- Email: [email protected]
Lateral tomographic reconstruction of multiple in-line holograms
Ultramicroscopy 85 (2000) 155–164
Lateral tomographic reconstruction of multiple in-line holograms Mark R.A. Shegelski*, Timothy A. Rothwell Department of Physics, University of Northern British Columbia, 3333 University Way, Prince George, British Columbia, Canada V2N 4Z9 Received 17 August 1999; received in revised form 26 June 2000
Abstract Calculated electron in-line holograms, for the low-energy electron point source (LEEPS) microscope and based on scattering theory, give reconstructions that exhibit atomic resolution perpendicular to the optical axis. The depth resolution is not as sharp, and spurious peaks also result. We investigate the possibility of obtaining overall atomic resolution in the reconstruction of LEEPS in-line holograms by using a tomographic approach. We examine a few object positions with displacements lateral to the optical axis. We inquire as to whether or not the reconstructions obtained from a small number of lateral tomographic holograms can be combined in a manner such that all spurious peaks are eliminated and only atomic peaks result. The experimental consequences of these inquiries are discussed. # 2000 Elsevier Science B.V. All rights reserved. PACS: 61.16.B; 61.14.N Keywords: Electron microscopy; Reconstruction; Tomography
1. Introduction In previous works [1,2], we presented improved reconstructions of in-line holograms for lowenergy electron point source (LEEPS) microscopy. Our principal result was to eliminate spurious peaks in the source–object direction by sampling over a few screen positions in conjunction with a weighted energy sampling. We showed that, by rotating a flat screen through a few angles away *Corresponding author. Tel.: +1-250-960-6663; fax: +1250-960-5545. E-mail address: [email protected] (M.R.A. Shegelski).
from the optical axis, enough information is recorded to significantly improve the depth resolution in reconstruction. Moving the screen around the source is equivalent to laterally shifting, and rotating, the object, and is thus akin to tomography. In a subsequent investigation [3], we examined holograms and reconstructions obtained by shifting the object position only laterally. We found that laterally shifted reconstructions did not improve the resolution. The same conclusion was found when averaging such laterally shifted reconstructions. We also compared the nature of laterally shifted reconstructions with unshifted
0304-3991/00/$ - see front matter # 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 3 9 9 1 ( 0 0 ) 0 0 0 5 5 - 3
156
M.R.A. Shegelski, T.A. Rothwell / Ultramicroscopy 85 (2000) 155–164
reconstructions. The principal feature of our previous investigation that we draw on here is the following. The reconstructions showed peaks at the locations of the atoms, but also showed several other, ‘‘spurious’’ peaks of comparable magnitude at locations where there were no atoms. This feature was common to all reconstructions for all lateral displacements. We noticed that, whereas the atomic peaks appeared, for any displacement, in almost the same locations relative to one another, the spurious peaks shifted about considerably, relative to one another as well as relative to the atomic peaks. The primary purpose of this paper is to exploit this feature of the shifting about of the spurious peaks. A principal objective is to obtain improved depth resolution by combining lateral displacements. Our ultimate goal is to be capable of identifying all atomic sites, using only a few holograms, and having minimal knowledge of the atomic cluster under investigation. We seek to develop methods such that we can deduce the sites of all atoms, with no spurious peaks, working with the holograms only, i.e. without knowing in advance what atomic configuration produced the holograms. With these goals in mind, instead of simply averaging over lateral displacements, we combine a few reconstructions, obtained for different lateral shifts, using methods specifically designed to eliminate as many spurious peaks as possible. Another important reason for examining lateral tomography is to provide guidance for the experimentalist. Our ultimate goal, again, is to obtain reconstructions exhibiting peaks at atomic sites only. We will concentrate exclusively on LEEPS microscopy although the transfer of these techniques to other electron holographies and to optical holography is straightforward. (References to related works are given in Ref. [1].) Other approaches have been used by other researchers upon encountering similar problems with reconstruction. What we refer to in this paper as ‘‘spurious peaks’’ in the reconstruction is often referred to in the literature as ‘‘the twin image’’ or ‘‘the conjugate image’’. The twin image problem is also inherent in, e.g., photoemission holography. For example, the problem of poor depth resolu-
tion has been encountered in photoelectron and LEED holography; a remedy has been suggested and successfully implemented using the superposition of reconstructed images for several wavelengths [4–6].
2. Reconstruction The purpose of reconstruction is to obtain the three-dimensional structure of the object from the two-dimensional hologram on the screen. This can be achieved via a Kirchhoff–Helmholtz transform [7]: Z d2 xIðnÞexpðikn r=xÞ; ð1Þ KðrÞ ¼ S
where the integration extends over the two– dimensional surface of the screen with coordinates n ¼ ðX; Y; LÞ, where L is the distance from the source to the (center of the) screen; k ¼ 2p=l is the wave number of the electrons and IðnÞ is the contrast image on the screen obtained by subtracting the images with and without the object present [7]: i Lh ð2Þ IðnÞ ¼ 3 2 Re fs ðnÞ þ jfs ðnÞj2 r with XX expðÿikri ^nÞtl Ylm ð^nÞ Flm ðri Þ; ð3Þ fs ðnÞ ¼ i
l;m
where tl ¼ kÿ1 sinðdl Þexpðidl Þ, and Flm ðri Þ ikri 4pYlm ð^ ri Þrÿ1 . The function KðrÞ is significantly i e structured and different from zero only in the spatial region occupied by the object. By reconstructing KðrÞ in the vicinity of the object, a threedimensional image can be built up. KðrÞ is a complex function and one usually plots its magnitude to represent the object, although phase information can also be extracted. The applicability of this transform to in-line holography has been demonstrated both for electrons and for photons and for simulated and experimental holograms. In many cases, the geometry is such that the electron waves arrive at the screen, to a very good approximation, as plane waves, even in cases
M.R.A. Shegelski, T.A. Rothwell / Ultramicroscopy 85 (2000) 155–164
where the electrons emerge from a point source. In such situations, Eq. (1) can be replaced by a Fourier transform. Specifically, write n ¼ L þ R, where L is from the point source to the center of the screen and R is from the center of the screen to a particular point on the screen. If R5L, then the waves arrive at the screen as almost plane waves, and Eq. (1) can be simplified to give Z ð4Þ KðrÞ expðikrz Þ d2 RIðRÞexpð2piq rÞ; S
where q is a two-dimensional vector with components qx ¼ kRx =ð2pxÞ ¼ sin yx =l, and qy ¼ kRy = ð2pxÞ ¼ sin yy =l, where yx and yy are the scattering angles. This formula corresponds to the standard case of Fourier transforming; see, e.g., Eqs. (2.6) and (2.21) of Ref. [8]. In theoretical LEEPS microscopy, the screen size is such that this standard treatment cannot be used. The waves arriving at the screen must be treated as spherical waves because we do not have R5L. For example, one typically begins theoretical LEEPS inquiries using L ¼ 10 cm and a screen half-width of 7 cm (details are given below). In standard treatments where a Fourier transform can be used, and where Fraunhoffer holography is done, the Fraunhoffer method (see Ref. [9] pp. 51–52) can be used to resolve the twin image problem, i.e. to remove image artifacts (‘‘the spurious peaks’’). In lensless LEEPS microscopy, the electron waves must be treated as spherical waves, and the full Kirchhoff–Helmholtz formula (Eq. (1)), must be employed, instead of a Fourier transform. Consequently, the Fraunhoffer method cannot, unfortunately, be used to resolve the twin image problem. As such, we must develop another method of removing the spurious peaks that inextricably accompany the reconstruction. The principal objective of this paper is to present the method we have developed that removes spurious peaks and gives a final reconstruction that shows the true atomic structure of the object. To explore the reconstruction procedure we will make use of simulated, rather than experimental, LEEPS holograms to avoid limitations and possible artifacts of the experiment.
