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Retrieving modulation parameters from HRTEM images of modulated structures
Ultramicroscopy 96 (2003) 181–190
Retrieving modulation parameters from HRTEM images of modulated structures . Thomas Hoche*, Wolfgang Neumann Lehrstuhl fur . Kristallographie, Institut fur . Physik, Humboldt-Universitat . zu Berlin, InvalidenstraX e 110, Berlin D-10115, Germany Received 24 July 2002; received in revised form 10 December 2002
Abstract Quantitative high-resolution transmission electron microscopy (qHRTEM) is methodologically extended towards the assessment of occupationally and positionally modulated structures. For this purpose, iterative digital image matching has been combined with data processing within the Rietveld refinement code JANA2000. In this approach, the number of free parameters is kept low and rather complicated modulated structures become assessable by qHRTEM. The feasibility of the improved methodology is demonstrated for the 1D modulated structure of Ba2TiGe2O8. r 2003 Elsevier Science B.V. All rights reserved. PACS: 68.37.Lp; 61.18.J; 91.60.Ed Keywords: High-resolution transmission electron microscopy (HRTEM)
1. Introduction Aperiodic crystals do not possess 3D translational symmetry and can be subdivided into three types: incommensurately modulated structures, incommensurate intergrowth compounds and quasicrystals [1]. An incommensurately modulated crystal can be thought of being a slightly disturbed periodic crystal in the sense that the periodic arrangement of atoms is superimposed by distinct modulation waves. Such modulation waves can lead to occupational and/or positional modulations and *Corresponding author. Leibniz-Institut fur . Oberfl.achenmodifizierung e.V., Permoserstrasse 15, D-04318 Leipzig, Germany. Tel.: +49-341235-3287; fax: +49-341-235-3189. . E-mail address: [email protected] (T. Hoche).
originate from either modulations in the electronic structure (e.g., charge–density waves) or competing short-range interactions. De Wolff et al. [2] introduced the superspace description for incommensurate crystals by assigning—in addition to three real space dimensions— up to three mutually independent dimensions to modulation vectors. Using this scheme, superspace groups were subsequently defined [3] and refinement procedures commonly used for solving ordinary structures could be transferred to crystals hosting positional and occupational modulations [4,5]. Using either X-ray diffraction or neutron diffraction, the latter routines have been successfully applied to a considerable number of modulated phases, including melilites [6] and fresnoites [7,8].
0304-3991/03/$ - see front matter r 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0304-3991(03)00006-8
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T. Hoche, W. Neumann / Ultramicroscopy 96 (2003) 181–190 .
Diffraction methods, integrating over at least a few thousand mm3, suffer from the inherent assumption that the crystal investigated is structurally and chemically homogeneous. Therefore, a spatially resolving method enabling the localised assessment of structural modulations would be highly desirable. Transmission electron microscopy (TEM) is the most promising method in this respect since—in addition to its outstanding spatial resolution—chemical homogeneity can be verified by applying analytical techniques including energy-dispersive X-ray analysis and electron energy-loss spectroscopy. Moreover, structural homogeneity of the modulation can be seen in high-resolution TEM images [9] or by applying electron microdiffraction [7]. Quantitative high-resolution TEM (qHRTEM) has been successfully applied to the study of grain boundaries in metals and ceramics [10–13]. The basic idea of qHRTEM is to compare simulated HRTEM images with one or more experimental images. Based on a measure of image similarity, the structural model is subsequently refined until an acceptable agreement between experiment and simulation is achieved. In general, structural parameters (atomic positions, translation state of rigid slabs, etc.) as well as imaging parameters (defocus, aberration coefficients, etc.) can be simultaneously refined. However, the number of parameters should be kept as low as possible. In this contribution, we will demonstrate that qHRTEM can be generalised towards making it applicable for the determination of modulation vectors for a known average structure. The methodology is applied to a test structure, the fresnoite-type framework germanate Ba2TiGe2O8 (BTG), since the latter hosts an extremely pronounced 1D modulation [8,14–16]. In the average structure of BTG (Fig. 1), Ge2O7 pyrogermanate groups and TiO5 pyramids are forming sheets perpendicular to [0 0 1] and large barium ions are interspersed between these sheets. At room temperature, the superspace group of BTG is Cmm2 (0,b;12) s00 with b ¼ 0:635 [8]. Based on a thorough analysis of extinction conditions, this superspace group was independently predicted by Withers et al. [16] from an extended series of electron diffraction patterns.
Fig. 1. Average structure of Ba2TiGe2O8 (BTG).
Fig. 2. HRTEM image of BTG in the projection along [0 0 1] showing the modulation wave propagating along the [1 0 0]direction recorded in a JEM 4000EX operated at 400 kV close to Scherzer focus. The modulation causes a superstructure of period B3a to become superimposed on the image.
Fig. 2 illustrates that the modulation wave can in fact be seen in HRTEM images of BTG taken along [0 0 1], but in this contribution, we shall restrict ourselves to methodological aspects
