ADVANCES IN ELECTRONICS AND ELECTRON PHYSICS VOLUME 32
CONTRIBUTORS TO THIS VOLUME A. E. Bell William C. Erickson David L. Feinstein V. L. Granatstein Frank J. Kerr D. L. Misell L. W. Swanson
Advances in
Electronics and Electron Physics EDITED BY L. MARTON Smithsonian Institution, Washington, D.C. Assistant Editor CLAIRE MARTON
EDITORIAL BOARD E. R. Piore T. E. Allibone M. Ponte H. B. G . Casimir W. G . Dow A. Rose L. P. Smith A. 0. C. Nier F. K. Willenbrock
VOLUME 32
1973
ACADEMIC PRESS
New York and London
COPYRIGHT 0 1973, BY ACADEMIC PRESS,INC. ALL RIGHTS RESERVED. N O 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 1N WRITING FROM THE PUBLISHER.
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PRINTED IN THE UNITED STATES OF AMERICA
CONTENTS CONTRIBUTORS TO VOLUME 32 .
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vii
ix
FOREWORD . . . . . . . . . . . . . . . . . . . . .
Technology and Observations in Radio Astronomy WILLIAM C . ERICKSON A N D FRANK J . KERR
I. Introduction . . . . . I1. Radio Astronomy Techniques III . Radio Observations . . . References . . . . .
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1 2 22 60
Introduction . . . . . . . . . . . . . . . . . The Angular and Energy Distributions for Electron Scattering . . . Image Formation by the Elastic Componcnt . . . . . . . . Image Formation by the Inelastic Component . . . . . . . V . Incoherent Theory of Image Formation . . . . . . . . . VI . Conclusions . . . . . . . . . . . . . . . . . Appendix: Deconvolution of Two-Dimensional Data . . . . . References . . . . . . . . . . . . . . . . . . . Addendum to References . . . . . . . . . . . . . .
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64 66 112 158 165 179 183 187 191
Image Formation in the Electron Microscope with Particular Reference to the Defects in Electron-Optical Images D . L . MISELL I. I1 . I11 . I V.
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Recent Advances in Field Electron Microscopy of Metals L . W . SWANSON A N D A E . BELL
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1. Introduction . . . . . . . . . . . . . . . . . . . I1 . Theoretical Considerations . . . . . . . . . . . . . . .
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. . . . . . . V. Surface Adsorption . . . . VI . Emitter Surface Rearrangement . VII . Technological Advances . . . Appendix . . . . . . . 111 Techniques
1V. Clean Surface Characteristics
References
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194 194 212 223 258 277 283 296 304
vi
CONTENTS
Multiple Scattering and Transport of Microwaves in Turbulent Plasmas V . L. GRANATSTEIN AND DAVID L. FEINSTEIN I . Introduction . . . . . . . . . . . . . . I1 Derivation of the Radiative Transport Equation . . . . 111 Applications and Model Calculations . . . . . . . 1V Comparison of Experimental Results and Model Calculations V . Briefsummary . . . . . . . . . . . . . Glossary . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . .
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AUTHORINDEX SUBJECT INDEX
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312 320 346 359 372 373 376 383 393
CONTRIBUTORS TO VOLUME 32 A. E. BELL,Linfield Research Institute, McMinnville, Oregon WILLIAMC. ERICKSON,Astronomy Program, University of Maryland, College Park, Maryland DAVID L. FEINSTEIN,*Cornell Aeronautical Laboratory, Inc., Buffalo, New York V. L. GRANATSTEIN,~ Bell Laboratories, Murray Hill, New Jersey
FRANK J. KERR,Astronomy Program, University of Maryland, College Park, Maryland D. L. MISELL,Department of Physics, Queen Elizabeth College, London University, London, England L. W. SWANSON, Linfield Research Institute, McMinnville, Oregon
* Present address: Department of Mathematics, University of Wisconsin, River Falls, Wisconsin 54022. t Present address: Division of Plasma Physics, Naval Research Laboratory, Washington, D.C. 20390. vii
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FOREWORD Again we have a volume appearing within less than a year of its predecessor. Furthermore, it will be separated only by months from publication of the next volume. This rather closely packed appearance of volumes reflects the abundance of extremely interesting material, all of which weare pleased to present in these Adrances. The first review, by Erickson and Kerr, deals with the technology and observations of the “passive ” branch of radio astronomy. This branch, much more important than the “active” one, offers a beautiful example of the interrelationships among astronomy, physics, chemistry, and, last but not least, advanced electronic technology. The next two reviews deal with widely different aspects of microscopy. Misell considers the role of image defects in the image formation of the transmission-type electron microscope, with particular attention to the influence of inelastic scattering in the specimen. This subject was of great interest to your editor many years ago and I welcome this thorough treatment of it. The author deals with the deterioration of contrast, rather than that of resolution, and points to the role of radiation damage in the specimen affecting the image quality. Swanson and Bell, in their review, discuss the present status of field emission microscopy of metals. The rapid growth in this area required several previous reviews; as our last one was published 12 years ago, the time appeared ripe to consider the advances since that time. Swanson and Bell examine the theoretical, experimental, and technological aspects of recent progress. The last review in this volume, by Granatstein and Feinstein, deals with the multiple scattering and transport of microwaves in turbulent plasmas. Some aspects of this subject were treated in earlier reviews in this series (Bowles, 1964; McLane, 1971). The importance of the subject, both for electromagnetic wave propagation and for controlled thermonuclear research, amply justifies this presentation. As has been our custom, we list again future reviews, together with their prospective authors: The Effects of Radiation in MIS Structures Small Angle Deflection Fields for Cathode Ray Tubes Sputtering Interpretation of Electron Microscope Images of Detcit5 in Cry3tal\ ix
Karl Zaininger R. G . E . Hutter and H . Dressel M. W. Thompson M. J . Whelan
X
FOREWORD
Optical Communication through Scattering Channels Hollow Cathode Arcs Channelling in Solids Physics and Applications of MIS-Varactors Ion Implantation in Semiconductors Self Scanned Solid State Image Sensors Quantum Magneto-Optical Studies of Semiconductors in the Infrared Gas Discharge Displays Photodetectors for the 1~ to 0 . 1 Spectral ~ Region High Resolution Nuclear Magnetic Resonance in High Superconducting Fields The Photovoltaic Effect The Future Possibilities for Neural Control Electron Bombardment Ion Sources for Space Propulsion Recent Advances in Hall-effect Research and Development Semiconductor Microwave Power Devices The Gyrator Electrophotography Microwave Device Technology Assessment The Excitation and Ionization of Ions by Electron Impact Whistlers and Echoes Experimental Studies of Acoustic Waves in Plasmas Multiphoton Ionization of Atoms and Molecules Auger Electron Spectroscopy
Robert S. Kennedy J. L. Delcroix R. Sizrnann and Constantin Varelas W. Harth and H. G. Unger S. Namba and Kohzoh Masuda Paul K. Weimer Bruce D. McCombe and Robert J. Wagner R. N. Jackson and K. E. Johnson David H. Seib and L. W. Aukerman H. Sauzade Joseph J. Loferski Karl Frank and Frederick T. Hambrecht Harold R. Kaufman D. Midgley S. Teszner K. M. Adams, E. Deprettkre, and J. 0. Voorman M. D. Tabak and J. L. Thourson Jeffrey Frey and Raymond Bowers John W. Hopper and R. K. Feeney Robert A. Helliwell J. L. Hirshfield J. Bakos N. C. Macdonald and P. W. Palrnberg
Suggestions about future reviews and authors proved to be most useful. We would like to repeat our invitation for more suggestions.
L. MARTON CLAIRE MARTON
Technology and Observations in Radio Astronomy WILLIAM C. ERICKSON
AND
FRANK J. KERR
Astronomy Program, University of Maryland, College Park, Maryland
I. Introduction ......................................................... A. Observational Considerations. .................
C. High Resolution Techniques.. .....
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A. Introduction.. ............ .............................................. B. Solar Radio Astronomy. ................................................................... C. The Planets .. ......................................................................... D. The Galaxy .......................................... .................................. E. Extragalactic Radiation.. ......................... ............................. F. Conclusion. ................... .......................................... References .........................
1 2 2 4 9 18 21 22 22 23 25 27 48 59 60
I. INTRODUCTION Radio astronomy is a branch of physics which deals with matter and energy under conditions that are unattainable on Earth. Several decades ago, when scientific work in radio astronomy began, the observed phenomena seemed very strange and unrelated to physics or optical astronomy. In the late 1950s and early 60s, optical identification of radio sources began, hydrogen line observations of galactic structure were related to optical observations, and radio data became an integral part of astronomical knowledge. In the past few years, the increased sophistication of radio techniques has led to solid, quantitative knowledge. Concurrent developments in plasma physics have provided a theoretical framework through which some of these quantitative data can be understood. The radio observations often provide the most crucial tests of a theory and the unexplained phenomena suggest fruitful avenues of theoretical research. Radio astronomy is therefore taking its place as a vital part of physics. In an analogous way, the discovery of new interstellar 1
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WILLIAM C. ERICKSON AND FRANK J. KERR
molecules and an understanding of the processes involved in their formation are closely tied to chemistry, and perhaps to elementary biology. These advances have been brought about primarily through technical developments. The technical advances which have been most important to radio astronomy are in the development of large aperture antennas, highly stable, low noise amplifiers, and data processing techniques. In this article we will attempt to show the ways in which technological developments have stimulated scientific discoveries. However, it should not be assumed that this is always a premeditated process ; important scientific discoveries have often occurred quite serendipitously when powerful new instruments became available. In this article we will attempt to give some discussion of most aspects of the science of radio astronomy. In our discussion of technological developments, we will confine ourselves to those subjects to which radio astronomers have been the most important direct contributors. Many of the techniques most vital to current research in radio astronomy were developed primarily for other purposes. The application of such techniques to astronomy will be mentioned but a discussion of their development is beyond the scope of this article. 11. RADIOASTRONOMY TECHNIQUES A , Observational Considerations
Before discussing the development of techniques for radio astronomy, we shall briefly outline the main observational factors which determine the types of equipment and technology that are required. In this paper, we consider only the “passive” branch of radio astronomy, in which radio energy emitted by celestial bodies is received at the Earth. The “active” branch, in which radar transmissions are used, is extremely important for solar system research but it is limited to the nearer bodies in the solar system because the strength of the received echo is inversely proportional to the fourth power of the distance of the echoing object (1). The main energy received is in the form of a continuum, extending over a wide frequency range, but a substantial number of spectral lines from interstellar atoms and molecules have also been detected. The observer’s problem is to measure the variation of intensity as a function of position in the sky and also of frequency. For a number of sources, variations on several time scales must also be taken into account, but usually the intensity and position appear to be constant to our present accuracy of measurement. Except at long wavelengths, the received signals are weak and the highest possible receiver sensitivity is required. The radiation has the character of
TECHNOLOGY A N D OBSERVATIONS IN RADIO ASTRONOMY
3
random noise and is similar to the noise generated inside a receiver or in the environment of an antenna. The astronomical signal produces an increase in the noise level at the output of the receiver, and has no other distinguishing characteristics. Under most circumstances the radiation is randomly polarized, but some sources show linear or circular polarization. Often the degree of polarization is quite low, and spurious polarization effects in the instruments must be kept to a minimum. Because the wavelengths are long in comparison with those common in optical or ultraviolet astronomy, adequate angular resolution can only be obtained with structures of large physical size or interferometers with large separations between the component parts. Attempts to increase resolving power have played an important part in the entire history of radio astronomy. Until recently there was no radio analog of the optical photograph, and it was necessary to work with point-by-point measurements. Systems now exist which can effectively form images, but they are necessarily complex. One type can scan rapidly over a strong source such as the Sun. In the other case a long period of observation is required to build up a detailed “picture” of some particular area of the sky. In an analogous way a wide spectral range can only be observed slowly. Even with a multichannel receiver a single observation can cover only a very small fractional bandwidth without replacement or extensive retuning of receiver components. Radio astronomers commonly express the strength of the received radiation in terms of temperatures. The antenna temperature corresponding to a cosmic source is defined as the temperature of a terminating resistor which could replace the antenna and which would produce the same receiver output as the source in the frequency band of the observation. The brightness temperature or surface brightness of that part of the sky under observation can be obtained from the antenna temperature if the antenna efficiency is known. The average brightness temperature of the radio sky varies from about 10,000 K at 30 MHz to a few kelvin above 1 GHz. The strength of a discrete source whose angular size is smaller than most antenna beamwidths is usually specified in terms of its flux density. This is a measure of the total power received at the Earth from the source in a unit area and bandwidth. The conventional “flux unit” (f.u.) is W m-2 (Hz)-’. The flux density, S, of a radio source can be characterized by a power law in frequency v S
= kv“,
(1)
where k and c1 are constants. For most radio sources, a is -0.7 to -0.8. If a is not constant, the source is said to have a “curved” spectrum. Since c1 is usually negative and the brightness temperature of the sky decreases with frequency, antenna temperatures generally fall with increasing
4
WILLIAM C. ERICKSON AND FRANK J. KERR
frequency. At high frequencies low noise techniques are extremely important while at longer wavelengths angular resolution is the most important consideration. In the past few years the greatest technical advances have been made in three areas: (1) The surface tolerances of parabolic telescopes with apertures in the 100 m range have been improved; this allows operation a t wavelengths of a few centimeters. A few years ago most telescopes of such sizes were limited to wavelengths greater than about 20 cm. These improvements, along with the development and common use of low noise preamplifiers, have resulted in an order of magnitude increase in the frequency range available for sensitive observation. This has led to the discovery of spectral lines from many interstellar molecules and to rapid development of the new field of interstellar chemistry. (2) Very high angular resolutions have been achieved. Radio sources can now be completely mapped with angular resolutions of a few arcseconds and some partial information can be obtained concerning fine structure to a limiting resolution of about 1O-j arcsec. A wealth of information concerning the detailed structure of radio sources is becoming available. (3) Observations from above the Earth's atmosphere have become available. The entire electromagnetic spectrum from zero hertz to the most energetic gamma rays can now be examined in at least a rudimentary fashion. This promises to revolutionize our knowledge of astrophysics. Already, correlations between the radio and the x-ray emissions from highly energetic electrons have been found. The shells of gases expelled from stars which have undergone supernova explosions have been observed optically, in the decimetric wavelength range, and at hectometric wavelengths. Millimeter wavelength and infrared observations provide clues concerning completely unforeseen processes in the nuclei of galaxies.
B. Single Telescopes and Spectroscopy A single radio telescope has the virtue of simplicity, easy steerability, and great flexibility in wavelength. These characteristics give rise to a wide range of applications. The angular resolution achievable with a single telescope is limited to about one arcminute. Deflections of the structure due to gravitational, thermal, or wind loading effects generally combine to limit either the useful aperture or the minimum wavelength, so that single steerable telescopes can be no more than a few thousand wavelengths in diameter. This implies beamwidths of an arcminute or more, and the most interesting structural features of many types of radio sources are unresolved by such beamwidths. Different techniques are therefore required to obtain higher resolution, and these will be described in the next section. On the other hand, high resolution instruments are usually restricted in their applications, each being designed for specialized purposes.
TECHNOLOGY AND OBSERVATIONS IN RADIO ASTRONOMY
5
Single telescopes have been widely used for mapping extensive regions of the sky at various wavelengths, both in the continuum and in the stronger lines. Aperture synthesis techniques are now taking over much of the mapping work. This is especially the case for continuum studies; the use of these techniques for mapping of spectral line sources is only just beginning. Most of the pulsar work has been done with single telescopes. However, the principal application of single dishes now and in the foreseeable future is in the field of spectroscopy, i.e. in the discovery and study of a wide range of interstellar radio lines. In fact, all the detection and searching for lines as well as the greater part of the work on the characteristics of spectral line sources has been done with single dishes. These studies require the combination of a large collecting area and the ability to operate over a large range of wavelength. The particular suitability of a single dish comes from the ease with which receiver changes can be made when only a single feed point is involved. In the 1960's, the largest steerable dishes were those at Jodrell Bank, England [250 ft (76 m) in diameter], and at Parkes, Australia [210 ft (64 m)]. Larger dishes with limited steerability were built at Arecibo, Puerto Rico [lo00 ft (305 m)], and Green Bank, West Virginia [300 ft (91 m)]. These were used in their original form down to wavelengths of 18, 6, 50, and 21 cm, respectively. All have been, or are being, resurfaced to extend their operating ranges to somewhat shorter wavelengths. The biggest steerable telescope has been recently completed at Bonn, with a diameter of 100 m and an expected minimum wavelength near 2 cm. Some large telescopes in current use are shown in Figs. 1 through 5. The present trend is towards the development of bigger millimeterwavelength telescopes, as the greatest interest in searching for new interstellar spectral lines is now in this wavelength range. The biggest mm-wave telescopes at present are the 36 ft (1 1 m) dish of the National Radio Astronomy Observatory at Kitt Peak, Arizona, and the 22 m instrument at the Crimean Astrophysical Observatory. The latter however has not yet been greatly used for spectral line work. Several countries are planning the construction of bigger and more precise mm-wave dishes, and the limiting factors involved in telescope accuracy are being very carefully examined. For example, the principle of homologous deformation (2),in which a structure maintains similarity of shape under gravitational loading while changing its zenith angle, has been already used to some extent in the Parkes and Bonn telescopes; it will be the fundamental basis of design in future installations. The status of singledish technology has recently been reviewed by Findlay (3). Single telescopes have normally been paraboloids, but the Bell Telephone Laboratories have had great advantages from a horn which they have used at 21 cm and other wavelengths. A well-designed horn picks up very little thermal radiation from the ground nearby, and so it can form part of a very
6
WILLIAM C. ERICKSON AND FRANK J. KERR
FIG.1 . The 100 m telescope of the Max-Planck-Institutfur Radioastronomie at Bonn. This recently completed antenna is the newest major facility in radio astronomy.
low noise system. Also, the performance of a horn is accurately calculable, which is important for absolute flux or brightness measurements. Historically, radio astronomy receivers have been developed for special needs at particular wavelengths, each covering only a small frequency range. Technical developments are pushing down the lower limits of wavelength, and the needs of spectral line investigators are closing the gaps in the available frequency coverage. Before long, receivers with good sensitivity should be available over the whole wavelength range accessible from the ground. In the quest for high sensitivity, parametric amplifiers have been extensively used in many frequency ranges. Masers have received less attention, in spite of their inherently low noise properties, partly because of their greater complexity under field conditions, and partly because the pickup of thermal emission from the ground and from feed system losses prevents the full potentiality of the maser from being realized. Experiments are now being carried out at the shortest wavelengths with detectors of the style used in the infrared, such as indium antimonide crystals and Josephson detectors.
TECHNOLOGY AND OBSERVATIONS IN RADIO ASTRONOMY
7
FIG.2. The Arecibo lo00 ft (305 m) antenna. In this antenna the spherical reflector surface is immobile and supported by the Earth. Correction for spherical aberration is performed by the feed and beam steering to zenith distances of 20"is accomplished by movement of the feed system only.
At millimeter wavelengths, the atmosphere is a limiting factor, and there are only a few spectral "windows" where the absorption due to water vapor and oxygen is low enough to allow successful ground-based observations (Fig. 6). Everywhere in this wavelength range dry climates and mountain sites offer an advantage, but observations between the windows or at even shorter wavelengths will have to be carried out from above the atmosphere. In this case, large structures will be required and an eventual observatory on the Moon might be utilized. Successul work in microwave spectroscopy depends on a good blend of telescope performance and electronics expertise. The great importance of the latter is indicated by the fact that most of the known lines (except for the early discoveries of HI, OH, NH, , and H,O) have been found with the facilities of one organization, the National Radio Astronomy Observatory, at Green Bank and Kitt Peak.
