- Email: [email protected]
An atomic model of AlCuFe, and its comparison with high-resolution electron microscope images
]OURNA
Journal of Non-Crystalline Solids 153&154(1993) 145-149 North-Holland
~ I l I ~
L OF
~LII~
An atomic model of A1-Cu-Fe, and its comparison with high-resolution electron microscope images D a v i d P. D i V i n c e n z o , W i l l i a m K r a k o w a n d P e t e r A. B a n c e l IBM Research Division, Thomas J. Watson Research Center, Yorktown Heights, N Y 10598, USA
Eric Cockayne and Veit Elser Department of Physics, Cornell University, Ithaca, N Y 14853, USA
We present an atomic-structure model for high-quality AI-Cu-Fe icosahedral quasicrystals which is simple and yet incorporates most of what is known about the microscopic structure of the Al-Cu-Fe alloy. Using realistic image simulations, we compare this model with actual lattice-image micrographsof thermodynamicallystable quasicrystal. We find good agreement between the micrographs and the models, although we find that lattice imaging is incapable of distinguishing small differences in the atomic structure. We have also determined the microscope defocus conditions that produce the most faithful images.
1. Atomic model [1] The construction of a theoretical structural model of a quasicrystal can be divided into three steps: 1) choose an appropriate atomic cluster as the basic microscopic motif; 2) permit these clusters to joint in a number of well-defined, discrete ways to form a limited number of intermediatescale structures, which can sometimes be viewed as 'tiles' (it may be necessary to include atoms in addition to those in the cluster, which are commonly referred to as 'glue' atoms); 3) construct a space-fitting 'tiling' with long-range quasiperiodic order. There are uncertainties at each of these three stages, but we believe that we have been able to make a set of reasonable choices, so that a plausible microscopic quasicrystal model is obtained. We employ certain simplifications such as not permitting relaxations of the atoms off ideal positions, and we have not assigned definite
Correspondence to: Dr D.P. DiVincenzo, IBM Research Division, Thomas J. Watson Research Center, Yorktown Heights, NY 10598, USA. Tel: + 1-914 945 3076. Telefax: + 1-914 945 4421.
atomic species to all the distinct sites in the quasicrystal. (This is acceptible for the present study since the images presented here do not distinguish between the atom species.) For step one we simply follow most of the previous work in using the M a c k a y icosahedron [2] as the basic cluster motif. For the second step, determining the near-neighbor connections between Mackay clusters, there are a variety of possible approaches. We have found a very natural approach (which fits very well with the third step of the modeling) which relies on overlap [3] of the Mackay clusters. This approach has the advantage that very few holes are left over, so that a minimal number of 'glue' atom [4] have to be inserted. Such a scheme is possible because there are several interpenetrations of the Mackay icosahedron that involve only minimal readjustment of atomic positions, and we have exploited these in our modelling. This approach is very similar in spirit to the model proposed recently by Burkov [5] for a decagonal quasicrystal phase. The interpenetrating near-neighbor connections between clusters are of three types: type-1 neighboring clusters are joined along a five-fold axis of the icosahedron, and the center of one
0022-3093/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
146
D.P. DiVincenzo
e t al. / A n
cluster coincides with a transition-metal secondshell site of th~ other. The two clusters share a distorted decagonal bipyramid volume. Type-2 neighbors are joined along a two-fold axis, with a small interpenetration of the two clusters; they share five two-fold sites and two five-fold sites in their second shells. Type-3 neighbors are separated by a vector lying in a mirror plane between a two-fold axis and a five-fold axis; the two clusters share four atoms. For type 2 and type 3 neighbors, it is necessary to coalesce a pair of atomic sites to avoid unphysical short interatomic separations. Given the clusters and their joining we must still perform the third step: place them into a network with long-range quasiperiodic translational order. There turns out to be a simple network which meets our requirements. It is obtained from the projection technique [6], and it is a subset of the vertices of the canonical 3D icosahedral tiling using two Penrose rhombohedra. It is known that the vertices of the 3D Penrose tiling are selected by slicing the 6D hypercubic lattice by a strip which is flat and infinite in the three physical dimensions, and has the cross-section of a rhombic triacontahedron in the other three (phason) dimensions. The