T. Hoche, W. Neumann / Ultramicroscopy 96 (2003) 181–190 .
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2. Results
normalised Euclidean distance, NED, was used as an image-agreement factor when comparing images I1 and I2 : vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u P 2 u k;l jI1 ðrkl Þ I1 ðrkl Þj ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q : NED ¼ u t P P 2 2 ðr Þ I I ðr Þ kl kl k;l 1 k;l 2
The consideration of modulations in qHRTEM is greatly facilitated when supercells of the modulated structure can be generated based on a set of modulation parameters. Due to the cooperativeness of Vaclav Petricek and Michal Dusek, the latest version of their crystallographic computing system JANA2000 [5] offers an option to generate a modulated supercell as required by Stadelmann’s EMS code [17] used in this work for HRTEM image simulation. Now, one has to define, govern, and finally refine modulation parameters. Since JANA2000 offers an elaborated scheme of introducing and handling modulation wave vectors, m40-files used by JANA2000 for storing and updating information on the modulation were adopted. In Figs. 3 and 4, flow charts for the initiation and course of the structural refinement are given. The whole routine is controlled by a scheduler written in IDL (Research Systems, Inc., Boulder, CO, USA), similar to a code used earlier [11,18]. After defining crystallographic parameters including (super) space group, atomic positions, and modulation parameters, corresponding files for data handling (*.m40, *.m50, etc. in JANA2000) are generated first. In the next step, within the scheduler, an n-dimensional parameter vector, p; containing variables to be optimised is defined and filled with data from the m40-file. Then, an (n n)-dimensional parameter space matrix, S; is spanned by first filling all columns with p’s and superimposing to this matrix a perturbation d (equal to 7s; with the starting step size s). For each column of S; an m40-file is written, an EMS supercell is generated by JANA2000, and an EMS image simulation is performed. The resulting simulated HRTEM images are consecutively compared with the experimental image and the corresponding image-agreement factors are stored in the n-dimensional target function f : The
In order to check the methodology, in this study the experimental image was replaced by a simulated image based on a 3 1 2 approximant structure generated from modulation parameters obtained by neutron powder diffraction (columns 3–5 in Table 1, [8]). Applying the very robust downhill-simplex algorithm developed by Nelder and Mead [19], the target function f (NED of ‘‘experimental’’ and model-structure based simulated HRTEM image) is optimised until satisfactory agreement between experiment and model structure is attained (Fig. 4). One refinement cycle takes about 60 s on a Compaq Alpha Server DS20E equipped with two 64 bit Alpha EV6.7 processors running at 667 MHz and 4GB RAM (resulting in a total performance of about 514 MFLOPs). For HRTEM image simulation, the following parameter were used: acceleration voltage=400 kV, cs ¼ 1:0 nm, defocus=50 nm (Scherzer focus corresponds to 49.66 nm), spread of focus=9 nm, beam semi-convergence=0.6 mrad, total thickness=10 1.0737 nm. Starting at NED ¼ 0:919; the refinement of 56 parameters readily converged and reached a NED value of less than 0.013 after about 3000 cycles (Fig. 5). In Figs. 6a–d, the starting simulated image (no modulation superimposed to the average BTG structure) is compared with the ‘‘experimental’’ image. In order to avoid aliasing effects, NED values were calculated for regions of slightly reduced size (as indicated in Figs. 6 and 7). In the difference image of ‘‘experimental’’ and simulated image (Fig. 6c), deviations, between +15% and 14.5% are found to be more or less evenly spread about the image (see intensity adapted image in Fig. 6d). After refinement (Figs. 7a–d), deviations have been reduced to between +0.8% and 0.3%. In Figs. 8a–d, structural changes associated with the refinement are illustrated. By removing the
without application to experimental data. This is mainly due to the fact that high-quality HRTEM images of BTG are difficult to obtain due to severe radiation damage under the electron beam.
T. Hoche, W. Neumann / Ultramicroscopy 96 (2003) 181–190 .
184
Define crystallographic parameters (space group, atomic positions, modulation parameters, etc.)
JANA2000 Write *.m40 file
Read *.m40 file
IDL
Define parameter vector, p = (p1, p2, p3, ..., pn) to become optimised
Span parameter space, S, to become optimised (IDL):
S=
p1+δ p2 p3 . . . pn
p1 p2 +δ p3 . . . pn
p1 p2 p3 +δ . . . pn
. . . . . . .
. . . . . . .
. . . . . . .
p1 p2 p3 . . . pn +δ
Write *.m40 file for each column
Generate supercell file from *.m40-file for each column
JANA2000
Simulate HRTEM image for each supercell
EMS
• Compare simulated HRTEM image for each supercell with experimental image n-dimensional target function f (vector consisting of n image agreement factors) • Search for optimum of target function f (down-hill simplex) until agreement between experiment and simulation is sufficient
IDL
Fig. 3. General flowchart of the modulation-parameter retrieval showing the interplay of JANA2000, the IDL scheduler, and EMS.
modulation Fourier amplitudes of all modulation waves, the starting supercell depicted in Fig. 8b is obtained from the structure shown in Fig. 8a. After HRTEM-image refinement, the supercell
shown in Fig. 8c ([0 0 1]-projection) and Fig. 8d (along [0 1 0]) and modulation-wave Fourier amplitudes compiled in columns 6–8 of Table 1 are obtained.
T. Hoche, W. Neumann / Ultramicroscopy 96 (2003) 181–190 .
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Determine refined parameter vector according to the downhill simplex method by either of the four steps in multidimensions: 1. Reflection away from the high point in multidimensional space, 2. Reflection and expansion away from the high point, 3. Contraction along one dimension from the high point, or 4. Contraction along all dimensions towards the low point
IDL
Write *.m40 file
Generate supercell file
JANA2000
Read *.m40 file
Write *.cel file
EMS
Simulate HRTEM image
Write *.ima file
Write *.ima file
f > predefined limit
• Compare simulated HRTEM image with experimental HRTEM image • Calculate image-agreement factor f
Write results files (*.cel, *.m40)
If (f)
f < predefined limit
IDL
Fig. 4. Detailed flowchart illustrating actions performed within each single refinement step.
