ADVANCES IN IMAGING AND ELECTRON PHYSICS VOLUME 109
EDITOR-IN-CHIEF
PETER W. HAWKES CEMESlLahoratoire d'Optique Elec...
23 downloads
588 Views
15MB Size
Report
This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form
ADVANCES IN IMAGING AND ELECTRON PHYSICS VOLUME 109
EDITOR-IN-CHIEF
PETER W. HAWKES CEMESlLahoratoire d'Optique Electronique du Centrz National de la Recherche Sc ientrfrqrte Toulouse. France
ASSOCIATE EDITORS
BENJAMIN KAZAN Xero.1 Corporation Palo Alto ResearcA Center Palo Alto. Califoinia
TOM MULVEY Department of Electronic, Engineering and Applied Physics Aston Uniwrsity Birminghanr, United Kingdom
Advances in
Imaging and Electron Physics EDITEDBY PETER W. HAWKES CEMESlLahor~~itnire (1’ Optique Elec~tr~otiiyue du Centre National de lu Recherche Scientifque Toulousr. F1.unc.e
VOLUME 93
ACADEMIC PRESS San Diego New York Boston London Sydncy Tokyo Toronto
This book is printed on acid-free paper.
@
Copyright 0 1995 by ACADEMIC PRESS, INC All Rights Reserved. No part of this publicarion may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher.
Academic Press, Inc. A Division of Harcourt Brace & Company 525 B Street, Suite 1900, San Diego, California 92101-4495 United Kingdon? Edition pihlislzed by Academic Press Limited 24-28 Oval Road, London NW 1 7DX
International Standard Serial Number: 1076-5670 International Standard Book Number: 0- 12-0 14735- I PRINTED IN THE UNITED STATES OF AMERICA 95 96 91 98 99 00 B C 9 8 7 6 5
4
3 2
1
CONTENTS CONTRIBUTORS . . . . . . . . . . . . . . . . . . . . . . . PREFACE. . . . . . . . . . . . . . . . . . . . . . . . . .
vii ix
Group Invariant Fourier Transform Algorithms R . TOLIMIERI. M . AN. Y. ABDELATIF. C. Lu. G . KECHRIOTIS. AND N . ANUPINDI I. I1 . 111. IV. V. VI . VII . VIII . IX .
Introduction . . . . . . . . . . . . Group Theory . . . . . . . . . . . FT of a Finite Abelian Group . . . FFT Algorithms . . . . . . . . . . Examples and Implementations . . . Affine Group RT Algorithms . . . . Implementation Results . . . . . . Affine Group CT FFT . . . . . . . Incorporating ID Symmetries in FFT References . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . .
. . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 3 14 16
21 30 42 46 53 55
Crystal-Aperture STEM JACOBUST. FOURIE I . Introduction . . . . . . . . . . . . . . I1 . Theoretical Considerations and Experimental 111. Experimental Results in Imaging . . . . IV. Summary and Conclusions . . . . . . . References . . . . . . . . . . . . . .
. . . . . . . . 57 . . . . 59 . . . . . . . . . 90 . . . . . . . . . 106 . . . . . . . . 107
Evidence
Phase Retrieval Using the Properties of Entire Functions N . NAKAJIMA I . Introduction . . . . . . . . . . . . . . . . . . . . . . I1 . Theoretical Background . . . . . . . . . . . . . . . . . I11. Extension to Two-Dimensional Phase Retrieval . . . . . . . V
109 112 131
vi
CONTENTS
IV. Application to Related Problems . . . . . . . . . . . V. Conclusions . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . .
. . 139 167 168
Multislice Approach to Lens Analysis GIULIOPozzr I . Introduction . . . . . . . . . . . . . . . . . . . . . . I1 . Standard Multislice and BPM Equations and First Applications . . . . . . . . . . . . . . . . . . . . . . 111. Application of the Multislice Equations to Round Symmetric Electron Lenses . . . . . . . . . . . . . . . . . . . . 1V. Improved BPM Equations and Application to Gradient Index Lenses . . . . . . . . . . . . . . . . . . . . . V. Beyond the Paraxial Approximation . . . . . . . . . . . . VI . Conclusions . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . .
202 207 215 216
Orientation Analysis and Its Applications in Image Analysis N . KEITH TOVEY.MARKW. HOUNSLOW. A N D JIANMINWANG Introduction . . . . . . . . . . . . . . . . . . . . . . Definition of the Task . . . . . . . . . . . . . . . . . . Image Acquisition . . . . . . . . . . . . . . . . . . . Image Processing and Analysis of Orientation . . . . . . . . Generalized Intensity Gradient Operators . . . . . . . . . . Enhanced Orientation Analysis-Domain Segmentation . . . . Applications of Orientation Analysis . . . . . . . . . . . . Implementation and Automation of Orientation Analysis . . . . Concluding Remarks . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . .
220 224 228 231 246 287 300 319 323 326
I. I1. I11 . 1V. V. V1. VII . VIII . IX .
INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . .
173 176 186
331
CONTRIBUTORS Numbers in parentheses indicate the pages on which the authors’ contributions begin.
Y.ARDELATIF ( I ) , AWARE Inc., Cambridge, Massachusetts 02142 M. AN ( I ) , AWARE Inc., Cambridge, Massachusetts 02142 N. ANUPINDI ( I ) , AWARE Inc., Cambridge, Massachusetts 02142 JACOBUS T. FOURIE(57), CSIR Division of Materials Science and Technology, Pretoria 0001, South Africa MARKW. HOUNSLOW(219), School of Environmental Sciences, University of East Anglia, Norwich NR4 7TJ, United Kingdom
G. KECHRIOTIS( I ) , AWARE Inc., Cambridge, Massachusetts 02142 C. Lu ( I ) , AWARE Inc., Cambridge, Massachusetts 02142 N. NAKAJIMA(109), College of Engineering, Shizuoka University, Hamamatsu 432, Japan
GIULIOPozzr (173), Department of Physics, University of Bologna, 40126 Bologna, Italy
R. TOLIMIERI ( 1 ), AWARE Inc., Cambridge, Massachusetts 02 142 N. KEITH TOVEY(219), School of Environmental Sciences, University of East Anglia, Norwich NR4 7TJ, United Kingdom JIANMIN WANG(2 l9), School of Environmental Sciences, University of East Anglia, Norwich NR4 7TJ, United Kingdom
This Page Intentionally Left Blank
PREFACE
The five chapters that make up this volume cover advanced topics in crystallographic computing, image restoration and analysis, particle optics, and a revolutionary new idea concerning the scanning transmission imaging mode. The volume opens with a chapter by a group of authors, most of whom are no strangers to these Advances. M. An and colleagues have already contributed a survey on discrete FFT algorithms; here they present in detail their work on group-invariant Fourier transform algorithms, which are of vital interest in crystallography. By linking the crystal symmetry to the algorithm itself, higher dimensional Fourier transforms can be performed very efficiently. The authors set out the underlying mathematics and its practical implementation fully and this account will no doubt be helpful for many users of these techniques. The second chapter is by J . T. Fourie, who has been publishing articles in the electron microscopy literature over the past few years on a revolutionary way of attaining high resolution information. He has not, however, previously prepared a long connected account of these ideas and the associated experimental tests; here, he has brought together both the theoretical background and the related experiments, which will excite widespread interest in his approach. N. Nakajima, author of the third contribution, has been working on phase retrieval for several years and has prepared a detailed account of this research; the theory is recapitulated with care and a variety of types of application are then examined. Despite the immense amount of thought that has been devoted to this problem, difficulties still remain, as Nakajima points out. Complementary contributions on this theme are planned for future volumes, notably from the school of the late Richard Bates, who contributed to these Advances in 1986. Electron lens properties have been very thoroughly studied for more than half a century, essentially by calculating trajectories through the lens fields and then evaluating the various cardinal elements and aberration coefficients. In the fourth chapter, G. Pozzi demonstrates that this is not the only way of analyzing lenses. For many years, it has been usual to calculate the propagation of electron waves through specimens by picturing the latter cut into very thin slices and then propagating the wave through the potential in each slice. Pozzi applies this idea to the calculation of lens properties. A full account of this new approach is presented, including aberrations, which had not previously been fully covered in this way. The volume ends with a magisterial account of orientation analysis and the associated image processing methods by N . K . Tovey, M. W. Houslow, and IX
X
PREFACE
J. Wang. The whole subject is reviewed: first restoration, enhancement, and edgedetection, then simple and more advanced applications in the specific domain of orientation analysis (for mineralogical samples in particular but of course the techniques are of much wider applicability). This is virtually a short monograph on the subject and will be heavily used in the specialist area in question. I am most grateful to all the authors for the trouble that they have taken, not only in preparing these surveys but also in ensuring that they are accessible to readers who are not specialists in the same subject area. 1 thank them all most sincerely and conclude as usual with a list of forthcoming articles. The volume numbers of those already in press are indicated. Peter W. Hawkes
FORTHCOMING ARTICLES Nanofabrication Use of the hypermatrix Image processing with signal-dependent noise The Wigner distribution Parallel detection Discontinuities and image restoration
Hexagon-based image processing Microscopic imaging with mass-selected secondary ions Modern map methods for particle optics Nanoemission Magnetic reconnection Cadmium selenide field-effect transistors and display
ODE methods Electron microscopy in mineralogy and geology The artificial visual system concept Projection methods for image processing Space-time algebra and electron physics The study of dynamic phenomena in solids using field emission Gabor filters and texture analysis
H. Ahmed D. Antzoulatos H. H. Arsenault M. J. Bastiaans P. E. Batson L. Bedini, E. Salemo, and A. Tonazzini S. B. M. Bell M. T. Bernius M. Berz and colleagues Vu Thien Binh A. Bratenahl and P. J. Baum T. P. Brody, A. van Calster, and J. E Farrell J. C. Butcher P. E. Champness J. M. Coggins P. L. Combettes C. Doran and colleagues M. Drechsler J. M. H. Du Buf
PREFACE
Group algebra in image processing Miniaturization in electron optics The critical-voltage effect Amorphous semiconductors Stack filtering Median filters RF tubes in space Mirror electron microscopy Relativistic microwave electronics Rough sets The quantum flux parametron The de Broglie-Bohm theory Contrast transfer and crystal images Morphological scale space operations Algebraic approach to the quantum theory of electron optics Signal representation Electron holography in conventional and scanning transmission electron microscopy Quantum neurocomputing Surface relief
Spin-polarized SEM Sideband imaging Ernst Ruska, a memoir Regularization Near-field optical imaging Vector transformation Seismic and electrical tomographic imaging SEM image processing Electronic tools i n parapsychology
xi D. Eberly (vol. 94) A. Feinerman A. Fox W. Fuhs M. Gabbouj N. C. Gallagher and E. Coyle A. S. Gilmour R. Godehardt (vol. 94) V. L. Granatstein J. W. GrzymalaBusse (vol. 94) W. Hioe and M. Hosoya F! Holland K. Ishizuka I? Jackway R. Jagannathan and S. Khan W. de Jonge and P. Scheuermann E Kahl and H. Rose (vol. 94) S. Kak (vol. 94) J. J. Koenderink and A. J. van Doom K. Koike W. Kmkow L. Lambert and T. Mulvey A . Lannes A. Lewis W. Li McCann and colleagues N. C. MacDonald R. L. Morris
xii
PREFACE
Image formation in STEM
C. Mory and
The Growth of Electron Microscopy
T. Mulvey (ed.) (vol. 95)
The Gaussian wavelet transform
R. Navarro, A. Taberno and G. Cristobal G. Nemes T. Oikawa and N. Mori S. J . Pennycook G. A. Peterson H. Rauch H. G. Rudenberg D. Saldin G. Schmahl J. I? E Sellschop J. Serra M. I. Sezan H. C. Shen T. Soma J. Toulouse J. K. Tsotsos Y. Uchikawa D. van Dyck L. Vincent L. Vriens, T. G. Spanjer, and R. Raue A. Zayezdny and I. Druckmann (vol. 94) A. Zeilinger, E. Rasel, and H. Weinfurter
C. Colliex
Phase-space treatment of photon beams Image plate Z-contrast in materials science Electron scattering and nuclear structure The wave-particle dualism Scientific work of Reinhold Rudenberg Electron holography X-ray microscopy Accelerator mass spectroscopy Applications of mathematical morphology Set-theoretic methods in image processing Texture analysis Focus-deflection systems and their applications New developments in ferroelectrics Knowledge-based vision Electron gun optics Very high resolution electron microscopy Morphology on graphs Cathode-ray tube projection TV systems
Signal description
The Aharonov-Casher effect
ADVANCES I N IMAGING AND ELECTRON PHYSICS. VOL . 93
Group Invariant Fourier Transform Algorithms' R . TOLIMIERI. M . AN. Y . ABDELATIF. C . LU. G . KECHRIOTIS. and N . ANUPINDI. A WARE Inc., One Memorial Drive. Cambridge. Massachusetts
. . . . . I . Introduction . . . . 11. GroupTheory . . . . . . . . . . A . Finite Abelian Group . . . . . . . B . Character Group . . . . . . . . C . Point Group . . . . . . . . . . D . Affine Group . . . . . . . . . E . Examples . . . . . . . . . . . 111. FT of a Finite Abelian Group . . . . . . A . Periodization-Decimation . . . . . 1V . FFTAlgorithms . . . . . . . . . . A . Introduction . . . . . . . . . . B . RT Algorithm . . . . . . . . . C . CT FFT Algorithm . . . . . . . . D . Good-Thomas Algorithm . . . . . . V . Examples and Implementations . . . . . A . RT Algorithm . . . . . . . . . B . CT FFT Algorithm . . . . . . . V1 . Affine Group RT Algorithms . . . . . . A . Introduction . . . . . . . . . . B . Point Group RT Algorithm . . . . . C . AffineGroup RT Algorithm . . . . . D . X'.l nvariant RT Algorithm . . . . . VII . Implementation Results . . . . . . . A . Complexity . . . . . . . . . . VIII . Affine Group CT FFT . . . . . . . . A . Extended CT FFT: Abelian Point Group . B . CT FFT with Respect to Pmmm . . . . C . Extended CT FFT: Abelian Affine Group D . C T FFT with Respect to Fmmm . . . . IX . Incorporating ID Symmetries in FFT . . . References . . . . . . . . . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . . . . . . . . . . . . . . . . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 3 3 6 9 10 11 14 15 16 16 16 17 19 21 21 21
30 30 31 39 41 42 45 46 41 48 49 52 53 55
'
This research was supported by the Advanced Research Projects Agency of the Department of Defense and was monitored by the Air Force Office of Scientific Research under contract number F49620.91.0098 . The views and conclusions contained in this document are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied. of the Advanced Research Projects Agency or the U.S. Government . 1
Copyright 0 1995 by Academic Press. Inc . All rights of reproduction in any form reserved .
