Advances in Imaging and Electron Physics, Volume 121 (Advances in Imaging and Electron Physics)

ADVANCES IN IMAGING AND ELECTRON PHYSICS VOLUME 121 Electron Microscopy and Holography EDITOR-IN-CHIEF PETER W. HAWK...

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ADVANCES IN IMAGING AND ELECTRON PHYSICS VOLUME 121 Electron Microscopy and Holography

EDITOR-IN-CHIEF

PETER W. HAWKES CEMES-CNRS Toulouse, France

ASSOCIATE EDITORS

BENJAMIN KAZAN Xerox Corporation Palo Alto Research Center Palo Alto, California

TOM MULVEY Department of Electronic Engineering and Applied Physics Aston University Birmingham, United Kingdom

Advances in

Imaging and Electron Physics Electron Microscopy and Holography

EDITED BY

PETER W. HAWKES CEMES-CNRS Toulouse, France

VOLUME 121

San Diego San Francisco New York London Sydney Tokyo

Boston

∞ This book is printed on acid-free paper. C 2002 by ACADEMIC PRESS Copyright

All Rights Reserved. No part of this publication 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. The appearance of the code at the bottom of the first page of a chapter in this book indicates the Publisher’s consent that copies of the chapter may be made for personal or internal use of specific clients. This consent is given on the condition, however, that the copier pay the stated per-copy fee through the Copyright Clearance Center, Inc. (222 Rosewood Drive, Danvers, Massachusetts 01923), for copying beyond that permitted by Sections 107 or 108 of the U.S. Copyright Law. This consent does not extend to other kinds of copying, such as copying for general distribution, for advertising or promotional purposes, for creating new collective works, or for resale. Copy fees for pre-2002 chapters are as shown on the title pages: If no fee code appears on the title page, the copy fee is the same as for current chapters. 1076-5670/02 $35.00 Explicit permission from Academic Press is not required to reproduce a maximum of two figures or tables from an Academic Press chapter in another scientific or research publication provided that the material has not been credited to another source and that full credit to the Academic Press chapter is given.

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CONTENTS

CONTRIBUTORS . . . . . . . . . . . . . . . . . . . . . . . . . . PREFACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . FUTURE CONTRIBUTIONS . . . . . . . . . . . . . . . . . . . . . .

vii ix xi

High-Speed Electron Microscopy O. BOSTANJOGLO

I. II. III. IV.

Introduction . . . . . . . . High-Speed Techniques . . . Time-Resolving Microscopes. Conclusions . . . . . . . . References . . . . . . . .

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53 55 59 81 87 87

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91 93 94 118 119 124 126 132 140 141

Applications of Transmission Electron Microscopy in Mineralogy P. E. CHAMPNESS

I. II. III. IV. V.

Introduction . . . . . . . . . . . . . . Analytical Electron Microscopy of Minerals Phase Separation (Exsolution) . . . . . . HRTEM and Defect Structures . . . . . . Concluding Remark . . . . . . . . . . References . . . . . . . . . . . . . .

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Three-Dimensional Fabrication of Miniature Electron Optics A. D. FEINERMAN AND D. A. CREWE

I. II. III. IV. V. VI. VII. VIII. IX

Introduction . . . . . . . . . . . . . . . Scaling Laws for Electrostatic Lenses . . . . Fabrication of Miniature Electrostatic Lenses . Fabrication of Miniature Magnetostatic Lenses Electron Source . . . . . . . . . . . . . Detector . . . . . . . . . . . . . . . . Electron-Optical Calculations . . . . . . . Performance of a Stacked Einzel Lens . . . . Summary and Future Prospects . . . . . . . References . . . . . . . . . . . . . . .

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CONTENTS

A Reference Discretization Strategy for the Numerical Solution of Physical Field Problems CLAUDIO MATTIUSSI

I. II. III. IV. V. VI.

Introduction . . Foundations . . Representations Methods . . . Conclusions . . Coda . . . . . References . .

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144 147 183 222 273 275 276

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281 282 288 290 294 302 305 330 330

INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

333

The Imaging Plate and Its Applications NOBUFUMI MORI AND TETSUO OIKAWA

I. II. III. IV. V. VI. VII. VIII.

Introduction . . . . . . . . . . . . . . . . . . Mechanism of Photostimulated Luminescence (PSL) . Imaging Plate (IP) . . . . . . . . . . . . . . . Elements of the IP System . . . . . . . . . . . . Characteristics of the IP System . . . . . . . . . Practical Systems. . . . . . . . . . . . . . . . Applications of the IP . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . .

CONTRIBUTORS

Numbers in parentheses indicate the pages on which the authors’ contributions begin.

O. BOSTANJOGLO (1), Optisches Institut, Technische Universität Berlin, D-10623 Berlin, Germany P. E. CHAMPNESS (53), Department of Earth Sciences, University of Manchester, Manchester M13 9PL, United Kingdom D. A. CREWE (91), Microfabrication Applications Laboratory, Department of Electrical and Computer Engineering, University of Illinois at Chicago, Chicago, Illinois 60607 A. D. FEINERMAN (91), Microfabrication Applications Laboratory, Department of Electrical and Computer Engineering, University of Illinois at Chicago, Chicago, Illinois 60607 CLAUDIO MATTIUSSI (143), Clampco Sistemi-NIRLAB, AREA Science Park, Padriciano 99, 34012 Trieste, Italy NOBUFUMI MORI (281), Fuji Photo Film Co., Ltd., 798, Miyanodai, Kaisei, Ashigarakami, Kanagawa, 258-8538 Japan TETSUO OIKAWA (281), JEOL Ltd., Shin-Suzuharu Bld. 3F, 2-8-3 Akebonocho, Tachikawa, Tokyo, 180-0012 Japan

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PREFACE

The founding editor of these Advances was Ladislaus (Bill) Marton, one of the pioneers of electron microscopy, who built early microscopes in Brussels in the 1930s and obtained the first biological micrographs. He was later involved in the first efforts to construct a commercial model in the USA during the 1940s. Articles on electron and, more recently, other forms of microscopy have hence appeared regularly in the series. It is thus very natural that this volume, the second of the two thematic volumes announced in volume 119, should contain a collection of recent chapters in the broad area of electron microscopy and holography. In fact, the selection that I originally made proved to occupy too many pages for a single book and three further chapters (by K. Hiraga on quasicrystals, G. Matteucci, G. F. Missiroli and G. Pozzi on electron holography and E. Oho on digital processing of the scanning electron microscope image) will be included in volume 122, together with a regular contribution by A. Khursheed. No further thematic volumes are planned. The five chapters reprinted here cover the very specialized techniques of high-speed electron microscopy, the study of minerals by electron microscopy, miniature electron lenses and microscopes and the imaging plate, which is now usefully complementing the more traditional recording media. In addition, there is a contribution by C. Mattiussi on numerical methods for field calculation. These all seemed to me important enough to deserve republication in this form, though I have to admit that many other contributions had arguably just as strong a claim. I am most grateful to the contributors to this volume for consenting to reappear here and for the work of revision. Their chapters first appeared in vol. 110 (O. Bostanjoglo), vol. 101 (P. E. Champness), vol. 102 (A. D. Feinerman and D. A. Crewe), vol. 113 (C. Mattiussi) and vol. 99 (N. Mori and T. Oikawa). Peter Hawkes

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FUTURE CONTRIBUTIONS

T. Aach Lapped transforms G. Abbate New developments in liquid-crystal-based photonic devices S. Ando Gradient operators and edge and corner detection A. Arnéodo, N. Decoster, P. Kestener and S. Roux A wavelet-based method for multifractal image analysis M. Barnabei and L. Montefusco Algebraic aspects of signal and image processing C. Beeli Structure and microscopy of quasicrystals I. Bloch Fuzzy distance measures in image processing G. Borgefors Distance transforms A. Carini, G. L. Sicuranza and E. Mumolo V-vector algebra and Volterra filters Y. Cho Scanning nonlinear dielectric microscopy E. R. Davies Mean, median and mode filters H. Delingette Surface reconstruction based on simplex meshes A. Diaspro Two-photon excitation in microscopy D. van Dyck Very high resolution electron microscopy R. G. Forbes Liquid metal ion sources xi

xii


E. Förster and F. N. Chukhovsky X-ray optics A. Fox The critical-voltage effect L. Frank and I. Mullerov´ ¨ a Scanning low-energy electron microscopy M. Freeman and G. M. Steeves Ultrafast scanning tunneling microscopy A. Garcia Sampling theory L. Godo & V. Torra Aggregation operators P. W. Hawkes Electron optics and electron microscopy: conference proceedings and abstracts as source material M. I. Herrera The development of electron microscopy in Spain J. S. Hesthaven Higher-order accuracy computational methods for time-domain electromagnetics K. Ishizuka Contrast transfer and crystal images I. P. Jones ALCHEMI W. S. Kerwin and J. Prince The kriging update model B. Kessler Orthogonal multiwavelets A. Khursheed (vol. 122) Recent accessories for scanning electron microscopes G. Kögel Positron microscopy W. Krakow Sideband imaging


xiii

N. Krueger The application of statistical and deterministic regularities in biological and artificial vision systems B. Lahme Karhunen–Loeve decomposition B. Lencová Calculation of the properties of electromagnetic fields and electron lenses C. L. Matson Back-propagation through turbid media P. G. Merli, M. Vittori Antisari and G. Calestani, eds (vol. 123) Aspects of Electron Microscopy and Diffraction S. Mikoshiba and F. L. Curzon Plasma displays M. A. O’Keefe Electron image simulation N. Papamarkos and A. Kesidis The inverse Hough transform M. G. A. Paris and G. d’Ariano Quantum tomography C. Passow Geometric methods of treating energy transport phenomena E. Petajan HDTV F. A. Ponce Nitride semiconductors for high-brightness blue and green light emission T.-C. Poon Scanning optical holography H. de Raedt, K. F. L. Michielsen and J. Th. M. Hosson Aspects of mathematical morphology E. Rau Energy analysers for electron microscopes H. Rauch The wave-particle dualism

xiv


R. de Ridder Neural networks in nonlinear image processing D. Saad, R. Vicente and A. Kabashima Error-correcting codes O. Scherzer Regularization techniques G. Schmahl X-ray microscopy S. Shirai CRT gun design methods T. Soma Focus-deflection systems and their applications I. Talmon Study of complex fluids by transmission electron microscopy M. Tonouchi Terahertz radiation imaging N. M. Towghi Ip norm optimal filters Y. Uchikawa Electron gun optics J. S. Walker Tree-adapted wavelet shrinkage C. D. Wright and E. W. Hill Magnetic force microscopy F. Yang and M. Paindavoine Pre-filtering for pattern recognition using wavelet transforms and neural networks M. Yeadon Instrumentation for surface studies S. Zaefferer Computer-aided crystallographic analysis in TEM

ADVANCES IN IMAGING AND ELECTRON PHYSICS, VOL. 121

High-Speed Electron Microscopy O. BOSTANJOGLO Optisches Institut, Technische Universität Berlin, D-10623 Berlin, Germany

I. Introduction . . . . . . . . . . . . . . . . . . II. High-Speed Techniques . . . . . . . . . . . . . A. Short-Time Exposure Imaging . . . . . . . . . 1. Laser-Driven Thermionic Gun . . . . . . . . 2. Laser-Driven Photoelectron Guns . . . . . . . B. Streak Imaging . . . . . . . . . . . . . . . . C. Image Intensity Tracking . . . . . . . . . . . III. Time-Resolving Microscopes . . . . . . . . . . . A. Time-Resolving Transmission Electron Microscopy 1. Instrumentation . . . . . . . . . . . . . . 2. Applications . . . . . . . . . . . . . . . 3. Space–Time Resolution . . . . . . . . . . . B. Flash Photoelectron Microscopy . . . . . . . . 1. Instrument for Short Exposure Imaging . . . . 2. Applications . . . . . . . . . . . . . . . 3. Limits . . . . . . . . . . . . . . . . . . C. Pulsed High-Energy Reflection Electron Microscopy D. Pulsed Mirror Electron Microscopy . . . . . . . IV. Conclusions . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . .

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I. Introduction Electron microscopy is used to investigate miscellaneous material properties with a high spatial resolution. The most familiar applications are imaging of the atomic structure of solids, of crystal defects, of magnetic and electric fields in solids, and of the chemical composition of thin films and surfaces (e.g., Murr, 1991; Reimer, 1985, 1993). Conventionally, a stationary electron beam either illuminates the whole specimen in a single exposure or scans the specimen. An image of the static distribution of a specific material property is produced in both cases. If time-varying effects are to be captured the microscope must be pulsed. Periodic variations of a material property are pinned down by synchronously pulsing the electron beam with the period of the time-varying material property and summing the signals within a selected acquisition time to produce the image. This sampling procedure reduces the superimposed noise to a low level 1 Volume 121 ISBN 0-12-014763-7

C 2002 by Academic Press ADVANCES IN IMAGING AND ELECTRON PHYSICS Copyright All rights of reproduction in any form reserved. ISSN 1076-5670/02 $35.00

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because of its statistical nature, and “images” with a joint submicrometer– picosecond resolution have been produced, for example, by Brunner et al. (1987). A fast nonrepetitive process is less easily uncovered, as all information about the transient state must be captured by a single short probing pulse. Nevertheless, these nonrepetitive processes have attracted considerable interest in fundamental and applied research in connection with material processing by laser pulses. Typical applications in which pulse lasers progressively replace established tools are localized cutting, drilling, ablating, patterning, alloying, and connecting of a wide variety of materials. The key condition for a precise local treatment, that is, for a minimum thermal and mechanical loading of the neighboring material, is that the required photon energy be deposited locally and in a short time. Thermal melting, melt flow, crystalline and noncrystalline solidification, and thermal evaporation are the main processes which determine the product of material treating with a laser pulse in excess of some 10 ps. Commonly, the pump-probe technique, exploiting, for example, light-optical microscopy, is used as a diagnostic tool to track laser-induced effects. The light-optical methods are very fast and reach a time resolution of several femtoseconds (e.g., Schönlein et al., 1987). Their drawbacks are a limited spatial resolution (>1 μm) and the fact that they primarily sense the electronic system, so that properties related to the atomic structure must be deduced with a suitable model. Material structure is better approached by electron microscopy, some modes of which directly probe the atomic packing. Furthermore, effects which are not accompanied by a large change of electronic states strongly interacting with visible light, such as phase transitions in metals, are not easily detected by light optics. They appear, however, with good contrast when they are imaged by electron microscopy based on Coulomb scattering of the probing electrons at the atomic structure. This article describes the various time-resolving electron-optical techniques which were developed to study fast transient effects in freestanding films and on surfaces of bulk materials down to the nanosecond time scale. Hydrodynamic instabilities in confined laser pulse–produced melts and the solidification and evaporation of these melts were investigated, as they are of major concern to micromachining with laser pulses. The mechanisms uncovered by high-speed electron microscopy are presented.

II. High-Speed Techniques There are three time-resolving techniques, which are distinguished by the number of spatial coordinates in the image: short-time exposure imaging, streak imaging, and image intensity tracking.

HIGH-SPEED ELECTRON MICROSCOPY

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A. Short-Time Exposure Imaging The short-time exposure imaging technique pins down a transient stage of a fast process by producing a two-dimensional image of the specimen with a short exposure time. This may be realized either by using a stationary illumination of the specimen and enabling the image detector for only a short time (Bostanjoglo, Kornitzky, et al., 1989; Bostanjoglo, Tornow, et al., 1987a, 1987b) or by illuminating/exciting the specimen with a short electron/photon pulse and recording the electron image with a stationary detector. The first method requires sophisticated pulse electronics and shielding precautions. Preferably, the image detector is a charge-coupled-device (CCD) camera backed by an image intensifier. A sealed intensifier may be gated by pulsing the moderate voltage between the photocathode and the first gain stage, which is a microchannel plate (MCP). An open MCP intensifier is enabled by pulsing the voltage across the channel plate. This voltage is the smallest one so that electromagnetic interference due to switching is minimized. In addition, the applied voltage may appreciably exceed the maximum safe dc voltage for a short period, giving a gain in the pulsed mode which surpasses the dc value by two orders of magnitude. The second technique is superior as it may provide a much brighter illumination than that of a stationary beam if the electrons are emitted by a pulsed source. Short electron pulses may be produced by a fast deflection of a constant current beam (Gesley, 1993), by pulsing the voltage of the Wehnelt electrode (Szentesi, 1972) or of a filter lens (Plies, 1982), or by exciting the electron emitter with a laser pulse. Only the last method yields the high current densities required for nonsampling short-time exposure imaging. The laser-driven gun used in the author’s group is distinguished by the fact that it can be operated both as a conventional dc thermal gun and as a highcurrent pulsed gun. It is a three-electrode-type gun, consisting of a hairpin emitter, a Wehnelt electrode, and an anode, which houses an aluminum mirror for directing the laser beam onto the tip of the hairpin. This gun may be pulsed in the thermionic or the photoelectron emission mode. As high current density guns are the key component for short exposure imaging they are considered in some detail next. 1. Laser-Driven Thermionic Gun If the emitter is heated by a nanosecond (or shorter) laser pulse, the emitter can attain a temperature well above the melting point, without being destroyed, and thermal electron pulses with current densities exceeding those produced by dc heating are attained (Bostanjoglo and Heinricht, 1987; Bostanjoglo, Heinricht, et al., 1990; Schäfer and Bostanjoglo, 1992). In addition, emitter atoms are evaporated. They are ionized by the accelerated thermal electrons and reduce their negative space charge, so that electron current densities exceeding the

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Child limit of genuine electron emitters by one order of magnitude can be generated. However, this gun has several serious drawbacks. First, as the surface is eroded by each laser pulse, its absorption coefficient and therefore the deposited laser fluence vary from pulse to pulse, which produces unpredictable electron pulse currents. In addition, the length of the electron pulse may exceed that of the laser pulse by more than 100% as a result of delayed emission of captured electrons as the plasma is diluted by expanding into the vacuum. This poor pulse-to-pulse stability makes the laser-driven thermionic gun unsuitable for multiframe imaging. Last, this mode of operation is hazardous, since the gun is driven to the threshold of laser-induced electric breakdown. A small up-deviation of the deposited laser fluence triggers a high-voltage breakdown, which in turn launches a high-amplitude traveling wave that may destroy electronic circuits of the microscope or of the attached high-speed diagnostic devices. 2. Laser-Driven Photoelectron Guns Photocathodes with work functions ranging from the lowest values of ≈2 eV up to 4 eV have been used in laser-excited guns. Data on a number of electron emitters are given, for example, by Anderson et al. (1992), Chevallay et al. (1994), and Travier (1994). Materials with low work functions (100 nm) is converted into a voltage signal by a fast plastic scintillator


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Figure 3. Transmission electron microscope for tracking laser pulse–induced transitions in thin films: 1 – 5, as in Figure 2; 6, circular field aperture; 7, plastic scintillator; 8, photomultiplier tube.

(Pilot U, 1.9-ns rise time) plus a photomultiplier (rise time, 2 ns). This signal is recorded with a storage oscilloscope (rise time, 0.35 ns). The resulting time resolution of the recording unit is (1.92 + 22 + 0.352)1/2 ≈ 3 ns. The illuminating electron pulse is generated as in the case of streak imaging. Intensity tracking continuously records changes of the electron scattering, which may be due to phase transitions or removal/accumulation of

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material from/at the probed region. This technique is therefore well suited to detect transient states and measure their lifetime and the period of phase transformations. 2. Applications Two typical applications of time-resolved transmission microscopy are reported in this section: hydrodynamic instabilities of metal melts subjected to high lateral thermal gradients (≈109 K/m) and ablation of metal films by laser pulses. These processes and metals as material were selected because they have a bearing on micromachining with laser pulses. A laser pulse, bell-shaped both in time (5–15 ns, FWHM) and in space (12 μm, 1/e2 diameter), is applied to a freestanding metal film with a typical thickness of 100 nm. The film contains impurities due to a preceding exposure to air. The existence of such impurities is typically the case in laser microprocessing. As the fluence of the treating laser pulse is increased, two regimes are encountered. In the lower fluence regime a local melt is produced which solidifies again. In the upper regime parts of the treated region are ablated. The details of the observed behavior of the treated metal considerably deviate from what naively is expected. a. Thermal Gradient–Driven Instabilities of Metal Melts The thickness D of the treated film must be smaller than the thermal diffusion length during the laser pulse (D < 200 nm for all metals and 10-ns pulses). In this case an in-plane bell-shaped distribution of the temperature T is produced in the film depending only on the radial coordinate r. The fluence of the laser pulse must be high enough to melt the film within a certain radius but too low to heat the film appreciably above the melting temperature. Then, radiation pressure as well as evaporation of metal atoms and their recoil pressure can be safely neglected for nanosecond pulses. The only force the originally flat melt is subject to after the laser pulse stems from a possible gradient dγ /dr of the surface tension γ, which is identical to a shear stress acting on both surfaces. There then exists a negative thermal gradient ∂T/∂r < 0 in the melt. Since the surface tension depends on temperature, and tabulated thermal coefficients are negative (about −3 × 10−4 N/m · K for many metals; e.g., Iida and Guthrie, 1988), the melt is expected to experience a positive shear stress at both surfaces: ∂γ ∂ T dγ = · >0 dr ∂T ∂r

(1)

This shear force monotonously drags the liquid to the cooler solid periphery, piling it up there and finally opening a hole at the center of the melt. The actual flow, however, is quite different (Bostanjoglo and Nink, 1997; Bostanjoglo and Otte, 1993; Nink et al., 1999). Figures 4 through 6 show


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Figure 4. Short exposure images of flow in a laser pulse–produced melt in an amorphous Ni0.8P0.2 film (60 nm). Exposure time was 10 ns. The moment of exposure was counted from the peak of the treating laser pulse (−∞ before, ∞ 10 s after the pulse) and is given at the upper right corners. The flow stopped about 1 μs after the laser pulse, whereas the melt crystallized within 4–10 μs after the pulse. (a) Centripetal flow after a low-energy laser pulse (1.2 μJ). There is no reversal of the flow direction. (b) Centripetal flow followed by centrifugal flow after a high-energy pulse (1.6 μJ). Flow direction is reversed ≈300 ns after the laser pulse.

the hydrodynamics of laser pulse–produced melts in different metal films, visualized by the three time-resolving techniques described in Section II. None of the liquids, which were subjected to an in-plane thermal gradient, was perforated, as was expected for a flow driven by negative thermocapillarity (∂γ /∂T < 0). Instead, the flow conspicuously depends on the starting temperature. At lower temperatures the liquid simply contracts within 100 ns and solidifies with a bump at the center. At higher temperatures flow starts with a fast contraction and continues with reversals of the flow direction. In addition to flow, crystallization of the melt of an “ordinary” metal with a high thermal diffusivity starts at its solid periphery and proceeds with an almost constant velocity of several meters per second toward the center of the melt (Fig. 7; Bostanjoglo and Nink, 1996). In the case that the melt accumulates at the center, a solid film with concentric modulations of the thickness is produced (Fig. 8). Melts which are produced with a pulse of a higher fluence are subdivided by an emerging concentric ring-shaped trench (Fig. 9; Niedrig and Bostanjoglo,

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Figure 5. Nonmonotonous flow in a laser pulse–produced melt pool in a polycrystalline cobalt film (60 nm). (a) Streak image of the melt flow. The melting 5-ns laser pulse was applied at the top edge. The slit aperture (width, 1 μm) passed the central region of the melt (lower edge in (b)). (b) Texture after crystallization of the melt.

1997). The inner zone contracts and finally separates, which forms a free disk that continues to contract as a result of surface tension and disappears in the end. The observed complicated flow can under no circumstances be explained with tabulated material parameters and the assumed shear stress in Eq. (1). Rigorous numerical simulation based on the Navier–Stokes and heat equation and simple physical arguments lead inevitably to a monotonous perforation of the melt within 100 ns. Figure 10 gives a hint to the decisive mechanism behind the actual flow. A melt in a gold film contracts after the first laser pulse. If a second pulse of similar fluence is applied after solidification but before a monolayer of gas is adsorbed from the high vacuum of the microscope, the melt then flows to the periphery. This reversal does not occur if the treated area is allowed to adsorb about a monolayer of air molecules (Bostanjoglo and Nink, 1996). Obviously the flow of “real” liquid metals is determined by surface active impurity atoms. These atoms accumulate at the surface by replacing metal atoms, which thereby decreases the surface tension according to the Gibbs isotherm: dγ = −kT Ŵ d(ln X )

(2)


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Figure 6. Oscillating flow in a laser pulse–produced melt pool in a polycrystalline iron film (60 nm). (a) Texture after crystallization of the melt. (b, c) Oscilloscope traces showing the bright-field image intensity within the circle in (a) at two time scales after the melting laser pulse (arrow). m and cr denote melting and crystallization, respectively. The final level of the intensity in (c) remains constant.

with k, Ŵ, and X the Boltzmann constant, the excess surface density of the surface active atoms adsorbed at the surface layer, and the atomic fraction of the surface active atoms in the bulk liquid, respectively. Thus, the surface tension decreases with increasing concentration of surface active impurities (∂γ /∂X < 0). The thermal coefficient ∂γ /∂T is also changed (Fig. 11; Ricci and Passerone, 1993; Vitol and Orlova, 1984). If the concentration of the impurities is high enough ∂γ /∂T becomes even positive below some temperature To. Above To the coefficient is again negative and approaches the value of the pure metal. Taking into account that the surface tension is a function of temperature and atomic fraction of the surface active impurities, γ = γ (T, X ), the shear

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Figure 7. Typical crystallization at nearly constant velocity of a melt pool produced by a focused laser pulse in a crystalline metal film (aluminum, 60 nm). (a) Streak image. The melting 5-ns laser pulse was applied at the upper edge. The dark triangle is liquid metal; the vertical dark stripes within the bright area are Bragg-scattering crystals in the crystalline material. Propagation velocity of the crystal/liquid boundary is 5 m/s. (b) Texture after solidification of the melt. The rectangle marks the location of the streak aperture.

Figure 8. Concentric thickness modulations of a solidified laser pulse–produced melt in a gold film (90 nm), imaged by backscattered electrons in the scanning microscope.


17

Figure 9. Chemocapillary flow in a laser pulse–produced melt in an aluminum film (90 nm), imaged by short exposure transmission microscopy. Exposure time was 5 ns. The moment of exposure was counted from the peak of the laser pulse and is indicated at the upper left corners. The applied laser pulse (15 ns, 3.5 μJ) produced a hole.

stress driving the melt flow must then be ∂γ ∂γ ∂T ∂X dγ = + dr ∂ T X ∂r ∂ X T ∂r

(3)

It is determined by the thermal and compositional gradients, which cause a thermo- and a chemocapillary flow, respectively. Oxygen atoms are known

Figure 10. Solidified melt pools in the same gold film (65 nm), showing that melt flow after one laser pulse is opposite that after two successive pulses. (a) Transmission microscope image of the solidified melt after one laser pulse of 1.6 μJ. The melt piled up at its center. (b) Structure after two successive pulses that are 4 μs apart and have the same energy as in (a). The melt solidified after the first pulse and piled up at its periphery after the second melting pulse. The melts solidified about 1 μs after a laser pulse.

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Figure 11. Typical dependence of the surface tension γ of a metal on temperature and atomic fraction X of surface active impurities in the bulk liquid. Tm and Tc are the melting and critical temperatures, respectively. With growing X a maximum of γ appears at To.

to be surface active in various metals (Ricci and Passerone, 1993; Vitol and Orlova, 1984), and they were abundant in the investigated films that were exposed to ambient atmosphere. With all this in mind the following scenario is expected. The melt, originally having a homogeneous distribution of impurities along r (∂X/∂r = 0), but being subjected to a thermal gradient (∂T/∂r < 0), starts to flow as a result of the thermocapillary shear stress: ∂γ ∂T ∂γ (4) = ∂r X ∂ T X ∂r The bulk liquid lags behind the near-surface layers, since it is dragged by them by means of viscosity and since it is driven by the Laplace pressure, which appears only as the surface deforms. Accordingly, the surface active atoms are redistributed by fast surface flow in such a way that their concentration is reduced in regions having a positive gradient of the surface velocity, and vice versa. A compositional gradient ∂X/∂r emerges, which has the same sign as that of the thermocapillary force, and which produces a chemocapillary shear stress: ∂γ ∂γ ∂X = (5) ∂r T ∂ X T ∂r Since (∂γ /∂X)T < 0, the chemocapillary force produced by a thermocapillarydriven flow always opposes the latter. Therefore, the original flow is either stopped or even reversed, in agreement with the observed flow dynamics. The different directions of the early stages of flow (i.e., centripetal after a low-energy and centrifugal after a high-energy laser pulse) follow from the


19

convex shape of the γ –T curve at high concentrations of surface active impurities (Fig. 11). Since the compositional gradient ∂X/∂r is zero at the beginning, the direction of the early melt flow is determined by the sign of the thermal coefficient ∂γ /∂T alone. If the maximum temperature produced by the laser pulse is below To in Figure 11, the thermocapillarity coefficient ∂γ /∂T is positive everywhere, and the liquid contracts in the negative thermal gradient according to Eq. (4). If, however, the maximum temperature of the liquid (which is at the center of the melt pool) exceeds To, then ∂γ /∂T is negative up to some radius ro, where the local temperature coincides with To, and positive beyond ro up to the solid rim. Then the liquid experiences an outward thermocapillary drag at the center up to a radius ro and an inward shear stress beyond ro. The melt starts to deplete at the center and at the periphery and produces a ring-shaped bulge somewhere between (Fig. 6). The appearance of a circular trench in aluminum films at higher temperatures (Fig. 9) cannot be explained as before with a positive thermocapillary coefficient ∂γ /∂T > 0, as the temperature after the used laser pulses is too high (T > To at the center). Instead, chemocapillarity presumably is operating. The surface active oxygen atoms, stemming from the disintegrated native oxide, are evaporated from the center (which is hottest) to a large extent during the laser pulse (see also Fig. 16). Thus, a positive gradient ∂X/∂r > 0 of the oxygen concentration is produced. Since the temperature has its maximum at the center of the melt, the thermal gradient is small near the center (∂T/∂r ≈ 0), so that the sign of the total shear stress in Eq. (3) may become negative there and force the central zone of the melt to contract. This physical picture has been substantiated by numerical simulations (Balandin, Otte, et al., 1995). The concentric ripples occurring in solidified melts produced by lower energy laser pulses (Fig. 8) cannot be explained by simple physical arguments. They certainly are not frozen capillary waves, as one might think at first. The large number of ripples would mean that they are due to a high-frequency mode, whose excitation, however, is very improbable. The formation of the observed ripples was reproduced by a numerical simulation which is based on the Navier–Stokes equation, comprising thermo- and chemocapillary shear stress, and which assumes that the surface active impurity atoms segregate at the moving crystallization front and accumulate in the adjacent melt (Balandin, Gernert, et al., 1997; Balandin, Nink, et al., 1998). These simulations give the following physical picture of the solidification process in a metal melt with surface active impurities. As the crystallization velocity exceeds a threshold of about 6 m/s (in gold), a front wave with a width of about 1 μm is produced ahead of the moving phase boundary. It pulsates and periodically emits steps of the impurity concentration, which in turn cause steps of the surface tension and these in turn steps of the flow velocity. All these abrupt changes propagate into the melt. As the crystallization front sweeps across the agitated liquid, ripples of the observed period are in fact frozen. A front wave moving

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Figure 12. Short exposure transmission electron microscopy images showing the crystallization of a laser pulse–produced melt in a germanium film (50 nm). Exposure time was 40 ns. The time of exposure after the laser pulse is indicated at the right top corner (∞ 10 s after the melting 30-ns laser pulse). Note the pileup of liquid at the moving crystallization front.

along with the phase boundary in a crystallizing germanium melt is shown in Figure 12 (Bostanjoglo, Marine, et al., 1992). b. Ablation of Metal Films If the deposited laser pulse energy exceeds the enthalpies of melting and evaporation, a certain amount of the hottest part of the melt will evaporate during the laser pulse. Figure 13 shows how evaporation and thermocapillarity compete in ablating an aluminum film after a pulse of medium fluence (Niedrig and Bostanjoglo, 1997). A circular trench emerges, as at lower fluences. In addition, the liquid is removed by evaporation at the center. A hovering liquid ring remains, which collapses as a result of the surface tension. Simultaneously, the hole is expanded by surface tension with a velocity v that can be estimated by equating the approximate change d(2πr2γ ) of the surface energy and the change d(2πrπ R2ρv 2/2) of the kinetic energy v ≈ (4γ/ρ D)1/2

(6)

Calculated and measured velocities are in the order of 100 m/s for films with a thickness of D ≈ 100 nm. As the fluence exceeds a threshold (e.g., ≈5 J/cm2 for a 90-nm Al film) ablation of the aluminum film proceeds exclusively by evaporation (Fig. 14). Liquid flow is reduced to a short radial expansion of the hole, curling up its rim and disrupting it by Rayleigh instabilities into spheres. At first sight, the ablation processes in Figures 13 and 14 seem to be selfexplanatory, but numerical simulations uncover some surprises (Balandin,


21

Figure 13. Double-frame short exposure imaging of the ablation of an aluminum film (90 nm) by volume evaporation and thermocapillary flow, caused by a 15-ns laser pulse of 4 μJ. Exposure time was 5 ns. The moment of exposure was counted from the peak of the laser pulse and is given at the left top corners of the frames. The double-frame series a–c were produced at three different regions of the same film. The final state was always a hole as in series a.