157
2.1. Reconstructions for lateral object displacements The first step in our method is to obtain holograms and reconstructions for selected lateral displacements of the atomic cluster. For the undisplaced object, the source–object distance is 1000 A˚, the source–screen distance is 10 cm, and the screen size is 14 cm14 cm. The source and the screen are fixed, and the object is displaced lateral to the optical axis. We will examine holograms and reconstructions for lateral displacements of 150 A˚. For lateral tomography, we use the following coordinate system. The (x,y,z) coordinates are defined by the z-axis directed from the point source through the geometrical ‘‘center’’ of the (undisplaced) object, and with the center of the screen on the z-axis. The x and y-axis are perpendicular to the optical axis. We will combine and analyze the reconstructions obtained for (a) the object on the optical axis, (b) the object shifted along the x-axis, and (c) the object shifted along the y-axis. We will denote these reconstructions, respectively, as K0 ðrÞ, Kx ðrÞ, and Ky ðrÞ. Most of our reconstructions are for a screen large enough to record the information in a cone of 358 (e.g. with a diameter of 14 cm at a distance of 10 cm from the source). We acknowledge that the emission cone of present day electron point sources is typically less than 108, i.e. much too small to realize the geometries proposed here. We will, however, also report results for a 7 cm 7 cm screen. The intention of our theoretical study is, partially, to serve as a guide to the experimentalist. 2.2. Combining selected lateral displacements We next combine reconstructions for selected lateral displacements. The essential idea in our method is to compare the spatial locations of local maxima, and the values of these maxima, in the three reconstructions K0 ðrÞ, Kx ðrÞ, and Ky ðrÞ. Consider, for example, a maximum located in all three reconstructions at very nearly the same spatial point, and having very nearly the same value of K in all three cases. Such a correspondence is expected for
158
M.R.A. Shegelski, T.A. Rothwell / Ultramicroscopy 85 (2000) 155–164
atomic peaks, i.e. the local maxima that occur at atomic sites. Spurious peaks, however, will have local maxima at significantly separated spatial points in the three reconstructions K0 ðrÞ, Kx ðrÞ, and Ky ðrÞ. This is clear from inspecting Figs. 1(a)– (c), which show, respectively, jK0 ðrÞj, jKx ðrÞj, and jKy ðrÞj for the x–z plane through the middle 5 5 chain of a 5 5 j 4 4 carbon cluster. The layers are in x2y planes: when the cluster is between the source and the screen, the 5 layer is a distance d ¼ 1000 A˚ from the point source, with the atoms arranged on a square lattice with spacing a ¼ 2:5 A˚, and with the 4 layer at d ¼ 1004:5 A˚. The spacing between the 5 layer and the 4 layer is
4:5 A˚, thus mimicking a BCC lattice. Note that the ‘‘atomic peaks’’ – the peaks that are at the atomic sites – are at almost the same locations in Figs. 1(a)–(c). Comparing Figs. 1(a) and (b), note that the spurious peaks are shifted in Fig. 1(b) in the x2z plane, so that they are at different locations in the two figures. Comparing Figs. 1(a) and (c), one sees that, for Ky ðrÞ [Fig. 1(c)], the spurious peaks are shifted, not in the x2z plane, as in Fig. 1(b), but in the y2z plane. The spatial separation of these local maxima serves as a signal that these are, indeed, spurious peaks. The idea of combining selected lateral displacements is not new. The first paper to use lateral
Fig. 1. Reconstructions for a 5 5 j 4 4 carbon cluster. jKðrÞj is shown for a cut in the x–z plane through the middle 5 5 string of atoms. In this plane, there is a row of 5 atoms along z ¼ 1000 A˚, spaced 2:5 A˚ apart in the x-direction. Note that the distance scale in the x-direction is expanded as compared with the z-direction; this facilitates examination of the atomic and spurious peaks. The lateral displacements of the cluster are as follows: (a) no displacement, (b) 150 A˚, in the x-direction, and (c) 150 A˚, in the y-direction. Note the shift in the locations of the spurious peaks, and that the atomic peaks exhibit a much smaller shift. These reconstructions were obtained from holograms generated using a 14 cm14 cm screen. See text for details.
M.R.A. Shegelski, T.A. Rothwell / Ultramicroscopy 85 (2000) 155–164
displacement to reconstruct in–line electron holograms [10] considered, as an example, the case of a two atom object, modeled by a Gaussian–shaped atoms (see Ref. [10], Fig. 2). By superposing the reconstructed images from 11 holograms with lateral shifts, Lin and Cowley were able to eliminate most of the conjugate image ‘‘noise’’, obtaining a clear two-atom, one-dimensional reconstruction (see Ref. [10], Fig. 10 and Section 6). In a previous work [3] we employed a similar superposition for a multi-atom cluster in LEEPS, but found that such superposition did not clean up the reconstruction adequately. In this present work, we examine the twodimensional and/or three-dimensional structure of 2 or 3 reconstructions, KðrÞ, obtained from lateral shifts of the atomic cluster. Our method thus goes well beyond superposition of reconstructions. By exploring the topologies of the 2 or 3 reconstructions, we develop a means of identifying and eliminating the peaks that are not true atomic peaks; i.e. the spurious peaks due to the twin image are identified and removed. The result of our method is that we arrive at a final reconstruction that gives a drastically improved picture of the atomic structure. The details of our method of distinguishing between atomic and spurious peaks are necessarily more complicated than just comparing the spatial locations of the various local maxima, especially when considering larger clusters. In order to give a reasonably complete picture of our method, we next present some of the salient details of our approach. At the end of this section, we will give a list of the various parameters that we use in applying the methods described below. We take the following steps in order to eliminate as many spurious peaks as possible. Our methods may be categorized as ‘‘three-dimensional’’ or ‘‘two-dimensional’’, depending on whether we search through the volume around the cluster, or, having found a plane of special interest, we examine characteristics of the reconstruction in this plane. In this paper, we focus primarily, but not exclusively, on BCC-type structures. Some results are given for random distributions of atoms. From our results, we see that generalizing to any atomic
159
arrangement is a project that is well beyond the scope of this first investigation. 2.3. Three-dimensional methods An outline of the principal methods used for three dimensions is as follows. (1) For a given atomic cluster (object), we first create three holograms, one with the object on the optical axis, one with the object laterally displaced, by 150 A˚, in the x-direction, and one laterally displaced, by 150 A˚, in the y-direction. The same electron energy and screen resolution is used in each case. (2) We generate the three reconstructions K0 ðrÞ, Kx ðrÞ, and Ky ðrÞ. In each case, K is calculated within the same, selected volume around the atomic cluster, and the Ks for lateral displacement are ‘‘mapped back’’ into the volume around the undisplaced cluster. Since the reconstructions are calculated numerically, we use a chosen spatial resolution for ‘‘points’’ inside the volume of interest. We thus determine the values of K in tiny volume elements DV ¼ DxDyDz. We can then compare the values of K for the same point in rspace for the three different reconstructions. (3) We take the reconstructions and renormalize each to the highest jKðrÞj value encountered within that reconstruction. We thus have the K values scaled so that 04jKðrÞj41 for all three cases. (4) We set to zero, i.e. we remove, all points r for which KðrÞ is below a given threshold value. For example, in order to circumvent inspection of the numerous local maxima having small magnitudes, we could remove all points for which jKðrÞj50:2. (5) We then determine the spatial locations of all the local maxima of jKðrÞj for the three reconstructions. This can be done, for example, by comparing the value of jKj at each (non-zero) point to the values of jKj at all neighbouring points. We will refer to these local maxima as ‘‘peaks’’, and to the corresponding values of jKj as ‘‘peak heights’’. (6) For each peak in the undisplaced reconstruction, K0 ðrÞ, we locate nearby peaks in each of the two displaced reconstructions, Kx ðrÞ, and Ky ðrÞ. We select a set of three peaks, one from each reconstruction, such that the set has minimal
160
M.R.A. Shegelski, T.A. Rothwell / Ultramicroscopy 85 (2000) 155–164