appropriate subset of these vertices is obtained by selecting only points within a smaller triacontahedral window, scaled down by ~- (the golden ratio). Each vertex in this network has either one, two, or three nearest neighbors in the five-fold directions; three is the maximum number of type-1 neighbors which a cluster may have without introducing additional unacceptable cluster-cluster separations. Thus we produce a large 'quasicrystallite' which is suitable for the lattice-image modelling. It is in the form of a right rhombic prism with its c-axis along a five-fold icosahedral axis. The purpose of the prismatic shape is to keep a uniform thickness in the five-fold direction, although we found that beyond a certain thickness the images should be independent of thickness. The model contains 59 567 atoms, and is about 80 A thick, with comparable dimensions in the plane. Based on the measured stoichiometry of A 1 - C u - F e , the model has a density of 5.2 g / c m [3]. This is significantly
atomic
model of AI-Cu-Fe
50 .',*~,'~I,~O,.*G..~,.,:j*,::
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:'.';i ~'.%"~,:~X,~r;,;~%";:i 3.-%"j;",;~ =i d,5.';: b£ .~.'.".:;.'~:i.Z.%;-:~'t~;~ 0
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--
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;
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t*t t.f
." *
'
lttl~*
I
ANGSTROMS Fig. |. A t o m positions o f the A I - C u - F e
mode] p r o j e c t e d
a l o n g t h e f i v e - f o l d axis. A t o m l o c a t i o n s c o r r e s p o n d dots.
to b l a c k
higher than the experimental density [7], of 4.41 ___0.04 g / c m 3. The model would need apporoximantly 15% of its sites vacant in order to give the observed density. Figure 1 shows a large subset of the structure (about a 100 x 100 × 80 .~-3 area) projected onto the five-fold axis; the atom positions correspond to black dots. The structure consists of two general kinds of atom columns: distinct, well-separated columns, and clusters of atom columns very close together which, taken together, we call 'fuzzy columns'. The well-separated columns correspond to atoms that run along the central axis of columns of Mackay icosahedra. The 'fuzzy columns' consists of atoms off the vertical axis of these clusters; the degree of lateral displacement in these columns is as large as ~ 1 A and the numbers of atoms in the fuzzy columns are about the same as those in sharp columns. It is apparent that ten-fold spoke patterns occur around well defined atom positions.
2. Experiments and image computations A I - C u - F e samples were prepared by one of the authors from elemental constituents ( >
D.P. DiVincenzo et al. / An atomic model of AI-Cu-Fe
99.99% pure) in an induction furnace under an atmosphere of high purity argon. The bulk atomic composition was A163.7Cu23.6Fea2.7. After alloying, ingots several centimeters in size were vacuum-encapsulated and annealed at 840°C for several days and at 812°C for a comparable time. U p o n removal from the furnace, the ampoules were rapidly cooled (within a few minutes) to room temperature. Faceted grains were removed which ranged in size from 100 Ixm to over a millimeter. Thin samples for high resolution microscopy were then p r e p a r e d by dispersing crushed samples onto thin carbon films or holey carbon films. The samples were very brittle, which allowed them to be p r e p a r e d in such a manner. The microscope used for high resolution observations was operated at a 400 kV accelerating voltage and had a point resolution of ~ 1.7 ,~. Usually, the shards were observed at their edges since the specimens were thinnest there. High magnification views of the specimen for
147
two different values of defocus are shown in fig. 2. We have carefully selected thin regions such that elastic scattering processes dominate and interpretable structure images are obtained. The images have been high-pass Fourier filtered to eliminate contrast variations due to thickness variations, low-pass filtered to eliminate photographic shot noise as well as contrast-stretched. We expect this procedure to preserve all the important microscopic details of the image. In both images, five sets of symmetry-related lattice planes are visible; there is a decided lack of the phason disorder which is present in abundance in materials processed at lower annealing temperatures [8]. Figure 2(A) is at the approximate Scherzer defocus condition and shows a ten-fold spoke pattern measuring ~ 14.5 A (see graphics circle in figure) with a small donut shaped region at the center. We know from previous work that the microscope transfer function under these conditions is almost entirely negative, so the pro-
Fig. 2. High magnification images of (A) Scherzer defocus image and (B) image which is ~ 300 A more under focus.