2
R . TOLIMIERI et al.
I . INTRODUCTION The design of algorithms for computing the crystallographic Fourier transform is a subject in applied group theory. In previous works (An et al., 1991; Tolimieri el al., 1993) we exploited several elementary results in finite abelian group theory and developed the basic abstract constructs underlying the class of divide and conquer algorithms for computing the multidimensional (MD) discrete Fourier transform (DFT). This setting provides a convenient landscape for introducing a class of divide and conquer crystallographic algorithms. In An et al. (1991), we outlined a systematic approach for classifying three-dimensional (3D) crystallographic groups. Applications to 3D crystallography require a detailed understanding of this classification. Similar classifications exist to some extent in higher dimensions and are equally important for applications to quasicrystallography. The theory developed in this work will operate within the abstract formulation presented in An et al. (1991), Tolimieri et af. (1993). Finite abelian groups will serve as data indexing sets. A class of affine group fast Fourier transform (FFT) algorithms will be introduced which fully use data invariance with respect to subgroups of the affine group of data indexing sets. The affine subgroup need not come from a crystallographic group. This approach removes dimension, transform size, and crystallographic group from algorithm design and serves to bring out fundamental algorithmic procedures rather than produce an explicit algorithm. These procedures provide tools for writing code which scales over dimension, transform size, and crystallographic group and which can be targeted to various architectures. In fact these methods apply to all 230 3D crystallographic groups and to composite transform sizes. We will show the power of these tools by way of an extensive list of implementation examples. We distinguish three algorithm strategies. The first is based on the well-known Good-Thomas (GT) or prime factor algorithm which breaks up an FT computation into a sequence of smaller size DFT computations determined by the relatively prime factors of the initial transform sizes. In An et al. (1991) we developed an abstract formulation of the GT and applied it as a tool for crystallographic algorithms. Our treatment here will be brief and mostly contained in examples. Reduced transform (RT) algorithms were considered in detail in An el al. (1991), Tolimieri et al. (1993). A simple generalization of the RT approach based on collections of subgroups will be presented, which provides a universal framework for affine group Fourier transform (FT) algorithms. In applications to 3D crystallography this class of algorithms replaces the problem of computing the FT of 3D group invariant data by that of computing in parallel the FT of a collection of 1D or 2D group-invariant
GROUP INVARIANT FOURIER TRANSFORM ALGORITHMS
3
data sets. The latter problem is substantially simpler and several efficient implementations are widely practiced. A third approach, based on a generalization of Cooley-Tukey fast FT (CT FFT), will be discussed which performs generalized periodizations (Tolimieri et al., 1993) with respect to affine subgroups. This method applies to abelian affine subgroup invariant data and hence to about 100 of the 230 3D crystallographic groups. A C T FFT algorithm associated to an abelian subgroup X of the affine group provides code for Y invariant data with respect to every subgroup Y of X . In applications, we choose X such that the associated CT FFT is easy to code and efficient and such that X contains a large collection of subgroups Y of interest. X itself need not be a crystallographic group. An example will be provided which shows how one code applies to 71 of the crystallographic groups. This work is organized as follows: In Section 11, we will review all the necessary group theory. Finite abelian group theory will be briefly considered as it is covered in many elementary texts. We reference Tolimieri et a / . (1993) as it contains all the necessary results. The affine group of a finite abelian group will be defined. Constructs related to the action of affine subgroups on data indexing sets will be introduced. In Section 111 we define the Fourier transform of an abelian group and study its fundamental role in interchanging periodization and decimation operations (duality). The RT, CT, FFT, and GT algorithms are presented in Section IV as applications of this duality to different global decomposition strategies. Affine group FFT algorithms based on the RT algorithm are discussed in Section VI, while those coming from the application of the affine group CT FFT are introduced in Section VIII. In Section IX, we briefly sketch a method of incorporating 1D symmetry into FFT computations, which calls on lower order existing FFT routines using the symmetry condition. Throughout this work, we will provide many examples. These examples have been chosen to reflect both the theory and our experience and others over several years in writing code for the 3D crystallographic FT. 11. GROUPTHEORY
A . Finite Abelian Group Denote by Z / N the group of integers modulo N consisting of the set (0, 1 ,
..., N
-
11,
with addition taken modulo N. Z / N is a cyclic group of order N and every cyclic group of order N is isomorphic to Z / N . For example, the
4
R. TOLlMlERI et
a/.
multiplicative group UN o f complex N t h roots of unity ( 1 , w,
WN- 1 - a * ,
= eZri/N
I,
9
is a cyclic group of order N a n d the mapping 0: Z/N
--t
UN,
defined by o ( n ) = wn, 0 In < N , is a group isomorphism from Z / N onto U,,. The direct product of two finite abelian groups A , xA 2
is the set of all pairs ( a l ,a,), a, E A , , a, E A , with componentwise addition. By the fundamental theorem of finite abelian groups, every finite abelian group A is isomorphic to a direct product of cyclic groups, A = Z/N, x
*.*
x Z/NR.
(1)
We call Eq. ( 1 ) a presentation of A . A finite abelian group can have several presentations which vary as to the number of cyclic group factors as well as the orders of the cyclic groups. For example, 2 / 3 0 = Z / 2 x Z / 1 5 = Z / 3 x Z/10 = Z / 5 x Z / 6 = Z/2 x Z / 3 x Z / 5 . In general, we have Theorem 11.1. The direct product of cyclic groups having relatively prime orders is a cyclic group. Theorem 11.1 is a special case of the Chinese remainder theorem (CRT). Theorem 11.2 (Chinese Remainder Theorem). Let N = N , N , . NR be a factorization of N into pairwise relatively prime integers. Then there exist uniquely determined integers 0
Ie,,e,,
..., eR < N
satisfying e, = 1 mod N,, e, = OmodN,,
1 5 r, s IR , r # s.
The set (el, e,, ..., e R )is called the complete system of idempotents for the factorization N = N , N , .. . NR.
GROUP INVARIANT FOURIER TRANSFORM ALGORITHMS
5
Let (el, e,, . .., eR1 be the complete system of idempotents for the factorization N = N, N, ...N R . By CRT,
= e, mod N,
e;
(2)
eres = OmodN,
1
I
r, s
IR ,
r#s
(3)
R
e, = 1 modN.
(4)
r= 1
It follows that every n n
= nlel
E
Z / N has a unique expansion of the form
+ n,e, + .--+ n R e R m o d N ,
n,EZ/N,.
In fact, n,
= nmodN,,
1 Ir
R.
I
CRT shows that the mapping X: Z / N
+
Z/N1
X
Z/N, x
X
Z/N,
defined by x(n)
=
( n l , n,,
n, = n mod N,,
n,),
Ir IR
(5)
+ n2e2 + ... + n R e R m o d N .
(6)
1
is an isomorphism having inverse ~ - ~ ( n , , n , , . - - n ,=) rile,
CRT is the basis for many theoretic and applied results in algorithm design. It is a major tool for interchanging between 1D and MD arrays which is the core of the GT algorithm. The use of idempotents in describing this interchange is most important in implementation (Tolimieri et al., 1993). CRT can be used to derive the primary factorization of a finite abelian group. Suppose A is a finite abelian group of order N, and we write N
where P I ,P,, e.g.,
= ppIp,"2
.. . P G M ,
. . .,PM are distinct A = Z/N,
X
X
a, 2 1,
(7)
primes. Choose any presentation of A , Z/NR,
N
=
Ni... N R
and write N, = PY~(') P;~"',
a,(r)
2
0, 1
Im
Then Z/N,
=
Z/PPI"' x
*
- x Z/PGM'",
I M.
(8)
6
R . TOLIMIERI ef al.
and we have, by rearranging factor, the primary factorization of A , where The primary factorization of A is unique as the factors A,,, can be described as the set of all elements in A having order which is a power of the prime P,,, . B. Character Group Consider a finite abelian group A of order N . The character group A * of A is the set of all group homomorphisms
a*:A
+
Or,
which group addition defined by
(a* + b*)(a) = a*(a)b*(a),
a*, b*
E
A*, a
E
A.
(10)
The character group A* is the natural indexing set for FT as we can view A as the time parameter space and A* as the frequency parameter space. We will usually write a*(a) as ( a , a*>. The mapping 4: Z / N ( Z / N ) * defined by 2?ri(mn/ N ) , Osn,m= e -+
establishes an isomorphism
Z/N = (Z/N)*.
More, generally, the mapping
4: Z/Nl x ... X Z/NR
+
(Z/N,X
X
Z/NR)*
defined by
( ( m l , * * * , m R ) , 4 ( n. l. -, , n R ) > = e
27ri(m,n,”,)
.. . e 2 n ; ( ~ R n n ” R )
(1 1)
establishes an isomorphism
Z/N, X
a * *
X
Z/NR = (Z/Nl X ... X Z/NR)*.
By the fundamental theorem, every finite abelian group A is isomorphic to its character group A * .
7
GROUP INVARIANT FOURIER TRANSFORM ALGORITHMS
1. Duality
Fix an isomorphism $ from A onto A * . The dual B' of a subgroup B of A is defined by B' = ( a E A : ( 6 , $(a)) = 1, for all b E Bj. (12) Since $ is an isomorphism, $(B')
=
( $ ( b ' ) :b'
E
B')
is the subgroup of all characters of A that act trivially on B. Consider the quotient group A/B of B-cosets
+ B = {a + b :b E B]
a with abelian group addition
(a + B)
+ (a' + B) = ( a + a ' ) + B.