spatial separation of peak locations. One way to determine the set is to calculate the perimeter of the triangle formed by the three peak locations. This perimeter is determined from the r-space coordinates of the volume elements DV ¼DxDyDz. (7) A peak is retained if the perimeter lies below a given threshold value. (We find, for best results, this threshold to be typically 3:5–5 A˚, for the clusters studied.) (8) If the perimeter of the trio of associated peaks lies between two selected threshold values, a second test is performed. These two threshold values are selected such that (i) only atomic peaks have perimeters that are smaller than the first threshold, and (ii) all atomic peaks and some spurious peaks have perimeters that are smaller than the second threshold. These two threshold values are, of course, readily found if the cluster is known; we will test this method on unknown clusters, in which case we choose the two threshold values using experience as a guide (see below). Trios of peaks having perimeters between the two threshold values are tested for a peak height match, as follows. The smallest jKðrÞj value of the three points is divided by the largest jKðrÞj value. This quotient is then compared to another threshold value (typically greater than or equal to 0.8). The (undisplaced) peak is retained if the value of the quotient lies above the threshold. The reason we sometimes need to make this test is to separate the atomic peaks from spurious peaks for those cases where the atomic peak’s perimeter is somewhat large as compared to other atomic peaks. (9) We sometimes also use a method that is especially effective at identifying spurious peaks that appear ‘‘in front’’ of the cluster, i.e. on the side of the cluster that is closer to the source. The method is as follows. The three reconstructions are renormalized as described above. All points below a given threshold are again set to zero. The peaks are located in each reconstruction by previously described methods. Each (non-zero) point in each of the 3-d reconstructions is assigned to an ‘‘island’’ determined by the nearest peak. Each island is independently renormalized to the highest value of jKðrÞj in that island, i.e. the value of each peak height is set to one, and the associated islands are all rescaled appropriately. The ‘‘final re-
construction’’ is obtained by multiplying these ‘‘renormalized reconstructions’’ together. This eliminates many spurious peaks because those areas which do not have good overlap of peaks are substantially reduced. This averaged reconstruction is also filtered for noise. In this case, all points that are less than a selected threshold value are set to zero. The final reconstruction is then searched for peaks; all peaks found are retained, and the set of such peaks constitutes the final result. As noted above, this method is often successful in removing spurious planes located on the source side of the cluster or object. To summarize: the undisplaced peak is retained if the value of the perimeter for the trio of associated peak positions lies below a threshold value. A peak is also retained if the perimeter is larger than this value but smaller than a second value and if the ratio of lowest peak height to highest peak height in the trio lies above a third threshold value. Having done this search in three dimensions, and having found planes within which the retained peaks’ maxima lay, we can then examine these planes of special interest. We next report the methods we use for study in these planes. 2.4. Two-dimensional methods We can obtain pictures of the atomic peaks identified by the 3-d methods described above by using a 2-d filter program on x–z or y–z planes. We take two plane reconstructions, one for the undisplaced cluster and one for the cluster displaced along the axis normal to the reconstruction plane (e.g. y-displacement for an x2z plane). This has the effect of removing many spurious peaks within the plane under consideration. The reason, recall, is that the spurious peaks tend to shift in the direction of the displacement of the cluster (see Figs. 1). For example, in the case of study in the x2z plane, the reconstructions considered would be the undisplaced reconstruction, K0 ðrÞ, and the y-displaced reconstruction, Ky ðrÞ, [see, e.g., Figs. 1(a) and (c)]. We first normalize the two reconstructions, and set to zero all KðrÞ values that have magnitudes below the ‘‘noise threshold’’, as described above.
M.R.A. Shegelski, T.A. Rothwell / Ultramicroscopy 85 (2000) 155–164
For study within a plane, we break the region of interest into blocks of area, e.g. DA ¼ DxDz for the x2z plane. We next locate the peaks in the two jKðrÞj using the methods described above for 3-d. We then assign each non-zero jKðrÞj element to an ‘‘island’’, as follows. A point r is assigned to the island associated with a given peak if r is closer to the position, within the plane, of the given peak, than it is to any other peak within the same plane. In this way, each peak has a set of points, an ‘‘island’’, associated with it. Each peak from the undisplaced cluster is paired with the nearest peak in the displaced reconstruction, and the distance between each such pair of peaks is noted. The largest such distance is used to renormalize the distance values. If the distance between nearest neighbour peaks is less than a certain selected value, they are retained, and the associated undisplaced island is not subjected to any further test. If the distance between the pair of peaks is larger than the selected value, then a further test is used. In this test we compare slopes and heights of each point in each island. Consider a particular point r in each of the two reconstructions. The direction of the slope, i.e. the sign of the derivative of jKðrÞj, is tested to see if they match. For example, for the x2z plane, both the sign of the x-slope and the zslope of corresponding islands must match if the peak is to be retained. Every other point is set to zero. Those points which survive this ‘‘slope test’’ must also have the same relative height to within a given tolerance to be included in the final reconstruction. The final reconstruction thus includes those points in the undisplaced reconstruction which have survived either the ‘‘nearest neighbour test’’ or the ‘‘slope and height test’’. 2.5. Parameters used We complete this section by defining the various parameters we use. (1) tn denotes the ‘‘noise threshold’’: all points r having jKðrÞj5tn are removed from the reconstruction, i.e. the value of jKðrÞj is set to zero.
161
(2) Pin denotes the ‘‘inner (or smaller) perimeter’’; all trios of peaks with a total perimeter P less than this value (P5Pin ) are retained without further testing. (3) Pout denotes the ‘‘outer (or larger) perimeter’’; trios of peaks with perimeters between this value and Pin (Pin 5P5Pout ) are subjected to a height match check to decide whether or not they will be retained. (4) ht denotes the ‘‘height match threshold’’; the ratio of the smallest peak in a given trio to the largest peak in the same trio must be larger than this value to order that the peak be retained. (5) dn denotes the ‘‘nearest-neighbor distance’’ in the 2-d method; if the distance between corresponding pairs of points in the two reconstructions is less than dn , then the point in the undisplaced reconstruction is retained. We will give the values of these parameters for the results reported next.
3. Results and discussion We first present results for a 5 5 j 4 4 j 5 5 j 4 4 cluster of carbon atoms. The layers are in x2y planes: when the cluster is between the source and the screen, the first 5 layer is a distance d ¼ 1000 A˚ from the point source, with the atoms arranged on a square lattice with spacing a ¼ 2:5 A˚, and with the first 4 layer at d ¼ 1004:5 A˚. The spacing between the 5 layers and the 4 layers is 4:5 A˚, thus mimicking a BCC lattice. In Fig. 2(a) we show a three-dimensional view of the spatial locations of all the peaks for the undisplaced reconstruction, i.e. all the local maxima in jK0 ðrÞj. In Fig. 2(b) we show a threedimensional view of the spatial locations of those peaks that remain after using the method described above. A striking feature of these figures is the efficacy with which such a seeming simple method removes spurious peaks. Whereas there are 203 peaks in Fig. 2(a), there are only 78 in Fig. 2(b). Only 4 of these peaks are spurious peaks. The 5 5 j 4 4 j 5 5 j 4 4 structure is manifest in Fig. 2(b). We emphasize that the atoms that are
162
M.R.A. Shegelski, T.A. Rothwell / Ultramicroscopy 85 (2000) 155–164
Fig. 2. The locations of local maxima in jKðrÞj in the volume around a 5 5 j 4 4 j 5 5 j 4 4 carbon cluster. Part (a) shows all such maxima, or ‘‘peaks’’, obtained from the reconstruction with the cluster on the optical axis. Part (b) shows the peaks that remain after being scrutinized by the methods reported in this paper. Only atomic peaks, and a few spurious peaks, are retained in (b). The following parameter values were used in part (b): tn ¼ 0:475, Pin ¼ 2:625A˚, Pout ¼ 6:0 A˚, and ht ¼ 0:5. Details are given in the text.
missing in the 5 5 layers are recovered when we apply the 2–d methods to obtain, e.g., x2z plane reconstructions. We thus regard this reconstruction as virtually completely successful. We also wish to emphasize that, in obtaining Fig. 2(b), we have treated the cluster as if it were an unknown cluster. In Figs. 3(a) and (b), we show x–z cuts through the middle of the 5 5 layer of a 5 5 j 4 4 cluster for (a) the undisplaced reconstruction, and (b) the ‘‘processed’’ reconstruction, after the spurious peaks have been removed. Again, the
Fig. 3. Reconstructions in the x2z plane through the middle of the 5 5 layer of a 5 5 j 4 4 carbon cluster. Part (a) shows the undisplaced reconstruction, and (b) shows the ‘‘processed’’ reconstruction, after removal of the spurious peaks. The following parameter values were used in part (b): tn 0 (no noise rejection), ht 1(perfect height match), and dn 0:2. Details are given in the text.
improvement is very good. A similar result is obtained for a 5 5 j 4 4 j 5 5 j 4 4 cluster. To further test our method, we arranged for a member of our research group to generate 3 holograms for a variety of square-planar-type clusters. For each cluster, we were provided holograms for no displacement (i.e. object on the optical axis), x- and y-displaced. We were also given the corresponding displacement distances, and the source–object distance (for the on–axis case). From the three in-line holograms, we generated the three reconstructions K0 ðrÞ, Kx ðrÞ, and Ky ðrÞ. Using the methods described above, we quite successfully identified various unknown clusters.