148
D.P. DiVincenzo et al. / A n atomic model of Al-Cu-Fe
jected potential of the atoms are expected to yield dark contrast. In other words, the white image features correspond to channels in the projected atomic structure. Alternately, fig. 2(B) corresponds to the condition where the reverse contrast occurs and atom positions are represented by the white image features. Here the spoke pattern measures ~ 16.7 A and radiates from a central white image feature. To relate these images features to atomic structure, image simulations, based upon a realistic model, are now presented.
3. Discussion
We have previously developed a set of computer programs which realistically simulate electron microscope images from digitized graphically represented model structures [9]; we have performed these simulations on the model of fig. 1. The resulting bright-field image focal series from the five-fold orientation model is displayed in figs. 3(A) to (D) for 400 kV electrons. Here ~ 55 A × 55 A squares are displayed and the focal
series values are A f = 200, 450, 700 and 1100 ~,. The spherical aberration coefficient used for these images was Cs = 1.0 mm corresponding to the microscope employed for the experiments. All the image computations include the effect of a finite incident beam divergence which is taken to be at the approximate experimental conditions employed. All the contrast ranges have been expanded to fill the full grey levels available on the display. Hence, some of the images have an inflated contrast. It is found, however, that the images taken at defocus values A f = 200, 450 and 700 ,~ have higher contrast than for any other values of Af. Our most important results is that image features in the simulations agree very well with the experimental micrographs of fig. 2 for the apl~ropriate values of the defocus Af. Af = 450 A is the approximate Scherzer defocus, and indeed this simulated lattice image in fig. 3(B) is in good agreement with the Scherzer-defocus micrograph of fig. 2(A). Comparison of both with the atomcolumn positions in fig. 1 indicates that these images faithfully depict the atom positions with white image features, in accord with the transfer-function theory. Likewise, the Af = 700 simulation (fig. 3(C)) agrees very well with fi~. 2(B), for which the focus is approximately 300 A from the Scherzer value; in this case, the images produce faithful lattice images with black contrast. In both cases, high-density features are seen which correspond to the sharp atom columns in the model, with more diffuse features corresponding with the 'fuzzy' atom columns. Figures 3(A) and (D) illustrate the fact that this agreement is highly non-trivial; many different images are possible as A f is varied (only a small sampoling is shown here), and many, like the Af = 1100 A simulation, look nothing like the experiment.
4. Conclusion
Fig. 3. (A)-(D) Computer generated bright-field axial illumination images of the five-fold model for 400 kV electrons for different defocus values listed in each figure.
We have shown that a model for the A l - C u - F e icosahedral alloy that respects the known features of the microscopic structure of this compound reproduce experimental lattice images of this compound for several different values of the mi-
D.P. DiVincenzo et al. / A n atomic model of AI-Cu-Fe
c r o s c o p e d e f o c u s . U n f o r t u n a t e l y this r e s u l t c a n n o t b e c o n s i d e r e d to b e d e f i n i t i v e ; w e h a v e f o u n d t h a t m a k i n g s m a l l b u t s i g n i f i c a n t v a r i a t i o n s in o u r theoretical model made no detectable difference in t h e s i m u l a t e d l a t t i c e i m a g e s . T h i s w o r k indic a t e s t h a t l a t t i c e i m a g i n g by i t s e l f will u l t i m a t e l y not be the best way of determining the correct atomic model of quasicrystals. Nevertheless, the a g r e e m e n t c o n f i r m s t h a t o u r m o d e l is t r u e in many important respects.
