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A new approach to structure amplitude determination from 3-beam convergent beam electron diffraction patterns
Ultramicroscopy 111 (2011) 841–846
Contents lists available at ScienceDirect
Ultramicroscopy journal homepage: www.elsevier.com/locate/ultramic
A new approach to structure amplitude determination from 3-beam convergent beam electron diffraction patterns Philip N.H. Nakashima a,b,c,n, Alexander F. Moodie b,c, Joanne Etheridge b,c a
ARC Centre of Excellence for Design in Light Metals, Monash University, Victoria 3800, Australia Department of Materials Engineering, Monash University, Victoria 3800, Australia c Monash Centre for Electron Microscopy, Monash University, Victoria 3800, Australia b
a r t i c l e i n f o
abstract
Available online 4 February 2011
The intensity distribution in three-beam CBED patterns from centrosymmetric crystals can be inverted analytically to enable the direct measurement of crystal structure amplitudes and three-phase invariants. The accuracy of the measurements depends upon the accuracy and precision with which specific loci within the discs can be identified. The present work exploits the equivalence in form of the intensity distribution along these loci to provide an algorithm for their automated location, enabling the rapid and unequivocal identification of their position. Moreover, it demonstrates how the loci can be used to determine directly the relative magnitudes of structure amplitudes with superior accuracy and without recourse to complex pattern-matching calculations. & 2011 Elsevier B.V. All rights reserved.
Keywords: Convergent beam electron diffraction Structure solution Structure amplitude determination Phase problem Three-beam diffraction
1. Introduction The direct determination of structural phases and magnitudes from the analytical inversion of 3-beam convergent beam electron diffraction (CBED) patterns has been demonstrated on several occasions for centrosymmetric crystals [1–3]. This experimental method, which is based on the measurement of distances in 3-beam CBED patterns and was developed from the theory established by Moodie et al. [4–6], has proven an extremely powerful approach to structure solution. In recent work [7], it was shown that the structure of a-Al2O3, with 30 atoms in its unit cell, could be determined to 0.1 A˚ precision by starting from the direct measurement of 3-phase-invariants and then adding measurements of 4 independent structure amplitudes, which were made using the 3-beam CBED techniques of [2,3]. Whilst such inversion of 3-beam CBED patterns has proved a quick and reliable way to determine three-phase invariants (often by inspection), the accuracy of the measurement of magnitudes of structure amplitudes can be limited, in part because of the potential for ambiguity in identifying the position of specific loci, which mark the distances to be measured. These loci have a known orientation and a known form of the intensity distribution but their lateral position within the CBED disc must be identified and this can sometimes be ambiguous, particularly in the
n Corresponding author at: ARC Centre of Excellence for Design in Light Metals, Rm G83, Building 27, Monash University, Victoria 3800, Australia. Tel.: + 61 3 99053827; fax: + 61 3 99054940. E-mail address: [email protected] (P.N.H. Nakashima).
0304-3991/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.ultramic.2011.01.044
presence of significant n-beam perturbations. In the present work, we present a new approach for locating one of the two loci, locus C, which enables it to be more rapidly, precisely and objectively located (locus C is the locus oriented perpendicular to the coupling vector between the two diffracted discs, g and h). This results in more accurate measurements of the 3 distances required to determine the absolute magnitudes of the structure amplitudes, namely, the distances to the Gjønnes–Høier (GH) point and the distance to the centre of symmetry in the intensity along locus C, as per the approach of [2,3]. More significantly, we show that this locus can be used to determine directly and rapidly the relative magnitudes of structure amplitudes from the relative intensities in the diffracted discs along locus C, providing an additional and more accurate means for acquiring this information from the 3-beam CBED pattern. Professor Spence [8] has fostered much research in the related area of structure amplitude refinement through iterative patternmatching of CBED patterns, including 3-beam patterns [9]. We have enjoyed enthusiastic and fruitful engagement with Professor Spence in this and many other areas of electron diffraction and we present this work in honour of him on the occasion of his birthday.
2. 3-Beam CBED revisited At this point, it is worth re-examining the main features of 3-beam CBED patterns from centrosymmetric crystals relevant to the present work. It has already been demonstrated [2,3] that the distances to particular features in 3-beam CBED patterns are
842
P.N.H. Nakashima et al. / Ultramicroscopy 111 (2011) 841–846
sufficient to determine the phase and magnitude of the relevant structure amplitudes without recourse to any pattern-matching refinement. As already described in [2,3], the sign of a three-phase invariant can be obtained by inspection of the corresponding 3-beam CBED pattern, from the direction of deflection of the rocking curve near the 3-beam Bragg condition. This corresponds to the displacement of the centres of symmetry of the two centrosymmetric loci in each diffraction disc. The magnitudes can be determined from the distances of the GH point and centres of symmetry of the centrosymmetric loci from the origin as in [2,3]. Alternatively, we show here that their relative magnitude can be determined from the locus C in the discs g and h alone and that the position of the locus C can be determined with considerable accuracy. This locus is found in every disc and is oriented perpendicular to the coupling vector, g–h, as shown schematically in Fig. 1. This new approach derives from one of the apparent redundancies in the three-beam derivations. In particular, as illustrated in Fig. 1 (see also Fig. 2), a two-beam distribution is generated in all three discs along the locus C whereas the locus has only to be detected in one to complete the inversion. Since the accuracy to which this locus can be defined is an important factor in establishing the accuracy to which the structure can be determined, it becomes relevant to compare the calculated intensity distributions along locus C in both the discs g and h. This can be done very simply using the reduction of the three-beam equations to two-beam form, with the pseudopotential VC ¼ Vg þ iVh