The HRTEM-based refinement shows a very reasonable agreement of x- and y-modulation amplitudes for the heavily scattering barium ions (maximum departure=2.5%, cf. columns 6–8 in Table 1), while the z-component of Ba2 is about 6.4% off. For germanium and titanium
ions, Fourier amplitudes are less precisely reproduced which is mainly due to the fact that absolute values of those amplitudes are rather small (the modulation is mainly hosted in the oxygen sublattice). For oxygen positions, modulation amplitudes are better reproduced than
T. Hoche, W. Neumann / Ultramicroscopy 96 (2003) 181–190 .
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Table 1 Fourier amplitudes of sine (s) and cosine (c) displacive modulation functions Site
Wave
Ba1
s,1 c,1
Ba2
s,1 c,1
Ge1
Refinement of neutron powder-diffraction data (target structure)
Refinement of HRTEM data
x
x
y 0.0085(11) 0.0064(11)
z
y
0 0
0 0
0 0.0066(9)
0.0123(10) 0
0 0.0146(9)
s,1 c,1
0.0024(6) 0.0121(6)
0 0
0 0
Ge2
s,1 c,1
0 0.0175(7)
0.0107(5) 0
0 0.0021(10)
0 0.013860
0.013179 0
0 0.004728
Ti1
s,1 c,1
0 0.0043(11)
0 0.0204(12)
0.0070(14) 0
0 0.006065
0 0.014455
0.00731 0
O1
s,1 c,1
0.0142(5) 0.0038(6)
0.0360(7) 0.0198(8)
0.0204(12) 0.0008(15)
0.015234 0.003738
0.035164 0.020139
0.014176 0.018415
O2
s,1 c,1
0 0.0478(10)
0 0.048056
0 0
0 0
O3
s,1 c,1
0.0256(6) 0.0173(6)
O4
s,1 c,1
0 0.0200(9)
0 0
0 0
0 0.019952
0 0
0 0
O5
s,1 c,1
0.0071(10) 0.0059(10)
0 0
0 0
0.008395 0.017234
0 0
0 0
O6
s,1 c,1
0 0.0084(9)
O7
s,1 c,1
0 0.0024(12)
0 0 0.0195(5) 0.0073(5)
0.0115(10) 0 0 0.0229(9)
expected with the exception of the z-component of O3 s; 1: At the first glance, the agreement in terms of Fourier amplitudes seems to be rather insufficient. However, closer inspection of Fig. 8 reveals that in fact the agreement in terms of [0 0 1]-projected atomic positions is not as bad. The statistical analysis of fractional coordinates in the original and refined supercells gives mean positional errors and corresponding standard deviations listed in
0 0 0.0050(8) 0.0051(9)
0.008553 0.006540
z 0 0
0 0
0 0.006765
0.012293 0
0 0.015541
0.001058 0.006553
0 0
0 0
0.025820 0.016765
0.019737 0.004614
0.128540 0.004105
0 0.0018(11)
0 0.018156
0.000536 0
0 0.022730
0.0055(11) 0
0 0.000794
0 0.034676
0.002863 0
Table 2. While perpendicular to the imaging direction (along [0 0 1]), the average error is below 0.1 pm, positions are not as precise parallel to the direction of projection. The standard deviation of non-fixed oxygen y- and z-coordinates is significantly larger than for the other coordinates. This is related to the z-component of the sine Fourier amplitude of O3 sites which is 1.5 orders of magnitude larger than the corresponding target value. In Fig. 8d, it can be
T. Hoche, W. Neumann / Ultramicroscopy 96 (2003) 181–190 .
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Euklidean norm (image agreement)
1
initial image agreement = 0.919
0.1
0.01 0
500
1000
1500
2000
2500
3000
number of refinement cycle
Fig. 5. NED versus refinement-cycle number.
Fig. 7. HRTEM images at the end of the refinement: (a) target image (used as experimental image); (b) HRTEM-image corresponding to the best-fitting modulated structure; (c) difference between (a) and (b); (d) intensity adapted image (c).
Fig. 6. HRTEM images at the beginning of the refinement: (a) target image (used as experimental image); (b) HRTEM-image corresponding to the non-modulated average structure; (c) difference between (a) and (b); (d) intensity adapted image (c).
clearly seen that there are GeO4 tetrahedra deviating significantly from the regular tetrahedral shape resulting in z-coordinate deviations as large
( This can be understood as follows: Due as 1.37 A. to the strongly forward-directed scattering in TEM, another combination of Fourier amplitudes has been found by the refinement algorithm yielding an almost perfect match in the direction of projection. Such effects can only be avoided by imposing constraints to the refinement by either setting limits for bond lengths and bond angles or defining polyhedral entities. Therefore, GeO4 groups in the average structure were treated as absolutely rigid units and modulation-wave Fourier amplitudes for single atoms (Ba1, Ba2, Ti, O7 [apical oxygen in TiO5 pyramids]) and two distinct germanate groups were considered. In the present case, such definition does not reduce the number of modulation parameters (56), since site-symmetry related parameter fixation of constituting atoms gets lost with the introduction of molecules. The refinement
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T. Hoche, W. Neumann / Ultramicroscopy 96 (2003) 181–190 .
Fig. 8. (3 1 2) BTG supercells with coordination polyhedra indicated: (a) [0 0 1]-projected approximant of the modulated structure as determined from neutron-powder diffraction data [8]; (b) non-modulated average structure (starting model) in [0 0 1]-projection; best-fitting modulated structure after refinement in [0 0 1]- (c) and [0 1 0]-projection.
of the modulated BTG structure using rigid GeO4 units, however, gave an unsatisfactory final NED value of 0.27. Since a similar finding is obtained with the constrained refinement of neutron
powder-diffraction data [8], it can be concluded that in BTG, a certain flexibility of the coordination polyhedra is required to adequately describe the modulated structure.