M.R.A. Shegelski, T.A. Rothwell / Ultramicroscopy 85 (2000) 155–164
Some of the cases we investigated are listed next. Comments are added as required. (1) We started with some simple clusters, such as 3 3 j 2 2 j 3 3. We easily found the atomic sites as well as the interatomic distance within a plane and the separation between planes. Only 3-d methods were needed. (2) We readily identified another unknown cluster as being a 2 2 j 3 3 j 4 4 j 5 5 BCC structure. In this case we used both 3-d and 2-d methods. (3) We successfully identified a 3 3 j 4 4j 3 3 cluster, with the four corners of the ‘‘4 4’’ layer having no atoms. We show in Fig. 4 the middle layer of this cluster. (4) Other unknown clusters we found, correctly, included random arrays of 6 atoms within a cube of edge length 6:0 A˚. We extended our attempts for higher numbers of atoms, keeping the volume per atom fixed (at about 45 A˚3 , which is the value for the 5 5 j 4 4 j 5 5 j 4 4 cluster). We were able to successfully identify up to about 15 atoms, randomly distributed, even without knowing the number of atoms present. (5) We have looked at clusters with missing atoms, again without knowing which atoms were missing. While we could easily identify missing atoms in small clusters (see, e.g., Fig. 4), for larger
163
clusters, such as the 5 5 j 4 4 j 5 5 j 4 4 cluster, we were unable to reliably locate missing atoms. (6) We have also investigated decreasing the screen size, and looking at larger clusters. In brief, our principal results are as follows. For the 14 cm 14 cm screen, the largest cluster for which we obtained high quality reconstruction was a 6 6 j 5 5 j 6 6 j 5 5 cluster. When we tried our methods on a 7 7 j 6 6 j 7 7 j 6 6 cluster, the reconstruction was rather poor: too many atoms were missing and too many spurious peaks survived. In the case of the 10 cm 10 cm screen, the largest cluster for which we obtained high quality reconstruction was a 5 5 j 4 4 j 5 5 j 4 4 cluster. Even with this cluster, however, there was a noticeable number of missing atoms and somewhat too many spurious peaks. When we used the 7 cm 7 cm screen, even the 5 5 j 4 4 j 5 5 j 4 4 cluster was quite poorly reconstructed. There were many missing atoms and many spurious peaks. (7) We repeated our investigation of limiting cluster size for the three different screen sizes, changing only the electron energy. The results given above were obtained using 95 eV electrons. At 195 eV, the corresponding largest clusters for which good reconstruction was obtained were as follows. For the 14 cm14 cm screen, the largest cluster giving good reconstruction was a 7 7 j 6 6 j 7 7 j 6 6 cluster. For the 10 cm 10 cm screen, the largest cluster was a 5 5 j 4 4 j 5 5 j 4 4 cluster. For the 7 cm 7 cm screen, the largest cluster was a 5 5 j 4 4 cluster. In future work, we aim to develop improved methods so that larger clusters can be more adequately reconstructed using a smaller screen size. Ultimately, our goal is to develop methods that will be directly applicable to experimental LEEPS holograms.
4. Summary and outlook Fig. 4. Reconstruction in the x2y plane through the middle layer ( the ‘‘4 4’’ layer) of a 3 3 j 4 4 j 3 3 carbon cluster, with no atoms at the corners of the 4 4 layer. This reconstruction is for the undisplaced cluster, and clearly shows the missing atoms.
The principal results of this paper are as follows. (1) We have developed methods which are quite successful for finding the locations of atoms in a specified three-dimensional volume surrounding
164
M.R.A. Shegelski, T.A. Rothwell / Ultramicroscopy 85 (2000) 155–164
an unknown atomic cluster. In the cases we have examined to date, we are able to locate the atomic peaks and distinguish them from spurious peaks, with significant accuracy. The principal idea underlying our approach is to compare the locations of local maxima in the reconstructions for 3 holograms: one with the geometric center of the cluster on the optical axis, and two others where the center of the cluster has been shifted a specified distance along directions mutually perpendicular to the optical axis. We have given results for a variety of clusters in this paper. (2) We have also developed methods to further scrutinize special planes of interest, with emphasis on investigation of reconstructions in depth. Again, we have been quite successful at removing spurious peaks and thereby obtaining significantly improved reconstructions, especially in depth. The basic idea here is not only to compare locations of local maxima in different reconstructions, but also to compare peak heights and slopes. Results for several interesting clusters have been given in this paper. (3) Our focus has been on reconstructions for square planar clusters, where the intraplanar and interplanar nearest-neighbor distances are unknown. We have also studied random atomic arrangements. Our successes to date have been encouraging. (4) We have investigated the limits of our methods, with emphasis on (a) missing atoms or extra atoms, (b) cluster size, and (c) screen size. Details are given above. We have identified some limitations of our approach. For example, our methods are not sensitive to missing atoms in clusters of appreciable size. We encounter difficulties when we restrict the screen size to be of order of the experimental screen size for LEEPS microscopy, or if the cluster is larger than about 100 atoms, with about 4 planes along the optical axis.
Further work is planned to address these present limits. It is of interest to also develop methods that are able to treat clusters consisting or two or more atomic types. Results to this point are encouraging. Acknowledgements This work was partially supported by the Natural Sciences and Engineering Research Council of Canada (NSERC). We thank Silas Faltus for useful discussions. We thank Prof. H.J. Kreuzer for useful conversations and constructive comments. We also thank the referee for a very helpful report. Matthew Reid provided useful conversations and valuable assistance with various aspects of this work. We thank the UNBC Mathematics and Computer Science Department for access to their computational facilities. We also thank Wilson Yeung for useful discussions regarding computer programming technique and for providing an element of the three-dimensional program. References [1] M.R.A. Shegelski, S. Faltus, T. Clark, H.J. Kreuzer, Ultramicroscopy 74 (1998) 169. [2] M.R.A. Shegelski, T. Clark, M. Reid, S. Faltus, Ultramicroscopy 77 (1999) 129. [3] M.R.A. Shegelski, S. Faltus, Ultramicroscopy 77 (1999) 135. [4] K. Heinz, Rep. Prog. Phys. 58 (1995) 637. [5] D.K. Saldin, G.R. Harp, B.L. Chen, B.P. Tonner, Phys. Rev. B 44 (1991) 2480. [6] C.M. Wei, T.C. Zhao, S.Y. Tong, Phys. Lett. 65 (1990) 2278. [7] H.J. Kreuzer, K. Nakamura, A. Wierzbicki, H.-W. Fink, H. Schmid, Ultramicroscopy 45 (1992) 381. [8] E. Volko, D.C. Jay, L.F. Allard (Eds.), Introduction to Electron Holography, Plenum, New York, 1999. [9] R.J. Collier, C.B. Burkhardt, L.H. Lin, Optical Holography, Academic Press, New York, 1971. [10] J.A. Lin, J.M. Cowley, Ultramicroscopy 19 (1986) 179.