References [1] A more complete report of this work may be found in: W. Krakow, D.P. DiVincenzo, P.A. Bancel, E. Cockayne and V. Elser, J. Mater. Res., in press.
149
[2] V. Elser, in: Extended Icosahedral Structures, eds. M.V. Jaric and D. Gratias (Academic Press, 1989) p. 105. [3] The overlap idea may also be found in: J.-L. VergerGaugry, J. de Phys. 1 1 (1991) 1303; and J,-L. Arag6n, D. Romeu and A. Gomez, Phys. Rev. B44 (1991) 584. [4] C.L. Henley, Phys. Rev. B43 (1991) 993. [5] S.E. Burkov, Phys. Rev. Lett. 67 (1991) 614. [6] V. Elser, Acta Cryst. A42 (1986) 36. [7] M. Cornier-Quiquandon, A. Quivy, S. Lefebvre, E. Eliakim, G. Heger, A Katz and D. Gratias, Phys. Rev. B44 (1991) 2071. [8] P.A. Bancel, in: Quasicrystals: The State of the Art, eds. D.P. DiVincenzo and P.J. Steinhardt (World Scientific, Singapore, 1991)p. 17. [9] W. Krakow, J. Electron Microsc. Techn. 17 (1991) 212; 19 (1991) 366.
Journal of Non-Crystalline Solids 153&154(1993) 145-149 North-Holland
~ I l I ~
L OF
~LII~
An atomic model of A1-Cu-Fe, and its comparison with high-resolution electron microscope images D a v i d P. D i V i n c e n z o , W i l l i a m K r a k o w a n d P e t e r A. B a n c e l IBM Research Division, Thomas J. Watson Research Center, Yorktown Heights, N Y 10598, USA
Eric Cockayne and Veit Elser Department of Physics, Cornell University, Ithaca, N Y 14853, USA
We present an atomic-structure model for high-quality AI-Cu-Fe icosahedral quasicrystals which is simple and yet incorporates most of what is known about the microscopic structure of the Al-Cu-Fe alloy. Using realistic image simulations, we compare this model with actual lattice-image micrographsof thermodynamicallystable quasicrystal. We find good agreement between the micrographs and the models, although we find that lattice imaging is incapable of distinguishing small differences in the atomic structure. We have also determined the microscope defocus conditions that produce the most faithful images.
1. Atomic model [1] The construction of a theoretical structural model of a quasicrystal can be divided into three steps: 1) choose an appropriate atomic cluster as the basic microscopic motif; 2) permit these clusters to joint in a number of well-defined, discrete ways to form a limited number of intermediatescale structures, which can sometimes be viewed as 'tiles' (it may be necessary to include atoms in addition to those in the cluster, which are commonly referred to as 'glue' atoms); 3) construct a space-fitting 'tiling' with long-range quasiperiodic order. There are uncertainties at each of these three stages, but we believe that we have been able to make a set of reasonable choices, so that a plausible microscopic quasicrystal model is obtained. We employ certain simplifications such as not permitting relaxations of the atoms off ideal positions, and we have not assigned definite
Correspondence to: Dr D.P. DiVincenzo, IBM Research Division, Thomas J. Watson Research Center, Yorktown Heights, NY 10598, USA. Tel: + 1-914 945 3076. Telefax: + 1-914 945 4421.