ð1Þ
and the effective excitation error
zC ¼ 2pzg þ
sVh Vhg Vg
as per [5] and using the same notation therein. Thus, ( ! ) ! Uð0Þ pzC sVC 1 ipzC z exp i z ¼e Uð1Þ sVC pzC 0
ð2Þ
ð3Þ
with the solution, sin Uð1Þ ¼ eipzC isVC
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðpzC Þ2 þ s2 VC VC z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðpzC Þ2 þ s2 VC VC
ð4Þ
Equating real and imaginary parts, the central result for this communication is obtained in the form, Vg2 IC,g ¼ 2: IC,h Vh
ð5Þ
Here the symbols for the intensities are italicised in order to emphasise that the ratio holds only along the locus C. It is important to note that this result derives from a dynamical calculation, which is exact within the three-beam approximation. In particular it is not related to the single scattering or kinematical approximation and so the concept of extinction cannot apply. To emphasise this important point, it is briefly elaborated upon. The kinematical or single scattering approximation is properly written as Ig p9Vg 9
2
ð6Þ
As is widely known, this has wide currency when the interaction is weak, for instance in X-ray and neutron scattering (though even here a number of restrictions and corrections are usually invoked). Extinction in X-ray diffraction is the effect of applying the kinematic scattering approximation to the analysis of the total integrated intensity within each reflection via Eq. (6). The lack of a treatment of all dynamic scattering effects gives rise to the concept of extinction and departures from the relativity of structure amplitudes that is suggested by Eq. (6). The fact that Eq. (5) applies to the intensity distribution along a unique locus (locus C), located at equivalent positions in the discs g and h in a 3-beam CBED pattern, and is a consequence of dynamic scattering, removes the concept of extinction all together. Given that Eq. (5) holds for the locus C, which traverses both discs g and h, a simple algorithm can be developed that not only locates the locus C very accurately within all discs, but also results in a more accurate approach to determining Vg/Vh compared with the measurement of amplitudes from distances described previously in [2,3]. A description of the algorithm follows in the next section, making use of a synthetic 3-beam CBED pattern calculated with Vh ¼1.0 V, Vg ¼ 2.0 V and Vg–h ¼3.0 and shown in Fig. 2. The location of locus C is shown in the discs g and h and the corresponding intensity profiles are shown to have the same 2-beam intensity distributions. The amplitudes of these distributions differ by a factor of 4, resulting from Vg/Vh ¼ 2 and evident from Eq. (5). As a consequence, the subtraction of disc h multiplied by 4 from disc g reveals a sharp line of zeroes along locus C across which, the sign of the difference map flips.
3. The algorithm
Fig. 1. A schematic illustration of a general (9g9a 9h9a 9g–h9) 3-beam pattern. The locus C, is perpendicular to the vector g–h, has a centrosymmetric intensity distribution and is located at the same position in each disc as described in [1–6]. The form of the intensity distribution is identical in the discs g and h as shown by the two graphs of intensity along Cg and Ch. The relative intensities along Cg and Ch are determined by the relative magnitudes of the structure amplitudes, as shown in the boxed equation.
Fig. 3 explains the algorithm using the synthetically calculated 3-beam CBED pattern of Fig. 2 as an example (a). The physics of 3-beam-scattering ensures that the locus C is always oriented perpendicular to the coupling vector, g–h, so the starting point is to rotate the pattern in order to make this vector horizontal (the locus C is then oriented vertically in the image) as has already been done in Fig. 2. The angle of rotation is found simply by drawing the tangent common to both discs g and h as shown in Fig. 3a. The discs g and h are then extracted from the pattern (b and c, respectively) with particular care taken in making sure that the sub-images containing each disc are correctly registered with respect to one another (i.e. each point in each of the sub-images b and c are exactly related to one another by the coupling vector g–h). Eq. (5) is the foundation of the algorithm and can be rewritten as: Ig ¼
Vg2
Ih Vh2
¼ mIh ,
m¼
Vg2 Vh2
ð7Þ
P.N.H. Nakashima et al. / Ultramicroscopy 111 (2011) 841–846
843
Fig. 2. An ideal synthetic 3-beam pattern (where 9g9 a9h9a 9g–h9) computed without absorption and with Vh ¼ 1.0 V, Vg ¼ 2.0 V and Vg–h ¼3.0 V. It follows from Eq. (5) that Ig ¼ 4Ih along the locus C. The subtraction of 4 times disc h from disc g reveals the position of locus C as a line of zeroes about which the phase of the difference map inverts.
Fig. 3. A schematic description of the algorithm developed here for determining Vg/Vh. The pattern (a) must first be rotated so that the coupling vector, g–h, is horizontal, as in Fig. 2. The discs g and h are then extracted from the rotated pattern and assigned to separate sub-images (b and c). The variable m increases from 0 to some userdefined value, N, and is used to calculate 9(disc g)–m(disc h)9 (d). The boxed region with dimensions X and Y (user-defined, see d) is used to generate the profile (e) of (d) integrated in the direction Y, which is the direction of the locus C. The minimum of the profile (e) is found and compared with the smallest minimum obtained so far, gmin. The process of calculating 9(disc g)–m(disc h)9 is repeated after increasing variable m by a small increment. The algorithm finds the smallest minimum of all profiles (e) obtained from 9(disc g)–m(disc h)9, for 0r mr N.
