview of threebeam electron diffraction
In this work, we will derive a method for using dynamic electron diffraction at and near `threebeam conditions' in CBED patterns to determine threephase invariants in both centrosymmetric and noncentrosymmetric crystals. A threephase invariant ϕ, is the sum of the phases φ, of three structure factors whose vectors form a closed loop, i.e. . It is therefore appropriate to first review threebeam electron diffraction here.
Under threebeam diffraction conditions, the crystal is oriented with respect to the incident beam so that two, and only two, reflections simultaneously satisfy their Bragg conditions and no other reflections are excited. At and near a threebeam condition, the eigenequation for dynamic Nbeam electron diffraction, which consists of an N × N matrix, can be approximated by an eigenequation involving a 3 × 3 matrix as follows
and can be approximately replaced by
where is the g, is the interaction constant, which depends on the accelerating voltage, is the excitation error for reflection g, is the eigenvalue for the ith Bloch state (for threebeam diffraction, i runs from one to three), and C_{g} is the excitation amplitude of a certain Bloch state for reflection g. In equations (1) and (2), only elastically scattered electrons are considered, so the matrices are Hermitian.
of the vectorEquation (2) describes the dynamic diffraction of electrons in a threebeam geometry. This is a valid approximation when two, and only two, reflections satisfy their Bragg conditions simultaneously while other beams are weakly excited. This threebeam condition can often be achieved approximately in an experiment. In the central disc of CBED or largeangle CBED (LACBED) patterns, threebeam conditions can be found at the intersections of two Braggcondition lines [for an introduction to CBED geometry see, for example, Spence & Zuo (1992)]. Threebeam diffraction tends to be more prevalent in smaller structures rather than large ones, as in the latter case, many beams can be strongly excited simultaneously.
Now we consider situations where equation (2) holds approximately. This will result in an intensity which can be expressed in terms of only three structurefactor magnitudes (), the thickness of the specimen (z), the incident angle (in the form of two excitation errors, ) and a threephase invariant (). Under these circumstances, the equations describing dynamic diffraction are greatly simplified and thus, the inverse problem (describing the structure factors in terms of intensities) is made much easier.
So far, a complete inversion of the intensities which allows for experimental determination of and at any thickness is achievable only for centrosymmetric crystals (Moodie, 1979; Moodie et al., 1996; Nakashima et al., 2007, 2008, 2013), where is either 0 or . For noncentrosymmetric crystals where , the expression for the intensities in threebeam electron diffraction is lengthy and complicated (Hurley et al., 1978, 1999; Moodie et al., 1998). To date, efforts to invert these equations have involved making further approximations to simplify the intensity expression so that threephase invariants can be determined in certain special cases. For example, in the cases of weak scattering where the specimen is very thin or only weak beams are included, threebeam diffraction can be treated as a perturbation of kinematic diffraction, where a Born series including only the first and secondorder terms is used (Bird & James, 1988). However, this approximation can fail, even for specimens as thin as ∼300 Å, particularly for threebeam cases consisting of strong reflections (Guo, 2017). In practice, it is not easy to judge whether this approximation holds for a CBED pattern from an unknown recorded at an unknown thickness, and the determination of the sign of threephase invariants can still be ambiguous (Marthinsen, 1993). Bethe's (1928) and Kambe's (1957a) formulations treated threebeam electron diffraction equations as perturbations of twobeam dynamic diffraction. However, these approximate formulations cannot distinguish the sign of (Zuo et al., 1989), even though the threebeam diffraction intensities indeed contain the sign. Therefore, none of the existing theories are applicable to general cases of threebeam electron diffraction in noncentrosymmetric crystals.
1.3. An outline of the current work
The current work starts from the equation describing the intensity arising from diffraction by an arbitrary crystal potential and makes two approximations: (i) that the threebeam approximation is valid and (ii) the scattered electrons do not lose or gain energy (inelastic scattering is ignored). From two sets of reduced forms of intensity expressions, we can derive rules for determining if and are >0, ≃0 or <0 (there are eight combinations), from different regions of the CBED patterns. Therefore, the octant in which the threephase invariant lies can be determined, i.e. the uncertainty of the phase measurement is ±22.5°. We show that phaseinvariants can be determined just from observations of indexed CBED patterns, without additional structural information or quantitative measurement of the thickness of the TEM specimen.