T. Hoche, W. Neumann / Ultramicroscopy 96 (2003) 181–190 . Table 2 Mean error, m:e:; and standard deviation, s:d:; of non-fixed atomic positions for the comparison of original and refined supercell. Numbers of non-fixed atoms considered are given in brackets
Ba
Ge
Ti
O
x-coordinate
y-coordinate
z-coordinate
( (m.e.) 0.0074 A ( (s.d.) 0.1024 A (22)
( (m.e.) 0.0044 A ( (s.d.) 0.0850 A (60)
( (m.e.) 0.0552 A ( (s.d.) 0.1028 A (25)
( (m.e.) 0.0084 A ( (s.d.) 0.1109 A (22)
( (m.e.) 0.0037 A ( (s.d.) 0.0968 A (49)
( (m.e.) 0.0017 A ( (s.d.) 0.0462 A (25)
( (m.e.) 0.0009 A ( (s.d.) 0.1146 A (24)
( (m.e.) 0.0000 A ( (s.d.) 0.0600 A (24)
( (m.e.) 0.0001 A ( (s.d.) 0.0510 A (24)
( (m.e.) 0.0003 A ( (s.d.) 0.0019 A (136)
( 0.0037 A ( (s.d.) 0.2219 A (192)
( 0.0011 A 0.5810 (s.d.) (145)
As the usage of modulation waves for polyhedral units instead of modulated individual atoms will allow to better control z-components and generally reduce the number of free parameters, the usage of the molecule option should be advantageous for more complicated structures with rather rigid coordination polyhedra. For such cases, it will not only speed up the calculation (or make it even feasible) but will improve the reliability of the refinement. In this study, the image analysis procedure is demonstrated for a single image, simulated close to Scherzer focus. When applied to real experimental images, the precision of modulation parameters obtained will be deteriorated due to noise superimposed to the image as well as imaging and electron–matter interaction parameters to be refined, too. Hence, the method should be ideally performed on reconstructed exit wave functions as this would admit to leave imaging parameters out of the refinement.
3. Conclusions The methodology introduced could be shown to allow to retrieve Fourier amplitudes for modula-
189
tion waves in modulated structure from iterative digital HRTEM image matching. The latter approach is not restricted to somewhat exotic incommensurately modulated structures, as found in BTG, but can be applied to all kinds of superstructures, e.g. polytypes, commensurate modulations, etc. In the present example, the refinement of Fourier amplitudes instead of individual atomic positions reduced the number of free parameters from 987 (x-, y-, and z-coordinates of all individual atoms in the approximant supercell) to 56. This significant reduction is achieved by just considering a priori information on structural symmetry. Parameter spaces with 56 dimensions can be still handled while about 1000-dimensional optimisations are not feasible at all. The consideration of coordination-polyhedra modulations generally keeps the number of free parameters small also for even more complicated structures. The latter model might give poorer results for structures with some inherent flexibility of the coordination polyhedra but z-components can stand to benefit from the constraints imposed.
Acknowledgements We would like to sincerely thank Vaclav Petricek and Michal Dusek, Institute of Physics, Academy of Sciences of the Czech Republic in Prague for mounting the HRTEM supercell option in JANA2000. T.H. is indebted to Uwe Richter for granting access to workstations of the Faculty of Mathematics & Computer Science at Friedrich-Schiller-Universit.at Jena and manifold technical support. The HRTEM image was recorded at the Institute for Microstructure Research in Julich . and visits to this laboratory were funded within the program ‘‘GroXger.ateinitiative Analytische HRTEM’’ by Deutsche Forschungsgemeinschaft Bonn (NE646/4-1).
References [1] S. van Smaalen, Cryst. Rev. 4 (1995) 79. [2] P.M. de Wolff, T. Janssen, A. Janner, Acta Cryst. A 37 (1981) 625.
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[3] T. Janssen, A. Janner, A. Looijenga-Vos, P.M. de Wolff, in: A.J.C. Wilson (Ed.), International Tables for Crystallography, Vol. C, Mathematical, Physical and Chemical Tables, Kluwer Academic Publishers, Dordrecht, 1995, p. 797–844. [4] V. Petricek, P. Coppens, Acta Cryst. A 44 (1988) 235. [5] V. Petricek, M. Dusek, The Crystallographic Computing System JANA2000, Institute of Physics, Praha, Czech Republic, 2000. [6] L. Bindi, P. Bonazzi, M. Dusek, V. Petricek, G. Chapuis, Acat Cryst. B 57 (2001) 739. . [7] T. Hoche, W. Neumann, S. Esmaeilzadeh, R. Uecker, M. Lentzen, C. Russel, . J. Solid State Chem. 166 (2002) 15. . [8] T. Hoche, S. Esmaeilzadeh, R. Uecker, S. Lidin, W. Neumann, Acta Cryst. B, 2002, accepted for publication.
[9] C. van Heurck, G. van Tendeloo, S. Amelinckx, Phys. Chem. Minerals 18 (1992) 441. [10] D. Hofmann, F. Ernst, Ultramicroscopy 53 (1994) 205. . [11] T. Hoche, P.R. Kenway, H.-J. Kleebe, M.W. Finnis, M. Ruhle, . J. Phys. Chem. Solids 55 (1994) 1067. [12] W.E. King, G.H. Campbell, Ultramicroscopy 56 (1994) 46. . [13] G. Mobus, R. Schweinfest, T. Gemming, T. Wagner, M. Ruhle, . J. Microscopy 190 (1998) 109. . [14] T. Hoche, C. Russel, . W. Neumann, Solid State Commun. 110 (1999) 651. . [15] T. Hoche, W. Neumann, Ferroelectrics 250 (2001) 67. . [16] R.L. Withers, Y. Tabira, Y. Liu, T. Hoche, Phys. Chem. Minerals 29 (2002) 624. [17] P.A. Stadelmann, Ultramicroscopy 21 (1987) 131. . [18] T. Hoche, M. Ruhle, . J. Am. Ceram. Soc. 79 (1996) 1961. [19] J.A. Nelder, R. Mead, Comput. J. 7 (1965) 308.