Multiplying the image of disc h by m and subtracting the product from disc g and taking the absolute value gives 9(disc g)– m(disc h)9 (see Fig. 3d). If the correct value of m is used, then a
vertical line of zeroes (to within the rounding error due to finite pixel size) should present along locus C, not only revealing the location of the locus but also giving the relative magnitudes of Vg
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and Vh. In the example of Fig. 3d, the wrong value of m has deliberately been applied (m¼1, Vg ¼2.0 V and Vh ¼1.0 V) and no vertical dark line is apparent. However, a horizontal intensity profile across Fig. 3d, X, integrated in the vertical axis, Y (the dimensions X and Y shown by the box in Fig. 3d) reveals the existence of a minimum in the integrated intensity profile, ‘‘min’’ (Fig. 3e). The proportionality constant, m, can be determined by varying m until the minimum, ‘‘min’’, in the integrated intensity profile (Fig. 3e) of 9(disc g)–m(disc h)9 (Fig. 3d) reaches a global minimum, ‘‘gmin’’. This is then taken as the final ratio, V2g/V2h. This optimisation approach was taken with the ideal calculated 3-beam CBED pattern of Figs. 2 and 3 in the absence of absorption (Fig. 4a) and the correct ratio of Vg/Vh ¼ 2 returned to better than 0.01%. This level of error is representative of the rounding associated with finite pixel size. In Fig. 4b, absorption was included in the 3-beam calculation of the pattern and a 3% error manifests in the determination of Vg/Vh. To examine a more realistic situation, the effect of n-beam scattering was included in parts c and d of Fig. 4 by including 126 beams in the calculation of the CBED pattern in the same 3-beam orientation as in Figs. 2 and 3. Absorption was excluded in Fig. 4c and included in Fig. 4d. In both cases, the ratio Vg/Vh was underestimated by about 13%, which is indicative of the perturbation caused by many-beam effects. However, in all cases, the position of locus C is very robustly identified. As expected for the 3-beam calculated patterns (Fig. 4a and b), the locus C manifests itself as a straight, unbroken and sharp line of near-zero intensities in 9(disc g)– m(disc h)9. For the many-beam calculated patterns (Fig. 4c and d), locus C still manifests itself quite clearly, though it vanishes very locally or is sharply kinked in the immediate vicinity of strong excitations of other beams. The integrated intensity profiles shown for the boxed regions in all cases show very sharp minima at the X position of locus C, showing that the position determination is robust even in the presence of many-beam effects. This in itself serves as a significant aid to the distance determination techniques of 3-beam CBED used in [1–3]. Application to real experimental 3-beam CBED patterns serves as a final test of the present technique.
4. Analysis of experimental 3-beam CBED data Experimental analysis can be complicated by the presence of the diffuse background due to inelastic scattering in CBED patterns, even if energy-filtering optics are used (due to thermal diffuse scattering). This will inevitably introduce error into any analysis of Ig/Ih as the derivation of Eq. (5) is solely based on elastic scattering theory. However, if one uses the thickness difference approach to CBED first demonstrated in [10], the complicating effects of the inelastic background can be almost completely negated in all forms of CBED. The use of patterns from different specimen thicknesses does not affect the validity of Eqs. (5) and (7) as is shown in the workings below. In these workings, Ig,1 and Ih,1 represent the intensities along the loci C in discs g and h, respectively, in the data collected from specimen thickness H1. Similarly, Ig,2 and Ih,2 represent the intensities along the loci C in discs g and h, respectively, at specimen thickness H2. Thus: m¼
Ig,1 Ig,2 ¼ Ih,1 Ih,2
ð8Þ
then Ig,1 Ig,2 mIh,1 mIh,2 ¼ ¼m Ih,1 Ih,2 Ih,1 Ih,2
ð9Þ
and therefore Eqs. (5) and (7) and the present algorithm are valid for thickness difference 3-beam CBED patterns. Sources of error stemming from the application of these 3-beam equations to an experimental 3-beam CBED pattern include deviations from the ideal 3-beam condition due to N-beam effects, errors in the alignment of disc g with disc h and noise. The use of two separate CBED patterns in the thickness difference approach will magnify the resulting uncertainties, as well as introduce additional errors that stem from the alignment of the two patterns from different thicknesses with one another. This error magnification will, however, be strongly offset by the correction of errors in the measured ratio, m, that arise due to the inelastic background present in single patterns.
Fig. 4. A comparison of 3 and N-beam calculations of the 3-beam CBED pattern of Figs. 2 and 3 and the capacity of the algorithm of Fig. 3 to find the locus C and the ratio Vg/Vh. Each set of output corresponds to Figs. 3d and 3e.
P.N.H. Nakashima et al. / Ultramicroscopy 111 (2011) 841–846
845
Fig. 5. An application of the present technique to experimental thickness difference CBED patterns [10] at the zone [16 -10 -1] in a-Al2O3. Two permutations of the same 3-beam triplet are examined (a and f). The algorithm developed here returns the optimum 9(disc g)–m(disc h)9 (b and g) with the corresponding integration profiles (c and h). The minimum in each case is used to locate the loci C in each of the discs g and h in both permutations and the intensity profiles along the loci C are shown (d, e, i and j). In the present cases, V1 2 4/V0 1 10 ¼ 0.97 and V1 1 6/V1 2 4 ¼ 1.66, were determined from the refined values of m in each case.
Fig. 5 shows two permutations of the same 3-beam triplet about the [16 -10 -1] zone in a-Al2O3. The first permutation will yield the ratio V1 2 4/V0 1 10 and the second permutation, the ratio V1 1 6/V1 2 4. These in combination will imply the ratio V1 1 6/V0 1 10. These structure amplitudes have been determined previously [3,10–13] in two different ways; in one case using structure amplitude refinement via quantitative CBED (QCBED) to achieve the highest possible accuracy [10–13] and in the other case, from a direct inversion of 3-beam CBED patterns via the measurement of distances [3]. Close agreement was achieved between these two approaches, making this an excellent example to test the present method. As in the case of the calculated intensities of Fig. 4, the optimum 9(disc g)–m(disc h)9 images (Fig. 5b and g) and corresponding integrated profiles (Fig. 5c and h) reveal the positions of locus C conclusively. In addition to finding the position of locus C in discs g and h for each permutation, the thickness difference intensity distribution along the loci are shown in components d, e, i and j of Fig. 5 at the determined positions marked in the difference patterns a and f. The resemblance in form of the distributions in Fig. 5d and e as well as the pair in Fig. 5i and j is further assurance of the correct identification of locus C in both patterns a and f. The differences in amplitude of the distribution within each pair ((d and e) and (i and j)) are indicative of the ratios V1 2 4/V0 1 10 and V1 1 6/ V1 2 4 that might be returned.