Niedrig, et al., 1995; Niedrig and Bostanjoglo, 1997). The observed time scales of the ablation, by the combined action of thermocapillary flow and evaporation, and by evaporation alone, require that the following two conditions hold: 1. The surface tension decreases with a constant tilt of −3 × 10−4 N/m · K from the melting temperature up to ≈3000 K. This coefficient is equal to the tabulated value of pure aluminum near the melting point (933 K). Above ≈3000 K the surface tension heads, with a very small coefficient of −0.2 × 10−4 N/m · K, toward zero at the critical temperature of ≈8500 K. 2. Surface evaporation is marginal when aluminum is heated by nanosecond laser pulses. Instead, evaporation proceeds by volume evaporation (i.e., boiling), which is calculated to set in at ≈6000 K, assuming that nucleation of critical bubbles is homogeneous in the freestanding films. Models which are based on equilibrium surface evaporation (the evaporation rate and pressure are given by the Hertz–Knudsen–Langmuir and Clausius– Clapeyron equations, respectively) have been advanced, for example, by Ho et al. (1995), Metev and Veiko (1998), and Pronko et al. (1995) to explain

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Figure 14. Volume evaporation of an aluminum film (90 nm) by a 15-ns laser pulse of 6.5 μJ. Exposure time was 5 ns. The moment of exposure was counted from the peak of the laser pulse and is indicated at the top left corners of the frames. The four double frames a–d were produced at different regions of the same film. The final state was always a hole coinciding in size with that in frame “45 ns” of d.

ablation of metals by short laser pulses. Although these models reproduce the ablated volume surprisingly well (Preuss et al., 1995; Singh et al., 1990), according to the preceding findings they cannot deal with the dynamics of evaporation of aluminum, and probably of other metals, by nanosecond laser pulses, and are therefore misleading. 3. Space–Time Resolution a. Short-Time Exposure Bright-Field Imaging As each electron image point requires a minimum electron dose to be registered in a single exposure, space and time resolution are not independent. The joint resolution is limited by shot noise in the electron beam and by the detector noise. A specimen area of diameter x be illuminated by n electrons during an exposure time t. A fraction ni of the scattered electrons is passed by the objective lens aperture and produces the bright-field image. An image detector with the gain G delivers nd = Gni signal electrons. Two adjacent areas of


23

equal diameter x, which produce different numbers ni1 and ni2 of image ¯ n¯ i1 − electrons, are distinguished by the detector if the mean difference |G ¯ n¯ i2 | of the signal electrons exceeds the root-mean-square noise amplitude G ((n d1 )2 + (n d2 )2 )1/2 by a minimum signal-to-noise ratio of about 3; that is, ¯ n¯ i1 − G ¯ n¯ i2 |/ (n d1 )2 + (n d2 )2 1/2 > 3 |G (7)

The overbar denotes the average value. The fluctuations nd of the number of the detected electrons comprise shot noise ni in the beam and detector noise expressed by G. The mean square of nd then is (n d )2 = G 2 (n i )2 + n i2 (G)2

(8)

Since the shot noise obeys the Poisson distribution one has (n i )2 = n¯ i and ¯ 2 + (G)2 , n i2 = n¯ i2 + n¯ i which gives, with G 2 = G ¯ 2 [1 + (2 + n¯ i )(G)2 /G ¯ 2] > ni G ¯2 (n d )2 = n¯ i G

(9)

The term within the brackets is of the order 1 as the detector gain G is usually very high and the number n i of electrons imaging the small area π(x)2/4 within the short time t is small. If one combines Eqs. (7) and (9), using the image contrast K = |n i1 − n i2 |/(n i1 + n i2 ) of the two adjacent regions and expressing the average number of image electrons by the current density j of the illuminating electrons, their charge e, and the average transmission factor ε of the objective lens aperture—that is, (n i1 + n i2 )/2 = επ(x)2 jt/4e—the relation between spatial resolution x and exposure time t becomes (x)2 t > 18e/πεK 2j

(10)

If one uses a laser-driven photoelectron gun which delivers 2-mA electron pulses into an area of the object of about 30 μm ø and takes ε ≈ 0.1 and K ≈ 1, Eq. (10) gives an ultimate joint resolution of (x)2 t ≥ 5 × 103 nm2 · ns. An image with an exposure time of, for example, 10 ns will have a spatial resolution of 20 nm at best. An additional limitation is imposed by electron beam heating of the specimen. On the one hand, the beam current density should be as high as possible to keep shot noise low. On the other hand, the probing electron beam should not induce any transitions in the specimen. Since heating by the illuminating electron pulse is adiabatic at the short exposure times, the total energy n¯ E deposited by the n¯ electrons of the pulse must obey n¯ E ≤ π ( x/2)2 Dρc T

(11)

where E, ρ, c, D, and T are average energy loss of a beam electron, density,

24

O. BOSTANJOGLO

specific heat, thickness, and the maximum allowed electron-induced rise of temperature of the film, respectively. If one inserts E = AρD with A ≈ 5 × 10−13 Jcm2/g according to the Bethe stopping power formula (e.g., Reimer, 1993) and uses Eq. (10), the resolution limit due to electron beam heating is given by x > (18A/πε K 2 c T )1/2

(12)

Taking, for example, iron and replacing the actual specific heat by its hightemperature value c = 3k/m (k, Boltzmann constant; m, atomic mass) and T by the melting temperature, one gets, with ε ≈ 0.1 and K ≈ 1, x ≈ 3 nm as the absolute spatial resolution in this case. b. Streak Imaging The time resolution t of a streak image may be defined as t = ts w/L

(13)

where ts is the streak period, w the width of the streak aperture, and L the streak distance, both measured in the object plane. The spatial resolution x along the streak aperture is determined as for short exposure imaging. Two adjacent rectangular areas of the specimen with width w and length x are distinguished by the detector within the time t if their signal-to-noise ratio exceeds a minimum value of about 3. Expressing the average number of the image electrons ni1 and ni2 from the two areas again by the illuminating current density j, that is, (ni1 + ni2)/2 = εwxj t/e, one gets an inequality similar to that in the preceding section: x t > 9e/2εK 2 wj

(14)

Taking typical values ts = 100 ns, w/L ≈ 0.1, ε ≈ 0.1, K ≈ 0.2, w ≈ 1 μm, and j ≈ 3 A /cm2, a one-dimensional space resolution of x ≈ 0.6 μm is calculated for a time resolution of t = 10 ns. This value approximately agrees with the actual resolution. c. Image Intensity Tracking The joint space–time resolution is derived in a similar way as before. The specimen is illuminated by an electron current of density j. A fraction ε of the scattered electrons passes the objective lens aperture and produces a brightfield image. If x is the diameter of the specimen area viewed by the scintillator/photomultiplier detector, the current picked up by the detector then is J = εjπ(x)2/4. The output signal current of the detector, having a gain G (≫1) is Js = GJ. This signal is superimposed by a noise current Jn with an average amplitude (Jn2 )1/2 . The noise is composed of fluctuations G of the gain plus


the amplified shot noise (2eJ f )1/2 of the image current J: 1/2 2 1/2 Jn2 ≈ (2e J f G 2 )1/2 = J (G)2 + 2e J f G 2

25

(15)

where f is the bandwidth of the detector and the processing electronic circuits. Since the detector is based on multiplication processes with very high gain one has (G)2 ≈ G >> 1, and since the image current J and its average shot noise amplitude are of the same magnitude near the resolution limit, Eq. (15) simplifies as indicated. A transition producing a change Js of the signal is resolved, if it exceeds the noise amplitude (Jn2 )1/2 by a factor of at least 3: 1/2 Js = G J ≥ 3 Jn2 ≈ 3G(2e J f )1/2 (16) If one inserts J and replaces the bandwidth f by the minimum detectable rise/fall time t ≈ 0.35/f the joint space–time resolution becomes (x)2 t ≥ 25e/πεj(J/J )2

(17)

Assuming typical values j ≈ 10 A/cm2 (from a conventional thermal tungsten hairpin gun) and ε ≈ 0.1, Inequality (17) states that a phase transition of, for example, 3-ns duration, which produces a change J/J ≈ 1 of the image current, can be detected in specimen areas with diameters down to x ≈ 0.2 μm.

B. Flash Photoelectron Microscopy Any electrons released from a surface (e.g., by ion, electron, or photon bombardment or by heating or high electric fields) can be used to image the surface. Photoelectrons ejected by laser pulses are particularly suited for short exposure imaging because r

r

high electron current densities can be produced without damaging the specimen the moment of exposure can be freely chosen

For decades photoelectron microscopy has been used as a powerful surfaceimaging technique. Very different material properties have been characterized: r

r

r

crystal texture and defects (Engel, 1966; Griffith and Rempfer, 1987; Griffith et al., 1991; Möllenstedt and Lenz, 1963) chemical reactions and pattern formation ( Ehsasi et al., 1993; Engel et al., 1991; Rotermund et al., 1991) p–n junctions, metal leads, and surface states on semiconductor devices (Giesen et al., 1997; Ninomiya and Hasegawa, 1995)

26

O. BOSTANJOGLO r r

surface diffusion (von Oertzen et al., 1992) biological tissue (De Stasio et al., 1998; Griffith, 1986)

The spatial resolution of photoelectron microscopy and related techniques, such as low-energy and mirror electron microscopy, was discussed, for example, by Möllenstedt and Lenz (1963) and by Rempfer and Griffith (1992). Photoelectrons are emitted after single- or multiphoton absorption. The former requires that the photon energy hf (h, Planck constant; f, frequency of the light) exceed the bond energy of the electron; that is, hf > WA for a metal with a work function WA. At nonzero temperatures thermally excited electrons can be emitted by lower energy photons. Two-photon absorption, as the simplest multiphoton process, produces a photoelectron by the simultaneous absorption of two photons. If they have equal frequencies their quantum energy must exceed only WA/2. However, the intensity of the light must be so high that on the average two photons interact with an electron within a time h/WA according to the uncertainty relation. As the absorption cross section is about σ ≈ 10−16 cm2, intensities of at least WA2/hσ ≈ 1013 W/cm2 are required for metals with WA ≈ 4 eV. Such high light intensities can be produced by laser pulses, but they inevitably damage most metals unless femtosecond pulses are used. Unfortunately, these ultrashort pulses produce far too small numbers of electrons per pulse for a short exposure image with an acceptable signal-to-noise ratio (at fluences below the damage threshold). Therefore, single-photon absorption has been exclusively exploited for photoelectron microscopy. The contrast is determined mainly by the local yield of photoelectrons. This yield depends on the true local work function, on the local thickness of possibly present dielectric (oxide) coating films, and on local variations of the electric field caused by surface geometry and by adsorbed molecules with high electric polarizability or with a permanent electric dipole. Such adsorbed molecules (e.g., water molecules) may enhance the photoelectron emission by more than one order of magnitude (Buzulutskov et al., 1997). All these effects merge to produce an effective work function WA with a local variation WA. Since the density of the photoelectron current (induced by one-photon absorption) is j = const(h f − W A )n

(18)

with n a positive constant, the contrast becomes j/j = −n W A /(h f − W A )

(19)

The contrast increases sharply as the photon energy approaches the work function. Conversely, the quantum efficiency decreases to zero and the photoelectron image is disguised by shot noise. For this reason the illuminating photons should have a large quantum energy. Most metals of technical interest have work functions around 4 eV, so a good compromise between contrast and shot noise is photons with hf ≈ 5 eV.


27

Short exposure photoelectron imaging is most easily realized by illuminating the specimen with an ultraviolet laser pulse. Suitable lasers are frequencymultiplied solid-state lasers and excimer lasers. The latter are preferable because of their smaller coherence length, which helps to avoid disturbing interference patterns in the image. A good choice is the KrF laser (wavelength, 248 nm; hf = 5.0 eV). 1. Instrument for Short Exposure Imaging All previous photoelectron microscopes had a time resolution limited to several milliseconds. Releasing the photoelectrons with a pulse from an excimer laser, having a short coherence length, and carefully avoiding parasitic reflections which cause interference patterns allowed a resolution of a few nanoseconds to be achieved. Figure 15 schematically shows the assembled flash photoelectron microscope that can image nonrepetitive changes of a surface on the

Figure 15. Flash photoelectron microscope with attached lasers for treating the specimen.

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O. BOSTANJOGLO

nanosecond time scale (Bostanjoglo and Weingärtner, 1997; Weingärtner and Bostanjoglo, 1998). The specimen is at a high negative potential (−25 to −30 kV). Imaging photoelectrons are released by a 4-ns (FWHM) pulse from a KrF excimer laser. The fluence of the ultraviolet pulse is kept so low that the surface is not damaged. The photoelectrons are accelerated with a field of 5–8 kV/mm toward a grounded stainless steel anode. They are focused by an electrostatic einzel lens to an intermediate image which is projected by a magnetic lens on a fiber plate transmission screen. The converted electron image is picked up with a fiber-coupled MCP image intensifier plus a CCD camera, digitized by a frame grabber, and stored in computer memory. A home-built trigger circuit allows one to make an “exposure” at any time relative to the processing visible laser pulse (wavelength, 532 or 620 nm). The aperture in the back focal plane of the electrostatic lens decreases the angular and energy spread of the imaging electrons, and therefore makes geometrical modulations of the surface become visible and increases the spatial resolution (Boersch, 1943; Möllenstedt and Lenz, 1963). Two adjustable aluminum mirrors, which are fixed at the anode, direct the illuminating ultraviolet and the processing visible laser pulse onto the specimen. A beam blanker passes electrons to the detector for 5 ns only during the ultraviolet laser pulse. The beam blanker consists of a low-impedance parallel plate capacitor which normally deflects the electrons beyond the intercepting aperture in the back focal plane of the electrostatic einzel lens, and which is switched by an avalanche transistor–based cable pulser. In this way disturbing long-lasting thermal electrons and delayed electrons liberated by excited atoms and ions are kept away from the image. Their contribution to the image during the acquisition time (i.e., “exposure”) is negligible if the fluence of the processing laser pulse is not excessive. The specimen can be heated by electron bombardment from the back side for cleaning purposes. The investigated fast processes were launched in the specimen by a focused pulse either from a Q-switched frequency-doubled Nd:YAG laser (pulse width, 10 ns; wavelength, 532 nm) or from a colliding pulse mode–locked dye laser (pulse width, 100 fs; wavelength, 620 nm). The laser beams were focused on the specimen to a spot with a 1/e2 diameter of 15 and 50 μm for the nano- and femtosecond pulses, respectively. For a controlled positioning of the processing laser beam, the specimen is illuminated with white light and imaged with reflected and scattered radiation. The accelerating voltage can be cut off and the specimen grounded within 20 ns with a fast switch consisting of cascaded transistors. In this way a laserinduced electric breakdown is avoided by interrupting the avalanche buildup. This technique is successful only if the breakdown is delayed by more than the fall time of the switch (20 ns) plus the acquisition time for the image (5 ns).


29

Figure 16. Photoelectron images of an aluminum film (100 nm) on (100) silicon, showing the removal of the native aluminum-oxide covering layer by a laser pulse (10 ns, 6 μJ, 20 μm ø). Exposure time was 5 ns. The moment of exposure was counted from the peak of the laser pulse and is given at the right top corners of the images (∞ 10 s after the pulse). The images were produced at previously untreated neighboring regions with equal laser pulses.

2. Applications Because photoelectron emission reflects the bonding of surface electrons, pulsed photoelectron microscopy is an excellent method for imaging local chemical reactions. Figure 16 shows as an example the reaction induced by a nanosecond laser pulse in aluminum covered with its native oxide (thickness, D ≈ 3–4 nm). The fluence was high enough to melt the surface of the metal but too low for appreciable evaporation of metal atoms (as no flashover was initiated). The photoelectron emission at first decreases during 5–10 ns after the laser pulse and then considerably increases, saturating after some 10 ns. If the surface is exposed to air the photoelectron emission returns to the low value of the untreated material. It is well known that liquid aluminum decomposes aluminum oxide, Al2O3, which produces a volatile suboxide, Al2O (Champion et al., 1969). The dielectric native oxide coating reduces the number of the ejected photoelectrons because only a fraction exp(−D/L) are transmitted, the mean free path of the photoelectrons in the oxide being L ≈ 1–3 nm (Buzulutskov et al., 1997). As the oxide coating disintegrates after the laser pulse, the photoelectron yield increases and its rise time reflects the time it takes to decompose the oxide and evaporate the products from the melt. There remains the puzzling early decrease of the photoelectron emission. Such a decrease was observed with all metal surfaces that were not cleaned by electron beam heating prior to the laser treatment. This decrease is therefore probably due to the removal of adsorbed polar molecules (e.g., water molecules), which add their dipole field to the cathode field, decreasing the

30

O. BOSTANJOGLO

work function by epn/εo ( p and n, dipole moment and surface density of adsorbed molecules; εo, vacuum permittivity). A particular benefit of photoelectron microscopy is the fact that the first top layers of a specimen are probed. It is therefore particularly suited to uncover incubation effects and early stages of radiation-induced material modifications. As an example with a bearing on laser microprocessing flash photoelectron microscopy is applied to visualize effects produced by nano- and femtosecond laser pulses with fluences near the ablation threshold. These two pulse lengths are much longer and much shorter, respectively, than the electron/lattice relaxation time, which is some picoseconds for typical metals (e.g., Elsayed-Ali et al., 1987). The laser pulse energy is primarily absorbed by the electrons. In the case of a nanosecond pulse the electrons are practically in equilibrium with the atomic lattice, and the laser power is fed directly to it, which gradually destabilizes it by ordinary heating. This is not so in the case of a femtosecond pulse. In this case the laser pulse energy is almost totally absorbed first by electrons, which excites them to high levels and destabilizes the atomic lattice. A metal is destabilized by the high pressure of the hot conduction electron gas, whereas bonds in semiconductors are weakened as the valence electrons are excited into the conduction band (Stampfli and Bennemann, 1992). If the electron excitation is high enough the lattice will collapse. At lower fluences just a destabilized lattice is produced which starts to sink the energy of the electrons either by mechanical work or by exchange of heat (Stampfli and Bennemann, 1992). Since the atomic lattice occupies two very different states when it sinks the energy of a nano- and a femtosecond laser pulse, respectively, its response on the thermodynamic time scale (some picoseconds and longer) is expected to be quite different. Both metals and semiconductors have been observed to respond in different ways to nano- and femtosecond pulses (Weingärtner et al., 1998). Figure 17 shows the completely different effects produced by a 10-ns and a 100-fs laser pulse on (100) silicon with a native oxide layer (thickness, ≈3 nm). The nanosecond pulse causes the silicon surface to melt, as is substantiated by the final smooth crater-like structure (Fig. 18). Photoelectron emission rises as the silicon surface is molten and remains high until the melt solidifies 100–200 ns after the laser pulse. Freezing is accompanied by a slight decrease of photoemission. Exposure of the surface to air returns the photoelectron yield to the low value of the untreated silicon. The oxide coating is decomposed as the laser pulse melts the silicon surface, and the photoelectrons can escape the liquid without crossing a solid coating. As the liquid silicon solidifies, oxygen atoms which were dissolved in the melt are segregated at the surface and a covering oxide layer is grown again. However, this layer is thinner than the original one, as part of the oxygen atoms were evaporated, and the


31

Figure 17. Photoelectron images of (100) silicon with a native oxide covering layer of 3 nm thickness, showing the completely different response to (a) a 10-ns and (b) a 100-fs laser pulse. Exposure time was 5 ns. The moment of exposure was counted from the peak of the laser pulse and is given at the top right corner of the images (∞ 10 s after the pulse). The images were produced at previously untreated neighboring regions. The energy was ≈6 μJ for the nanosecond pulse and ≈0.9 μJ for the femtosecond pulse.

photoemission is higher after the laser pulse. Thus, a nanosecond laser pulse effects a partial removal of the oxide from silicon by decomposition, transient storage of some oxygen dissolved in the melt, and regrowth of a thinner coating within 200 ns. This partial cleaning by a melting nanosecond laser pulse, but not the time scale of the process, was previously documented by Auger spectroscopy (Larciprete et al., 1996). The pileup of the melt, freezing at the periphery after a nanosecond pulse (Fig. 18), is not caused by recoil pressure from evaporating atoms. Evaporation was marginal as no flashover occurred.

Figure 18. Typical smooth flat crater produced by a 10-ns laser pulse (≈5 μJ ) on (100) silicon with native oxide and imaged by scanning electron microscopy with secondary electrons.

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Figure 19. Typical rough patch produced by a 100-fs laser pulse (≈0.9 μJ ) on (100) silicon with native oxide and imaged by scanning electron microscopy with secondary electrons at grazing incidence (80◦ against the normal of the surface).

Since very similar final structures were produced on (100) silicon without an oxide coating, the crater-like distribution of the melt is not affected by chemocapillary forces but must be caused by thermocapillary forces. The 100-fs laser pulse has a very different effect on (100) silicon covered by a native oxide (Figs. 17b and 19). The photoelectron yield is heavily reduced during ≈100 ns after the laser pulse within the laser spot. An irregular small zone with increased photoelectron emission develops from the dark area. The final structure consists of a weakly corrugated surface which is barely visible in the scanning electron microscope (Fig. 19). It is invisible to light-optical microscopy, even to such surface-sensitive techniques as dark-field and interference microscopy. The transient phase produced by a femtosecond laser pulse on an oxidecoated silicon surface has a very low photoelectron yield and effectively suppresses evaporation of silicon atoms. Probably, it is a foam consisting of oxygen from disintegrated oxide mixed with liquid silicon. This foam settles to a blistered surface with partially removed oxide after ≈100 ns. If a femtosecond laser pulse of equal fluence is applied to a silicon surface having no oxide layer, a heavy ablation occurs. This leads to an electrical breakdown if the high voltage is not switched off within 50 ns. The response of a metal covered by a transparent oxide to a nanosecond laser pulse depends on the thermal stability of the oxide. Either the oxide is thermally destroyed or decomposed by the liquid metal, or the oxide is stable at the melting temperature of the metal, as in the case of cobalt oxide, CoO, on cobalt. Then the coating oxide may increase in thickness after a nanosecond laser pulse, which melts the metal, by gathering oxygen atoms originally dissolved in the crystal. These atoms are abundant in the liquid after the crystal is molten and segregate at the floating oxide as the melt freezes again. This scenario


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Figure 20. Photoelectron images showing the completely different response of cobalt to (a) a 10-ns and (b) a 100-fs laser pulse (fluence, ≈1 J/cm2). Exposure time was 5 ns. The moment of exposure was counted from the peak of the laser pulse. The arrow in a2 shows the fast shrinking zone with unimpeded photoelectron emission in the solidifying melt. The arrow in b3 shows the crystal defect produced by the 100-fs laser pulse (already visible in b2).

explains the decrease of photoelectron emission of nanosecond laser-treated cobalt during cooldown (Fig. 20a). The reduction of the photoelectron yield was not caused by desorption of adsorbed polar molecules (e.g., water), as adsorbed layers were removed by electron beam heating. A femtosecond pulse of a simular fluence as that of the chemically active nanosecond pulse typically produces dark lines within a crystal (Fig. 20b), which probably are slip lines, bundles of stacking faults, or grain boundaries. There is a transient increase of the electron emission during 20 ns after the laser pulse, where the linear crystal defect later appears. This emission occurs also without photostimulation. Melting does not occur within the laser spot as the crystals remain visible, so that the actual temperature is too low to account for the electron emission as thermal emission. A nanosecond laser pulse with a fluence high enough to melt the treated metal starts chemical reactions between the metal and a coating oxide. When the same metal is treated by a femtosecond pulse of equal fluence (additionally being below the threshold for ablation) it experiences plastic deformations, which

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proceed on the nanosecond time scale and are accompanied by emission of exoelectrons. 3. Limits Flash photoelectron microscopy is subject to the usual limitations of the resolution, which originate from lens aberrations and shot noise and are shared with other imaging techniques. However, there are additional constraints as the specimen is located in a high electric field. a. Limits of the Resolution The space–time resolution is restricted by the aberration of the uncorrected accelerating field at the specimen, by the space charge of the imaging electrons, and by their shot noise. Assuming all lenses except the cathode lens to be ideal (which is a good approximation), the spatial resolution xL is then that of the used two-electrode cathode lens, which is given by Möllenstedt and Lenz (1963) as x L = 1.2E/eF

(20)

where E and F are energy spread of the photoelectrons and electric field at the specimen, respectively. The space charge produced by the photoelectrons reduces the applied accelerating field and blurs the image. There exists no simple relation between the resolution and the electron current density. The actual blurring is considerably larger than that predicted by model calculations (Massey, 1983; Massey et al., 1981). In any case space charge effects can be neglected if the current density jp of the photoelectrons is less than the space charge–limited Child current density jCh by one order of magnitude: j p < jCh /10 = C F 3/2 /10a 1/2

(21)

with C = 2.34 × 10−6 A/V3/2 and a the spacing of the two accelerating electrodes of the cathode lens. The joint space–time resolution, limited by shot noise, is given by Eq. (10) with j replaced by the current density jp of the emitted photoelectrons: (x N )2 t > 18e/πεK 2 j p

(22)

If one combines Inequalities (21) and (22), the space–time resolution, limited by the combined action of shot noise and space charge, is found to obey (x N,s )2 t > 180ea 1/2 /πCεK 2 F 3/2

(23)

The spatial resolution is improved by reducing the distance a between the electrodes and by increasing the accelerating field F. The former is limited to


35

a > 3 mm to provide convenient access for the laser beams, whereas the electric field should not exceed a safe value of ≈10 kV/mm. When these limits are used and ε ≈ 0.1, K ≈ 0.2, t = 5 ns, and E ≈ 1.5 eV, a spatial resolution of xL + xN,s ≈ 0.8 μm is calculated. b. Limitation of the Laser Treatment In situ material processing by the laser is constrained by the requirement that thermal electron emission and evaporation should not interfere with photoelectron imaging. Heating by the treating laser pulse must be such that the current density jth of the thermal electrons stays below that of the photoelectrons jp; that is, jth = AT 2 exp(−W A /kT ) < j p < C F 3/2 /10a 1/2

(24)

Inequality (21) is applied and the Richardson–Dushman expression is used for the current density of the thermal electron emission with A < 120 A/cm2K2, k the Boltzmann constant, and T the absolute temperature. Inserting the values for electrode spacing (a = 5 mm) and electric field (F = 5 × 106 V/m), permits maximum allowed temperatures of 2400–2800 K to be calculated for metals with work functions in the range 3.6–4.5 eV. Pulsed photoelectron microscopy can be applied up to and even above the melting temperature of most materials without interference from thermionic emission, as was actually observed. The treating laser pulse also causes ablation of the specimen, and its fluence must be kept so low that formation of a laser-induced plasma is avoided. However, even if the laser pulse produces only neutral atoms, these are ionized by the thermal and photoinduced electrons which gain abundant energy in the accelerating field. These ions may cause troubling secondary electrons. Photoionization can be neglected as at least two photons of the used quantum energies must be absorbed for ionization of free atoms, and two-photon processes are very improbable at the restricted fluences. In fact, photoionization was not observed. The number of ions ni produced by electron collisions during the imaging time t is estimated to be n i = nσi ( j p + jth )t/e

(25)

with n and σ i the number of evaporated atoms (during the imaging time) and the ionization cross section averaged over the electron energies, respectively. The positive ions are accelerated toward the specimen, which is at a negative potential, and release ηni electrons (η, secondary electron yield). The number of these secondary electrons must stay below the number of the imaging photoelectrons. This requirement and Inequality (24), jth < jp, limit the allowed

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number na of evaporated atomic layers (during the imaging time) according to n a < d 2 /2σi η

(26)

2

where d is the area per atom within the processed surface. If relevant values are inserted (σ i ≈ 10−20 m2, d2 ≈ 6 × 10−20 m2, η ≈ 10), Inequality (26) requires that less than one third of a monolayer be evaporated during imaging, so that ion-induced secondary electrons can be neglected. The vapor pressure of most metals is so low up to several 100 K above the melting temperature that one atomic layer is evaporated during the imaging time of 5 ns. Accordingly, most metals can be pulse molten without disturbing photoelectron imaging, but adsorbates and oxides which decompose can be a problem in short exposure imaging. The reaction products contain excited molecules and atoms, which may liberate electrons from the contacting metal by an Auger process.

C. Pulsed High-Energy Reflection Electron Microscopy In high-energy reflection electron microscopy the surface of a bulk specimen is illuminated by a collimated electron beam at grazing incidence, and specularly scattered electrons are used to image the surface. Reflection electron microscopy was invented by Ruska (1933), who exploited electrons scattered by 90◦ , however. von Borries (1940) introduced a decisively improved technique, concerning chromatic aberration and image intensity, by using glancing incidence illumination and electrons scattered into low angles to image the surface. Reflection microscopy was abandoned with the advent of the scanning electron microscope. It was revived, however, in the early 1980s. The use of improved electron optics, on-axis dark-field imaging with Bragg-reflected “loss-less” electrons, drove the resolution to the atomic scale. Prominent applications have since been the imaging of reconstructing single crystal surfaces (Tanishiro et al., 1983), atomic steps (Cowley and Peng, 1985), structures of submonolayer deposits on silicon surfaces (Osakabe et al., 1980), and surface migration of atoms (Yamanoka and Yagi, 1989). A review of techniques and studies of surface structures and slow dynamic processes is given byYagi (1993). Despite its enormous potential as a surface probe, reflection microscopy based on Bragg diffraction is not very suitable for short exposure imaging of the surface. Usually only a small fraction of the electrons are passed by the objective lens aperture, and the image is buried beneath shot noise. Brightfield imaging with grazing incident and exit angles is more promising. A considerable disadvantage is the almost one-dimensional image of the surface. However, this technique is the only one that visualizes the space above the


37

Figure 21. Pulsed reflection electron microscope: 1, laser pulse–driven thermal electron gun; 2–5, as in Figure 1; 6, fiber plate transmission phosphor screen; 7, MCP image intensifier; 8, CCD sensor.

surface of a specimen that is at ground potential (in contrast to emission and mirror microscopes) so that massive evaporation and plasma formation are accessible to investigations. Figure 21 shows a reflection electron microscope for short exposure imaging of laser-induced processes (Bostanjoglo and Heinricht, 1990). The setup is

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similar to that of the transmission microscope in Figure 1, except for the electron illumination system which can be tilted against the specimen and some minor deviations. For reasons of intensity a laser-driven thermionic electron gun is used, which delivers only one, but intense, electron pulse. This allows to produce one shorttime exposure image with an exposure time of 20 ns. The bulk specimen can be rotated about an axis that is orthogonal to the electron and the treating laser beam. Incident and exit angles of the electrons are about 5◦ as measured against the surface. Because of these grazing angles the image of a geometrical structure on the surface is extremely shortened in the direction of the incident electrons. A laser-produced circular crater appears as a very slender ellipse. Any particle ejected from the laser-processed region has two images which appear symmetrically to the slender image of the eroded crater (Fig. 22). The two images are due to the absorption of incident and reflected (at the surface) electrons, respectively. The reflection microscope was used to visualize the evaporation of semiconductors and the ablation of metal films on semiconductors (Bostanjoglo and Heinricht, 1990; Heinricht and Bostanjoglo, 1992). Figure 23 shows, for example, the detachment of a gold film from a silicon wafer by a low-energy laser pulse which melts only the metal. The film was produced by evaporation on a silicon surface covered by native oxide and adsorbed molecules from the ambient atmosphere. As the gold film is molten by the laser pulse the adsorbed layers evaporate and lift the liquid film within 340 ns after the laser

Figure 22. Generation of the double image of a shadow-casting particle above a plane specimen in the reflection electron microscope. e−, illuminating electron beam.


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Figure 23. Short exposure reflection electron images showing the liftoff process of a laserpulsed 100-nm gold film on a silicon wafer. Exposure time was 20 ns. The moment of exposure was counted from the peak of the laser pulse and is given below the images. These were produced at neighboring, previously untreated regions with equal laser pulses with an energy and a fluence of 1.3 μJ and 0.6 MW/cm2, respectively.

pulse. About 300 ns later the liquid has separated from the wafer and contracted to a drop, which is driven back to the substrate by electrostatic forces (Fig. 24). Such processes occur whenever a light-absorbing coating produces a nonwetting liquid film on the substrate. Separation of the liquid may be due to true nonwetting, or to an isolating gas produced by desorbed molecules or to volatile products from a disintegrated oxide. Laser-based cleaning and restoration methods rely on these and similar ablation effects. The joint spatial (x) and time (t) resolution of the pulsed reflection microscopy is determined mainly by shot noise of the imaging electrons and is derived as for transmission microscopy. It is given by a relation identical in form to Inequality (10): (x)2 t > 18e/πεK 2 j

(27)

As before, j is the current density of the electrons illuminating the specimen for a time t, K the contrast between two distinguished adjacent areas with diameter x, and ε the fraction of electrons passing the aperture of the objective lens. The difference to the relation for bright-field transmission microscopy is in the physical meaning of the passed fraction of electrons, which in this case are all scattered electrons. Conversely, the fraction ε in Inequality (10) contains for not too thick films mostly unscattered electrons and is therefore much larger.

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Figure 24. Scanning electron image of the rest of the gold film after the liftoff process shown in Figure 23.

Assuming values of the parameters typical for the assembled pulsed reflection microscope ( j ≈ 80 A/cm2, ε ≈ 10−3, t ≈ 20 ns) and choosing as a specimen an opaque shadow-casting particle on an ideally flat surface (i.e., K = 1) yields a spatial resolution perpendicular to the electron beam of x ≈ 0.3 μm. This is in the order of what actually was achieved. The resolution along the direction of the electron beam is x/sin α, with α ≈ 5◦ the angle which the illuminating electron beam makes with the imaged surface.