The isomorphism $ induces isomorphisms B'
-+
(A/B)*,
I&: A/BL
-+
B*,
by the formulas
(a
+ B$,(b'))
=
(a,$(b')),
+ B'))
=
(6, $(a)>,
( b , &(a
a E A , b' a
E
A, b
E
E
B',
(13)
B.
(14)
The characterization of $(B') by Eqs. (13) and (14) implies both induced isomorphisms are well defined, i.e., independent of coset representation. The induced isomorphisms 4, and $* play fundamentral roles in the description of divide and conquer FT algorithms. 2. The Vector Space L ( X )
Denote the space of all complex valued functions on a finite set X by L ( X ) . L ( X ) is a vector space over C with addition and scalar multiplication defined by
(f + g)(x) = f ( 4+ g(x), (af)(x)= 4 f ( X ) ) ,
Q!
f,g E U X ) ,x E X, E c,f E U X ) ,x E x.
Consider a finite abelian group A and a subgroup B of A . For f define
PerLtf(4
=
c f(a + b)
beB
E
L(A) (15)
8
R . TOLIMIERI et al.
and
The periodization operator Per, and the decimation operator Dec, are fundamental operators on L ( A ) . Suppose A has order N.L ( A ) has dimension N.The evaluation basis of L(A) is the collection of functions
(e, : a E A ) defined by
We will denote the evaluation basis by A . The character basis of L(A) is the collection A* of characters of A . Relative to the inner product on L ( A ) defined by (f9g)
=
c f(a)g(a),
f,g
E
UA),
(18)
O€A
where s(a)denotes the complex conjugate of g(a), the evaluation basis is an orthonormal basis of L ( A ) . Since for a*, b* E A * ,
N,
(a*,b*) =
0,
a* = b*, a* # b*,
the set 1
-A*
JN
is an orthonormal basis of L ( A ) . 3 . Canonical Isomorphism
The evaluation basis A and the character basis A* are canonical in the sense that they depend solely on group structures and not on presentation. Although the groups A and A* are isomorphic, there is no canonical isomorphism. Duality is defined relative to a particular choice of isomorphism from A onto A * . By extension, the groups A and A * * , the dual of A * , are also isomorphic, and in fact a canonical isomorphism can be defined. The canonical isomorphism, as we will see in Section 111, defines the FT of A .
9
GROUP INVARIANT FOURIER TRANSFORM ALGORITHMS
For a E A , the mapping @(a)of A* a*
@(a)(a*)= ( a , a*>,
(19)
A*,
E
is a character of A * . The mapping @ : A + A**
(20)
is a canonical isomorphism, since it is defined without reference to presentation. Consider the evaluation basis A of L ( A ) and the character basis A** of L(A*). The canonical isomorphism 0 of A onto A** defines a linear isomorphism L(@)from L ( A ) onto L(A*). C. Point Group
Denote the automorphism group of a finite abelian group A by Aut(A). Subgroups of Aut(A) are called point groups. For a point group H a n d a point a E A , the isotopy subgroup Ha of a in H is defined by Ha = ( a E H : a(a) = a). (21)
H, is a subgroup of H . A point a E A is called a fixed point of H if H = H,. The H-orbit of a, denoted by H(a), is defined by H(u)
( a ( ~: a)
=
H).
E
The mapping a
+
a(a):H
-+
A
induces a bijection from the space of right cosets aH,, a E H , onto H(a). Fix a group isomorphism 9: A + A * . For a E Aut(A),define the adjoint a+ E Aut(A) by (a, d(a+(c))>=
Set a'
=
(44,$(c)>,
a, c
E
A.
(24)
( a + ) - ' ,and observe that
(cup)# = a'p',
( a - y = (a')-'.
For a point group H, define
H'
=
(a' : a
E
H).
The H-orbit H(B) of a subgroup B of A is the collection of subgroups
H ( B ) = (a@): a
E
H).
(25)
10
R. TOLIMIERI et al.
Under duality
H # ( B * ) = (H(B))'. A collection G3 of subgroups of A is called H-invariant if
h E H, B
h(B) E 63,
E
63.
if G3 is H-invariant, the action of H partitions 63 into disjoint H-orbits. Define a complete system of H-orbit representatives in 63 as any collection of subgroups in G3
B , , ..-,BR such that 63 is the disjoint union of the collection of H-orbits
H(B,),
9
H(BR).
A covering of A is a collection of subgroups 63 of A such that
A = U B . B€63
Set
63'
=
(B' : B
E
631.
We say that G3 is a dual covering of A if 63' is a covering of A . We can always construct an H-invariant covering 63 of A .
D. Affine group The affine group of A ,
Aff(A) = A QAut(A), is the set of all (a, a), a
EA
,a
E
Aut(A), with group composition
(a, a)(a',a ' ) = (a + a(a'),aa'). A f f ( A ) acts on A by ( a ,a)(c)= a
+ a(c),
a, c
E
A , a E Aut(A).
(29)
For x E A f f ( A ) , we write x = (a,, a,), a, E A , a, E Aut(A). We define two actions of Af f ( A ) on L ( A ) .For f E L ( A ) and x E Af f ( A ) , define xfW
= f(x(a)),
a EA,
x#f(a)= ( a , , $(c',d(c)>, is a group isomorphism. Relative to 4 C=B',
b=CL.
b, 6'
E
B, c, c'
E
C
20
R. TOLIMIERI e t a / .
Since A/B = B' and A/B' = B, 4; = $ B ~and 4; the notation of the previous section, we can take
=
In particular, in
b* E B*,
z(b*) = 4i'(b*),
which amounts to taking B as a complete system of B'-coset representatives in A. Under these assumptions, the CT FFT takes the form
F, f ( b
+ b')
b E B, 6' E B L .
F,,igg,(b)(b'),
=
Compute
b
g,,(b) E W ' ) ,
E
B.
Compute F+BL(gdB(b))E L(B'),
The second stage is a collection of FT computations over B I . We will see that the first stage is a collection of FT computations over B . By definition
which equals F,Bfb'
(b) 9
where
fbi(b) = f ( b + b'),
b E B, b' E B'.
The precise statement of the stages of the GT can now be given as follows:
GT algorithm Form the slices
fbL
E
L(B),
b'
E
B'.
Compute the collection of FT over B Fg, fbl
E
L(B),
b' E B'
Form the functions
gg,(b) E L(B'),
b
E
B
This step requires data transpose (or permutation).
21
GROUP INVARIANT FOURIER TRANSFORM ALGORITHMS
Compute the collection of FT over B'
bE
F4HLgr$B(b) E L(B'), Set
Fr$f(b +, b*) = F,,I gg,Cb)(b*)* This step requires data transpose (or permutation).
V. EXAMPLES AND IMPLEMENTATIONS
For applications to X-ray crystallography, we will take a 3D case to illustrate the theory presented here. In particular, the smallest nontrivial case, 2 / 1 2 x 2 / 1 2 x 2 / 1 2 is used in many of the examples, while Z / 3 N x Z / 3 N x Z / 6 M and Z / 2 N , x Z / 2 N 2 x Z / 2 N 3 are used in the implementation for several natural numbers. In all the examples, we will take the fixed isomorphism $I given in Eq. ( 1 1). To simplify notation, especially in presenting covering subgroups, we will use the following definition and notation. Let A be a finite abelian group. For a E A denote by ( a ) , the subgroup of A generated by a,
( a ) = { a ,2a, 3a, ..., ( K - l ) a ) , where K is the smallest positive integer such that Ka the order of a.
=
0 E A . K is called
A . RT Algorithm Two forms of RT algorithm wil be derived for A = Z / 3 x Z / 3 x Z / 3 . Using CRT, we will extend our current example to groups of the form Z / 3 * 2N x Z / 3 * 2N x Z / 6 M for integers N a n d M .
Example V.2. RT algorithm I for A = Z / 3 x Z / 3 x Z / 3 . Set A Z / 3 x Z / 3 . The following four subgroups cover A : ((0,1 ) ) x Z / 3 ,
B:
= ((1,
B t = ((2, 1 ) ) x Z/3,
Bt
=
B:
=
1)) x Z/3,
((1,O)) x Z / 3 ,
=
Z/3 x
22
R . TOLIMIERI et a/.
c2=o c , = o
Example V.2. RT algorithm I1 for A = Z/3 x Z/3 x Z/3. We list a collection of 13 covering subgroups along with their dual groups. Each of the covering subgroups is of order 3, while the dual group is a subgroup
23
GROUP INVARIANT FOURIER TRANSFORM ALGORITHMS
of order 9. For a
=
0, 1 , 2 and b l , b2 = 0 , 1,2,
p ( u l , a 3 ,u3) = 1 for all ( a l ,a2, u3) E A , except p(O,O, 0) = 13. We will show two of the computations explicitly. The rest follows in exactly the same way. To index the periodizations with respect to D,, set
A103 : ((O,O, 01, (1,0,0), (2,O,O)J,
(52)
Usually, coset representatives are not unique. Note that although the collection in Eq. (52) can be used as AID5 as well as A / D 3 ,Eq. (53) cannot be used for A / D , . For a, c = 0 , 1,2, 2
2
per,, f(c, 0 , 0 ) =
C C f(bl bl=0 bl=0
PerD5f(0,0 , c)
C C f ( b l , b 2 , h + b2 + d. bl=O b , = o
2
=
+ c, 2bl, b2),
2
2
F + , , J ~ ( ~a,, 0 ) =
c f3(c,
0,0)~(-2*i/3)oc,
c=o 2
F+,,,f5(2a,2a, a)
=
C f 5 ( 0 ,0 , ~ c=o
) e ( - ~ ~ ~ / ~ ) ~ ~ .
24
R. TOLIMIERI et al.
Remaining cases follow in the same way, and the induced FT computations are implemented by 13 independent 3-point FTs. The above two derivations show uniform decomposition of a 3D problem into 2D and 1D problems, respectively. However, the above two cases can be combined to provide various decompositions.
Example V.3. RT algorithm for A = Z/2N x Z/2N. We will list a collection of covering subgroups of A and their dual subgroups of order 2N by listing their generators. A is covered by the 2N + 2N-' subgroups shown in Table I. To organize the periodizations, we will set
The collection of induced FT is implemented by 2N + 2N-' independent 2N-point FT computation. For the dual RT algorithm, we list the values of the function p on A with respect to the collection of covering subgroups given in Table I. Denote by Uo the multiplicative units of Z/2N, i.e.,
U,
= (a E
Z/2N : a = 1 mod 2).
For 1 5 n 5 N - 1, set
U,
= (a E
Z/2N: GCD(a,2N) = 2"). TABLE I.
COVERING SUBGROUPS OF
~
/ x ~2
/~
2
~
GROUP INVARIANT FOURIER TRANSFORM ALGORITHMS
25
Then
u u,.
N- 1
Z/2N
=
n=O
For a,, E U,,, a,, # 0 ,
o Ij < 2 N , o II
=
(e,L:
+ e,M;)
x A,,
where L: and M; are a collection of covering subgroups in Z/2N x Z/2N and Z/3 x Z/3, respectively, as listed in Tables VIII and IX. For easier reference, we repeat the tables here. It is straightforward to show that B is a P6/mmmu-invariant dual covering of A . We will give the P6/mmm'-orbit decomposition of B. Recall Pu = /3 and y u = y. TABLE VIII. ~
COVERING SUBGROUPS OF
/ x2
Table Note: We will denote this collection by 03
~
GROUP INVARIANT FOURIER TRANSFORM ALGORITHMS
37
TABLE IX. COVERING SunGKouPs OF Z/3 x 2/3
k
Subgroup
Generator
Dual group generator
P6/mmm#-orbit structure in 213 x Z/3 is the same as that of P3#, since actions by or y do not change the orbit structure. P6/mmmu(L,) W,)
=
L3
P6/mrnmu(L,)
{Lo,L2,L,},
=
PW,)
I
=
L,,
P(L2)
=
=
[L,].
L2.
P6#-orbit of ( ( j ,l ) ) , p6'((j9 1))
=
{x A , ,
((1, e l l ) ) x A , ,
((-ell
((ell, -ell + e2)>x A 3 .