atomic species to all the distinct sites in the quasicrystal. (This is acceptible for the present study since the images presented here do not distinguish between the atom species.) For step one we simply follow most of the previous work in using the M a c k a y icosahedron [2] as the basic cluster motif. For the second step, determining the near-neighbor connections between Mackay clusters, there are a variety of possible approaches. We have found a very natural approach (which fits very well with the third step of the modeling) which relies on overlap [3] of the Mackay clusters. This approach has the advantage that very few holes are left over, so that a minimal number of 'glue' atom [4] have to be inserted. Such a scheme is possible because there are several interpenetrations of the Mackay icosahedron that involve only minimal readjustment of atomic positions, and we have exploited these in our modelling. This approach is very similar in spirit to the model proposed recently by Burkov [5] for a decagonal quasicrystal phase. The interpenetrating near-neighbor connections between clusters are of three types: type-1 neighboring clusters are joined along a five-fold axis of the icosahedron, and the center of one
0022-3093/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
146
D.P. DiVincenzo
e t al. / A n
cluster coincides with a transition-metal secondshell site of th~ other. The two clusters share a distorted decagonal bipyramid volume. Type-2 neighbors are joined along a two-fold axis, with a small interpenetration of the two clusters; they share five two-fold sites and two five-fold sites in their second shells. Type-3 neighbors are separated by a vector lying in a mirror plane between a two-fold axis and a five-fold axis; the two clusters share four atoms. For type 2 and type 3 neighbors, it is necessary to coalesce a pair of atomic sites to avoid unphysical short interatomic separations. Given the clusters and their joining we must still perform the third step: place them into a network with long-range quasiperiodic translational order. There turns out to be a simple network which meets our requirements. It is obtained from the projection technique [6], and it is a subset of the vertices of the canonical 3D icosahedral tiling using two Penrose rhombohedra. It is known that the vertices of the 3D Penrose tiling are selected by slicing the 6D hypercubic lattice by a strip which is flat and infinite in the three physical dimensions, and has the cross-section of a rhombic triacontahedron in the other three (phason) dimensions. The appropriate subset of these vertices is obtained by selecting only points within a smaller triacontahedral window, scaled down by ~- (the golden ratio). Each vertex in this network has either one, two, or three nearest neighbors in the five-fold directions; three is the maximum number of type-1 neighbors which a cluster may have without introducing additional unacceptable cluster-cluster separations. Thus we produce a large 'quasicrystallite' which is suitable for the lattice-image modelling. It is in the form of a right rhombic prism with its c-axis along a five-fold icosahedral axis. The purpose of the prismatic shape is to keep a uniform thickness in the five-fold direction, although we found that beyond a certain thickness the images should be independent of thickness. The model contains 59 567 atoms, and is about 80 A thick, with comparable dimensions in the plane. Based on the measured stoichiometry of A 1 - C u - F e , the model has a density of 5.2 g / c m [3]. This is significantly
atomic
model of AI-Cu-Fe
50 .',*~,'~I,~O,.*G..~,.,:j*,::
.. "..
...:'.,'.',~'~¢. _-'..?:~-.%ke2e~%'d/='.%'~.,,"......... ... .....'.,'.',,'p2e, V,h'.?,'.,.~-'b~,L',.'.:i,~:i~;'.>'~.',~4,'.,',:,d;:;,:'.: ".,."
:'.';i ~'.%"~,:~X,~r;,;~%";:i 3.-%"j;",;~ =i d,5.';: b£ .~.'.".:;.'~:i.Z.%;-:~'t~;~ 0
- "o
(t) ¢.9
,°rz ",~'~.".%La,'e",%'d?'~k~'d,%'d.::',¢'~.".%~ *","e':%'~':%,'-'2.°.%'~%'-: ".
t---
-'--~-"
....
"---~'.
~4111"tt1.~*~*1¢~ ? l t l . f # ~ * l t . t
-50
--
.':,;-o
l*4fll*t
-'--'~".