D. Pulsed Mirror Electron Microscopy In the mirror microscope the specimen is biased slightly negative with respect to the electron gun. Accordingly, the incoming electrons are reflected by a near-surface equipotential plane. As the latter is a replica of the geometrical and electrical roughness of the specimen surface, the reflected electrons carry information on surface morphology and local electric fields. These may be due to contact potentials, spontaneous electric polarization, p–n junctions, or


41

nonuniform electric conductivity. Furthermore, magnetic stray fields, such as from ferromagnetic domains, influence the trajectories of nonaxial electrons and can also be imaged. Two inconveniences are associated with the mirror microscope. First, the specimen is constrained by the fact that it is at high potential and therefore an integral part of the electron optics. Second, illumination and magnification cannot be chosen independently if the relevant electron beams are not separated. However, there also are merits to using this type of microscope. Since the electrons travel slowly near their point of inflection, they are very susceptible to lateral and axial electric fields, they have a high depth resolution, and they are effectively scattered by gases, which are emitted from the surface. Finally, this type of microscope has the unique property that the electrons can probe the specimen without touching it. The theory of electron mirrors and associated devices is covered by Rempfer and Griffith (1992) and by Hawkes and Kasper (1996). The design of mirror microscopes and applications to stationary specimen, slowly varying and periodic processes are described with numerous references by Bethge and Heydenreich (1987). Previous mirror microscopes were not suited for studying fast nonrepetitive processes. Since the mirror lens requires illuminating beams with a small divergence the electron gun must emit high-current pulses and have a high brightness. Figure 25 shows a mirror microscope which allows short exposure imaging of fast nonrepetitive processes on laser-treated surfaces (Kleinschmidt and Bostanjoglo, 2000). The setup has components in common with the transmission (Fig. 1) and photoelectron (Fig. 15) microscopes. The specimen stage of a transmission microscope was replaced by an electromagnetic prism, which bends the trajectories of the illuminating and reflected electrons by 90◦ , and an electron mirror. The specimen is the decelerating electrode of the mirror which may be biased negatively against the electron gun. As in the flash photoelectron microscope the mirror is a two-electrode lens in order to minimize accumulation of space charge near the specimen when high-current electron pulses are used. A beam blanker passes electrons to the detector for 8 ns only during the illumination in order to minimize blurring of the image by thermal and Auger electrons emitted by the laser-treated material. A 90◦ prism was chosen to separate the electron trajectories as this design allows one to treat the specimen with an expanded laser beam, which is focused to a spot of 20 μm in diameter. Very high thermal gradients of about 108 K/m can be produced and the material subjected to unusual chemical and mechanical processes. For cleaning purposes the specimen is heated by electron bombardment from the back side. Figure 26 demonstrates the sensitivity of the mirror microscope to space– time variations of contact potentials (i.e., work functions of the contacting

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Figure 25. Pulsed mirror electron microscope with attached laser for treating the specimen: 1, laser pulse–driven photoelectron gun; 2, magnetic prism; 3, pulse laser for treating the specimen (25 ns, FWHM); 4, electron mirror; 5, beam blanker; 6, fiber plate transmission phosphor screen; 7, MCP image intensifier; 8, CCD sensor.

materials) and for gases evaporating from a heated surface. The series shows the response of a (100) silicon surface, passivated by a monolayer of atomic hydrogen (Miyata et al., 1998; Yablonovitch et al., 1986), to a laser pulse. During and shortly after the laser pulse the treated region is obscured by a cloud of evaporated hydrogen, which expands into the microscope vacuum with a velocity of about 1000 m/s. As the cloud clears, the cleaned region emerges as a dark patch with a bright rim about 100 ns after the peak of the treating laser pulse. The contrast reverses a few seconds after the laser pulse, showing the treated area as a bright spot, and gradually disappears during several hours’ exposure of the surface to the microscope vacuum. No geometrical modification of the treated surface could be detected by scanning electron and light interference microscopy. This variation of contrast can be explained as follows. The cleaned region has a lower work function than that of the passivated surface, as was directly observed in the flash photoelectron microscope. Consequently, the cleaned surface is more positive than the passivated periphery (Babout et al., 1977). The associated localized drop of the decelerating field immediately above the treated region represents a convex microlens having a much smaller focal


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Figure 26. Mirror electron images of a Si crystal passivated with hydrogen. The series shows the evaporation of the hydrogen monolayer (0–50 ns) after a heating laser pulse (25 ns, ≈6 μJ, 20 μm ø) and adsorption of a monolayer of gas molecules from the microscope vacuum (5–18 s). The exposure time was 5 ns. The pictures were taken at the indicated times, counted from the peak of the treating laser pulse. They were produced at fresh neighboring regions being equally treated.

length than that of the macroscopic concave mirror lens (Orthuber, 1948). Therefore the microlens is expected to produce a reduced intensity in the projected image. As gas molecules from the microscope vacuum are absorbed, the work function of the treated area and the focal length of the associated microlens increase. When the focusing of the microlens compensates the defocusing of the mirror lens the contrast disappears. This occurs a few seconds after the laser pulse, which is just the time it takes the clean area to be covered by a monolayer of gas molecules adsorbed from the vacuum of the used microscope (5 × 10−6 mbar). As the adsorption of the gas molecules continues, the work function, and accordingly the focal length of the microlens, further grows, reaching a stage where the reflected electrons are focused on the detector plane giving the transient bright spot. Finally, the adsorption saturates and the work function approaches the value of the untreated periphery, so that the contrast disappears permanently. As with the other time-resolving microscopes, the joint space–time resolution of the pulsed mirror microscope is determined by the shot noise of the imaging electrons. As a result of the projection-type imaging by the twoelectrode mirror, the divergence of the illuminating electron beam imposes a second restriction. The resolution is derived as follows. Two regions of diameter d in a homogeneous specimen, which is the reflecting potential plane, are considered. They are judged as equally bright in successive exposures of time t, if the mean number n¯ of electrons, which

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each region reflects during an exposure, exceeds the root-mean-square of the √ shot noise amplitude n¯ by a minimum signal-to-noise ratio r: n¯ √ >r n¯

(28)

Since n¯ = (π/4e)d 2j t, with e and j the electron charge and mean current density at the reflecting plane, respectively, and since the latter can be expressed by the half angle α of the illuminating beam and by the brightness R of the used electron gun as j = Rπ α 2, Eq. (28) gives (π 2/4e)Rα 2 td 2 > r 2

(29)

The two circular regions are clearly separated, if the distance of their centers x, which denotes the spatial resolution, exceeds their radius d/2. This gives as a preliminary result (π 2/e)Rα 2 t(x)2 > r 2

(30)

Now, a nonparallel electron beam with half-angle α entering the mirror produces a disk as the projected image of an object point. This aberration is easily determined, to a good approximation for electrons paraxially entering the mirror. The latter may be replaced by a thin concave lens, produced by the field step at the mirror anode and a homogeneous electric field (e.g., Rempfer and Griffith, 1992). This field decelerates the incoming electrons and accelerates the reflected electrons, respectively. Two object points, that is, two points on the reflecting plane, with a distance x have projected images spaced by (2L/f ) x, where f is the focal length of the concave lens and L is the distance of the image plane from the mirror anode (L being much larger than f and the spacing of the two mirror electrodes). The image of a point is found to be a disk with a diameter of 2Lα. Accordingly, two object points can be distinguished only if (2L/f ) x > 2Lα, or if their spacing obeys x > f α

(31)

Eliminating α with this relation in Eq. (30) gives for the joint space–time resolution (x)4 t >

r 2e f 2 π2R

(32)

With the relevant values f ≈ 16 mm and R ≈ 7 × 106 A/cm2 · sr and assuming a minimum signal-to-noise ratio of r ≈ 5, a spatial resolution of x ≈ 0.7 μm is computed during the used exposure time of t = 5 ns, which is in the order of what was achieved.


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The diameter xC of the chromatic aberration disk is computed in a similar way to Relation (31) as E (33) E where x, a, E, and E are the distance of the object point from the optical axis, the spacing of the mirror electrodes, the kinetic energy of the illuminating electrons, and the kinetic energy spread. For typical values x < 300 μm, a ≈ 4 mm, α ≈ 10−3 rad, E ≈ 20 keV, and E < 1 eV, the chromatic aberration is xC < 15 nm and therefore negligible as compared with the resolution limit due to shot noise in nanoseconds exposure images. xC ≈ (x + 2aα)

IV. Conclusions Electron microscopy is an indispensable method for characterization and analysis of materials down to the atomic scale. A very useful application is the in situ investigation, which allows imaging of the dynamics of miscellaneous processes. The time scale of four types of electron microscopes was pushed down to a few nanoseconds for nonrepetitive processes by implementing a high-current laser pulse–driven thermal- and photoelectron gun, fast electron beam shifting, and electronic image registration. The extended electron microscopes were of the transmission, photoemission, and reflection and mirror types, which give access to the volume of the specimen, its surface, and the space above the surface, respectively. Three complementary high-speed techniques were realized: multiframe imaging, streak imaging, and image intensity tracking. The potential of the new time-resolving probes was demonstrated by tracing fast laser-triggered effects as phase transitions, melt instabilities, chemical reactions, and mechanical deformations. Melt flow driven by large thermal and compositional gradients, evaporation of superheated liquid metals, and decomposition and precipitation of oxide surface layers were investigated. High-speed electron microscopy has uncovered effects to which conventional methods, based on light optics, have no easy access. Femtosecond laser pulses, depositing their energy in the electronic system, which then destabilizes the atomic lattice, were found to produce extraordinary effects on a “thermodynamic” (nanosecond) time scale. These effects were completely different from those initiated on the same time scale by the exclusively “thermal” nanosecond laser pulses. Modeling the dynamics, visualized by transmission microscopy, with computer-based numerical simulations allows one to extract material parameters

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relevant at temperatures up to the critical point, at thermal gradients up to several 103 K /μm, and at stresses up to the theoretical yield point. Photoelectron and mirror microscopy were found to be well suited to uncover on the nanosecond time scale early stages of material modifications, such as removal or addition of monolayers, where rival high-speed light-optical methods fail because of lacking contrast. The resolution of the described highspeed microscopes is currently limited by shot noise in the electron image to several hundred nanometers and a few nanoseconds for nonrepetitive processes. A higher space–time resolution of the photoelectron microscope can be reached only if the buildup of negative space charge at the electron emitters is reduced. Brighter electron guns would improve the resolution of transmission, reflection, and mirror microscopy. They can possibly be realized by locally increasing the electric field with suitably corrugated emitters, or perhaps by exploiting the very high electric fields of ultrashort laser pulses in completely new designs. Adverse space charge effects at the surface of specimens in the photoelectron microscope could be overcome by pulsing the accelerating voltage. Voltage levels significantly exceeding the presently used safe dc value could be applied during the short imaging time without causing an electric breakdown. So that blurring due to the inevitable oscillations in the voltage pulse at the cathode is avoided, the emission microscope must be all electrostatic and the voltage of all lenses must be derived from the cathode voltage by fast capacitive/resistive dividers.

Acknowledgments Sincere thanks are due to F. Rohn-Schwarz, H. Dömer, H. Kleinschmidt, T. Nink, and M. Weingärtner for helping to produce this article. The high-speed research was generously supported by the Deutsche Forschungsgemeinschaft and by the Alexander von Humboldt Stiftung.

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Applications of Transmission Electron Microscopy in Mineralogy P. E. CHAMPNESS Department of Earth Sciences, University of Manchester, Manchester M13 9PL, United Kingdom

I. Introduction . . . . . . . . . . . . . . . II. Analytical Electron Microscopy of Minerals . . III. Phase Separation (Exsolution) . . . . . . . . A. Alkali Feldspars . . . . . . . . . . . . B. Amphiboles . . . . . . . . . . . . . . 1. Exsolution in Monoclinic Amphiboles . . 2. Exsolution in Orthorhombic Amphiboles . IV. HRTEM and Defect Structures . . . . . . . A. Biopyriboles and Polysomatic Defects . . . 1. New Biopyriboles . . . . . . . . . . 2. Chain-Width Disorder in Pyriboles . . . 3. Polysomatic Reactions in Pyriboles . . . V. Concluding Remark . . . . . . . . . . . . References . . . . . . . . . . . . . . . .

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53 55 59 60 68 71 77 81 81 82 83 84 87 87

I. Introduction Although transmission electron microscopy (TEM) became a routine tool for the physical metallurgist in the 1960s and the theory of image formation from crystalline materials was well established by then, it was not until the 1970s that the TEM was adopted to any great extent by workers in the earth sciences. The main reason for the long delay was that there was no reliable method for preparing thin foils of nonmetallic materials; studies were restricted to cleavage fragments of layered structures or to powdered fragments sedimented onto carbon films. The latter technique allowed examination of only microstructural features smaller than about 1 μm, and spatially related information on a larger scale than this was largely lost. The advent of reliable, commercial, beam-thinning devices in the early 1970s solved the problem of specimen preparation. Foils in which hundreds of square microns are transparent to the electron beam can now be prepared almost routinely. Disks 3 mm in diameter can be drilled from petrographic thin sections that are approximately 25 μm thick and thinned with a beam of energetic ions or atoms (usually argon) until perforation. The thin sections 53 Volume 121 ISBN 0-12-014763-7


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Figure 1. Partial projections of the linkages of the Si–O tetrahedra in pyroxenes (singlechain silicates), amphiboles (double-chain silicates), micas (sheet silicates), and feldspars (framework silicates).

can be studied beforehand in the petrographic optical microscope, the scanning electron microscope (SEM), or the electron-microprobe analyzer (EMPA), and regions of interest for TEM study can be chosen. At almost the same time that beam-thinning machines came on the market, the first moon-rock samples started arriving on Earth as a result of the Apollo space missions. For a time in the early 1970s, more moon-rock samples had been studied in the TEM than terrestrial samples. My first ion-thinned mineral specimen was a pyroxene (a single-chain silicate, Fig. 1) from Apollo 11. Since those days, the TEM has become an integral part of much of mineralogical research. In this review I highlight just a few examples that illustrate the impact that TEM has had in mineralogy in the last 25 years. I have chosen to concentrate on two of the commonest silicate groups—the alkali (Na–K) feldspars (framework aluminosilicates, Fig. 1) and the amphiboles (double-chain silicates, Fig. 1)— although I also describe the important contribution that high-resolution transmission electron microscopy (HRTEM) has made to our understanding of mixed-chain structures.

APPLICATIONS OF TEM IN MINERALOGY

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II. Analytical Electron Microscopy of Minerals The advent of X-ray analysis in the TEM has allowed us to identify fine-scale mineral phases that would have been impossible or extremely tedious to identify by electron diffraction, given the large unit cells, complex chemistry, and low symmetries that are involved in most cases. As will be seen in Section III.B, investigations of phase separation (or exsolution) in the amphibole group have relied very heavily on analytical electron microscopy (AEM), so it is worth outlining here some of the procedures that need to be adopted in the AEM of minerals and some of the precautions that need to be taken. The basis of quantification of mineral analyses is the thin-film criterion of Cliff and Lorimer (1975) in which X-ray absorption and secondary X-ray fluorescence are assumed to be negligible to a first approximation and the ratio of the concentrations of two elements C A /C B is related to the ratio of their measured X-ray intensities I A /I B by the equation IA CA = k AB CB IB

(1)

where k AB is a sensitivity factor that accounts for the relative efficiency of production and detection of the X-rays. For silicates, the reference element, B, is silicon. Because silicates are composed predominantly of oxygen and specific gravities are normally between 2.5 and 3.5, the thickness, tmax, at which Eq. (1) breaks down and corrections for absorption and fluorescence must be made is larger than that for metallic systems. Nord (1982) calculated the value of tmax for Mg/Si, Ca/Si, and Fe/Si for members of the pyroxene quadrilateral. Figure 2 shows a compilation of the minimum value of tmax for all three elemental ratios and indicates that analysis must be carried out in areas that are less than 130–300 nm thick, depending on the bulk composition, if absorption effects are to be insignificant. As it happens, the maximum thickness for which microstructures in silicates can be observed when 100-kV electrons are used is about 200 nm, so if microscopy can be carried out in an area of the foil at ∼100 kV, it can be assumed that the foil fulfills the thin-film criterion for elements Z ≥ 11. For higher voltages or lighter elements, this rule of thumb cannot be used and care must be taken to work in suitably thin areas, or, alternatively, to correct for absorption. Silicates are composed predominantly of oxygen, which cannot be reliably quantified by AEM, even with detectors with ultrathin windows. The method adopted for quantification of anhydrous silicate (or other oxide) phases is to assume that all cations are present as oxides and that the sum of the oxides is 100%. The chemical formula is then recalculated to a suitable number of oxygens: for example, six in the case of the pyroxene (a single-chain silicate,

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Figure 2. Maximum thickness (in nanometers) of Ca–Mg–Fe pyroxenes for which absorption corrections can be ignored. (Source: After Nord, 1982.)

Fig. 1) in Table 1 because the general formula for pyroxene is M2Si2O6, where M stands for cations other than silicon. Problems obviously arise in the case of cations such as Fe that can take a number of valences and when there are elements other than oxygen that cannot be detected. The most common of these is hydrogen, as many silicates are hydrated (Fig. 1). For hydrated samples, if all other cations can be detected, a total can be assumed for the oxide analysis that is appropriate to the mineral type (e.g., 95 wt % for the sheet silicate mica, Fig. 1) or the formula can be normalized to an appropriate number of oxygens [22 for micas, Table 1, as the general formula for mica is X2Y4–6 Z8 O20(OH)4, where X and Y are nontetrahedral cations and Z is Si or Al]. In the general case it is recommended that, when possible, normalization be carried out on the basis of the known number of cations in a particular crystallographic site (Peacor 1992). For instance, in Table 1 the tetrahedral sites in pyroxene and mica have been assigned 2 and 8 (Si + Al), respectively. For the mica, the cations known to occupy the X and Y crystallographic sites have been grouped to give totals of 1.78 and 5.71, respectively. A more complex assignment of cations to particular sites will be encountered in Section III.B, where the amphibole group is considered in some detail. Perhaps the severest problem encountered in the AEM of silicates is that of specimen damage during analysis. Silicates are known to suffer from radiolysis (i.e., electronic excitation leading to atomic displacement) during electron

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APPLICATIONS OF TEM IN MINERALOGY TABLE 1 AEM Analyses of Two Silicatesa Pyroxene

Mica (biotite)

1 SiO2

48.01

Al2O3

4.88

TiO2 FeO MnO MgO Na2O CaO K2 O

0.00 29.91 1.89 15.31 0.00 0.00 0.00

Total

100.00

2

3

Si

1.87

AlIV AlVI Ti Fe2+ Mn Mg Na Ca K

0.13 ⎫ 0.09 ⎪ ⎪ ⎪ 0.00 ⎪ ⎪ ⎪ ⎪ 0.97 ⎪ ⎪ ⎬ 0.06 0.89 ⎪ ⎪ ⎪ ⎪ 0.00 ⎪ ⎪ ⎪ ⎪ 0.00 ⎪ ⎭ 0.00

O

6

2.00

2.01

4

38.13

Si

5.31

23.20

AlIV AlVI Ti Fe2+ Mn Mg Na Ca K

2.69 ⎫ 1.12 ⎪ ⎪ 0.16 ⎪ ⎬ 1.60 ⎪ ⎪ 0.00 ⎪ ⎭ 2.83

0.16 0.00 1.62

O

22

1.58 13.75 0.00 13.58 0.62 0.00 9.14

8.00

5.71

1.78

100.00

Source: Champness (1995); reproduced by permission of Chapman & Hall. The oxide weight percentages in columns 1 and 3 were derived assuming a total of 100%. The atomic formulas in columns 2 and 4 were calculated assuming a total of 6 and 22 oxygens and a total of 2 and 8 (Si + Al) for the pyroxene and the biotite, respectively. All iron has been assumed to be Fe2+. a

irradiation. The high current densities used for high-resolution AEM can lead to significant structural and chemical changes which ultimately limit the accuracy of analyses. The degree of sensitivity to damage depends on a number of factors, among which are the type of linkage of the Si–O tetrahedra, the nature of the cations (Na and K being the most vulnerable to loss), and the presence or absence of hydroxyl ions (Champness and Devenish, 1992; Hobbs, 1984; Veblen and Buseck, 1983). Champness and Devenish (1992) and Devenish and Champness (1993) have shown that all silicates experience some mass loss at the highest current densities used in AEM, but that there is a threshold of the current density for each element in a particular structure for which no loss occurs. For instance, the threshold values of the current density for which no loss occurs for any element is ≈105 A/m2 for calcic pyroxene (diopside) and about 3 × 104 A/m2 for calcic mica (margarite). Notice that both these values are lower than the current density in a focused beam from a LaB6 gun. At the highest current densities available [i.e., those obtainable with a field emission gun (FEG) plagioclase (Na–Ca) feldspar is reduced to the composition of SiO2 after 200 s (Fig. 3).

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Figure 3. Energy-dispersive X-ray spectra from plagioclase feldspar: (a) defocused beam rastered over specimen for 200 s; (b) beam focused at an approximate current density of 1.8 × 108 A/m2 in a dedicated scanning transmission electron microscope (STEM). The accelerating voltage was 100 kV.

It is clearly important that, when possible, the analyst operate below the current density at which damage occurs if quantitative results are required. Because of the dependence of the rate of damage on the current density, rather than on the total dose, defocusing the electron beam is more effective in minimizing mass loss than is rastering a focused beam over the same area. The effect of mass loss may also be minimized by using the highest voltage available (Fig. 4a) and by using a cooling stage (Fig. 4b).

Figure 4. Semilog plot for the loss of Na from plagioclase feldspar at a current density of 1.8 × 103 A/m2: (a) dependence on voltage; (b) dependence on temperature. (Source: After Devenish and Champness, 1993; reproduced by permission of the Institute of Physics.)


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III. Phase Separation (Exsolution) It is in the field of phase transformations that TEM has probably had the widest influence in mineralogy. It had long been known from the study of petrographic thin sections in the polarizing microscope that phase separation (or exsolution) is common in the pyroxenes, amphiboles, and feldspars from slowly cooled rocks such as large igneous intrusions. In the 1950s and 1960s, studies by single-crystal X-ray diffraction (XRD) (e.g., Bown and Gay, 1959; Smith and MacKenzie, 1955) were able to indicate the lattice orientations of these intergrowths and to show that exsolution was present in many minerals from more quickly cooled rocks, although the intergrowth was below the resolution of the light-optical microscope. XRD could not, however, give any indication of the mechanisms of exsolution, nor, in general, of the size of the precipitates or the orientation of their interfaces. These areas are where TEM has come into its own. During the early days of the investigation of exsolution in silicates by TEM, it became apparent that two mechanisms that are extremely rare in metallic systems are very common in silicates: spinodal decomposition (the gradual evolution of sinusoidal compositional waves, without a nucleation stage) and homogeneous nucleation and growth of the equilibrium phase (nucleation without the aid of structural defects). The reasons for this difference in behavior between metals and silicates lies in the fact that whereas the crystal structures of the matrix and equilibrium product phases are different in metallic systems, in most cases the structures of the two silicate phases involved in exsolution are identical (Aaronson et al., 1974). Added to this, in silicate systems the equilibrium solubility at high temperatures is relatively small and the volume change involved in the transformation is small. These factors result in the depression of the coherent spinodal below the equilibrium solvus being small enough that relatively rapid diffusion can take place when the temperature drops below that of the coherent spinodal. The factors that favor spinodal decomposition also favor homogeneous nucleation, although homogeneous nucleation is the more difficult process. However, because the equilibrium phases in silicates usually have a common structure of Si–O tetrahedra and only second, third, or even higher nearest neighbors need be in the “wrong” positions across the interphase interface, the chemical interfacial energy term is small. In addition, the appreciable decrease in solubility with temperature that occurs in silicates provides a high driving force for nucleation and growth. Nevertheless, the cooling rate needs to be extremely slow, as it is in many plutonic and metamorphic rocks, for homogeneous nucleation to occur before the coherent spinodal is reached.

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My examples of exsolution come from the alkali (Na–K) feldspars and the amphiboles and nicely illustrate the diversity of microstructures in the mineral kingdom. They also provide some very spectacular textures.

A. Alkali Feldspars The feldspars are the commonest silicates in the earth’s crust, making up some 54%. They largely belong to the ternary system NaAlSi3O8 (albite)–KAlSi3O8 (orthoclase)–CaAl2Si2O8 (anorthite), the NaAlSi3O8–KAlSi3O8 series being known as the alkali feldspars and the NaAlSi3O8–CaAl2Si2O8 being known as the plagioclase feldspars. The alkali feldspars show (almost) complete solid solution at temperatures above 660◦ C, but there is a solvus at lower temperatures which extends to almost pure albite and orthoclase at low temperatures (Fig. 5). For most of the

Figure 5. Simplified subsolidus phase diagram for the alkali feldspar binary NaAlSi3O8 (albite)–KAlSi3O8 (orthoclase) at 1 kbar as calculated by Robin (1974). The dashed line is the coherent solvus and the dotted line is the coherent spinodal. (Source: Champness and Lorimer, 1976; reproduced by permission of Springer-Verlag.)


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composition range, the alkali feldspars are monoclinic C2/m above the solvus, but both end members undergo a transition to triclinic C 1¯ ∗ symmetry at lower temperatures. For the sodic phase the transition is the result of distortion of the Si/Al–O framework and is rapid (it is classed as displacive by mineralogists and may well be martensitic), whereas the transition in the potassic phase is slow because it involves Si/Al ordering. The alkali feldspars show coarser precipitation structures (called perthites) than those of any other silicates; lamellae can be several millimeters wide in plutonic (slowly cooled) rocks. This fact can be attributed to the relatively high diffusivities of K and Na ions within the Si/Al–O framework and the fact that, unlike the plagioclase feldspars, precipitation does not require diffusion of the Si and Al. McConnell (1969) was the first to examine the microstructure of a volcanic alkali feldspar (compositional, 36% K-feldspar) in the TEM. He showed that it consisted of coherent compositional modulations with a wavelength of about 10 nm approximately parallel to (100). The diffraction pattern showed a single reciprocal lattice with strong streaks approximately parallel to a.∗ This was the first direct evidence that spinodal decomposition is an important mechanism of phase transformation in the alkali feldspars, as had first been suggested by Christie (1968). Since then, natural samples have been homogenized and heat treated to reproduce the modulated structures (Fig. 6; Owen and McConnell, 1971; Yund et al., 1974). Owen and McConnell were able to show that the wavelength of the modulation was characteristic of the annealing temperature and was larger for higher temperatures, as predicted by spinodal theory. Yund et al. (1974) annealed an initially homogeneous alkali feldspar for several days at 600◦ C and found that the modulations eventually developed into two separate ¯ lamellar phases approximately parallel to (601). Calculations by Willaime and Brown (1974) of the elastic energy at the boundary between two alkali feldspars where both are monoclinic, or where the Na-feldspar is average monoclinic due ¯ to periodic twinning, showed that a minimum occurs at approximately (601). Hence the orientation of the interphase boundary is determined predominantly by minimization of elastic strain. The chemical component of the interphase boundary energy is much less important because the Si/Al–O framework is unchanged across the interface. Although exsolution textures that can be attributed to nucleation and growth (including homogeneous nucleation) have been identified in natural alkali feldspars (e.g., Brown and Parsons, 1988; Snow and Yund, 1988), the interdiffusion of Na and K is too slow to allow nucleation of exsolution lamellae to occur in alkali feldspars in the laboratory. To circumvent this problem, Kusatz ∗ The nonstandard space group is used so that the monoclinic and triclinic phases have the same unit cells.

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Figure 6. Natural alkali feldspar (36 mol % K-feldspar) that has been homogenized and annealed at 540◦ C for 48 h at 1 kbar to produce a modulated structure approximately parallel ¯ to (601). Inset is an enlargement of a diffraction spot that shows satellites in a direction perpendicular to the modulations. (Sources: Owen and McConnell, 1971; reproduced by permission of Nature.)

et al. (1987) carried out exsolution experiments on alkali feldspars in which some of the Si had been substituted with Ge∗ to give compositions along the binary NaAlGe2.1Si0.9O8–KAlGe2.1Si0.9O8. This substitution causes the incoherent and coherent solvi to rise (to almost 900◦ C for the critical composition of the incoherent solvus), the solidus to be depressed, and the displacive transformation to move toward the K-rich side of the phase diagram. Kusatz et al. (1987) found two types of textures in their experiments. Short, widely spaced, lens-shaped lamellae were produced between the incoherent solvus and the coherent spinodal and were ascribed to nucleation and growth, whereas thin, closely spaced, and branching lamellae formed only in the central part of the solvus and were ascribed to spinodal decomposition. ∗ This is a trick that mineralogists often employ. For instance, Ge has been substituted for Si in olivine, Mg2SiO4, so that the olivine → spinel transition that occurs at a depth of about 400 km in the earth can be studied in the laboratory (e.g., Rubie and Champness, 1987). The transition occurs at a lower pressure in the germanate because Ge has a smaller ionic radius than that of Si.

63


In a detailed study of the coarsening of spinodal textures in alkali feldspars, Yund and Davidson (1978) found that the lamellar spacing could be described as being proportional to the cube root of the annealing time at constant temperature by the relation λ = λ0 + kt 1/3

(2)

where λ0 is the spacing at zero time and k is a rate constant for each temperature. An Arrhenius plot of the natural logarithm of k against 1/T , where T is the temperature, showed a linear relationship within experimental error. However, as Yund and Davidson (1978) acknowledged, the t 1/3 law applies to the coarsening of spherical particles and is not appropriate to the coarsening of lamellae. Brady (1987) proposed that the principal mechanism for coarsening in this case is diffusional exchange between the wedge-shaped terminations of exsolution lamellae as seen in the TEM (Fig. 6) and the large, flat sides of adjacent lamellae. Having derived a formula for the chemical potential gradient due to interfacial energy effects, Brady extended the work of Cline (1971) on the coarsening and stability of lamellar eutectics, to show that the appropriate rate law for lamellar coarsening in silicates is given by λ2 = λ20 + kt

(3) 2

Brady replotted Yund and Davidson’s (1978) data on a graph of λ versus t (Fig. 7) and found an excellent fit which gave an activation energy for

Figure 7. Plot of λ2 versus time, t, for the coarsening experiments of Yund and Davidson (1978) on alkali feldspars. λ is the lamellar wavelength. (Source: Brady, 1987; reproduced by permission of the Mineralogical Society of America.)

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coarsening of 33 kcal/mol. Further evidence for the correctness of Brady’s model was provided by the fact that the values of λ0 , the lamellar wavelength at the beginning of coarsening, as derived from the graphs, increased systematically with temperature, as predicted by the theory of spinodal decomposition. The λ0 values obtained by Yund and Davidson from the t 1/3 rate law did not increase in this way. Equations (2) and (3) give different values of predicted lamellar wavelengths for long coarsening times (a difference of more than an order of magnitude for coarsening for 106 years at 500◦ C) but give comparable results for rapidly cooled rocks ( Brady, 1987). However, attempts to determine the cooling history of relatively quickly cooled rocks from the spacing of the lamellae have met with mixed success. There was good agreement between the lamellar spacings observed in a 5.2-m-wide dike and those predicted from heat-flow calculations and Eq. (2) (Christoffersen and Schedl, 1980), but less good agreement for lamellar spacings in a lava flow (Yund and Chapple, 1980) and in a large rhyolitic ash flow (Snow and Yund, 1988). It is also apparent that Si/Al ordering and twinning inhibit coarsening in more slowly cooled rocks (Brown et al., 1983). In some more slowly cooled alkali feldspars, the two-phase lamellar intergrowths have coarsened to the scale of visible light, with the consequence that the scattering of light from their regular interfaces produces iridescence. It was a TEM study by Lorimer and Champness (1973) of two gem-quality varieties of these feldspars, known as moonstones, that led to an understanding of the later stages of coarsening. Fleet and Ribbe (1963) were the first to examine a moonstone in the TEM, using crushed grains. They showed that it contained coherent, lamellar precipitates of triclinic Na-feldspar and mono¯ clinic K-feldspar approximately parallel to (601), the plane of iridescence. The Na-feldspar contained regularly spaced Albite twins,∗ as had been predicted by Laves (1952) from the presence of superlattice reflections parallel to b∗ in X-ray diffraction patterns. (The regularity of the twins, Laves suggested, reduces the strain energy of the interface between the two phases, a suggestion that was subsequently verified from calculations of the strain energy by Willaime and Gandais, 1972.) Lorimer and Champness’s samples had similar compositions (57.3 and 53.7 wt % K-feldspar) but showed markedly different phase distributions. The first sample, which exhibits a blue iridescence, was shown to contain wavy lamellae of regularly Albite-twinned Na-feldspar approximately parallel to ¯ (601), together with apparently monoclinic K-feldspar (Fig. 8a). The other moonstone, which shows a white iridescence, has a coarser microstructure ∗ Albite twins arise during the triclinic → monoclinic transition in Na-feldspar. They are normal twins with (010) as the twin and composition plane.