Retrieving modulation parameters from HRTEM images of modulated structures . Thomas Hoche*, Wolfgang Neumann Lehrstuhl fur . Kristallographie, Institut fur . Physik, Humboldt-Universitat . zu Berlin, InvalidenstraX e 110, Berlin D-10115, Germany Received 24 July 2002; received in revised form 10 December 2002
Abstract Quantitative high-resolution transmission electron microscopy (qHRTEM) is methodologically extended towards the assessment of occupationally and positionally modulated structures. For this purpose, iterative digital image matching has been combined with data processing within the Rietveld refinement code JANA2000. In this approach, the number of free parameters is kept low and rather complicated modulated structures become assessable by qHRTEM. The feasibility of the improved methodology is demonstrated for the 1D modulated structure of Ba2TiGe2O8. r 2003 Elsevier Science B.V. All rights reserved. PACS: 68.37.Lp; 61.18.J; 91.60.Ed Keywords: High-resolution transmission electron microscopy (HRTEM)
1. Introduction Aperiodic crystals do not possess 3D translational symmetry and can be subdivided into three types: incommensurately modulated structures, incommensurate intergrowth compounds and quasicrystals [1]. An incommensurately modulated crystal can be thought of being a slightly disturbed periodic crystal in the sense that the periodic arrangement of atoms is superimposed by distinct modulation waves. Such modulation waves can lead to occupational and/or positional modulations and *Corresponding author. Leibniz-Institut fur . Oberfl.achenmodifizierung e.V., Permoserstrasse 15, D-04318 Leipzig, Germany. Tel.: +49-341235-3287; fax: +49-341-235-3189. . E-mail address: [email protected] (T. Hoche).
originate from either modulations in the electronic structure (e.g., charge–density waves) or competing short-range interactions. De Wolff et al. [2] introduced the superspace description for incommensurate crystals by assigning—in addition to three real space dimensions— up to three mutually independent dimensions to modulation vectors. Using this scheme, superspace groups were subsequently defined [3] and refinement procedures commonly used for solving ordinary structures could be transferred to crystals hosting positional and occupational modulations [4,5]. Using either X-ray diffraction or neutron diffraction, the latter routines have been successfully applied to a considerable number of modulated phases, including melilites [6] and fresnoites [7,8].
0304-3991/03/$ - see front matter r 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0304-3991(03)00006-8
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T. Hoche, W. Neumann / Ultramicroscopy 96 (2003) 181–190 .
Diffraction methods, integrating over at least a few thousand mm3, suffer from the inherent assumption that the crystal investigated is structurally and chemically homogeneous. Therefore, a spatially resolving method enabling the localised assessment of structural modulations would be highly desirable. Transmission electron microscopy (TEM) is the most promising method in this respect since—in addition to its outstanding spatial resolution—chemical homogeneity can be verified by applying analytical techniques including energy-dispersive X-ray analysis and electron energy-loss spectroscopy. Moreover, structural homogeneity of the modulation can be seen in high-resolution TEM images [9] or by applying electron microdiffraction [7]. Quantitative high-resolution TEM (qHRTEM) has been successfully applied to the study of grain boundaries in metals and ceramics [10–13]. The basic idea of qHRTEM is to compare simulated HRTEM images with one or more experimental images. Based on a measure of image similarity, the structural model is subsequently refined until an acceptable agreement between experiment and simulation is achieved. In general, structural parameters (atomic positions, translation state of rigid slabs, etc.) as well as imaging parameters (defocus, aberration coefficients, etc.) can be simultaneously refined. However, the number of parameters should be kept as low as possible. In this contribution, we will demonstrate that qHRTEM can be generalised towards making it applicable for the determination of modulation vectors for a known average structure. The methodology is applied to a test structure, the fresnoite-type framework germanate Ba2TiGe2O8 (BTG), since the latter hosts an extremely pronounced 1D modulation [8,14–16]. In the average structure of BTG (Fig. 1), Ge2O7 pyrogermanate groups and TiO5 pyramids are forming sheets perpendicular to [0 0 1] and large barium ions are interspersed between these sheets. At room temperature, the superspace group of BTG is Cmm2 (0,b;12) s00 with b ¼ 0:635 [8]. Based on a thorough analysis of extinction conditions, this superspace group was independently predicted by Withers et al. [16] from an extended series of electron diffraction patterns.
Fig. 1. Average structure of Ba2TiGe2O8 (BTG).
Fig. 2. HRTEM image of BTG in the projection along [0 0 1] showing the modulation wave propagating along the [1 0 0]direction recorded in a JEM 4000EX operated at 400 kV close to Scherzer focus. The modulation causes a superstructure of period B3a to become superimposed on the image.
Fig. 2 illustrates that the modulation wave can in fact be seen in HRTEM images of BTG taken along [0 0 1], but in this contribution, we shall restrict ourselves to methodological aspects