8
WILLIAM C. ERICKSON AND FRANK J. KERR
FIG. 3. The NRAO 140 ft (43 m) telescope at Green Bank, West Virginia. This telescope has been operated to wavelengths of 1 cm for the discovery of numerous molecular lines.
Radio astronomers must share an already overcrowded radio spectrum with many other users. The appropriate international organizations have recognized radio astronomy as a “ service,” entitled to receive some protected bands in the frequency allocation tables. In this way, a varying amount of protection has been obtained at some widely separated places in the continuum, and in narrow bands containing some of the most important lines. Good cooperation has generally been achieved in keeping these bands clear,
TECHNOLOGY AND OBSERVATIONS IN RADIO ASTRONOMY
9
FIG.4. The NRAO 36 ft (11 m) telescope at Kitt Peak, Arizona. This instrument operates to 3 mm wavelength. It has yielded observations of many millimeter wavelength radio sources.
on a worldwide, national, or sometimes a local basis. However, as the number of detected lines increases there will be growing difficulty in avoiding interference. Geographical isolation of observatories can help a great deal, but the greatest potential threat comes from the growing number of transmissions from Earth satellites or other space vehicles. Frequencies used for these services will be essentially unusable for radio astronomy anywhere on Earth, and there will come a time when some radio interstellar lines will be detectable only from the far side of the Moon. Radio astronomy needs the cooperation of all other groups who make use of the radio spectrum in order to survive as the vigorous science that it is today (4).
C. High Resolution Techniques The attainment of higher angular resolution has occurred through the development of four radically different types of technique ; aperture synthesis,
10
WILLIAM C. ERICKSON AND FRANK J. KERR
FIG.5. The parabolic cylinder antenna of the Tata Institute of Fundamental Research at Ootacamund,India. The antenna has its long dimensionin the north-south direction, and is very conveniently equatorially mounted by utilizing the slope of a south-facing hillside.
lunar occultations, interplanetary and interstellar scintillation, and very long baseline (independent local oscillator) interferometry. To overcome the size limitations of single telescopes, it was quickly recognized that the Earth is extremely rigid; a radio telescope can be broken into components that are well separated from each other and the relative positions of the different elements can be maintained with great precision. Christiansen and Hogbom (5) have described the evolution of telescopes from simple interferometers to complex arrays. A pair of antennas operated as an interferometer measures the mutual coherence between the signals received by the two elements. Apart from the properties of the source and the propagation medium, the mutual coherence is a function only of the length and orientation of the interferometer baseline. It is related to the apparent angular intensity distribution of the source by a Fourier transform (6). In the case of a single parabolic antenna, the mutual coherence throughout the aperture plane is transformed after reflection of the radiation by the antenna surface into a real image of the source in the focal plane. This real image is then examined by the feed system, one point at a time.
TECHNOLOGY AND OBSERVATIONS IN RADIO ASTRONOMY
0.01
1
1
1
L
I ,I,
I
1
I
I
I I , ,
10
100 Frequency
500
(GHz)
FIG.6. Zenith attenuation for a very dry and a normal atmosphere, showing the various “windows” and regions of heavy absorption (3).
However, it is unnecessary to make the transformation to the image at the time of observation. The mutual coherence function itself could be determined interferometrically in the aperture plane and the image can be formed later by computation. If the intensity distribution of the source and, therefore, the mutual coherence which it generates in the aperture plane is independent of time, the coherence at each interferometer baseline length and orientation can be measured in succession by changing the relative positions of the interferometer elements. This is the principle of aperture synthesis (7). If the source were to vary during the period of observation, this method could not be employed and all interferometer spacings and orientations would have to be measured simultaneously. The aperture synthesis method is extremely powerful. Through the use of small moveable elements, a large aperture can be synthesized. Radio source regions can be mapped with the angular resolution and sensitivity of a filled aperture equal in radius to the longest interferometer spacings employed. For the observation of a single point in the sky, a filled aperture is more sensitive than an aperture synthesis instrument; however, if an area of sky is to be mapped, an aperture synthesis instrument is generally more sensitive. This occurs because the small aperture synthesis elements collect radiation continuously from the whole area of sky under observation (the beam area of a single element), while a filled aperture with a single feed can observe only one point at a time.
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WILLIAM C. ERICKSON AND FRANK J. KERR
Ryle and his co-workers (8,9) have led the development of aperture synthesis methods. The first aperture synthesis instruments used large linear arrays which were correlated with smaller moveable elements. During the course of the observing program, every spacing and orientation that could be found between any two elements in the synthesized aperture is represented by a line joining the movable element with some element in the linear array (see Fig. 7). c
LINEAR ARRAY
1-7 I- -1
ci
L
_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _j
FIG.7. Aperture synthesis employing a large fixed array and a small moveable array. The aperture that is synthesized after the movable array has been placed at each station is shown within the dashed lines.
In their most highly developed form, aperture synthesis instruments use the rotation of the Earth to vary the orientation of the elements as shown in Fig. 8. The radio source under observation is tracked by a pair of steerable elements for a period of 12 hr, during which period the orientation of the interferometer baseline rotates through 180". Since an interferometer's response is unaffected by exchange of its elements, it is unnecessary to track the source for more than 12 hr because the information obtained with a baseline rotation of over 180" would be redundant. One observation generates a synthesized aperture in the form of an elliptical ring. The spacing of the interferometer is then changed by moving one element, and the observation is repeated to generate a second ring of different radius. This process is continued until a complete aperture is synthesized. A three-element synthesis instrument using Earth rotation, the " One Mile Telescope," has been in operation at Cambridge, England, since 1965 (see Fig. 9). Obviously, observations with the One Mile Telescope are time
TECHNOLOGY AND OBSERVATIONS IN RADIO ASTRONOMY
13
NORTH WLE
AXIS OF EARTH'S ROTATION
FIG.8. Aperture synthesis employing Earth rotation to vary the antenna orientation. The diagram illustrates the change in relative positions of two antennas on an east-west axis during a 12 hr observing period and how observations with such an instrument provide one elliptical ring of the complete aperture.
consuming because each element must occupy forty different stations, each for a 12 hr observation period, before all spacings have been employed and a one mile aperture has been uniformly synthesized. Much higher data rates are obtained with the Westerbork Synthesis Telescope that commenced operation in 1970 (Fig. 10). This Dutch instrument employs ten 25 m parabolic dishes placed along a line at 144 m intervals. Each of these elements is connected as a separate interferometer with each of two moveable 25 m elements mounted on railroad tracks at the end of the fixed array. Twenty interferometer pairs are operated simultaneously to generate a synthesized aperture consisting of 20 elliptical rings. Four 12 hr tracks are required to cover all spacings and synthesize a complete aperture. At 20 cm wavelength, a source region can then be mapped with an angular resolution of 20 arcsec. Earth rotation synthesis with east-west interferometers breaks down for sources near the celestial equator where the interferometer elements trace out nearly linear paths rather than ellipses. For this reason, American designs of synthesis telescopes have included both north-south as well as east-west baselines (10). Design studies have been carried out at the California Institute of Technology of a seven-element array of 36 m elements on 5000 m baselines
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WILLIAM C. ERICKSON AND FRANK J. KERR
FIG.9. The “One Mile Telescope” at the Cavendish Laboratory, Cambridge, England. Three 60 ft (18 m) antennas are employed. The moving antenna and rails can be seen in the foreground.
and at the National Radio Astronomy Observatory of a 37 element array of 25 m elements on 30 km baselines. No existing instruments can produce complete maps of radio source regions with angular resolutions greater than a few arcseconds. For higher angular resolution, special techniques are employed which permit delineation of certain structural features of radio sources, while other features remain unobserved. Lunar occultations provide one such technique. When the Moon’s limb passes in front of a radio source region, a one-dimensional scan of the
TECHNOLOGY AND OBSERVATIONS IN RADIO ASTRONOMY
15
FIG.10. The Westerbork Synthesis Radio Telescope in the Netherlands. This instrument consists of twelve paraboloids of 25 meter aperture.Ten are fixed at 144 meter spacing and two are moveable on rails.
source brightness distribution is obtained with an effective strip resolution of about 0.5 arcsec. The angular resolution which can be achieved depends upon the signal-to-noise ratio, so large collecting areas are required. The group at the Tata Institute in Bombay are exploiting this technique with their new Ootacamund telescope. If Nature provides a number of occultations with the Moon's limb oriented in various directions with respect to the source region, a reasonably complete picture of that source can be developed. Obviously, this technique is limited to those sources which lie on the Moon's apparent path in the sky. Radio sources of small angular size were first discovered by the fact that they exhibit ionospheric scintillations ( 2 2 ) . If a source is observed through a medium which possesses refractive irregularities, and if the observer is sufficiently distant from the medium, the irregularities will alternately focus or deflect the radiation. The source will appear to scintillate in intensity unless its angular size is much larger than the apparent angular size of the refractive irregularities. For a large source, the random intensity fluctuations of each portion of the source will average out the scintillations. In the case of ionospheric, interplanetary, or interstellar scintillations, we are dealing with ionized media whose refractivity varies as ,I2. Consequently, the " Fresnel
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WILLIAM C. ERICKSON AND FRANK J. KERR
length” or the distance from the scattering medium at which “focusing” occurs and at which scintillations build up varies as l / i . Ionospheric scintillations occur at the Earth’s surface for wavelengths greater than a few meters in the case of radio sources less than 10 arcmin in angular diameter. Hewish (12) showed that sources of less than a few tenths of an arcsecond diameter will show scintillations due to interplanetary scattering. The Fresnel length for interplanetary scintillation is a strong function of the distance of the ray path from the Sun, since the density and scattering power of the interplanetary medium fall rapidly with increasing distance from the Sun. This Fresnel length is less than 1 A.U. (the mean distance of the Earth from the Sun), and scintillations are observed at 1 = 11 cm for ray paths very close to the Sun (13). Most interplanetary scintillation observations are made at meter wavelengths and at angles 30” to 60” from the Sun (14). At longer wavelengths, scintillations can be observed at even larger elongation angles, and at 11 m wavelength they have been observed in the antisolar direction (15). Through observations at various times, solar elongation angles, and wavelengths, a sizable body of data has been accumulated for the study of source structure and the structure of the interplanetary medium. The interplanetary medium casts a diffraction pattern on the Earth which involves a convolution of the brightness distribution of the radio source with the “response function” of the medium. In principle, if the response function of the medium were determined by measurement of the pattern cast by a point source, the image of a complex source could be restored after measurement of its diffraction pattern. In practice, this is not possible because the diffraction pattern is incompletely sampled and the response function of the medium varies with time. Only gross features of source regions have been determined in this way. The method has been used to learn which sources possess fine angular structure and to estimate the angular sizes of such structure. The interplanetary scintillation technique is most applicable to the determination of structure in the 0.02 to 0.5 arcsec range. Interstellar scintillations have also been recently observed (16). Only pulsars appear to exhibit this phenomenon because interstellar scintillations apparently disappear for radio sources of larger angular diameter. Pulsars are the only intense objects with sufficiently small angular diameters (< arcsec) to scintillate. The other vigorously exploited method of obtaining high angular resolution is that of Very Long Baseline Interferometry. VLBI has come to mean interferometry in which there is no electrical/real-time connection between the elements of the interferometer. The radio frequency signal from each element is mixed with an independent local oscillator and heterodyned to a video band which is recorded on magnetic tape. The recorded signals are later correlated to produce the interferometer response. Since there is no electrical connection between the elements, transcontinental and intercontinental baselines are
TECHNOLOGY AND OBSERVATIONS IN RADIO ASTRONOMY
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practicable. Very close synchronization is required in the two local oscillator frequencies and in the recording of time. The first system of this nature was operated by the Canadian group (17) employing analog tape recorders. This group has used various antennas to make extensive observations with the system in the 400 MHz range. Fine structure in radio sources has been determined to a resolution limit of 0.01 arcsec. An American VLBI system was developed simultaneously with the Canadian system (18). I n this system, the video signal is clipped and recorded digitally. Data correlation is performed by a general purpose computer. The system has been used for a wide variety of experiments at wavelengths ranging from 1.3 cm to 11.3 m over many baselines in the United States and to Australia or to the Crimea (14). A new American system has recently become operational. In this system the clipped signal is recorded on a video recorder and processing is performed by a special purpose computer. The principal advantages of the new system are increased bandwidth and integration times. The highest angular resolution obtained to date, arcsec, was achieved in May 1971 by this system operating between the Haystack Observatory and the Crimean Astrophysical Observatory at the H,O line wavelength of 1.3 cm. VLB interferometers synthesize apertures which consist of small portions of the elliptical rings shown in Fig. 8. Twelve-hour tracks are often impossible for various practical reasons such as the source being below the horizon at one of the observatories. The antennas cannot be moved to all intermediate locations to synthesize an aperture the size of the Earth. Nevertheless, after observations have been combined from observatories at various geographical positions, a number of elliptical segments are synthesized and a reasonable “picture” of the source region emerges. It has been suggested that interferometer baselines longer than those obtainable on the Earth’s surface may not be particularly useful for radio source observations (19). At centimeter wavelengths, interferometers with Earth radius baselines resolve all of the structure that is to be expected under fairly general assumptions. At longer wavelengths, longer baselines would be needed to resolve the source structure, but such structural details are probably obscured by interplanetary and/or interstellar scattering. We need to place radio interferometers in space primarily for the observation of wavelength ranges that are blocked by the Earth’s atmosphere. Aperture synthesis requires that the source structure remain time-invariant during the period of observation. The fine structure in quasars is frequently variable* and VLBI observations taken many months apart cannot be * In fact, the quaser 3C-279 displays the images of two point sources which appear to be flying apart at 3 to 10 times the velocity of light! (20). Several resolutions of this apparent contradiction of the theory of relativity are possible.
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WILLIAM C. ERICKSON AND FRANK J. KERR
combined. For highly variable sources, such as the Sun, aperture synthesis methods break down completely. In this case it is necessary to observe all interferometer spacings and orientations simultaneously. Especially designed for solar studies, a ring-shaped telescope has been built by the Australian Commonwealth Scientific and Industrial Research Organization at Culgoora, N.S.W. It consists of 96 antennas placed in a circle 3 km in diameter. The signals from the 96 elements are cleverly combined to form a picture of the Sun in each circular polarization once each second. The telescope operates at 80 MHz with 4 arcmin angular resolution. Although this angular resolution is modest by present standards, it is an important achievement to produce the radio picture on a one-second time scale compared with the weeks or months required to produce aperture synthesis maps. D. Data Processing
Data processing methods in radio astronomy are becoming sophisticated and promise to become increasingly important in the future. As we process information with complex structure in the two angular coordinates, the frequency domain, and/or the time domain, a major emphasis must be placed on data processing and display techniques.
I . Continuum Systems Most of the signals observed in radio astronomy have the character of wideband noise. Time variations, when present in radio galaxies or quasars, occur over periods of many days or months. Therefore, the signals are timeinvariant over the period of observation and essentially frequency-invariant over the bandpass of the receiving system. In these cases we have to deal only with the structure of the signals in the two angular coordinates. When a single or multiple beam telescope is employed, the radio source region is simply scanned by the response pattern of the telescope, the data are smoothed by appropriate averaging or convolution with the beam profile of the system, and the data are plotted on a coordinate grid to form a two-dimensional map of the source region. In the case of aperture synthesis, more complex processing is required. The raw data are smoothed, and all observations with a given interferometer spacing and orientation are averaged together. When all spacings and orientations have been observed, the amplitude and phase of the response at each point yields a two-dimensional map of the mutual coherence function. This complex two-dimensional function is the two-dimensional Fourier transform of the brightness distribution of that region of sky under observa-
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tion, i.e. that region of sky within the primary response patterns of the interferometer elements. Using digital techniques, such as the Cooley-Tukey algorithm, the mutual coherence function is Fourier transformed to yield the angular brightness distribution of the source region. Radio sources often display polarization when observed with high angular resolution, and this process must be carried out separately for each state of polarization. The data processing required by very long baseline interferometry is rather involved (18). The radio frequency signal accepted by each antenna element is converted to the video band by mixing it with a stable local oscillator. The resulting video band is recorded on magnetic tape. The recorded signals are later replayed and correlated. Before correlation, it is necessary to compute all geometrical time delays between the wavefronts arriving at the two elements, to account for Earth motion, and to account for all known local oscillator frequency offsets and timing errors. Because the latter errors are not known precisely, the exact interferometer fringe frequency and relative time delay at which the correlated signal will appear is unknown. The average correlation coefficients are determined at short time intervals (e.g. 0.2 sec) throughout the observing period for a variety of relative time delays. A fringe rate spectrum is computed for each relative time delay and a sharp peak in the spectrum is normally found at the proper time delay and fringe rate. Once this correlated signal is found, it can be used to determine the local oscillator frequency offsets and relative times to high accuracy. Frequency offsets of 0.01 Hz and timing errors of 0.1 psec are easily determined. Most of the scientific conclusions from VLBJ work are drawn from determinations of the amplitude of the correlation coefficient. Phase information is difficult to obtain due to small local oscillator offsets or other sources of phase errors. For this reason, higher accuracy is obtainable in measuring source diameters than in absolute positions. Phase information has been used in a few experiments such as observation of the gravitational deflection of radiation where careful phase calibrations were obtained. 2. Spectral Line Systems
In spectral line work we must deal with three coordinates: two angular dimensions and one frequency dimension. The frequency coordinate yields the radial velocity of the matter under observation through the Doppler effect, provided the rest frequency of the line is accurately known. Good frequency and radial velocity resolution requires the use of narrow effective bandwidths. As a result, the fluctuation level at the radiometer output is high when compared with that of a wideband continuum system. Integration times of many hours are required for the highest sensitivity. For this reason,
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WILLIAM C. ERICKSON AND FRANK J. KERR
multichannel receivers for the simultaneous observation of 10 to 400 separate frequency bands have been developed. Multichannel spectral-line receivers take two general forms, filter systems and autocorrelation systems (21). The broadband intermediate-frequency signal may be supplied to a bank of narrowband filters that are each tuned to adjacent center frequencies in the IF band. The output of each filter is detected and integrated so that the power spectrum of the input signal is obtained directly across the filter bank outputs. Alternatively, the IF signal can be correlated with itself after various time delays. This can be done with a bank of correlators and a series of equally spaced time delays. When the output of each correlator is averaged over the observation period, the autocorrelation function of the input signal is obtained. This autocorrelation function is the Fourier transform of the power spectrum of the input signal. Therefore, the power spectrum (the line profile) can be derived by Fourier transforming the observed autocorrelation function. The overall time delay is approximately equal to the reciprocal of the required frequency resolution. It is generally more convenient to employ analog techniques in filter systems while digital techniques are usually employed in autocorrelation systems. The latter have the usual advantages of digital systems, flexibility and stability; their greatest advantage is the ease with which the bandwidth can be selected from a large number of alternative values. However, the maximum signal band that can be analyzed is limited by the maximum sampling rate of present digital techniques. Analog filter systems are therefore built for wideband applications. The display of three-dimensional data presents a severe problem. Various two-dimensional contour plots have been used to represent different cuts through the data along various axes. The use of color to represent one coordinate is becoming increasingly common. As our data become more and more complex, with increasing angular and frequency resolution, our display problems grow and they must receive more and more attention. 3. Rapidly Varying Phenomena
Time variability in astronomical sources is always accompanied by the observation of complex frequency structure. This spectral structure is usually generated by the source, but even in the cases of broadband emission the dispersion of the intervening media produces complex spectral structure at the Earth. Therefore, we confront the problem of processing and displaying data in four coordinates : two angular coordinates, frequency, and time. Radio astronomical technology is really only starting to recognize and cope with this problem.