The results of the optimisation of the proportionality constant, m, for each permutation yield the following ratios: V1 2 4 =V0 1 10 ¼ 0:970 70:003 and V1 1 6 =V1 2 4 ¼ 1:663 70:001; implying V1 1 6 =V0 1 10 ¼ 1:613 7 0:006: These structure amplitudes have been refined previously from QCBED pattern-matching analyses [10–13]. Using the values published in [10–13], the corresponding accurate QCBED structure amplitude magnitudes and their ratios are: ðV1 1 6 ¼ 3:7097 0:010 V; V0 1 10 ¼ 2:241 7 0:007 V and V1 2 4 ¼ 2:231 7 0:022 VÞ V1 2 4 =V0 1 10 ¼ 0:996 7 0:013; V1 1 6 =V1 2 4 ¼ 1:662 70:021 and V1 1 6 =V0 1 10 ¼ 1:655 7 0:010: Results of previous 3-beam CBED determinations based on the distance technique [3] give: ðV1 1 6 ¼ 3:80 7 0:42 V; V0 1 10 ¼ 2:45 7 0:55 V and V1 2 4 ¼ 2:23 7 0:53 VÞ V1 2 4 =V0 1 10 ¼ 0:91 7 0:34; V1 1 6 =V1 2 4 ¼ 1:70 70:48 and V1 1 6 =V0 1 10 ¼ 1:55 70:42:
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P.N.H. Nakashima et al. / Ultramicroscopy 111 (2011) 841–846
The present results compare favourably with those from QCBED pattern-matching [10–13] and fall well within the larger uncertainties associated with the measurement of structure magnitudes from distances in 3-beam patterns [3]. The present technique has less redundancy and thus less multiplicity of measurements of the same structure amplitudes than the distance approach to 3-beam CBED, however, it is much more robust and much more precise.
5. Conclusions A new approach to determining the location of the locus C and the relative magnitudes of structure amplitudes in 3-beam CBED patterns is demonstrated. The associated algorithm removes all aspects of subjective judgement in locating the loci and is very robust and precise. Futhermore, the proportionality of the intensities along the locus C in discs g and h gives the ratio of the corresponding magnitudes of the structure amplitudes, Vg/Vh, with surprising accuracy and precision in the present demonstration with experimental data, even in the presence of many-beam perturbations of the ideal 3-beam condition. This new approach offers the prospect of determining unknown structures to a much higher resolution using the analytical inversion of 3-beam CBED patterns from centrosymmetric crystals. Furthermore, the present method of comparing discs g and h may provide an avenue for developing the analytical inversion of 3-beam CBED patterns from non-centrosymmetric crystals.
Acknowledgements This work was supported by the Australian Research Council grant DP0346828. The data used in this work was obtained in the Monash Centre for Electron Microscopy using equipment funded
by the Australian Research Council Grant number RIEFP 99. PN is grateful for the support of the Australian Research Council’s Centre of Excellence for Design in Light Metals. References [1] J. Etheridge, A.F. Moodie, C.J. Humphreys, Direct measurement of structure amplitudes from three beam interactions, electron microscopy, in: Proceedings of the 14th International Congress on Electron Microscopy, vol. III, 1998, pp. 737. [2] P.N.H. Nakashima, A.F. Moodie, J. Etheridge, Structural phase and amplitude measurement from distances in convergent beam electron diffraction patterns, Acta Crystallogr. A 63 (2007) 387. [3] P.N.H. Nakashima, A.F. Moodie, J. Etheridge, A practical guide to the measurement of structure phases and magnitudes by three-beam convergent beam electron diffraction, Ultramicroscopy 108 (2008) 901. [4] A.F. Moodie, Some problems in imaging and inversion, Chem. Scr. 14 (1978-79) 21. [5] A.F. Moodie, J. Etheridge, C.J. Humphreys, The symmetry of three-beam scattering equations: inversion of three-beam diffraction patterns from centrosymmetric crystals, Acta Crystallogr. A 52 (1996) 596. [6] A.F. Moodie, J. Etheridge, C.J. Humphreys, The Electron, IOM Communications Ltd, London, UK, Ch, The Coulomb Interaction and Direct Measurement of Structural Phase, 1998, pp. 235. [7] P.N.H. Nakashima, A.F. Moodie, J. Etheridge, Locating atoms by observing structural phase in centrosymmetric crystals II. Atom location using a Lonsdale-like approach. Acta Crystallogr. A, submitted for publication. [8] J.C.H. Spence, On the accurate measurement of structure-factor amplitudes and phases by electron diffraction, Acta Crystallogr. A 49 (1993) 231. [9] J.M. Zuo, R. Høier, J.C.H. Spence, Three-beam and many-beam theory in electron diffraction and its use for structure-factor phase determination in non-centrosymmetric crystal structures, Acta Crystallogr. A 45 (1989) 839. [10] P.N.H. Nakashima, Thickness difference—a new quantitative filtering tool for electron diffraction, Phys. Rev. Lett. 99 (12) (2007) 125506. [11] P.N.H. Nakashima, B.C. Muddle, Differential convergent beam electron diffraction: experiment and theory, Phys. Rev. B 81 (2010) 115135. [12] V.A. Streltsov, P.N.H. Nakashima, A.W.S. Johnson, A combination method of charge density measurement in hard materials using accurate, quantitative electron and X-ray diffraction: the a-Al2O3 case. [13] P.N.H. Nakashima, Accurate measurements of charge density in a-Al2O3 by combined electron and X-ray diffraction, Ph.D. Thesis, The University of Western Australia, 2002.