Lateral tomographic reconstruction of multiple in-line holograms Mark R.A. Shegelski*, Timothy A. Rothwell Department of Physics, University of Northern British Columbia, 3333 University Way, Prince George, British Columbia, Canada V2N 4Z9 Received 17 August 1999; received in revised form 26 June 2000
Abstract Calculated electron in-line holograms, for the low-energy electron point source (LEEPS) microscope and based on scattering theory, give reconstructions that exhibit atomic resolution perpendicular to the optical axis. The depth resolution is not as sharp, and spurious peaks also result. We investigate the possibility of obtaining overall atomic resolution in the reconstruction of LEEPS in-line holograms by using a tomographic approach. We examine a few object positions with displacements lateral to the optical axis. We inquire as to whether or not the reconstructions obtained from a small number of lateral tomographic holograms can be combined in a manner such that all spurious peaks are eliminated and only atomic peaks result. The experimental consequences of these inquiries are discussed. # 2000 Elsevier Science B.V. All rights reserved. PACS: 61.16.B; 61.14.N Keywords: Electron microscopy; Reconstruction; Tomography
1. Introduction In previous works [1,2], we presented improved reconstructions of in-line holograms for lowenergy electron point source (LEEPS) microscopy. Our principal result was to eliminate spurious peaks in the source–object direction by sampling over a few screen positions in conjunction with a weighted energy sampling. We showed that, by rotating a flat screen through a few angles away *Corresponding author. Tel.: +1-250-960-6663; fax: +1250-960-5545. E-mail address: [email protected] (M.R.A. Shegelski).
from the optical axis, enough information is recorded to significantly improve the depth resolution in reconstruction. Moving the screen around the source is equivalent to laterally shifting, and rotating, the object, and is thus akin to tomography. In a subsequent investigation [3], we examined holograms and reconstructions obtained by shifting the object position only laterally. We found that laterally shifted reconstructions did not improve the resolution. The same conclusion was found when averaging such laterally shifted reconstructions. We also compared the nature of laterally shifted reconstructions with unshifted
0304-3991/00/$ - see front matter # 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 3 9 9 1 ( 0 0 ) 0 0 0 5 5 - 3
156
M.R.A. Shegelski, T.A. Rothwell / Ultramicroscopy 85 (2000) 155–164
reconstructions. The principal feature of our previous investigation that we draw on here is the following. The reconstructions showed peaks at the locations of the atoms, but also showed several other, ‘‘spurious’’ peaks of comparable magnitude at locations where there were no atoms. This feature was common to all reconstructions for all lateral displacements. We noticed that, whereas the atomic peaks appeared, for any displacement, in almost the same locations relative to one another, the spurious peaks shifted about considerably, relative to one another as well as relative to the atomic peaks. The primary purpose of this paper is to exploit this feature of the shifting about of the spurious peaks. A principal objective is to obtain improved depth resolution by combining lateral displacements. Our ultimate goal is to be capable of identifying all atomic sites, using only a few holograms, and having minimal knowledge of the atomic cluster under investigation. We seek to develop methods such that we can deduce the sites of all atoms, with no spurious peaks, working with the holograms only, i.e. without knowing in advance what atomic configuration produced the holograms. With these goals in mind, instead of simply averaging over lateral displacements, we combine a few reconstructions, obtained for different lateral shifts, using methods specifically designed to eliminate as many spurious peaks as possible. Another important reason for examining lateral tomography is to provide guidance for the experimentalist. Our ultimate goal, again, is to obtain reconstructions exhibiting peaks at atomic sites only. We will concentrate exclusively on LEEPS microscopy although the transfer of these techniques to other electron holographies and to optical holography is straightforward. (References to related works are given in Ref. [1].) Other approaches have been used by other researchers upon encountering similar problems with reconstruction. What we refer to in this paper as ‘‘spurious peaks’’ in the reconstruction is often referred to in the literature as ‘‘the twin image’’ or ‘‘the conjugate image’’. The twin image problem is also inherent in, e.g., photoemission holography. For example, the problem of poor depth resolu-
tion has been encountered in photoelectron and LEED holography; a remedy has been suggested and successfully implemented using the superposition of reconstructed images for several wavelengths [4–6].
2. Reconstruction The purpose of reconstruction is to obtain the three-dimensional structure of the object from the two-dimensional hologram on the screen. This can be achieved via a Kirchhoff–Helmholtz transform [7]: Z d2 xIðnÞexpðikn r=xÞ; ð1Þ KðrÞ ¼ S
where the integration extends over the two– dimensional surface of the screen with coordinates n ¼ ðX; Y; LÞ, where L is the distance from the source to the (center of the) screen; k ¼ 2p=l is the wave number of the electrons and IðnÞ is the contrast image on the screen obtained by subtracting the images with and without the object present [7]: i Lh ð2Þ IðnÞ ¼ 3 2 Re fs ðnÞ þ jfs ðnÞj2 r with XX expðÿikri ^nÞtl Ylm ð^nÞ Flm ðri Þ; ð3Þ fs ðnÞ ¼ i
l;m
where tl ¼ kÿ1 sinðdl Þexpðidl Þ, and Flm ðri Þ ikri 4pYlm ð^ ri Þrÿ1 . The function KðrÞ is significantly i e structured and different from zero only in the spatial region occupied by the object. By reconstructing KðrÞ in the vicinity of the object, a threedimensional image can be built up. KðrÞ is a complex function and one usually plots its magnitude to represent the object, although phase information can also be extracted. The applicability of this transform to in-line holography has been demonstrated both for electrons and for photons and for simulated and experimental holograms. In many cases, the geometry is such that the electron waves arrive at the screen, to a very good approximation, as plane waves, even in cases
M.R.A. Shegelski, T.A. Rothwell / Ultramicroscopy 85 (2000) 155–164
where the electrons emerge from a point source. In such situations, Eq. (1) can be replaced by a Fourier transform. Specifically, write n ¼ L þ R, where L is from the point source to the center of the screen and R is from the center of the screen to a particular point on the screen. If R5L, then the waves arrive at the screen as almost plane waves, and Eq. (1) can be simplified to give Z ð4Þ KðrÞ expðikrz Þ d2 RIðRÞexpð2piq rÞ; S
where q is a two-dimensional vector with components qx ¼ kRx =ð2pxÞ ¼ sin yx =l, and qy ¼ kRy = ð2pxÞ ¼ sin yy =l, where yx and yy are the scattering angles. This formula corresponds to the standard case of Fourier transforming; see, e.g., Eqs. (2.6) and (2.21) of Ref. [8]. In theoretical LEEPS microscopy, the screen size is such that this standard treatment cannot be used. The waves arriving at the screen must be treated as spherical waves because we do not have R5L. For example, one typically begins theoretical LEEPS inquiries using L ¼ 10 cm and a screen half-width of 7 cm (details are given below). In standard treatments where a Fourier transform can be used, and where Fraunhoffer holography is done, the Fraunhoffer method (see Ref. [9] pp. 51–52) can be used to resolve the twin image problem, i.e. to remove image artifacts (‘‘the spurious peaks’’). In lensless LEEPS microscopy, the electron waves must be treated as spherical waves, and the full Kirchhoff–Helmholtz formula (Eq. (1)), must be employed, instead of a Fourier transform. Consequently, the Fraunhoffer method cannot, unfortunately, be used to resolve the twin image problem. As such, we must develop another method of removing the spurious peaks that inextricably accompany the reconstruction. The principal objective of this paper is to present the method we have developed that removes spurious peaks and gives a final reconstruction that shows the true atomic structure of the object. To explore the reconstruction procedure we will make use of simulated, rather than experimental, LEEPS holograms to avoid limitations and possible artifacts of the experiment.