65

Figure 8. Microstructure of two moonstones: (a) feldspar with bulk composition 57.3 wt % K-feldspar contains wavy lamellae of regularly Albite-twinned Na-feldspar approximately paral¯ lel to (601); (b) feldspar with bulk composition 53.7 wt % K-feldspar has a coarser microstructure ¯ with lozenge-shaped particles of Na-feldspar with boundaries approximately parallel to (6¯ 61) ¯ and smaller, zigzag lamellae parallel to approximately (601). (Source: Lorimer and Champness, 1973; reproduced by permission of Philosophical Magazine.)

containing discrete lozenge-shaped particles of regularly twinned Na-feldspar ¯ (Fig. 8b). Significantly, this with boundaries approximately parallel to (6¯ 61) sample also contained zigzag lamellae of Na-feldspar that were smaller in size than the lozenge-shaped particles and therefore must have predated them. Detailed investigation of the K-feldspar showed that it was triclinic and mostly twinned on the diagonal association (basically Albite-twinned, but slightly deformed). The preceding observations suggest the sequence shown in Figure 9 for the evolution of the microstructure in the coarser moonstone. After coarsening of the spinodal modulations has produced distinct lamellae approximately ¯ parallel to (601), the Na-feldspar becomes triclinic and twins on the Albite law. The periodic twinning relieves the strain at the interphase interface and the Na-feldspar remains monoclinic, on average. As the K-feldspar becomes ¯ triclinic, however, the lowest-energy interface becomes approximately (6¯ 61) (as shown in calculations by Willaime and Brown, 1974) and the interface

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Figure 9. Sequence of evolution of the microstructure in the moonstones in Figure 8 (Source: Putnis, 1992; reproduced by permission of Cambridge University Press.)

gradually changes during the coarsening process, producing, first, wavy lamellae and, later, discrete, lozenge-shaped particles. Examination of the phase distribution in the coarser of the two samples examined by Lorimer and Champness (1973) shows that rafting of the Narich particles has taken place (Fig. 8b) as a result of interaction of their strain fields during coarsening. This phenomenon has been reported in metallic systems (Ardell et al., 1966). Although the presence of a fluid phase is known not to have an affect on lattice diffusion (Yund, 1983) or on the coarsening of coherent lamellae in alkali feldspars (Yund and Davidson, 1978), it has a dramatic effect on the coarsening of alkali feldspar intergrowths as coherency is lost. Almost all plutonic, igneous rocks are affected to a greater or lesser extent by water derived from the magma (deuteric alteration) at temperatures 3 keV). The design of the test structure was dictated by the need to secure, align and electrically insulate the source from an extractor electrode. As recommended by the manufacturer, the source is placed 500 μm from an extractor electrode containing a commercially available 500-μm-diameter Pt-Ir aperture. We chose to machine a bulky stainless-steel extractor electrode fit with a commercially available aperture for the purpose of absorbing most of the emitted electrons from the source. The source and extractor are placed 1 cm before the silicon lens. The test assembly consists of alternating stainless steel and Macor (a machinable glass ceramic made by Corning Inc.) rings. From the bottom up the structure consists of a Faraday cup to collect electrons; a sample holder designed to house a commercial 3-mm gold grid; a parallel plate deflector assembly, which must electrically isolate the deflectors from each other as well as the elements above and below; the micromachined electrostatic lens, which is mounted to a 16-pin Airpax header; an extractor electrode; and the FEI source. The assembly is stacked one ring above another and is held together under compression in a mu-metal exterior can, which provides both the structural integrity of the assembly and magnetic shielding of the optical column (Fig. 32). The critical alignment necessary in the structure is the alignment of the lens electrode apertures to one another and the alignment of the electron source to the lens apertures. The electrode-to-electrode alignment is accomplished through our micromachining technique and the electron source alignment is accomplished by means of two insulated linear-motion feedthroughs, which push on the FEI source at 90◦ with a return spring. This allowed the majority of the pieces in the assembly to be machined to fairly low tolerances (tolerances were specified to ∼±50 μm), which kept the machining cost low. The entire assembly is inserted into a commercially available 6-in. UHV vacuum chamber containing a 30-liter/s nonevaporable getter pump that is mounted to a 120-liter/s ion pump. The motion feedthroughs are attached, electrical connections are made, and the system is evacuated. A base pressure of 1 × 10−9 torr is achieved in 48 h. The silicon lens was fabricated from 380-μm-thick silicon chips separated with 250-μm gaps. The performance of a three-element lens using these physical parameters has been calculated and the results are shown in Figure 33. These calculations indicate that the lens can produce a high-quality focus from a position near the exit aperture of the lens to a working distance of up to a

3D FABRICATION OF MINIATURE ELECTRON OPTICS

135

Figure 33. Solid and dash–dot lines represent 4- and 0.5-mm working distances. (1) Current MSEM operating point; expected resolution of 425 nm; (2) 4-mm working distance; expected resolution of 6.2 nm; (3) 0.5-mm working distance; expected resolution of 2.3 nm.

few centimeters with potentials on the focusing electrode(s) that are allowed by the die-to-die gaps. The extractor aperture is optically aligned to the silicon apertures in the micro-machined lens by placing the assembly under a microscope, using bottom illumination to view the bright circular spot formed by the apertures in the silicon, centering the 500-μm extractor aperture over that spot, and securing the extractor in place. Typical operating potential differences between the source and the extractor electrode are in the range of 2.5–3.75 kV for an emission current of 1–25 μA. The FEI source also contains a suppressor electrode, which is biased negative with respect to the tip to prevent thermally generated electron emission from escaping the source. Initially our micromachined silicon apertures were only 3.5 μm thick, which was probably not thick enough to take the bombardment of ∼30 μA of 3-keV emission. However, we subsequently improved the silicon process to give 100-μm-thick apertures and will later remove the extractor from the assembly. With the stainless-steel electrode in the system, the first two silicon electrodes can be operated in parallel as one optically long focusing electrode. This has been calculated to produce a higher-quality probe as well as to provide more flexibility in operation (Feinerman, Crewe, Perng, Spindt et al., 1994). Calculations indicate that a stacked lens with 150-μm-diameter apertures will produce a 425-nm focus with a 2.5-kV beam at a working distance of 4 mm and a field-emission source 1 cm above the lens (Fig. 33). If the final angle of convergence is reduced from 10 to 2.6 mrad, the focus improves to 6.2 nm. If the working distance is reduced to 0.5 mm, a 2.3-nm

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A. D. FEINERMAN AND D. A. CREWE

resolution can be achieved at a final angle of convergence equal to 6.5 mrad. The efficiency of the electron detector will have to be increased, however, since the probe current is inversely proportional to the square of the convergence angle. Images of a 200- and 1000-mesh gold TEM wire grid at a working distance of 4 mm have been obtained in transmission. The beam is scanned over the sample by using parallel plate deflectors. The silicon lens is 1.64 mm long and consists of three silicon die separated by Pyrex optical fibers as shown in Figure 2. Images of the grid at magnifications above 7000× are now being obtained.

B. MSEM Operation and Image Formation The potentials applied to the source and lens electrodes and the filament heating current are supplied by a computer-controlled set of electronics (Fig. 34). Three high-voltage power supplies and a constant current supply are floated with their

Figure 34. Flowchart of MSEM control and image-acquisition system. The use of two PCs is redundant and will be reduced to one computer controlling both the high-voltage gun control unit and the scan-generator/image-acquisition electronics.


137

virtual ground at the beam potential. Isolation from earth ground is achieved through optical couplers. The suppressor and focus potentials, and the filament heating current are controlled through an RS232 serial connection to a personal computer (PC). The beam potential is manually set on an externally regulated high-voltage power supply. After initial conditioning of the extractor electrode to allow for electronstimulated desorption of gas ions, the total emission is increased to ∼3 μA and the source-to-silicon aperture alignment is performed. Once a beam is brought through the lens, the focus electrode potential is optimized by comparing successive line scans over the gold grid. The optimal focusing potential agrees well with calculated values, differing by less than 10%. The deflection potential signals and the image data are generated and received by data-acquisition boards in a PC. The low-voltage deflection ramps are the input to a high-speed, high-voltage amplifier capable of generating −500- to +500-V signals at a rate of 10 kHz. The faces of the deflectors that are perpendicular to the electron beam measure 1.5 × 1.5 mm and are spaced 1.25 mm apart. A simple time-of-flight deflection calculation predicted a beam deflection of 0.5 μm/V of applied deflection signal. Experimentally, we have observed that one volt of deflection potential yields approximately 0.4 μm of beam deflection. Typical deflection signals are staircase ramps in the range −150 to +150 V (for a field of view 120 by 120 μm) generated at a line rate of 10 Hz. Imageacquisition time for a 512 × 512-pixel image is then 51.2 s. The image data consist of the Faraday cup current (for a dark-field image) or the sample current (for a bright-field image) that has been put through a current-to-voltage amplifier with a gain of approximately 1010 and a maximum pixel rate of 100 kHz. This 0- to 1-V signal is the input to a 12-bit analog-to-digital converter that acquires the image data synchronously with the deflection ramp generation. The raw image data are then normalized and imported into a commercially available image-processing software package for viewing. The initial and final voltages of the X and Y deflection ramps can be software selected, and the magnitude of the amplified deflection signal can be varied, which allows the user to perform a direct current offset high-magnification scan of a region of interest that is not in the center of a low-magnification image. Low- and high-magnification images of a 1000-mesh gold wire grid are shown in Figures 35–38. The 10–90% rise time of the line scan shown in Figure 39 covers a lateral distance of 2.1 μm. This indicates that if the probe is Gaussian, it has a sigma of 0.75 μm. This is a worst case estimation of the beam probe size, since the grid wires in reality have a finite slope, but does give a value that agrees well with calculations.

138


Figure 35. This image obtained with test apparatus demonstrates the ability of the micromachined silicon electron lens to focus on a 1000-mesh gold TEM grid. The grid wires are 6 μm wide and are spaced 19 μm apart, and the signal is from the Faraday cup current.

Figure 36. Magnification is ∼2000 of 6 μm grid, and the signal is from the Faraday cup current. Defect in center of image is from screen saver turning on.

Figure 37. Magnification is ∼3500 of 6-μm grid, and the signal is from the Faraday cup current.

Figure 38. High-magnification image of defect on wire grid. Image has been electronically rotated to bring wire to a nearly vertical position. The cross wire is not at a right angle to the vertical wire, possibly as a result of a deformation of the sample when it was fit into the test assembly. The defect is approximately 0.5 μm wide.

140


Figure 39. Line scan data from a high-magnification image of one period of the 1000-mesh grid. The scan signals were electrically rotated so that the beam was deflected perpendicular to the wires. The 10–90% rise time of 2.1 μm corresponds to a Gaussian probe sigma of 0.75 μm. In its present configuration the MSEM is spherical-aberration limited, so the Gaussian probe is a good approximation to the actual beam.

IX. Summary and Future Prospects Microfabrication techniques have advanced to the point where conductors, semi-conductors, and insulators can be positioned in complex threedimensional arrangements with very high precision. This is equivalent to a conventional machinist’s operating miniature milling machines and lathes with micron-sized bits. This flexible machining capability allows electric and magnetic fields to be created that can accelerate, focus, steer, and/or align charged particles, because the fields occupy a volume of space rather than simply existing next to a surface. Specific fabrication techniques developed at UIC include stacking silicon chips with Pyrex fibers, selective anodic bonding (slicing), and using a LIGA lathe. These techniques are being used to integrate chargedparticle sources, electrodes, and detectors into various miniature instruments including a subcentimeter SEM, a 10-cm time-of-flight mass spectrometer, a 10-cm nuclear magnetic resonance instrument, and a 5-m linear accelerator/undulator capable of producing hard X-rays. Analytical instruments of this size will allow the analytical laboratory to be brought to the sample, which will be essential when the sample must be observed in situ (e.g., at a toxic waste site or in outer space).


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A Reference Discretization Strategy for the Numerical Solution of Physical Field Problems CLAUDIO MATTIUSSI∗ Clampco Sistemi-NIRLAB, AREA Science Park, Padriciano 99, 34012 Trieste, Italy

I. Introduction . . . . . . . . . . . . . . . . . . . II. Foundations . . . . . . . . . . . . . . . . . . . A. The Mathematical Structure of Physical Field Theories B. Geometric Objects and Orientation . . . . . . . . 1. Space–Time Objects . . . . . . . . . . . . . C. Physical Laws and Physical Quantities . . . . . . . 1. Local and Global Quantities . . . . . . . . . . 2. Equations . . . . . . . . . . . . . . . . . . D. Classification of Physical Quantities . . . . . . . . 1. Space–Time Viewpoint . . . . . . . . . . . . E. Topological Laws . . . . . . . . . . . . . . . F. Constitutive Relations . . . . . . . . . . . . . . 1. Constitutive Equations and Discretization Error . . G. Boundary Conditions and Sources . . . . . . . . . H. The Scope of the Structural Approach . . . . . . . III. Representations . . . . . . . . . . . . . . . . . . A. Geometry . . . . . . . . . . . . . . . . . . . 1. Cell Complexes . . . . . . . . . . . . . . . 2. Primary and Secondary Mesh . . . . . . . . . 3. Incidence Numbers . . . . . . . . . . . . . . 4. Chains . . . . . . . . . . . . . . . . . . . 5. The Boundary of a Chain . . . . . . . . . . . B. Fields . . . . . . . . . . . . . . . . . . . . 1. Cochains . . . . . . . . . . . . . . . . . . 2. Limit Systems . . . . . . . . . . . . . . . . C. Topological Laws . . . . . . . . . . . . . . . 1. The Coboundary Operator . . . . . . . . . . . 2. Properties of the Coboundary Operator . . . . . 3. Discrete Topological Equations . . . . . . . . . D. Constitutive Relations . . . . . . . . . . . . . . E. Continuous Representations . . . . . . . . . . . 1. Differential Forms . . . . . . . . . . . . . . 2. Weighted Integrals . . . . . . . . . . . . . .

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∗ Current affiliation: Evolutionary and Adaptive Systems Team, Institute of Robotic Systems (ISR), Department of Micro-Engineering (DMT), Swiss Federal Institute of Technology (EPFL), CH-1015 Lausanne, Switzerland.

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3. Differential Operators . . . . . . . . . . . . 4. Spread Cells . . . . . . . . . . . . . . . . 5. Weak Form of Topological Laws . . . . . . . . IV. Methods . . . . . . . . . . . . . . . . . . . . . A. The Reference Discretization Strategy . . . . . . . 1. Domain Discretization . . . . . . . . . . . . 2. Topological Time Stepping . . . . . . . . . . 3. Strategies for Constitutive Relations Discretization 4. Edge Elements and Field Reconstruction . . . . . B. Finite Difference Methods . . . . . . . . . . . . 1. The Finite Difference Time-Domain Method . . . 2. The Support Operator Method . . . . . . . . . 3. Beyond the FDTD Method . . . . . . . . . . C. Finite Volume Methods . . . . . . . . . . . . . 1. The Discrete Surface Integral Method . . . . . . 2. The Finite Integration Theory Method . . . . . . D. Finite Element Methods . . . . . . . . . . . . . 1. Time-Domain Finite Element Methods . . . . . 2. Time-Domain Edge Element Method . . . . . . 3. Time-Domain Error-Based FE Method . . . . . V. Conclusions . . . . . . . . . . . . . . . . . . . VI. Coda . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . .

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I. Introduction One of the fundamental concepts of mathematical physics is that of field; that is, naively speaking, of a spatial distribution of some mathematical object representing a physical quantity. The power of this idea lies in that it allows the modeling of a number of very important phenomena—for example, those grouped under the labels “electromagnetism,” “thermal conduction,” “fluid dynamics,” and “solid mechanics,” to name a few—and of the combinations thereof. When the concept of field is used, a set of “translation rules” is devised, which transforms a physical problem belonging to one of the aforementioned domains—a physical field problem—into a mathematical one. The properties of this mathematical model of the physical problem—a model which usually takes the form of a set of partial differential or integrodifferential equations, supplemented by a set of initial and boundary conditions—can then be subjected to analysis in order to establish if the mathematical problem is well posed (Gustafsson et al., 1995). If the result of this inquiry is judged satisfactory, it is possible to proceed to the actual derivation of the solution, usually with the aid of a computer. The recourse to a computer implies, however, a further step after the modeling step described so far, namely, the reformulation of the problem in discrete

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terms, as a finite set of algebraic equations, which are more suitable than a set of partial differential equations to the number-crunching capabilities of present-day computing machines. If this discretization step is made by starting from the mathematical problem in terms of partial differential equations, the resulting procedures can logically be called numerical methods for partial differential equations. This is indeed how the finite difference (FD), finite element (FE), finite volume (FV), and many other methods are often categorized. Finally, the system of algebraic equations produced by the discretization step is solved, and the result is interpreted from the point of view of the original physical problem. More than 30 years ago, while considering the impact of the digital computer on mathematical activity, Bellman (1968) wrote Much of the mathematical analysis that was developed over the eighteenth and nineteenth centuries originated in attempts to circumvent arithmetic. With our ability to do large-scale arithmetic . . . we can employ simple, direct methods requiring much less old-fashioned mathematical training. . . . This situation by no mean implies that the mathematician has been dispossessed in mathematical physics. It does signify that he is urgently needed . . . to transform the original mathematical problems to the stage where a computer can be utilized profitably by someone with a suitable scientific training. . . . Good mathematics, like politics, is the art of the possible. Unfortunately, people quickly forget the origins of a mathematical formulation with the result that it soon acquires a life of its own. Its genealogy then protects it from scrutiny. Because the digital computer has so greatly increased our ability to do arithmetic, it is now imperative that we reexamine all the classical mathematical models of mathematical physics from the standpoints of both physical significance and feasibility of numerical solution. It may well turn out that more realistic descriptions are easier to handle conceptually and computationally with the aid of the computer. (pp. 44–45)

In this spirit, the present work describes an alternative to the classical partial differential equations–based approach to the discretization of physical field problems. This alternative is based on a preliminary reformulation of the mathematical model in a partially discrete form, which preserves as much as possible the physical and geometric content of the original problem, and is made possible by the existence and properties of a common mathematical structure of physical field theories (Tonti, 1975). The goal is to maintain the focus, both in the modeling step and in the discretization step, on the physics of the problem, thinking in terms of numerical methods for physical field problems, and not for a particular mathematical form (e.g., a partial differential equation) into which the original physical problem happens to be translated (Fig. 1).

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Figure 1. The alternative paths leading from a physical field problem to a system of algebraic equations. p.d.e., partial differential equation.

The advantages of this approach are various. First, it provides a unifying viewpoint for the discretization of physical field problems, which is valid for a multiplicity of theories. Second, by basing the discretization of the problems on the structural properties of the theory to which they belong, this approach gives discrete formulations which preserve many physically significant properties of the original problem. Finally, being based on very intuitive geometric and physical concepts, this approach facilitates both the analysis of existing numerical methods and the development of new ones. The present work considers both these aspects, introducing first a reference discretization strategy directly inspired by the results of the analysis of the structure of physical field theories. Then, a number of popular numerical methods for partial differential equations are considered, and their workings are compared with those of the reference strategy, in order to ascertain to what extent these methods can be interpreted as discretization methods for physical field problems. The realization of this plan requires the preliminary introduction of the basic ideas of the structural analysis of physical field theories. These ideas are simple, but unfortunately they were formalized and given physically unintuitive names at the time of their first application, within certain branches of advanced


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mathematics. Therefore, in applying them to other fields, one is faced with the dilemma of inventing for these concepts new and, one would hope, more meaningful names, or maintaining the names inherited from mathematical tradition. After some hesitation, I chose to keep the original names, to avoid a proliferation of typically ephemeral new definitions and in consideration of the fact that there can be difficult concepts, not difficult names; we must try to clarify the former, not avoid the latter (Dolcher, 1978). The intended audience for this article is wide. On the one hand, novices to the field of numerical methods for physical field problems will find herein a framework which will help them to intuitively grasp the common concepts hidden under the surface of a variety of methods and thus smooth the path to their mastery. On the other hand, the ideas presented should also prove helpful to the experienced numerical practitioner and to the researcher as additional tools that can be applied to the evaluation of existing methods and the development of new ones. Finally, it is worth remembering that the result of the discretization must be subjected to analysis also, in order to establish its properties as a new mathematical problem, and to measure the effects of the discretization on the solution when it is compared with that of nondiscrete mathematical models. This further analysis will not be dealt with here, the emphasis being on the unveiling of the common discretization substratum for existing methods, the convergence, stability, consistency, and error analyses of which abound in the literature.

II. Foundations A. The Mathematical Structure of Physical Field Theories It was mentioned in the Introduction that the approach to the discretization that will be presented in this work is based on the observation that physical field theories possess a common structure. Let us, therefore, start by explaining what we mean when we talk of the structure of a physical theory. It is a common experience that exposure to more than one physical field theory (e.g., thermal conduction and electrostatics) aids the comprehension of each single one and facilitates the quick grasping of new ones. This occurs because there are easily recognizable similarities in the mathematical formulation of theories describing different phenomena, which permit the transfer of intuition and imageries developed for more familiar cases to unfamiliar realms.∗ Building in a systematic way on these similarities, one can fill a correspondence ∗ One may say that this is the essence of explanation (i.e., the mapping of the unexplained on something that is considered obvious).

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table that relates physical quantities and laws playing a similar role within different theories. Usually we say that there are analogies between these theories. These analogies are often reported as a trivial, albeit useful curiosity, but some scholars have devoted considerable efforts to unveiling their origin and meaning. In these scholars’ quest, they have discovered that these similarities can be traced to the common geometric background upon which the “physics” is built. In the book that, building on a long tradition, took these enquiries almost to their present state, Tonti (1975) emphasized the following: r

r

r r

The existence within physical theories of a natural association of many physical quantities, with geometric objects in space and space-time∗ The necessity to consider as oriented the geometric objects to which physical quantities are associated The existence of two kinds of orientation for these geometric objects The primacy and priority, in the foundation of each theory, of global physical quantities associated with geometric objects, over the corresponding densities

From this set of observations there follows naturally a classification of physical quantities, based on the type and kind of orientation of the geometric object with which they are associated. The next step is the consideration of the relations held between physical quantities within each theory. Let us call them generically the physical laws. From our point of view, the fundamental observation in this context relates to r

The existence within each theory of a set of intrinsically discrete physical laws

These observations can be given a graphical representation as follows. A classification diagram for physical quantities is devised, with a series of “slots” for the housing of physical quantities, each slot corresponding to a different kind of oriented geometric object (see Figs. 7 and 8). The slots of this diagram can be filled for a number of different theories. Physical laws will be represented in this diagram as links between the slots housing the physical quantities (see Fig. 17). The classification diagram of physical quantities, complemented by the links representing physical laws, will be called the factorization diagram of the physical field problem, to emphasize its role in singling out the terms in the governing equations of a problem, according to their mathematical and physical properties. The classification and factorization diagrams will be used extensively in this work. They seem to have been first introduced by Roth (see the discussion ∗

For the time being, we give the concept of oriented geometric object an intuitive meaning (points, and sufficiently regular lines, surfaces, volumes, and hypervolumes, along with time instants and time intervals).


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in Bowden, 1990, who calls them Roth’s diagrams). Branin (1966) used a modified version of Roth’s diagrams, calling them transformation diagrams. Tonti (1975, 1976a, 1976b, 1998) refined and used these diagrams—which he called classification schemes—as the basic representational tool for the analysis of the formal structure of physical theories. We will refer here to this last version of the diagrams, which were subsequently adopted by many authors with slight graphical variations and under various names (Baldomir and Hammond, 1996; Bossavit, 1998a; Palmer and Shapiro, 1993; Oden and Reddy, 1983) and for which the name Tonti diagrams was suggested.∗ The Tonti classification and factorization diagrams are an ideal starting point for the discretization of a field problem. The association of physical quantities with geometric objects gives a rationale for the construction of the discretization meshes and the association of the variables to the constituents of the meshes, whereas singling out in the diagram the intrinsically discrete terms of the field equation permits us both to pursue the direct discrete rendering of these terms and to focus on the discretization effort with the remaining terms. Having found this common starting point for the discretization of field problems, one might be tempted to adopt a very abstract viewpoint, based on a generic field theory, with a corresponding generic terminology and factorization diagram. However, although many problems share the same structure of the diagram, there are classes of theories whose diagrams differ markedly and consequently a generic diagram would be either too simple to encompass all the cases or too complicated to work with. For this reason we are going to proceed in concrete terms, selecting a model field theory and referring mainly to it, in the belief that this could aid intuition, even if the reader’s main interest is in a different field. Considering the focus of the series in which this article appears, electromagnetism was selected as the model theory. Readers having another background can easily translate what follows by comparing the factorization diagram for electromagnetism with that of the theory they are interested in. To give a feeling of what is required for the development of the factorization diagram for other theories, we discuss the case of heat transfer, thought of as representative of a class of scalar transport equations. It must be said that there are still issues that wait to be clarified in relation to the factorization diagrams and the mathematical structure of physical theories. This is true in particular for some issues concerning the position of energy quantities within the diagrams and the role of orientation with reference to ∗ In fact, the diagrams used in this work (and in Mattiussi, 1997) differ from those originally conceived by Tonti in their admitting only cochains within the slots, whereas the latter had chains in some slots and cochains in others (depending on the kind of orientation of the subjacent geometric object). This difference reflects our advocating the use of the chain–cochain pair to distinguish the discrete representation of the geometry (which is always made in terms of chains) from that of the fields (which is always based on cochains).

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time. Luckily this touches only marginally on the application of the theory to the discretization of physical problems finalized to their numerical solution.

B. Geometric Objects and Orientation The concept of geometric object is ubiquitous in physical field theories. For example, in the theory of thermal conduction the heat balance equation links the difference between the amount of heat contained inside a volume V at the initial and final time instants Ti and Tf of a time interval I, to the heat flowing through the surface S, which is the boundary of V, and to the heat produced or absorbed within the volume during the time interval. In this case, V and S are geometric objects in space, whereas I, Ti , and Tf are geometric objects in time. The combination of a space and a time object (e.g., the surface S considered during the time interval I, or the volume V at the time instant Ti, or Tf) gives a space– time geometric object. These examples show that by “geometric object” we mean the points and the sufficiently well-behaved lines, surfaces, volumes, and hypervolumes contained in the domain of the problem, and their combination with time instants and time intervals. This somewhat vague definition will be substituted later by the more detailed concept of the p-dimensional cell. The preceding example also shows that each mention of an object comes with a reference to its orientation. To write the heat balance equation, we must specify if the heat flowing out of a volume or that flowing into it is to be considered positive. This corresponds to the selection of a preferred direction through the surface.∗ Once this direction is chosen, the surface is said to have been given external orientation, where the qualifier “external” hints at the fact that the orientation is specified by means of an arrow that does not lie on the surface. Correspondingly, we will call internal orientation of a surface that which is specified by an arrow that lies on the surface and that specifies a sense of rotation on it (Fig. 2). Note that the idea of internal orientation for surfaces is seldom mentioned in physics but is very common in everyday objects and in mathematics (Schutz, 1980). For example, a knob that must be rotated counterclockwise to ensure a certain effect is usually designed with a suitable curved arrow drawn on its surface, and in plane affine geometry, the ordering of the coordinate axes corresponds to the choice of a sense of rotation on the plane and defines the orientation of the space. ∗ Of course it must be possible to assign such a direction consistently, which is true if the geometric object is orientable (Schutz, 1980), as we will always suppose to be the case. Once the selection is made, the object acquires a new status. As pointed out by MacLane (1986): “A plane with orientation is really not the same object as one without. The plane with an orientation has more structure—namely, the choice of the orientation” (p. 84).


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Figure 2. (a) External and (b) internal orientations for surfaces.

In fact, all geometric objects can be endowed with two kinds of orientations but, for historical reasons, almost no mention of this distinction survives in physics.∗ Since both kinds of orientation are needed in physics, we will show how to build the complete orientation apparatus. We will start with internal orientation, using the preceding affine geometry example as inspiration. An n-dimensional affine space is oriented by fixing an order of the coordinate axes: this, in the three-dimensional case, corresponds to the choice of a screw-sense, or that of a vortex; in the two-dimensional case, to the choice of a sense of rotation on the plane; and in the one-dimensional case, to the choice of a sense (an arrow) along the line. These images can be extended to geometric objects. Therefore, the internal orientation of a volume is given by a screw-sense; that of a surface, by a sense of rotation on it; and that of a line, by a sense along it (see Fig. 5). Before we proceed further, it is instructive to consider an example of a physical quantity that, contrary to common belief, is associated with internally oriented surfaces: the magnetic flux φ. This association is a consequence of the invariance requirement of Maxwell’s equations for improper coordinate transformations; that is, those that invert the orientation of space, transforming a right-handed reference system into a left-handed one. Imagine an experimental setup to probe Faraday’s law, for example, verifying the link between the magnetic flux φ “through” a disk S and the circulation U of the electric field intensity E around the loop Ŵ which is the border of S. If we suppose, as is usually the case, that the sign of φ is determined by a direction through the disk, and that of U by the choice of a sense around the loop, a mirror reflection through a plane parallel to the disk axis changes the sign of U but not that of φ. Usually the incongruence is avoided by using the right-hand rule to define B and invoking for it the status of axial vector (Jackson, 1975). In other words, we are told that for space reflections, the sense of the “arrow” of the B vector ∗ However, for example, Maxwell (1871) was well aware of the necessity within the context of electromagnetism of at least four kinds of mathematical entities for the correct representation of the electromagnetic field (entities referred to lines or to surfaces and endowed with internal or with external orientation).

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Figure 3. Orientational issues in Faraday’s law. The intervention of the right-hand rule, required in the classical version (a), can be avoided by endowing both geometric objects Ŵ and S with the same kind of orientation (b).

does not count; only the right-hand rule does. It is, however, apparent that for the invariance of Faraday’s law to hold true without such tricks, all we have to do is either to associate φ with internally oriented surfaces and U with internally oriented lines, or to associate φ with externally oriented surfaces and U with lines oriented by a sense of rotation around them (i.e., externally oriented lines, as will soon be clear). Since the effects of an electric field act along the field lines and not around them, the first option seems preferable (Schouten, 1989; Fig. 3). This example shows that the need for the right-hand rule is a consequence of our disregarding the existence of two kinds of orientation. This attitude seems reasonable in physics as we have become accustomed to it in the course of our education, but consider that if it were applied systematically to everyday objects, we would be forced to glue an arrow pointing outward from the aforementioned knob, and to accompany it with a description of the right-hand rule. Note also that the difficulties in the classical formulation of Faraday’s law stem from the impossibility of comparing directly the orientation of the surface with that of its boundary, when the surface is externally oriented and the bounding line is internally oriented. In this case, “directly” means “without recourse to the right-hand rule” or similar tricks. The possibility of making this direct comparison is fundamental for the correct statement of many physical laws. This comparison is based on the idea of an orientation induced by an object on its boundary. For example, the sense of rotation that internally orients a surface induces a sense of rotation on its bounding curve, which can be compared with the sense of rotation which orients the surface internally. The same is true for the internal orientation of volumes and of their bounding surfaces. The reader can check that the direct comparison is indeed possible if the object and its boundary are both endowed with internal orientation as defined previously for volumes, surfaces, and lines. However, this raises an interesting issue, since our list of internally oriented objects does not so far include points, which nevertheless form the boundary of a line. To make inner orientation a coherent system, we must, therefore, define internal orientations for points (as in algebra we extend the definition of the nth power of a number to include the case n = 0). This can be done by means of a pair of symbols


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Figure 4. Each internally oriented geometric object induces an internal orientation on the objects that constitute its boundary.

meaning “inward” and “outward” (e.g., defining the point as a sink or a source, or drawing arrows pointing inward or outward), for these images are directly comparable with the internal orientation of a line which starts or ends with the point (Fig. 4). This completes our definition of internal orientation for geometric objects in three-dimensional space, which we will indicate with the terms P, L, S, and V. Let us now tackle the definition of external orientation for the same objects. We said before that in three-dimensional space the external orientation of a surface is given, specifying what turned out to be the internal orientation of a line which does not lie on the surface. This is a particular case of the very definition of external orientation: in an n-dimensional space, the external orientation of a p-dimensional object is specified by the internal orientation of a dual (n − p)dimensional geometric object (Schouten, 1989). Hence, in three-dimensional space, external orientation for a volume is specified by an inward or outward symbol; for a surface, it is specified by a direction through it; for a line, by a sense of rotation around it; for a point, by the choice of a screw-sense. To distinguish internally oriented objects from externally oriented ones, we will ˜ L, ˜ S, ˜ and V˜ for externally add a tilde to the terms for the latter, thus writing P, oriented points, lines, surfaces, and volumes, respectively (Fig. 5). The definition of external orientation in terms of internal orientation has many consequences. First, contrary to internal orientation, which is a combinatorial concept∗ and does not change when the dimension of the embedding ∗ For example, a line can be internally oriented by selecting a permutation class (an ordering) of two distinct points on it, which become three nonaligned points for a surface, four noncoplanar points for a volume, and so on.

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Figure 5. (a) Internal and (b) external orientations for geometric objects in threedimensional space. The disposition of objects reflects the pairing of reciprocally dual geometric objects.

space varies, external orientation depends on the dimension. For example, external orientation for a line in two-dimensional space is assigned by a direction through it and not around it as in three-dimensional space.∗ Another consequence is the inheritance from internal orientation of the possibility of comparing the orientation of an object with that of its boundary, when both are endowed with external orientation. This implies once again the concept of induced orientation, applied in this case to externally oriented objects (Fig. 6). The duality of internal and external orientation gives rise to another important pairing, that between dual geometric objects; that is, between pairs of geometric objects that in an n-dimensional space have dimensions p and (n − p), respectively, and have differents kinds of orientation (Fig. 5). Note that also in this case the orientation of the objects paired by the duality can be directly compared. However, contrary to what happens for a geometric object and its boundary, the objects have different kinds of orientation. In the context of the mathematical structure of physical theories, this duality plays an ∗ Note, however, that the former can be considered the “projection” onto the surface of the latter.