2. Theory
Commencing from the formulation derived from projection operators (Hurley et al., 1978), we are able to derive the intensity expression for threebeam electron diffraction (see S1 of the supporting information). Then, we reduce the intensity expression to,
and
where is the difference between two eigenvalues, such that , . The sign convention of the threephase invariant, , is defined in this paper such that the g, h−g and −h form a closed loop in the counterclockwise direction. The symbols ζ_{g} and ζ_{h}, represent the excitation errors for reflections g and h, respectively, which together specify the angle of incidence of the electron beam with respect to the crystal. Since every point within a CBED disc corresponds to a certain angle of incidence, a coordinate system in which the two axes are and can be constructed in each disc (see Fig. 1), and the coordinate of a point can be written as (). It is important to emphasize that () denotes the exact threebeam condition and g and form a The symbol, , denotes the intensity of reflection g near its Bragg condition on the negative side of ζ_{h}. The difference between two eigenvalues μ_{i}, is a function of the three structurefactor magnitudes, two excitation errors and cos ϕ. As a convention in this paper, the three branches of the dispersion surface (which is a threedimensional plot of the eigenvalue λ_{i} versus the angle of incidence; a cut view of the dispersion surfaces or dispersion curves is given in Fig. S1) are labelled as λ_{1}, λ_{2} and λ_{3}, such that λ_{1} > λ_{2} > λ_{3}, and thus μ_{1} > 0, μ_{3} > 0 and μ_{2} = −(μ_{1} + μ_{3}) < 0.
vectorsEquations (3) and (4) expose the nature of the dependence of the threebeam scattered intensity distribution on the threephase invariant and enable the identification of simple methods to measure this, as described step by step in the following sections.
2.1. Qualitative measurement of sin ϕ from equation (3)
We first consider how to measure sin ϕ from equation (3), which is identical to equation (23) in the work by Hurley et al. (1999) except for the righthand side which is factorized. The lefthand side represents the intensity difference between a of reflections, g and (which can also be h and ), at the same thickness. Two Friedel pairs, one with g and and the other with h and , form two separate threebeam conditions, 0/g/h and . If the is set perpendicular to the plane formed by the reciprocal vectors g and h (which is also formed by −g and −h), the centres of the Laue circles for the threebeam conditions 0/g/h and are on the opposite side of each other (see Fig. 1). The two threebeam conditions are typically tens of mrad (or equivalently ∼0.5 Å^{−1} at 200 kV) apart. To achieve this experimentally, two CBED patterns from different incident angles or a single largeangle CBED pattern are/is needed. The righthand side of equation (3) is the product of three structurefactor magnitudes, sin ϕ, and the product of three thicknessdependent terms. The benefit of factorizing the righthand side is to determine the maximum thickness, below which, the sign of the intensity difference depends only on the sign of sin ϕ, i.e. when
we have
In the vicinity of the threebeam conditions, (), inequality (5a) can be satisfied for a broad range of specimen thicknesses, which makes the measurement feasible. In this case,
where we define ξ_{3beam} as the `threebeam extinction distance', that is the thickness z = ξ_{3beam} when the intensity difference at the threebeam condition vanishes. The range of thicknesses that satisfy the inequality (6) is very large in many cases (to gain an impression of typical values of the threebeam extinction distances, see Appendix A). Nevertheless, we have derived a rule (which will be stated in the following section) for checking if the inequality (6) is satisfied for an unknown thickness.
Instead of comparing a hkl and hl. In this case, only a single threebeam CBED pattern is needed to determine the sign of sin ϕ. If sin ϕ = 0, Friedel's law will be preserved for all thicknesses according to equation (3).
from two CBED patterns, we can compare the two diffracted beams within a single threebeam CBED pattern if they have the same structurefactor magnitudes, for example, if they form a such as2.2. Qualitative measurement of cos ϕ from equation (4)
Equation (4) is formulated to enable the practical identification of the sign of cos ϕ. Equation (4) holds only approximately for the regions that are close to the twobeam condition, however, away from the threebeam condition (as discussed in section S1.3 of the supporting information), the approximation is valid for the purpose of determining the sign of cos ϕ.