Contents lists available at ScienceDirect
Ultramicroscopy journal homepage: www.elsevier.com/locate/ultramic
A new approach to structure amplitude determination from 3-beam convergent beam electron diffraction patterns Philip N.H. Nakashima a,b,c,n, Alexander F. Moodie b,c, Joanne Etheridge b,c a
ARC Centre of Excellence for Design in Light Metals, Monash University, Victoria 3800, Australia Department of Materials Engineering, Monash University, Victoria 3800, Australia c Monash Centre for Electron Microscopy, Monash University, Victoria 3800, Australia b
a r t i c l e i n f o
abstract
Available online 4 February 2011
The intensity distribution in three-beam CBED patterns from centrosymmetric crystals can be inverted analytically to enable the direct measurement of crystal structure amplitudes and three-phase invariants. The accuracy of the measurements depends upon the accuracy and precision with which specific loci within the discs can be identified. The present work exploits the equivalence in form of the intensity distribution along these loci to provide an algorithm for their automated location, enabling the rapid and unequivocal identification of their position. Moreover, it demonstrates how the loci can be used to determine directly the relative magnitudes of structure amplitudes with superior accuracy and without recourse to complex pattern-matching calculations. & 2011 Elsevier B.V. All rights reserved.
Keywords: Convergent beam electron diffraction Structure solution Structure amplitude determination Phase problem Three-beam diffraction
1. Introduction The direct determination of structural phases and magnitudes from the analytical inversion of 3-beam convergent beam electron diffraction (CBED) patterns has been demonstrated on several occasions for centrosymmetric crystals [1–3]. This experimental method, which is based on the measurement of distances in 3-beam CBED patterns and was developed from the theory established by Moodie et al. [4–6], has proven an extremely powerful approach to structure solution. In recent work [7], it was shown that the structure of a-Al2O3, with 30 atoms in its unit cell, could be determined to 0.1 A˚ precision by starting from the direct measurement of 3-phase-invariants and then adding measurements of 4 independent structure amplitudes, which were made using the 3-beam CBED techniques of [2,3]. Whilst such inversion of 3-beam CBED patterns has proved a quick and reliable way to determine three-phase invariants (often by inspection), the accuracy of the measurement of magnitudes of structure amplitudes can be limited, in part because of the potential for ambiguity in identifying the position of specific loci, which mark the distances to be measured. These loci have a known orientation and a known form of the intensity distribution but their lateral position within the CBED disc must be identified and this can sometimes be ambiguous, particularly in the
n Corresponding author at: ARC Centre of Excellence for Design in Light Metals, Rm G83, Building 27, Monash University, Victoria 3800, Australia. Tel.: + 61 3 99053827; fax: + 61 3 99054940. E-mail address: [email protected] (P.N.H. Nakashima).
0304-3991/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.ultramic.2011.01.044
presence of significant n-beam perturbations. In the present work, we present a new approach for locating one of the two loci, locus C, which enables it to be more rapidly, precisely and objectively located (locus C is the locus oriented perpendicular to the coupling vector between the two diffracted discs, g and h). This results in more accurate measurements of the 3 distances required to determine the absolute magnitudes of the structure amplitudes, namely, the distances to the Gjønnes–Høier (GH) point and the distance to the centre of symmetry in the intensity along locus C, as per the approach of [2,3]. More significantly, we show that this locus can be used to determine directly and rapidly the relative magnitudes of structure amplitudes from the relative intensities in the diffracted discs along locus C, providing an additional and more accurate means for acquiring this information from the 3-beam CBED pattern. Professor Spence [8] has fostered much research in the related area of structure amplitude refinement through iterative patternmatching of CBED patterns, including 3-beam patterns [9]. We have enjoyed enthusiastic and fruitful engagement with Professor Spence in this and many other areas of electron diffraction and we present this work in honour of him on the occasion of his birthday.
2. 3-Beam CBED revisited At this point, it is worth re-examining the main features of 3-beam CBED patterns from centrosymmetric crystals relevant to the present work. It has already been demonstrated [2,3] that the distances to particular features in 3-beam CBED patterns are
842
P.N.H. Nakashima et al. / Ultramicroscopy 111 (2011) 841–846
sufficient to determine the phase and magnitude of the relevant structure amplitudes without recourse to any pattern-matching refinement. As already described in [2,3], the sign of a three-phase invariant can be obtained by inspection of the corresponding 3-beam CBED pattern, from the direction of deflection of the rocking curve near the 3-beam Bragg condition. This corresponds to the displacement of the centres of symmetry of the two centrosymmetric loci in each diffraction disc. The magnitudes can be determined from the distances of the GH point and centres of symmetry of the centrosymmetric loci from the origin as in [2,3]. Alternatively, we show here that their relative magnitude can be determined from the locus C in the discs g and h alone and that the position of the locus C can be determined with considerable accuracy. This locus is found in every disc and is oriented perpendicular to the coupling vector, g–h, as shown schematically in Fig. 1. This new approach derives from one of the apparent redundancies in the three-beam derivations. In particular, as illustrated in Fig. 1 (see also Fig. 2), a two-beam distribution is generated in all three discs along the locus C whereas the locus has only to be detected in one to complete the inversion. Since the accuracy to which this locus can be defined is an important factor in establishing the accuracy to which the structure can be determined, it becomes relevant to compare the calculated intensity distributions along locus C in both the discs g and h. This can be done very simply using the reduction of the three-beam equations to two-beam form, with the pseudopotential VC ¼ Vg þ iVh