T. Hoche, W. Neumann / Ultramicroscopy 96 (2003) 181–190 .
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2. Results
normalised Euclidean distance, NED, was used as an image-agreement factor when comparing images I1 and I2 : vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u P 2 u k;l jI1 ðrkl Þ I1 ðrkl Þj ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q : NED ¼ u t P P 2 2 ðr Þ I I ðr Þ kl kl k;l 1 k;l 2
The consideration of modulations in qHRTEM is greatly facilitated when supercells of the modulated structure can be generated based on a set of modulation parameters. Due to the cooperativeness of Vaclav Petricek and Michal Dusek, the latest version of their crystallographic computing system JANA2000 [5] offers an option to generate a modulated supercell as required by Stadelmann’s EMS code [17] used in this work for HRTEM image simulation. Now, one has to define, govern, and finally refine modulation parameters. Since JANA2000 offers an elaborated scheme of introducing and handling modulation wave vectors, m40-files used by JANA2000 for storing and updating information on the modulation were adopted. In Figs. 3 and 4, flow charts for the initiation and course of the structural refinement are given. The whole routine is controlled by a scheduler written in IDL (Research Systems, Inc., Boulder, CO, USA), similar to a code used earlier [11,18]. After defining crystallographic parameters including (super) space group, atomic positions, and modulation parameters, corresponding files for data handling (*.m40, *.m50, etc. in JANA2000) are generated first. In the next step, within the scheduler, an n-dimensional parameter vector, p; containing variables to be optimised is defined and filled with data from the m40-file. Then, an (n n)-dimensional parameter space matrix, S; is spanned by first filling all columns with p’s and superimposing to this matrix a perturbation d (equal to 7s; with the starting step size s). For each column of S; an m40-file is written, an EMS supercell is generated by JANA2000, and an EMS image simulation is performed. The resulting simulated HRTEM images are consecutively compared with the experimental image and the corresponding image-agreement factors are stored in the n-dimensional target function f : The
In order to check the methodology, in this study the experimental image was replaced by a simulated image based on a 3 1 2 approximant structure generated from modulation parameters obtained by neutron powder diffraction (columns 3–5 in Table 1, [8]). Applying the very robust downhill-simplex algorithm developed by Nelder and Mead [19], the target function f (NED of ‘‘experimental’’ and model-structure based simulated HRTEM image) is optimised until satisfactory agreement between experiment and model structure is attained (Fig. 4). One refinement cycle takes about 60 s on a Compaq Alpha Server DS20E equipped with two 64 bit Alpha EV6.7 processors running at 667 MHz and 4GB RAM (resulting in a total performance of about 514 MFLOPs). For HRTEM image simulation, the following parameter were used: acceleration voltage=400 kV, cs ¼ 1:0 nm, defocus=50 nm (Scherzer focus corresponds to 49.66 nm), spread of focus=9 nm, beam semi-convergence=0.6 mrad, total thickness=10 1.0737 nm. Starting at NED ¼ 0:919; the refinement of 56 parameters readily converged and reached a NED value of less than 0.013 after about 3000 cycles (Fig. 5). In Figs. 6a–d, the starting simulated image (no modulation superimposed to the average BTG structure) is compared with the ‘‘experimental’’ image. In order to avoid aliasing effects, NED values were calculated for regions of slightly reduced size (as indicated in Figs. 6 and 7). In the difference image of ‘‘experimental’’ and simulated image (Fig. 6c), deviations, between +15% and 14.5% are found to be more or less evenly spread about the image (see intensity adapted image in Fig. 6d). After refinement (Figs. 7a–d), deviations have been reduced to between +0.8% and 0.3%. In Figs. 8a–d, structural changes associated with the refinement are illustrated. By removing the
without application to experimental data. This is mainly due to the fact that high-quality HRTEM images of BTG are difficult to obtain due to severe radiation damage under the electron beam.
T. Hoche, W. Neumann / Ultramicroscopy 96 (2003) 181–190 .
184
Define crystallographic parameters (space group, atomic positions, modulation parameters, etc.)
JANA2000 Write *.m40 file
Read *.m40 file
IDL
Define parameter vector, p = (p1, p2, p3, ..., pn) to become optimised
Span parameter space, S, to become optimised (IDL):
S=
p1+δ p2 p3 . . . pn
p1 p2 +δ p3 . . . pn
p1 p2 p3 +δ . . . pn
. . . . . . .
. . . . . . .
. . . . . . .
p1 p2 p3 . . . pn +δ
Write *.m40 file for each column
Generate supercell file from *.m40-file for each column
JANA2000
Simulate HRTEM image for each supercell
EMS
• Compare simulated HRTEM image for each supercell with experimental image n-dimensional target function f (vector consisting of n image agreement factors) • Search for optimum of target function f (down-hill simplex) until agreement between experiment and simulation is sufficient
IDL
Fig. 3. General flowchart of the modulation-parameter retrieval showing the interplay of JANA2000, the IDL scheduler, and EMS.
modulation Fourier amplitudes of all modulation waves, the starting supercell depicted in Fig. 8b is obtained from the structure shown in Fig. 8a. After HRTEM-image refinement, the supercell
shown in Fig. 8c ([0 0 1]-projection) and Fig. 8d (along [0 1 0]) and modulation-wave Fourier amplitudes compiled in columns 6–8 of Table 1 are obtained.
T. Hoche, W. Neumann / Ultramicroscopy 96 (2003) 181–190 .
185
Determine refined parameter vector according to the downhill simplex method by either of the four steps in multidimensions: 1. Reflection away from the high point in multidimensional space, 2. Reflection and expansion away from the high point, 3. Contraction along one dimension from the high point, or 4. Contraction along all dimensions towards the low point
IDL
Write *.m40 file
Generate supercell file
JANA2000
Read *.m40 file
Write *.cel file
EMS
Simulate HRTEM image
Write *.ima file
Write *.ima file
f > predefined limit
• Compare simulated HRTEM image with experimental HRTEM image • Calculate image-agreement factor f
Write results files (*.cel, *.m40)
If (f)
f < predefined limit
IDL
Fig. 4. Detailed flowchart illustrating actions performed within each single refinement step.