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Perhaps we are fortunate that many dynamic sources, such as pulsars, have intrinsic and apparent sizes that are far smaller than the angular resolution limits of our instruments. This fact simplifies our information handling problem. Several systems have been developed to compensate for the dispersion impressed by the interstellar medium on pulsar signals. These allow sensitive wideband observation of highly dispersed pulses provided that we have previous knowledge of the amount of dispersion to be expected. Searching techniques for the discovery of new pulsars with unknown position, dispersion, and pulse repetition frequencies are still rather primitive. The radio sun represents the most challenging data processing problem. I t exhibits an exceedingly complex brightness distribution that is a rapidly varying function of time and frequency. Most solar radio spectrographs utilize small antennas that accept radiation from the whole Sun and yield little or no angular information. Some spectrographs employ interferometers, and one employs a linear antenna for high angular resolution in one dimension. The Australian radioheliograph produces two-dimensional maps of the solar brightness distribution as a function of time at one frequency. but the system is being extended to include an additional frequency. The full problem of four-dimensional data handling and display has not been yet attacked.
E. Space Radio Astronomy With the advent of space radio astronomy, an additional three decades of frequency range, from about 10 kz to 10 MHz, have been opened to astronomical investigation. In the 1960’s, a number of rocket and satellite experiments with simple dipole antennas were flown above the ionosphere by Soviet, Canadian, British, and American groups (22). These observations showed that the galactic continuum peaks in the 3 MHz range and falls in brightness at lower frequencies. In addition, the low frequency properties of Type I I I solar bursts were determined. The early experiments also illustrated the problems concerned with the separation of geophysical and astronomical emission, the modification of the receiving antenna’s properties by the ambient plasma, and the development of radio-quiet spacecraft. Much of our information concerning low frequency emission is still derived from experiments involving simple dipole antennas. A more ambitious experiment, the Radio Astronomy Explorer satellite, was launched in July, 1968. This satellite possesses oppositely directed “ V ” antennas, each 229 m in length. The satellite is gravity-gradient stabilized in a circular orbit of 6000 km altitude. Radiometers aboard the satellite operate from 0.2 to 9.2 MHz and, above 3 MHz, the “ V ” antennas provide moderate angular resolution. The galactic continuum emission at various hectometric wavelengths is gradually being mapped with this instrument.
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WILLIAM C. ERICKSON AND FRANK J. KERR
A more ambitious experiment is underway. The second Radio Astronomy Explorer will be placed in lunar orbit. At that distance from the Earth, interference due to terrestrial emissions should normally be low and will be entirely absent when the satellite is on the far side of the Moon. The lowfrequency limit of the system will be set by the interplanetary plasma frequency of about 20 kHz. The Moon will also be used as an occulting disk to assist in determining the positions of radio sources. French astronomers are developing a stereographic technique for determining the positions of solar bursts through measurement of the time of arrival of solar emissions at the Earth and at a distant spacecraft. Radiophysics experiments will also be conducted in the 10 Hz to 3 MHz range on the joint German/U.S. HELIOS project. Several instruments involving very large ( x 20 km) apertures and/or interferometers have been proposed. These instruments would permit observations with angular resolutions limited only by interplanetary scattering (x 1" at 500 kHz). None of these instruments has yet been authorized, and their concepts are likely to change radically before they are developed for flight. 111. RADIOOBSERVATIONS
A . Introduction
Radio astronomy has come a long way since its early days in the thirties and forties when crude low angular resolution maps of the radio sky existed at a small number of wavelengths. Observations are now available with considerable detail, and many new and unexpected discoveries have been made. Some of the objects were known previously from optical astronomy, but many of them are invisible optically or were discovered first from their radio emission and then subsequently observed optically. A great variety of spectral lines have now been found in the radio region. The expansion of knowledge in all branches of the subject has been closely tied to instrumental advances. Sometimes the introduction of better antennas or receivers has led immediately to new discoveries; in other cases the posing of astrophysical problems has provided the incentive for some particular instrumental development. We will present an overview of observational radio astronomy. The field is now so vast that it is only possible to develop a few themes, but we have tried to give an impression of the wide-ranging nature of the subject. The varying degrees of emphasis given to the different topics inevitably reflect the special interests of the authors. In this type of review it is not possible to give individual attribution for all the results and interpretations discussed here. Instead, we list one or more review papers for each branch of the subject where the reader can find greater descriptive detail and extensive lists of references.
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B. Solar Radio Astronomy The Sun is the only star whose disk can be resolved and studied in detail at radio wavelengths.* Its atmosphere is a fully ionized magnetized plasma which provides a wealth of intriguing physical phenomena. Solar flares result in a wide variety of related electromagnetic and particle emissions. These emissions are of scientific importance and, when they react with the Earth’s atmosphere, are of practical importance as well. The whole literature concerning solar radio emission cannot be described here (23). The solar corona. at a temperature of roughly one million kelvin, continuously emits thermal radio waves. Observation of this undisturbed solar emission has provided data concerning the temperature and density distribution of the quiescent corona and chromosphere. Disturbances in the chromosphere and corona above solar active regions generate additional radio emission by a variety of mechanisms. At centimeter and decimeter wavelengths, disturbed emission is generated primarily through thermal or gyro processes. At longer wavelengths, the disturbed emission occurs in the form of bursts possessing extremely complex time and frequency structure. Meter wavelength bursts are generated through plasma oscillation and synchrotron processes. At every wavelength, the observed emission is strongly influenced by the transmission properties of the intervening solar atmosphere. Until the development of the Australian radioheliograph, two-dimensional mapping of the Sun was restricted to decimeter and centimeter wavelengths (24, 25) and each map required several hours of observation. The rapidly varying emissions at meter wavelengths which are closely related to plasma instabilities could not be studied. Solar radio bursts of spectral types 11 and 111 are short-lived narrowband emissions that drift from high to low frequencies. They occur at the local plasma frequency of the corona. After an explosion deep in the Sun’s atniosphere. particles and shock waves propagate outwards. As these disturbances reach successively lower coronal densities they generate emission at correspondingly lower plasma frequencies. Type I I I bursts are caused by electron streams traveling at nearly the velocity of light. Type 11 bursts are caused by more slowly moving shock waves and require about 30 min to drift across the same frequency range. Groups of Type 111 bursts usually precede a Type I 1 burst. Heliograph observations at 80 MHz indicate that the Type I 1 source often occurs in the same region as the preceding Type I1 I source. The Type 1 I source is initially unpolarized but after reaching its peak intensity it breaks up
* A number of other stars have now been observed i n the centimeter wavelength range. The study of their brightness distributions by VLBl techniques is on the verge of technical capability.
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WILLIAM C. ERICKSON AND FRANK J. KERR
and exhibits complex polarization structure. Weak explosions produce Type 11 emission that is restricted to the centers of coronal streamers where the velocity of magnetohydrodynamic waves, the Alfven velocity, is low; strong explosions cause Type 11 shock waves across extended regions as large as the solar diameter. Spectacular evidence of the interaction between widely separated centers of activity on the Sun has emerged. Bursts are triggered or other interactions occur between centers separated by more than a solar radius (- lo6 km). In one case, Wild (26) traced a sequence of events in which shock waves from a flare produced a Type I1 coronal source directly above it and a few minutes later triggered the eruption of a prominence more than lo6 km away; this eruption in turn appeared to cause high energy particles and further shock waves which produced a second large radio source. Another kind of interaction has been observed in which two distant radio sources appear within a few seconds of each other. The widely spaced sources are closely correlated in time and even their circular polarization is of the same sense. Wild suggested that there must exist some central source of excitation high in the corona. Perhaps, at an instability where two bipolar magnetic field configurations cross each other, electron streams are accelerated and travel down the magnetic field lines to excite plasma waves that result in radio emission. Moving Type IV solar bursts are caused by clouds of high energy electrons that emit synchrotron radiation as they spiral around solar magnetic field lines. The emission regions lie well above the appropriate plasma frequency level and occasionally travel rapidly out through the corona. The ability to observe these phenomena in two dimensions has revealed their physical nature and their relation to optical activity for the first time. In some cases the radio source appears soon after a prominence eruption and the source moves with the same direction and velocity as the matter ejected from the prominence. The velocity of the Moving Type IV has been identified with the local Alfven velocity in several cases. The coronal magnetic fields are occasionally outlined by the radio emission, which displays looped structures above eruptive prominences. The stationary part of the Type IV emission is found to be a separate highly polarized source lying above the projected position of the solar flare that triggered the prominence activity. From heliograph observations, the unstable nature of the corona emerges. Previously, we had known that it was unstable in the sense that it constantly evaporates into space in the form of the solar wind. Explosions in the form of solar flares have long been observed. Heliograph observations prove that these explosions, the most energetic events in the solar system, are often triggered by other events at large distances. Huge amounts of energy are apparently stored in unstable magnetic field configurations. Only relatively
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minor disturbances are required to release this energy. The full development of these phenomena will make a fascinating story. By means of satellites, the long wavelength emissions from the Sun are being studied. Below 10 MHz, the Sun appears to emit almost nothing but Type 111 bursts. At times of solar decametric activity, these bursts are produced copiously. One active region may emit hundreds of thousands of bursts over a few day period. Since low frequencies are generated at plasma levels high in the corona, long wavelength bursts can be traced to great altitudes in the corona. Most of the disturbances are dissipated within 20 solar radii of the Sun, but some have been followed out through the interplanetary medium to the orbit of the Earth (215 solar radii) and beyond.
C. The Planets The planets would be expected to appear as thermal radio sources, in which the radio frequency emission is proportional to the temperature of the surface. According to the Rayleigh-Jeans law, the flux should be proportional to the square of the frequency. Thermal emission in the radio region is weak, and the planetary diameters are small ; therefore large antennas and sensitive receivers are needed to detect this type of emission. All the planets from Mercury to Neptune have been detected as radio sources. Radio observations can contribute to studies of planetary heating through measuring the brightness temperature as a function of the radio frequency, and as a function of the phase of the solar illumination cyc:e. In the latter case, the temperature at a particular phase depends on the amount of solar heating, and also on the thermal properties of the planetary surface and atmosphere. The existence or otherwise of a phase effect for Venus is the subject of controversy at the present time. A recent high resolution map of Venus, obtained by aperture synthesis techniques at 6 cm, is shown in Fig. 1 1. Jupiter is the most intriguing of the planets as far as radio emission is concerned. Burke and Franklin (28) discovered that Jupiter emits intense bursts at decametric wavelengths. These bursts have been the subject of intensive study in several countries. They occur sporadically, but are most frequent when certain regions of the planet are facing the Earth; this effect enables a precise determination to be made for the planet's rotation period, especially when observations extending over a number of years are used. The resultant period is 9h55"29s.37, which is slightly different from rotation periods determined optically, presumably because it refers to a different part of the somewhat fluid planet. The bursts also show a variation with the solar cycle, and they appear to be influenced by the position of Jupiter's satellite, lo. Simultaneous observations at different places on the Earth give results that vary with time, but they
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WILLIAM C. ERICKSON A N D FRANK J. KERR
I
’
8
\
I
‘I,’ .
FIG.1 1 . Aperture synthesis map of Venus at 6 cm wavelength at epoch 1969.2 (27); the contour interval is 60 K and the circle represents the limb of the planet. The halfpower beamwidth and the portion of the planet illuminated by the Sun are shown in the corners of the map.
show that substantial correlation can sometimes be obtained at spacings as large as 100 km. These observations have demonstrated that the short duration (millisecond) bursts are produced at the source, but variations on a longer time scale (seconds) are caused by inhomogeneities in the ionosphere and interplanetary medium. Jupiter also produces excess emission at decimeter wavelengths, but this is a relatively small increase above the thermal level and is almost constant with time. This radiation is attributed to synchrotron emission from relativistic electrons in a radiation belt of magnetically trapped particles. Interferometric and aperture synthesis observations have shown that the emission comes from a region which is several times the diameter of the planetary disk. Also, the radiation is polarized, and the polarization “wobbles” with the same rotation period as that indicated by the decametric bursts. The varia-
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tions of the plane of polarization show that Jupiter’s magnetic axis is tilted 10”from the rotation axis. The magnetospheric structure of Jupiter is very similar to that of the Earth. Both planets possess strong magnetic fields and Van Allen belts. The synchrotron emission from Jupiter’s Van Allen belts is clearly shown in Fig. 12. lo revolves within Jupiter’s magnetosphere and perturbs the magnetic field causing particles to be dumped on the upper atmosphere of the planet. When this occurs, the decametric bursts are generated.
FIG.12. Maps of Jupiter at 21 cm wavelength for central meridian longitudes of 15”, 135”, and 255” respectively (29). In each map north is at the top and east at the left. The divisions on the axes are 30 arcsec apart, and the contour interval is 47 K . The planet is at a distance of 4.8 A.U., and the approximate directions of the magnetic and rotational axes are indicated at the bottom right-hand corner of each map. The dark circles represent the planet’s optical limb.
Due to the lower magnetic field, synchrotron emission from the Earth’s Van Allen belts is faint. The orbit of the Moon is well outside the Earth’s magnetosphere and does not appreciably perturb the magnetic field. However, perturbations of the field produced by disturbances in the solar wind cause dumping of Van Allen particles on to the Earth’s ionosphere and result in the emission of low frequency bursts observed from satellites, the aurora, and magnetic storms.
D. The Galaxy 1. Contitiirum
Galactic “noise” can be observed from the Earth’s surface over a spectral range from about 10 MHz to 100 GHz, with the (variable) cutoff at the lower frequency end produced by the ionosphere, and at the upper end by the gases of the lower atmosphere. The galactic disk stands out in the distributions at all frequencies as a narrow band, narrower than its visible counterpart, the
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WILLIAM C. ERICKSON A N D FRANK I. KERR
Milky Way. The Sun is the brightest source in the sky, but it is not as dominant as it is in the visible sky. There is almost no correlation between the brightest objects in the radio and optical ranges, in keeping with the fact that different emission processes are responsible. At low frequencies, the radiation is strong and detectable over the whole sky as a general distribution, with many localized sources superposed; some of these are of small angular diameter while others are more extended. At the higher frequencies, the radiation is much weaker and limited to portions of the Milky Way, again with localized concentrations. An example of a low frequency map for the whole sky is given in Fig. 13; this is due to Landecker and Wielebinski (30) and is mainly derived from observations at Parkes and Cambridge near 150 MHz. It is plotted in galactic coordinates, in which the equator is the mean plane of the Galaxy. The map has the fairly low angular resolution of about 2", but shows the general distribution quite well. There is a concentration towards the Milky Way and the galactic plane, as well as a concentration towards the center of the Galaxy (in the middle of the diagram), with the principal emission extending over the longitude range 275" to 100". There are also a number of prominent "spurs" at a high angle to the galactic plane concentration. The nature of these spurs is a very interesting problem, as they have no optical counterpart. They may be related to the geometry of the interstellar magnetic field in our vicinity; an alternative suggestion is that they might be part of the shells produced by one or more nearby supernova explosions at some time in the past. A portion of a higher frequency map is shown in Fig. 14, indicating the much greater concentration to the galactic plane. The fact that the continuum distribution over the sky is different in shape at different wavelengths shows that the spectrum varies from place to place. Consequently there must be more than one emission mechanism operating, with different spectral characteristics. The two important mechanisms are free-free emission (bremsstrahlung) and the synchrotron process. In the former, an electron radiates classically as it passes near a proton. This process occurs in ionized hydrogen, and is detectable from discrete HI1 regions such as the localized sources in Fig. 14; some of these are visible optically, while others are more distant and are optically obscured by dust. The synchrotron process, in which relativistic electrons spiral around magnetic field lines, occurs in the main disk of the Galaxy, in the spurs, and in supernova remnants, many of which can be recognized as individual objects. The two mechanisms can be distinguished by their spectral indices; free-free emission is more important at high frequencies, and synchrotron emission at low frequencies. In addition, if polarization is observed, we can conclude that synchrotron emission must have been responsible, but the converse does not necessarily apply, as polarization
FIG.13. Map of the sky at 150 MHz in galactic coordinates, from observations at Parkes and Cambridge (30). The contour units are brightness temperatures in kelvin.
h)
W
Y
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which is present in the radiation at the source is progressively reduced in strength by propagation through the galactic disk. This depolarization effect results from the fact that the interstellar medium is highly irregular, leading to different amounts of Faraday rotation for different elements of solid angle within the antenna beam-eventually, the average polarization over the whole beam becomes randomized. At the lowest frequencies, the synchrotron emission is very strong, and the ionized hydrogen appears only in absorption against the nonthermal background. In fact, at the frequencies observable only from satellites (< 10 MHz), this absorption is so great that radiation is being received only from the nearer portions of the Galaxy. A useful consequence is that by changing the frequency, one can actually probe to different distances in the ionized hydrogen disk. The intensity and spectral distribution of the radiation provide valuable information about the mean density of ionized hydrogen in the solar neighborhood, and also about its spatial variation. Statistical data are obtainable on the cloudiness of the ionized material; also, prominent nearby features, such as the Gum nebula and the concentration of ionized gas in the Cygnus region, can be studied in depth. The possibilities for detailed mapping are however quite limited at the very low frequencies, because of the poor angular resolution associated with the very long wavelength antennas. Polarization studies provide information about the interstellar magnetic field in the vicinity of the Sun (32). Three methods are available: (i) Much of the galactic background radiation (underlying the discrete sources) shows weak linear polarization, with somewhat stronger effects in a few regions such as the north polar spur. The polarization pattern over the sky presents a more ordered appearance as we go to higher frequencies (> 1000 MHz) where Faraday rotation and depolarization effects are less important; the polarization vector is then an indicator of the transverse component of the magnetic field. By contrast, the polarization cannot be seen at all at frequencies below 300 MHz or so. Most of the areas with observable polarization appear to lie in a band about 60" wide, containing a great circle through the galactic poles, which is presumably normal to the direction of the local field. (ii) The Faraday rotation of the radiation from extragalactic radio sources gives evidence on the mean longitudinal component of the field through the nearby galactic disk. As the rotation effect is greatest near the galactic equator, shows a longitude effect, and varies fairly smoothly over the sky, it must occur mainly in the Galaxy, rather than being due to intrinsic properties of the more distant sources. FIG.14. Twenty-one cni map of portion of the Southern Milky Way (31). The unit is brightness temperature in 1.74kelvin, and the antenna halfpower beamwidth was 10 arcmin.
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WILLIAM C. ERICKSON AND FRANK J. KERR
(iii) The longitudinal field can also be investigated through Zeeman splitting in the 21 cm hydrogen line (see below). The effect has only been seen in a few dense clouds, where high values of up to 5 0 p G have been found, possibly because the field lines are compressed as a cloud collapses. The strength of the general field is only a few microgauss or less. This result covers the general vicinity of the Sun, and no information can be obtained about the field at distances beyond a few hundred parsecs.* The older observations of the optical polarization of starlight and the radio observations of Faraday rotation have generally been interpreted in terms of a magnetic field running longitudinally along the local spiral arm. An alternative model has been gaining ground in which the field is wound in a rather open helix around the spiral arm, with the geometry possibly connected with the local structural feature known as the Gould belt. The helical model was first proposed by Hornby (33) to account for the distribution of the radio polarization vectors, and has been most extensively developed by Mathewson (34), mainly based on a study of recent optical polarization measures. Mathewson suggests that the helical structure is local to the solar vicinity becoming more longitudinal further out, but even so it is difficult to reconcile this model with the Faraday rotation observations. It is probably quite significant that the longitudinal field model is based on observations of the transverse component of the field, while the helical model comes from observations of the longitudinal component. The true field geometry may be more complex than either model proposed so far. A very active area in radio astronomy at present is the study of the structures and physical conditions in HI1 regions, localized regions in which the interstellar hydrogen is ionized by ultraviolet radiation from one or more hot stars. The recent increases in angular resolution have made detailed radio mapping possible for the first time. For example, an aperture synthesis map of a portion of W3, obtained by Wynn-Williams (3.9, is shown in Fig. 15, with a photograph in the Ha line given for comparison. A detailed comparison of such a pair of maps can indicate the distribution of absorbing dust in the HI1 region, as it is believed that the radio and Ha emissions are produced in a constant ratio, and the dust affects the Ha alone. Studies of various radio lines also give important information on the excitation conditions, density, and constituents of an HI1 region. Supernova remnants form another group of discrete continuum sources in the disk of our Galaxy (36). They can be distinguished from the HI1 regions
* The parsec, the common unit of distance in present-day astronomy, is the distance at which the parallax angle subtended by the mean radius of the Earth's orbit (one astronomical unit) is one arcsecond: 1 pc = 3.26 light yr = 2.06 x los A.U.= 3.09 x lo'* cm.