By solving equation (2) for λ_{i} (which gives a cubic equation), it can be proven analytically that the sign of cos ϕ has a direct dependence on the relative magnitudes of μ_{3}(−ζ_{h}) and μ_{1}(ζ_{h}), which are on opposite sides of the threebeam condition in each CBED disc, such that
and
Therefore, for a specimen that is not very thick, i.e. , one can decide the sign of cos ϕ by direct comparison of the intensities between the negative and the positive sides of ζ_{h} within reflection g or 0 (Figs. 1 or 5). For a thicker specimen, we are also able to determine the sign of cos ϕ by comparing the same two regions as mentioned above: the factorized form of has a thicknessdependent factor of {[sin^{2}(μ_{i}z/2)]/μ_{i}^{2}}, which is similar in form to the twobeam intensity expression. Therefore, in the regions where equation (4) is valid, oscillating `thickness fringes' can be observed and the spacings of the thickness fringes are different on different sides of the threebeam condition. The difference in the spacing depends on the relative magnitudes of μ_{3}(−ζ_{h}) and μ_{1}(ζ_{h}), and thus depends on the sign of cos ϕ.
When cos ϕ = 0, a centre of inversion is present in the threebeam condition at all thicknesses.
2.3. Determination of ϕ
Once we have determined if sin ϕ and cos ϕ are positive, negative or approximately zero, we are then able to determine the octant in which the threephase invariant ϕ lies (i.e. the uncertainty is ±22.5°) according to Table 1.

3. Practical procedures for determining threephase invariants (ϕ)
In the previous section, we explained the theoretical basis for the phase determination with equations and schematics. These theories lead to simple criteria for determining the signs of sin ϕ and cos ϕ (and also whether they are zero) by qualitative observations of the CBED patterns. In particular, these signs can be determined without the need for quantitative measurement of diffracted intensities or numerical simulation. In this section, we state these criteria explicitly and illustrate with examples using simulated CBED patterns in threebeam orientations. These criteria are collated into a flowchart which provides stepbystep instructions for using threebeam CBED to determine threephase invariants by inspection (the flowchart is shown in Fig. 7 after summarizing all of the criteria). In order to emulate realistic experimental data, the simulations of CBED patterns were performed using JEMS (Stadelmann, 2004) using the Bloch wave formalism, incorporating more than 100 beams and including absorptive potentials to model the effect of (i.e. the approximations used in deriving the analytical expressions were not employed in the simulations). For practical purposes, users of the present threebeam method can refer to the current section alone without concerning themselves with the mathematics in the previous section.
From equation (3), we can directly establish the following criterion.
Criterion 1. To decide whether V_{h−g} sin ϕ = 0
At any thickness, if , i.e. Friedel's law is preserved, then .
To determine the sign of sin ϕ by inspection only, it needs to be ensured that the thickness cannot exceed a certain value, the socalled threebeam extinction distance ξ_{3beam}.
Criterion 2. To decide if z < ξ_{3beam}
When both of the following conditions are satisfied, then z < ξ_{3beam}.
Condition I. In disc g, the excess Bragg line away from the threebeam condition has a central bright fringe that is more than 1.6× wider than the neighbouring bright fringes (Fig. 2a).
Condition II. In disc 0, the intensity profile along the locus has no maximum but a local minimum at or near the exact threebeam condition, i.e. (Figs. 2c and 2e).
The detailed derivations of Criterion 2 can be found in section S2 of the supporting information.
By comparing a g and , from the couple of threebeam conditions that consist of two Friedel pairs, 0/g/h and , the sign of can be determined as follows.
of reflections,Criterion 3. To determine the sign of sin ϕ
If z < ξ_{3beam}, and , then (Figs. 3a, 3b and 4a); if z < ξ_{3beam}, and , then (Figs. 3c, 3d and 4b).
If a Criterion 3 to determine the sign of sin ϕ, with the exception of replacing with , see examples in Fig. 4.
exists, we can also applyNow we come to the criteria for the determination of cos ϕ.
Criterion 4. To determine if V_{h−g} cos ϕ = 0
At any thickness, if , i.e. a centre of inversion is present in the threebeam condition, then (Fig. 4). Otherwise, .
By comparing the two regions, and (), labelled as rectangles with dashed (green) and solid lines (red) in both Figs. 5 and 6, one can determine the sign of for both thin and thick specimens by checking Figs. 5 and 6, respectively.