ð1Þ
and the effective excitation error
zC ¼ 2pzg þ
sVh Vhg Vg
as per [5] and using the same notation therein. Thus, ( ! ) ! Uð0Þ pzC sVC 1 ipzC z exp i z ¼e Uð1Þ sVC pzC 0
ð2Þ
ð3Þ
with the solution, sin Uð1Þ ¼ eipzC isVC
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðpzC Þ2 þ s2 VC VC z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðpzC Þ2 þ s2 VC VC
ð4Þ
Equating real and imaginary parts, the central result for this communication is obtained in the form, Vg2 IC,g ¼ 2: IC,h Vh
ð5Þ
Here the symbols for the intensities are italicised in order to emphasise that the ratio holds only along the locus C. It is important to note that this result derives from a dynamical calculation, which is exact within the three-beam approximation. In particular it is not related to the single scattering or kinematical approximation and so the concept of extinction cannot apply. To emphasise this important point, it is briefly elaborated upon. The kinematical or single scattering approximation is properly written as Ig p9Vg 9
2
ð6Þ
As is widely known, this has wide currency when the interaction is weak, for instance in X-ray and neutron scattering (though even here a number of restrictions and corrections are usually invoked). Extinction in X-ray diffraction is the effect of applying the kinematic scattering approximation to the analysis of the total integrated intensity within each reflection via Eq. (6). The lack of a treatment of all dynamic scattering effects gives rise to the concept of extinction and departures from the relativity of structure amplitudes that is suggested by Eq. (6). The fact that Eq. (5) applies to the intensity distribution along a unique locus (locus C), located at equivalent positions in the discs g and h in a 3-beam CBED pattern, and is a consequence of dynamic scattering, removes the concept of extinction all together. Given that Eq. (5) holds for the locus C, which traverses both discs g and h, a simple algorithm can be developed that not only locates the locus C very accurately within all discs, but also results in a more accurate approach to determining Vg/Vh compared with the measurement of amplitudes from distances described previously in [2,3]. A description of the algorithm follows in the next section, making use of a synthetic 3-beam CBED pattern calculated with Vh ¼1.0 V, Vg ¼ 2.0 V and Vg–h ¼3.0 and shown in Fig. 2. The location of locus C is shown in the discs g and h and the corresponding intensity profiles are shown to have the same 2-beam intensity distributions. The amplitudes of these distributions differ by a factor of 4, resulting from Vg/Vh ¼ 2 and evident from Eq. (5). As a consequence, the subtraction of disc h multiplied by 4 from disc g reveals a sharp line of zeroes along locus C across which, the sign of the difference map flips.
3. The algorithm
Fig. 1. A schematic illustration of a general (9g9a 9h9a 9g–h9) 3-beam pattern. The locus C, is perpendicular to the vector g–h, has a centrosymmetric intensity distribution and is located at the same position in each disc as described in [1–6]. The form of the intensity distribution is identical in the discs g and h as shown by the two graphs of intensity along Cg and Ch. The relative intensities along Cg and Ch are determined by the relative magnitudes of the structure amplitudes, as shown in the boxed equation.
Fig. 3 explains the algorithm using the synthetically calculated 3-beam CBED pattern of Fig. 2 as an example (a). The physics of 3-beam-scattering ensures that the locus C is always oriented perpendicular to the coupling vector, g–h, so the starting point is to rotate the pattern in order to make this vector horizontal (the locus C is then oriented vertically in the image) as has already been done in Fig. 2. The angle of rotation is found simply by drawing the tangent common to both discs g and h as shown in Fig. 3a. The discs g and h are then extracted from the pattern (b and c, respectively) with particular care taken in making sure that the sub-images containing each disc are correctly registered with respect to one another (i.e. each point in each of the sub-images b and c are exactly related to one another by the coupling vector g–h). Eq. (5) is the foundation of the algorithm and can be rewritten as: Ig ¼
Vg2
Ih Vh2
¼ mIh ,
m¼
Vg2 Vh2
ð7Þ
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Fig. 2. An ideal synthetic 3-beam pattern (where 9g9 a9h9a 9g–h9) computed without absorption and with Vh ¼ 1.0 V, Vg ¼ 2.0 V and Vg–h ¼3.0 V. It follows from Eq. (5) that Ig ¼ 4Ih along the locus C. The subtraction of 4 times disc h from disc g reveals the position of locus C as a line of zeroes about which the phase of the difference map inverts.
Fig. 3. A schematic description of the algorithm developed here for determining Vg/Vh. The pattern (a) must first be rotated so that the coupling vector, g–h, is horizontal, as in Fig. 2. The discs g and h are then extracted from the rotated pattern and assigned to separate sub-images (b and c). The variable m increases from 0 to some userdefined value, N, and is used to calculate 9(disc g)–m(disc h)9 (d). The boxed region with dimensions X and Y (user-defined, see d) is used to generate the profile (e) of (d) integrated in the direction Y, which is the direction of the locus C. The minimum of the profile (e) is found and compared with the smallest minimum obtained so far, gmin. The process of calculating 9(disc g)–m(disc h)9 is repeated after increasing variable m by a small increment. The algorithm finds the smallest minimum of all profiles (e) obtained from 9(disc g)–m(disc h)9, for 0r mr N.