From the orbit of (( 1, 1)) and L o , we obtain
((-el + 2e2,e1 + 1 ) ) x A 3 ,
((el,1))xA3, ((-2e,
+ e2,el)> x A 3 .
In a, there are 4 ... 2N-1 P6mmm#-orbits, four of which contain three subgroups; the rest contain six subgroups. For completeness, we list the values of idempotents as follows: 1. If 2N = 1 mod3, then
el
=
2N- 1
+ 1,
e2 = 2N.
2. If 2N = 2 mod 3, then el
=
2N + 1,
e,
=
2N-1.
39
GROUP INVARIANT FOURIER TRANSFORM ALGORITHMS
Choose a P6/mmm-invariant function f E L(A). By the invariance, the induced FT computation only on a collection of P6/mmm#-orbit representatives determines the FT off. As in Example VI.4, the periodized functions are invariant under one of the two subgroups of P6/mmm, H,,, or H I , , . Specifically, a periodized function F D is HI,,-invariant if the P6/mmm# orbit of D contains six subgroups, while fD is H,,,-invariant if the P6/mmm# orbit of D contains three subgroups.
C . Affine Group R T Algorithm
Choose a subgroup X of Aff(A) and denote the point group of X by For X-invariant f E L(A) we have F+f(a!a)
=
a E A, x E X .
(ax,4(a!a)>F4f(a),
k.
(60)
F+f i.s not invariant under Xi."but F+f(a) determines F+f at each point in the X#-orbit of a. Choose an k-invariant dual covering 63 of A and a complet? system a, of k-orbit representatives in 63. 63; is a complete system of X' representatives in the covering 63' of A . In the presence of X-invariance, the RT algorithm can be implemented by first computing the induced FT Fi(Per, f ),
B
E
630,
The remaining induced FT computations can be determined by complex multiplications implied by Theorem VI. 1. the X-invariance off reduces the number of required induced FT computations. For any subgroup B < A, define X,
= (X E
X : a,@) = B ) .
X , is a subgroup of X and acts on L(A/B). Theorem VI.3.
Iff is X-invariant then Per, f E L(A/B) is X,-invariant.
By the theorem the induced FT computations
are taken on X,-invariant data. To make full use of the X-invariance o f f we must provide a code which makes full use of the X,-invariance of Per, f, B E 63,. In 1D or 2D, affine group invariant FFT algorithms are substantially simpler because of the restricted class of 1D or 2D affine group actions.
40
R . TOLIMIER1 ef al.
X-Invariant RT Algorithm. Choose an k-invariant dual covering 63 of A and a complete system a0of k-orbit representatives in 63. Form the periodizations Per, f
E
L(A/B),
B
E
a,,.
Compute X,-invariant FT
%.
Ft(PerBf),
B
F;(Per,f),
B E 63,
E
Compute by Eq. (60).
Example VZ.6. Affine group-invariant RT. There are five affine crystallographic groups whose point group is P 6 (see Table X). RT algorithm proceeds as in the case of P6. Now the invariance condition on FT is given by Eq. (60). For 0 II I5, a P6,-invariant f E L ( A ) , the induced FT of the Dj,,-periodization o f f determines f^ on DLk E a,,. To determine ?on P6#-orbits of Djtk set ((el, c 2 ,c,), ~ ( o , oM , ))= w ?(el,
~ 2~,3 = )
=
e-2ai’6.
wC’!f(a#(c1 , c2, ~ 3 ) )
, c 2 ,c,))
=
wzc3~((LyZ)#(cI
=
~ ~ ‘ ~ ! f ( (, ca2~,c,)) )#(~~
= w4‘3!f((a4)#(c1,
c2,
c,))
= w5C3!f((CY5)#(c1,
c2,
c,)),
1 5 1 I5 .
The group that contains all of the 48 tetragonal crystallographic groups is P4/rnmrn. As in the case of P6/rnrnrn, once a P4/rnrnrn#-invariant covering subgroup is partitioned into P4/rnrnrn#-orbits, a code for the RT TABLE X . AFFINEGROUPSWITH POINTGROUPP6 Group
Generator
GROUP INVARIANT FOURIER TRANSFORM ALGORITHMS
41
algorithm with respect to this partitioning contains codes for FT computation of functions invariant under subgroups of P4/mmm. One can also choose a group that contains all the crystallographic point groups; this group need not be a crystallographic group.
D. X#-Invariant RT Algorithm Consider a subgroup X of A f f ( A ) . In many applications we will have to compute the inverse FT of X#-invariant data. Up to index reversal, this problem is equivalent to computing the FT of X'hvariant data. We will embed this problem in the second form RT algorithm. In problems requiring several stages of FT and inverse FT, it makes sense to follow the first form RT algorithm which outputs decimated data by the second form RT algorithm which inputs decimated data and conversely, removing the necessity of data rearrangement steps at each cycle. In the second form of RT algorithm we compute F+f , f E L (A ) by first computing the collection of induced FT
Ft(Dec; f ) , Theorem VI.4.
B
For a subgroup B < A , i f f
F+(Dec, f ) ( - a )
=
63.
E
E
L ( A ) is X#-invariant, then a
F+(Dec,:, f ) ( - x a ) ,
E
A , x E X.
(61)
Proof. F+(DecBf)(-c) =
C
f ( b ) ( b ,6 ( C ) )
C
f(a 'b )(b , d
beB
=
c - ai'a,))
bEB
=
C,
f ( b ) ( b ,6(a,c - a,))
b E a,B
=
F+(Decaf,f (-
XC).
Choose an k'#-invariant covering 63 of A and a complete system cR0 of k'#-orbit representatives in a.It suffices to compute the collection of induced FT F;(Dec,f), B E (Ro. The remaining induced FT computations can be computed from the theorem. Set X B = [ X E X : a,@) = B ) .
42
R. TOLIMIERI et a/.
Theorem VI.5. in variant.
For X#-invariant f E L(A) and B < A , Dec, f is
Dec, f ( b ) = (a,, 4(a:b))DeC, f(a,#b),
b
E
B, X
E
X,
i'-
.
In 3D crystallographic applications, specialized routines as described in the preceding two subsections can be applied to these induced FT computations.
VII. IMPLEMENTATION RESULTS We have implemented symmetrized 3D crystallographic FFTs for the case of P6 symmetric data. The data is assumed to be defined on the Z/3N x Z/3N x Z/6M lattice, where N and M are powers of two.
Algorithm 1 1. Use CRT to re-index the data set such that the problem is transformed to an equivalent 5D computation:
Z/3N x Z/3N x Z/6M
+
Z / 3 x Z / 3 x Z / N x Z / N x Z/6M.
Although this step is computationally expensive, involving irregular accessing of the data stored in the main memory, it should be noted that in many applications where a large number of iterations of the forward and inverse FFT are required, the CRT re-indexing can be carried out only once and then the optimization can be performed in the 5 D domain. 2. Apply the RT algorithm to the Z / 3 x Z / 3 to compute the periodized data on two out of the total four subgroups. The periodization results in two distinct data sets, A , and A , , each defined on Z / 3 x Z / N x Z / N x Z/6M. 3 . Perform two 4D FFTs on the data sets A , and A , to implement the induced FT. The sets A , and A , are P2 and P6 symmetric correspondingly, such that efficient symmetrized FFT code can be used for the computations. If symmetrized FFT code is not used in step 3 , the computational savings are roughly on the order of 1/2. In Fig. 1 we plot the speed up over the nonsymmetrized FFT versus the size of the data set. The second implementatioon results in even more speedups over the nonsymmetrized FFT:
43
GROUP INVARIANT FOURIER TRANSFORM ALGORITHMS Speedup
2.6
1.6'
I 1
0.5
1.5
Data Size
2
lo5
FIGURE 1. Speedup of the P6 symmetrized FFT over the nonsymmetrized FFT versus the data size. Symmetrized RTA on Z/3 x Z/3.
Algorithm 2 1. Use the CRT to re-index the data set such that the problem is transformed to an equivalent 5D computation:
Z/3N x Z/3N x Z/6M
--t
Z/3 x Z/3 x Z / N x Z/N x Z/6M.
2. Apply the RT algorithm on Z/3 x Z/3 x Z / N x Z / N and compute the periodized data on one-third of the total 4 x (3/2)N subgroups. The periodization results in 2 N distinct data sets, each defined on Z/6M. 3. Perform 2 N independent 1D FFTs on data of length 6M. These distinct data sets are P 2 symmetric, so that efficient P2-symmetrized FFT code can be used. If symmetrized FFT code is not used in step 3, the computational savings are roughly on the order of 1/3. In Fig. 2 we plot the speedup over the nonsymmetrized FFT versus the size of the data set. If P2-symmetrized FFT code is used, the computational savings are roughly on the order of 1/6, which is the theoretical maximum since the original data are P 6 symmetric. The P 6 symmetrized RT algorithm-based FFTs share the highly parallelizable structure of the general RT algorithm. A variety of choices of a
44
R . TOLIMIER1 et al.
Speedup
2.510
2
4
6
8
10 Data Size
12
lo4
FIGURE 2. Speedup of the P6 symmetrized FFT over the nonsyrnrnetrized FFT versus the data size.
multiprocessor algorithm are available allowing for efficient implementations depending on the characteristics of the particular platform. Consider for example Algorithm 1. If two processors are available and all of the 2 * 3 N N 6M data set is stored in each processor, no interprocessor communication is needed since each processor can independently compute the periodization and 4D FFT. If only half of the data is stored in the memory of each processor, then in order to compute the periodizations, each processor has to send its data to the other, resulting in a total amount of communication (number of processors x size of messages) equal to 2 * 3 * N * N * 6M. If P > 2 processors are available, the data can be divided along the last dimension into sets of size 2 * 3 N N * 6 M / P , each set being stored into the local memory of one processor. After the computation of the periodizations, each processor keeps 3 N N * 6 M / P of local data, and then performs local FFTs along the first three dimensions. To complete the computation, FFTs along the last dimension have to be performed. Since the data are distributed among the processors along the last dimension, a global transposition is required: Each processor keeps 1/ P of its local data, and sends ( P - 1 ) / P data to other processors. The total communication
- - -
-
- -
GROUP INVARIANT FOURIER TRANSFORM ALGORITHMS
45
requirements are then ( P - 1) x local data size = ( P - 1) x 3 * N * N 6 M / P . In an alternative implementation, P processors are being divided into P / 2 clusters of two processors, with local data being duplicated within each cluster. In this implementation, each node stores twice as much data as before, but the efficiency can be increased in certain multiprocessor networks since now the global transposition step is replaced with two independent global transpositions each involving only P / 2 nodes.
A . Complexity 1. Row-Column Algorithm
Set
A
=
Z/3N x Z/3N x Z/3M.
The computation of the 3D FT using a conventional row-column algorithm of processing the data dimension at a time on many parallel systems exacts a considerably higher price on interprocessor communication than FT computation. RT algorithm offers an alternate data movements in MD FT computation. We list some performance results here.
2 . GT-RT Algorithm I Using CRT,
A = A , x A 2 = (Z/3 x Z/3) x (Z/3 x Z/N x Z / N x Z/M). Data reduction (periodization) stage costs 4 x 2 x 3N 2M additions, which can be combined with data loading operation in a broadcasting mode; on some parallel systems it is given for free. In a 4-processor system, each processor carries out 2 x 3N2Madditions, while receiving input data, followed by a local 5D 3 x 3 x n x N x M FT computation. This algorithm eliminates interprocessor communication completely, and each processor has a balanced load with uniform computation format. 3. GT-RT Algorithm 11
A = A , x A 2 = (Z/3 x Z/3 x Z/3) x (Z/N x Z / N x Z/M). In this decomposition, each processor carries out ( 2 x 3) x N2Madditions to implement periodization while receving input data, followed by a local 4D 3 x N x N x M FT Computation. This decomposition is well suited on a 13-processor system. Both reduction and FT computation are carried out in parallel.