TECHNOLOGY AND OBSERVATIONS IN RADIO ASTRONOMY
33
because they show nonthermal spectra and they are often highly polarized, both of which are characteristic of synchrotron emission. These objects are the remnants of stellar explosions. The best known example is the Crab nebula, in Taurus, which was produced by an explosion observed by Chinese astronomers in 1054 A.D. It is a powerful continuum source over the whole electromagnetic spectrum, and it also contains a pulsar which has been detected at radio, optical, and x-ray wavelengths. Other sources identifiable as supernova remnants appear to have been expanding for periods up to 50,000 yr. Many of them show a ring-shaped brightness distribution, with a polarization pattern which implies a fairly regular magnetic field distribution. The Sun is the most powerful source in the sky because of its proximity, but for a long time radio emission was not detectable from any other star. Because the Sun’s radio output increases so dramatically at times of solar flares, attempts were made to detect “flare stars,’’ some dwarf M stars whose light output increases many times during sporadically occurring flares. Radio flares have been detected from several of these stars, at time approximately coincident with the visual Aares. Also it has been shown that, as for the Sun, the burst occurs slightly later at lower frequencies. This work has been mainly done with the big dishes at Jodrell Bank and Parkes, as a large collecting area is needed for such weak signals. The other requirement is patience, in waiting for infrequent events. More recently, weak radiation has been detected from several novae and infrared objects by the use of long integration on the Green Bank interferometer.
2. Atomic Lines a. 21 cm line. The first radio-interstellar line detected (37) was the hyperfine transition line from neutral hydrogen atoms at 1420.406 MHz (21.1 cm). This line is detectable from galactic hydrogen all over the sky, with a concentration towards the Milky Way, and no increase of intensity towards the direction of the galactic center. The observed line profiles are often complex in shape. The natural width of the line is very small, and the broadening of the profiles is essentially Doppler in origin, due in part to thermal motions in the gas clouds, but mainly arising from galactic rotation and mass motions of the hydrogen. The peak brightness temperatures in the profiles go up to 100-150 K. The 21 cm line was discovered as a result of an astrophysical prediction. In this case, the physics of the emission process is well understood, no anomalous effects have been observed and no variable sources. The line is therefore very suitable for estimating the physical parameters in HI regions, and for use as a tracer for studies of galactic structure (38).The main peaks of
34
WILLIAM C. ERICKSON AND FRANK J. KERR 4995MHz
w3 I
I
56'
00'
02h22m
I
52'
I
48'
44 '
40' 02h 2 P
FIG.15. (A)Aperture synthesis map of portion of W3at 4995 MHz (3.5). The beam size for the Cambridge " One-Mile telescope" is shown towards the bottom right. The contour interval is 220 K. (B) The small rectangle near the center of the photograph (taken in red light) indicates the area covered by the map.
the profiles are closely related to the spiral structure and the main concentrations of young material in the Galaxy. The primary difficulty in mapping the spiral pattern is that there is no direct method for estimating the distances of the hydrogen concentrations. It is necessary to use the fact that the main line broadening results from the differential galactic rotation, and to assume or derive a model for the velocity field in the Galaxy. One map derived in this way is shown in Fig. 16. However, this is a controversial subject, as there are differing views on the interpretation of the features indicated by the 21 cm profiles. Recent theoretical developments, in which spiral arms are regarded as quasi-stable density waves, give hope for a better understanding of the problems involved. Several extensive surveys of 21 cm radiation have been carried out in recent years, largely in the region of the Milky Way, but also in other parts of the sky. Although the strength of the radiation is usually well above the sensitivity limits now available, such surveys take a long time when high angular resolution is used and also the frequency structure of the profiles is being studied in detail. The needs of these surveys have been an important contribution towards the development of autocorrelation receivers with large numbers of channels (c. 400-1000). In these receivers, the Fourier transform of a line profile is studied in the time domain. Very large bodies of data have been accumulated, and highly efficient methods of data handling have been
TECHNOLOGY AND OBSERVATIONS IN RADIO ASTRONOMY
35
FIG.1.5(B). See Fig. 15(A) for legend.
developed. For example, Fig. 17 shows " maps " of the hydrogen distribution in velocity-right ascension space for particular values of declination. These maps were generated in a computer from observations taken on a number of different occasions, and then photographed from a cathode-ray tube display. The large number of peaks and valleys in this one map gives some indication of the great amount of detail coming out of present-day radio line work, a far cry from the simple maps of the early days of radio astronomy.
36
WILLIAM C. ERICKSON AND FRANK J. KERR 240’
210’
\
\
\
\
I
180’
I 1
150’
120-
/
/
1
so’
60’
/ 330.
I
I
I 0
\
\
\ 30’
FIG.16. A sketch of the main features of the neutral hydrogen spiral structure, as interpreted by Kerr (38). using 21 cm data and a kinematic model of the rotational motion in the Galaxy. Structural details are not shown in the inner region, owing to the large uncertainty in the distance. Regions of low hydrogen density are indicated by L.
In addition to the large scale surveys, the properties of individual hydrogen clouds are being studied (40).Some of these produce very narrow lines, and seem to be quite cold (<50 K). When seen in front of hotter hydrogen, they appear in absorption as narrow dips in a profile. Such cases occur fairly often in the direction of dark clouds, indicating that some of the cold hydrogen in dust clouds remains atomic. One of the mysteries in the 21 cm observations is the character and location of the “high velocity clouds” of hydrogen ( I VI > 70 km/sec), which have been found in fairly large numbers. The high velocity clouds near the galactic center are certainly connected with the rapid motions and general turmoil in the central region, but the rest of these clouds are more puzzling. They have FIG.17. Computer-processed contour maps of the 21 cm line radiation for three declinations, as a function of right ascension (right-hand scale) and radial velocity with respect to the local standard of rest (horizontal scale) (39). The observations were taken with the NRAO 300-ft (91-m)telescope. The corresponding galactic coordinates for each scan are given at the left. The contour unit is 5 K in antenna temperature.
TECHNOLOGY AND OBSERVATIONS IN RADIO ASTRONOMY
37
38
WILLIAM C. ERICKSON AND FRANK J. KERR
been attributed to material falling into the Galaxy from outside (because most of the clouds have a component of motion towards us), to material ejected in large explosions and now falling back, or to intergalactic objects, but their distances and origin are still not known. Absorption effects are also seen at the frequency of the 21 cm line in directions of discrete continuum sources, due to absorption of the continuum emission in hydrogen clouds along the line of sight. These observations can be used to investigate the relative distances of the sources and the hydrogen clouds, and also to get better estimates of the excitation temperature of the hydrogen, which cannot be determined from emission observations alone. The Zeeman splitting observations mentioned above have been carried out only on absorption lines, which are considerably sharper in frequency than most emission features. These measurements depend on the fact that the two outer Zeeman components are circularly polarized in opposite senses. The split is difficult to detect, however, because the separation of the two components is only 2.8 Hz/pG, i.e. the maximum separation is only 140 Hz at a frequency of 1420 MHz, and in most of the cases investigated the separation is considerably smaller. The observation is carried out by switching rapidly between the two circular polarizations, and studying the small difference in voltage as a function of frequency across the absorption-line profile. Measurements of this type were carried out for nearly a decade before improvements in antennas and receivers made the instrumental effects small enough for the Zeeman splitting to be unambiguously identified (41). Absorption observations require the greatest possible antenna gain, as the size of the absorption dip depends on the antenna temperature due to the background source, which is usually small in angular size, On the other hand, resolving power is generally the more important antenna requirement in 21 cm emission studies. b. Recombination lines. The other atomic radio lines that are being widely studied in the Galaxy are the high-level recombination lines from hydrogen and other atoms (42). These transitions are primarily observed in HIT regions; they occur when electrons cascade down through the long series of levels during atomic recombinations. For example, a change of the principal quantum number from 110 to 109, conventionally known as the H109a transition, occurs at 5008.9 MHz. Many Ha lines have been reported over the range from n = 56 at 36,446 MHz to n = 253 at 404 MHz. In addition, a number of hydrogen transitions have been observed in which the quantum number changes by 2, 3, 4, or 5 (p, y , 6, or E transitions), as well as some helium recombination lines, and a possible carbon line. See Fig. 18. The recombination lines from HIT regions normally show a simple profile shape and the velocities agree with those determined optically in cases where comparisons are possible. However the radio observations have the advantage
39
TECHNOLOGY AND OBSERVATIONS IN RADIO ASTRONOMY
0.8T(70 0.7
-
0.6
-
05
-
0.4
-
-.-
H I10
109
IC 1795
(W3)
n
I
03-
-
0.2 0.1
-
0-
H
139-137 He 11O-lO9
..A ... *.......... I -4
A...
*
...:a
I
-3
I -2
I I
C(?) I0-m
I
0
I I
1
I
2
3
FREOUEmY (MHx)
FIG.18. Portion of the recombination line spectrum for the HI1 region W3 = IC1795 (43).
that they enable HI1 regions to be seen all over the Galaxy for the first time, and their distribution and kinematics investigated. Comparisons between 21 cm and recombination-line velocities have shown that the neutral and ionized hydrogen display essentially the same kinematics, so that the forces acting on the gas and the young stars cannot be very different. In the case of the neutral hydrogen, we pointed out that no unusual physics was involved, even though the densities are exceedingly small by terrestrial standards (- 1 atom ~ m - ~ We ) . have a rather different situation for the highly excited atoms in HI1 regions. The atomic radius is so large for these atoms (e.g., 0.5 p for n = 100) that their environment is quite different from anything that can be produced in a laboratory vacuum. Even for such large atoms, no Stark-broadening effects have yet been demonstrated with certainty, and the observed line broadening appears to be entirely of Doppler origin. The variety of recombination lines that have been detected provide an excellent tool for studying the astrophysical conditions in HI1 regions. Some departures from local thermodynamic equilibrium have been found, but they do not appear to be very large. Considerable attention has been given to the derivation of electron temperatures, the population of the atomic levels by dielectronic recombination, and other processes.
40
WILLIAM C. ERICKSON AND FRANK J. KERR
3. Molecular Lines One of the most striking developments in radio astronomy in recent years has been the detection of a rapidly increasing number of molecular lines and the use of these emissions in studying the physics and chemistry of the interstellar gas (44,45). The whole development has been very closely tied to advances in techniques. The first serious discussion of transitions that might be detectable by radio astronomers was given by Townes in 1957 (46). Because most interstellar space is cold. and the low density results in very infrequent collisions, we would expect a priori that only the simplest (diatomic) molecules would have a reasonable chance to form, and these would substantially all be in the ground state. The first molecule looked for was the hydroxyl radical, OH, and specifically the lambda-doubling transitions at 1667 and 1665 MHz (47). However, the search was hindered by lack of precise knowledge of the transition frequencies. A new laboratory measurement of the frequencies by Ehrenstein et al. (48) led to a positive detection of these two lines by Weinreb et al. (49), and soon afterwards the weaker satellite lines of the quadruplet were also found. The OH lines have been observed in absorption in many places in the Galaxy, and also in emission of two different types. Low intensity radiation has been received from the cold gas in dense dust clouds, but more surprising is the high-level and time-variable radiation from isolated spots in HI1 regions. The latter type has characteristics which imply that a masering process is responsible (see below). After the detection of OH, unsuccessful searches were made for several other hydrides, and then the next molecules found were ammonia (50) and water (51), demonstrating the presence of triatomic molecules in the interstellar medium. Shortly afterwards, formaldehyde, H,CO, was added to the list (52), implying a further stage of complexity, and since then new discoveries have been frequent as the search has extended to higher frequencies. The majority of the recent discoveries have been in the millimeter wave region, following the completion of the NRAO 36 ft (1 1 m) radio telescope in Arizona and the availability of more sensitive receivers in this wavelength region. Some sample profiles are shown in Figs. 19 and 20. A list of the 27 molecules from which radio interstellar lines have been detected (up to October 1971) is given in Table I, including the number of transitions observed so far, and their frequency range. In general terms, electronic transitions in molecules tend to occur in the ultraviolet and optical range, vibrational transitions in the infrared, and rotational ones in the radio range. Most of the observed transitions are rotational, but there are also a few cases in which splitting of the rotational energy levels is
TECHNOLOGY AND OBSERVATIONS IN RADIO ASTRONOMY
41
0TA( O K )
-
-10-
-20 -
-200
-150
-100 -50 0 RADIAL VELOCITY (krn/sec)
50
100
FIG.19. Absorption profile in the 1667 MHz OH line in the direction of Sagittarius A, the strong continuum source at the galactic center (53).
r
40
+
-- 30 -
+ +
Y
[I 4:
10
+
I
1
+
+
+
+
+
+
+
++
+++++:+++
+++
I
-10
0
t++++ 10
FREQUENCY DISPLACEMENT (MHz)
FIG.20. The 2.6 mm emission line from CO in the Orion nebula, recorded with the National Radio Astronomy Observatory 36 ft (1 1 m) dish and &-channel receiver (544).The center frequency is 115, 267.2 MHz. Taken from Wilson, Jefferts, and Penzias, Astrophys. J. L e r r . 161, L43 (1970). Courtesy of University of Chicago Press.
42
WILLIAM C. ERICKSON AND FRANK J. KERR
TABLE I FROM WHICHRADIOINTERSTELLAR MOLECULES LINESHAVEBEENREPORTED (TO OCTOBER 1971) (45)
Molecule Inorganic 016H-hydroxyl 0'"H SO-silicon monoxide H20-water NH3-ammonia Organic CN-cynanogen C12016-carbonmonoxide C13016
clzo'8
Number of transitions observed
10 2 1
I 7 1
1 1 1 1
CS-carbon monosulfide HC12N14-hydrogen 3 cyanide 3 HCl3NI4 1 OCS-carbonyl sulfide 6 HzC12016-formaldehyde 1 HzC13016 H2C12018 1 2 HNCO-isocyanic acid 1 H2CS-thioformaldehyde 1 HCOOH-formic acid 2 HC3N-cyanoacetylene 7 CH,OH-methyl alcohol CH3CN-methyl cyanide 5 4 HCONH2-formamide 1 CH3C2H-methylacetylene 1 HCOCH3-acetaldehyde 1 " x-ogen" (unidentified) "HNC"-hydrogen isocyanide (not yet known 1 in laboratory)
Frequency range (MHz)
1612-1 3,441 1638-1,639 130,246 22,235 22,834-25,056 1 13,492 115,271 110,201 109,782 146,969
88,630-88,634 86,339-86,342 109,463 4830-1 50,498 4593 4389 21,982-87,925 3139 1639 9097-9098 83485,521 110,331-1 10,383 15404620 85,457 1065 89,190
90,665
TECHNOLOGY A N D OBSERVATIONS IN RADIO ASTRONOMY
43
important. These include the lambda doubling in OH, and inversion doubling in NH, and H,CO. Because it is difficult to search over a wide frequency range at high sensitivity in a finite time, the usual procedure has been to look for specific lines whose frequencies have been measured in the laboratory, or predicted (generally with lower accuracy) from other laboratory measurements. In practice, a frequency must be known to within 0.1 % or better before a search is likely to be successful. One unidentified line (“x-ogen”) at 89 GHz is included in the table, but so far very little attention has been given to exploratory searching in any spectral range. In several cases, once a line has been detected, astronomical measurements of its frequency have been able to improve on the laboratory value. Such determinations depend on assuming that the Doppler velocity shift for the line under consideration is the same as that for a line of well-known frequency (e.g. an OH line) in the same source; in other words, we have to assume that the two molecules are distributed in the same way and have the same motions in that particular region. This assumption seems to be fairly good if the circumstances are chosen appropriately. Table 1 indicates that there are several cases where a number of different lines have been detected from the same molecule, sometimes from both the ground and excited states. In such cases, the identification with a particular molecule can be made with greater confidence. In addition, a number of isotope lines have been observed, involving the less abundant isotopes, l8O, ” C . Counting all the transitions for each molecule, and all isotopes, about 70 molecular lines have so far been reported. Many other molecules have been looked for and not yet found, perhaps around 50 in all. Unfortunately, unsuccessful searches are not always reported in the literature. Both the discovered and undetected lines tend to cluster in particular frequency groups, clearly related to the availability of receivers, which do not yet cover the whole frequency range of interest at adequate sensitivity. Many of the molecules have been seen in absorption in front of discrete continuum sources, such as HI1 regions. Almost all the molecules have been seen in absorption or emission in the direction of the source Sagittarius A, which is believed to be at the center of our Galaxy. This line of sight gives a long path through the Galaxy, and samples a wide range of conditions. The general characteristics of the observations, and in particular the OH, H,CO, and CO emission from some particular dust clouds, suggest that all the interstellar molecules may be connected with dust. The interior of a dust cloud would provide a low temperature environment where molecules could form more easily on the surfaces of dust grains. Also, after evaporation from the grains, the molecules would be shielded by the dust against dissociation by ultraviolet photons, which cannot easily penetrate the cloud.