Multiplying the image of disc h by m and subtracting the product from disc g and taking the absolute value gives 9(disc g)– m(disc h)9 (see Fig. 3d). If the correct value of m is used, then a
vertical line of zeroes (to within the rounding error due to finite pixel size) should present along locus C, not only revealing the location of the locus but also giving the relative magnitudes of Vg
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and Vh. In the example of Fig. 3d, the wrong value of m has deliberately been applied (m¼1, Vg ¼2.0 V and Vh ¼1.0 V) and no vertical dark line is apparent. However, a horizontal intensity profile across Fig. 3d, X, integrated in the vertical axis, Y (the dimensions X and Y shown by the box in Fig. 3d) reveals the existence of a minimum in the integrated intensity profile, ‘‘min’’ (Fig. 3e). The proportionality constant, m, can be determined by varying m until the minimum, ‘‘min’’, in the integrated intensity profile (Fig. 3e) of 9(disc g)–m(disc h)9 (Fig. 3d) reaches a global minimum, ‘‘gmin’’. This is then taken as the final ratio, V2g/V2h. This optimisation approach was taken with the ideal calculated 3-beam CBED pattern of Figs. 2 and 3 in the absence of absorption (Fig. 4a) and the correct ratio of Vg/Vh ¼ 2 returned to better than 0.01%. This level of error is representative of the rounding associated with finite pixel size. In Fig. 4b, absorption was included in the 3-beam calculation of the pattern and a 3% error manifests in the determination of Vg/Vh. To examine a more realistic situation, the effect of n-beam scattering was included in parts c and d of Fig. 4 by including 126 beams in the calculation of the CBED pattern in the same 3-beam orientation as in Figs. 2 and 3. Absorption was excluded in Fig. 4c and included in Fig. 4d. In both cases, the ratio Vg/Vh was underestimated by about 13%, which is indicative of the perturbation caused by many-beam effects. However, in all cases, the position of locus C is very robustly identified. As expected for the 3-beam calculated patterns (Fig. 4a and b), the locus C manifests itself as a straight, unbroken and sharp line of near-zero intensities in 9(disc g)– m(disc h)9. For the many-beam calculated patterns (Fig. 4c and d), locus C still manifests itself quite clearly, though it vanishes very locally or is sharply kinked in the immediate vicinity of strong excitations of other beams. The integrated intensity profiles shown for the boxed regions in all cases show very sharp minima at the X position of locus C, showing that the position determination is robust even in the presence of many-beam effects. This in itself serves as a significant aid to the distance determination techniques of 3-beam CBED used in [1–3]. Application to real experimental 3-beam CBED patterns serves as a final test of the present technique.
4. Analysis of experimental 3-beam CBED data Experimental analysis can be complicated by the presence of the diffuse background due to inelastic scattering in CBED patterns, even if energy-filtering optics are used (due to thermal diffuse scattering). This will inevitably introduce error into any analysis of Ig/Ih as the derivation of Eq. (5) is solely based on elastic scattering theory. However, if one uses the thickness difference approach to CBED first demonstrated in [10], the complicating effects of the inelastic background can be almost completely negated in all forms of CBED. The use of patterns from different specimen thicknesses does not affect the validity of Eqs. (5) and (7) as is shown in the workings below. In these workings, Ig,1 and Ih,1 represent the intensities along the loci C in discs g and h, respectively, in the data collected from specimen thickness H1. Similarly, Ig,2 and Ih,2 represent the intensities along the loci C in discs g and h, respectively, at specimen thickness H2. Thus: m¼
Ig,1 Ig,2 ¼ Ih,1 Ih,2
ð8Þ
then Ig,1 Ig,2 mIh,1 mIh,2 ¼ ¼m Ih,1 Ih,2 Ih,1 Ih,2
ð9Þ
and therefore Eqs. (5) and (7) and the present algorithm are valid for thickness difference 3-beam CBED patterns. Sources of error stemming from the application of these 3-beam equations to an experimental 3-beam CBED pattern include deviations from the ideal 3-beam condition due to N-beam effects, errors in the alignment of disc g with disc h and noise. The use of two separate CBED patterns in the thickness difference approach will magnify the resulting uncertainties, as well as introduce additional errors that stem from the alignment of the two patterns from different thicknesses with one another. This error magnification will, however, be strongly offset by the correction of errors in the measured ratio, m, that arise due to the inelastic background present in single patterns.
Fig. 4. A comparison of 3 and N-beam calculations of the 3-beam CBED pattern of Figs. 2 and 3 and the capacity of the algorithm of Fig. 3 to find the locus C and the ratio Vg/Vh. Each set of output corresponds to Figs. 3d and 3e.
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Fig. 5. An application of the present technique to experimental thickness difference CBED patterns [10] at the zone [16 -10 -1] in a-Al2O3. Two permutations of the same 3-beam triplet are examined (a and f). The algorithm developed here returns the optimum 9(disc g)–m(disc h)9 (b and g) with the corresponding integration profiles (c and h). The minimum in each case is used to locate the loci C in each of the discs g and h in both permutations and the intensity profiles along the loci C are shown (d, e, i and j). In the present cases, V1 2 4/V0 1 10 ¼ 0.97 and V1 1 6/V1 2 4 ¼ 1.66, were determined from the refined values of m in each case.
Fig. 5 shows two permutations of the same 3-beam triplet about the [16 -10 -1] zone in a-Al2O3. The first permutation will yield the ratio V1 2 4/V0 1 10 and the second permutation, the ratio V1 1 6/V1 2 4. These in combination will imply the ratio V1 1 6/V0 1 10. These structure amplitudes have been determined previously [3,10–13] in two different ways; in one case using structure amplitude refinement via quantitative CBED (QCBED) to achieve the highest possible accuracy [10–13] and in the other case, from a direct inversion of 3-beam CBED patterns via the measurement of distances [3]. Close agreement was achieved between these two approaches, making this an excellent example to test the present method. As in the case of the calculated intensities of Fig. 4, the optimum 9(disc g)–m(disc h)9 images (Fig. 5b and g) and corresponding integrated profiles (Fig. 5c and h) reveal the positions of locus C conclusively. In addition to finding the position of locus C in discs g and h for each permutation, the thickness difference intensity distribution along the loci are shown in components d, e, i and j of Fig. 5 at the determined positions marked in the difference patterns a and f. The resemblance in form of the distributions in Fig. 5d and e as well as the pair in Fig. 5i and j is further assurance of the correct identification of locus C in both patterns a and f. The differences in amplitude of the distribution within each pair ((d and e) and (i and j)) are indicative of the ratios V1 2 4/V0 1 10 and V1 1 6/ V1 2 4 that might be returned.