46
R. TOLIMIERI et al. TABLE XI. TIMINGRESULTSON iPSC/860 (3D) (4 NODES) GT-RT (4 nodes) Size 48 x 48 x 48 48 x 48 x 96 48 x 96 x 96
Row-Column (4 nodes)
Time (ms)
Size
Time (ms)
3 60 512 980
64 x 64 x 64 6 4 x 64 x 128 64 x 128 x 128
566 1122 2202
TABLE XI1. TIMINGRESULTSON iPSC/860 (3D) (4 NODES) GT-RT (4 nodes) Size 48 x 48 x 48 x 96 x
Row-Column (8 nodes) Size
Time (ms) 48 x 48 x 96 x 96 x
48 96 96 96
360 512 980 2029
64x 64 x 64 x 128 x
64x 64 x 128 x 128 x
Time (ms) 64 128 128 128
282 585 1152 2216
The RT Algorithms I and I1 show uniform decomposition of a 3D problem into subsets. The combination of RT algorithms with other fast algorithms will provide a highly scalable feature that can be matched to various degrees of parallelism and granularity of a parallel system. The RT algorithm partitions input data at the global level to match each subset into node processors, carrying out loading and reduction operations concurrently at each node; then FT computations are performed in parallel. In Tables XI and XII, timing results on the Intel iPSC/860 with 4- and 8-node implementations are given. The timing results of the next power of 2 sizes of Intel FFT library are also included for comparison. (Non-power of 2 routines are not available in the standard library.) The GT-RT algorithm I was implemented on the 4-node hypercube architecture. The periodization (reduction stage) is coded in standard Fortran, whereas the FFT and 3-point FT calls on the Kuck & Associates optimized assembly routines and our own vectorized 3-point FT routines, respectively. VIII. AFFINEGROUPCT FFT The global decompostion stage of a CT FFT algorithm computes pseudoperiodizations relative to a subgroup B of the indexing group A . In this section we present a CT FFT algorithm whose pseudoperiodizations
47
GROUP INVARIANT FOURIER TRANSFORM ALGORITHMS
are taken relative to an abelian subgroup X c A f f ( A ) . In the classic case, X consists of pure translations. If Y is a subgroup of X , the CT FFT algorithm associated to X can easily be adopted to produce an FFT algorithm for Y-invariant data. The code which implements this CT FFT produces, by a process of disabling, Y-invariant FFT code for every subgroup Y of X. For applications, the choice of X is motivated by two factors. First, the code for the CT FFT associated to X should be simple to write, scalable, and efficient. Second, X should contain a large collection of subgroups of interest in applications. A . Extended CT FFT: Abeliun Point Group Choosef E L ( A ) and an abelian subgroup G of Aut(A).For y* the pseudoperiodizations fy* E L ( A ) by fy*(a)
=
c f(Y)(Y, Y*),
0
E
G* define
€A.
(62)
yeG
Since
o(G),
y = identity map,
otherwise,
y * E G*
we can write
We can compute F+f by computing the collection of FTs fy*,
y* E G*.
(65)
We have replaced a single FT computation by a collection of FT computations. However, the pseudoperiodizations satisfy the following group invariance property:
Theorem VIII.1.
For y*
E
G*,
f,*Ow)= ( Y , y*>f,*(a), F+fy*(y"(a))= ( Y ,
Y*>F+fT*(U),
U E A ,~ E G . a E A , Y E G.
We will say thatf, is G-invariant with character. The CT FFT associated to G decomposes the computation of F+f into a collection of FT computations on G-invariant with character data which can be implemented by simple modifications of the point group RT algorithm.
48
R. TOLIMIERI et al.
Suppose K is a subgroup of G . If we begin with a K-invariant data, we can reduce the number of FT computations. Set K,
= (y* E
G* : ( K , y*>
= 1,
for all
K E K).
(66)
K, is a subgroup of G* isomorphic to the character group (G/K)*. Choose a complete set of representatives of K-cosets in G Y o , Y1, Then every g
E
(67)
YL-I.
G can be written uniquely in the form y = KY/,
K
EK, 0
I < L.
I
(68)
Theorem VIII.2. I f f E L ( A ) is K-invariant then the pseudoperiodization f,. vanishes unless y* E K, .
Proof. L-1
f,*(a)
=
c c f(KY/a)(KY/,Y*)
/ = 0K E K L-l
=
c f~Yra)(rr,Y*>c
I=O
(K,Y*>
KEK
by K-invariance. Since C, ( K , y * > vanishes unless y* E K, , the proof of the theorem is complete. Code f o r the CT FFT algorithm associated to G applies to the computation of the FT of the K-invariant data, K < G , by disabling all the pseudoperiodizations corresponding to y* B K, .
B. CT FFT with Respect to Pmmm For p, p
E
Pmmm, p = p;'ppp;3,
T = p;1ppp:3,
define ( p , r*> = (-
1)rltl+r2t2+r3f3
Associate with the function f E L ( A ) , the column vector fo of length K = 8NML by listing f ( a l ,a 2 ,a,), antilexicographic ordering of (a, ,a 2 ,a,) E A . Also define the vectors f , , 0 Ij 5 7 by listing f(s,(a, , a 2 ,a3),in order of ( a , ,a 2 ,a,) E A . The generalized periodizations off with respect t o Pmmm can be implemented by the vector additions
GROUP INVARIANT FOURIER TRANSFORM ALGORITHMS
where F(2) denotes the 2-point FT matrix, F(2)
=
[
1 - 1
49
'1
and I, is the K x K identity matrix. Crystallographic group P 2 (Henry and Lonsdale, 1952) is a subgroup of Pmmm. P2
=
(1,s24].
p2*
=
(
9
s 2 4 9 s32
9
s56)*
I f f € L(A) is P2-invariant, then four of the periodizations vanish. Each of the non-vanishing periodizations are Pmmm-invariant up to multiplication by k 1, and FT is computed with this invariance. Another crystallographic subgroup of Pmmm is P222: p222 P222,
= (
3
s24
9
s40 9 s481,
= ( 1, S 56).
For P222-invariant f , all the periodizations except f,; and f s f 6 vanish. Iff is Pmmm-invariant, then computation is carried out only for f S ; . C. Extended CT FFT: Abelian Affine Group
The discussion of Section A will be extended to abelian subgroups X of Aff(A) of the form X = B x K where B is a subgroup of A and K is a subgroup of Aut(A). The CT FFT algorithm associated to X combines features of the standard CT FFT associated to B and the abelian point group CT FFT associated to K . The pseudoperiodizations are now taken with respect t o the affine subgroup X . The motivation is to unify the writing of FT code for affine group invariant data.
50
R. TOLIMIERI et a/.
Choose an abelian subgroup X of A ff (A)of the form X = B x K. Then X * = B* x K * . We will usually write bk for (b,k) and b*k* for (b*,k*). Denote a complete set of B'-coset representatives by
z(b*) = 4i1(b*),
b* E B*.
For f E L ( A ) , define the pseudoperiodizations fx*
fx*(a)=
(70)
E L ( A ) ,x* E X*, by
a E A , x* E x*.
f(xa)<x,x*),
(71)
x EX
a
fx*(x(a))= (x,x*)fx*(a),
E
A , x*
E
x*
Since 1
c
f=-
(73)
fX*l
O ( X ) x* EX*
we can compute F+f by the collection of FT computations
F+fx*,
EX*.
X*
A direct computation shows that f,. satisfies the group invariance with character condition. In particular,
f,*(b Define g,
E
+ a) = ( b , b*)fx*(a),
b
E
B,X*
=
b*k* E X * .
(74)
L ( A ) ,x* E X*,by
a
g,*(a) = fx*(a)(a,4(z(b*))),
E
A , x*
=
b*k*.
(75)
g,. is B-invariant and can be viewed as a function in L ( A / B ) .
Theorem VIII.3. and we have
For x*
=
b*k*
F+fx*(z(b*)+ b')
=
E
X*,F+fx* vanishes off of z(b*) + B'
o(B)F+lgxo(b'), 6'
E B'.
Proof. Choose a complete system of representatives for the B-cosets in A
m ,, ...,m,. Setting
c
=
~ ( b f+) b',
a = mj
in
+ b,
bf
E
B*, b' E B'
1 Ij s J , b E B ,
GROUP INVARIANT FOURIER TRANSFORM ALGORITHMS
51
we have, applying Eq. (74),
which vanishes unless b: = b*, proving F+f vanishes off of z(b*) + B I . Then by Theorem IV.2, J
F+f(z(b*)+ b') = o(B) C g,*(mj)(mj, d(b*)), j= 1
completing the proof of the theorem. For b* E B* define S(b*) = ( g b t k s : k* E K * ] . By Theorem VIII.3,
(76)
b* E B ' ,
(77)
which implies that F+f on the coset z(b*) + B',
b* E B * ,
is determined by the induced FT of functions in S(b*). The pseudoperiodization operations introduce data redundancies which we will now describe. Set C = A / B . K acts by the identity mapping on B and induces a group of automorphisms of C denoted also by K . For b* E B* and k E K , there exists a unique C b * ( k ) E B' such that
Theorem VIII.4. For x*
=
b*K* E X * and
K E
K,
52
R. TOLIMIER1 et
al.
Proof. By Eqs. (72), ( 7 9 , and (78) g,*(K(a)) =
(K,
K*>(K(a),4(z(b*))>fx*(a), a E A ,
= (K,
K*>(a,W # ( z ( b * ) ) ) > f x * ( a )
= (K,
K*)F&, 4(Cb*(K)))gx*(a).
K EK
The second statement can be proved by usual arguments. A modified RT algorithm can be applied to the induced FT computations. For a subgroup Y of X,set
Y*
=
(x* E X * :( y , x * ) = 1, for ally
E
Y).
(83)
Arguing as in Theorem VIII.2, we have the following theorem:
Theorem VIII.5. I f X is a subgroup o f A f f ( A )and Y is a subgroup of X, then f o r Y-invariant f E L ( A ) , the pseudoperiodizations f x * , x* E X* vanishes unless x* E Y, . Affine group CT FFT code for X can be used to compute the FT of Y-invariant data, for any subgroup Y of X . In several important applications, the group X can be chosen such that the corresponding CT FFT algorithm can be implemented by simple 1D routines, while more complicated code is required for a direct implementation of the FT of Y-invariant data Y.
D. CT FFT with Respect to Fmmm We will continue with the notations established in Example 11.4:
Fmmm
=
B x Pmmm.
We will use the B-periodization computation of Example V.7 as the first stage of the two-stage pseudoperiodizations with respect to Fmmm. Recall the ordering of the elements of Fmmm given in Example 11.4: B
= (SO, sl,s 2 ,
Pmmm
=
Fmmm
= (SS/+k:
s3
9
Iso,ss,s16, $ 4 ,
s4,
s5, s6, s 7 ) ,
s32, 3 4 0 , s48,
~
~
~
0 5 k, 1 571.
For (ala2a3) E A , observe that
~ ~ ~ ( a ~ = , sas /~+,/ (aa l~, a)z , a 3+)s I ,
~ E B .
1
,
GROUP INVARIANT FOURIER TRANSFORM ALGORITHMS
53
In Example V.7, periodizations 0 5 15 7
fb,\
are made on the collection of B-coset representatives
c = ( ( a l , a , , a , ) : O I a i I N ; , i =1 , 2 , 3 ) . 7
f,?+,(a) =
7
c c c c
f(S6na
-k S m ) ( S m v
Sk*)(s8n9 $/)
n = O m=O
7
=
fb;f(s6na)(s6n
9
$/)
n=O
7
=
fbtf(s8n+na)<sn
9
bz)(s6n3
s&>*
n=O
CT FFT with respect to F m m m was implemented on a Sun4 station (Abdelatif, 1994).
IX. INCORPORATING1D SYMMETRIES IN FFT We have developed various FFT algorithms incorporating certain 1D symmetry. In this appendix, we give an example of incorporating invariance conditions in data without giving up the use of highly efficient FFT routines. Set A = Z/N, for a natural number N . For f E L ( A ) , the invariance conditions we will consider here are
f ( a ) = * f ( - a).