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Figure 6. Each externally oriented geometric object induces an external orientation on the objects that constitute its boundary.

important role; for example, it is used in the definition of energy quantities and it accounts for some important adjointness relationships between differential operators. We have now at our disposal all the elements required for the construction of a first version—referring to the objects of three-dimensional space—of the classification diagram of physical quantities. As anticipated, it consists of a series of slots for the housing of physical quantities, each slot corresponding to an oriented geometric object. As a way to represent graphically the distinction between internal and external orientation, the slots of the diagram are subdivided between two columns. So that the important relationship represented by duality is reflected, these two columns—for internal and external orientation, respectively—are reversed with respect to each other, which thus makes dual objects row-adjacent (Fig. 7). 1. Space–Time Objects In the heat balance example that opens this section, it was shown how geometric objects in space, time, and space–time make their appearance in the foundation of a physical theory. Until now, we have focused on objects in space; let us extend our analysis to space–time objects. If we adopt a strict space–time viewpoint—that is, if we consider space and time as one, and our objects as p-dimensional objects in a generic

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Figure 7. The Tonti classification diagram of physical quantities in three-dimensional space. Each slot is referred to an oriented geometric object; that is, points P, lines L, surfaces S, and volumes V. The left column is devoted to internally oriented objects, and the right column to externally oriented ones. The slots are paired horizontally so as to reflect the duality of the corresponding objects.

four-dimensional space—the extension from space to space–time requires only that we apply to the four-dimensional case the definitions given previously for oriented geometric objects. However, one cannot deny that in all practical cases (i.e., if a reference frame has to be meaningful for an actual observer) the time coordinate is clearly distinguishable from the spatial coordinates. Therefore, it seems advisable to consider, in addition to space–time objects per se, the space–time objects considered as Cartesian products of a space object by a time object. Let us list these products. Time can house zero- and one-dimensional geometric objects: time instants T and time intervals I. We can combine these time objects with the four space objects: points P, lines L, surfaces S, and volumes V. We obtain thus eight combinations that, considering the two kinds of orientation they can be endowed with, give rise to the 16 slots of the space–time classification diagram of physical quantities (Tonti, 1976b; Fig. 8). Note that the eight combinations correspond, in fact, to five space–time geometric objects (e.g., a space–time volume can be obtained as a volume in space considered at a time instant, that is, as the topological product V × T, or as a surface in space considered during a time interval, which corresponds to S × I). This is reflected within the diagram by the sharing of a single slot by the combinations corresponding to the same oriented space–time object. To distinguish space–time objects from merely spatial ones, we will use the symbols P , L, S , V , and H for the former and the symbols P, L, S, and V for the latter. As usual, a tilde will signal external orientation.


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Figure 8. The Tonti space–time classification diagram of physical quantities. Each slot is referred to an oriented space–time geometric object, which is thought of as obtained in terms of a product of an object in space by an object in time. The space objects are those of Figure 7. The time objects are time instants T and time intervals I. This diagram can be redrawn with the slots referring to generic space–time geometric objects; that is, points P , lines L, surfaces S , volumes V , and hypervolumes H (see Fig. 11).

C. Physical Laws and Physical Quantities In the previous sections, we have implicitly defined a physical quantity (the heat content, the heat flow, and the heat production, in the heat transfer example) as an entity appearing within a physical field theory, which is associated with one (and only one) kind of oriented geometric object. Strictly speaking, the individuation within a physical theory of the actual physical quantities and the attribution of the correct association with oriented geometric objects should be based on an analysis of the formal properties of the mathematical entities that appear in the theory (e.g., considering the dimensional properties of those entities and their behavior with respect to coordinate transformations). Given that formal analyses of this kind are available in the literature (Post, 1997; Schouten, 1989; Truesdell and Toupin, 1960), the approach within the present work will be more relaxed. To fill in the classification diagram of the physical quantities of a theory, we will look first at the integrals which appear within the theory, focusing our attention on the integration domains in space and time. This will give us a hint about the geometric object that a quantity is associated with. The attribution of orientation to these objects will be based on heuristic considerations deriving from the following fundamental property: the sign of a global quantity associated with a geometric object changes when

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the orientation of the object is inverted. Further hints would be drawn from physical effects and the presence of the right-hand rule in the traditional definition of a quantity, as well as from the global coherence of the orientation system thus defined. The reader can find in Tonti (1975) an analysis based on a similar rationale, applied to a large number of theories, accompanied by the corresponding classification and factorization diagrams. 1. Local and Global Quantities By their very definition, our physical quantities are global quantities, for they are associated with macroscopic space–time domains. This complies with the fact that actual field measurements are always performed on domains having finite extension. When local quantities (densities and rates) can be defined, it is natural to make them inherit the association with the oriented geometric object of the corresponding global quantity. However, it is apparent that the familiar tools of vector analysis do not allow this association to be represented. This causes a loss of information in the transition from the global to the local representation, when ordinary scalars and vectors are used. For example, from the representation of magnetic flux density with the vector field B, no hint at internally oriented surfaces can be obtained, nor can an association to externally oriented volumes be derived from the representation of charge density with the scalar field ρ. Usually the association with geometric objects (but not the distinction between internal and external orientations) is reinserted while one is writing integral relations, by means of the “differential term,” so that we write, for example, B · ds (1) S

and

ρ dv

(2)

V

However, given the presence of the integration domains S and V, which accompany the integration signs, the terms ds and dv look redundant. It would be better to use a mathematical representation that refers directly to the oriented geometric object that a quantity is associated with. Such a representation exists within the formalism of ordinary and twisted differential forms (Burke, 1985; de Rham, 1931). Within this formalism, the vector field B becomes an ordinary 2-form b2 and the scalar field ρ a twisted 3-form ρ˜ 3 , as follows: B ⇒ b2

ρ ⇒ ρ˜ 3

(3) (4)


159

The symbols b2 and ρ˜ 3 explicitly refer to the fact that magnetic induction and charge density are associated with (and can be integrated only on) internally oriented two-dimensional domains and externally oriented three-dimensional domains, respectively. Thus, everything seems to conspire for an early adoption of a representation in terms of differential forms. We prefer, however, to delay this step in order to show first how the continuous representation tool they represent can be founded on discrete concepts. Waiting for the suitable discrete concepts to be available, we will temporarily stick to the classical tools of vector calculus. In the meantime, the only concession to the differential-form spirit will be the systematic dropping of the “differential” under the integral sign, so that we write, for example, B (5) S

and

ρ

(6)

V˜

instead of Eqs. (1) and (2). 2. Equations After the introduction of the concept of oriented geometric objects, the next step would ideally be the discussion of the association of the physical quantities of the field theory (in our case, electromagnetism) with the objects. This would parallel the typical development of physical theories, in which the discovery of quantities upon which the phenomena of the theory may be conceived to depend precedes the development of the mathematical relations that link those quantities in the theory (Maxwell, 1871). It turns out, however, that the establishment of the association between physical quantities and geometric objects is based on the analysis of the equations appearing in the theory itself. In particular, it is expedient to list all pertinent equations for the problem considered, and isolate a subset of them, which represent physical laws lending themselves naturally to a discrete rendering, for these clearly expose the correct association. We start, therefore, by listing the equations of electromagnetism. We will first give a local rendition of all the equations, even of those that will eventually turn out to have an intrinsically discrete nature, since this is the form that is typically considered in mathematical physics. The first pair of electromagnetic equations that we consider represent in local form Gauss’s law for magnetic flux [Eq. (7)] and Faraday’s induction

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law [Eq. (8)]: div B = 0 (7) ∂B curl E + =0 (8) ∂t where B is the magnetic flux density and E is the electric field intensity. We will show next that these equations have a counterpart in the law of charge conservation [Eq. (9)]: ∂ρ =0 (9) ∂t where J is the electric current density and ρ is the electric charge density. Similarly, Eqs. (10) and (11), which define the scalar potential V and the vector potential A, div J +

curl A = B (10) ∂A −grad V − =E (11) ∂t are paralleled by Gauss’s law of electrostatics [Eq. (12)] and Maxwell– Ampère’s law [Eq. (13)]—where D is the electric flux density and H is the magnetic field intensity—which close the list of differential statements: div D = ρ (12) ∂D =J (13) curl H − ∂t Finally, we have a list of constitutive equations. A very general form for the case of electromagnetism, accounting for most material behaviors, is t D(r, t) = Fε (E, r′ , τ ) (14) B(r, t) = J(r, t) =

t0

D

t0

D

t0

D

t

t

Fμ (H, r′ , τ )

(15)

Fσ (E, r′ , τ )

(16)

but, typically, the purely local relations D(r, t) = f ε (E, r, t)

(17)

J(r, t) = f σ (E, r, t)

(19)

B(r, t) = f μ (H, r, t)

(18)


161

or the even simpler relations D(r) = ε(r)E(r)

(20)

J(r) = σ (r)E(r)

(22)

(21)

B(r) = μ(r)H(r)

adequately represent most actual material behaviors. We will now consider all these equations, aiming at their exact rendering in terms of global quantities. Integrating Eqs. (7) through (13) on suitable spatial domains, writing ∂D for the boundary of a domain D, and making use of Gauss’s divergence theorem and Stokes’s theorem, we obtain the following integral expressions: B= 0 (23) ∂V

E+

d J+ dt

∂S

∂ V˜

−

d dt

∂L

d dt

V˜

∂S

L

A=

∂ V˜

D=

S˜

D=

V

0

(24)

0

(25)

S

V˜

A=

d H− dt ∂ S˜

ρ=

S

V−

B=

B

(26)

E

(27)

S

L

ρ

(28)

J

(29)

V˜

S˜

Note that in Eqs. (23), (24), and (25) we have integrated the null term on the right-hand side. This was done in consideration of the fact that the corresponding equations assert the vanishing of some kind of physical quantity, and we must investigate what kind of association it has. Moreover, in Eqs. (25), (28), and (29) we added a tilde to the symbol of the integration domains. These are the domains which will turn out later to have external orientation.

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In Eqs. (24), (25), (27), and (29) a time derivative remains. A further integration can be performed on a time interval I = [T1, T2] as a way to eliminate this residual derivative. For example, Eq. (24) becomes

T2 T1

∂S

E+

B S

T2 T1

=

T2 T1

0

(30)

S

We adopt a more compact notation, which uses I for the time interval. Moreover, we will consider as an “integral on time instants,” a term evaluated at that instant, according to the following symbolism: def ·= · (31) · = S

S

T

T

S

Correspondingly, since the initial and final instants of a time interval I are actually the boundary ∂I of I, we write boundary terms as follows: T2 def · = · S

T1

∂I

(32)

S

Remark II.1 The boundary of an oriented geometric object is constituted by its faces endowed with the induced orientation (Figs. 4 and 6). For the case of a time interval I = [T1, T2], the faces that appear in the boundary ∂I correspond to the two time instants T1 and T2. If the time interval I is internally oriented in the direction of increasing time, T1 appears in ∂I oriented as a source, whereas T2 appears in it oriented as a sink. However, as time instants, T1 and T2 are endowed with a default orientation of their own. Let us assume that the default internal orientation of all time instants is as sinks; it follows that ∂I is constituted by T2 taken with its default orientation and by T1 taken with the opposite of its default orientation. We can express this fact symbolically, writing ∂I = T2 − T1, where the “minus” sign signals the inversion of the orientation of T1. Correspondingly, if there is a quantity Q associated with the time instants, and Q1 and Q2 are associated with T1 and T2, respectively, the quantity Q2 − Q1 will be associated with ∂I. We will give these facts a more precise formulation later, using the concepts of chain and cochain. For now, this example gives a first idea of the key role played by the concept of orientation of space–time geometric objects, in a number of common mathematical operations such as the T increment of a quantity and the fact that an expression like T12 d f corresponds to ( f |T2 − f |T1 ) and not to its opposite. In this context, we alert the reader to the fact that if the time axis is externally oriented, it is the time instants that are oriented by means of a (through) direction, whereas the time instants themselves are oriented as sources or sinks.


163

With these definitions [Eqs. (31) and (32)], Eqs. (23) through (29) become B= 0 (33) T

E+

J+

I

∂S

I˜

−

∂ V˜

I˜

∂I

∂ I˜

V˜

V−

∂L

∂ S˜

H−

I˜

∂I

L

A=

∂ I˜

S

A=

∂ V˜

D=

S˜

D=

I

ρ=

∂S

T˜

V

B=

T

T

S

I

∂V

0

(34)

0

(35)

V˜

T

S

I

L

T˜

V˜

I˜

S˜

B

(36)

E

(37)

ρ

(38)

J

(39)

The equations in this form can be used to determine the correct association of physical quantities with geometric objects. D. Classification of Physical Quantities In Eqs. (33) through (39), we can identify a number of recurrent terms and deduce from them an association of physical quantities with geometric objects. From Eqs. (33) and (34) we get E ⇒ (L × I ) (40) I

L

T

S

B ⇒ (S × T )

(41)

where the arrow means “is associated with.” The term in Eq. (41) confirms the association of magnetic induction with surfaces and suggests a further one with time instants, whereas Eq. (40) shows that the electric field is associated with lines and time intervals. These geometric objects are endowed with internal orientation, as follows from the analysis made previously for the orientational issues in Faraday’s law.

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The status of electric current and charge as a physical quantity can be deduced from Eq. (35), which gives the terms J ⇒ ( S˜ × I˜) (42) S˜

I˜

T˜

V˜

ρ ⇒ (V˜ × T˜ )

(43)

which show that electric current is associated with surfaces and time intervals, whereas charge is associated with volumes and time instants. Since the current is due to a flow of charges through the surface, a natural external orientation for surfaces follows. Given this association of electric current with externally oriented surfaces, the volumes to which charge content is associated must also be externally oriented to permit direct comparison of the sign of the quantities in Eq. (35). The same rationale can be applied to the terms appearing in Eqs. (38) and (39); that is, H ⇒ ( L˜ × I˜) (44) I˜

L˜

T˜

S˜

D ⇒ ( S˜ × T˜ )

(45)

This shows that the magnetic field is associated with lines and time intervals and the electric displacement with surfaces and time instants. As for orientation, the magnetic field is traditionally associated with internally oriented lines but this choice requires the right-hand rule to make the comparison, in Eq. (39), of the direction of H along ∂ S˜ with the direction of the current flow through the ˜ Hence, so that the use of the right-hand rule can be dispensed with, surface S. the magnetic field must be associated with externally oriented lines. The same argument applies in suggesting an external orientation for surfaces to which electric displacement is associated. Finally, Eqs. (36) and (37) give the terms V ⇒ (P × I ) (46) I

P

T

L

A ⇒ (L × T )

(47)

which show that the scalar potential is associated with points and time intervals, whereas the vector potential is associated with lines and time instants. From the association of the electric field with internally oriented lines, it follows that for the electromagnetic potentials, the orientation is also internal.


165

Figure 9. The Tonti classification diagram of local electromagnetic quantities.

The null right-hand-side terms in Eqs. (33) through (35) remain to be taken into consideration. We will see subsequently that these terms express the vanishing of magnetic flux creation (or the nonexistence of magnetic charge) and the vanishing of electric charge creation, respectively. For now, we will simply insert them as zero terms in the appropriate slot of the classification diagram for the physical quantities of electromagnetism, which summarizes the results of our analysis (Fig. 9). 1. Space–Time Viewpoint The terms T˜ V˜ ρ and I˜ S˜ J in Eqs. (42) and (43) refer to the same global physical quantity: electric charge. Moreover, total integration is performed in both cases on externally oriented, three-dimensional domains in space–time. We can, therefore, say that electric charge is actually associated with externally oriented, three-dimensional space–time domains of which a three-dimensional space volume considered at a time instant, and a three-dimensional space surface considered during a time interval, are particular cases. To distinguish these two embodiments of the charge concept, we use the terms charge content, referring to volumes and time instants, and charge flow, referring to surfaces and time intervals. A similar distinction can be drawn for other quantities. For example, the terms I L E and T S B in Eqs. (40) and (41) are both magnetic fluxes associated with two-dimensional space–time domains of which we could say that the electric field refers to a “flow” of magnetic flux tubes which cross internally oriented lines, while magnetic induction refers to a surface “content”

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of such tubes. Since the term content refers properly to volumes, and the term flow to surfaces, it appears preferable to distinguish the two manifestations of each global quantity by using an index derived from the letter traditionally used for the corresponding local quantity, as in ρ = Q ρ (V˜ × T˜ ) (48) V˜

T˜

I˜

and

S˜

J = Q j ( S˜ × I˜)

(49)

(50) (51)

T

S

B = φ b (S × T )

I

L

E = φ e (L × I )

The same argument can be applied to electric flux, D = ψ d ( S˜ × T˜ ) T˜

S˜

I˜

L˜

H = ψ h ( L˜ × I˜)

and to the potentials in global form, A = U a (L × T ) T

L

I

P

V = U v (P × I )

(52) (53)

(54) (55)

With these definitions we can fill in the classification diagram of global electromagnetic quantities (Fig. 10). Note that the classification diagram of Figure (10) emphasizes the pairing of physical quantities which happen to be the static and dynamic manifestations of a unique space–time entity. We can group these variables under a single heading, obtaining a classification diagram of the space–time global electromagnetic quantities U , φ, ψ, and Q (Fig. 11), which corresponds to the one that could be drawn for local quantities in four-dimensional notation. Note also that all the global quantities of a column possess the same physical dimension; for example, the terms in Eqs. (48), (49), (52), and (53) all have the physical dimension of electric charge. Nonetheless, quantities appearing in different rows of a column refer to different physical quantities since, even


167

Figure 10. The Tonti classification diagram of global electromagnetic quantities.

if the physical dimension is the same, the underlying space–time oriented geometric object is not. This fact is reflected in the relativistic behavior of these quantities. When an observer changes his or her reference frame, his or her perception of what is time and what is space changes and with it his or her method of splitting a given space–time physical quantity into its two “space plus time” manifestations. Hence, the transformation laws, which account for

Figure 11. The Tonti classification diagram of global electromagnetic quantities, referring to space–time geometric objects.

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CLAUDIO MATTIUSSI

the change of reference frame, will combine only quantities referring to the same space–time oriented object. In a four-dimensional treatment such quantities will be logically grouped within a unique entity (e.g., the charge–current vector; the four-dimensional potentials; the first and second electromagnetic tensor—or the corresponding differential forms—with groupings E and B, and H and D, respectively; and so on). E. Topological Laws Now that we have seen how to proceed to the individuation and classification of the physical quantities of a theory, there remains, as a last step in the determination of the structure of the theory itself, the establishment of the links existing between the quantities, accompanied by an analysis of the properties of these links. As anticipated, the main result of this further analysis—valid for all field theories—will be the singling out of a set of physical laws, which lend themselves naturally to a discrete rendering, opposed to another set of relations, which constitute instead an obstacle to the complete discrete rendering of field problems. It is apparent from the definitions given in Eqs. (48) through (55), that Eqs. (33) through (39) can be rewritten in terms of global quantities only, as follows: e

φ b (∂ V × T ) = 0(V × T ) b

(56)

φ (∂ S × I ) + φ (S × ∂ I ) = 0(S × I )

(57)

Q j (∂ V˜ × I˜) + Q ρ (V˜ × ∂ I˜) = 0(V˜ × I˜)

(58)

v

U a (∂ S × T ) = φ b (S × T ) a

e

(59)

−U (∂ L × I ) − U (L × ∂ I ) = φ (L × I )

(60)

ψ d (∂ V˜ × T˜ ) = Q ρ (V˜ × T˜ ) ψ h (∂ S˜ × I˜) − ψ d ( S˜ × ∂ I˜) = Q j ( S˜ × I˜)

(61) (62)

Note that no material parameters appear in these equations, and that the transition from the local, differential statements in Eqs. (7) through (13) to these global statements was performed without recourse to any approximation. This proves their intrinsic discrete nature. Let us examine and interpret these statements one by one. Gauss’s magnetic law [Eq. (56)] asserts the vanishing of magnetic flux associated with closed surfaces ∂V in space considered at a time instant T. From


169

Figure 12. Faraday’s induction law admits a geometric interpretation as a conservation law on a space–time cylinder. The (internal) orientation of geometric objects is not represented.

what we said previously about space–time objects, there must be a corresponding assertion for timelike closed surfaces. Faraday’s induction law [Eq. (57)] is indeed such an assertion for a cylindrical closed surface in space–time constructed as follows (Fig. 12): the surface S at the time instant T1 constitutes the first base of a cylinder; the boundary of S, ∂S, considered during the time interval I = [T1, T2], constitutes the lateral surface of the cylinder, which is finally closed by the surface S considered at the time instant T2 [remember that T1 and T2 together constitute the boundary ∂I of the time interval I, hence the term S × ∂I in Eq. (57) represents the two bases of the cylinder] (Bamberg and Sternberg, 1988; Truesdell and Toupin, 1960). This geometric interpretation of Faraday’s law is particularly interesting for numerical applications, for it is an exact statement linking physical quantities at times T < T2 to a quantity defined at time T2. Therefore, this statement is a good starting point for the development of the time-stepping procedure. In summary, Gauss’s law and Faraday’s induction law are the space and the space–time parts, respectively, of a single statement: the magnetic flux associated with the boundary of a space–time volume V is always zero: φ(∂ V ) = 0(V )

(63)

(Remember that the boundary of an oriented geometric object must always be thought of as endowed with the induced orientation.) Equation (63), also called the law of conservation of magnetic flux (Truesdell and Toupin, 1960), gives to its right-hand-side term the meaning of a null in the production of magnetic flux. From another point of view, the right-hand side of Eq. (56) expresses

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CLAUDIO MATTIUSSI

the nonexistence of magnetic charge and that of Eq. (57) the nonexistence of magnetic charge current. The other conservation statement of electromagnetism is the law of conservation of electric charge [Eq. (58)]. In strict analogy with the geometric interpretation of Faraday’s law, a cylindrical, space–time, closed hypersurface is constructed as follows: the volume V˜ at the time instant T˜1 constitutes the first base of a hypercylinder; the boundary of V˜ , ∂ V˜ , considered during the time interval I˜ = [T˜1 , T˜2 ], constitutes the lateral surface of the hypercylinder, which is finally closed by the volume V˜ considered at the time instant T˜2 . The law of charge conservation asserts the vanishing of the electric charge associated with this closed hypercylinder. This conservation statement can be referred to the boundary of a generic space–time hypervolume H˜ , which yields the following statement, analogous to Eq. (63): (64) Q(∂ H˜ ) = 0(H˜ ) In Eq. (64) the zero on the right-hand side states the vanishing of the production of electric charge. Note that in this case a purely spatial statement, corresponding to Gauss’s law of magnetostatics [Eq. (56)] is not given, for in four-dimensional space–time a hypervolume can be obtained only as a product of a volume in space multiplied by a time interval. The two conservation statements [Eqs. (63) and (64)] can be considered the two cornerstones of electromagnetic theory (Truesdell and Toupin, 1960). de Rham (1931) proved that from the global validity of statements of this kind [or, if you prefer, of Eqs. (33) through (35)] in a homologically trivial space follows the existence of field quantities that can be considered the potentials of the densities of the physical quantities appearing in the global statements. In our case we know that the field quantities V and A, defined by Eqs. (10) and (11), are indeed traditionally called the electromagnetic potentials. Correspondingly, the field quantities H and D defined by Eqs. (12) and (13) are also potentials and can be called the charge–current potentials (Truesdell and Toupin, 1960). In fact the definition of H and D is a consequence of charge conservation, exactly as the definition of V and A is a consequence of magnetic flux conservation; therefore, neither is uniquely defined by the conservation laws of electromagnetism. Only the choice of a gauge for the electromagnetic potentials and the hypothesis about the media properties for charge–current potentials removes this nonuniqueness. In any case, the global renditions [Eqs. (59) through (62)] of the equations defining the potentials prove the intrinsic discrete status of Gauss’s law of electrostatics, of Maxwell–Ampère’s law, and of the defining equations of the electromagnetic potentials. A geometric interpretation can be given to these laws, too. Gauss’s law of electrostatics asserts the balance of the electric charge contained in a volume with the electric flux through the surface that bounds


171

Figure 13. Maxwell–Ampère’s law admits a geometric interpretation as a balance law on a space–time cylinder. The (external) orientation of geometric objects is not represented.

the volume. Similarly, Maxwell–Ampère’s law defines this balance between the charge contained within a space–time volume and the electric flux through its boundary, which is a cylindrical space–time closed surface analogous to the one appearing in Faraday’s law, but with external orientation (Fig. 13). This geometric interpretation, like that of Faraday’s law, is instrumental for a correct setup of the time stepping within a numerical procedure. Equations (61) and (62) can be condensed into a single space–time statement that asserts the balance of the electric charge associated with arbitrary space– time volumes with the electric flux associated with their boundaries: ψ(∂ V˜ ) = Q(V˜ ) (65) Analogous interpretations hold for Eqs. (59) and (60), relative to a balance of magnetic fluxes associated with space–time surfaces and their boundaries:

U (∂ S ) = φ(S )

(66)

We can insert the global space–time statements [Eqs. (63) through (66)] in the space–time classification diagram of the electromagnetic physical quantities (Fig. 14). Note that all these statements appear as vertical links. These links relate a quantity associated with an oriented geometric object with a quantity associated with the boundary of that object (which has, therefore, the same kind of orientation). What is shown here for the case of electromagnetism applies to the great majority of physical field theories. Typically, a subset of the equations which form a physical field theory link a global quantity associated with an oriented geometric object to the global quantity that, within

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Figure 14. The position of topological laws in the Tonti classification diagram of electromagnetic quantities.

the theory, is associated with the boundary of that object (Tonti, 1975). These laws are intrinsically discrete, for they state a balance of these global quantities (or a conservation of them, if one of the terms is zero) whose validity does not depend on metrical or material properties, and is, therefore, invariant for very general transformations. This gives them a “topological significance” (Truesdell and Toupin, 1960), which justifies our calling them topological laws. The significance of this finding for numerical methods is obvious: once the domain of a field problem has been suitably discretized, topological laws can be written directly and exactly in discrete form. F. Constitutive Relations To complete our analysis of the equations of electromagnetism, we must consider the set of constitutive equations, represented, for example, by Eqs. (14) through (16). We emphasize once again that each instance of this kind of equation is only a particular case of the various forms that the constitutive links between the problem’s quantities can take. In fact, while topological laws can be considered universal laws linking the field quantities of a theory, constitutive relations are merely definitions of ideal materials given within the framework of that particular field theory (Truesdell and Noll, 1965). In other words, they are abstractions inspired by the observation of the behavior of actual materials. More sophisticated models have terms that account for a wider range of observed material behaviors, such as nonlinearity, anisotropy, nonlocality,


173

Figure 15. The Tonti factorization diagram of electromagnetism in local form. Topological laws are represented by vertical links within columns, whereas constitutive relations are represented by transverse links bridging the two columns of the diagram.

hysteresis, and the combinations thereof (Post, 1997). This added complexity implies usually a greater sophistication of the numerical solvers, but does not change the essence of what we are about to say concerning the discretization of constitutive relations. If we consider the position of constitutive relations in the classification diagram of the physical quantities of electromagnetism, we observe that they constitute a link that connects the two columns (Fig. 15). This fact reveals that, unlike topological laws, constitutive relations link quantities associated with geometric objects endowed with different kinds of orientation. From the point of view of numerical methods, the main differences with topological laws are the observation that constitutive relations contain material parameters∗ and the fact that they are not intrinsically discrete. The presence of a term of this kind in the field equations is not surprising, since otherwise—given the intrinsic ∗ In some cases material parameters seemingly disappear from constitutive equations. This is the case, for example, with electromagnetic equations in empty space when we adopt Gaussian units and set c = 1. This induces the temptation to identify physical quantities—in this case E and D, and B and H, respectively. However, the approach based on the association with oriented geometric objects reveals that these quantities have a distinct nature.

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discreteness of topological laws—it would always be possible to exactly discretize and solve numerically a field problem, and we know that this is not the case. Constitutive relations can be transformed into exact links between global quantities only if the local properties do not vary in the domain where the link must be valid. This means that we must impose a series of uniformity requirements on material and field properties for a global statement to hold true. On the contrary, since, aside from discontinuities, these requirements are automatically satisfied in the small, the local statement always applies. The uniformity requirement is in fact the method used to experimentally investigate these laws. For example, we can investigate the constitutive relation D = εE

(67)

examining a capacitor with two planar parallel plates of area A, having a distance l between them and filled with a uniform, linear, isotropic medium having relative permittivity ε r. With this assumption, Eq. (67) corresponds approximately to V ψ =ε (68) A l where ψ is the electric flux and V the voltage between the plates. Note that to write Eq. (68), besides using the material parameter ε, we invoke the concepts of planarity, parallelism, area, distance, and orthogonality, which are not topological concepts. This shows that, unlike topological laws, constitutive relations imply the recourse to metrical concepts. This is not apparent in Eq. (67), for—as explained previously—the use of vectors to represent field quantities tends to hide the geometric details of the theory. Equation (67) written in terms of differential forms, or a geometric representation thereof, reveals the presence, within the link, of the metric tensor (Burke, 1985; Post, 1997). The local nature of constitutive relations can be interpreted by saying that these equations summarize at a macroscopic level something going on at a subjacent scale. This hypothesis may help the intuition, but it is not necessary if we are willing to interpret them as definitions of ideal materials. By so doing, we can avoid the difficulties implicit in the creation of a convincing derivation of field concepts from a corpuscular viewpoint. There is other information about constitutive equations that can be derived by observing their position in the factorization diagram. These are not of direct relevance from a numerical viewpoint but can help us to understand better the nature of each term. For example, it has been observed that when the two columns of the factorization diagram are properly aligned according to duality, constitutive relations linked to irreversible processes (e.g., Ohm’s law linking E and J in Fig. 15) appear as slanted links, whereas those representing reversible processes appear as horizontal links (Tonti, 1975).


175

1. Constitutive Equations and Discretization Error We anticipated in the preceding discussion that, from our point of view, the main consequence of the peculiar nature of constitutive relations lies in their preventing, in general, the attainment of an exact discrete solution. By “exact discrete solution,” we mean the exact solution of the continuous mathematical model (e.g., a partial differential equation) into which the physical problem is usually transformed. We hinted in the Introduction at the fact that the numerical solution of a field problem implies three phases (Fig. 1): 1. The transformation of the physical problem into a mathematical model 2. The discretization of the mathematical model 3. The solution of the system of algebraic equations produced by the discretization (The fourth phase represented in Fig. 1, the approximate reconstruction of the field function based on the discrete solution, obviously does not affect the accuracy of the discrete solution.) Correspondingly, there will be three kinds of errors (Fig. 16; Ferziger and Perić, 1996; Lilek and Perić, 1995): 1. The modeling error 2. The discretization error 3. The solver error

Figure 16. The three kinds of errors associated with the numerical solution of a field problem.

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Modeling errors are a consequence of the assumptions about the phenomena and processes, made during the transition from the physical problem to its mathematical model in terms of equations and boundary conditions. Solver errors are a consequence of the limited numerical precision and time available for the solution of the system of algebraic equations. Discretization errors act between these two steps, preventing the attainment of the exact discrete solution of the mathematical model, even in the hypothesis that our algebraic solvers were perfect. The existence of discretization errors is a well-known fact, but it is the analysis based on the mathematical structure of physical theories that reveals where the discretization obstacle lies; that is, within constitutive relations, topological laws not implying in themselves any discretization error. As anticipated in the Introduction, this in turn suggests the adoption of a discretization strategy in which what is intrinsically discrete is included as such in the model, and the discretization effort is focused on what remains. It must be said, however, that once the discretization error is brought into by the presence of the constitutive terms, it is the joint contribution of the approximation implied by the discretization of these terms and of our enforcing only a finite number of topological relations in place of the infinitely many that are implied by the corresponding physical law that shapes the actual discretization error. This fact will be examined in detail subsequently.

G. Boundary Conditions and Sources A field problem includes, in addition to the field equations, a set of boundary conditions and the specification that certain terms appearing in the equations are assigned as sources. Boundary conditions and sources are a means to limit the scope of the problem actually analyzed, for they summarize the effects of interactions with domains or phenomena that we choose not to consider in detail. Let us see how boundary conditions and sources enter into the framework developed in the preceding sections for the equations, with a classification that parallels the distinction between topological laws and constitutive relations. When boundary conditions and sources are specified as given values of some of the field quantities of the problem, they correspond in our scheme to global values assigned to some geometric object placed along the boundary or lying within the domain. Hence, the corresponding values enter the calculations exactly, but for the possibly limited precision with which they are calculated from the corresponding field functions (usually by numerical integration) when they are not directly given as global quantities. Consequently, in this case these terms can be assimilated with topological prescriptions.