The results of the optimisation of the proportionality constant, m, for each permutation yield the following ratios: V1 2 4 =V0 1 10 ¼ 0:970 70:003 and V1 1 6 =V1 2 4 ¼ 1:663 70:001; implying V1 1 6 =V0 1 10 ¼ 1:613 7 0:006: These structure amplitudes have been refined previously from QCBED pattern-matching analyses [10–13]. Using the values published in [10–13], the corresponding accurate QCBED structure amplitude magnitudes and their ratios are: ðV1 1 6 ¼ 3:7097 0:010 V; V0 1 10 ¼ 2:241 7 0:007 V and V1 2 4 ¼ 2:231 7 0:022 VÞ V1 2 4 =V0 1 10 ¼ 0:996 7 0:013; V1 1 6 =V1 2 4 ¼ 1:662 70:021 and V1 1 6 =V0 1 10 ¼ 1:655 7 0:010: Results of previous 3-beam CBED determinations based on the distance technique [3] give: ðV1 1 6 ¼ 3:80 7 0:42 V; V0 1 10 ¼ 2:45 7 0:55 V and V1 2 4 ¼ 2:23 7 0:53 VÞ V1 2 4 =V0 1 10 ¼ 0:91 7 0:34; V1 1 6 =V1 2 4 ¼ 1:70 70:48 and V1 1 6 =V0 1 10 ¼ 1:55 70:42:
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The present results compare favourably with those from QCBED pattern-matching [10–13] and fall well within the larger uncertainties associated with the measurement of structure magnitudes from distances in 3-beam patterns [3]. The present technique has less redundancy and thus less multiplicity of measurements of the same structure amplitudes than the distance approach to 3-beam CBED, however, it is much more robust and much more precise.
5. Conclusions A new approach to determining the location of the locus C and the relative magnitudes of structure amplitudes in 3-beam CBED patterns is demonstrated. The associated algorithm removes all aspects of subjective judgement in locating the loci and is very robust and precise. Futhermore, the proportionality of the intensities along the locus C in discs g and h gives the ratio of the corresponding magnitudes of the structure amplitudes, Vg/Vh, with surprising accuracy and precision in the present demonstration with experimental data, even in the presence of many-beam perturbations of the ideal 3-beam condition. This new approach offers the prospect of determining unknown structures to a much higher resolution using the analytical inversion of 3-beam CBED patterns from centrosymmetric crystals. Furthermore, the present method of comparing discs g and h may provide an avenue for developing the analytical inversion of 3-beam CBED patterns from non-centrosymmetric crystals.
Acknowledgements This work was supported by the Australian Research Council grant DP0346828. The data used in this work was obtained in the Monash Centre for Electron Microscopy using equipment funded
by the Australian Research Council Grant number RIEFP 99. PN is grateful for the support of the Australian Research Council’s Centre of Excellence for Design in Light Metals. References [1] J. Etheridge, A.F. Moodie, C.J. Humphreys, Direct measurement of structure amplitudes from three beam interactions, electron microscopy, in: Proceedings of the 14th International Congress on Electron Microscopy, vol. III, 1998, pp. 737. [2] P.N.H. Nakashima, A.F. Moodie, J. Etheridge, Structural phase and amplitude measurement from distances in convergent beam electron diffraction patterns, Acta Crystallogr. A 63 (2007) 387. [3] P.N.H. Nakashima, A.F. Moodie, J. Etheridge, A practical guide to the measurement of structure phases and magnitudes by three-beam convergent beam electron diffraction, Ultramicroscopy 108 (2008) 901. [4] A.F. Moodie, Some problems in imaging and inversion, Chem. Scr. 14 (1978-79) 21. [5] A.F. Moodie, J. Etheridge, C.J. Humphreys, The symmetry of three-beam scattering equations: inversion of three-beam diffraction patterns from centrosymmetric crystals, Acta Crystallogr. A 52 (1996) 596. [6] A.F. Moodie, J. Etheridge, C.J. Humphreys, The Electron, IOM Communications Ltd, London, UK, Ch, The Coulomb Interaction and Direct Measurement of Structural Phase, 1998, pp. 235. [7] P.N.H. Nakashima, A.F. Moodie, J. Etheridge, Locating atoms by observing structural phase in centrosymmetric crystals II. Atom location using a Lonsdale-like approach. Acta Crystallogr. A, submitted for publication. [8] J.C.H. Spence, On the accurate measurement of structure-factor amplitudes and phases by electron diffraction, Acta Crystallogr. A 49 (1993) 231. [9] J.M. Zuo, R. Høier, J.C.H. Spence, Three-beam and many-beam theory in electron diffraction and its use for structure-factor phase determination in non-centrosymmetric crystal structures, Acta Crystallogr. A 45 (1989) 839. [10] P.N.H. Nakashima, Thickness difference—a new quantitative filtering tool for electron diffraction, Phys. Rev. Lett. 99 (12) (2007) 125506. [11] P.N.H. Nakashima, B.C. Muddle, Differential convergent beam electron diffraction: experiment and theory, Phys. Rev. B 81 (2010) 115135. [12] V.A. Streltsov, P.N.H. Nakashima, A.W.S. Johnson, A combination method of charge density measurement in hard materials using accurate, quantitative electron and X-ray diffraction: the a-Al2O3 case. [13] P.N.H. Nakashima, Accurate measurements of charge density in a-Al2O3 by combined electron and X-ray diffraction, Ph.D. Thesis, The University of Western Australia, 2002.