44
WILLIAM C. ERICKSON AND FRANK J. KERR
In cases where straightforward " thermal " processes appear to be operating, the strengths of the various lines can give information on the relative abundances of the different molecular species, the temperature of the gas in various locations, and the excitation conditions for the observed transitions. One of the surprises has been the high abundance of CO, which shows a density 5 or 6 orders of magnitude greater than might be anticipated. Several other molecules have densities larger than expected, but CO is the outstanding case, as the line is very strong, and the molecule shows a very wide distribution throughout the Galaxy. The OH and H2C0 molecules are also distributed widely; the other molecules have so far been found only in a limited number of locations, but this may be due primarily to a lack of sensitivity, especially for the species whose abundances are fairly low. Observations of the widely spread molecules provide new ways to map the Galaxy. Studies of CO are likely to be of particular interest for this, because the short wavelength results in very high angular resolution [e.g. one arcminute for a 36 ft (1 1 m) telescope], although of course the use of high resolution carries with it the need to spend a very long time in surveying a region. The detection of a series of complex molecules has aroused wide interest. None of the molecules discovered so far necessarily involves the existence of life, but they include some molecules which might be found in a prebiotic stage, and therefore the philosophical possibility is increased that life may exist elsewhere in the Galaxy. Meanwhile, searches are going on for progressively more complex molecules, such as amino acids, which may eventually show a connection with life. The masering phenomena are amongst the most surprising observations in the molecular work. Several dozen OH and H 2 0 sources show behavior which cannot possibly be accounted for by any thermal processes (Fig. 21). These profiles are typically complex, with many narrow components at different velocities, most of which vary with time-the characteristic times are months for OH and weeks for H 2 0 .In addition, in the case of OH sources, the relative strengths of the lines in the quadruplet are anomalous, and very complex polarization phenomena are seen (55). The argument for masering is that the line components are quite narrow, implying low temperatures of around 100 K for thermally-broadened lines, and yet the brightness temperatures of the sources must be in the region of 1012-10'3K, because the rapidity of the time variations and VLB interferometry measurements both indicate that the sources must be very small (<0!'001 in angular diameter, or 2 A.U. in physical size, for someof the water sources). The combination of narrow lines and high surface brightness can be well accounted for by a high gain maser. These sources are all connected with HI1 regions, but not with any specially notable locations in them. A single HI1 region may show a few small bright spots, which suggests that the
TECHNOLOGY A N D OBSERVATIONS I N RADIO ASTRONOMY
45
KM/S
-LO
0
10
20
FIG.21. Some maser emission profiles for the distant HI1 region W49 (44). The first four profiles show the time variations in the water line, while the last two are for the main OH transitions.
radiation may be strongly beamed towards us, with large numbers of other masers beamed in other directions. The details of the maser process are not clear, probably because it is taking place in an irregular region without the sharp boundaries of a laboratory maser. The problems are discussed for example by Litvak (56) and Sullivan (57). For a maser, one first needs radiation to be amplified; this may come from the continuum generated in the HI1 region, from spontaneous emission in the line concerned, or from the cosmic microwave background. A pump source is also required, to supply energy to invert the populations of the levels concerned in the transition. Many possibilities have been suggested for the pump, including various forms of ultraviolet, infrared, and chemical pumping. An HI1 region is full of activity, so that ample energy must be available in one form or another. The shape of the masering region is another question : an elongated or sheet-like form could be appropriate for a highlybeamed maser, and a more spherical form for an isotropic maser, which has been discussed but seems less likely. The small size of the maser requires a
46
WILLIAM C. ERICKSON A N D FRANK J. KERR
correspondingly high density of molecules to produce sufficient amplification, and a possible association with protostars has often been suggested. This would favor infrared pumping. An apparent antimasering process has also been found; weak absorption has been observed in the 4830 MHz formaldehyde line in the directions of several dark clouds, even though there were no background discrete sources to provide the radiation to be absorbed. It appears that the 3 K cosmic background was being absorbed, indicating that the formaldehyde in the dust clouds was effectively cooler than 3 K, as any material in thermal equilibrium with the general radiation field would itself be at a temperature of 3 K. Hence by some unknown process the formaldehyde was being " refrigerated " (to 1.8 K), giving an excess population in the lower level over that for an equilibrium distribution, which is clearly the inverse of a maser effect. N
4. Pulsars
Pulsating radio sources in the Galaxy with a highly constant repetition frequency were discovered accidentally at Cambridge, England, in 1967 during the testing of an array which had been set up for studying scintillations of radio sources (58). This was the first time anyone had searched for radio sources with a sufficiently high time resolution. About 55 pulsars are now known, with pulse periods ranging from 0.033 to 3.7 sec, and emitting radiation over a wide spectral range (59). Although the periods are highly constant in comparison with those of any other astronomical variation, they are found to be slowly lengthening when measured with precision. The pulse widths are typically 5 % of the pulse periods, but many pulsars have more complicated structure in the arrangement of pulses and subpulses. The detailed shape is characteristic of the particular pulsar (Fig. 22). The generally accepted interpretation is that a pulsar is a neutron star, which is a star in a highly collapsed state in a late stage of its evolution. The star, with a diameter 15 km, is considered to be rotating rapidly with a period equal to that of the radio pulse. The observed slowdown of the pulse period corresponds to a spindown of the rotation on a time scale of lo9 yr, which is theoretically reasonable for a neutron star. The radiation is thought to originate in the surrounding plasma which is dragged around with the rotation; one main pulse is beamed towards the Earth in each rotation, and the fine pulse structure is related in some way to the electromagnetic or geometrical configuration of the envelope of the neutron star. A large number of detailed theories have been proposed for the way in which rotational energy is converted into electromagnetic radiation.
-
TECHNOLOGY AND OBSERVATIONS IN RADIO ASTRONOMY
47
FIG.22. (a) Some typical mean pulse envelopes for a group of pulsars, designated by their positions in right ascension (60). (b) The mean pulse envelope of NP0532, the Crab nebula pulsar, at radio, optical, and x-ray wavelengths (60).
A discontinuity was observed at one stage in the period and the slowdown rate for the pulsar PSR083345 in the constellation Vela (61, 62). Between February 24 and March 3, 1969, the period changed from 0 S 089209204738 to 0 S 089209070988, and the slowdown rate ( A P / P ) from 1.242619 x to 1.25264 x This effect could have resulted from a change in the effective diameter of the neutron star by about 1 cm, due to some internal rearrangement of material in the star, or from a sudden small change in the strength of the external magnetic field. This particular pulsar is probably associated with the supernova remnant Vela X, which is a well-known radio source. Another one is associated with the central star of the Crab nebula, another important supernova remnant, and in this case optical and x-ray pulsations are observed with the period of the radio pulses. No other pulsars have been identified with objects already known optically. The interstellar medium affects the pulses in two ways as they travel from the source to the observer. Firstly, the pulse amplitude varies on time scales of minutes, hours, days, and even months, due to interstellar scintillations. These arise because the medium is highly irregular, and variable refraction effects are produced as the individual clouds move across the line of sight. Secondly, the pulses are delayed through interstellar dispersion, arriving later the longer the wavelength. Identification of individual pulses over a large spectral range requires a series of careful comparisons over smaller
48
WILLIAM C. ERICKSON AND FRANK J. KERR
wavelength steps, but it has been shown for example that a total differential delay in one pulsar between 430 and 40 MHz is about 32 sec. The dispersion can be measured very precisely, usually by comparisons at two fairly close frequencies, and this gives a very good value for the total number of electrons along the line of sight, e.g. 1.754 x 10’’ cm-‘ for the Crab nebula pulsar. Although the total number can be accurately measured, the distribution of the electrons in space cannot be worked out, in particular the way they are arranged in clouds, spiral arms, etc. Distances to pulsars can be estimated from their dispersion measures, based on an average value for the interstellar electron density, but the interstellar medium is so irregular that these estimates have limited accuracy for individual pulsars. In a few cases, better distances have been derived from 21 cm absorption measurements, but these observations are very difficult for any but the strongest pulsars, because of the small duty cycle. From the available distance estimates, and the fact that the pulsars are spread over a moderate range of galactic latitude, it is deduced that the observed pulsars are all fairly close, within 2-3 kpc. In searching for pulsars, the sensitivity can clearly be increased if the receiver is gated at the pulse frequency. Methods have therefore been developed in which a computer looks for periodicities in the received waveform, recorded over a time corresponding to a substantial number of pulses. The interstellar dispersion also affects the sensitivity, because the pulse arrival time varies appreciably over the receiver bandwidth. At low frequencies, in fact, the arrival time can vary so much over the band that the pulse pattern is completely washed out in the total receiver output. If the dispersion measure is known, this effect can in principle be allowed for by putting in a compensating delay which is an appropriate function of frequency across the band. In searching, the pulse period and the dispersion measure must both be regarded as variables, and the greatest sensitivity will be obtained if the record can be searched in terms of both of these quantities. Various applications have been proposed in the fields of astronomy, geodesy, and navigation for the very precise measurement possibilities which are inherent in pulsar observations. E. Extragalactic Radiation
Large numbers of external galaxies have been detected as radio sources. These include many nearby galaxies, whose radio emission shows a “ normal ” (though sometimes complex) relationship to the optically observed characteristics. In most extragalactic sources, however, the ratio of radio emission to optical emission is much higher than in normal galaxies; these are the so-called “ radio galaxies ” and “ quasi-stellar sources ” or “ quasars.” In
TECHNOLOGY AND OBSERVATIONS IN RADIO ASTRONOMY
49
addition, there is a low-level background, best observed at high frequencies, which is substantially isotropic and is believed to be extragalactic in origin. 1. Normal Galaxies
Continuum emission has been observed from a substantial number of nearby spiral and elliptical galaxies (63), and from the Magellanic Clouds, which are believed to be satellites of our own Galaxy. For spirals, the angular size of the radio source is usually equal to, or smaller than, that of the optical galaxy, with the emission distributed in a disk and a small bright nucleus. There is no definite evidence for a radio halo surrounding any spiral galaxy, contrary to earlier interpretations of lower resolution data. There is no clearly established relationship between the radio and optical emission from spirals in the case of the continuum. Recent high resolution work has started to show the radio spiral structure, for example in the galaxies M31, M33, NGC4631, and M51, and is indicating a good correlation between the positions of the radio and optical spiral arms. An aperture synthesis map of M51 at 1415 MHz, obtained at Westerbork, shows a strong tendency for the radio continuum arms to lie slightly inside the optically brightest parts of the arms (see Fig. 23). This result is regarded as the best evidence so far for the densitywave theory of spiral structure, in which the arms are regarded as the instantaneous position of a spiral density wave which is rotating more slowly than the material in the galaxy. When the material catches up on the density wave, a shock front develops, leading to a sharp increase of density, which initiates a burst of star formation. On this view, the peak of the (nonthermal) radio emission would come from the position of the shock front, whereas the new massive stars would reach their brightest after an appropriate lapse of time, during which rotation would carry the material away from the shock front, corresponding to the separation seen in Fig. 23. Many spirals and a few irregulars have also been seen in the 21 cm hydrogen line (6.5).In this case, the neutral hydrogen distribution extends beyond the usually measured optical dimensions, and for spirals the peak of the HI radial distribution is well out from the center of the galaxy. Many spirals are found to show hydrogen concentrations in the outer region which may represent hydrogen " companions " (unaccompanied by much visible matter), as in Fig. 24. Hydrogen " bridges " have also been found, for example between the two Magellanic Clouds. There is a clear relationship between the fractional mass of hydrogen and the galaxy type for spirals and irregulars. The proportion of hydrogen increases through the sequence Sa-Sb-Sc-Sd-Irr, which is the direction along the sequence in which the spiral is less tightly wound and has less angular momentum.
50
WILLIAM C. ERICKSON AND FRANK J. KERR
FIG.23. The galaxies M51 and NGC5195. The 1415 MHz isophotes from awesterbork aperture synthesis study are superposed on a photograph reproduced from a blue plate taken by Humason with the Palomar 200 inch (5 m) telescope (64). The contour unit is 0.8 K brightness temperature. The radio ridge lines of maximum intensity tend to lie along the inner edges of the stellar spiral arms in M51.
51
TECHNOLOGY AND OBSERVATIONS IN RADIO ASTRONOMY
M 33
I
I
HI I
BRIGHTNESS I
DISTRIBUTION I
I
I
I
I
I
lh30m
31"'
32"'
33"'
I
I
1
.I 34"'
I
1
lh35m
. .
1
I lh28"'
I
29"'
1
+
a (1950) FIG.24. Integrated 21 cm brightness distribution over the spiral galaxy M33 (66), superposed on an optical photograph. Large extensions are seen in the hydrogen in the southeast and northwest quadrants. The 21 crn brightness distribution is from Gordon (66). From Gordon, Astrophys. J., 169,235 (1971). Courtesyof Universityof Chicago Press.
Observations with line radiation (at 21 cm) make possible the study of internal motions inside galaxies, and rotation curves have been derived for many spirals and for the Magellanic Clouds. These rotation curves are asymmetrical in most cases, with the asymmetry of motion sometimes corresponding to asymmetries of structure. In some cases, these effects suggest a warping of the thin disk of the galaxy, in others they indicate the presence of substantial noncircular motion.
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WILLIAM C. ERICKSON AND FRANK J. KERR
Spectral lines other than the 21 cm line are difficult to observe in external galaxies because they are intrinsically much weaker. Hydrogen recombination lines have been detected from a very bright nebula in the Large Magellanic Cloud, 30 Doradus, but no molecular lines have yet been detected from either of the Magellanic Clouds. The OH molecule has however been weakly detected in three spirals (67). 2. Radio Galaxies Certain radio galaxies emit up to W in the radio region. This is approximately a million times the power emitted by normal galaxies. The strongest radio galaxies are usually giant ellipticals or D systems. They often show optical emission line spectra characteristic of high excitation. Galaxies with extremely bright, compact nuclei, such as Seyfert or N Galaxies, are approximately as powerful as the D systems. Double systems, galaxies which appear to have double nuclei, often are found to be radio emitters. Radio emission appears to be correlated with a dense concentration of matter in the nucleus of a galaxy and with optical emission spectra. It should not be assumed that there exist immutable rules relating the optical and radio properties of galaxies. Many galaxies, which by their optical appearance would be expected to be strong radio emitters, are found to radiate only weakly in the radio region. We find only a correlation between certain optical properties and radio emission. The angular distributions of brightness in the optical and radio region are usually completely different. Optically, galaxies almost always show a more or less uniform decrease in brightness from their centers outward. The radio source is frequently double and larger than the optical object. The optical object is normally situated somewhere between the two radio emission regions. This is illustrated in Fig. 25. Although frequent, double radio sources do not always occur. Some are single objects a fraction of an arcsecond in diameter, some show bright cores and a large, diffuse halo, and some are extremely complex, triple or quadruple with bridges connecting the various components. The optical and radio appearances are practically unrelated. The radio emission is apparently generated when gigantic eruptions occur in the nuclear regions of galaxies. Some type of instability results when large amounts of mass are concentrated in galactic nuclei. An explosion takes place and clouds of particles with relativistic energies are ejected. These clouds of high energy particles radiate synchrotron emission in the radio range. Time variations over periods of a few months are observed in certain compact radio sources. The structure of the time variations is consistent with a model which assumes a rapidly expanding dense cloud of relativistic particles (69).
T
I
36' 30"
?-
z
3b' OOa
U
35'rn'
2
m
2 P
4 6 35'00'
?1 505
I
485
I
-
4b5
1
I
44
1
42'
I
I
1
I
385
40' 19"
57"'
FIG.25. A 5 GHz aperture synthesis map of Cygnus A given by Mitton and Ryle (68). The optical galaxy is in the center, and. the components of the double radio source are extended in the direction of the line which joins them. Each radio contour interval corresponds to 4800 K and the angular resolution is 10 arcsec.
E!
0 ?-
Y
54
WILLIAM C. ERICKSON AND FRANK J. KERR
3. Quasi-Stellar Sources The quasi-stellar sources or quasars continue to be an enigma. After a decade of intensive study, there is still no general agreement concerning their distances and structure. The quasars are radio sources optically identified with starlike objects whose angular diameters are less than the resolution limits of any telescope. About 200 quasars have been identified. They are blue in color and possess broad emission lines which have been identified with the highly redshifted emission lines of ionized hydrogen, oxygen, and neon. The redshifts imply recession velocities up to 0.84~.These lines are characteristic of a highly excited, rarefied gas. lntensity variations over periods of days to months are often observed at both optical and radio wavelengths. There exists a much larger class of objects which are optically identical to the quasars, but are not radio emitters. These are called quasi-stellar objects or blue stellar objects. The quasars are apparently a radio emitting subset of this class of object (70). The basic controversy concerning quasars involves their distances. If the redshift of their spectra is cosmological, i.e. due to general expansion of the Universe, then the quasars are extremely distant. This is the most reasonable explanation of the redshifts. However, the intensity variations indicate that the objects have relatively small dimensions on the order of the distance that light travels during the period of the variations. The different parts of larger objects would vary independently, and the variations would tend to average to zero. On the other hand, to produce the flux observed at the Earth, the quasars must be among the most luminous objects in the Universe. They must produce 100 to 1000 times the power output of a normal galaxy. No mechanism for the production of such amounts of power in a small object which consists largely of a rarefied gas has yet been suggested. Arp (71) has shown that the quasars often seem to be multiple and to be associated with eruptive galaxies. This would indicate that the quasars are situated at distances equivalent to those of other galaxies. If this were true, the power generation requirements would become more moderate, but no satisfactory alternative to the cosmological explanation of the large redshifts in the spectra of quasars has been found. 4. Properties of Extragalactic Radio Sources
Although quite speculative and lacking in detail, a general picture of extragalactic radio sources is emerging. All galaxies apparently contain relativistic electrons and magnetic fields to some extent. The nonthermal continuum emission is synchrotron radiation from these particles.
TECHNOLOGY AND OBSERVATIONS IN RADIO ASTRONOMY
55
In the weaker sources, some fraction, perhaps a major fraction, of the relativistic particles are accelerated in supernovae explosions or by interaction with interstellar magnetic fields (Fermi acceleration). These objects tend to have straight spectra (see Section II,A), with spectral indices in the range -0.7 to -1.2. They are transparent to radio waves and, for a constant magnetic field, a power law radio spectrum of index a implies a power law electron energy spectrum of index (2a - 1). No time variations have been found in these rather quiescent objects. Their fairly steep spectra suggest a relative deficiency of high energy electrons. The lower energy electrons radiate their energy at a low rate, and are long-lived. A continuous progression exists from the weak sources to the violently erupting, powerful radio sources. Our own Galaxy is an intermediate type. It is relatively stable and quiescent, yet there are definite indications of activity in its nuclear region. Radio galaxies are extraordinarily massive systems of stars. They apparently possess a massive core that has generated large fluxes of relativistic particles through some mechanism that is not yet understood. These particles permeate the whole system and cause it to be a large and powerful radio source. Radio galaxies usually have straight spectra with indices between -0.6 and -0.9. Unless the galaxy contains a compact core, no time variations occur. In some cases, the relativistic particles and the radio emission are confined to the approximate region of the visible galaxy. Jn other cases, the particles have escaped from this core into the surrounding intergalactic medium and a large " halo " of radio emission exists (see Fig. 26). The halo generally has a straight steep spectrum which indicates older lower energy particles than those in the core. Often, the relativistic particles are expelled from the central region in the form of two large clouds. In these cases, a double radio source is observed. The parent galaxy then lies approximately midway between the two sources. Whether a radio galaxy develops into a double radio source, a core-halo source, or a simple-core source must depend upon the structure of its magnetic field and the nature of the particle acceleration mechanism. The morphology of these sources is not understood in detail. The compact radio sources are identified with Seyfert galaxies, N galaxies, or quasi-stellar objects (73). Seyfert and N galaxies possess very bright, starlike nuclei. Quasi-stellar objects are similar to these nuclei but have no visible galaxy surrounding them (74). They are among the most powerful radio sources. Apparently dense massive objects develop and are powerful radio emitters. Often the clouds of relativistic particles associated with the objects are so dense that they self-absorb their own synchrotron emission. These self-absorbed components modify the radio spectrum so that the spectra of compact sources are often curved. Their spectra steepen at high frequencies and also reach a maximum and fall at lower frequencies (see Fig. 27).
56
WILLIAM C. ERICKSON AND FRANK J. KERR
I
I
IC 340
! 03h47m
03h16m
03h15m
I 03h14m
03h13m
FIG.26. An aperture synthesis map of Perseus A derived from 408 MHz observations with the " One-Mile Telescope" (72). The dashed contours represent a large halo surrounding the complex radio source regions. Because of the wide range of surface brightnesses in the core sources, different contour intervals are used for the various components. Contour intervals: NGC 1275 (3C84A)-1350 K; NGC 1265 (3C83.1B) and 3C83.1A-250 K ; IC310-67 K. Dashed contours-2.9 K. Positions of optical objects are indicated by crosses.
The active centers generate successive outbursts of relativistically expanding clouds of particles. At any given frequency, one of these clouds is initially opaque. As the cloud expands, its angular size increases and its total emission rises. Eventually, the cloud expands to such an extent that it becomes transparent, and the total emission falls. Thus, over periods of weeks and months, time variations occur. The variations are most observable at centimeter and millimeter wavelengths (69). These sources are frequently powerful in the infrared region of the spectrum. Since these objects are compact, the synchrotron mechanism may not be the only radiation mechanism of importance ; various coherent radiation mechanisms are conceivable. As yet, it has not been shown that any other mechanism is operating (74). Such an observation would be extremely important to our understanding of the nature of these objects since high fluxes
TECHNOLOGY A N D OBSERVATIONS IN RADIO ASTRONOMY
57
1
\
FIG.27. Radio source spectra given by Kellermann and Pauliny-Toth (75). Frequencies in MHz areon the abscissa and intensities in flux units areon theordinate. 3C2 is an example of a straight spectrum; 3C123 is slightly curved; 3C48 and 3C295 are strongly curved; 3C84 (Perseus A) and 3C279 display complex time-variable spectra possessing several components.