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In other cases boundary and source terms are assigned in the form of equations linking a problem’s field variable to a given excitation. In these cases, these terms must be considered as additional constitutive relations to which all the considerations made previously for this kind of equation apply. In particular, within a numerical formulation, such terms must be subjected to a specific discretization process. For example, this is the case for convective boundary conditions in heat transfer problems. In still other cases boundary conditions summarize the effects on the problem domain of the structure of that part of space–time which lies outside the problem domain. Think, for example, about radiative boundary conditions in electrodynamics, and inlet and outlet boundary conditions in fluid dynamics. In these cases, one cannot give general prescriptions, for the representation depends on the geometric and physical structure of this “outside.” Physically speaking, a good approach consists of extending the problem’s domain, enclosing it in a (thin) shell whose properties account, with a sufficient approximation, for the effect of the whole space surrounding the domain, and whose boundary conditions belong to one of the previous kinds. This shell can then be modeled and discretized by following the rules used for the rest of the problem’s domain. However, devising the properties of such a shell is usually not a trivial task. In any case, the point is that boundary conditions and source terms can be brought back to topological laws and constitutive relations by physical reasoning, and from there they require no special treatment with respect to what applies to these two categories of relations.

H. The Scope of the Structural Approach The example of electromagnetism, examined in detail in the previous sections, shows that to approach the numerical solution of a field problem by taking into account its mathematical structure, we must first classify the physical quantities appearing in the field equations, according to their association with oriented geometric objects, and then factorize the field equations themselves to the point of being able to draw the factorization diagram for the field theory to which the problem belongs. The result will be a distinction of topological laws, which are intrinsically discrete, from constitutive relations, which admit only approximate discrete renderings (Fig. 17). Let us examine briefly how this process works for other theories and the difficulties we can expect to encounter. From electromagnetism we can easily derive the diagrams of electrostatics and magnetostatics. If we drop the time dependence, the factorization diagram for electromagnetism splits naturally into the two distinct diagrams of electrostatics and magnetostatics (Figs. 18 and 19).

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Figure 17. The distinction between topological and constitutive terms of the field equations, as it appears in the Tonti factorization diagram. Topological laws appear as vertical links and are intrinsically discrete, whereas constitutive relations appear as transverse links and in general permit only approximate discrete renderings.

Given the well-known analogy between stationary heat conduction and electrostatics (Burnett, 1987; Maxwell, 1884), one would expect to derive the diagram for this last theory directly from that of electrostatics. An analysis of physical quantities reveals, however, that the analogy is not perfect. Temperature, which is linked by the analogy to electrostatic potential V, is indeed associated, like V, to internally oriented points and time intervals, but heat flow density, traditionally considered analogous with electric displacement D, is in fact associated with externally oriented surfaces and time intervals, whereas D is associated with surfaces and time instants. In the stationary case, this distinction makes little difference, but we will see later, in Fig. 20, that this results in a slanting of the constitutive link between the temperature gradient g and the diffusive heat flux density qd , whereas the constitutive link between E and D is not slanted. This reflects the irreversible nature of the former process, as opposed to the reversible nature of the latter. Since the heat transfer equation can be considered a prototype of all scalar transport equations, it is worth examining in detail, including both the nonstationary and the convective terms. A heat transfer equation that is general enough for our purposes can be written as follows (Versteeg and Malalasekera, 1995): ∂(ρcθ) + div(ρcθu) − div(k grad θ) = σ ∂t

(69)


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Figure 18. The Tonti factorization diagram for electrostatics in local form.

where θ is the temperature, ρ is the mass density, c is the specific heat, u is the fluid velocity, k is the thermal conductivity, and σ is the heat production density rate. Note that we always start with field equations written in local form, for these equations usually include constitutive terms. We must first factor out these terms before we can write the topological terms in their primitive, discrete form. Disentangling the constitutive relations from the topological laws, we

Figure 19. The Tonti factorization diagram for magnetostatics in local form.

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obtain the following set of topological equations, grad θ = g

(70)

∂qc + div qu + div qd = σ ∂t and the following set of constitutive equations,

(71)

qu = ρcθu

(72)

qd = −kg

(73)

qc = ρcθ

(74)

To write Eqs. (70) through (74), we have introduced four new local physical quantities: the temperature gradient g, the diffusive heat flow density qd , the convective heat flow density qu , and the heat content density qc. Note that of the three constitutive equations, Eq. (72) appears as a result of a driving source term, with the parameter u derived from an “external” problem. This is an example of how the information about interacting phenomena is carried by terms appearing in the form of constitutive relations. Another example is given by boundary conditions describing a convective heat exchange through a part ∂ Dv of the domain boundary. If θ∞ is the external ambient temperature, h is the coefficient of convective heat exchange, and we denote with qv and θv the convective heat flow density and the temperature at a generic point of ∂Dv , we can write qv = h(θv − θ∞ )

(75)

An alternative approach is to consider this as an example of coupled problems, where the phenomena that originate the external driving terms are treated as separate interacting problems, which must also be discretized and solved. In this case, a factorization diagram must be built for each physical field problem intervening in the whole problem, and what is treated here as driving terms become links between the diagrams. In these cases, a preliminary classification of all the physical variables appearing in the different phenomena is required, so that we can select the best common discretization substratum, especially for what concerns the geometry. Putting the topological laws, with the new boundary term [Eq. (75)], in full integral form, we have θ= g (76) I˜

∂ V˜

qv +

I˜

∂ V˜

qu +

I˜

∂ V˜

qd +

I

∂L

∂ I˜

V˜

qc =

I

L

I˜

V˜

σ

(77)


181

We can define the following global quantities θ = (P × I )

(78)

g = G(L × I )

(79)

I

P

I

L

S˜

qu = Q u ( S˜ × I˜)

(80)

I˜

S˜

qv = Q v ( S˜ × I˜)

(81)

I˜

S˜

qd = Q d ( S˜ × I˜)

(82)

I˜

V˜

qc = Q c (V˜ × T˜ )

(83)

T˜

σ = F(V˜ × I˜)

(84)

V˜

I˜

with the temperature impulse associated with internally oriented points and time intervals; the thermal tension G associated with internally oriented lines and time intervals; the convective and diffusive heat flows Qu, Qv , and Qd associated with externally oriented surfaces and time intervals; the heat content Qc associated with externally oriented volumes and time instants; and the heat production F associated with externally oriented volumes and time intervals. The same associations hold for the corresponding local quantities. This permits us to write Eqs. (76) and (77) in terms of global quantities only: (∂ L × I ) = G(L × I ) Q v (∂ V˜ × I ) + Q u (∂ V˜ × I ) + Q d (∂ V˜ × I ) + Q c (V˜ ×∂ I˜) = F(V˜ × I˜)

(85) (86)

Note that Eq. (86) is the natural candidate for the setup of a time-stepping scheme within a numerical procedure, for it links exactly quantities defined at times which precede the final instant of the interval I to the heat content Qc at the final instant. This completes our analysis of the structure of heat transfer problems represented by Eq. (69) and establishes the basis for their discretization. The corresponding factorization diagram in terms of local field quantities is depicted in Fig. 20. Along similar lines one can conduct the analysis for many other theories. No difficulties are to be expected for those that happen to be characterized— like electromagnetism and heat transfer—by scalar global quantities. More complex are cases of theories in which the global quantities associated with

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Figure 20. The Tonti factorization diagram for the heat transfer equation in local form. Note the presence of terms derived from the diagrams of other theories or other domains.

geometric objects are vectors or more complex mathematical entities. This is the case of fluid dynamics and continuum mechanics (in which vector quantities such as displacements, velocities, and forces are associated with geometric objects). In this case, the deduction of the factorization diagram can be a difficult task, for one must first tackle a nontrivial classification task for quantities that have, in local form, a tensorial nature, and then disentangle the constitutive and topological factors of the corresponding equations. Moreover, for vector theories it is more difficult to pass silently over the fact that to compare or add quantities defined at different space–time locations (even scalar quantities, in fact), we need actually a connection defined in the domain. To simplify things, one could be tempted to write the equations of fluid dynamics as a collection of scalar transport equations, hiding within the source term everything that does not fit in an equation of the form of Eq. (69), and to apply to these equations the results of the analysis of the scalar transport equation. However, it is clear that this approach prevents the correct association of physical quantities with geometric objects and is, therefore, far from the spirit advocated in this work. Moreover, the inclusion of too many interaction terms within the source terms can spoil the significance of the analysis, for example,


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hiding essential nonlinearities.∗ Finally, it must be said that, given a field problem, one could consider the possibility of adopting a Lagrangian viewpoint in place of the Eulerian one that we have considered so far. The approach presented here applies, strictly speaking, only to a Eulerian approach. Nevertheless, the benefits derived from a proper association of physical quantities to oriented geometric objects extend also to a Lagrangian approach. Moreover, the case of moving meshes is included without difficulties in the space–time discretization described subsequently, and in particular in the reference discretization strategy that will be introduced in the section on numerical methods (Section IV). III. Representations We have analyzed the structure of field problems, aiming at their discretization. Our final goal is the actual derivation of a class of discretization strategies that comply with that structure. To this end, we must first ascertain what has to be modeled in discrete terms. A field problem includes the specification of a space–time domain and of the physical phenomena that are to be studied within it. The representation of the domain requires the development of a geometric model to which mathematical models of physical quantities and material properties must be linked, so that physical laws can finally be modeled as relations between these entities. Hence, our first task must be the development of a discrete mathematical model for the domain geometry. This will be subsequently used as a support for a discrete representation of fields, complying with the principles derived from the analysis of the mathematical structure of physical theories. The discrete representation of topological laws, then, follows naturally and univocally. This is not the case for constitutive relations, for the discretization of which various options exist. In the next sections we will examine a number of discrete mathematical concepts that can be used in the various discretization steps. A. Geometry The result of the discretization process is the reduction of the mathematical model of a problem having an infinite number of degrees of freedom into one with a finite number. This means that we must find a finite number of entities ∗ As quoted by Moore (1989), Schrödinger, in a letter to Born, wrote: “ ‘If everything were linear, nothing would influence nothing,’ said Einstein once to me. That is actually so. The champions of linearity must allow zero-order terms, like the right side of the Poisson equation, V = −4πρ. Einstein likes to call these zero-order terms ‘asylum ignorantiae’” (p. 381).

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which are related in a known way to the physical quantities of interest. If we focus our attention on the fields, and think in terms of the usual continuous representations in terms of scalar or vector functions, the first thing that comes to mind is the plain sampling of the field functions at a finite number of points— usually called nodes—within the domain. This sampling produces a collection of nodal scalar or vector values, which eventually appear in the system of algebraic equations produced by the discretization. Our previous analysis reveals, however, that this nodal sampling of local field quantities is unsuitable for a discretization which aims at preserving the mathematical structure of the field problem, since such a discretization requires the association of global physical quantities with geometric objects that are not necessarily points. From this point of view, a sound discretization of geometry must provide all the kinds of oriented geometric objects that are actually required to support the global physical quantities appearing within the problem, or at least, those appearing in its final formulation as a set of algebraic equations. Let us see how this reflects on mesh properties. 1. Cell Complexes Our meshes must allow the housing of global physical quantities. Hence, their basic building blocks must be oriented geometric objects. Since we are going to make heavy use of concepts belonging to the branch of mathematics called algebraic topology, we will adopt the corresponding terminology. Algebraic topology is a branch of mathematics that studies the topological properties of spaces by associating them with suitable algebraic structures, the study of which gives information about the topological structure of the original space (Hocking and Young, 1988). In the first stages of its development, this discipline considered mostly spaces topologically equivalent to polytopes (polygons, polyhedra, etc.). Many results of algebraic topology are obtained by considering the subdivisions in collections of simple subspaces, of the spaces under scrutiny. Understandably, then, many concepts used within the present work were formalized in that context. In the later developments of algebraic topology, much of the theory was extended from polytopes to arbitrary compact spaces. The concepts involved became necessarily more abstract, and the recourse to simple geometric constructions waned. Since all our domains are assumed to be topologically equivalent to polytopes, we need and will refer only to the ideas and methods of the first, more intuitive version of algebraic topology. With the new terminology, what we have so far called an oriented p-dimensional geometric object will be called an oriented p-dimensional cell, or simply a p-cell, since all cells will be assumed to be oriented, even if this is not explicitly stated. From the point of view of algebraic topology, a p-cell τ p in a domain D can be defined simply as a set of points that is homeomorphic to a closed p-ball B p = {x ∈ R p : x ≤ 1} of the Euclidean p-dimensional


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Figure 21. (a) Improper and (b) proper joining of cells.

space (Franz, 1968; Hocking and Young, 1988; Whitney, 1957). To model our domains as generic topological spaces, however, would be entirely too generic. We can assume, without loss of generality, that the domain D of our problem is an n-dimensional differentiable manifold of which our p-cells are p-dimensional regular subdomains∗ (Boothby, 1986). With these hypotheses a p-cell τ p is the same p-dimensional “blob” that we adopted as a geometric object. The boundary ∂τ p of a p-cell τ p is the subset of D, which is linked by the preceding homeomorphism to the boundary ∂ B p = {x ∈ R p : x = 1} of Bp. A cell is internally (externally) oriented when we have selected as the positive orientation one of the two possible internal (external) orientations for it. According to our established convention, we will add a tilde to distinguish externally oriented cells τ˜ p from internally oriented cells τ p . To simplify the notation, in presenting new concepts we will usually refer to internally oriented cells. The results apply obviously to externally oriented objects as well. In assembling the cells to form meshes, we must follow certain rules. These rules are dictated primarily by the necessity of relating in a certain way the physical quantities that are associated with the cells to those that are associated with their boundaries. Think, for example, of two adjacent 3-cells in a heat transfer problem; these cells can exchange heat through their common boundary, and we want to be able to associate this heat to a 2-cell belonging to the mesh. So that this goal can be achieved, the cells of the mesh must be properly joined (Fig. 21). In addition to this, since the heat balance equation for each 3-cell implies the heat associated with the boundary of the cell, this boundary must be paved with a finite number of 2-cells of the mesh. Finally, ∗ In actual numerical problems p-cells are usually nothing more than bounded, convex, oriented polyhedrons in Rn .

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to avoid the association of a given global quantity to multiple cells, we should ensure that two distinct cells do not overlap. A structure that complies with these requirements is an n-dimensional finite cell complex K. This is a finite set of cells with the following two properties: 1. The boundary of each p-cell of K is the union of lower-dimensional cells of K (these cells are called the proper q-dimensional faces of τ p, with q ranging from from 0 to p − 1; it is useful to consider a cell an improper face of itself). 2. The intersection of any two cells of K is either empty or a (proper or improper) face of both cells. This last requirement specifies the property of two cells’ being “properly joined.” We can, therefore, say that a finite cell complex K is a finite collection of properly joined cells with the property that if τ p is a cell of K, then every face of τ p belongs to K. Note that the term face without specification of the dimension usually refers only to the (p −1)-dimensional faces. We say that a cell complex K decomposes or is a subdivision of a domain D (written |K | = D), if D is equal to the union of the cells in K. The collection of the p-cells and of all cells of dimension lower than p of a cell complex is called its p-skeleton. We will assume that our domains are always decomposable into finite cell complexes and assume that all our cell complexes are finite, even if this is not explicitly stated. The requirement that the meshes be cell complexes may seem severe, for it implies proper joining of cells and covering of the entire domain without gaps or overlapping. A bit of reflection reveals, however, that this includes all structured and most nonstructured meshes, excluding only a minority of cases such as composite and nonconformal meshes. Nonetheless, this requirement will be relaxed later or, better, the concept of a cell will be generalized, so as to include structures that can be considered as derived from a cell complex by means of a limit process. This is the case in the finite element method and in some of its generalizations, for example, meshless methods. For now, however, we will base the next steps of our quest for a discrete representation of geometry and fields on the hypothesis that the meshes are cell complexes. Note that for time-dependent problems we assume that the cell complexes subdivide the whole space–time domain of the problem. 2. Primary and Secondary Mesh The requirement of housing the global physical quantities of a problem implies that both objects with internal orientation and objects with external orientation must be available. Hence, two logically distinct meshes must be defined, one with internal orientation and the other with external orientation. Let us denote them with the symbols K and K˜ , respectively. Note that this requirement does


187

not necessarily imply that two staggered meshes must always be used, for the two can share the same nonoriented geometric structure. There are, however, good reasons usually to also differentiate the two meshes geometrically. In particular, the adoption of two dual cell complexes as meshes endows the resulting discrete mathematical model with a number of useful properties. In an n-dimensional domain, the geometric duality means that to each p-cell τ pi i ˜ of K there corresponds a (n − p)-cell τñ− p of K , and vice versa. Note that in this case we are purposely using the same index to denote the two cells, for this not only is natural but facilitates a number of proofs concerning the relation between quantities associated with the two dual complexes. We will denote with n p the number of p-cells of K and with n˜ p the number of p-cells of K˜ . If the two n-dimensional cell complexes are duals, we have n p = n˜ n− p . The names primal and dual meshes are often adopted for dual meshes. To allow for the case of nondual meshes, we will call primary mesh the internally oriented one and secondary mesh the externally oriented one. Note that the preceding discussion applies to the discretization of domains of any geometric dimension. Figure 22 shows an example of the two-dimensional case and dual grids, whereas Fig. 33 represents the same situation for the three-dimensional case.

Figure 22. The primary and secondary meshes, for the case of a two-dimensional domain and dual meshes. Note that dual geometric objects share a common index and the symbol which assigns the orientation. All the geometric objects of both meshes must be considered as oriented.

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3. Incidence Numbers Given a cell complex K, we want to give it an algebraic representation. Obviously, the mere list of cells of K is not enough, for it lacks all information concerning the structure of the complex; that is, it does not tell us how the cells are assembled to form the complex. Since in a cell complex two cells can meet at most on common faces, we can represent the complex connectivity by means of a structure that collects the data about cell-face relations. We must also include information concerning the relative orientation of cells. This can be done as follows. Each oriented geometric object induces an orientation on its boundary (Figs. 4 and 6); therefore, each p-cell of an oriented cell complex induces an orientation on its (p −1)-faces. We can compare this induced orientation with the default orientation of the faces as (p −1)-cells in K. Given the ith j p-cell τ pi and the jth (p −1)-cell τ p−1 of a complex K, we define an incidence j number [τ pi , τ p−1 ] as follows (Fig. 23): ⎧ j 0 if τ p−1 is not a face of τ pi i j def ⎨ j τ p , τ p−1 = +1 if τ p−1 is a face of τ pi and has the induced orientation ⎩ −1 as above, but with opposite orientation (87) This definition associates with an n-dimensional cell complex K a collection of n incidence matrices j (88) D p, p−1 = τ pi , τ p−1

where the index i runs over all the p-cells of K, and j runs over all the (p −1)˜ p, p−1 the incidence matrices of K˜ . In the particular cells. We will denote by D case of dual cell complexes K and K˜ , if the same index is assigned to pairs of

Figure 23. Incidence numbers describe the cell-face relations within a cell complex. All the other 3-cells of the complex have 0 as their incidence number corresponding to the 2-cell τ˜2k .


189

dual cells, the following relations hold: ˜ p, p−1 = DT D n− p+1,n− p

(89)

It can be proved with simple algebraic manipulations (Hocking and Young, 1988) that for an arbitrary p-cell τ p, the following relationship holds among incidence numbers: i j i (90) τ p , τ p−1 τ p−1 , τ p−2 = 0 i

j

Even if at first sight this relation does not convey any geometric ideas, from it there follow many fundamental properties of the discrete operators that we shall introduce subsequently. The set of oriented cells in K and the set of incidence matrices constitute an algebraic representation of the structure of the cell complex. Browsing through the incidence matrices, we can know everything concerning the orientation and connectivity of cells within the complex. In particular, we can know if two adjacent cells induce on the common face opposite orientations, in which case they are said to have compatible or coherent orientation. This is an important concept, for it expresses algebraically the intuitive idea of two adjacent p-cells’ having the same orientation (Figs. 23 and 24). Conversely, given an oriented p-cell, we can use this definition to propagate its orientation to neighboring p-cells [on orientable n-dimensional domains it is always possible to propagate the orientation of an n-cell to all the n-cells of the complex (Schutz, 1980)].

Figure 24. Two adjacent cells have compatible orientation if they induce on the common face opposite orientations. The concept of induced orientation can be used to propagate the orientation of a p-cell to neighboring p-cells.

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4. Chains Now that we know how to represent algebraically the cell complex, which discretizes the domain, we want to construct a machinery to represent generic parts of it. This means that we want to represent an assembly of cells, each with a given orientation and weight of our choice. A first requirement for this task is the ability to represent cells with the default orientation and cells with the opposite one. This is most naturally achieved by denoting a cell with its default orientation with τ p and one with the opposite orientation with −τ p. We can then represent a generic p-dimensional domain cp composed by p-cells of the complex K as a formal sum, cp =

np

wi τ pi

i=1

τ pi ∈ K

(91)

where the coefficient wi can take the value 0, +1, or −1, to denote a cell of the complex not included in cp, or included in it with the default orientation or its opposite, respectively. This formalism, therefore, allows the algebraic representation of discrete subdomains as “sums” of cells. We now make a generalization, allowing the coefficients of the formal sum [Eq. (91)] to take arbitrary real values wi ∈ R. To preserve the representation of the orientation inversion as a sign inversion, we assume that the following property holds true: wi −τ pi = −wi τ pi (92)

With this extension, we can represent oriented p-dimensional domains in which each cell is weighted differently. This entity is analogous, in a discrete setting, to a subdomain with a weight function defined on it; thus it will be useful in order to give a geometric interpretation to the discretization strategies of numerical methods, such as finite elements, which make use of weight functions. In algebraic topology, given a cell complex K, a formal sum like Eq. (91), with real weights satisfying Eq. (92), is called a p-dimensional chain with real coefficients, or simply a p-chain cp (Fig. 25). If it is necessary to specify explicitly the coefficient space for the weights wi and the cell complex on which a particular chain is built, we write c p (K , R). We can define in an obvious way an operation of addition of chains defined on the same complex, and one of multiplication of a chain by a real number λ, as follows: wi + wi′ τ pi (93) c p + c′p = i wi′ τ pi + wi τ pi = i

λc p = λ

i w τ = (λwi )τ pi i p i

i

(94)


191

Figure 25. Given an oriented cell complex (top), a p-chain (bottom) represents a weighted sum of oriented p-cells. The weights are represented as shades of gray. Notice that negative weights make the corresponding cell appear in the chain with its orientation reversed with respect to the default orientation of the cell in the cell complex.

With these definitions the set of p-chains with real coefficients on a complex K becomes a vector space C p (K , R) over R, often written simply as C p (K ) or C p . The dimension of this space is the number n p of p-cells in K. Note that each p-cell τ p can be considered an elementary p-chain 1 · τ p. These elementary p-chains constitute a natural basis in Cp, which permits the representation of a chain by the n p -tuple of its weights: c p = (w1 , w2 , . . . , wn p )

(95)

Working with the natural basis, we can easily define linear operators on chains as linear extensions of their action on cells. In particular, this is the case for the definition of the boundary of a chain. 5. The Boundary of a Chain The boundary ∂τ p of a cell τ p is by definition the collection of its faces, endowed with the induced orientation (Figs. 4 and 6). Remembering the definition of the incidence numbers, we can write ∂τ p =

n p−1 j j τ p , τ p−1 τ p−1

(96)

j=1

where the index j runs on all the (p −1)-cells of the complex. Note that Eq. (96) gives to a geometric operation an algebraic representation based uniquely on incidence matrices. Since the p-cells constitute a natural basis

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for the space of p-chains, we can extend linearly the definition of ∂ to an operator—the boundary operator—acting on arbitrary p-chains, as follows: i ∂c p = ∂ wi ∂τ pi (97) wi τ p = i

i

Thus the boundary of a p-chain is a (p −1)-chain, and ∂ is a linear mapping ∂ : C p (K ) → C p−1 (K ) of the space of p-chains into that of (p −1)-chains. It can be proved (Hocking and Young, 1988), by using Eq. (90), that for any chain cp the following identity holds true: ∂(∂c p ) = 0

(98)

That is, the boundary of a chain has no boundary, a result that, when applied to elementary chains (i.e., to p-cells), satisfies our geometric intuition. The boundary of a cell defined by Eq. (96) coincides practically with the usual geometric idea of the boundary of a domain, complemented by the fact that the faces are endowed with the induced orientation. The calculation of the boundary of a chain defined by Eq. (97) can instead give a nonobvious result. Let us consider p-chains built with a set of cells that form a p-dimensional domain (Fig. 26). For some chains of this kind, it may happen that the result of the application of the boundary operator includes (p−1)-cells that we typically do not consider as belonging to the boundary of the domain. In fact, it turns out

Figure 26. Given a p-chain c p (top), its boundary ∂c p is a ( p − 1)-chain (bottom) that usually includes internal “vestiges” with respect to what we are used to considering the boundary of the domain spanned by the p-cells appearing in the p-chain. The weights of 2-cells are represented as shades of gray and those of 1-cells by the thickness of lines.


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that this represents the rule, not the exception, since each “internal” (p −1)-cell of the domain cp appears in Eq. (97), unless the sum of the weights received by it from the p-cells of which it is a face [the so-called cofaces of the (p −1)-cell] vanishes. Obviously, this vanishing is true only for particular sets of weights; that is, for particular chains. Later, we shall build a correspondence between chains and weighted domains. In that context, the boundary of a weighted domain will be defined, and the result will turn out to be confined to the traditional boundary only for particular weight functions.

B. Fields A consequence of our traditional mathematical education is that when we hear the word field we tend to think immediately of its representation in terms of some kind of field function; that is, of some continuous representation. If we refrain from this premature association, we can easily recognize that the transition from what is observed to this kind of representation requires a nontrivial abstraction. In practice, we can measure only global quantities; that is, quantities related to macroscopic p-dimensional space–time subdomains of a given domain. It is, however, natural to imagine that we could potentially perform an infinite number of measurements for all the possible subdomains. We then conceive this collection of possible measurements as a unique entity, which we call the field, and we represent this entity mathematically in a way that permits the modeling of these measurements, for example, as a field function that can be integrated on arbitrary p-dimensional subdomains. Consider now a domain in which we have built a mesh, say, a cell complex K. By so doing, we have selected a particular collection of subdomains, the cells of the complex K. Consequently we must (and can) deal only with the global quantities associated with these subdomains. The fields will manifest themselves on this mesh as collections of global quantities associated with these cells only. Of course, this association will be sensitive to the orientation and linear on cell assembly. This, in essence, is the idea behind the representation of field on discretized domains in terms of cochains. 1. Cochains Given an oriented cell complex K and an (algebraic) field F , consider a function c p which assigns to each cell τ pi of K (thought of as an elementary chain) an element ci of F , written i p (99) τ p , c = ci

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and is linear on the operation of cell assembly represented by chains; that is, it satisfies wi τ pi , c p wi τ pi , c p = (100) (c p , c p ) =

This function c p is called a p-dimensional cochain, or simply p-cochain c p . It can be written as c p (K , F ) or c p (K) to designate explicitly the cell complex and the algebraic field involved in the definition [when the complex is externally oriented, we will write c p ( K˜ ) if the complex is explicitly mentioned, and c˜ p if it is not]. We will call ordinary cochains those defined on an internally oriented cell complex, and twisted those defined on an externally oriented one (Burke, 1985; Teixeira and Chew, 1999b). We can readily see that this definition contains the essence of what we said previously concerning the action of physical fields on domains partitioned into cell complexes. The cochain, like a field, associates a value with each cell, and the association is additive on cell assembly. Note that from Eq. (100) it follows that (−τ p , c p ) = −(τ p , c p )

(101)

That is, as expected, the value assumed by a cochain on a cell changes sign with the inversion of the orientation of the cell. Thus, the only thing that must be added to the mathematical definition of a cochain to make it suitable for the representation of fields is the attribution of a physical dimension to the values associated with cells. With this further attribution the values can be interpreted as global physical quantities (which—we stress again—need not be scalars) and the corresponding entity can be called a physical p-cochain. All cochains considered in this work must be considered physical cochains, even if the qualifier “physical” is omitted. From Eq. (100) we see that a cochain c p is actually a linear mapping c p : C p (K ) → F of the space of chains Cp(K) into the algebraic field F , which assigns to each chain cp a value (c p , c p )

(102)

This representation emphasizes the equal role of the chain and of the cochain in the pairing. To assist our intuition, we can think of Eq. (102) as a discrete counterpart of the integral of a field function on a weighted domain, and this can suggest the following alternative representation for the pairing (Bamberg and Sternberg, 1988): cp (103) (c p , c p ) ≡ cp


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We can define the sum of two cochains and the product of a cochain by an element of F , as follows: ′

′

(c p , c p + c p ) = (c p , c p ) + (c p , c p )

(104)

(λc p , c p ) = λ(c p , c p )

(105) p

This definition transforms the set of cochains in a vector space C (K , F ) over F , usually written simply as C p (K) or C p . A natural basis for this vector space is constituted by the elementary p-cochains which assign the unity of F to a p-cell and the null element of F to all other p-cells of the complex. The dimension of C p (K) is, therefore, the number n p of p-cells in K, and on the natural basis we can represent uniquely a cochain as the n p -tuple of its values on cells: c p = (c1 , c2 , . . . , cn p )T ci = τ pi , c p ∈ F (106) With this representation, and with the corresponding one for a chain [Eq. (95)], the pairing of a chain and a cochain is given by (c p , c p ) =

np

wi ci

(107)

i=1

In the case of a physical cochain, the natural representation would be an n p -tuple of global physical quantities associated with p-cells. For example, in ˜ 3 is represented by the a heat transfer problem the heat content 3-cochain Q c n˜ 3 -tuple of the heat contents of the 3-cells τ˜3 of the cell complex K, which discretizes the domain: ˜ 3 = Q 1 , Q 2 , . . . , Q n˜ 3 T Q (108) c c c c where

˜3 Q ic = Q c τ˜3i = τ˜3i , Q (109) c n˜ 3 The heat Qc associated with a chain c˜ 3 = i=1 wi τ˜3i corresponds, therefore, to n˜ 3 ˜3 = Q c = c˜ 3 , Q wi Q ic c

(110)

i=1

Note the similarity with a weighted integral: wqc Qc =

(111)

V˜

Using the concept of cochain, we can redraw the classification diagrams of physical quantities for a discretized domain, substituting the field functions

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Figure 27. The Tonti classification diagram of global electromagnetic physical quantities in terms of cochains. Note the presence of two null cochains, corresponding to the absence of magnetic flux production and to the absence of electric charge production.

with the corresponding cochains. For example, in electromagnetism we have ˜ 2 of the 1-cochain U1 of electromagnetic potential; the 2-cochains 2 and 3 ˜ magnetic flux and electric flux, respectively; and the 3-cochain Q of electric charge (to which we must add the null 3-cochain 03 of magnetic flux production and the null 4-cochain 0˜ 4 of electric charge production). The corresponding classification diagram is depicted in Figure 27. Remark III.1 It is sometimes argued that on finite complexes, cochains and chains coincide, since both associate numbers with a finite number of cells (Hocking and Young, 1988). Even disregarding that the numbers associated by chains are dimensionless multiplicities whereas those associated by cochains are physical quantities, the two concepts are quite different. Chains can be seen as functions which associate numbers with cells. The only requirement is that the number changes sign if the orientation of the cell is inverted. Note that no mention is made of values associated with collections of cells, nor could it be made, for this concept is still undefined. Before the introduction of the concept of chain we have at our disposal only the bare structure of the complex—the set of cells in the complex and their connectivity as described by the incidence matrices. It is the very definition of chain which provides the concept of an assembly of cells. Only at this point can the cochains be defined, which associate numbers not only with single cells, as chains do, but also with assemblies of cells. This association is required to be not only orientation dependent, but also linear with respect to the assembly of cells represented by


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chains. This extension from weights associated with single cells to quantities associated with assemblies of cells is not trivial and makes cochains a very different entity from chains, even on finite cell complexes. 2. Limit Systems The idea of the field as a collection of its manifestations in terms of cochains on the cell complexes that subdivide the domain of a problem, finds a representation in certain mathematical structures called limit systems. The basic idea is that we can consider in a domain D the set K of all the cell complexes that can be built on it (with the kind of orientation that suits the field at hand). We can then form a collection of all the corresponding physical p-cochains on the complexes in K. This collection can be considered intuitively the collection of all the possible measurements for all possible field configurations on D. Next we want to partition this collection of cochains into sets, with each set including only measurements that derive from a given field configuration. We define for this task a selection criterion based on the additivity of global quantities. This criterion is the relation that links the cochains within each set and allows our considering each of these sets a new entity, which in our interpretation is a particular field configuration thought of as a collection of its manifestations in terms of cochains. We can define operations between fields, and operators acting on them, deriving naturally from the corresponding ones defined for cochains. For example, we can define addition of fields and the analogous of traditional differential operators (gradient, curl, and divergence) in intuitive discrete terms. This allows an easy transition from the discrete, observable properties to the corresponding continuous abstractions. The reader is warned that the rest of this section is abstract, as compared with the prevailing style of the present work. The details, however, can be skipped at first reading, since only the main ideas are required in the sequel. The point is not to give a sterile formalization to the ideas presented so far, but to provide conceptual tools for the representation of the link existing between discrete and continuous models. Let us now address the mathematics. Consider the set K = {K α } of all cell complexes which subdivide a domain D. In this case, the complexes are internally oriented, but they could be externally oriented ones as well. We will say that a complex Kβ is a refinement of Kα—written Kα < Kβ —if each cell of Kα is a union of cells of Kβ . The set K is partially ordered by the relation A).

more together with a high-power semiconductor laser of 680 nm. This is because the semiconductor laser is preferable for reliability and reducing the system size. Figure 3 shows a comparison of the stimulation spectra of the two phosphors. The advantage of the BaFBr0.85I0.15 : Eu2+ is apparent for the 680-nm laser. The particle size of the phosphor affects the resolution and the noise of the image. The IP in the 1980s was made from about 7-μm phosphor particles, but in the 1990s, 4-μm phosphor particles were made. Using a smaller particle size improves the resolution and reduces the noise of the phosphor grain. Therefore, the direction of phosphor development is to obtain the phosphor of higher luminescent intensity and smaller particle size. Figure 6 also illustrates

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schematically how the reading light scatters in the phosphor layer. The degree of scattering determines the resolution in position and the detection intensity of the luminescence. The thicker phosphor layer increases the absorption efficiency of ionizing radiation and luminescent intensity, especially for X-rays; however, scattering degrades the resolution. To improve the resolution, a blue pigment, which absorbs only reading light, is some-times useful. As the phosphor particle scatters light, it may be useful to use a transparent phosphor layer such as a single crystal; however, the reading light is reflected at the other side of the layer, and this reflection occurs back and forth many times. Thus, the resolution should become much worse, and another technology is needed to reduce the broadening of the laser beam. The protective layer is important for the durability of using the IP many times; however, the thickness of the layer affects the resolution of the image. Generally, a thicker protective layer is better for durability but worse for resolution. Another point concerning the protective layer is the attenuation effect for ionizing radiation. The penetration depth of the electron at the protective layer estimated by the equation of Katz and Penfold (1952) is 70 μm for 100-keV, 2 μm for 10-keV, and 0.07 μm for 1-keV electrons. The maximum energy of an electron of tritium is about 10 keV; more than half its energy will be dissipated even for the 1-μm protective layer. Thus, the Ip for tritium has no protective layer. The phosphor does not degrade in normal humidity, but it does decompose on contact with water. In the field of autoradiography, where the surface is in contact with the sample, water contained in the sample often permeates the phosphor through the protective layer. Thus, sample dryness is important for durability of the IP. These features provide a survey of the structure; however, there are many commercially available types of the IP similar in outward appearance. Size, thickness, and flexibility may vary. There are high-resolution types and highsensitivity types. In practice, some are better for X-rays, others for TEM and autoradiography. Each of them combines almost exclusively with a particular reading system. Thus, in selecting an IP for a specific purpose, one should study all the characteristics of the system.