Criterion 5. To determine the sign of for thin specimens
I_{g}(− ζ_{h}) > I_{g}(ζ_{h}) or I_{0}(− ζ_{h}) < I_{0}(ζ_{h}) ↠ cos ϕ > 0 (Fig. 5a)
I_{g}(− ζ_{h}) < I_{g}(ζ_{h}) or I_{0}(− ζ_{h}) > I_{0}(ζ_{h}) ↠ cos ϕ < 0 (Fig. 5b)
Criterion 6. To determine the sign of cos ϕ for thick specimens
The thickness fringes, which are located near the Bragg condition lines for g but away from the threebeam condition in disc g, are inspected.
If the fringe spacing on the negative side is less than that on the positive side, then (Fig. 6a), otherwise (Fig. 6b).
Criterion 6 can be applied to thick specimens, typically up to 2000 Å thick for most inorganic crystals (2000 Å is generally much thicker than most TEM specimens of inorganic crystals). The six criteria are assembled into a flowchart in Fig. 7 which summarizes the procedure for determining threephase invariants.
4. Experimental demonstration
In this section, we demonstrate the procedures in Section 3, with experimental data as a proof of concept. Electron diffraction experiments are carried out on a centrosymmetric crystal (Si) and a noncentrosymmetric crystal (GaAs) to demonstrate the feasibility of direct measurement of threephase invariants.
The experiments were conducted on a JEOL 2100F transmission electron microscope (TEM) and the patterns were recorded on a Gatan Ultrascan 1000 CCD camera. Commercial software, QED (HREM Research Inc., 2012) as a plugin for DigitalMicrograph (Gatan Inc., Pleasanton, CA), was used to generate largeangle rockingconvergentbeam electron diffraction (LARCBED) patterns to measure the threephase invariants, as the large angular range increases the opportunities for capturing several different threebeam conditions in a single pattern. LARCBED is similar to largeangle rockingbeam electron diffraction (LARBED) (Koch, 2011) except that convergent illumination is used instead of parallel illumination. In LARCBED, the incident beam is tilted about a spot on the specimen and CBED patterns are recorded sequentially over a grid of different beam tilts. Through cutting and stitching CBED discs for each reflection from the patterns recorded at all of the beam tilts, a largeangle CBED pattern can be reconstructed. An analogous approach for achieving largeangle CBED patterns from nanosized areas has also been demonstrated by Beanland et al. (2013). Unlike LACBED (Tanaka et al., 1980), where the probe is not focused in the specimen plane, a reconstructed LARCBED pattern can be obtained from a smaller specimen volume because all the CBED patterns that contribute to the LARCBED pattern are obtained from a probe focused on the same position on the specimen. The tiltinduced beam shift is compensated for by the application of QED software to keep the probe on the same specimen area. Nevertheless, there is still a residual beam shift, the size of which depends on the electronoptics of a given microscope and was approximately 10 nm in the present case. The large angular range provided by these LARCBED patterns significantly improves the likelihood of satisfying the threebeam conditions. Usually, a LARCBED pattern can cover several threebeam diffraction conditions, among which, a pair of threebeam conditions that involve two Friedel pairs (0/g/h and ) can be found. This allows for the comparison of a satisfying threebeam conditions that are tens of mrad (equivalent to ∼0.5 Å^{−1} at 200 kV) apart at the same thickness, which makes it experimentally feasible to apply Criteria 1 and 3.
4.1. The experimental procedures
First, standard alignments for CBED were performed using a convergence angle that avoided overlap between adjacent discs. Then, the deflection system was calibrated and the aberrationinduced beam shifts were compensated for in CBED mode by using QED software to minimize beam shift on the specimen as the beam was tilted. Third, the data collection was initiated by QED. When the data collection was finished, a LARCBED pattern could be reconstructed from the set of CBED patterns.
The LARCBED patterns in Figs. 8(a) and 8(b) were reconstructed from 121 CBED patterns recorded from Si near the [118] and GaAs near the [510] respectively.