The HRTEM-based refinement shows a very reasonable agreement of x- and y-modulation amplitudes for the heavily scattering barium ions (maximum departure=2.5%, cf. columns 6–8 in Table 1), while the z-component of Ba2 is about 6.4% off. For germanium and titanium
ions, Fourier amplitudes are less precisely reproduced which is mainly due to the fact that absolute values of those amplitudes are rather small (the modulation is mainly hosted in the oxygen sublattice). For oxygen positions, modulation amplitudes are better reproduced than
T. Hoche, W. Neumann / Ultramicroscopy 96 (2003) 181–190 .
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Table 1 Fourier amplitudes of sine (s) and cosine (c) displacive modulation functions Site
Wave
Ba1
s,1 c,1
Ba2
s,1 c,1
Ge1
Refinement of neutron powder-diffraction data (target structure)
Refinement of HRTEM data
x
x
y 0.0085(11) 0.0064(11)
z
y
0 0
0 0
0 0.0066(9)
0.0123(10) 0
0 0.0146(9)
s,1 c,1
0.0024(6) 0.0121(6)
0 0
0 0
Ge2
s,1 c,1
0 0.0175(7)
0.0107(5) 0
0 0.0021(10)
0 0.013860
0.013179 0
0 0.004728
Ti1
s,1 c,1
0 0.0043(11)
0 0.0204(12)
0.0070(14) 0
0 0.006065
0 0.014455
0.00731 0
O1
s,1 c,1
0.0142(5) 0.0038(6)
0.0360(7) 0.0198(8)
0.0204(12) 0.0008(15)
0.015234 0.003738
0.035164 0.020139
0.014176 0.018415
O2
s,1 c,1
0 0.0478(10)
0 0.048056
0 0
0 0
O3
s,1 c,1
0.0256(6) 0.0173(6)
O4
s,1 c,1
0 0.0200(9)
0 0
0 0
0 0.019952
0 0
0 0
O5
s,1 c,1
0.0071(10) 0.0059(10)
0 0
0 0
0.008395 0.017234
0 0
0 0
O6
s,1 c,1
0 0.0084(9)
O7
s,1 c,1
0 0.0024(12)
0 0 0.0195(5) 0.0073(5)
0.0115(10) 0 0 0.0229(9)
expected with the exception of the z-component of O3 s; 1: At the first glance, the agreement in terms of Fourier amplitudes seems to be rather insufficient. However, closer inspection of Fig. 8 reveals that in fact the agreement in terms of [0 0 1]-projected atomic positions is not as bad. The statistical analysis of fractional coordinates in the original and refined supercells gives mean positional errors and corresponding standard deviations listed in
0 0 0.0050(8) 0.0051(9)
0.008553 0.006540
z 0 0
0 0
0 0.006765
0.012293 0
0 0.015541
0.001058 0.006553
0 0
0 0
0.025820 0.016765
0.019737 0.004614
0.128540 0.004105
0 0.0018(11)
0 0.018156
0.000536 0
0 0.022730
0.0055(11) 0
0 0.000794
0 0.034676
0.002863 0
Table 2. While perpendicular to the imaging direction (along [0 0 1]), the average error is below 0.1 pm, positions are not as precise parallel to the direction of projection. The standard deviation of non-fixed oxygen y- and z-coordinates is significantly larger than for the other coordinates. This is related to the z-component of the sine Fourier amplitude of O3 sites which is 1.5 orders of magnitude larger than the corresponding target value. In Fig. 8d, it can be
T. Hoche, W. Neumann / Ultramicroscopy 96 (2003) 181–190 .
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Euklidean norm (image agreement)
1
initial image agreement = 0.919
0.1
0.01 0
500
1000
1500
2000
2500
3000
number of refinement cycle
Fig. 5. NED versus refinement-cycle number.
Fig. 7. HRTEM images at the end of the refinement: (a) target image (used as experimental image); (b) HRTEM-image corresponding to the best-fitting modulated structure; (c) difference between (a) and (b); (d) intensity adapted image (c).
Fig. 6. HRTEM images at the beginning of the refinement: (a) target image (used as experimental image); (b) HRTEM-image corresponding to the non-modulated average structure; (c) difference between (a) and (b); (d) intensity adapted image (c).
clearly seen that there are GeO4 tetrahedra deviating significantly from the regular tetrahedral shape resulting in z-coordinate deviations as large
( This can be understood as follows: Due as 1.37 A. to the strongly forward-directed scattering in TEM, another combination of Fourier amplitudes has been found by the refinement algorithm yielding an almost perfect match in the direction of projection. Such effects can only be avoided by imposing constraints to the refinement by either setting limits for bond lengths and bond angles or defining polyhedral entities. Therefore, GeO4 groups in the average structure were treated as absolutely rigid units and modulation-wave Fourier amplitudes for single atoms (Ba1, Ba2, Ti, O7 [apical oxygen in TiO5 pyramids]) and two distinct germanate groups were considered. In the present case, such definition does not reduce the number of modulation parameters (56), since site-symmetry related parameter fixation of constituting atoms gets lost with the introduction of molecules. The refinement
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T. Hoche, W. Neumann / Ultramicroscopy 96 (2003) 181–190 .
Fig. 8. (3 1 2) BTG supercells with coordination polyhedra indicated: (a) [0 0 1]-projected approximant of the modulated structure as determined from neutron-powder diffraction data [8]; (b) non-modulated average structure (starting model) in [0 0 1]-projection; best-fitting modulated structure after refinement in [0 0 1]- (c) and [0 1 0]-projection.
of the modulated BTG structure using rigid GeO4 units, however, gave an unsatisfactory final NED value of 0.27. Since a similar finding is obtained with the constrained refinement of neutron
powder-diffraction data [8], it can be concluded that in BTG, a certain flexibility of the coordination polyhedra is required to adequately describe the modulated structure.