IV. Elements of the IP System Figure 7 shows the typical configuration of the IP system. The reader reads out the IP after exposure to ionizing radiation. Luminescence from the IP is photoelectrically detected and converted to a digitized electrical signal to be processed by the computer system. The eraser then exposes the IP to visible light to erase stored data and the IP becomes reusable. The details of this procedure are as follows.

THE IMAGING PLATE AND ITS APPLICATIONS

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Figure 7. The configuration of the IP system. The IP system comprises the IP, the reader, the eraser, and the processor. After exposure to ionizing radiation, the IP is fed into the reader where the IP is scanned with the visible laser. The IP emits blue light, the intensity of which is proportional to the dose of ionizing radiation. The luminescence is detected by a photomultiplier tube and converted to an electrical signal. In the eraser, data are processed to be enhanced or analyzed to measure intensity and so on. After reading, the IP is irradiated with light to erase data stored in the IP.

A. Exposure The IP is a two-dimensional sensor for ionizing radiation. When the IP is exposed to ionizing radiation, the Ip stores the radiation energy as a latent image. Since stored energy on the IP disappears with light exposure, we must ensure that the ionizing radiation falls on the IP in the dark. For diagnostic X-ray imaging, it is convenient to use the light-shield case known as the cassette. This cassette is almost the same as that used in a conventional film screen system. For autoradiography, when the sample is very thin, like a membrane, we can expose the IP by contact with the sample in the cassette. However, when the sample is too thick to use the cassette, we need a box to shield light. The IP system is so sensitive a detector of environmental radioactivity, that the latter causes a noiselike fog level of photographic film. So that we can avoid this, it is preferable to erase just before an exposure to eliminate any prior stored

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activity and, furthermore, to use a shield box made of lead for long exposures as in autoradiography. After exposure, it is better to read the plate as soon as possible because the stored energy gradually escapes even in a dark place. We call this phenomenon fading.

B. Reading In the reader, the IP is scanned with a red-light beam that is focused on the surface of the IP. Luminescence is about 390 nm for BaFX : Eu2+ and comes to the detector with excitation light. Thus, the optical filter in front of the photodetector is used for cutting off laser light. There are many possible ways of scanning (Fig. 8). In the case of a flat-bed scanner, the IP is held on the flat bed and transported. A rotating mirror reflects laser beam light and focused beam spots move in a straight line on the IP. An F-θ lens is used to achieve uniform velocity scanning on the IP. The plate moves along the perpendicular direction to that of the spot motion. In the spinner-type scanner, the IP is fixed along the inner surface of the cylinder. Laser light passes through the dichromatic mirror (A), is reflected by the mirror (B), and is then focused on the IP surface. Luminescence of the IP is collected by a lens; the lens and mirror rotate as indicated by the dashed line in the figure. The pair of lenses forms a confocal configuration, which is used for PIXsysTEM. However, in the FDL-5000 system, lens (C) is not used. In the drum-type scanner, the IP is fixed on the cylindrical drum and the reading head moves parallel to the axis of the drum. In the disk-type scanner, the IP rotates and the reading head moves along the radial direction. In this type of scanner, the spatial density of reading must be kept the same between inner and outer positions of the plate. The flat-bed type is the most popular for medical applications or biotechnology; the spinner type is used for TEM.

C. Erasing After reading, exposing the IP to visible light erases the data stored. The light source is an ordinary fluorescent lamp or sodium lamp, chosen for its electrical power efficiency. The erasing level restricts the lowest detected level of the next measurement. High sensitivity means essentially competition with detection of unwanted environmental activity. Although the film has a one-way characteristic of storing information, the IP has a reset procedure by erasing. This is one of the reasons that the IP system can achieve highsensitivity detection.


293

Figure 8. Various types of scanners. (a) Flat-bed-type scanner: The IP is held on the flat bed. The laser beam reflected by a rotating or turning mirror scans the IP. The luminescence from the IP is guided to the photomultiplier tube (PMT) through the light guide. (b) Spinner-type scanner: The IP is held on the inner side of the cylinder and moves along the direction of the axis of the cylinder while the reading head (spinner) rotates. (c) Drum-type scanner: The IP is put on the rotating cylinder (drum). The reading head moves along the direction parallel to the axis of the cylinder. (d) Disk-type scanner: The IP turns around. The reading head, which irradiates it with laser beam light and collects luminescence, moves along the radial direction.

D. Image Processor The latest technology of computer and memory devices makes it possible to execute complicated image-processing tasks much more rapidly. Image processing is useful to distinguish patterns or to measure the quantity of activity and pattern shape, gradation processing, narrowing of the range, and enhancement contrast of the image. By broadening the range, we can easily

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observe whole patterns of large dynamic range such as diffraction patterns. Since the IP system has good linearity, direct reliable quantization is possible with image data, and displaying the profiles of image data is also useful to compare activity. Fast Fourier transform (FFT) or contour map processing is useful to improve distinguishing power. The “superimpose” function allows one to write letters or arrow marks on the recorded image, and this is useful for presentation. In the field of diagnosis, image processing may indicate the point that doctors should note. These types of processing enhance the value of the image and this is one of the merits of the IP system. Image data can be stored on a large-capacity memory device such as an optical magnetic disk; thus, we can archive image data and retrieve images quickly.

V. Characteristics of the IP System In this section, we will discuss mainly the sensitivity and dynamic range, the resolution, fading, and noise. The noise characteristic is important to assess the efficiency of the detector, although it is difficult to calculate. These characteristics will be discussed by using the data of the TEM system. However, this discussion should be applicable to other fields, if one takes into consideration any differences of ionizing radiation.

A. Sensitivity Sensitivity is the luminescent intensity detected. Thus, the flow of image data is important for any discussion of the sensitivity factor (Fig. 9). Let N be the initial number of quanta of ionizing radiation. This number is multiplied by efficiency factors. The efficiency of the IP is represented by α(I). Alpha includes the absorption efficiency of ionizing radiation, electron- and holecreating efficiency, and readout efficiency. Alpha depends on the intensity I of reading light; the dependence of α on laser intensity I is gradually saturated in a practical system. In the case of X-rays of 80 kVp, α is estimated to be about 10–200 in practical systems. Beta is the light-collecting efficiency, including the transmission characteristics of optical elements such as filters, light-collecting guides, or lenses. This is normally 0.1–0.5. Chi is the quantum efficiency of the photodetector. As for the PMT, it is the quantum efficiency of the photocathode, typically 0.1–0.3. Delta is the amplifying factor of the PMT or electrical circuit, normally 102–107. If we use this notation, detected luminescent intensity becomes Luminescent intensity = N α(I )βχ δ


295

Figure 9. The flow of image carriers. N quanta of ionizing radiation fall on the IP, which absorbs ionizing radiation and emits photons with efficiency α(I), when stimulated with light of intensity I. Photons from the IP reach the photodetector with efficiency β, which is defined by the light-collecting efficiency and transmission coefficient of the optics. Photons are converted to electrons by the photodetector with efficiency χ , and the number of electrons increases in both the photodetector and the electrical amplifier by a factor δ. The final number is the product of these efficiencies.

The signal intensity of the reader of PIXsysTEM as a function of electron dose is shown in Figure 10 along with the data of photographic film (Mori, Oikawa, Harada, et al., 1990). The figure shows good linearity of signal intensity to electron dose over five decades. The IP is used for many other types of ionizing radiation, and the linearity of the PSL intensity to the dose of radiation is generally observed. This is because ionizing radiation creates electrons and holes in the phosphor without any nonlinear process irrespective of the kind of radiation, although the efficiency will be different. The vertical axis for film is optical density. Although it may be possible to obtain a straight line by using another unit or calibrated data, the drawbacks of using film are its narrow dynamic range of about two decades and slightly poorer reproducibility, since the density changes by chemical conditions like the concentration or the temperature of the developer. Thus, using the IP improves the precision compared with that of the photographic film method. Figure 11 shows the dependence of the sensitivity of PIXsysTEM on accelerating voltage ( Mori, Oikawa, Harada, et al., 1990). The IP system shows its maximum intensity at about 150 keV. Ogura and Nishioka (1995) measured the dependence of the sensitivity for 40–200 keV for the FDL-5000 system and obtained similar results to those of Figure 11. The origin of the decrease below 100 keV is thought to be due to electron absorption by the protective layer.

296


Figure 10. Sensitivity characteristics of the IP system (PIXsysTEM). The signal intensity of the IP is plotted. The density curve of FG film, developed by D-19 for 2 min, is also plotted as a reference (Mori, Oikawa, Harada, et al., 1990).

The interpretation of the decrease in the higher-energy region is as follows: As the energy of the electrons goes up, the penetration depth of the incident electrons increases and electron energy is mainly dissipated in the deeper part of the phosphor layer. However, the intensity of the light for reading becomes weaker in the deeper part of the phosphor layer because of the absorption and diffusion of light. Luminescence from the deeper part of the phosphor layer is also diffused and weakened. As a result, the detected intensity of luminescence becomes weakened. Electrons of much higher energy will pass through the phosphor layer, and the intensity will then decrease substantially. B. Resolution The IP itself does not have discrete pixels, but a pixel is created as the electrical signal by the reader. Thus, signal response is very important for resolution.


297

Figure 11. Dependence of sensitivity on accelerating voltage. The signal intensity of the IP reader (PIXsysTEM) was measured by changing the accelerating voltage of the electrons (Mori, Oikawa, Harada, et al., 1990).

One of the factors determining the resolution is the scattering of the laser beam in the phosphor layer, as discussed in the IP section (Section III). Another factor is the time response of the luminescence and the photodetecting system. The decay characteristic of the luminescence, the time in which the luminescence declines to 1/e intensity, is about 0.6 μs in the case of BaFBr0.85I0.15 : Eu2+; the reading time for one pixel should be longer than this time. The response of the electrical system, which converts luminescence to a digital electrical signal, should be shorter than the time for one pixel. Of the many ways of evaluating resolution, some researchers select the method in which the lattice image of a gold crystal of graphitized carbon is used. This way is very practical, but the result is affected by the characteristics of the TEM and the operating conditions when one is taking images. A method using a metal wire has been examined (Burmester et al., 1994; Isoda et al., 1992). The wire was directly fixed on the IP; uniform electron radiation created a shadow of the wire on the IP. The resolution as MTF (modular transfer function) was determined by the frequency analysis of the difference between the theoretical image and the observed image: MTF(q) = Fobs (q)/Ftheo (q), where q was spatial

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Figure 12. The resolution of the IP system. The results of the response measured with the metal plate method are summarized. Squares: HR-II IP and CR-101 systems; closed circles: UR-III IP and PIXsysTEM (Mori, Oikawa, Harada, et al., 1990). Open circles: FDL-URV and FDL-5000 system (Ogura et al., 1994).

frequency, Fobs(q) was the amplitude of the Fourier spectrum of the observed shadow profile, and Ftheo(q) was the amplitude of the spectrum of the theoretical square-well profile. Instead of Fourier analysis, one may use a metal mask that has a pattern of openings of various spatial frequencies (Mori, Oikawa, Harada, et al., 1990). Uniform exposure made a square wave pattern on the IP; the readout amplitude of the wave pattern declined at a higher spatial frequency. Thus, the resolution was expressed by Response (q) = A(q)/A(0), where A(q) was the amplitude of the image profile at spatial frequency q. This corresponds to the contrast transfer function (CTF). The MTF and the CTF give almost the same result; however, the MTF is more suitable for treatment of theoretical analysis. Figure 12 shows the improvement of the resolution by comparing the resolutions of the three systems, measured with the metal mask method. Squares indicate the result of flat-bed scanning, with a pixel size of 100 μm; closed circles that of PIXsysTEM, with a pixel size of 50 μm; and open circles


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that of FDL-5000, with a pixel size of 25 μm. The improvement in resolution is important for TEM systems, since we can take images covering a wider area and at lower TEM magnification.

C. Fading The intensity of the stored image on the IP decreases with the passage of time. Figure 13 shows the fading characteristics of PIXsysTEM (Oikawa, Shindo, and Hiraga, 1994). The degree of fading depends on temperature, however, and is generally larger as the temperature is higher. This characteristic depends on the phosphor itself and on the wavelength of the reading light. There is no precise comparison of the dependence on the various types of the IP, but

Figure 13. Fading characteristics. Intensity change with the passage of time is plotted at 0 and 25◦ C. The measurement was made with doses of 1 × 10−10, 10−11, 10−12 C/cm2 (Oikawa, Shindo, and Hiraga, 1994).

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there is not much difference. Oikawa, Shindo, and Hiraga proposed empirical equations of fading as a function of temperature. This is useful for estimating the degree of fading. The fading characteristic does not depend on the dose; this is very important as it is possible to compare the intensity even after fading. The fading is negligible provided that the IP is kept in cool conditions, but this is not practical for TEM use. For autoradiography, however, it should help to increase the sensitivity because of its long time exposure. D. Granularity and Uniformity Image noise (granularity) is directly related to the perceptivity of the image. In this sense, noise is another aspect of the sensitivity of the system. Granularity is the deviation of the intensity of each pixel, composed of mainly two components. One is dependent on the number of image carriers, while the other is not and has a fixed value. The former follows a statistical deviation, the Poisson distribution: (Noise)2 = 1/n, where n is the number of image carriers (Dainty and Shaw, 1974). This number of image carriers changes as the detection process proceeds (Fig. 9). The fixed noise is electrical noise or the fixed noise of the IP. Total noise [reciprocal of signal-to-noise ratio (S/N )] is expressed as the sum of the individual types of noise: 1 1 1 1 1 1 1+ + Noise2fix + + + = (S/N )2 N α(I ) α(I )β α(I )βχ α(I )βχ δ On the basis of the preceding equation, the fixed noise appears at high dose (N is large) and determines the lower limit of the signal-to-noise ratio of the system. Conversely, the 1/N term appears at low dose (N is small). The multiplier is composed of α, β, χ, and δ. Alpha, the efficiency of the IP, is contained except for the first term. Beta and χ are important, since they are less than unity and may become the dominant part of the noise at low dose. Figure 14 shows noise characteristics of the FDL-5000 system (Ogura and Nishioka, 1995). The noise becomes better as the electron dose increases. The noise power is inversely proportional to the number of electrons exposed; however, improvement saturates because of the fixed noise. This figure shows that noise follows the relation just given. The efficiency of detectors is often discussed using a term called detective quantum efficiency (DQE), related to the noise characteristics, because it does not depend on the method of detection. The DQE is expressed as 2 $ 2 So Si DQE = No Ni where as usual S is signal and N is noise. Subscripts o and i are output and


301

Figure 14. Noise characteristics of the IP system (FDL-5000; Ogura and Nishioka, 1995).

input, respectively. The denominator is the number of quanta of ionizing radiation. The numerator is generally expressed by the equation discussed in the last paragraph. However, when one is calculating the numerator for DQE, it is false to take the noise of granularity directly because the resolution characteristic reduces the noise in appearance. With compensation for this factor, frequency analysis of noise becomes important together with resolution. In the case of 80-kVp X-rays at 1 mR, Ogawa et al. (1995) reported a DQE of 0.2 at 1 line-pair per mm for the FCR 9000/ST-V system. This is an accurate way of characterizing a system, but it is difficult because it needs resolution data (MTF) and the data must be processed by FFT (Dainty and Shaw, 1974). A convenient way of calculating DQE with larger pixels such as 3 × 3 is sometimes used for minimizing the effect of the response, although information about the frequency dependence is lost. Thus, in the case, the value should be discussed together with resolution. Burmester et al. (1994) estimated by this convenient way that the DQE of their IP system is about 0.9 for 120-keV electrons at about 10−13 C/cm2. Ogura and Nishioka (1995) also calculated the DQE of the FDL-5000 using the data of Figure 14 and found a value of almost unity for 100-keV electrons at about the same dose region, taking care to measure

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the electron dose accurately. The difference is large between the DQE result of Ogawa et al. and that of Burmester et al. We suppose that this is due to the difference in ionizing radiation (X-rays and electrons). Electrons in this energy region will all be stopped as is predicted by the Katz–Penfold equation; for X-rays, however, more than 50% of the X-ray photons escape from the phosphor layer. Thus, the efficiency α will be very different between the two sources. Another factor is involved when the continuous signal is converted to a digital signal: the density resolution of the signal. When the density resolution is not as small as the noise level of the image data, the pattern will have artifacts such as contours, or the precision of quantitative analysis will become degraded. However, too small a density resolution leads to a waste of memory resources or time for image processing. Normally, data are logarithmically transformed, as expressed by the following equation: I = A · 10(L · (Q/M)) where L is the dynamic range of the image, m is the density resolution expressed by the bit number p : m = 2 p, and Q denotes the digital data. The change of fraction between Q and Q +1 is D = L/m (ln 10), which is sometimes called the error of quantization. The value D should be almost the same as that of the image noise. For example, in the case of L = 4 and noise = 0.4%, then m should be 1000, which means that the density resolution should be 10 bit (1024). This density resolution should be selected depending on the application field because the necessary signal-to-noise ratio depends on the application field. Uniformity of sensitivity is important for quantitative analysis. In the flat-bed scanner, the uniformity of laser light intensity and the light-collective efficiency govern the uniformity characteristics. Uniformity is always the same and can in principle be calibrated in the system. In some systems, the calibration is executed automatically and the user does not need to recognize this factor. The uniformity of the IP originates mainly from the uniformity of thickness of the phosphor layer. Amemiya et al. (1988) reported that the uniformity error is about 1.3%. They concluded that this degree of uniformity is sufficient for X-ray diffraction analysis for their purpose.

VI. Practical Systems In the previous section, we discussed the principles of the IP system and dealt with the basic ideas. In this section, we will consider the practical system.


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A. Transmission Electron Microscope (TEM) System Figure 15 shows the layout and components of the TEM system of the FDL5000 (Ogura et al., 1994). We can use the IP in the TEM just like photographic film because we can use an ordinary film cassette for the IP, together with the film magazine of the TEM. After removal from the cassette, exposed IPs are put into the magazine for reading the system. After the information such as operation conditions of TEM, sample names, and reading parameters are set, the reader reads all the IPs automatically. The data of the IP are stored on the digital data storage (DDS) unit simultaneously while the reader is reading. When the printer is connected to the IP reader, the image hard copies are also available at the same time. The image data in the DDS are transferred to the processor and processed and displayed. Since the processor is independent of the reader, image capture and image analysis can be performed separately. The size of the IP used is about 94 × 75 mm. The pixel size is 25 μm. The data volume is about 23 M bytes. In the TEM, photographic film or a TV camera system has been used (Reimer, 1984). Burmester et al. (1994) summarized the DQE of image devices: less than 0.35 for photographic film and 0.4–0.7 for slow-scan chargecoupled devices; (SS-CCDs; Kujawa and Krahl, 1992). They also reported that

Figure 15. Transmission electron microscope (TEM) system (FDL-5000). In this configuration, the IP is used with the TEM cassette in the TEM and with an IP magazine in the reader. Data from the IP are transferred to the computer system by the data storage media of DDS. This is because the quantity of data in the system is several 10 M bytes, so the data transfer time is not negligible. The separation of data processing and reading makes the best of the independent operation of each step (Ogura et al., 1994).

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the DQE of their own IP system was about 0.9, as discussed in Section V.D. This high efficiency is one of the merits of the IP system. The high sensitivity is useful not only for saving the sample from damage by the electron beam, but also for making it possible to use a high-speed shutter, which is helpful for avoiding the deterioration of the image quality by the vibration of the sample. The pixel size is 25 μm; thus, the image enlarged 16 times by area is not unnatural because the resolution limit of the naked eye is 100 μm. This digital enlargement contains no distortion factor caused by the optical system of enlarging equipment, as in the case of photographic systems.

B. Computed Radiography and Radio Luminography System The IP system was first used in the medical field for X-ray imaging. In this field the technique was called computed radiography (Tateno et al., 1987). High sensitivity is good for reducing the dose of the patient. The digital image enables us to make a picture archiving and communication system (PACS) and allows comparative diagnosis between isolated hospitals by the transmission of digital images. The IP system is widely used in this field and various systems are now available. A built-in system, in which the system circulates the IP and exposure and the reading and erasing process is executed in one system, is very convenient for examination. TEM application, autoradiography, X-ray diffraction, and so on are called radio luminography. In these fields the scanner most popularly used is the flatbed type and for high resolution, the spinner type. The IP system was evaluated in 1986 in the field of X-ray diffraction (Miyahara et al.). The high DQE and wide dynamic range of the system, together with its absence of count-rate limitation, resulted in a significant reduction of exposure time. Thus, the IP has helped protein crystallographers to obtain accurate measurements in a shorter time. This saves the sample from beam damage, so full data can be obtained with the use of only one sample. In the case of photographic film, many samples are needed to get full data and this degrades the accuracy of the data. This is the reason why the IP system has led to much progress in this field (Amemiya et al., 1988; Sakabe, 1991). In the field of X-ray diffraction, the combination with a synchrotron-radiation source is most successful; in addition, the IP system should be promising for use with a conventional laboratory-scale X-ray source (Sato et al., 1993). In the biotechnology industry, autoradiography is commonly used to analyze gene and protein sequences. Since the exposure time ranges from a day to a month in the conventional way of using photographic film, a reduction of exposure time by a factor of more than 10 by the IP system is very useful (Amemiya and Miyahara, 1988). In addition, one can measure the radioactivity of part of the sample by image processing, without taking off the part of the


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sample and measuring by liquid scintillation counter. These merits raise the importance of the autoradiography method. Neutron radiography is used for nondestructive testing, such as inspection of organic material in a metal vessel, or neutron diffraction analysis to investigate the position of hydrogen in a protein. However, the conventional IP is not sensitive to neutrons. Niimura et al. (1994) developed an IP that contains a Gd or Li compound in the phosphor layer. Since Gd or Li atoms have large cross sections for neutrons, absorb neutrons, and emit gamma rays or electrons, these can be detected by the phosphor. These researchers justified the merits of this system and demonstrated neutron radiography with the IP. Katto et al. (1993) measured the beam profile of an ultraviolet (UV) laser with the IP for tritium. Since BaFX : Eu2+ phosphor is sensitive to UV-VUV (UV–vacuum ultraviolet) light (Iwabuchi, Mori, et al., 1994), the IP is a valuable image device in this region. Nishikawa, Akimoto, et al. (1994) examined field-emission and field-ion microscopies with the IP; that is, images of He+ or Ne+. They showed the possibility of a quantitative analysis of electron tunneling and a field ionization probability over individual surface atoms. It is the combination of all these characteristics—sensitivity, dynamic range, resolution, and large effective area—that generates the superiority of the IP system. In some characteristics, another image system is better than the IP system. For example, the film system has good resolution and a wide effective area, but its sensitivity and dynamic range are not sufficient. The TV camera system has good sensitivity, spatial resolution, and time resolution; however, the effective area is small. The IP system does not suffer from the drawbacks of the film system and is suitable for the detection of images of ionizing radiation. Furthermore, it is important to comment on the easiness of handling of the IP system. The IP itself does not need any electric power. It is merely a thin plate and the only essential precaution is to exclude stray light. On reading, we need a large precision system; however, this is not an obstacle at exposure. This easiness is another merit of the IP system. Thus, we can apply the IP to many fields of imaging—electromagnetic waves from the UV region to the gammaray region, electrons, ion beams, and neutrons. Its characteristics overcome the drawbacks of conventional image sensors. With the development of new types of IPs like those for neutron imaging, this new technology called radio luminography will expand the field and make itself more valuable.

VII. Applications of the IP In this section, application data obtained by many researchers are introduced, which illustrate the advantages of the IP. The application fields in which the IP is expected to exhibit its performance are listed in Table 1. In these fields, there

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NOBUFUMI MORI AND TETSUO OIKAWA TABLE 1 Likely Fields of Application of the IP No.

Advantages of the IP

1

High sensitivity

2

Wide dynamic range

3

Linear sensitivity

4

High-precision digital image

5

Dry system and others

a

Application fields a) Observation of beam-sensitive specimen b) Data acquisition with high-speed shutters (low- and high-temperature stages, etc.) c) Dark-field and weak beam method d) High-contrast images e) Electron diffraction and CBEDa patterns f) Electron intensity measurement g) Quantitative image analysis h) Image processing i) Image contrast enhancement j) Image filing and retrieving k) Reduction of personnel

CBED, Convergent-beam electron diffraction.

have been limitations to observation with conventional photographic film. Use of the IP is expected to break through those limitations.

A. High Sensitivity In this section, application data, illustrating the high-sensitivity performance of the IP, are introduced. For example, the IP was applied to TEM observation of silver bromide microcrystals, which are typical of the electron-sensitive materials, byAyato et al. (1990). Silver bromide (AgBr) microcrystals are so susceptible to beam irradiation damage that they are destroyed during room temperature recording using conventional photographic film, which makes recording difficult. Figure 16a shows AgBr microcrystals destroyed during exposure with conventional photographic film. The authors therefore reduced the electron dose by a factor of 100 by using the IP and thus succeeded in recording AgBr microcrystals without destroying them (Fig. 16b). The high sensitivity of the IP allowed us to record images of the silver bromide microcrystals at room temperature with very little irradiation damage by reducing the electron dose at the specimen. In low-dose observation, the IP is of great use for recording an image with good image contrast even at low-electron intensity. This is because the IP has a linear response to exposure even at low-exposure levels. Another example is a measurement of electron irradiation damage to a polyethylene single crystal (Oikawa, Shindo, Kudoh, et al., 1992). The degree of specimen damage was evaluated from the degree of intensity fading of an


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Figure 16. Electron micrographs of silver bromide microcrystals taken at room temperature (direct magnification: ×15,000). (a) Recorded with conventional photographic film (Fuji FG). Electron dose: 700 electrons/nm2. (b) Recorded with the IP. Electron dose: 7 electrons/nm2.

electron diffraction spot from the specimen (Kobayashi and Sakaoku, 1964). Figure 17 shows electron diffraction patterns of a polyethylene single crystal. These diffraction patterns were obtained at an accelerating voltage of 200 kV and an extremely low-electron dose rate, 1 electron/(nm2 · s). Moreover, the exposure time was set to 0.1 s in order to improve the time resolution per image during the exposure. Figure 17a shows an electron diffraction pattern taken by irradiating a fresh field of view with an electron beam. The image clearly shows even higher-order diffraction spots. Figure 17b shows a pattern taken after a dose of 600 electron/nm2. The logarithms of the intensity distributions of the two patterns are shown along the horizontal lines in the figures. Figure 18 shows background subtraction of the intensity of an electron diffraction with three-dimensional distributions. The spots are (200) irradiated with 200-kV electrons at doses of 250 and 480 electrons/nm2. Figures 18a and 18d show the original intensity distributions of the diffraction spots (200). Figure 18b and 18e show background intensity distributions obtained by a background fitting method (Shindo, Hiraga, Iijima, et al., 1993). Figures 18c and 18f show the net intensity distributions of the spots (200) after background intensity subtraction. Figure 19 shows the net intensity distribution changes after background subtraction of diffraction spots (200) after irradiation with

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Figure 17. Electron diffraction patterns of a polyethylene single crystal with a thickness of about 10 nm, and their intensity distributions obtained by the IP (Oikawa, Shindo, Kudoh, et al., 1992) (200 kV, room temperature). (a) Electron dose: 0.1 electrons/nm2 (fresh field of view). (b) Electron dose: 600 electrons/nm2.

0.1, 250, and 480 electrons/nm2. Integrating the spot intensity allowed measurement of the change of the diffraction intensity with electron irradiation. Figure 20 shows the change of the integrated (200) reflection spots for 200 and 100 kV. In this case, the incident electron intensity was obtained as the whole intensity of the diffraction pattern, and the integrated (200) reflection intensities were normalized relative to the incident intensity. In the same electron irradiation condition, the reflection intensity at 100 kV fades more rapidly than that at 200 kV. At 200 kV, the reflection intensity at 730 electrons/nm2 irradiation faded to one twentieth of the original value, and at 100 kV, the intensity at 480 electrons/nm2 faded to one tenth of the original value. Because of its wide dynamic range, the IP records both high intensities (diffraction spots) and weak intensities (halo rings) in a single image. In addition, its linear response characteristic allows quantitative measurement of


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Figure 18. Background subtraction process of the intensity of electron diffraction spots (200) irradiated with 200-kV electrons at doses of (a–c) 250 electrons/nm2 and (d–f) 480 electrons/nm2.

the beam intensity. Furthermore, using the high sensitivity of the IP allows the exposure to be carried out with a very low dose, by using a high-speed shutter. The intensity fading of the diffraction spots of polyethylene with electron irradiation had already been measured by the X-ray diffraction method (Kawaguchi, 1979). However, the electron diffraction method is more useful

Figure 19. Change of intensity distribution of diffraction spots (200) of polyethylene irradiated with 200-kV electrons: (a) 0.1 electrons/nm2, (b) 250 electrons/nm2, (c) 480 electrons/nm2.

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Figure 20. Change of diffraction intensity of the integrated (200) reflection spots irradiated with 200-kV (closed circles) and 100-kV (open circles) electrons.

than the X-ray diffraction method because the electron diffraction intensity is recorded simultaneously from the same specimen field of view, during electron irradiation.

B. Wide Dynamic Range In this section, application data illustrating the wide dynamic range performance of the IP are introduced. Since a convergent-beam electron diffraction (CBED) pattern has an intensity range covering about three orders of magnitude, the entire pattern cannot be recorded in a single image with conventional photographic film. With the IP, the dynamic range covers four orders of magnitude on a single image, which allows all the intensities of a CBED pattern to be covered. Figure 21a shows a CBED pattern recorded using the IP. Figure 21b shows a line profile (intensity distribution) along the center position of Figure 21a (indicated by the horizontal line). This profile shows that the pattern was recorded without saturation or loss, from the center to the periphery of the CBED pattern, which indicates the large width of the dynamic range. Figure 22 is a kind of a contour map presentation, obtained by dividing the intensity range of the image of Figure 21a into 16 parts and rendering the intensity steps of each part white and black alternately (Oikawa,


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Figure 21. CBED patterns taken with the IP and a JEM-2000FX II TEM at 100 kV. The specimen was a silicon (111) single crystal. (a) Low-contrast print. (b) Line profile (the intensity distribution) of part (a).

Mori, et al., 1990). It is seen that this presentation not only allows the pattern of the entire image to be recognized, but also is effective for extracting the features of the fine structures. With the IP, which has high-intensity resolution (4096 gray levels), contrast enhancement and image analysis applications can be carried out with high precision. Electron diffraction patterns of a Cu3Pd alloy were quantitatively analyzed by making good use of the wide dynamic range and good linearity of the IP by Shindo, Hiraga, Oikawa, et al. (1990). intensities of both fundamental and superlattice reflections of the alloy having a one-dimensional, long-period superstructure were measured in situ as a function of the temperature. The intensity changes of the superlattice reflections quantitatively evaluated clearly show the characteristic disordering process of the Cu3Pd alloy. It was demonstrated that quantitative structure analysis by electron diffraction patterns is possible with the use of the IP if the dynamical diffraction effect is taken into account. In this study, by measuring the intensities of the superlattice reflections and short-range-order diffuse scattering, the researchers quantitatively investigated the order–disorder transition of the Cu3Pd alloy, using the advantages of the IP, that is, a wide dynamic range and good linearity for the electron beam.

312


Figure 22. A contour map of the data in Figure 21a, showing that the intensity is recorded well over the whole pattern (Oikawa, Mori, et al., 1990).