(84)
An efficient algorithm was given by Cooley et al. (1970) and Rabiner (1979) which reduced the computation to that for an N/2-point FFT with preprocessing and postprocessing. The procedures are summarized as follows:
(a) Compute N/4- I
c
V(0) = 2
f(2a
+ 1).
a=O
(b) For a
=
1,2, ..., N/4 - 1 , formulate the sequence g(a) as g(a)
=f(24
+ [fW + 1) - f(2a
g(N/2 - a) =f(2a) - [f(2a g(0) = f ( O ) , g(N/4)
= f(N/2).
+ 1) -f(2a
-
1)1,
- l)],
54
R. TOLIMIER1 el a/.
(c) Take the N/2-point FFT of g(a); call this result G(b). (d) Form two sequences b
U(b) = Re[G(b)],
V(b) = (e) For b
=
Im [G(b)l 2 sin(2nb/N) '
0, 1,2, ..., N / 4 ,
b = 1 , 2) . . . )N / 4 - 1.
1,2, . . . , N / 4 , the transformed data sequence F(b) is given as
=
F(b) = U(b) + V(b), F(N/2
-
b) = U(b) - V(b),
F(0) = U(0) + V(O),
F(N/2)
=
U(0) - V(0).
Notice that in step (d), the computation involves division by {sin(2nb/N)J.This may case a stability problem for large size N . We summarize here an algorithm proposed in Lu and Tolimieri (1992) to overcome the stability problem. (a) Form two sequences h(a) = f ( a )
+ f(N/2
g(a) = [ f ( a )- f(N/2
a = 0, 1,2,
- a), -
a)[ cos(2na/N),
..., N / 4 , a = 0, 1,2, . . ., N/4,
where both h(a) and g(a) have invariance conditions. (b) Take the N/2-point (half size) symmetric FT of h(a) and g(a). (c) The transformed data sequence F(b) is given as F(2b) = H(b),
b
=
0, 1,2, ..., N / 4 - 1,
F(1) = G(O), F(2b
+ 1) = 2G(b) - F(2b - l),
b = 1, 2,
...,N/4
-
1.
This algorithm can be recursively used for transform size of N = 2'" or > 1 and I is an odd number. In step (a), multiplications by (cos(2na/N)) are required to formulate g(a). If, however, n is twice an odd number, then an alternative procedure, based on the Good-Thomas prime factor algorithm (Good, 1958; Thomas, 1963), can be used to avoid these multiplications. In this case, n = 2ml, where rn
GROUP INVARIANT FOURIER TRANSFORM ALGORITHMS
55
the computational procedures can be stated as (a) Take the N/2-point (half size) symmetric FFT of fl(a)= f(2a) and f 2 ( a )= f(N/2 + 2a); call them F,(b) and F2(b)respectively. (b) For b = 0, 1,2, ..., (N/2 - 1)/2, the transformed data sequence F(b) is given as F(2b) = F(N - 2b) = FI(2b) + F,(2b), F(N/2
+ 26) = F(N/2
- 26) =
FI(2b) - F2(26).
If the data is real, the same algorithm can be used with half size real FFTs. The saving in FFT computation will be approximately 50% in comparison with complex data. REFERENCES Abdelatif, Y. (1994). Periodization and Decimation for FFTs and crystallographic FFTs. Ph.D Thesis, CCNY, CUNY. An, M., Gertner, I . , Rofheart, M.. and Tolimieri, R. (1991). Discrete fast Fourier transform algorithms: A tutorial survey. In “Advances in Electronics and Electron Physics” (P. Hawkes, Ed.), Vol. 80. Academic Press, New York. An, M., Cooley, J . W., and Tolimieri, R. (1990). Factorization method for crystallographic Fourier transforms, A d v . Appl. Math. 11, 358-371. An, M., Lu, C., Prince, E., and Tolimieri, R. (1992a). Fast Fourier transform algorithms of real and symmetric data. Acta Cryst. A48, 415-418. An, M., Lu, E., Prince, E., and Tolimieri, R. (1992b). Fast Fourier transforms for space groups containing rotation axes of order three and higher. Acta Cryst. A48, 346-349. Anupindi, N., and Prabhu, K. M. (1990). Split-radix FHT algorithm for real-symmetric data. Electron. Leu. 26, 1973-1975. Bricogne, G. (1974). Geometric sources of redundency in intensity data and their use of phase determination. Acta Cryst. A30, 395-405. Bricogne, G., and Tolimieri, R. (1990). Symmeterized FFT Algorithms. “The IMA Volumes in Mathematics and Its Applications,” Vol. 23. Springer-Verlag, New York/Berlin. Burrus, C. S. (1977). Index mappings for multidimensional formulation of the DFT and convolution. IEEE Trans. ASSP ASSP-25, 239-242. Cooley, J . W., Lewis, P. A,, and Welch, P . D. (1970). The fast Fourier transform algorithms: programming considerations in the calculation of sine, cosine and Laplace transforms. J . Sound Vib. 12, 315-337. Gertner, I. (1988). A new efficient algorithm to compute the two-dimensional discrete Fourier transform. IEEE Trans. ASSP 37(7), 1036-1050. Good, I . J . (1958). The interaction algorithm and practical Fourier analysis. J . R . Statis. SOC. B. 20(2), 000-000. Henry, N. F. M., and Londsdale, K. (ed.) (1952). “International Tables for X-Ray Crystallography,” Vol. I. The Kynoch Press, England. Kechriotis, G . , An, M., Bletsas, M., Manolakos, E., and Tolimieri, R. (1993). A hybrid approach for computing multidimensional DFTs on parallel machines and its implementation on the iPSC/860 hypercube. IEEE Trans. Signal Proc. 00, 000-000.
56
R. TOLIMIERI et a / .
Lu, C., and Tolimieri, R. (1992). New algorithms for the FFT computation of symmetric and translational complex conjugate sequences. Proc. IEEE 1992 Int. Conf. ASSP, 23-26. Rabiner, L. (1979). On the use of symmetry in FFT computation. IEEE Trans. ASSSP, ASSSP-27, 000-OOO. Ten Eyck, L. F. (1973). Crystallographic fast Fourier transforms, ACTA Crystullogr. Sect. A 29, 183-191. Thomas, L. H. (1963). “Using a Computer to Solve Problems in Physics, Application of Digital Computers.” Ginn, Waltham, MA. Tolimieri, R., An, M., and Lu, C. (1993). “Mathematics of Multidimensional Fourier Transform Algorithms.” Springer-Verlag, New York/Berlin. Tolimieri, R., An, M., and Lu, C. (1989). “Algorithms for Discrete Fourier Transform and Convolutions.” Springer-Verlag, New York/Berlin.
ADVANCES IN IMAGING A N D ELECTRON PHYSICS. VOL. 93
Crystal-Aperture STEM JACOBUS T. FOURIE Division of Materials Science and Technology, CSIR. Pretoria, South Africa
I. Introduction . . . . . . . . . . . . . . 11. Theoretical Considerations and Experimental Evidence
. . . . . . . . . . . . . . . . .
A. Strong Absorption of Electron Waves and the Nature of Transmitted Radiation B. Crystal-Aperture Optical Systems of Atomic Dimensions . . . . . . . C. Predictions on Zone Axis Patterns from Electron-Ray Simulation . . . . D. Atomic Structure of Zone Axis Tunnels through a (110) Foil . . . . . . E. Electron-Source Requirements and the Virtual Source in [3 101 Field-Emission F. Auto-Magnification Effects in Direct Imaging of the Nucleus . . . . . . I l l . Experimental Results in Imaging . . . . . . . . . . . . . . . . A. Experimental Method in Crystal-Aperture STEM . . . . . . . . . . B. Improved Resolution in Crystal-Aperture STEM . . . . . . . . . . C. Imaging of Single Adatoms of Gold . . . . . . . . . . . . . . D. Imaging of Subatomic Detail . . . . . . . . . . . . . . . . IV. Summary and Conclusions . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . .
57 59 59 63 66 73 79 87 90 90 91 94 100
106 107
I . INTRODUCTION The science of electron microscopy has progressed, in terms of resolution, by about one order of magnitude since the middle 1940s. For example, the RCA EMU commercial electron microscope of that period, as described by Hall (1953), provided a resolution of slightly less than 2 nm, whereas modern microscopes can resolve about 0.1 nm. The method of crystal-aperture scanning transmission electron microscopy (STEM) is directed toward the obtaining of resolutions that are considerably better than the present optimum level. To this end, an attempt has been made to obtain images under conditions where electron optical diffraction would be absent. Under such circumstances, the incident aperture could be reduced to obtain minimal spherical and chromatic aberration, without incurring the usual diffraction broadening associated with a reduction in the magnitude of the aperture. At this point it should be stated emphatically, that there is no intention, within this chapter, to call into question the validity of the Heisenberg uncertainty principle which forms the basis of diffraction effects. Instead, the exploring of the crystal-aperture STEM method is simply an empirical procedure to establish whether, along 57
Copyright 6 1995 by Academic Press, Inc. All rights of reproduction in any form reserved.
58
JACOBUS T. FOURIE
crystal zone-axis directions in thin foils less than 20 nm in thickness and within atomic size optical systems, conditions might exist under which the diffraction effects might be absent. This empirical approach is guided by the information already known about electron wave propagation through crystal lattices, as discussed, for example, by Whelan (1979). Thus, the dynamical theory of electron diffraction dictates that for s < 0, where s is the diffraction error, the incident wave is strongly absorbed (Whelan, 1979). Consequently, when a high-intensity transmission of electrons occurs in spite of conditions existing where s < 0, such as in the centers of (110) or (100) zone axis patterns (ZAPs), a strong probability may exist that the particle nature of the electron could become a dominant factor in the transmission of radiation. A further aspect to consider, and one which may have an influence, is that the crystalaperture optical system is of near atomic size and thus of a magnitude which is many orders smaller than standard systems. The following aspects of imaging through a crystal aperture by STEM are carefully considered within this chapter: (i) The bent-foil ZAP forms the basis of the practical application of the crystal-aperture STEM method. In such patterns, the condition s < 0 exists for the sets of reflection planes involved in producing the configuration. Thus, in the present chapter, a detailed analysis will be made of ZAPs within the region where s < 0. This will be done in terms of computer-simulated straight line trajectories, as well as in experiemental electron microscopy at 100 to 200 kV. (ii) The cold field-emission electron source is considered at length and particular attention is given to field-emission along the [310] axis in tungsten. The tips used in commercial field-emission guns have this axial orientation and are positioned so that the tip axis coincides with the optical axis of the microscope. Attention is drawn in the discussion to the unique aspect of [310] emission, which is at least an order of magnitude brighter than along any other crystal axis in bodycentered-cubic (bcc) tungsten. It is pointed out, firstly, that this bright emission is incorporated within a finite electron current which is largely paraxial. Hence, this current may be related to a virtual point source at infinity, where the virtual current density of that point source would tend toward infinity. These aspects of the source are of cardinal importance in explaining the subatomic resolution that is shown in the results. Secondly, the paraxial nature of the radiation is essential for focusing an adequate electron current into the small aperture which the zone axis tunnel presents to the incoming focussed cone of rays, which, at focus, will form the probe. Thirdly, an analysis of the paraxial state of the radiation suggests that the system would
CRY STAL-APERTURE STEM
59
be insensitive to transverse or longitudinal vibrations of the source relative to the objective lens. Lastly, it follows from the crystalaperture configuration that the spherical aberration error of the objective lens would be reduced considerably by the narrow aperture, that is, as the cube of the aperture, whereas the chromatic aberration error would be reduced in direct proportion to that aperture. (iii) Experiments on the imaging by crystal-aperture STEM of thin gold deposits on ( 1 10) copper foils are discussed. These experiments involve the imaging of gold particles by means of STEM systems based on three different types of electron source, namely, heated tungsten, heated lanthanum hexaboride and a cold field-emission, [310] orientated, single crystal tungsten tip. In the discussion of these experiments, it is pointed out that the heated tungsten source, because of a lack of brightness, showed a reduced resolution when used in the crystal-aperture mode. On the other hand, the lanthanum hexaboride source, which is 10 times brighter than the tungsten source, produced images where the resolution was improved over that normally obtainable on the instrument. For the cold field-emisison STEM source, a greatly improved resolution was demonstrated when the related STEM machine was used in the crystal-aperture mode. Furthermore, there were strong experimental indications that resolutions better than 0.01 nm are obtainable. Consequently, the possibility exists, not only of imaging the positions of single adatoms on surfaces but also of resolving the structure within a given adatom. (iv) In the final application of the method to be discussed in this chapter, an attempt was made to resolve the structure within the gold atom itself. The relevant images were obtained at an instrument magnification of lo’, where the scan line density in object space was sufficiently high to allow the resolution of structure within the atom. These images suggest that a hexagonal, orbitlike structure is present in the gold atom, and this conclusion could be confirmed by the Fourier transform of a digitized image of that atom.