58
WILLIAM C. ERICKSON AND FRANK J. KERR
of relativistic particles might not then be required in the early stages of expansion of the source. It is difficult to maintain these high fluxes because the particles lose energy not only by synchrotron radiation but also by inverse Compton scattering in the radiation field of the source. Double radio sources are also identified with compact optical objects in many cases. Some of these double sources are themselves amazingly compact. The ratio of the distance between them to their diameters is sometimes well over a hundred. It is hard to understand how a cloud of relativistic particles can remain so compact in presumably empty intergalactic space. This lends credence to the idea that compact sources can expel coherent objects and that many quasi-stellar objects might be coherent projectiles hurled from the parent source in some sort of fission process. Unfortunately, this idea does not explain the redshifts through the Doppler effect; blueshifts should be at least as probable as redshifts, but none has been observed. Compact sources are exotic objects, and they have led to exotic theories (76). Some of these theories suggest that they are superdense clusters of stars with chain reactions of supernovae, massive objects undergoing gravitational collapse, regions of matter-antimatter interaction, massive objects in which time is so distorted that they are observed to be in the primeval stage of the Universe, or points where matter and energy are streaming into our observable Universe from some unobservable dimension. Speculation abounds, and we can only hope that further observations will lead to definitive theories. Extragalactic sources have been the most fruitful objects of study by high resolution techniques. Without these techniques, the field would be in a most primitive state. Progress in this field is paced primarily by technology. 5. lsotropic Background Radiation
In 1965, Penzias and Wilson (77) showed that there exists an isotropic component of the background emission at 3 cm wavelength which must be extragalactic in origin. Subsequent observations at a variety of wavelengths from 1 to 20 cm have shown that the spectrum of this emission is consistent with that of a blackbody (Fig. 28). No departures from isotropicity on a large angular scale have been found to a level of less than one percent, except for a small effect which is interpreted as arising from a motion of the solar system relative to the local supergalaxy (79). The background emission away from the discrete sources also has a galactic component, but this falls with increasing frequency. The isotropic component, however, appears to be rising towards a maximum near 1 mm wavelength. and therefore dominates at the shorter wavelengths. Away from the sources and the Milky Way, the two components combine to yield a broad minimum near 20 cm wavelength in the general emission level.
59
TECHNOLOGY AND OBSERVATIONS IN RADIO ASTRONOMY 1
10
I
I
I
10-1
1
10-2
IO-~
WAVELENGTH (cm)
FIG.28. Spectrum of the isotropic background radiation (78), based upon observations by various groups. The curves are for a 2.7 K blackbody, and a 16 K gray body diluted by a factor of six.
The isotropic component agrees well with predictions based upon a " big bang" cosmology. If the expanding Universe was created in the explosion of an initial fireball some 10 billion years ago, the original flux of x-radiation associated with this explosion would have become decoupled from the matter and would appear with continuously decreasing intensity and increasing wavelength as the Universe expands. At the present point in the evolution of the Universe, this initial fireball radiation would have the observed properties of the isotropic background component. The presence of a maximum near one millimeter has not yet been established observationally, so that the blackbody character of the radiation is not yet beyond doubt. Alternative interpretations have been discussed, postulating a large number of unresolved sources of previously unknown type. These suggestions have become more plausible with the discovery of very large amounts of infrared radiation from some types of galaxy. The question is unresolved, but general opinion at present favors the blackbody interpretation. F. Conclusion
In this review, it has only been possible to sketch the present state of radio astronomy. The subject has grown to cover an enormous range of topics, many of which show interactions between astronomy, physics, chemistry, and
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WILLIAM C. ERICKSON AND FRANK J. KERR
electronics. We have tried to give a general perspective on this very active branch of science, paying special attention to the interrelationships between scientific and technical developments.
REFERENCES I. 2. 3. 4.
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38. F. J. Kerr, Annu. Rev. Astron. Astrophys. 7, 39 (1 969). “ Maryland-Green Bank Galactic 21-cm Line Survey,” 2nd ed. Univ. Maryland, College Park, Maryland, 1969. 40. F. J. Kerr, in “Nebulae and Interstellar Matter” (B. M. Middlehurst and L. H. Aller, eds.), p. 575. Univ. of Chicago Press, Chicago, 1968. 41. G. L. Verschuur, Phys. Rev. Lett. 21, 775 (1968). 42. A. K. Dupree and L. Goldberg, Annu. Rev. Astron. Astrophys. 8, 231 (1970). 43. B. Zuckerman and P. Palmer, in press. 44. B. Zuckerman and P. Palmer, Atmu. Rev. Asfron. Astrophys. In press. 45. L. E. Snyder, MTP Znt. Rev. Sci.,Bieti. Rev. Chem., Specfrosc. 1, in press. 46. C. H. Townes, Znt. Asfron. Union Symp. 4, 92 (1957). 47. A. H. Barrett and A. E. Lilley, Astron. J. 62, 5 (1957). 48. G. Ehrenstein, C. H. Townes, and M. J. Stevenson, Phys. Rev. Lett. 3,40 (1959). 49. S. Weinreb, A. H. Barrett, M. L. Meeks, and J. C. Henry, Nature (London) 200, 829 (1 963). 50. A . C. Cheung, D. M. Rank, C. H. Townes, D. D. Thornton, and W. J. Welch, Phys. Rev. L e t f . 21, 1701 (1968). 51. A. C . Cheung, D. M. Rank, C. H. Townes, D. D. Thornton, and W. J. Welch, Nature (London) 221, 626 (1969). 52. L. E. Snyder, D. Buhl, B. Zuckerman, and P. Palmer, Phys. Rev. Lett. 22, 679 (1969). 53. A . Sandqvist, Astron. J. 75, 135 (1970). 54. R. W. Wilson, K. B. Jefferts, and A. A. Penzias, Astrophys. J . Lett. 161, L43 (1970). 55. P. Palmer and B. Zuckerman, Asfrophys. J. 148, 727 (1967). 56. M. M. Litvak, Science 165, 855 (1969). 57. W. T. Sullivan 111, Asfrophys. J. 166, 321 (1971). 58. A. Hewish, S. J. Bell, J. D. H. Pilkington, P. F. Scott, and R. A. Collins, Nature (London) 217, 709 (1968). 59. S. P. Maran and A. G. W. Cameron, Earth Exfraterr. Sci. 1, 3 (1969). 60. A. Hewish, Annu. Rev. Asiron. Astrophys. 8, 265 (1970). 61. P. E. Reichley and G. S. Downs, Nature (London) 222, 229 (1969). 62. V. Radhakrishnan and R. N. Manchester, Nature (London) 222, 228 (1969). 63. Y. Terzian, Inf. Astron. Union Symp. 44,15 (1972). 64. D. S. Mathewson, P. C. van der Kruit, and W. N. Brouw, Asfron. Asfrophys.17, 468 (1972). 65. M. S. Roberts, Inr. Asfron. Union Symp. 44, 12 (1972). 66. K. J. Gordon, Astrophys. J . 169, 235 (1971). 67. L. Weliachew, Astrophys. J. Lett. 167, 147 (1971). 68. S. Mitton and M. Ryle, Mon. Nofic. Roy. Astron. Soc. 146, 221 (1969). 69. K. I. Kellermann and I. I. K. Pauliny-Toth, Annu. Rev. Asfron.Asfrophys.8, 369 (1970). 70. M. Schmidt, Annu. Rev. Asfron. Asfrophys.7 , 527 (1969). 71. H. C. Arp, Astrophys. J. 148, 321 (1967). 72. M. Ryle and M. D. Windram Mon. Notie. Roy. Astron. SOC. 138, 1 (1968). 73. J. D. Wyndham, Astrophys. J. 144,459 (1966). 74. G . R. Burbidge, Annu. Rev. Astron. Astrophys. 8, 369 (1970). 75. K. I. Kellermann and I. I. K. Pauliney-Toth, Astrophys. J. Lett. 155, L71 (1969). 76. G . R. Burbidge and E. M. Burbidge, “ Quasi-Stellar Objects.” Freeman, San Francisco, 1967. 77. A. A. Penzias and R. W. Wilson, Asfrophys.J. 142, 419 (1965). 78. F. M. Ipavich and A. M. Lenchek, Phys. Rev. D 2, 266 (1970). 79. E. K. Conklin, Nature (London) 222,971 (1969). 39. G. Westerhout,
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Image Formation in the Electron Microscope with Particular Reference to the Defects in Electron-Optical Images D . L. MISELL Department of Physics. Queen Elizabeth College. London University. London. England
I . Introduction .................................. ................. ................... I1. The Angular and Energy Distributions A . Single Elastic Electron Scattering......................................................... B . Multiple Elastic Electron Scattering........ .................................. C . Single Inelastic Electron Scattering ...................................................... D . Multiple Inelastic Electron Scattering ...... E . Combined Inelastic-Elastic Electron Scattering...................................... F . Numerical Results for the Angular-Energy Distrib G . Localization and Coherence of Electron Scattering .................................. 111. Image Formation by the Elastic Component .............................................. A . Coherent Illumination ....................................................................... B . Spatially Incoherent Illumination ........................................................ C . Chromatically Incoherent Illumination .... D . Spatial and Chromatic Incoherence..................................................... 1V. Image Formation by the Inelastic Component ....... A . Coherent Illumination ..................................................................... B . Effect of Chromatic Aberration .......................................................... V . Incoherent Theory of Image Formation .................................................... A . Calculation of Scattering Contrast Images............................................ B. Convolution Integral ... ................................................. C . The Spherical Aberration Function ..................................................... D. The Chromatic Aberration Function ............ E. The Combined Aberration Function .................. F. Effect of Chromatic and Spherical Aberration on the Image VI . Conclusions ........................................................................................ Appendix: Deconvoiution of Two-Dimensional Data ...................................
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Addendum to References.......................................................................
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64 66 67 74 79 83 85 90 106 112 112 137 143 155
158 158 162 165 166 167 167 169 171 172 179 183 187 191
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D. L. MISELL “Yes I have a pair of eyes,” replied Sam, “and that’s just it. If they wos a pair 0’ patent double million magnifyin’ gas microscopes of hextra power, p’raps I might be able to see through a flight 0’ stairs and a deal door; but bein’ only eyes, you see, my wision’s limited.”-from Pickwick Papers by Charles Dickens.
I. INTRODUCTION In the transmission electron microscope information on the structure of a specimen, via its electron scattering properties, is transmitted to an image plane. This information is transferred from the specimen plane to an image plane by a set of electron lenses; in the scanning transmission electron microscope the lens system preceeds the specimen. The problem set is then to infer, from an intensity distribution (electron micrograph), information on the structure (e.g. mass thickness variation or electron density distribution) of the specimen. The three stages for the solution of the structure are then: (i) correction of the electron micrograph for lens aberrations, (ii) the calculation of the electron scattering properties of the specimen, and (iii) the calculation of the specimen structure from its electron scattering properties. The solution of the problem posed by each of these three steps has not yet been determined and even the calculation in the reverse direction, from structure to image, has not been convincingly evaluated. The optimistic electron microscopist is not daunted by the magnitude of the problem, and two approximate calculations have been used in an attempt to establish the specimen structure from the image intensity: (i) The weak phase or weak amplitude object (1-3). The specimen is characterized by an object wavefunction t+bo(x0,yo) = t,ho(ro) for a position coordinate ro = (xo, yo) in the object plane. $o(ro) is written in terms of a small phase shift term or a small amplitude attenuation term. A linear relationship is obtained between the image intensity j i ( r i )and Go(ro)(see Section HI). Explicitly neglected in this approximation is inelastic electron scattering and its role in image formation. (ii) The “incoherent” or geometrical approximation (4, 5). This approach represents an opposite extreme to (i) ; the phase information in the scattered electron beam is neglected and the final image is considered to be formed by a superposition of intensities, thus avoiding the problems encountered in the wave theory with respect to phase terms (Section V). This approximation would appear to be a valid description for fairly thick specimens (-500 A with incident electrons of energy 20-100 keV) of biological or polymeric specimens, where inelastic scattering processes are predominant (see Sections I1 and V). Without approximation (i) or (ii), calculations of $o(ro) from ji(rj) are not possible; at best it is possible to calculate the image wavefunction qi(rj)or intensity j i ( r i )= I qi(ri)1 from a detailed knowledge of the electron scattering properties of the specimen !
DEFECTS IN ELECTRON-OPTICAL IMAGES
65
This review will attempt to present the theoretical aspects of the calculation of an electron microscope image from the object and to indicate the conditions under which the approximations (i) and (ii) may be applied. Particular attention will be given to the calculation and the correction of the effect of lens aberrations on the final image. In the conventional transmission electron microscope the objective lens is of prime importance and any defects in the image formed by this lens are merely magnified by the projector lenses. Since the objective lens is placed immediately behind the specimen, the spherical and chromatic aberration in the final image will be closely related to the angular and energy distributions of the transmitted electron beam (6-10). As the specimen thickness increases the effect of aberrations on the image, due to angular-energy broadening of the emergent electron beam, increases (8). In the scanning transmission electron microscope the lens system produces a scanning electron beam of 3-5 A diameter on the specimen (Il-lj),and the three components of the transmitted electron beam, namely, the unscattered, the elastic, and inelastic components, can be separated to give component images. Since there is no lens system behind the specimen, the resolution of these images is not much affected by the extent of the angularenergy distribution of the transmitted electron beam. Aberrations in the final image are a result of the finite scan spot size, which is limited by the condenser lens defects. In particular, chromatic aberration, which is a dominant feature in the conventional microscope, is almost negligible in the scanning instrument (energy spread of the incident electron beam is only 0.1 eV). Explicitly omitted from this review are image reconstruction techniques by holography and other optical methods; this topic has been reviewed in detail by Hanszen (3). The transfer theory of image formation in the electron microscope will be given in Section 111; this provides a general mathematical procedure for relating the object wave $o(ro) to the image wave $i(ri). Except in such approximations as the phase grating approximation, the transfer theory does not readily lend itself to numerical evaluation. However, in a formal way, it is possible to include in this theory all types of electron-specimen interactions (elastic-Section 111, inelastic-Section IV), illumination conditions (spatial and chromatic incoherence), and lens aberrations. The transfer theory is used to develop the wave theory of image formation in the form of the Kirchhoff diffraction integral ( 1 4 1 9 , which is considered to be more suited to numerical evaluation (16, 17). On the basis of the diffraction integral formulation, referred back to the transfer theory, it is possible to include the effects on the final image of: the lens aberrations (including defocusing of the objective lens), lens apertures, and the angular-energy characteristics of the incident electron beam (see Section 111). In Section V a detailed account of the “incoherent” theory of image formation is presented; this approach 1s classical and as such limited in
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application. This theory neglects completely the phase information carried by the scattered electron beam. However, it is useful as a method of estimating the effect of spherical and chromatic aberration on the electron microscope image, and only a minimal amount of information on the electron scattering properties of the specimen is required. In addition, experimentally determined angular-energy distribution may be used in the theory, without requiring the phase information of the scattered electron beam (4). Before considering image formation in the electron microscope, it is necessary to investigate the electron scattering properties of a specimen. Although a detailed account of electron scattering theory is outside the scope of this review, a brief review of the main methods available for the calculation of angular-energy distributions for elastic and inelastic electron scattering would seem relevant in context (Section 11). In the conventional transmission electron microscope, the aberrations in the final image will be related to these distributions. Section I1 will be concerned particularly with electron scattering by carbonaceous materials (e.g. carbon, organic, and polymeric materials) serving as a possible model for unstained polymeric and biological specimens (618). 11. THEANGULAR AND ENERGY DISTRIBUTIONS FOR ELECTRON SCATTERING In the conventional transmission electron microscope the relationship between the electron scattering properties of the specimen and the final image is clear; the scattering properties of the specimen determine the scattered electron wave in the back focal plane (Fourier plane) of the objective lens. This wave is referred to as the diffraction or scattering pattern. The intensity distribution, with respect to the angular deviation and the energy (wavelength) change of the incident electron beam, may be determined experimentally, but unfortunately the information required on the amplitude and phase of the scattered electron wave may be inferred only by a detailed comparison of theory and experiment. The formation of an image from the scattered wave may be expressed mathematically in terms of a Fourier transformation (see Section 111); phase shifts introduced by objective lens aberrations and defocusing, and the effect of an objective aperture (normally positioned in the back focal plane) may be included in the Fourier integral. Clearly the angular and energy characteristics of the transmitted electron beam will be important in a determination of the effects of phase shifts introduced by the spherical aberration and the chromatic aberration of the objective lens. Electrons of energy E, are incident on a specimen of thickness t ; the interaction of the electron beam in the specimen may be elastic or inelastic in character. The former causes a change in direction of the incident electron
67
DEFECTS IN ELECTRON-OPTICAL IMAGES
and an associated phase shift (19), while the latter type of interaction deflects the incident electron with a change in energy (usually as an energy loss E ) . I f the interaction is weak then the scattering may be treated by a kinematical approximation (single scattering conditions, the criterion for the validity of which is that the specimen thickness t be much less than the mean free path for electron scattering A]. For both elastic and inelastic electron scattering a parallel development is made: one considers electron scattering by a single isolated atom, an array of single atoms (thin crystal or thin film), and a normal electron microscope specimen ( t = 100-1000 A, E, = 20-100 keV), where multiple electron scattering is an important modification to the angularenergy distribution of the transmitted electron beam. A . Single Elastic Electron Scattering
I . Scattering by a Single Atom Consider a monochromatic parallel electron beam incident on the specimen; the incident electron is represented by a plane wave exp(iK, * r) = $&), where the wave vector KO is defined by [KOI = 2 4 2 , . The incident electron is scattered by a single atom characterized by a potential V(r). The scattered electron wave may be represented by $(r), where $(r) = +")(r> $(')(r) + .. (1)
+
The first Born approximation corresponds to a calculation of $(')(r) neglecting higher order terms (20, 21). The Born series corresponds to a calculation of
f o r r - t co. 8 represents the vector angle of scattering (0, 4), where 0 and 4 are respectively the polar and azimuthal angles of scattering. f(O)exp[irl(O)] represents the complex scattering factor for electron scattering by a single atom. It may be shown that the +(")(r) are related by the integral equation (21)
-
Since KO r = KOr in the direction of the incident beam and at large distances 1 r 1 .> 1 r' 1, the equation for $(I)@) is
2nme exp(iKo r ) $( 1 )(r) = - hZ r
jato,,, V(r')exp(iK'
= exp(iKo r)(l/r)fB(0).
r') dr' (4)
D. L. MISELL
68
fB(0) is the atomic scattering factor in the first Born approximation. 1 K' 1 = Iql = q is the difference between the incident wave vector KO and the wave vector K of the scattered electron. From Eq. (4) it is seen that fB(0) is the Fourier transform of the potential V(r). For a spherically symmetric V(r) Eq. (4) becomes 8n2me exp(iKo r) $I (1)(r) = - h2 r
r r 2dr',
where q = 2K0 sin(8/2) = (4n/Ao)sin(8/2). The scattered wave $(')(r) = exp(iKor ) f B ( 0 ) / rrepresents a spherical wave and therefore a 4 2 phase difference between the incident and scattered electron waves. Hence the phase change (delay) on elastic scattering by an atom is, within the first Born approximation, n/2, and the addition of the unscattered wave with the elastically scattered component necessitates a factor of exp(in/2) (22). The numerical calculation of the complex electron scattering factor (Eq. 2), from the Born series (Eq. 3) suffers from several problems, notably the slow convergence of the series. A reformulation of the problem for a spherically symmetric potential leads to the concept of partial waves, represented by partial phase shifts 6, (23): m
f(O)exp[iq(B)] = (2X0)-' C(2/+ l)[exp(2i 6,) - ~]P,(cos6). I=O
(6)
P,(cos 0) are the Legendre polynomials of order I. The phase shifts 6, may be calculated from either of two approximations depending on the magnitude of d l (23), where V ( r ) is described by a potential of the form z e i=3 V(r) = - - ai exp( - 1.13 Z'/jb, r/aH). 4ne0 r i =
(7)
aH is the first Bohr radius = h2eo/nme2= 0.529 A; parameters a, and bi have been tabulated for the Hartree and the Thomas-Fermi potentials (24). More precise calculations of 6, have been given (25, 26), but it is doubtful whether these more complex calculations are justified. The integrations, Eqs. (3) and (S), extend over all r, although in practice the integrals converge to 1 part in lo4 for r 4 A. However, r = 4 A represents a long-range potential in the context of a solid; the atomic potential V(r) must be considerably modified when in the potential field of other atoms. Examining the effect of terminating the potential integrals at r = r,,, demonstrates clearly the dependence of fB(0) and f ( 0 ) on the range of the potential (24,27), particularly a t 0 = 0 (Tables I and 11). The normal electron microscope objective aperture size (semi-angle c( subtended at the specimen) corresponds to q < 4 A-' (a < 0.08 rad, Eo = 10 keV; c ( < 0.02 rad, Eo = 100 keV). Presented in
-
69
DEFECTS IN ELECTRON-OPTICAL IMAGES
TABLE I FREEATOMELECTRON SCATTERING FACTORS. CARBON z = 6
rInax(A)
f(O)(A)
(rad)
f’(4)(A)
7&4) (rad)
TF“ Eo = 10 keV
0.71 0.77 1.16 20.0
1.681 1.848 2.871 7.443
0.136 0.131 0.107 0.055
0.979 0.990 0.845 0.803
0.189 0.193 0.250 0.280
HBb Eo = 10 keV
0.71 0.77 1.16
1.550 1.660 2.137
0.142 0.137 0.117
0.952 0.959 0.898
0.192 0.195 0.221
20.0 0.71 0.77 1.16 20.0
2.449 1.994 2.191 3.397 8.172
0.105 0.049 0.047 0.038 0.019
0.876 1.166 1.178 I .005 0.956
0.230 0.068 0.069 0.092 0.106
0.71 0.77 1.16 20.0
1.840 1.971 2.533 2.902
0.051 0.049 0.042 0.037
1.135 1.143 1.070 1.043
0.069 0.070 0.079 0.083
T(0)
TF Eo = 100 keV
HB Eo = 100 keV
a
Thomas-Fermi potential (TF). Hartree-Byatt potential (HB).