In Figure 23, an electron diffraction pattern of Cu3Pd obtained with the IP is shown. The original signal intensities of 4096 gray levels were simply converted to 256 gray levels for the output; that is, each of the 16 gray levels of the original data were converted into 1 gray level in the output print of a diffraction pattern. The electron diffraction pattern shows sharp superlattice reflections, labeled A1, A2, B1, B2, and C. These superlattice reflections indicate a one-dimensional, long-period superstructure. In the single-crystal film, superlattice reflections from three variants are usually observed. The spots A, B, and C indicated in the pattern correspond to the three variants. The reflections A1 and B1 correspond to the periodicity of the basic ordered structure of the L12-type whereas A2, A3, B2, and B3 correspond to the periodicity of a long-period superstructure along each direction. By measuring the separation of superlattice reflections such as A2 and A3, the researchers obtained the period of the one-dimensional, long-period superstructure as M = 3.6. Figures 24a and 24b are electron diffraction patterns observed with the IP after the alloy was heated in the electron microscope at 823 K. Figure 24a is a pattern output in the same manner as in Figure 23, whereas in Figure 24b, only the gray levels below gray-level 1400 in the original signal intensity were converted into 256 gray levels; the gray levels above gray-level 1400 were


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Figure 23. Electron diffraction pattern of a single-crystal Cu3Pd observed by the IP. An , Bn (n = 1–3) and C indicate the superlattice reflections corresponding to three variants.

set to the value 256 for the output. It should be noted that in Figure 24a the superlattice reflections sharply observed in Figure 23 become faint. However, in Figure 24b, the diffuse scattering broadening at the positions of reflections such as A2 and A3 is clearly observed, which suggests the existence of a shortrange-ordered state, although the intensity of the transmitted beam and the fundamental reflections are saturated in this case.

Figure 24. (a) Electron diffraction pattern of Cu3Pd after heating to 823 K in an electron microscope. The conversion of the original intensity into the output is the same as in Figure 23. (b) The same electron diffraction pattern as that in part (a), but only the gray levels less than level 1400 of the original intensity were converted into 256 gray levels in the output print.

314


Figure 25. The change of electron diffraction patterns as a function of temperature. The intensity distribution of the superlattice reflection (or diffuse scattering) and that of the fundamental reflection are represented as a contour map. The area of the electron diffraction patterns corresponds to the square of that of the electron diffraction pattern shown in Figure 23.

In Figure 25, the intensity distribution of the electron diffraction patterns was plotted as a contour map in order to make clear the change of the intensity distribution with the increase of temperature. It should be noted that even the intensity of the fundamental reflection is not saturated owing to the wide dynamic range of the IP. Although reflections such as A2 and B1 correspond to the different regions with different variants, it was possible to compare these two reflections quantitatively to examine the disordering process, assuming that the thicknesses of these regions in each of these two variants are almost equal. This is because, during heating of the sample, a small drift of the sample was noticed and so the intensity variations due to the change of the excitation errors may be considerable when the intensities of superlattice reflections A2 and A1 situated relatively far from each other are compared. It is interesting to point


315

out that the intensity of superlattice reflections such as the one indicated by A2, which corresponds to the periodicity of the one-dimensional, long-period superstructure, decreases first, and above 790 K, the intensity of the superlattice reflections B1 decrease next. The different rates of decrease of the intensities in these superlattice reflections with the increase of temperature are consistent with the another report (Hirabayashi and Ogawa, 1957), which indicates that the disordering process preferentially occurs at the antiphase boundary of the long-period superlattice, leaving a fairly highly ordered state between the boundaries below 790 K. By utilizing the IP, the researchers quantitatively analyzed the disordering process of Cu3Pd by measuring the intensities of both superlattice reflections and fundamental reflections. The characteristic disordering process and the transition to the short-range-order state were quantified from the in situ experiment by using the IP. It was demonstrated that the IP can be used for quantitative analysis by taking account of the dynamical factor.

C. Quantitative Image Analysis In this section, the application data of quantitative image analysis illustrating the linear response of the IP are introduced. For instance, high-resolution electron microscope (HREM) images of W8Ta2O29 were observed quantitatively by using the IP with a 400-kV electron microscope, by Shindo, Hiraga, Oku, et al. (1991). Figure 26 is an example of an HREM image taken with the IP. The specimen used was W-Ta-O; the image was recorded with an HREM, the JEM-4000EX, at an accelerating voltage of 400 kV, a direct magnification of ×1,500,000, a current density of 10 pA/cm2, and an exposure time of 2 s. The image data were subjected to contrast adjustment and ×2 magnification, by using the image-processing software of the IP processor (Oikawa, Mori, et al., 1990). An original print that was magnified ×1.8 (finally ×3.6) with the IP printer was used directly for printing. Figure 27 shows a three-dimensional presentation of the electron intensity distributions in areas a and b of Figure 26, which were measured from the IP. In area a (where the specimen is thin), the measured intensity is least at heavy atomic columns (indicated by arrows H in Fig. 27a), which shows a good agreement with the projected potential of the atoms in the structure model (the inset in Fig. 26). In area b (where the specimen is a little thicker), the intensity is greatest in the low potential region (indicated by arrows L in Fig. 27). It was thus clear from this quantitative measurement that the region was subjected to a strong dynamical diffraction effect. Likewise, an HREM image of the high-Tc superconductor Tl2Ba2Cu1Oy was quantitatively observed by using the IP, by Shindo, Oku, et al. (1994). In order to evaluate quantitatively the difference between the intensity of the

316


Figure 26. Example of an HREM image (Shindo, Hiraga, Oku, et al., 1991). Specimen was W-Ta-O and accelerating voltage was 400 kV.

observed image and that of calculated images, the researchers calculated a residual index RHREM for 743 sampling points in the unit cell projected along the [010] direction. Although it has a rather complicated layered structure, RHREM = 0.0473 was obtained by choosing the experimental parameters and taking into account the partial occupancy of Tl atoms. On the basis of the analysis of the HREM image of Tl2Ba2Cu1Oy, several requirements for further refinement of crystal structure analysis by quantitative HREM were discussed. The observed intensity of the HREM image was compared with the calculated intensity by changing the experimental parameters such as the crystal thickness and defocus value. So that the difference between the intensity of an observed image and the of calculated images could be evaluated quantitatively, a residual index RHREM, which should show the accuracy of the simulated images, was


317

Figure 27. Three-dimensional presentation of intensity distribution, measured from the image data in Figure 26 (Shindo, Hiraga, Oku, et al., 1991).

introduced and evaluated. In the final refinement to reduce the value of RHREM, partial occupancy of Tl atoms was taken into account. On the basis of a quantitative analysis of the HREM image of Tl2Ba2Cu1Oy, some requirements for quantitative HREM were pointed out and were briefly discussed in comparison with those for the standard X-ray and neutron diffraction methods. An HREM study was carried out with a JEM-4000EX electron microscope. HREM image were recorded on the IP and were converted into digital data (2048 × 1536 pixels, 4096 gray levels) at the JEOL Laboratory. After investigation of the image intensity in the image-processing system (PIXsysTEM) (Oikawa, Mori, et al., 1990), the digital data were transferred to Tohoku University on magnetic tapes and were there analyzed with an engineering workstation (Sun: Argoss 5230) and a mainframe (NEC: ACOS-2020). An HREM image of Tl2Ba2Cu1Oy is shown in Figure 28. The incident electron beam was parallel to the [010] direction. The image was taken with a 2-s exposure and a direct magnification of ×1,500,000. It was noted that the image was observed with a defocus value which was rather smaller than the so-called Scherzer focus value (i.e., ∼48 nm). Although the image was recorded with 2048 × 1536 pixels and 4096 gray levels, only a part of the 1024 × 1024 pixels was output with 256 gray levels in Figure 28. In the image, small dark dots show heavy atom positions projected along the incident electron beam. In Figure 29, the number of pixels used for recording this HREM image is shown as a function of the gray level. Although the number of gray levels

318


Figure 28. HREM image of Tl2Ba2Cu1Oy recorded with the IP. The small rectangle shows a unit cell of Tl2Ba2Cu1Oy (Shindo, Oku, et al., 1994).

needed for recording HREM images seems to be much smaller than that for electron diffraction patterns, it is seen that about 1000 gray levels were used for recording the HREM image. A model of the atomic arrangement of Tl2Ba2Cu1Oy is presented in Figure 30a, which was proposed earlier by an X-ray diffraction study (Parkin et al., 1988). In Figure 30b, the intensity distribution of a part of the image near the crystal edge is shown as a contour map. The rectangles in the model of Figure 30a and in the intensity distribution of Figure 30b indicate unit cells of Tl2Ba2Cu1Oy, which has a tetragonal structure with the lattice constants a = 0.3866 nm and c = 2.324 nm. So that the noise such as quantum noise could be removed, the contour map was produced by smoothing the data with 2 × 2 sampling points and averaging the intensity after displacing the image


319

Figure 29. Number of pixels as a function of the gray level used for recording the HREM image of Figure 28.

by +a and −a. Even after the averaging process, there is a small asymmetry around metal atom positions in the contour map. The asymmetry is considered to come from the crystal thickness change. The observed intensity of the HREM image was divided by the intensity of the incident electron beam, which was measured at the vacuum region near the crystal edge. Thus, the normalized observed intensity can be directly compared with the calculated intensity without any scaling factor. Although the contour map reveals the detailed intensity distribution of the HREM image, it is not easy to distinguish the intensity maxima from the minima, since both intensity maxima and minima appear as similar dense contour lines. As a way to make a detailed investigation of both high intensity and low intensity, which may correspond to low and high potential regions, respectively, the contour map of Figure 30b was separated into two contour maps as shown in Figures 30c and 30d. In Figure 30c, the grid

320


Figure 30. (a) Structure model of Tl2Ba2Cu1Oy. (b) Contour map showing the intensity distribution of the HREM image of Tl2Ba2Cu1Oy in Figure 28. (c) High-intensity region of the contour map (b). The grid corresponds to the sampling points at which the observed and calculated intensities were compared to evaluate a residual index RHREM. (d) Low-intensity regions of the contour map (b).

indicates the positions where the observed intensities were measured with the IP. The number of sampling points on the grid in the unit cell was 743. The observed intensities at these sampling points were compared with the calculated ones. In the contour map of Figure 30d, which shows low intensity, the heavier atomic columns of Tl and Ba can be easily distinguished from those of Cu. It should be noted that there is no marked difference between the density of the contour lines at the Tl site and those at the Ba site, although the potential of Tl atoms is much larger than that of Ba atoms. This will be taken into account for the refinement of the computer simulation that follows. An image calculation based on a structure model suggested by an X-ray diffraction study was carried out, which is shown in Figure 30a. So that the difference between the observed intensity and the calculated one could be


evaluated, a residual index RHREM was calculated: % R HREM = |Iobs − Ical | Iobs

321

(1)

RHREM is the index for the observed and calculated image intensity and is basically different from the so-called R-factor or the residual index generally used in diffraction studies, where the factor or the index is evaluated for the absolute value of the structure factor. In Eq. (1) indicates the summation for the sampling points in the unit cell, which number 743 in this study and correspond to the grids of Figure 30c. As a way to get smaller values of RHREM, parameters, which depended on the experimental conditions (i.e., crystal thickness, defocus, and chromatic aberration), were changed. With the structure model of Figure 30a, RHREM = 0.0506 was obtained with the experimental parameters shown in Table 2, where the parameters which were changed to get a smaller RHREM in the calculation are indicated with asterisks. Images simulated with RHREM = 0.0506 are shown in Figure 31, where three types of contour maps (i.e., whole intensity, higher intensity, and lower intensity) are presented in Figures 31a through 31c in a similar manner to that of the observed images shown in Figures 30b through 30d, respectively. So that one could see the variation of RHREM with the change of the parameters in the calculation, RHREM was plotted as a function of crystal thickness t and of defocus f, as shown in Figures 32 and 33. In the calculation of Figure 32, all parameters except crystal thickness were set to be equal to those in Table 2. It is noted that RHREM is smaller than 0.07 in the crystal thickness range 4–6 slices. Figure 33a indicates the variation of RHREM as a function of defocus f in the range 5–75 nm. It is seen that RHREM is smaller than 0.06 in the range 15–45 nm. In Figure 33b, fine variation of RHREM as a function of f is indicated in the range 14–35 nm.

TABLE 2 Parameters Used for the Calculation of the HREM Image in Figure 31 Wavelength Spherical aberration constant Thickness of one slice Number of beams ∗ Defocus of objective lens ∗ Defocus due to chromatic aberration ∗ Crystal thickness a

0.00164 nm 1.0 nm 0.3866 nm 32 × 128 23.0 nm 24 nm 5 slice (=1.93 nm)

Asterisks indicate the parameters that were changed to obtain a smaller RHREM in the calculation.

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Figure 31. (a) Calculated image of Tl2Ba2Cu1Oy shown with a contour map. Parameters used for the calculation are listed in Table 2. (b) High-intensity region. (c) Low-intensity region.

The difference of the intensity of the observed images from that of the calculated image with RHREM = 0.0506 is shown with a contour map in Figure 34a. It is seen that there are small peaks such as those indicated by A and B, where the calculated intensity deviates widely from the observed intensity. Region A corresponds to the positions around the Tl atomic columns. As pointed out in the observed image of Figure 30d, the contrast of Tl atoms is similar to that of Ba atoms despite its much larger atomic number. It is thus reasonable to say that the discrepancy may be attributed to the fact that the concentration of Tl atoms is lower than the nominal concentration. This was noticed by Shindo, Hiraga, Oku, et al. (1991) in their previous HREM experiment of Tl2Ba2Cu1Oy. They therefore took into account the partial occupancy of Tl atoms and made new image calculations. It was found that RHREM became smaller if the partial occupancy of Tl atoms was taken into account. As a result, RHREM = 0.0473 was obtained with an 87% occupancy of Tl atoms, as shown in Table 3. The parameters with asterisks indicate those changed to get a small value of RHREM in the calculation. Figure 35 indicates the variation of RHREM


323

Figure 32. Variation of RHREM as a function of crystal thickness.

as a function of occupancy of Tl atoms. In Figure 34b, the low intensity of the calculated image with RHREM = 0.0473 is plotted as a contour map. It is noted that the density of the contour lines at the Tl position is slightly lower than that in Figure 31c, which was calculated with full occupancy of Tl atoms. In Figure 34c, the difference between the observed and the calculated images is shown as a contour map. Some of the contour lines around the Tl atom positions observed in Figure 30a disappear. However, there is still a fairly large difference at the positions indicated by B. These positions correspond to the interstices among oxygen atoms and Ba atoms in Figure 30a. As pointed out previously, there is some oxygen deficiency in the quenched samples. Thus,

324


Figure 33. Variation of RHREM as a function of defocus. The other parameters except the defocus value are the same as those in Table 2. (a) The range of defocus values is 5–75 nm. (b) The range is 14–35 nm.

the difference between the observed and calculated intensities in the preceding refinement may be attributed to some oxygen deficiency. In summary, in the analysis of an HREM image of Tl2Ba2Cu1Oy, a residual index RHREM of 0.0473 was obtained by changing the experimental parameters and introducing the partial occupancy of Tl atoms. By the refinement of the computer simulation, deficient oxygen positions were also detected. It was pointed out that a smaller residual index RHREM and a higher resolution limit are indispensable for obtaining more accurate atomic arrangements from HREM images observed with the IP. D. Image Processing Since the IP generates digital image data, it is convenient for digital image processing. In this section, two types of application data of the image processing are introduced.


325

TABLE 3 Parameters Used for the Calculation of Final Refinement Corresponding to the Contour Map of Figure 34B Wavelength Spherical aberration constant Thickness of one slice Number of beams ∗ Defocus of objective lens ∗ Defocus due to chromatic aberration ∗ Crystal thickness ∗ Occupancy of Tl atoms

0.00164 nm 1.0 nm 0.3866 nm 32 × 128 24.5 nm 24 nm 5 slice (=1.93 nm) 87%

a Asterisks indicate the parameters that were changed to obtain a smaller RHREM in the calculation.

Figure 34. (a) Difference between observed and calculated intensities of HREM images with RHREM = 0.0506. (b) Lower-intensity distribution of the calculated images taking into account 87% occupancy of Tl atoms. (c) Difference between observed and calculated intensities of HREM images with RHREM = 0.0473. Note that there are still some peaks at positions indicated by B.

326


Figure 35. Variations of RHREM as a function of occupancy of Tl atoms. The other parameters except the occupancy of Tl atoms are the same as those in Table 3.

One is the simple contrast enhancement of an image. Figure 36 shows an example of the image contrast enhancement of a biological specimen (a thin section of a dragonfly). The image contrast was enhanced by the look-up-table (LUT) as shown in Figure 37. Here, the image contrast γ is defined as in Eq. (2): γ =

Wo Wi

(2)

where Wi is the dynamic range of input data and Wo is the dynamic range of output data.


327

Figure 36. Contrast enhancement of an image from a thin section of a dragonfly. The contrast enhancement was carried out by the look-up-table (LUT) as shown in Figure 37.

The other is spatial frequency filtering. Figure 38 shows an example of the Fourier transformation of an HREM image (Si3N4 single crystal taken with the JEM-2010F 200-kV field-emission HREM). Figure 38a shows an original image, Figure 38b shows the Fourier-transformed two-dimensional power spectrum pattern (diffractogram), and Figure 38c shows an image reconstructed by selecting periodic spots in the spectrum, as indicated by the circles in 38b. The IP has a wide dynamic range and high intensity resolution (16,384 gray levels). Contrast enhancement and image analysis applications can hence be carried out with high precision.

E. Other Fields of Application of the IP The IP has begun to be used in the reflection high-energy electron diffraction (RHEED) field (Miura et al., 1995). In this field as well as in electron diffraction, the superior characteristics of the IP are valuable. Originally, the IP was developed as a highly sensitive image-recording device for X-ray images. The IP is widely used today in the fields of clinical medical science (Sonoda et al., 1983) and medicine and bioscience (Nakajima, 1993). The IP has also begun to be used in the field of X-ray crystallography (Fuji and Kozaki, 1993). Since the IP has good sensitivity for ultraviolet rays and ions (Nishikawa, Kimoto, et al., 1995), applications in these fields have also been started.

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Figure 37. Look-up-table (LUT) used for contrast enhancement in Figure 36. A gray-level histogram of the original image data is also shown in the figure.


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Figure 38. Image processing of spatial frequency filtering. (a) HREM image of an Si3N4 single crystal taken with the JEM-2010F field-emission TEM and the IP. (b) Fourier-transformed two-dimensional power spectrum pattern of (a). (c) Image reconstructed (spatial frequency filtered) by selecting periodic spectral spots indicated by the circles in (b).

Figure 39. Comparison of some characteristics for the image detection devices widely used today.

330


VIII. Conclusion The TEM is an instrument for observing magnified images of microscopic objects and outputs experimental results in the form of images. Moreover, the TEM outputs not only the morphology of the specimen but also the result of interaction between the incident electron and the specimen. In this sense, the TEM image is not a mere “picture,” but a “message from the microscopic world.” Of course, imaging with the TEM is modulated by instrumental factors such as lens aberrations. Image detection devices also have specific characteristics. Figure 39 shows comparisons of some characteristics for the image detection devices widely used today. These devices have both advantages and disadvantages, and they have very different characteristics. Among these devices, it is hoped that the IP, which has high sensitivity and high quantitative precision for beam intensity and which is also suited for image processing, will be widely used and assist in new research using the TEM.

Acknowledgments Among the application data introduced in this article, Figure 1 was obtained in a joint research project by Dr. Hiroshi Ayato of the Ashigara Research Laboratory of Fuji Photo Film Co., Ltd., and the authors. Many of the application data in this article were obtained in a joint research project by Professor Daisuke Shindo of the Institute for Advanced Materials Processing, Tohoku University, and one of the authors (T. O.). We hereby express our gratitude to them for allowing us to use the data included in this article.

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Index

A Ablation of metal films, 20–22 Albite twins, 64, 65 Algebraic topology, 184 Alkali feldspars, phase separation of, 60–68 Amphiboles, phase separation of, 68–71 monoclinic, 71–77 orthorhombic, 77–81 structure and classification, 68–69 Analogies between theories, 148 Analytical electron microscopy (AEM), 55–58, 74, 76–77 Atomic force microscope (AFM), 92 Atomic scattering, 7

B BaFX, 282–288 Bethe stopping power formula, 24 Biopyriboles, 81 chain-width disorder in, 83–84 new, 82–83 polysomatic reactions in, 84–87 Boundary conditions and sources, 176–177 Bragg scattering, 7

C Cassette, 291 Cell complexes, 184–186 Chains, 190–191

boundary of, 191–193 co-, 193–197 Chain-width disorder in, 83–84 Charge content, 165 Charge-current potentials, 170 Charge flow, 165 Chesterite, 82–83 Classification diagrams, 148–149 Classification schemes, 149 Clausius-Clapeyron equation, 21 Coboundary operator, 200–204, 214–215 Cochains, 193–197 field function approximation, 239–240 Computed radiography, 282, 304–305 Constant electric field scaling, 93–94 Constant potential scaling, 93–94 Constitutive equations/relations, 160–161, 172–176 discrete representation, 205–207 strategies for discretization, 231–239 Contour mapping, 294 Contrast transfer function (CTF), 298 Convergent-beam electron diffraction (CBED), 310–311 Coordinate maps, 211 Crystallographic shear planes, 83–84

D Dark-field technique, 81 Deflector 333

334

INDEX

Deflector (Cont.) sliced, 105–106 stacked, 102–104 De Rham functor, 236 Detective quantum efficiency (DQE), 300–302 Detector, 124–126 Differential forms, 210–211 Differential operators, 214–217 Discrete Green’s formula, 219 Discrete surface integral (DSI), 256–259 Discretization error, 175–176, 237–239 Discretization of the Hodge star operator, 232 Discretization strategy, reference, 222 constitutive relations, 231–239 domain discretization, 223–225 edge elements and field reconstruction, 239–246 error-based, 237–239 field function reconstruction and projection, 233–237 global application of local constitutive statements, 232–233 topological time stepping, 225–231 Domain discretization, 223–225 Dynamic random access memory (DRAM), 92

E Edenitic substitution, 77 Edge elements and field reconstruction, 239–246 Einzel lens. See Stacked einzel lens Electric charge, law of conservation of, 170

Electromagnetic potentials, 170 Electron-microprobe analyzer (EMPA), 54, 76 Electron microscopy applications, 1–2 Electron microscopy, high-speed applications, 45 flash photoelectron, 25–36 pulsed high-energy reflection, 36–40 pulsed mirror electron, 40–45 techniques, 2–6 time-resolving, 6–45 transmission electron, 7–25 Electron-optical calculations, 126–132 Electron source silicon, 121–124 spindt, 119–121 Electrostatic lenses See also Fabrication of miniature electrostatic lenses scaling laws for, 93–94 Erasing, imaging plate, 292 Error, discretization, 175–176, 237–239 Error-based finite element method, time-domain, 271–273 Euclidean space, 211–212 Exposure, imaging plate, 291–292 Exsolution (phase separation), 55, 59 alkali feldspars, 60–68 amphiboles, 68–81 Exterior differential, 215–216

F Fabrication of miniature electrostatic lenses detector, 124–126 electron-optical calculations, 126–132

INDEX

electron source, 119–124 future for, 140 LIGA lathe, 108–118 review, 94–95 slicing, 104–108 stacked einzel lens, 132–140 stacking, 95–104 Fabrication of miniature magnetostatic lenses, 118–119 Factorization diagrams, 148–149 Fading, 292, 299–300 Faraday cup, 124, 134 Faraday’s induction law, 151–152, 159–160, 163–164, 169, 170, 171, 225–226, 257 Fast Fourier transform (FFT), 294 F-centers, 283–288 Field, concept of, 144 Field function reconstruction and projection, 233–237 Field reconstruction, edge elements and, 239–246 Fields, discrete representation cochains, 193–197 limit systems, 197–199 Finite difference (FD), 145 methods, 246–255 support operator method (SOM), 252–254 Finite difference time-domain method (FDTD), 246–252, 254–255 Finite element (FE), 145, 219 methods, 264–273 time-domain, 267–269 time-domain edge, 269–271 time-domain error-based, 271–273 Finite integration theory (FIT), 260–264 Finite volume (FV), 145

335

discrete surface integral (DSI), 256–259 finite integration theory (FIT), 260–264 methods, 207–209, 219, 255–264 Flash photoelectron microscopes. See Photoelectron microscopes, flash

G Galerkin method, 267 Gauss’s divergence theorem, 161 Gauss’s law for electrostatics, 160, 170–171, 229–230 for magnetic flux, 159–160, 168–169 for magnetostatics, 170, 228–229 Geometric objects and orientation, 150–157 Geometry, discrete representation, 183 boundary of a chain, 191–193 cell complexes, 184–186 chains, 190–191 incidence numbers, 188–189 primary and secondary mesh, 186–187 Granularity and uniformity, imaging plate, 300–302 Green’s formulas, 219 Guinier-Preston (CP) zones, 72–73

H Hertz-Knudsen-Langmuir equation, 21 High-resolution electron microscope (HREM), quantitative image analysis, 315–324

336

INDEX

High-resolution TEM (HRTEM), 54 biopyriboles and polysomatic defects, 81–87 Hodge star operator, 232

exposure, 291–292 image processor, 293–294 reading, 292 Incidence numbers, 188–189

I

J

Image intensity tracking, 2, 6 space-time resolution, 24–25 in transmission microscopes, 10–12 Image processing, 293–294, 324–327 Imaging plate (IP) advantages of, 282 computed radiography and radio luminography systems, 304–305 configuration of, 290–291 description of layers, 288–290 development of, 281 fading, 292, 299–300 granularity and uniformity, 300–302 photomultiplier tube (PMT), 282 photostimulated luminescence (PSL), 282–288 resolution, 296–299 sensitivity, 294–296, 306–310 transmission electron microscope, 303–304 Imaging plate (IP), applications, 281, 305 high sensitivity, 306–310 image processing, 324–327 miscellaneous areas, 327–329 quantitative image analysis, 315–324 wide dynamic range, 310–315 Imaging plate (IP), elements erasing, 292

Jimthrompsonite, 82–83

L Laser-driven guns photoelectron, 4–5 thermionic, 3–4 Law of conservation of electric charge, 170 of magnetic flux, 169–170 LIGA (lithography and galvo-forming or electroplating) lathe, 94, 108 dose calculation, 111–118 processing, 109–111 Light-optical microscopy, 2 Limit systems, 197–199 Lucite, 109–111, 115–118

M Magnetic flux φ, 151–152 Gauss’s law for, 159–160, 168–169 law of conservation of, 169–170 Magnetostatic lenses, fabrication of miniature, 118–119 Material parameters, 173 Maxwell-Ampère’s law, 160, 170, 171, 225, 227–228, 257, 262 Maxwell grid equations, 262 time-domain edge element method, 269–271

INDEX

time-domain error-based finite element method, 271–273 time-domain finite element methods, 267–269 Meshes primary and secondary, 186–187, 223 Metal films, ablation of, 20–22 Metal melts, hydrodynamic instabilities of, 12–20 Microchannel plate, 126 Miniature electron optics, use of term, 91 Miniature scanning electron microscope (MSEM) See also under Fabrication applications of, 91–93 electron source, 119–124 stacked assembly, 100–102 stacked electrostatic deflector and stigmator, 102–104 tilted, 130–132 Mirror electron microscopy, pulsed, 40–45 Modular transfer function (MTF), 297–298 Moon-rock samples, 54 Moonstone, 64–65 Multivectors, 212–214

N Noise, imaging plate, 300–302 Nucleation, homogeneous, 59 in alkali feldspars, 61 in amphiboles, 73, 80

O Orientation compatible or coherent, 189 external, 150, 153–154, 164

337

geometric objects and, 150–157 internal, 150, 151, 164 propagate, 189

P p-dimensional cell, 150, 155 differential forms, 210–211 oriented, 184–186 p-dimensional cochains, 193–197 incipient, 210 Perthites, 61 Petrographic optical microscope, 54 Phase separation (exsolution), 55, 59 alkali feldspars, 60–68 amphiboles, 68–81 Photoelectron gun, laser-driven, 4–5 Photoelectron microscopes, flash, 25–27 applications, 29–34 limitations, 34–36 short-time exposure imaging, 27–29 Photoionization, 35 Photomultiplier tube (PMT), 282 Photostimulated luminescence (PSL), 282–288 Physical field problems, continuous representations, 207–208 compared with discrete, 209 differential forms, 210–211 differential operators, 214–217 spread cells, 217–220 weak form of topological laws, 220–222 weighted integrals, 211–214 Physical field problems, discrete representations compared with continuous, 209 constitutive relations, 205–207 fields, 193–199

338

INDEX

Physical field problems (Cont.) geometry, 183–193 topological laws, 199–205 Physical field problems, methods finite difference methods, 246–255 finite element methods, 264–273 finite volume methods, 255–264 reference discretization strategy, 222–246 Physical field problems, numerical solutions alternative methods, 145–147 boundary conditions and sources, 176–177 classification of physical quantities, 163–168 constitutive equations, 172–176 discretization step, 145 geometric objects and orientation, 150–157 mathematical structure of theories, 147–150 modeling step, 144 physical laws and quantities, 148, 157–163 scope of structural approach, 177–183 topological laws, 168–172 Physical laws and quantities, 148, 157 equations, 159–163 local and global quantities, 158–159 Physical quantities classification of, 163–168 Plagioclase feldspars, 60 Poly(methyl methacrylate) (PMMA), 109–111, 115–118 Polysomatic defects, 81–87 Polysomatic series, 81 Polysomatism, 81 Polysome, 81

Polytype, 81 Potentials charge-current, 170 electromagnetic, 170 Pullback, 211–212 Pump-probe technique, 2 Push forward, 214 p-vector, 212–214 Pyrex fiber processing, 100 Pyriboles, 81 chain-width disorder in, 83–84 polysomatic reactions in, 84–87

Q Quantitative image analysis, 315–324

R Radio luminography, 282, 304–305 Reading, imaging plate, 292 Reference discretization strategy. See Discretization strategy, reference Reflection electron microscopy, 36–40 Reflection high-energy electron diffraction (RHEED), 327 Residual equations, 265 Resolution, imaging plate, 296–299 Reversed-biased p-n, 125 Richardson-Dushman expression, 35 Riemann integral, 211–214 Roth’s diagrams, 149

S Scaling laws for electrostatic lenses, 93–94

INDEX

Scanning electron microscope (SEM), 54 miniature, 91–93 Scanning tunnel microscope (STM), 92 Schottky junction, 125 Sensitivity, imaging plate, 294–296, 306–310 Shape functions, 240–241, 265–266 Short-time exposure imaging, 2, 3–5 bright-field, 22–24 flash photoelectron microscopy and, 27–29 in transmission microscopes, 7–9 Silicates alkali feldspars, 60–68 amphiboles, 68–81 analytical electron microscopy of (AEM), 55–58 phase separation, 59 Silicon die processing, 98–99 Silicon source, 121–124 Slicing, 104–106 processing, 106–108 Space-time discretization, 223–225 objects, 155–157 viewpoint, 165–168 Space-time resolution image intensity tracking, 24–25 photoelectron microscopes and, 34–35 short-time exposure bright-field imaging, 22–24 streak imaging, 24 Spindt source, 119–121 Spinodal decomposition, 59 in alkali feldspars, 61, 62–63, 64 Spread cells, 217–220 Stacked einzel lens

339

MSEM construction, 132–136 MSEM operation and image formation, 136–140 Stacking, 95–97 MSEM assembly, 100–102 MSEM electrostatic deflector and stigmator, 102–104 pyrex fiber processing, 100 silicon die processing, 98–99 Stokes’s theorem, 161 Streak imaging, 2, 5–6 space-time resolution, 24 in transmission microscopes, 9–10 Structure of a physical theory, 147–148 Subdomain method, 266 Summation by parts formula, 219 Support operator method (SOM), 252–254

T Thermionic gun, laser-driven, 3–4 Thin-film criterion, 55 Tilted MSEM, 130–132 Time-domain edge element method, 269–271 Time-domain error-based finite element method, 271–273 Time-domain finite element methods, 267–269 Time-harmonic fields, 268 Time-resolving microscopes, 6 flash photoelectron, 25–36 pulsed high-energy reflection, 36–40 pulsed mirror electron, 40–45 transmission electron, 7–25 Tonti diagrams, 149

340

INDEX

Topological laws, 168–172 coboundary operator, 200–204 discrete representation, 199–205 weak form of, 220–222 Topological time stepping, 225–231 Transformation diagrams, 149 Transformation laws, 167–168 Transmission electron microscope (TEM), imaging plate, 303–304 Transmission electron microscopy, applications in mineralogy alkali feldspars, 60–68 amphiboles, 68–81 analytical electron microscopy (AEM), 55–58 high-resolution (HRTEM), 54, 81–87 phase separation (exsolution), 55, 59–81 specimen preparation problem, initial, 53–54 Transmission electron microscopy, time-resolving applications, 12–22 image intensity tracking, 10–12

instrumentation, 7–12 short-time exposure imaging, 7–9 space-time resolution, 22–25 streak imaging, 9–10 Tschermakite substitution, 76

V Variational approach, 264 Vector elements, 243

W Wadsley defects, 83–84 Wehnelt bias, 7 Weighted integrals, 211–214 Weighted multivectors, 213 Weighted residual approach, 264 Weight functions, 265 Whitney functor, 236 Wide dynamic range, imaging plate, 310–315

X X-ray diffraction (XRD), 59, 74


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