11. THEORETICAL CONSIDERATIONS AND EXPERIMENTAL EVIDENCE
A . Strong Absorption of Electron Waves and the Nature of Transmitted Radiation In this section, the basis of the present method is considered, and for this purpose, the transmission of electrons through a crystal lattice is analyzed. A clear distinction is made between situations where electrons are expected to demonstrate wave properties and where their particle properties
60
JACOBUS T. FOURIE
would be dominant. Firstly, the origin of bend extinction contours and related bent-foil ZAPS are considered. Secondly, the likelihood of electron transmission, as particles, through the center of zone axis tunnels is discussed. The well-known bend extinction contour, relevant to a single set of diffraction planes, has been analyzed (Whelan, 1979) in terms of the dynamical theory of electron diffraction. For further discussion it is necessary to define the expression w = s&, where s is the diffraction error, and $I is the extinction distance relating to the diffraction vector, g, of a set of ( h k l ) reflecting planes. In regard to the extinction-contour, it is significant that the dynamical theory predicts that this phenomenon will occur where w is negative. Thus, if the incident direction of the electron wave on the crystal is such that this condition is met, the Bloch wave that is most strongly absorbed is also the one that is excited predominantly. To interpret the appearance of the bend extinction-contour, a pair of rocking curves are placed back-toback (Whelan, 1979; Fourie and Terblanchk, 1992), as in Fig. 1. It is clear from this figure (Fourie and Terblanchk, 1992) that the maximum absorption of scattered waves would occur around the direction where 6 = 0 for a given set of reflection planes. Here, 6 is the angle between the incident direction and the reflection planes. A further classification of the problem is obtained by a consideration of the electron ray diagram in Fig. 2. The argument there is particularly significant in regard to the electron particle model that forms the basis of the present method. With reference to Fig. 2, then, the vertical lines are envisaged to represent a set of (200) reflecting planes in copper. The rays C’R’and CR are incident at exactly the Bragg angle, BB, which at 100 k V is 10.2 mrad, and for which direction, w = 0. Referring to Fig. 1, it will be noted that the transmitted intensity at 8, is considerable, but that, for B = 5.0mrad, where w = -0.5, this intensity is about zero. Similarly, for
>-,:
;
e
c (I)
s o
3
2
1
294
230
166
0 -1-16-1 1 0 2 ~ 1 ~ 3o 8 38 8 (mrad)
0
1
ioz(e,ps
2 250
~ 294
3
FIGURE1. Rocking curves based on the dynamical theory of electron diffraction. The curves are placed back-to-back for the purpose of representing bend extinction-contours in a crystal foil. Courtesy Fourie and Terblanche (1992).
CRYSTAL-APERTURE STEM
61
‘q \
R’
/
FIGURE2. A representation of directions within a cone of electron rays incident on a copper crystal, where the top surface has a {OOl) orientation.
6 = 3.8 mrad and w = -1, the intensity is about zero. The latter incident direction would correspond approximately with that of the rays B‘R’ and BR, whereas for rays A‘R’ and AR, 0 = 0 and w assumes the maximum negative value of - 1.6. Here, also, the transmitted intensity is about zero. The regions of incident angles between B’R’ and A’R’, on the left, and from BR to AR, on the right, correspond to regions where the strongly absorbed Bloch wave is primarily excited, as argued earlier. On the other hand, for incident directions D’R’ to C’R’ and DR to CR (in Fig. 2), the strongly transmitted Bloch wave is primarily excited. It ensues from Fig. 1 that the transmitted intensity reaches a maximum within these latter angular regions or at the position where w = 0.5. If the cone of rays O’PO’’ in Fig. 2 is considered, it is obvious that the ray directions within that cone would fall within the low-intensity transmittance regions of B’R’A’ and BRA, as discused previously. However, this prediction of low transmittance, on the basis of electron wave theory, apparently does not hold for the centers of bent-foil zone axis patterns, for zone axes such as (110). This situation exists even though the ray directions there would coincide with those within O’PO’’ and even though the bent foil ZAP is a combination of bend extinction contours related to a number of sets of reflection planes, such as the (002), ( l i l ) , and (711) planes for a [110] ZAP. Experimentally it is found that the brightest
62
JACOBUS T. FOURIE
FIGURE3 . A bent-foil ( 1 10) zone axis pattern in a copper single crystal foil, covered in gold particles on one side.
transmittance occurs, in fact, exactly along the ( 1 10) zone axis, as is clearly demonstrated in Fig. 3 for a ( 1 10) ZAP. On the basis of the arguments and experimental observations presented earlier, the following assumptions are made, which, within the experiments of crystal-aperture STEM, are shown,empirically, to be valid for the results obtained. Firstly, if the electrons with incident directions corresponding to the cone O’PO” exhibited a wave nature, they would be strongly absorbed by the crystal lattic, and, hence, the transmitted intensity would approach zero, as in Fig. 1, for those incident directions. Secondly, if the electrons, exclusively, exhibited particle properties, they would be transmitted with maximum intensity through the zone axis tunnel. Thirdly, for such electrons of an exclusively particle nature, there would be no manifestation, within the atom-size zone axis tunnel, of those electron-optical diffraction phenomena which are normally observed in the focusing of electron beams. On the basis of these assumptions, then, the following empirical conclusion can be made: within the crystal-aperture formed by a zone axis tunnel, electron optical conditions would be of such a nature that the point focusing of electrons originating from a point source might be approached. This achievement would be made possible by the absence of diffraction combined with the smallness, in size and angle, of the crystal aperture, which, thus, would minimize the spherical and chromatic aberration errors.
CRYSTAL-APERTURE STEM
63
B. Crystal-Aperture Optical Systems of Atomic Dimensions The method of crystal-aperture STEM differs markedly from other more conventional modes of imaging in three main aspects. The first is the fact that the final objective aperture is a zone axis tunnel within a crystal. The second is the fact that the sample, an atom, is mounted (or, more specifically, adsorbed) on the bottom surface of the crystal and in the center of the aperture (or zone axis tunnel). The third is the fact that the volume of the final aperture system is about 16 orders of magnitude smaller than that of a conventional STEM system. These three aspects are discussed in this section. A brief consideration of the zone axis tunnel is presented here, with reference to Fig. 4. The detailed structure will be discussed further in Sections II,D and II,E. For the zone axis tunnel there are two options, as shown in Figs. 4a and 4b. In Fig 4a, filled circles indicate copper atoms
FIGURE4. A simplified representation of a crystal-aperture, of magnitude a,, in the form of a zone axis tunnel. (a) In a copper foil of thickness i,, with a gold atom (open circle) adsorbed in the mouth of the tunnel; (b) in a copper foil of thickness f,, coincident with an equivalent tunnel, in a gold particle of thickness f,.
64
JACOBUS T. FOURIE
which line the tunnel for the full thickness, t,, of the foil. Within the exit mouth of this tunnel, a gold atom (open circle) has been adsorbed in a stacking fault position. The incident beam is limited to an aperture of a,by the zone axis tunnel, and is focused exactly on the adsorbed gold atom at the exit end of the tunnel. In Fig. 4b, a similar situation is depicted, except that, now, the presence of a thin, epitaxially-grown, gold particle of thickness t,, is present on the exit surface of the copper foil. The diagram portrays a position where the lattices of copper and gold, which differ parameter-wise, are in phase. Thus, the zone axis tunnel in the copper foil is extended by an additional three atomic spacings by the gold crystal. The atom upon which the beam is focused, in Fig. 4b, is a gold atom which is presumed to have been adsorbed on the exit surface of the gold crystal during the process of vapor-deposition of gold onto the copper foil, as discussed in detail in Section II1,A. As in Fig. 4a, the atom was adsorbed in a stacking fault position in the center of the zone axis tunnel. The optical characteristics and volume of a standard electron-optical system will now be compared with that of the crystal-aperture STEM system. In Fig. 5a is shown the classic broadening of a parallel beam of particles (electrons for example) which has been directed to pass through a slit of A y . According to the Heisenberg principle, the individual electron will undergo upward or downward deflection at the slit. Thus, it will acquire component momentum, perpendicular to its original direction of flight, of amount A p , with the resultant momentum, p , remaining constant. The well-known Heisenberg relation A p Ay I h , where h is Planck’s constant, is then valid. This process may be described as the diffraction of electrons at a slit. The electron-optical system, of atomic dimensions, used by Fourie (1992b, 1993) is shown diagrammatically in Fig. 5b. The design criteria of this system has been discussed in detail by Fourie (1993). The essential elements for subatomic resolution are (i) the cold field-emission electron source forms a virtual source of vanishing dimensions, as discussed later; (ii) the final probe formation occurs within a (1 10) zone axis tunnel of gold; and (iii) the adatom requiring study is placed centrally, at S, within the exit mouth of the zone tunnel, a position which must coincide with the image plane of the STEM system. It is noted from Fig. 5b that the standard optics of the field-emission STEM system is envisaged to focus a beam (outer cone unhatched, inner cone hatched) onto the crystal, with the focal point at S, a position which coincides with the exit surface of the crystal and the sample position. For the purpose of subatomic resolution, it is probably necessary that the diameter of the standard beam at the entrance surface, E, does not exceed 0.4 nm. The zone axis tunnel, with an effective apertureopening Ay, will then select the central (hatched) cone of the beam from the standard beam. Since the aperture involved will be about 1 mrad, the
65
CRYSTAL-APERTURE STEM
-
Lb
-
FIGURE5 . (a) Diffraction broadening of a beam of rays through a rectangular slit; (b) the suggested electron-ray paths through a crystal-aperture, in the absence of diffraction phenomena.
objective lens will be able to focus the beam to a spot of subatomic dimensions at S. Thus, focusing is still performed by the lens and it is not believed that the zone axis tunnel is involved in the focusing process. The only function of the tunnel is in providing an aperture for the objective lens. The combination of components involved in this last event of the focusing process may be seen as an optical system which contains an aperture, Ay, and an image plane at a distance, L b , from the aperture, where the sample is situated. The total volume of this ultramicro optical system would be Vb = (Ay)’Lb. Similarly the volume of the macroscopic system in Fig. 5a would be V, = (Ay)’L,. For standard STEM systems, as in Fig. 5a, the objective aperture t o focal plane distance would be about La = 10 mm, and the aperture opening, Ay, about 0.02mm, from which it follows that V, = 4 x mm3. However, for Fig. l b, where the (110) tunnel width
66
JACOBUS T. FOURIE
for copper is about 1.8 x lo-’ mm and L b , the foil thickness, is about mm, Vb = 3 x mm3. Thus & / V , = 8 x lo-’’. The latternumber emphasizes the smallness of the crystal-aperture optical system and its difference in volume relative to standard macroscopic systems by 16 orders of magnitude. As discussed earlier, the method requires that diffraction phenomena be absent within the final probe formation. The following are unique aspects of that system that either individually, or in combination, may be responsible for the absence of diffraction: (i) The crystal aperture and the sample are coherently connected by a single crystal atomic lattice. Thus, the sample is adsorbed in a stacking fault site on the exit surface of the crystal, whereas the aperture is situated on the entrance surface of an underlying, epitaxially related zone axis tunnel of copper which leads into a coincident tunnel of the adsorbent gold crystal. (ii) The volume size of the crystal aperture system is 16 orders of magnitude smaller than standard systems. (iii) The vanishingly small virtual source aspect of the cold field-emission tip probably only begins to have significance within the crystalaperture system, as in Fig. 4(b). Within standard systems, the aberrations probably override any benefits which otherwise might be associated with the virtual source concept. C. Predictions on Zone Axis Patterns From Electron-Ray Simulation In Section II,A a case was made for regarding the electron transmission through the centers of ZAPS to be essentially that of particles, with the transmission of waves being suppressed by strong absorption along the zone axis. In the present section, a two-dimensional arrangement of atoms along a [loo] zone axis is considered, to establish what configuration of pattern may be expected when the electron interaction with the atomic structure is of a purely particle nature. Thus the simplified model is, first, described together with the method of computer simulation. Second, results of the computer simulation are presented in graphical form and an indication is given of the qualitative configuration of the patterns that might be expected. Third, experimental observations (Fourie, 1992a) on ZAPS are presented, which appear t o confirm the predictions. For the computer simulation (Fourie and TerblanchC, 1992) of the rectilinear transmission of electrons through a crystal, a two-dimensional lattice as shown in Fig. 6 is used. Here, the (100) plane of the face-centered cubic (fcc) copper lattice is shown, and the plane of the figure bisects the
67
CRYSTAL-APERTURE STEM
3'
2'
M' 1'1
0
0
0 0
0
0
0 0 0 0
0 0
0
n'
c ' o 0'0 B' -9
1
D'
/ \
-
9
FIGURE6 . A diagrammatic representation of the proposed rectilinear paths of electrons through the fcc crystal lattice of a thin foil of copper. Dark circles indicate atoms involved in electron-atom encounters. Courtesy Fourie and Terblancht (1992).