--
-"
~.(tl*tl*.Y~l~.¢
;
"
".
t*t t.f
." *
'
lttl~*
I
ANGSTROMS Fig. |. A t o m positions o f the A I - C u - F e
mode] p r o j e c t e d
a l o n g t h e f i v e - f o l d axis. A t o m l o c a t i o n s c o r r e s p o n d dots.
to b l a c k
higher than the experimental density [7], of 4.41 ___0.04 g / c m 3. The model would need apporoximantly 15% of its sites vacant in order to give the observed density. Figure 1 shows a large subset of the structure (about a 100 x 100 × 80 .~-3 area) projected onto the five-fold axis; the atom positions correspond to black dots. The structure consists of two general kinds of atom columns: distinct, well-separated columns, and clusters of atom columns very close together which, taken together, we call 'fuzzy columns'. The well-separated columns correspond to atoms that run along the central axis of columns of Mackay icosahedra. The 'fuzzy columns' consists of atoms off the vertical axis of these clusters; the degree of lateral displacement in these columns is as large as ~ 1 A and the numbers of atoms in the fuzzy columns are about the same as those in sharp columns. It is apparent that ten-fold spoke patterns occur around well defined atom positions.
2. Experiments and image computations A I - C u - F e samples were prepared by one of the authors from elemental constituents ( >
D.P. DiVincenzo et al. / An atomic model of AI-Cu-Fe
99.99% pure) in an induction furnace under an atmosphere of high purity argon. The bulk atomic composition was A163.7Cu23.6Fea2.7. After alloying, ingots several centimeters in size were vacuum-encapsulated and annealed at 840°C for several days and at 812°C for a comparable time. U p o n removal from the furnace, the ampoules were rapidly cooled (within a few minutes) to room temperature. Faceted grains were removed which ranged in size from 100 Ixm to over a millimeter. Thin samples for high resolution microscopy were then p r e p a r e d by dispersing crushed samples onto thin carbon films or holey carbon films. The samples were very brittle, which allowed them to be p r e p a r e d in such a manner. The microscope used for high resolution observations was operated at a 400 kV accelerating voltage and had a point resolution of ~ 1.7 ,~. Usually, the shards were observed at their edges since the specimens were thinnest there. High magnification views of the specimen for
147
two different values of defocus are shown in fig. 2. We have carefully selected thin regions such that elastic scattering processes dominate and interpretable structure images are obtained. The images have been high-pass Fourier filtered to eliminate contrast variations due to thickness variations, low-pass filtered to eliminate photographic shot noise as well as contrast-stretched. We expect this procedure to preserve all the important microscopic details of the image. In both images, five sets of symmetry-related lattice planes are visible; there is a decided lack of the phason disorder which is present in abundance in materials processed at lower annealing temperatures [8]. Figure 2(A) is at the approximate Scherzer defocus condition and shows a ten-fold spoke pattern measuring ~ 14.5 A (see graphics circle in figure) with a small donut shaped region at the center. We know from previous work that the microscope transfer function under these conditions is almost entirely negative, so the pro-
Fig. 2. High magnification images of (A) Scherzer defocus image and (B) image which is ~ 300 A more under focus.
148
D.P. DiVincenzo et al. / A n atomic model of Al-Cu-Fe
jected potential of the atoms are expected to yield dark contrast. In other words, the white image features correspond to channels in the projected atomic structure. Alternately, fig. 2(B) corresponds to the condition where the reverse contrast occurs and atom positions are represented by the white image features. Here the spoke pattern measures ~ 16.7 A and radiates from a central white image feature. To relate these images features to atomic structure, image simulations, based upon a realistic model, are now presented.