Tables I and I1 are free atom electron scattering factors for carbon ( Z = 6) and gold ( Z = 79) for various rmaxvalues. In the case of carbon, the rmsx values were based on half the value of the C-C distance in graphite (0.71 A), diamond (0.77 A), and evaporated carbon ( I . 16 A); the dependence of f ( 0 ) ( N , f Bto one part in 10’) on rmaxis particularly striking. For gold rmaxvalues were based on the Au-Au distance in the metal crystal ( I .44A). The deviation off(0) fromf’(8) is significant for atoms with large Z and for low E, (20). It is evident that any calculations of electron microscope images of atoms will depend on the choice of V ( r ) and an effective cutoff parameter r,,,. It is surprising that to date few calculations have been made o n f B taking account of the periodic potential V(r) that exists in a crystalline specimen (28). In consideration of electron microscope images calculated for atoms of large Z , the phase shift q ( O ) , which measures the additional deviation of the phase shift on elastic scattering from n/2, should be included (29, 30). The elastic wave YE(0)in the back focal plane of the objective lens is described by the equation
D. L. MISELL
70
TABLE 11
FREE ATOMELECTRON SCATTERING FACTORS. GOLDZ = 79
5.659 5.569 5.546 5.653
3.288 3.249 3.239 3.279
1.137 1.168 1.179 1.159
0.431 0.389
5.329 5.271 5.256 5.323
3.092 3.061 3.053 3.085
1.029 1.050 1.057 1.044
15.438 16.213 16.837 19.106
0.253 0.243 0.236 0.211
6.637 6.532 6.505 6.630
5.615 5.526 5.504 5.607
0.567 0.579 0.584 0.574
12.603 13.107 13.506 14.871
0.259 0.250 0.244 0.223
6.250 6.182 6.164 6.242
5.288 5.229 5.214 5.281
0.527 0.535 0.538 0.532
1.30 1.44 1.58 20.0
14.463 15.144 15.692 17.675
10.704 11.321 11.821 13.648
0.462
1.30 1.44 1.58 20.0
11.889 12.333 12.684 13.880
8.639 9.040 9.360 10.457
0.460
TF Eo = 100 keV
1.30 1.44 1.58 20.0
16.963 17.762 1 8.404 20.730
HB Eo = 100 keV
1.30 1.44 1.58 20.0
13.944 14.465 14.877 16.279
TF" Eo = 10 keV
HBb Eo = 10 keV
0.444
0.429 0.378 0,444
Thomas-Fermi potential (TF). Hartree-Byatt potential (HB).
In contrast to the formality of scattering theory given above, the interaction of the incident electron with an atom may be viewed as a modification of the incident wave front by a potential field (19,31-33). The scattered wave $(r), including the unscattered component, is represented by
M9 = $o(r)exp[iaV(x, ~11,
(9)
where V ( x , y ) is the projection of the three-dimensional potential in the z direction, m
and CT = - 2nmeAO/fi2. The diffraction pattern can be described by the amplitude in the back focal plane of the objective lens, that is, ( 1 1)
v is the two-dimensional vector (v,, and v * r = v,x + vyy.
vy), with K' = q = 2nv or 2nv = K o 8
71
DEFECTS IN ELECTRON-OPTICAL IMAGES
If the effect of the potential field on the incident electron wave is small (phase grating approximation), then exp[iaV(x, y)] may be expanded as
$09 = $o(r” + icV(x, Y)l.
(12)
The scattered wave is then given by s,(v)
j
= a(v) - ( 2 n i r n e ~ , / ~ ~ )
-
( xy)exp(2niv , r) dr,
(13)
and from the definition offB, Eq. (4),
or withgB(v)= fB(e)(e= A,v). S(0) represents the unscattered wave and [ f B ( 0 the ) elastically scattered wave; the i directly reflects the n/2 phase difference between the two waves. Higher terms in the expansion of exp[iaV(x, y ) ] lead to a complex scattering factor which is equivalent to the scattering formulation, Eq. (2), that is, ?,,2
So(v) = 6(v) + iA,gB(v) - 2
JV.gB(v
- v’)yB(v’)dv’ -
...
or
1 we) = ace) -tp ( e ) - j fB(e- ey-B(e’)def - . . . 21,
e‘
= S(0) -t f(e)exp[iq(O)]. intensity distribution I YE(@)I
The differential cross section, doE/dQ,
(1 5 )
is classically related to the elastic
(daE/dQ)atom =
I f(e>I
(16)
and the elastic cross section is obtained by an integration over all scattering angles (element of solid angle dQ = de = sin 0 d0 d&),
The mean free path for elastic scattering AE = (NoE)-’ for N atoms in unit volume. If R , @ ) is defined as the elastic angular distribution for scattering by atoms with R,(O) = a;’ If@) 1 then R,(O) represents a probability distribution of elastic electron scattering into the element of solid angle AQ,
’,
72
D. L. MISELL
For a hypothetical specimen consisting of an array of randomly distributed single atoms, the scattered wave is yE(e)
= (t/AE)l/’Rl(e)l/’ exp{i[q(8)
+ n/2]>
(18)
’
for a specimen of thickness t 4 AE . Experimentally one measures I YE(@)1 and the phase factors of Eq. (18) are indeterminate, with
I ’de = ( t / l \ E )
IeI
1
e
Rl(e>
de (19)
= (t/AE)*
2. Scattering by a Single Crystal
In a single crystal the atoms form a regular array and the potential field is periodic. Corresponding to the first Born approximation, Eq. (l), the elastically scattered wave in the kinematical approximation is (22) 27cme exp(iK, r) $(l)(r) = - hZ r
V(r’)exp(iK‘* r’) dr‘,
(20)
and in the tight binding approximation (atomic potentials assumed unchanged in a crystal lattice), $(l)(r) =
exp(iKor ) r
-
zfiB(B)exp(iK’ ri), i
The crystal equivalent offB is the structure factor Fg
q e ) = C fiB(@w(2xig i
- rj),
(22)
where g is the reciprocal lattice vector (h, k, I ) , with 2xg = 27cv = K’ and rj represent the atom coordinates in the unit cell. More frequently the diffracted wave is defined in terms of the Fourier coefficients V,( U,or 0,) of the lattice potential, V(r)
=
C V, exp( - 2xig
r),
B
that is,
Rc is the unit cell volume and h2/27cme = 47.87 eV A’. The coefficients v,
DEFECTS IN ELECTRON-OPTICAL IMAGES
73
and U, are related to V, by 8n2meV, = 0.2625V, hZ
=-
A- ’,
V u, = g2me - 0.00665 V, A - 2
hZ
The argument o f f B is given in most tables in terms of the parameter s = sin O B / l o = q/4n (for sin 8 N 8, 6, is the Bragg angle = 8/2). In general V, is complex unless the unit cell has a centre of symmetry. Strictly speaking, Eqs. (22) and (24) should include a term exp( -BI g12) which takes account of the thermal vibrations of the atoms in the unit cell (34). For low order reflections the thermal effect causes a 10 % decrease in V,; for example, C (diamond) I V, I = 7.17 eV (no temperature factor), 1 V I l l I = 6.95 eV ( T = 300 K); A1 1 V,,, 1 = 6.12 eV, 1 V,,, 1 = 5.10 eV ( T = 300 K); Au I Vlll I = 18.61 eV, I V,,, I = 16.40 eV ( T = 300 K). Equations (22) and (24) represent the elastically scattered wave for a single unit cell orientated in the exact Bragg position. The diffracted wave from a crystal of thickness t (assumed infinite in the xy plane) is (22, 35)
-
,
Y,
= YE(q = (in/t,)[sin(nrs,)/.s,,I
exp( - inrs,)exp(icr,),
(25)
which represents a first Born approximation applied to a single crystal. t, is the extinction distance defined by t,
I V, I ) = loE~ cos e,/ 1 v,1, e, = 812, = (h2Ko/4nme)(cosO,/
(26)
s, is the deviation (in reciprocal space) from the exact Bragg position,
s,
= A%, sin
28,/a0
= t,/27~,
(27)
and a, is defined by V, = 1 V, 1 exp(icr,). In the case of a crystal orientated in the exact Bragg position, 118, = 0 and Y, = in(t/t,)exp(ia,),
I Y, I
= nzt2/tg2.
(28)
The diffracted wave is n/2 out of phase with the unscattered wave as indicated by the factor i in the equation for Y,, Eq. (28). Equation (28) indicates that for s, = 0, the diffracted intensity increases as t z ; thus for t 1, , 1 Y, 1 is comparable with unity and the kinematic approximation used in the derivation of Eq. (25) is invalid. The limit of validity for the kinematic
-
’
74
D. L. MISELL
-
approximation is t t,/3 [C(diamond)t, = 500 A, A1 t, = 600 A, Au t, = 160 A with Eo = 100 keV] (22). In the case of s,# 0, the crystal thickness for which the kinematic approximation is valid can be significantly larger than tJ3. The phase relation between the unscattered wave and Y, is represented in Eq. (25) by i exp( -ints,)exp(icc,); cc, = 0 for a centrosymmetric crystal. In the case of a polycrystalline specimen, the crystallites are randomly orientated and Eq. (22) is appropriate in describing the scattering from the specimen (36).
3. Scattering by an Amorphous Specimen In an amorphous specimen the degree of order is described by a distribution function p(r), defined such that 4nr2p(r)dr represents the average number of atoms lying at distances between r and r dr from the center of a given atom; po is the number of atoms in unit volume. The phase relation between electrons scattered from different parts of the amorphous specimen depends on the various atom-atom distances; the summation, Eq. (22), obtained for a single crystal [p(r) = 6(r - rj)] is replaced by an integration to be weighted with thedistribution functionp(r) (37).The intensitydistribution I YE(€)) I is, for a specimen composed of a single type of atom,
+
or for a spherically symmetric p(r)
The first term in Eq. (30), NfB(€))2, represents the incoherent scattering from N atoms and the second term represents the correlated atom-atom scattering effect. Evidently the phase relationships required for the calculation of YE(@)are not readily calculated. The concept of a Fourier coefficient of the potential, analogous to V, for the crystalline specimen, is not meaningful for an amorphous specimen.
B. Multiple Elastic Electron Scattering
-
In the case of a normal electron microscope specimen t A E , and the probability of multiple (dynamic) interactions is not negligible; the single scattering (kinematic) approximation of Section II,A must then be appropriately modified to reflect the more complex nature of electron scattering in the specimen.
75
DEFECTS IN ELECTRON-OPTICAL IMAGES
1. Scattering by Atomic Species As an approximation to the real situation, the atoms in the specimen are assumed to scatter the incident electron beam such that there is incoherence between electrons scattered from different centers. In the Born series approach to this problem (38), the electron scattering by each atom in the specimen is considered in the first Born approximation but the scattering by the specimen as a whole is treated by a Born series, Eq. (I),
$(r)
= $(')(r)
+ $(')(r) + . . + $tn)(r),
(3 1)
*
where $(")(r) represents the electron scattered n times. The elastic wave is then given by $(")(r)
gn(v) 1 = - j g , ( v - vn-l)gl(vn1 - v,- 2 ) n!
' *
.g l( v , - v l )dv, dv, * . ' dV,- 1,
(32) where g l ( v ) is identical to i&gB(v) in Eq. (15). The nth scattered wave is 4 2 out of phase with the unscattered wave, that is, j=n
Y!E(e)
=
C exp(~~/2)fj(e)/(3-lj!).
j= 1
(33)
The &(O) are calculated by the successive convolution of fi(0) = fB(6). An identical result to Eqs. (32) and (33) may be derived using the phase grating approximation. An alternative method for the calculation of the intensity distribution of the elastically scattered electrons is to consider the successive folding of the distribution R l ( 0 )with itself, that is, for n elastic scattering events the probability distribution is given by Rn(e)=
je'Rl(e - e')R,-,(e') de',
(34)
and the resulting intensity distribution is calculated from
I YE(0) I = R(0) = eXp( - t / l \ ~ ) 1(t/AE)n&(0)/n!. (35) For t/AE< 1 , Eq. (35) reduces to R(0) = (t/&)&(@). The scattering has n=
been assumed to obey Poisson statistics. Evidently in Eq. (35) all phase correlations between multiply scattered waves have been neglected and the final intensity distribution is found from a superposition of the probability distributions for n elastic interactions. The relationship between the I YEI calculated from Eq. (33) and I YEI calculated from Eq. (35) is not readily derived (9).
76
D. L. MISELL
2. Scattering by a Single Crystal The interaction of an electron beam in a single crystal has been treated by a pseudo kinematical approach, that is,fi"(O) in Eqs. (22) and (24) of the kinematical theory is replaced by the complex fj(0)exp[iqi(O)J(40). This modification to the kinematical theory takes account of the strong interaction between the incident electron and a single atom but does not consider dynamic (multiple) interactions in the crystal as a whole. Evidently the pseudokinematical theory leads to a complex Fourier coefficient V, even for a centrosymmetric crystal. The simplest dynamical formulation for electron scattering in a single crystal is the two-beam dynamical theory (41-43) ; in this approximation only the unscattered wave and a single diffracted wave g are considered. This is evidently a good approximation for a thick crystal (t 1000 A, Eo = 100 keV) (44). For a centrosymmetric crystal, the unscattered and diffracted waves respectively are calculated from (43)
-
iw
Y,=
i
(1 + w2)"2
sin[:
(1 + w2)1/2]exp(T), - intw
where w = r,s,. The common phase factor exp(-intwlt,) in Eq. (36) is irrelevant and may be omitted. It is seen from Eq. (36) that Yo and Y, vary periodically with t, corresponding to diffraction back into the zero order beam; for t = t,/2, Yo-,0 and Y, -+ l(w = 0). For all crystal thicknesses the total intensity distribution lYo12+ (Y,(' is unity and, in particular, for the exact Bragg position (w = t,s, = 0), Yo = cos(nt/t,) and Y, = i sin(nt/t,). Diffraction contrast in the images of single crystal specimens arises from the use of an objective aperture to allow the zero order beam to be transmitted whilst stopping the diffracted beam g (bright field) or the reverse procedure (dark field). Then the two-beam theory predicts the various diffraction contrast effects such as thickness fringes, bend contours (22). Equation (36) reduces to the kinematic equation for t < t,, Eq. (25). In the case of a polycrystalline specimen Eq. (36) for I Y,J is averaged over all crystallite orientations w to give for the integrated intensity (45)
'
where A, = 2nmetlo(V,(/h2N nr/r,; f ( A , ) N exp(-0.34A;) for A, I2 ( t I 200 A). In thelimit of 1, r -,0,Jdyn -'&in and for ilot -+ 00, Jdyn = &,/2A@ cc V, . Figure 1 shows the variation off(A,) with A,; the curve denoted by A
77
DEFECTS IN ELECTRON-OPTICAL IMAGES I
.o
08
--<
0.6
c
0A
0.2
I
(
2
3
4
A
FIG. I . The dependence of the Blackman dynamical function f(A) on the parameter A = -of 1 Vg I. The dashed curve (A,) shows the dependence off(A) on A for an averaging over all crystallite orientations.
is for the A , defined above and the curve denoted by A , corresponds to an averaging of A , over all angles between the diffracted wave and the normal to the surface of the crystallite (45). As an improvement to the two-beam approximation other diffracted beams (sometimes referred to as weak beams) may be included ; the Bethe dynamic potentials are an effective correction to the kinematic V,, that is, (41). ( VOg)dyn
= ( vgO)dyn
=V,+AV,
for a centrosymmetric crystal. replaces V, in the calculation of the diffracted intensity. A criticism of Eq. ( 3 8 ) is the absence of any dependence
D. L. MISELL
78
of AV, on I , t ; in particular AV, is nonzero for I. t + 0 (kinematic approximation) (46). A detailed consideration of this problem (47) leads to a correction term T(K,,, KO) in the summation of Eq. (38); T(K,., K,)+O as I , t + 0 and Eq. (38) reduces to the kinematic theory (AV, = 0). Beyond the two-beam dynamical theory, when multiple diffraction effects are included, an analytic solution for 1 Y, 1 is not possible. The numerical character of these many-beam theories means that a simple relationship between the specimen structure and the diffracted intensity no longer exists. The possibility of deriving structural information using a many-beam theory has been examined (48).The dynamic theory of electron scattering, as applied to crystalline specimens, has appeared in several different formulations, for example: the scattering matrix formulation (49-51), the Born series development (52, 38), and the phase grating method (19); the equivalence of these approaches to the n-beam dynamical theory of electron diffraction has been demonstrated (44). In the phase grating approximation, the crystal is divided into thin sections ( 5 A) and the diffraction by successive layers considered in the kinematical approximation (19, 31). The effect of the potential distribution on the incident electron beam is described by considering a section Az centered at z = z1 ; V ( x , y ) is the projection of the potential distribution onto the ( x , y ) plane, that is, N
The diffracted wave Q,(v) for the crystal element Az is then calculated from the Fourier transform
1:
-
J [ l + iaV(x, y)]exp(2niv r) dr.
(40)
-
The first term l.exp(2niv r) in Eq. (40) represents the unscattered component 6(v). The diffracted wave Ql(v) is then incident on a second section of crystal and the effect of the potential field (for z = z 2 ) on Ql(v) is evaluated; Fresnel diffraction effects between crystal elements may be included. The equation for the wave incident on the nth element of crystal is (33) F,(v) =
1F, ,(v’)Q, -
- ,(v - v’)P(v) dv’,
(41)
V’
with F,(v) = 6(v) representing the incident electron wave and Q,-,(v) representing the diffraction at the (n - 1)th element of the crystal with F2(v) = Ql(v). P(v) exp[in22 Az(v,2 + v,2)/Ko] represents the phase changes in the
-
79
DEFECTS IN ELECTRON-OPTICAL IMAGES
transmitted wave due to Fresnel diffraction between successive sections. Iteration of Eq. (41) allows the diffracted beam amplitude for a crystal specimen to be computed for any crystal orientation. In the particular case of a single crystal specimen (19) Q(h, k ) =
/J [ 1 + ia V(x,y)]exp[2ni(hx + ky)] dx dy,
(42)
where the integration is over the unit cell in the ( x ,y) plane. V(x,y) is evaluated from the Fourier series
where
Vh,k = C V, exp[ - 2ni(Iz)].