4.2. Application of the reconstructed LARCBED patterns
The reconstructed central beam is useful for searching for threebeam conditions, which can be found at the intersection of lines defining two Bragg conditions (usually two dark lines at low thicknesses) where no other Bragg condition lines for strong reflections lie in the neighbourhood. Then, the corresponding threebeam conditions in the diffracted beams can be located as they lie in the same position as in the central beam. For example, in Fig. 8(a), a pair of threebeam conditions (including the neighbourhoods), and , are labelled in the reconstructed LARCBED pattern with green and red circles, respectively. All the three green circles enclose the same threebeam condition for and its neighbourhood. In the same place in the other diffracted beams such as reflection , the diffraction pattern is comparatively dark and featureless, simply because the threebeam interaction amongst 000, and dominates in this orientation.
4.3. Practical examples of determining the threephase invariants
Once we have circled the regions of threebeam conditions, we can proceed to determine the threephase invariants following the flowchart in Fig. 7.
4.3.1. Example 1, Si
First, consider the experimental LARCBED pattern from Si (Fig. 8a) as an example.
(i) According to Criterion 2, the thickness of the specimen satisfies the condition, z < ξ_{3beam}, which means we are allowed to determine the sign of the threephase invariant from inspection.
(ii) By comparing the Friedelpair reflections, and , there is centrosymmetry/twofold symmetry between the intensity distributions within the green circle in the disc and within the red circle in the disc. According to Criterion 1, .
(iii) According to Criterion 4, , and thus .
(iv) According to Criterion 5, intensity asymmetry about the threebeam conditions reveals .
(v) From Table 1 we can conclude that . This agrees with the known value for Si of 0°.
4.3.2. Example 2, GaAs
Here we take the experimental LARCBED pattern from GaAs (Fig. 8b), as a noncentrosymmetric example.
(i) According to Criterion 2, the thickness of the specimen satisfies the condition z < ξ_{3beam}.
(ii) By comparing the Friedelpair reflections and , we can see that > 0 from the intensity difference according to Criterion 3.
(iii) According to Criterion 4, since there is a centre of inversion at the exact threebeam condition, .
(iv) According to Table 1, . The result of this qualitative measurement agrees approximately with the known value for GaAs of +88°.
5. Potential applications
5.1. Improvement in ab initio phasing
If the `guessed' threephase invariants in ; Hauptman, 1991), which are based on probability, are replaced by threephase invariants measured directly from threebeam diffraction, then the success rate and accuracy of ab initio phasing can be greatly improved: a small set of measured threephase invariants with a mean error of ± 22.5° (Mo et al., 1996) (i.e. which is equivalent to finding the octant of the threephase invariants as achieved in this article) or even 40° (Weeks et al., 2000) may enable a structure solution where it would otherwise be impossible. In such cases, fewer structurefactor magnitudes would need to be measured.
(Woolfson, 1971Therefore, one application of the current threebeam method is to combine it with Xray diffraction data or with electrondiffraction data based on quasikinematic diffraction such as PED (Vincent & Midgley, 1994) to improve ab initio phasing. Importantly, combining this threebeam method with PED may help to solve structures for nanosized crystals which can be most readily studied using the <1 nm probes that are routinely available in TEMs.
5.2. Resolution of enantiomorphs
For a pair of enantiomorphically related structures (L and R), the same threephase invariants have opposite signs, i.e. , and the threephase invariants for the opposite threebeam conditions have the same sign, i.e. . Therefore, to identify the of a structure, both the signs of the indices (of reflections g and h) and the sign of the threephase invariant, , have to be determined consistently. Since threebeam diffraction alone involves three vectors that are coplanar, it is impossible to identify which is threedimensional, unless a fourth beam that is not in the same (noncoplanar) is present in the field of view (Spence et al., 1994). Therefore, at least one threebeam condition and one HOLZ reflection should be present in the same diffraction pattern.