157
2.1. Reconstructions for lateral object displacements The first step in our method is to obtain holograms and reconstructions for selected lateral displacements of the atomic cluster. For the undisplaced object, the source–object distance is 1000 A˚, the source–screen distance is 10 cm, and the screen size is 14 cm14 cm. The source and the screen are fixed, and the object is displaced lateral to the optical axis. We will examine holograms and reconstructions for lateral displacements of 150 A˚. For lateral tomography, we use the following coordinate system. The (x,y,z) coordinates are defined by the z-axis directed from the point source through the geometrical ‘‘center’’ of the (undisplaced) object, and with the center of the screen on the z-axis. The x and y-axis are perpendicular to the optical axis. We will combine and analyze the reconstructions obtained for (a) the object on the optical axis, (b) the object shifted along the x-axis, and (c) the object shifted along the y-axis. We will denote these reconstructions, respectively, as K0 ðrÞ, Kx ðrÞ, and Ky ðrÞ. Most of our reconstructions are for a screen large enough to record the information in a cone of 358 (e.g. with a diameter of 14 cm at a distance of 10 cm from the source). We acknowledge that the emission cone of present day electron point sources is typically less than 108, i.e. much too small to realize the geometries proposed here. We will, however, also report results for a 7 cm 7 cm screen. The intention of our theoretical study is, partially, to serve as a guide to the experimentalist. 2.2. Combining selected lateral displacements We next combine reconstructions for selected lateral displacements. The essential idea in our method is to compare the spatial locations of local maxima, and the values of these maxima, in the three reconstructions K0 ðrÞ, Kx ðrÞ, and Ky ðrÞ. Consider, for example, a maximum located in all three reconstructions at very nearly the same spatial point, and having very nearly the same value of K in all three cases. Such a correspondence is expected for
158
M.R.A. Shegelski, T.A. Rothwell / Ultramicroscopy 85 (2000) 155–164
atomic peaks, i.e. the local maxima that occur at atomic sites. Spurious peaks, however, will have local maxima at significantly separated spatial points in the three reconstructions K0 ðrÞ, Kx ðrÞ, and Ky ðrÞ. This is clear from inspecting Figs. 1(a)– (c), which show, respectively, jK0 ðrÞj, jKx ðrÞj, and jKy ðrÞj for the x–z plane through the middle 5 5 chain of a 5 5 j 4 4 carbon cluster. The layers are in x2y planes: when the cluster is between the source and the screen, the 5 layer is a distance d ¼ 1000 A˚ from the point source, with the atoms arranged on a square lattice with spacing a ¼ 2:5 A˚, and with the 4 layer at d ¼ 1004:5 A˚. The spacing between the 5 layer and the 4 layer is
4:5 A˚, thus mimicking a BCC lattice. Note that the ‘‘atomic peaks’’ – the peaks that are at the atomic sites – are at almost the same locations in Figs. 1(a)–(c). Comparing Figs. 1(a) and (b), note that the spurious peaks are shifted in Fig. 1(b) in the x2z plane, so that they are at different locations in the two figures. Comparing Figs. 1(a) and (c), one sees that, for Ky ðrÞ [Fig. 1(c)], the spurious peaks are shifted, not in the x2z plane, as in Fig. 1(b), but in the y2z plane. The spatial separation of these local maxima serves as a signal that these are, indeed, spurious peaks. The idea of combining selected lateral displacements is not new. The first paper to use lateral
Fig. 1. Reconstructions for a 5 5 j 4 4 carbon cluster. jKðrÞj is shown for a cut in the x–z plane through the middle 5 5 string of atoms. In this plane, there is a row of 5 atoms along z ¼ 1000 A˚, spaced 2:5 A˚ apart in the x-direction. Note that the distance scale in the x-direction is expanded as compared with the z-direction; this facilitates examination of the atomic and spurious peaks. The lateral displacements of the cluster are as follows: (a) no displacement, (b) 150 A˚, in the x-direction, and (c) 150 A˚, in the y-direction. Note the shift in the locations of the spurious peaks, and that the atomic peaks exhibit a much smaller shift. These reconstructions were obtained from holograms generated using a 14 cm14 cm screen. See text for details.
M.R.A. Shegelski, T.A. Rothwell / Ultramicroscopy 85 (2000) 155–164
displacement to reconstruct in–line electron holograms [10] considered, as an example, the case of a two atom object, modeled by a Gaussian–shaped atoms (see Ref. [10], Fig. 2). By superposing the reconstructed images from 11 holograms with lateral shifts, Lin and Cowley were able to eliminate most of the conjugate image ‘‘noise’’, obtaining a clear two-atom, one-dimensional reconstruction (see Ref. [10], Fig. 10 and Section 6). In a previous work [3] we employed a similar superposition for a multi-atom cluster in LEEPS, but found that such superposition did not clean up the reconstruction adequately. In this present work, we examine the twodimensional and/or three-dimensional structure of 2 or 3 reconstructions, KðrÞ, obtained from lateral shifts of the atomic cluster. Our method thus goes well beyond superposition of reconstructions. By exploring the topologies of the 2 or 3 reconstructions, we develop a means of identifying and eliminating the peaks that are not true atomic peaks; i.e. the spurious peaks due to the twin image are identified and removed. The result of our method is that we arrive at a final reconstruction that gives a drastically improved picture of the atomic structure. The details of our method of distinguishing between atomic and spurious peaks are necessarily more complicated than just comparing the spatial locations of the various local maxima, especially when considering larger clusters. In order to give a reasonably complete picture of our method, we next present some of the salient details of our approach. At the end of this section, we will give a list of the various parameters that we use in applying the methods described below. We take the following steps in order to eliminate as many spurious peaks as possible. Our methods may be categorized as ‘‘three-dimensional’’ or ‘‘two-dimensional’’, depending on whether we search through the volume around the cluster, or, having found a plane of special interest, we examine characteristics of the reconstruction in this plane. In this paper, we focus primarily, but not exclusively, on BCC-type structures. Some results are given for random distributions of atoms. From our results, we see that generalizing to any atomic
159
arrangement is a project that is well beyond the scope of this first investigation. 2.3. Three-dimensional methods An outline of the principal methods used for three dimensions is as follows. (1) For a given atomic cluster (object), we first create three holograms, one with the object on the optical axis, one with the object laterally displaced, by 150 A˚, in the x-direction, and one laterally displaced, by 150 A˚, in the y-direction. The same electron energy and screen resolution is used in each case. (2) We generate the three reconstructions K0 ðrÞ, Kx ðrÞ, and Ky ðrÞ. In each case, K is calculated within the same, selected volume around the atomic cluster, and the Ks for lateral displacement are ‘‘mapped back’’ into the volume around the undisplaced cluster. Since the reconstructions are calculated numerically, we use a chosen spatial resolution for ‘‘points’’ inside the volume of interest. We thus determine the values of K in tiny volume elements DV ¼ DxDyDz. We can then compare the values of K for the same point in rspace for the three different reconstructions. (3) We take the reconstructions and renormalize each to the highest jKðrÞj value encountered within that reconstruction. We thus have the K values scaled so that 04jKðrÞj41 for all three cases. (4) We set to zero, i.e. we remove, all points r for which KðrÞ is below a given threshold value. For example, in order to circumvent inspection of the numerous local maxima having small magnitudes, we could remove all points for which jKðrÞj50:2. (5) We then determine the spatial locations of all the local maxima of jKðrÞj for the three reconstructions. This can be done, for example, by comparing the value of jKj at each (non-zero) point to the values of jKj at all neighbouring points. We will refer to these local maxima as ‘‘peaks’’, and to the corresponding values of jKj as ‘‘peak heights’’. (6) For each peak in the undisplaced reconstruction, K0 ðrÞ, we locate nearby peaks in each of the two displaced reconstructions, Kx ðrÞ, and Ky ðrÞ. We select a set of three peaks, one from each reconstruction, such that the set has minimal
160
M.R.A. Shegelski, T.A. Rothwell / Ultramicroscopy 85 (2000) 155–164