T. Hoche, W. Neumann / Ultramicroscopy 96 (2003) 181–190 . Table 2 Mean error, m:e:; and standard deviation, s:d:; of non-fixed atomic positions for the comparison of original and refined supercell. Numbers of non-fixed atoms considered are given in brackets
Ba
Ge
Ti
O
x-coordinate
y-coordinate
z-coordinate
( (m.e.) 0.0074 A ( (s.d.) 0.1024 A (22)
( (m.e.) 0.0044 A ( (s.d.) 0.0850 A (60)
( (m.e.) 0.0552 A ( (s.d.) 0.1028 A (25)
( (m.e.) 0.0084 A ( (s.d.) 0.1109 A (22)
( (m.e.) 0.0037 A ( (s.d.) 0.0968 A (49)
( (m.e.) 0.0017 A ( (s.d.) 0.0462 A (25)
( (m.e.) 0.0009 A ( (s.d.) 0.1146 A (24)
( (m.e.) 0.0000 A ( (s.d.) 0.0600 A (24)
( (m.e.) 0.0001 A ( (s.d.) 0.0510 A (24)
( (m.e.) 0.0003 A ( (s.d.) 0.0019 A (136)
( 0.0037 A ( (s.d.) 0.2219 A (192)
( 0.0011 A 0.5810 (s.d.) (145)
As the usage of modulation waves for polyhedral units instead of modulated individual atoms will allow to better control z-components and generally reduce the number of free parameters, the usage of the molecule option should be advantageous for more complicated structures with rather rigid coordination polyhedra. For such cases, it will not only speed up the calculation (or make it even feasible) but will improve the reliability of the refinement. In this study, the image analysis procedure is demonstrated for a single image, simulated close to Scherzer focus. When applied to real experimental images, the precision of modulation parameters obtained will be deteriorated due to noise superimposed to the image as well as imaging and electron–matter interaction parameters to be refined, too. Hence, the method should be ideally performed on reconstructed exit wave functions as this would admit to leave imaging parameters out of the refinement.
3. Conclusions The methodology introduced could be shown to allow to retrieve Fourier amplitudes for modula-
189
tion waves in modulated structure from iterative digital HRTEM image matching. The latter approach is not restricted to somewhat exotic incommensurately modulated structures, as found in BTG, but can be applied to all kinds of superstructures, e.g. polytypes, commensurate modulations, etc. In the present example, the refinement of Fourier amplitudes instead of individual atomic positions reduced the number of free parameters from 987 (x-, y-, and z-coordinates of all individual atoms in the approximant supercell) to 56. This significant reduction is achieved by just considering a priori information on structural symmetry. Parameter spaces with 56 dimensions can be still handled while about 1000-dimensional optimisations are not feasible at all. The consideration of coordination-polyhedra modulations generally keeps the number of free parameters small also for even more complicated structures. The latter model might give poorer results for structures with some inherent flexibility of the coordination polyhedra but z-components can stand to benefit from the constraints imposed.
Acknowledgements We would like to sincerely thank Vaclav Petricek and Michal Dusek, Institute of Physics, Academy of Sciences of the Czech Republic in Prague for mounting the HRTEM supercell option in JANA2000. T.H. is indebted to Uwe Richter for granting access to workstations of the Faculty of Mathematics & Computer Science at Friedrich-Schiller-Universit.at Jena and manifold technical support. The HRTEM image was recorded at the Institute for Microstructure Research in Julich . and visits to this laboratory were funded within the program ‘‘GroXger.ateinitiative Analytische HRTEM’’ by Deutsche Forschungsgemeinschaft Bonn (NE646/4-1).
References [1] S. van Smaalen, Cryst. Rev. 4 (1995) 79. [2] P.M. de Wolff, T. Janssen, A. Janner, Acta Cryst. A 37 (1981) 625.
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T. Hoche, W. Neumann / Ultramicroscopy 96 (2003) 181–190 .
[3] T. Janssen, A. Janner, A. Looijenga-Vos, P.M. de Wolff, in: A.J.C. Wilson (Ed.), International Tables for Crystallography, Vol. C, Mathematical, Physical and Chemical Tables, Kluwer Academic Publishers, Dordrecht, 1995, p. 797–844. [4] V. Petricek, P. Coppens, Acta Cryst. A 44 (1988) 235. [5] V. Petricek, M. Dusek, The Crystallographic Computing System JANA2000, Institute of Physics, Praha, Czech Republic, 2000. [6] L. Bindi, P. Bonazzi, M. Dusek, V. Petricek, G. Chapuis, Acat Cryst. B 57 (2001) 739. . [7] T. Hoche, W. Neumann, S. Esmaeilzadeh, R. Uecker, M. Lentzen, C. Russel, . J. Solid State Chem. 166 (2002) 15. . [8] T. Hoche, S. Esmaeilzadeh, R. Uecker, S. Lidin, W. Neumann, Acta Cryst. B, 2002, accepted for publication.
[9] C. van Heurck, G. van Tendeloo, S. Amelinckx, Phys. Chem. Minerals 18 (1992) 441. [10] D. Hofmann, F. Ernst, Ultramicroscopy 53 (1994) 205. . [11] T. Hoche, P.R. Kenway, H.-J. Kleebe, M.W. Finnis, M. Ruhle, . J. Phys. Chem. Solids 55 (1994) 1067. [12] W.E. King, G.H. Campbell, Ultramicroscopy 56 (1994) 46. . [13] G. Mobus, R. Schweinfest, T. Gemming, T. Wagner, M. Ruhle, . J. Microscopy 190 (1998) 109. . [14] T. Hoche, C. Russel, . W. Neumann, Solid State Commun. 110 (1999) 651. . [15] T. Hoche, W. Neumann, Ferroelectrics 250 (2001) 67. . [16] R.L. Withers, Y. Tabira, Y. Liu, T. Hoche, Phys. Chem. Minerals 29 (2002) 624. [17] P.A. Stadelmann, Ultramicroscopy 21 (1987) 131. . [18] T. Hoche, M. Ruhle, . J. Am. Ceram. Soc. 79 (1996) 1961. [19] J.A. Nelder, R. Mead, Comput. J. 7 (1965) 308.