Within the brightfield component of a reconstructed LARCBED pattern, a pair of threebeam conditions that involve two Friedel pairs and deficit HOLZ lines can be found simultaneously. Based on the positions of the deficit HOLZ lines relative to the threebeam conditions in the bright field, a
can be indexed without ambiguity. The sign of threephase invariants can be determined from direct observations following the present method. Therefore, can be clearly identified from just a single LARCBED pattern (with both the central and diffracted beams) where a Friedel or of reflections satisfies threebeam conditions.6. Challenges and limitations
6.1. Applicability to large structures
The threebeam approximation in equation (2) can hold very well if the excitation errors of the reflections other than reflections g and h are much larger than any structurefactor magnitude, i.e.
and thus,
It is easy to find threebeam diffraction conditions in structures with small unit cells (say, with cell volumes less than a few thousand Å^{3}) because their points are sparsely spaced and reflections other than g and h tend to be weakly excited near the threebeam condition for 0/g/h. In contrast, the projection of the for large structures is dense and other reflections g′, h′…, tend to be strongly excited at the same time (their excitation errors, tend to be small). Threebeam conditions can be difficult to isolate in large structures and the influence of manybeam diffraction effects is difficult to avoid. Nevertheless, if the structurefactor magnitudes relevant to a particular threebeam diffraction condition, , are much larger than those of the other excited reflections (like , etc.), then manybeam diffraction can still be treated as a perturbation of the threebeam approximation. This has been shown by studies in threebeam Xray diffraction (Weckert & Hümmer, 1997). Threebeam diffraction of Xrays has been demonstrated in some small protein crystals such as myoglobin (Hümmer et al., 1991), tetragonal lysozyme and catalase oxidoreductase (Weckert et al., 1993).
Since the
for highenergy electron diffraction is much `flatter' than that of soft Xray diffraction, which results in many more reflections being excited simultaneously, the applicability of threebeam electron diffraction to large structures like protein crystals may be very limited. Nevertheless, the applicability of threebeam electron diffraction to moderately complex structures (a with dozens to a few hundred atoms) can be expected. Feasibility may be increased with the use of the new generation of very low voltage TEMs (>30 kV), offering large electron wavelengths and hence increased curvature in the Ewald sphere.6.2. Applicability to beamsensitive structures
Under standard operating conditions, with no effort to minimize electron dose, the total dose can be large and unsuitable for beamsensitive structures. For example, the set of 121 CBED patterns at 200 kV (which gave the LARCBED patterns in Fig. 8) was estimated to use a total dose of 10^{5} e Å^{−2}. However, it is important to note that the threebeam method described here relies only on a qualitative inspection of features in the CBED patterns, rather than quantitative measurement of absolute intensities. It can therefore tolerate a high noise level in the diffraction data, so that lowdose CBED methods, such as those used by Wu & Spence (2003), can be used. Coupled with a new generation of highsensitivity and fast detectors, lowdose threebeam measurements of threephase invariants in beamsensitive materials is likely to be very feasible.
7. Clarifying points
7.1. Crystallographic phase versus phase of the exiting electron wave
The current threebeam method provides direct measurement of threephase invariants (which is the phase information of structure factors) rather than the phase of exiting electron waves, which can be measured by electron holography (Gabor, 1948), through focal series methods (Schiske, 1968, 2002; Kirkland, 1984) or electron ptychography (Rodenburg, 2008; Humphry et al., 2012). Under dynamical scattering conditions, there is no direct or general analytical relationship between the phase of the exit wavefunction and the phase of the structure factors.
7.2. Comparison with electron ptychography for structurefactor phase determination
Apart from the current threebeam CBED approach, electron ptychography can also provide a measurement of structurefactor phases (Nellist et al., 1995) and thus, threephase invariants, provided the single scattering condition prevails. Therefore, it requires extremely thin and weakly scattering specimens. Electron ptychography uses the coherent interference in overlapping CBED discs and hence requires an effective source that is spatially coherent over the angular range of the disc and a probeforming lens system that does not introduce significant coherent aberrations within this range. In contrast, threebeam CBED is based on dynamical scattering and is thus valid for thicker specimens. Furthermore, as the discs do not overlap, it is independent of coherent aberrations [such an independence can be simply proven by deriving the dynamical intensities in CBED without coherent interference, from the exit wavefunction which contains the initial phase of the probe, such as equation 14.87 in the book by Zuo & Spence (2017); for an earlier reference, see Spence & Cowley (1978)].
7.3. Comparison with quantitative CBED
The current threebeam CBED method is an ab initio approach where no structural model is assumed. This is in contrast to quantitative CBED for refining structure factors and threephase invariants within a given structural model (Goodman & Lehmpfuhl, 1967; Spence, 1993; Nakashima, 2017). A quantitative analysis of experimental threebeam CBED patterns has been used to refine threephase invariants to an accuracy of within one degree (for example, Høier et al., 1999). However, the quality of an experimental pattern for quantitative CBED needs to be much higher than required by the current method, where only qualitative inspection is involved.