spatial separation of peak locations. One way to determine the set is to calculate the perimeter of the triangle formed by the three peak locations. This perimeter is determined from the r-space coordinates of the volume elements DV ¼DxDyDz. (7) A peak is retained if the perimeter lies below a given threshold value. (We find, for best results, this threshold to be typically 3:5–5 A˚, for the clusters studied.) (8) If the perimeter of the trio of associated peaks lies between two selected threshold values, a second test is performed. These two threshold values are selected such that (i) only atomic peaks have perimeters that are smaller than the first threshold, and (ii) all atomic peaks and some spurious peaks have perimeters that are smaller than the second threshold. These two threshold values are, of course, readily found if the cluster is known; we will test this method on unknown clusters, in which case we choose the two threshold values using experience as a guide (see below). Trios of peaks having perimeters between the two threshold values are tested for a peak height match, as follows. The smallest jKðrÞj value of the three points is divided by the largest jKðrÞj value. This quotient is then compared to another threshold value (typically greater than or equal to 0.8). The (undisplaced) peak is retained if the value of the quotient lies above the threshold. The reason we sometimes need to make this test is to separate the atomic peaks from spurious peaks for those cases where the atomic peak’s perimeter is somewhat large as compared to other atomic peaks. (9) We sometimes also use a method that is especially effective at identifying spurious peaks that appear ‘‘in front’’ of the cluster, i.e. on the side of the cluster that is closer to the source. The method is as follows. The three reconstructions are renormalized as described above. All points below a given threshold are again set to zero. The peaks are located in each reconstruction by previously described methods. Each (non-zero) point in each of the 3-d reconstructions is assigned to an ‘‘island’’ determined by the nearest peak. Each island is independently renormalized to the highest value of jKðrÞj in that island, i.e. the value of each peak height is set to one, and the associated islands are all rescaled appropriately. The ‘‘final re-
construction’’ is obtained by multiplying these ‘‘renormalized reconstructions’’ together. This eliminates many spurious peaks because those areas which do not have good overlap of peaks are substantially reduced. This averaged reconstruction is also filtered for noise. In this case, all points that are less than a selected threshold value are set to zero. The final reconstruction is then searched for peaks; all peaks found are retained, and the set of such peaks constitutes the final result. As noted above, this method is often successful in removing spurious planes located on the source side of the cluster or object. To summarize: the undisplaced peak is retained if the value of the perimeter for the trio of associated peak positions lies below a threshold value. A peak is also retained if the perimeter is larger than this value but smaller than a second value and if the ratio of lowest peak height to highest peak height in the trio lies above a third threshold value. Having done this search in three dimensions, and having found planes within which the retained peaks’ maxima lay, we can then examine these planes of special interest. We next report the methods we use for study in these planes. 2.4. Two-dimensional methods We can obtain pictures of the atomic peaks identified by the 3-d methods described above by using a 2-d filter program on x–z or y–z planes. We take two plane reconstructions, one for the undisplaced cluster and one for the cluster displaced along the axis normal to the reconstruction plane (e.g. y-displacement for an x2z plane). This has the effect of removing many spurious peaks within the plane under consideration. The reason, recall, is that the spurious peaks tend to shift in the direction of the displacement of the cluster (see Figs. 1). For example, in the case of study in the x2z plane, the reconstructions considered would be the undisplaced reconstruction, K0 ðrÞ, and the y-displaced reconstruction, Ky ðrÞ, [see, e.g., Figs. 1(a) and (c)]. We first normalize the two reconstructions, and set to zero all KðrÞ values that have magnitudes below the ‘‘noise threshold’’, as described above.
M.R.A. Shegelski, T.A. Rothwell / Ultramicroscopy 85 (2000) 155–164
For study within a plane, we break the region of interest into blocks of area, e.g. DA ¼ DxDz for the x2z plane. We next locate the peaks in the two jKðrÞj using the methods described above for 3-d. We then assign each non-zero jKðrÞj element to an ‘‘island’’, as follows. A point r is assigned to the island associated with a given peak if r is closer to the position, within the plane, of the given peak, than it is to any other peak within the same plane. In this way, each peak has a set of points, an ‘‘island’’, associated with it. Each peak from the undisplaced cluster is paired with the nearest peak in the displaced reconstruction, and the distance between each such pair of peaks is noted. The largest such distance is used to renormalize the distance values. If the distance between nearest neighbour peaks is less than a certain selected value, they are retained, and the associated undisplaced island is not subjected to any further test. If the distance between the pair of peaks is larger than the selected value, then a further test is used. In this test we compare slopes and heights of each point in each island. Consider a particular point r in each of the two reconstructions. The direction of the slope, i.e. the sign of the derivative of jKðrÞj, is tested to see if they match. For example, for the x2z plane, both the sign of the x-slope and the zslope of corresponding islands must match if the peak is to be retained. Every other point is set to zero. Those points which survive this ‘‘slope test’’ must also have the same relative height to within a given tolerance to be included in the final reconstruction. The final reconstruction thus includes those points in the undisplaced reconstruction which have survived either the ‘‘nearest neighbour test’’ or the ‘‘slope and height test’’. 2.5. Parameters used We complete this section by defining the various parameters we use. (1) tn denotes the ‘‘noise threshold’’: all points r having jKðrÞj5tn are removed from the reconstruction, i.e. the value of jKðrÞj is set to zero.
161
(2) Pin denotes the ‘‘inner (or smaller) perimeter’’; all trios of peaks with a total perimeter P less than this value (P5Pin ) are retained without further testing. (3) Pout denotes the ‘‘outer (or larger) perimeter’’; trios of peaks with perimeters between this value and Pin (Pin 5P5Pout ) are subjected to a height match check to decide whether or not they will be retained. (4) ht denotes the ‘‘height match threshold’’; the ratio of the smallest peak in a given trio to the largest peak in the same trio must be larger than this value to order that the peak be retained. (5) dn denotes the ‘‘nearest-neighbor distance’’ in the 2-d method; if the distance between corresponding pairs of points in the two reconstructions is less than dn , then the point in the undisplaced reconstruction is retained. We will give the values of these parameters for the results reported next.
3. Results and discussion We first present results for a 5 5 j 4 4 j 5 5 j 4 4 cluster of carbon atoms. The layers are in x2y planes: when the cluster is between the source and the screen, the first 5 layer is a distance d ¼ 1000 A˚ from the point source, with the atoms arranged on a square lattice with spacing a ¼ 2:5 A˚, and with the first 4 layer at d ¼ 1004:5 A˚. The spacing between the 5 layers and the 4 layers is 4:5 A˚, thus mimicking a BCC lattice. In Fig. 2(a) we show a three-dimensional view of the spatial locations of all the peaks for the undisplaced reconstruction, i.e. all the local maxima in jK0 ðrÞj. In Fig. 2(b) we show a threedimensional view of the spatial locations of those peaks that remain after using the method described above. A striking feature of these figures is the efficacy with which such a seeming simple method removes spurious peaks. Whereas there are 203 peaks in Fig. 2(a), there are only 78 in Fig. 2(b). Only 4 of these peaks are spurious peaks. The 5 5 j 4 4 j 5 5 j 4 4 structure is manifest in Fig. 2(b). We emphasize that the atoms that are
162
M.R.A. Shegelski, T.A. Rothwell / Ultramicroscopy 85 (2000) 155–164
Fig. 2. The locations of local maxima in jKðrÞj in the volume around a 5 5 j 4 4 j 5 5 j 4 4 carbon cluster. Part (a) shows all such maxima, or ‘‘peaks’’, obtained from the reconstruction with the cluster on the optical axis. Part (b) shows the peaks that remain after being scrutinized by the methods reported in this paper. Only atomic peaks, and a few spurious peaks, are retained in (b). The following parameter values were used in part (b): tn ¼ 0:475, Pin ¼ 2:625A˚, Pout ¼ 6:0 A˚, and ht ¼ 0:5. Details are given in the text.
missing in the 5 5 layers are recovered when we apply the 2–d methods to obtain, e.g., x2z plane reconstructions. We thus regard this reconstruction as virtually completely successful. We also wish to emphasize that, in obtaining Fig. 2(b), we have treated the cluster as if it were an unknown cluster. In Figs. 3(a) and (b), we show x–z cuts through the middle of the 5 5 layer of a 5 5 j 4 4 cluster for (a) the undisplaced reconstruction, and (b) the ‘‘processed’’ reconstruction, after the spurious peaks have been removed. Again, the
Fig. 3. Reconstructions in the x2z plane through the middle of the 5 5 layer of a 5 5 j 4 4 carbon cluster. Part (a) shows the undisplaced reconstruction, and (b) shows the ‘‘processed’’ reconstruction, after removal of the spurious peaks. The following parameter values were used in part (b): tn 0 (no noise rejection), ht 1(perfect height match), and dn 0:2. Details are given in the text.
improvement is very good. A similar result is obtained for a 5 5 j 4 4 j 5 5 j 4 4 cluster. To further test our method, we arranged for a member of our research group to generate 3 holograms for a variety of square-planar-type clusters. For each cluster, we were provided holograms for no displacement (i.e. object on the optical axis), x- and y-displaced. We were also given the corresponding displacement distances, and the source–object distance (for the on–axis case). From the three in-line holograms, we generated the three reconstructions K0 ðrÞ, Kx ðrÞ, and Ky ðrÞ. Using the methods described above, we quite successfully identified various unknown clusters.