3. Discussion
We have previously developed a set of computer programs which realistically simulate electron microscope images from digitized graphically represented model structures [9]; we have performed these simulations on the model of fig. 1. The resulting bright-field image focal series from the five-fold orientation model is displayed in figs. 3(A) to (D) for 400 kV electrons. Here ~ 55 A × 55 A squares are displayed and the focal
series values are A f = 200, 450, 700 and 1100 ~,. The spherical aberration coefficient used for these images was Cs = 1.0 mm corresponding to the microscope employed for the experiments. All the image computations include the effect of a finite incident beam divergence which is taken to be at the approximate experimental conditions employed. All the contrast ranges have been expanded to fill the full grey levels available on the display. Hence, some of the images have an inflated contrast. It is found, however, that the images taken at defocus values A f = 200, 450 and 700 ,~ have higher contrast than for any other values of Af. Our most important results is that image features in the simulations agree very well with the experimental micrographs of fig. 2 for the apl~ropriate values of the defocus Af. Af = 450 A is the approximate Scherzer defocus, and indeed this simulated lattice image in fig. 3(B) is in good agreement with the Scherzer-defocus micrograph of fig. 2(A). Comparison of both with the atomcolumn positions in fig. 1 indicates that these images faithfully depict the atom positions with white image features, in accord with the transfer-function theory. Likewise, the Af = 700 simulation (fig. 3(C)) agrees very well with fi~. 2(B), for which the focus is approximately 300 A from the Scherzer value; in this case, the images produce faithful lattice images with black contrast. In both cases, high-density features are seen which correspond to the sharp atom columns in the model, with more diffuse features corresponding with the 'fuzzy' atom columns. Figures 3(A) and (D) illustrate the fact that this agreement is highly non-trivial; many different images are possible as A f is varied (only a small sampoling is shown here), and many, like the Af = 1100 A simulation, look nothing like the experiment.
4. Conclusion
Fig. 3. (A)-(D) Computer generated bright-field axial illumination images of the five-fold model for 400 kV electrons for different defocus values listed in each figure.
We have shown that a model for the A l - C u - F e icosahedral alloy that respects the known features of the microscopic structure of this compound reproduce experimental lattice images of this compound for several different values of the mi-
D.P. DiVincenzo et al. / A n atomic model of AI-Cu-Fe
c r o s c o p e d e f o c u s . U n f o r t u n a t e l y this r e s u l t c a n n o t b e c o n s i d e r e d to b e d e f i n i t i v e ; w e h a v e f o u n d t h a t m a k i n g s m a l l b u t s i g n i f i c a n t v a r i a t i o n s in o u r theoretical model made no detectable difference in t h e s i m u l a t e d l a t t i c e i m a g e s . T h i s w o r k indic a t e s t h a t l a t t i c e i m a g i n g by i t s e l f will u l t i m a t e l y not be the best way of determining the correct atomic model of quasicrystals. Nevertheless, the a g r e e m e n t c o n f i r m s t h a t o u r m o d e l is t r u e in many important respects.
References [1] A more complete report of this work may be found in: W. Krakow, D.P. DiVincenzo, P.A. Bancel, E. Cockayne and V. Elser, J. Mater. Res., in press.
149
[2] V. Elser, in: Extended Icosahedral Structures, eds. M.V. Jaric and D. Gratias (Academic Press, 1989) p. 105. [3] The overlap idea may also be found in: J.-L. VergerGaugry, J. de Phys. 1 1 (1991) 1303; and J,-L. Arag6n, D. Romeu and A. Gomez, Phys. Rev. B44 (1991) 584. [4] C.L. Henley, Phys. Rev. B43 (1991) 993. [5] S.E. Burkov, Phys. Rev. Lett. 67 (1991) 614. [6] V. Elser, Acta Cryst. A42 (1986) 36. [7] M. Cornier-Quiquandon, A. Quivy, S. Lefebvre, E. Eliakim, G. Heger, A Katz and D. Gratias, Phys. Rev. B44 (1991) 2071. [8] P.A. Bancel, in: Quasicrystals: The State of the Art, eds. D.P. DiVincenzo and P.J. Steinhardt (World Scientific, Singapore, 1991)p. 17. [9] W. Krakow, J. Electron Microsc. Techn. 17 (1991) 212; 19 (1991) 366.