8. Conclusions
This article has introduced an ab initio method using threebeam convergentbeam electron diffraction (CBED) for the practical determination of threephase invariants in noncentrosymmetric as well as centrosymmetric crystals. It can readily be applied to nanosized crystals. Starting from the exact solutions to threebeam electron diffraction, we have derived a theory which allows for the inversion of the diffracted intensities to determine the signs of sin ϕ and cos ϕ (and also whether they are close to zero). Based on this theory, we have provided instructions for determining the octant of threephase invariants (i.e. the uncertainty in the determination of threephase invariants is ±22.5°) by qualitative inspection of indexed CBED or LARCBED patterns.
Importantly, only qualitative inspection of the diffraction patterns is required, without any need for quantitative intensity measurement or numerical pattern matching or
No additional knowledge about the structure or the specimen thickness, is needed (except for the pattern indices).LARCBED experiments on a centrosymmetric crystal of Si and a noncentrosymmetric crystal of GaAs have been carried out to demonstrate the current method.
Threebeam CBED can be combined with Xray and precession electron diffraction data to improve ab initio phasing. Also, the ambiguity can be eliminated by observations of the LARCBED patterns where HOLZ reflections and threebeam conditions are present in the same pattern. Furthermore, the threebeam method may have the potential to measure threephase invariants in beamsensitive structures using the latest generation electron detectors with high speed and sensitivity.
Note added in proof. It is with great sadness that we learned of the passing of Professor Alexander Moodie FAA on 8 July 2018, a pioneer in the field of electron crystallography. Among his many distinguished contributions, is the unique analytical inversion of threebeam electronscattering equations. We dedicate this work to a brilliant scientist, inspiring colleague and generous mentor.
9. Related literature
The following references are cited in the supporting information: Blackman (1939); Wolfram Research, Inc. (2014).
APPENDIX A
Typical values for the threebeam extinction distance, ξ_{3beam}
Here we show that the validity of the condition, z < ξ_{3beam}, is not restricted to very thin specimens. The threebeam extinction distance, ξ_{3beam}, decreases with increasing structurefactor magnitudes, so the range of the thickness that satisfies the condition, z < ξ_{3beam}, becomes narrower with larger structurefactor magnitudes. One may think that the specimen needs to be very thin for a threebeam case with large structure factors. However, it can be shown that even if the threebeam case involves large structure factors, the threebeam extinction distance is still large. For example, a threebeam case, , in zinc blende ZnTe, where = 4.2 V and = 2.5 V, may serve as an example of threebeam cases with three fairly large structurefactor magnitudes since a fairly heavy element (Te) is present in the structure, and and are among the largest structurefactor magnitudes in this structure. In this case, the threebeam extinction distance, ξ_{3beam}, at 200 kV is about 650 Å, which is sufficiently large to allow specimen thicknesses less than ξ_{3beam} to be accessed practically in TEM. In general, a typical value of the threebeam extinction distance, ξ_{3beam}, at 200 kV for inorganic crystal structures is above 1000 Å and for organic crystal structures it is above 2000 Å.
This relatively large range of validity may seem surprising in the context of previous methods based on the special weak scattering case [(Bird et al., 1987), see comment by Marthinsen (1993)]. However, the present general threebeam method is not restricted to weak scattering, ensuring it has validity over a much wider range of thicknesses. To get a sense of the difference in the validity ranges between the kinematic approximation and current approach, the Mathematica code from the website https://github.com/DrYGuo/3beamproject can be used.
Supporting information
Derivations for equations (3) and (4), and Criterion 2. DOI: https://doi.org/10.1107/S2052252518012216/gq5009sup1.pdf
Acknowledgements
We thank Professor Alexander Moodie, Professor Andrew Johnson and Dr Andrew Stewart for their helpful and insightful discussions. The assistance of Professor Christoph Koch and HREM Research Inc. on using the QED software are gratefully acknowledged. The authors acknowledge use of facilities within the Monash Centre for Electron Microscopy.
Funding information
The following funding is acknowledged: Australian Research Council (grant No. DP150104483 to Joanne Etheridge; grant No. FT110100427 to Philip Nakashima).
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