M.R.A. Shegelski, T.A. Rothwell / Ultramicroscopy 85 (2000) 155–164
Some of the cases we investigated are listed next. Comments are added as required. (1) We started with some simple clusters, such as 3 3 j 2 2 j 3 3. We easily found the atomic sites as well as the interatomic distance within a plane and the separation between planes. Only 3-d methods were needed. (2) We readily identified another unknown cluster as being a 2 2 j 3 3 j 4 4 j 5 5 BCC structure. In this case we used both 3-d and 2-d methods. (3) We successfully identified a 3 3 j 4 4j 3 3 cluster, with the four corners of the ‘‘4 4’’ layer having no atoms. We show in Fig. 4 the middle layer of this cluster. (4) Other unknown clusters we found, correctly, included random arrays of 6 atoms within a cube of edge length 6:0 A˚. We extended our attempts for higher numbers of atoms, keeping the volume per atom fixed (at about 45 A˚3 , which is the value for the 5 5 j 4 4 j 5 5 j 4 4 cluster). We were able to successfully identify up to about 15 atoms, randomly distributed, even without knowing the number of atoms present. (5) We have looked at clusters with missing atoms, again without knowing which atoms were missing. While we could easily identify missing atoms in small clusters (see, e.g., Fig. 4), for larger
163
clusters, such as the 5 5 j 4 4 j 5 5 j 4 4 cluster, we were unable to reliably locate missing atoms. (6) We have also investigated decreasing the screen size, and looking at larger clusters. In brief, our principal results are as follows. For the 14 cm 14 cm screen, the largest cluster for which we obtained high quality reconstruction was a 6 6 j 5 5 j 6 6 j 5 5 cluster. When we tried our methods on a 7 7 j 6 6 j 7 7 j 6 6 cluster, the reconstruction was rather poor: too many atoms were missing and too many spurious peaks survived. In the case of the 10 cm 10 cm screen, the largest cluster for which we obtained high quality reconstruction was a 5 5 j 4 4 j 5 5 j 4 4 cluster. Even with this cluster, however, there was a noticeable number of missing atoms and somewhat too many spurious peaks. When we used the 7 cm 7 cm screen, even the 5 5 j 4 4 j 5 5 j 4 4 cluster was quite poorly reconstructed. There were many missing atoms and many spurious peaks. (7) We repeated our investigation of limiting cluster size for the three different screen sizes, changing only the electron energy. The results given above were obtained using 95 eV electrons. At 195 eV, the corresponding largest clusters for which good reconstruction was obtained were as follows. For the 14 cm14 cm screen, the largest cluster giving good reconstruction was a 7 7 j 6 6 j 7 7 j 6 6 cluster. For the 10 cm 10 cm screen, the largest cluster was a 5 5 j 4 4 j 5 5 j 4 4 cluster. For the 7 cm 7 cm screen, the largest cluster was a 5 5 j 4 4 cluster. In future work, we aim to develop improved methods so that larger clusters can be more adequately reconstructed using a smaller screen size. Ultimately, our goal is to develop methods that will be directly applicable to experimental LEEPS holograms.
4. Summary and outlook Fig. 4. Reconstruction in the x2y plane through the middle layer ( the ‘‘4 4’’ layer) of a 3 3 j 4 4 j 3 3 carbon cluster, with no atoms at the corners of the 4 4 layer. This reconstruction is for the undisplaced cluster, and clearly shows the missing atoms.
The principal results of this paper are as follows. (1) We have developed methods which are quite successful for finding the locations of atoms in a specified three-dimensional volume surrounding
164
M.R.A. Shegelski, T.A. Rothwell / Ultramicroscopy 85 (2000) 155–164
an unknown atomic cluster. In the cases we have examined to date, we are able to locate the atomic peaks and distinguish them from spurious peaks, with significant accuracy. The principal idea underlying our approach is to compare the locations of local maxima in the reconstructions for 3 holograms: one with the geometric center of the cluster on the optical axis, and two others where the center of the cluster has been shifted a specified distance along directions mutually perpendicular to the optical axis. We have given results for a variety of clusters in this paper. (2) We have also developed methods to further scrutinize special planes of interest, with emphasis on investigation of reconstructions in depth. Again, we have been quite successful at removing spurious peaks and thereby obtaining significantly improved reconstructions, especially in depth. The basic idea here is not only to compare locations of local maxima in different reconstructions, but also to compare peak heights and slopes. Results for several interesting clusters have been given in this paper. (3) Our focus has been on reconstructions for square planar clusters, where the intraplanar and interplanar nearest-neighbor distances are unknown. We have also studied random atomic arrangements. Our successes to date have been encouraging. (4) We have investigated the limits of our methods, with emphasis on (a) missing atoms or extra atoms, (b) cluster size, and (c) screen size. Details are given above. We have identified some limitations of our approach. For example, our methods are not sensitive to missing atoms in clusters of appreciable size. We encounter difficulties when we restrict the screen size to be of order of the experimental screen size for LEEPS microscopy, or if the cluster is larger than about 100 atoms, with about 4 planes along the optical axis.
Further work is planned to address these present limits. It is of interest to also develop methods that are able to treat clusters consisting or two or more atomic types. Results to this point are encouraging. Acknowledgements This work was partially supported by the Natural Sciences and Engineering Research Council of Canada (NSERC). We thank Silas Faltus for useful discussions. We thank Prof. H.J. Kreuzer for useful conversations and constructive comments. We also thank the referee for a very helpful report. Matthew Reid provided useful conversations and valuable assistance with various aspects of this work. We thank the UNBC Mathematics and Computer Science Department for access to their computational facilities. We also thank Wilson Yeung for useful discussions regarding computer programming technique and for providing an element of the three-dimensional program. References [1] M.R.A. Shegelski, S. Faltus, T. Clark, H.J. Kreuzer, Ultramicroscopy 74 (1998) 169. [2] M.R.A. Shegelski, T. Clark, M. Reid, S. Faltus, Ultramicroscopy 77 (1999) 129. [3] M.R.A. Shegelski, S. Faltus, Ultramicroscopy 77 (1999) 135. [4] K. Heinz, Rep. Prog. Phys. 58 (1995) 637. [5] D.K. Saldin, G.R. Harp, B.L. Chen, B.P. Tonner, Phys. Rev. B 44 (1991) 2480. [6] C.M. Wei, T.C. Zhao, S.Y. Tong, Phys. Lett. 65 (1990) 2278. [7] H.J. Kreuzer, K. Nakamura, A. Wierzbicki, H.-W. Fink, H. Schmid, Ultramicroscopy 45 (1992) 381. [8] E. Volko, D.C. Jay, L.F. Allard (Eds.), Introduction to Electron Holography, Plenum, New York, 1999. [9] R.J. Collier, C.B. Burkhardt, L.H. Lin, Optical Holography, Academic Press, New York, 1971. [10] J.A. Lin, J.M. Cowley, Ultramicroscopy 19 (1986) 179.