atoms in that (100) plane. The (020) planes are perpendicular to the plane of the figure and are assumed to bisect the atoms along the columns AA', BB', etc. The optical axis of the probe, 00', is assumed to be parallel to [OOl], to coincide with the plane of the figure and to be positioned at 1/4a with respect to the column of atoms DD'. That is, the optical axis is positioned symmetrically between the atom columns DD' and EE'. Under these conditions, the probe will be bisected by the plane of the figure. The outermost rays within this section of the probe are 3 ' 0 ' and 30', as shown in Fig. 6.
68
JACOBUS T. FOURIE
In the simplified model which was used for the simulation (Fourie and Terblanche, 1992), it was assumed that the interaction of the probe rays within the section 3 ’ 0 ’ 3 with the bisected atoms within the plane of the figure, may be equated with the interaction of the equivalent probe rays in the (020) plane with the atoms in the (020) planes. That is, the twodimensional situation depicted in Fig. 6 was assumed to be comparable with the three-dimensional situation, where the (020) planes would extend above and below the plane of the diagram and where the probe rays would form a solid cone. For the computer simulation, the ray 00’ is tilted from the direction it occupies parallel to [OOl] through prescribed angular increments, e.g., 68, , 68,, and 60,, to positions 1, 2, and 3, respectively, as in Fig. 6. At every position the number of encounters with atoms is recorded. Clearly, the number of encounters will be a function of 0, r, and t , where 8 is the angular position of the ray, r is the atomic radius, and t is the foil thickness. Note that there would be no encounters with atoms for rays falling only within the central section, M’O’M, for r a n d t as in Fig. 6. It is clear that a variation in r in the simulation is equivalent to a variation in V , the accelerating voltage. This follows from the formulae for the elastic cross section of atoms in relation to the electron velocity (see, for example Reimer (1984), pp. 21 and may be deduced. The results 150) from which the relationship r a obtained from the computer simulation will now be discussed. In Fig. 7, for constant r = 0.025 nm, the number of encounters as a function of 0 is plotted for foils of different thicknesses, where t = 25, 50, and 120 nm for curves A, B, and C, respectively. The results show that for thin crystals, such as for curve A, there is a wide angular region around the zone axis, where no electron-atom interaction occurs, that the peaks of encounters are widely spaced, and that there is little contrast between the zero-encounter regions and the peaks of encounter. For thicker crystals, such as curve B or C, the region of zero encounters contracts, the peaks of encounters lie closer together and the contrast between the zero-encounter region and the peaks increases. In Fig. 8, curves for constant t = 50 nm, and for r = rl , r,, and r 3 ,where r, = 0.025 (curve A), r, = 0.050 (curve B) and r3 = 0.075 nm (curve C), are plotted. On the basis of the formula r a it follows that r I / r 2= Thus, for the values given, it follows that r l / r 2 = 0.5, and thus that V,/V, = 0.25. Thus if 6 is set equal to 200 kV, V, would be 50 kV. It is clear from Fig. 8, therefore, that the diameter of the central, encounterfree region, would increase with decreasing atomic radius, or equivalently, with increasing voltage. It follows that, in terms of the electron-ray or particle model, the results in Figs. 7 and 8 provide definite predictions on how the central bright region
m
m.
m,
69
CRYSTAL-APERTURE STEM
150
120
cn
L
90
Q)
4J
C
1
30
0
- 15
- 10
-5
Theta
0
5
10
15
(rnradl
FIGURE7. The effect of the thickness, t , in computer-simulated electron-atom encounters, as a function of 8, for the (020) atomic column model, for the atomic radius r = 0.025 nm. For curves A, B, C , respectively, t = 25, 50, and 120 nm. Courtesy Fourie and Terblanche (1992).
of the ZAP would react to variations in foil thickness or in the acclerating voltage. Subsequently, it was possible to carry out experiments which confirmed these predictions. These experiments will now be discussed. For the purpose of obtaining real space ZAPs in transmission electron microscopy (TEM), it is necessary that dome- or cup-shaped dimples should be present in the foil (Reimer, 1984). If these dimples are not present from a chance bending of the foil, the foil may be purposely deformed by a slight plastic bending, in order to introduce such dimples. Thus, the (110) bent foil ZAP could easily be obtained and then used in assessing the conclusions drawn from theory. In this regard it was necessary to obtain ZAPs at various t’s and V’s. This was achieved, for t , by tilting the sample around an appropriate crystallographic direction at constant V, causing the ZAP to shift to either larger or smaller 1. For altering V , at constant t, a ZAP at an appropriate t was photographed, and then, while maintaining the tilt and position of the sample constant, V was altered before taking another photograph.
70
JACOBUS T. FOURIE
120
ln
L a,
t
90
c, C
3 0
u
c
W
60
30
0 -15
- 10
-5
Theta
0
5
10
15
(mradl
FIGURE8. The effect of the atomic radius, r, in computer-simulated electron-atom encounters, as a function of 0 for the (020) atomic column model, for t = 50 nm. For curves A, B, C, respectively, r = 0.025, 0.050, and 0.075 nm. Courtesy Fourie and Terblanche (1992).
In the following discussion, observations in TEM on zone-axis patterns will be described in which the variation of such patterns, as a function of t or V , were recorded. These results will be discussed concerning their significance in relation to the crystal-aperture STEM method. A good example of the effect of t on the diameter of the central bright region of a ZAP, is shown in Figs. 9a and 9b. Here, ( 1 10) ZAPS were recorded at 200 kV for t approximately equal to 30 and 50 nm, respectively. The corresponding microdensitometer traces taken along X’X, are shown in Figs. 10a and lob, respectively. These curves should be compared with the theoretical curves in Fig. 7, where it should be noted that the condition of zero encounters represents the highest intensity and thus that the curves are inverted with respect to Figs 10a and lob. Thus the diameter of the central bright region, in an experimental study, is seen to decrease with increasing thickness. That is, with reference to Figs. 10a and Fig. lob, the FWHM (full width at half maximum) of the central peak decreases from Fig. 1Oa to Fig. lob. Also, the ratio of the central peak height t o that of the neighboring peak height increases with increasing thickness. This observation
CRYSTAL-APERTURE STEM
71
FIGURE9. (a) A ( I 10) ZAP at 200 k V in a foil thickness, I , , estimated at about 30 nm; (b) a ( I 10) Z A P at 200 k V in a position near to that of Fig. 9a, but where the foil thickness, t,, was estimated to be about 50 nrn. Courtesy Fourie (1992a).
CRYSTAL-APERTURE STEM
73
is clearly supported by the same trend in the corresponding ratio of the theoretical curves in B and C in Fig. 7. The variation of the central peak diameter with a variation in voltage can clearly be assessed from the experimental images in Figs 1l a and 11b, where the (1 10) ZAP was photographed under identical orientation conditions for accelerating voltages of 150 and 200 kV, respectively. From these images it is clear that the diameter of the central bright region has increased markedly with increased voltage. This result should be compared with the theoretical curves A and B (for example) in Fig. 8. Note that the width of the region of zero encounters (or the highest intensity of transmission) increases from curve B to curve A, where the value of r decreases from 0.050 nm to 0.025 nm, respectively. Since a decrease in r represents an increase in voltage, the theoretical curves in Fig. 8, support qualitatively the experimental observations in Figs 1 l a and 1 lb. In summary, concerning the electron particle theory of ZAPS and the associated experimental results, it may be asserted that there is a close correspondence of the central regions of simulated zone axis patterns with experimental patterns. This fact lends strong support to the underlying assumption for the simulation procedure; namely, that within a certain narrow aperture, electrons will penetrate the crystal zone axis tunnels along rectilinear paths and, in the process, will demonstrate particle properties.
D. Atomic Structure of Zone Axis Tunnels through a ( 1 10) Foil The predictions concerning electron-ray trajectories and the associated interactions with atoms surrounding (100) zone axis tunnels, were considered in II,C. The selection of (100) tunnels for the computer simulation procedure had been decided upon since the two-dimensional approach used was physically more reasonable for the (100) tunnels than for (1 10) tunnels. However, all of the experimental work, thus far, has been confined to the exclusive use of (1 10) tunnels. In this section then, a detailed consideration of the (1 10) zone axis tunnel is presented. This will include an empirical consideration of the electronic structure between atoms. In Fig. 12, ABCDEF are atoms at the corners of unit cells in the facecentered cubic lattice, with G and H representing atoms in the face centers. Atoms at corners of unit cells are, for convenience, represented by larger circles than those at face centers. The rest of the structure is built up in an obvious manner, from this starting structure, by the addition of unit cells and by the sectioning of the structure along the planes DIJK and U'UQTT', resulting in surfaces of a [ 110) orientation. The (1 10) tunnel of interest is along the direction QM. That is, it is a tunnel in a (110) direction and
74
JACOBUS T. FOURIE
FIGURE11. (a) A (110) ZAP at 150 k V in a foil thickness estimated at about 30nm; (b) a ( 1 10) ZAP at 200 k V in exactly the same position and tilt orientation as for Fig. 1 la. Courtesy Fourie (1992a).
CRYSTAL-APERTURE STEM
FIGURE12. Perspective diagram of zone axis tunnels in the fcc crystal lattice. The 1101 faces DIJK and U’UQTT’, are on opposite sides of the thin foil. A ( 1 10) zone axis tunnel is shown, with triangle QRS as the entrance mouth and MNP the exit mouth of the tunnel, which extends along the apices QVCM. The crosses I , 2, and 3 represent gold atoms adsorbed at tunnel positions on the exit surface.
is perpendicular to the plane of the foil DIJK. The entrance to the tunnel is defined atomically by the triangle of atoms QRS on the (110) entrance surface, upon which the STEM probe is incident. The internal tunnel within the bulk is defined by repeating, identical triangles with apices Q, V , C, and M, with M the apex of the triangle NMP on the exit surface, DIJK. Alternatively, an identical but inverted tunnel could be defined by NPW. Furthermore, the tunnels defined by DNE, or the inverse, KPY, could also function as crystal apertures. The crosses 1 , 2, and 3 on the exit surface DIJK, represent gold atoms adsorbed in the mouths of tunnels. When the entrance and exit surfaces are within close proximity of each other as, for example, for a 20-nm-thick foil, it is envisaged that the internal electronic structure of the foil would assume a surfacelike state and that a cylinderlike tunnel, in the electronic sense, would come into existence, as indicated by the broken circles along the direction QVCM. It is further envisaged that high-energy electrons may travel along this electronic tunnel without significant interaction with the electric field within the bulk of the material. This would correspond with the conditions for forming a 110) bent-foil ZAP in TEM, with the very marked bright center, as in Fig. 3 . Also, it is clear from Fig. 3 that the central bright region consists, roughly, of two concentric regions, with the central region showing extreme brightness.