(44)
I
+
The final FJv) calculated gives the diffracted wave "(0) = p S(0) YE@); pZ represents the fraction of the incident beam that has not been scattered. It is noted that the information required to evaluate Y(0) amounts to a detailed knowledge of the specimen structure and its electron scattering properties. 3. Scattering by an Amorphous Specimen
In consideration of the importance of electron scattering by amorphous specimens, particularly in relation to polymeric and biological specimens, there is a dearth of theoretical work on this topic. The only theoretical work on this problem would appear to be a Born series approach to multiple scattering(38), although in principle the phase grating method may be applied to the problem (53).A basic requirement of a theoretical model for the calculation of YE@)in amorphous specimens is the knowledge of the distribution function p(r), which may be determined by x-ray diffraction techniques (37). In the absence of any tenable theory, the incoherent approximation given in Section II,B,I is adopted as a viable approach to multiple scattering by an amorphous specimen.
C. Single Inelastic Electron Scattering Inelastic electron scattering processes cause a change in the wavelength of the incident electron beam, more commonly referred to as an electron energy loss, E, which is taken to be a positive quantity; that is, the energy of the scattered electron beam is defined by E, - E. Processes in which the incident electron beam gains energy are considered to be negligible for
80
D. L. MISELL
medium energy electrons (Eo = 10-100 keV) in thin specimen films ( t = 100-1000 A).
I . Scattering by a Single Atom As with elastic electron scattering, a first approach to inelastic electron scattering involves the concept of free atoms in a solid; in so far as core excitations are involved, the atomic model is a valid description for inelastic electron scattering. The differential cross section for inelastic electron scattering by a single atom may be written as (nonrelativistic)
where Sincis the incoherent scattering factor for X-rays; Sin,is represented by a sum over all one-electron (atomic) excitations (54, 55). The parameter q, = K0(B2+ OE2)l/’and OE = E/2E0 is taken as a measure of the energy loss by the incident electron beam; in the free atom formulation a single value for E cannot be chosen in a meaningful context. In a solid the free atom formulation cannot be a valid description of the “valence” electron scattering; the free atom theory is then used to describe the interaction between the core electrons (assumed unaffected by binding in the solid state) and the incident electron beam using a suitably reduced (56). expression for the Waller-Hartree Sinc The cross section for inelastic electron scattering is given by
and A, = (Nol)-’; qmin= KO0, and qmaxdescribes the maximum angle of scattering with qmax= KOOmax. The probability distribution Q, (0) for inelastic electron scattering is defined as o;’(dq/dQ); for a specimen of thickness t( $ A,), the intensity distribution for inelastic scattering is
I wde) I
I
= ( t / 4 ) +l(e) I = (t/AdQl(e),
(47)
(t/Ad1’24i(e) = ( t / A ~ ) ” ~ Q i ( e ) exp[i~i(@, ”~ Ell.
(48)
or wi(e) =
PI(& E ) represents the phase term for a scattering angle 8 and energy loss E ; the behavior of pi@, E ) determines the coherence of the inelastically scattered wave (see Section 11,G).
DEFECTS IN ELECTRON-OPTICAL IMAGES
81
The free atom theory in the form given above is not adequate to describe the energy distribution of the inelastic component of the transmitted electron beam, and in order to do so it is necessary to examine a “solid state” theory. 2. Scattering in a Solid
The valence electrons in a solid constitute a potential with a long range, which gives rise to the predominant form of small-angle inelastic electron scattering. The consideration of valence electron scattering represents a more realistic approach to the treatment of small-angle inelastic scattering than given by the free atom formulation. The valence electron scattering can be divided into two components: (i) collective excitations of a “free electron gas” (plasma oscillations) (57, 58) which is the predominant mechanism for small-angle electron scattering for 8 < Oc (“cutoff” angle) and (ii) oneelectron excitations of the valence electrons for 6 > 8, (59). The Bohm-Pines plasma oscillation theory leads to the following expression for the differential cross section da,/dQ (expressed per unit volume of solid):
E is now defined in terms of the excitation energy of the electron gas (plasmon energy). The concept of a “cutoff” angle arises from an upper limit to q1; q1 represents a reciprocal length in the specimen and a lower limit exists on 4’; (a distance) since the concept of a collective (plasma) excitation becomes meaningless when q;’ is comparable with the interelectronic spacing in the specimen. The cross section for plasmon excitation is obtained by the integration of Eq. (49) over all angles of scattering, that is,
and A, = 0;’;qc = Ko(Oc2+ 6E’)1/2. Corresponding to q1 > qc , the electron-electron interactions become of a short-range character with da,/dQ oc F 3(59). The addition of the free electron and one-electron expressions for the differential cross section describes completely the valence electron scattering, but omits core excitations. The contribution from core excitations to da,/dQ is taken from a suitably reduced form of the free atom theory; the total inelastic differential cross section which includes plasmon excitation, one-electron (valence) excitations, and core excitations will be referred to as the composite theory. The inelastically scattered wave, Y,(O), is defined in an analogous way to the definition
D. L. MISELL
82
used in the free atom theory, Eq. (48). It is noted that the free electron theory as given describes only the angular characteristics of the scattered electron and does not give information on the energy distribution. A more complete formulation for the inelastic electron scattering is the dielectric theory, which relates the differential cross section d201/dildE to the dielectric constant ~ ( q ,E, ) (60-62). The inelastic wave Y@, E ) may be written as Yde, E ) = (l/AI)l’z$l(e, E)exP[iP,(% Ell,
(51)
where (60, 63)
I $l@,
I
= D,(% E ) =
- ( 1/2n2~,E, oI)[l/(e2 + eE2)]1m[I/&,, E ) ] .
(52)
The imaginary part (Im) of E ( q l , E ) - ’ determines the energy loss distribution of the inelastic component for single scattering ( t -4 AI). Equation (52) omits surface excitations and retardation effects (64) which are important only for very small angles of scattering (6’ rad) and small energy loss values ( E 5 eV). Although in principle information on ~(0E , ) may be determined from optical measurements in the vacuum ultraviolet region of the electromagnetic spectrum, information on the q, dependence of E(ql, E ) is not readily available. The assumption that E(qlr E ) = ~ ( 0E , ) enables the 0 dependent and E dependent terms in equation (52) to be separated, that is,
-
N
I
E ) I = Di(8, E ) = Qi<e>fi(E>,
(53)
where f , ( E ) = Im[l/e(O, E)] is the profile of the energy loss distribution obtained under single scattering conditions. Dl(8, E ) is normalized such that
and Dl(8, E ) then represents a probability distribution for an energy loss E with an angular deflection 8. Q,(O) and f , ( E ) are separately normalized and f , ( E ) = 0 for E 5 0 (that is, energy loss processes only are included). As an alternative (or additional) approach to the consideration of inelastic electron scattering, the interband transition theory (65) deserves more attention then has been given in the literature. The interband theory is a viable approach to the description of inelastic electron scattering in the transition metals, where the plasma oscillation theory fails to predict completely observed energy loss peaks. The close relation between the interband transition theory and the band structure of the material, via the Fourier coefficients of the potential distribution, is a feature of this theory. Experimentally it is not possible to distinguish between the free atom theory, the plasma oscillation
83
DEFECTS IN ELECTRON-OPTICAL IMAGES
theory, and the interband transition theory by measurements on the angular dependence of da,/dR; all three theoretical models predict a small-angle behavior for da,/dR of the form c(02 + OE2)-’ (56).It is difficult to apply the interband transition theory to inelastic electron scattering by amorphous specimens (Fourier coefficients I/e not defined). Omitted from this section on inelastic electron scattering are phonon excitations and bremsstrahlung ; phonon excitations give rise to energy losses below 0.1 eV and the contribution of phonon scattering to the total inelastic scattering is significant only for medium angles of scattering (0 0.02 rad). The cross section for bremsstrahlung is significant only for electrons of incident energy greater than 1 MeV.
-
D . Multiple Inelastic Electron Scattering
Plural inelastic electron scattering not only modifies the angular distriof the transmitted electron beam but also the energy loss bution, distribution, fi( E ) . Unlike elastic scattering, inelastic electron scattering does not appear to depend critically on the detailed structure of the specimen; it is then possible to calculate the angular energy distribution D,(e, E ) after n inelastic interactions by a statistical approach, that is,
el(€)),
I 4n(@,
E ) I = On(@,E )
s, s,. +m
=
D , ( e - w, E - E’)D,-
l(er,E’) def d ~ ’ ,
(55)
and the energy distribution for n inelastic events is given by
From the dielectric formulation D2(8, E ) is calculated using the convolution of 1 $,(& E ) 1 with I E ) 1 2, that is, (66, 64)
I $203,
E ) I = Dz(% E )
where elastic electron scattering is omitted (see Section 11,E); C = - (2nza, Eo q)-’.Equations (55) and (57) represent double integrations (the 8 integration may be simplified by an integration over the azimuthal angle of scattering for a cylindrically symmetric angular term). As pointed
D. L. MISELL
84
out in Section II,C a knowledge of Im[l/&(O,E ) ] is not readily available and two further approximations are made in order to simplify the calculation of the I &(8, E ) I ', namely, &(Of, E ) = ~(0,E ) over the 8' integral and OEo is a constant over the E' integral. These two approximations enable the 8' and E' integrations to be separated, that is,
I Cbn(e, E ) I
I4n(e) I 'fn(E).
= Dn(Q,E ) =
(58)
'
1 $,(8) I = Qn(8)represents the angular distribution of the scattered electron and is dependent on 8 only. From Eq. (55) it may be verified that if D,(O, E ) is a separable function of 8 and E then +m
M e ,E) =
-a
f d E - E'lfn-
I@')
dE'
je'Ql(e- e')Qn-,(e') de',
(59)
which is of the form D,(8, E ) =fn(E)Qn(8),Eq. (58). The resultant inelastic angular-energy distribution 1 Y,(8, E ) I is calculated from the superposition of the I &(8, E ) 1 weighted with the appropriate Poisson coefficient, that is,
Iw e , E ) I
=
~ ( eE,) m
(60)
or m
I LO, E ) I
=
c pn I4n(e, E )
n= 1
12 =
2 I Qn(8, E )
12.
n= 1
The angular distribution for inelastic electron scattering, Q(8), is calculated from an integration of Eq. (60) over all E [thef,(E) are defined as zero for E 5 01:
QW
2
= exp( - ~ / A J (t/AI)nQn(e)/n 1, n= 1
(61)
and the resultant energy loss distribution, F(E), is calculated from the integration of Eq. (60) over 8: m
2
F(E) = ~ X P-(~ / A J ( t / W f n (E)/ n n= 1
(62)
The distributions D(8, E ) , Q(8),and F(E) may be experimentally determined; in principle D,(8, E ) andf,(E) may be determined from the observed distributions D(8, E ) (67) and F(E) (68),respectively. The inelastic wave Qn(8,E ) , Eq. (60), may be written as
Q,,(e, E ) = [pn On(&E)P2 exp[ipn(8,E)I,
(63)
where the p,,express the phase relationships between electrons with different n (see Section 11,G).
DEFECTS IN ELECTRON-OPTICAL IMAGES
85
E. Combined Inelastic-Elastic Electron Scattering For materials of biological significance the cross section for inelastic electron scattering is greater than that for elastic electron scattering. Thus the effect of inelastic scattering on the elastic component represents a major modification to the original elastic distributions R,,(O).In its simplest form the effect of inelastic scattering represents an attenuation of the elastic wave. The effect of elastic electron scattering on the inelastic component represents only a modification to the angular distributions Qn@) ;the original small-angle behavior of inelastic electron scattering is affected by the medium-angle elastic scattering.
I . Incoherent Approximation Following the analysis of Sections II,B,l and II,D, the effect of elasticinelastic interactions on the angular-energy distributions of the transmitted electron beam is considered. In a statistical approach the probability of n inelastic, m elastic interactions is given by the Poisson distribution:
The unscattered component is represented by
I v,,(e)I
s(e).
= R,(e) = exp( - t/AT)
(65)
Due to the finite source size, the incident electron beam has an angular divergence and S(0) should be replaced by Zo(0). Formally the incident electron beam has an energy distribution N ( E ) due to the energy distribution of emitted electrons (e.g. thermal distribution). Under normal operating conditions of a heated filament electron gun, N ( E ) may be represented by an exponential form (69),that is,
with a maximum at E = 0 and N ( E ) = 0 at E = p / p ;r(p+ 1) is the gamma function ( = p ! for p integer). The energy half-width of N ( E ) (typically 0.5-1 .O eV for a tungsten filament) is determined by both p and P ; p determines the asymmetry of N ( E ) about E = 0. In order to conform with the work on electron energy loss, E > 0 corresponds to an energy less than E , , E, - E (that is, the reverse of convention). Equation (65) may then be rewritten as an energy dependent distribution (4)
I yo(e,E ) I
= D,@, E ) = exp( - t/AT)Io(e)N(E).
(67)
86
D. L. MISELL
a. Multiple elastic scattering. The probability of elastic electron scattering is obtained by a summation of Eq. (64) over m for n = 0 (probability of no inelastic interactions). The angular distribution, R(6), for elastic scattering is then given by
IyE(6) I
= m
= exP( - f/&)
(t/AdmRm(6)/m1,
m= 1
(68)
which differs from Eq. (35) only by the factor exp( - t/A,). Formally the elastic component should include the energy distribution N(E), that is, ( 4 )
m= 1
The effect of the finite source size, which is represented by a convolution integral, on the elastic distribution Rm(6),may also be formally included in the equation (69). b. Multiple inelastic scattering. The modification to the inelastic angular distributions Q,(6) is described by the convolution of the elastic distributions Rm(6)with the Qn(6)(4).The probability distribution for n inelastic, m elastic interactions S,,,,(6) is given by
.
for n = 1,2,. . . and m = 0, I , 2, . . with Ro(6) = S(6). The resultant inelastic angular distribution S(6) is obtained by a summation of the Sn,,(6) over all m and n ( # O ) weighted with the appropriate Poisson coefficient, Eq. (64), that is,
I 'YmIZ = S(6)
(tty Je
Q,(6 - 6')Rm(6')d6 .
(71)
The order of the m summation and the 6' integration is reversed to give, using Eq. (68),
I 'Yl(6) I * = S(6) =
2 {exp(- t/AT) 6(6) + R(8)) * (t/A,)"Qn(6)/n!
n= 1
DEFECTS IN ELECTRON-OPTICAL IMAGES
87
where the asterisk * represents a convolution of two functions. The first term of Eq. (72), exp( - t/A,)Q,(e), represents the inelastic component of the scattered intensity that is not scattered elastically, and the second term, R(8) * Qn(0),represents the modification to Qn(0)by elastic electron scattering. The probability distribution s,,@)is defined for n inelastic interactions by the equation
Id
mI
= sn(@ = ~ x P ( -t/AdQn(e)
+ ex~(t/Ai)R(o)* Q n ( e ) ,
(73)
where (8 ,:/)
and
d8 = 1
e():/
d e = 1 - exp(-t/A,).
Inserting in Eq. (73) the energy loss distributions f , ( E ) , the following equation is obtained for the angular-energy distribution Q(8, E )
I YI(8, E ) I
=
w, E)
2. Absorption
The word absorption is used to describe the effect of inelastic scattering on the elastic scattering. The elastic scattering is effectively attenuated from [ 1 - exp( - t / h ~ ) to ] [ 1 - exp( - ?/AE)]~XP( - t/AJ by the inelastic scattering. In the context of diffraction contrast the effect of inelastic scattering on the diffracted wave is described by the introduction of a complex potential V(r) = V(r) + iV("(r) (70, 71); the Fourier coefficients of the potential V, = VB + iVii)arecomplex. The imaginarycomponent of V B ,iV$)represents the effect of absorption; the Vji)may be calculated for the appropriate inelastic process, e.g. phonon excitations (thermal diffuse scattering) (72-74), plasmon excitations (74), and one-electron core excitations (71, 75). The effect of multiple inelastic scattering on Vii)has not been evaluated. In relation to electron microscopy, the two important absorption processes are plasmon excitation (76)
88
D. L. MISELL
with V?) = 0 for g # 0 since the free electron density is not modulated by the periodic lattice potential, and one-electron core excitations (71,77, 75)
where SinC(q, q - 2ng) is a modified form of the incoherent scattering factor for X-rays; in the literature on the calculation of Vii),s is used in place of q but defined identically and g is used to denote 2ng. The concept of a complex Fourier coefficient leads to an absorption coefficient p, = -(8n'me/h2Ko)V,(');
for g = 0, po = oI. Detailed calculations of the Vf) show that Vi') 0.05 V, for C (diamond) and Vf) 0.15 V, for Au (Eo = 100 keV). This increase in the ratio V f ) / V , with Z is similar to the behavior of the complex V, that arises from the use of a complex scattering factor, f(O)exp[iq(O)], in the expression for V, , Eq. (24), e.g. Vj') 0.07 V, for C (diamond) and V;') 0.5 V, for Au; the increase in V i i )with decreasing incident electron energy is also similar. A noncentrosymmetric crystal will also give complex Fourier coefficients V, . Evidently it is not possible experimentally to differentiate between the phenomenological l'?)that represents absorption and a complex V, that arises from, for example, the failure of the first Born approximation ; theoretically the distinction is clear (78). In view of the insensitivity of "theoretical" diffraction contrast effects to the Vf)values and thecritical effect of the choice of the V, (real part) (79),it would seem realistic to concentrate on the " accurate" evaluation of the V, using a crystal potential. N
N
-
N
3. Scattering by a Single Crystal
In order to include the effect of absorption on the elasticelectron scattering in a single crystal, the n-beam dynamical theory is formulated with a complex lattice potential with corresponding complex Fourier coefficients, that is, V(r) -,V(r) + iV")(r) with I
The calculation of the diffracted wave in the two-beam dynamical theory leads to the equations (80, 22, 43) Y,, = exp( - nt/ty)){cos x - [iw/(I Y, = exp(-nt/tA'))[i sin X / ( I
+ w2)'/']sin x},
+ w2)'/'I,
(79)
89
DEFECTS IN ELECTRON-OPTICAL IMAGES
where ty) = h’Ko cos 0,/4rmze I V:i)I
and nt x=(1 +
w2)”2
4
+ ty)(]+int
w2)1/2
*
Equation (79) may be derived directly from the simple two-beam theory by allowing the extinction distance rg to be complex, that is, t i 1 -,r;’ + it:)-’. If tf’ -+ co ( V r )+ 0) then Eq. (79) reduces to the two-beam equation (36). The term exp( - nt/t:)), which represents the effect of absorption, may be rewritten as exp( - ta,/2) (tr) N 2n/p0]; thus the attenuation of I Yo and I Y, I is represented by exp( - to1)or exp( - r/A1),which is identical to the factor in Eq. (68) for the modification of the elastic scattering in the incoherent approximation. The evaluation of the diffracted intensities for the n-beam case may be considered using the phase grating approximation (19). The effect of a crystal element (thickness Az centered at z = z , ) on the incident wave $o(r) may be described by q(x, y ) , where
I
’
q(x, y ) = exp[ia ~ ( xy,) - a ~ ( ” ( xy,) ] =I iaV(x, y ) - aV(”(x,y ) ,
+
(80)
and the diffracted wave is given by
for a single crystal. The analysis of Section II,B,2 (Eqs. 39-44) may then be directly applied to the consideration of elastic scattering in the presence of absorption effects; the projected potential V(x, y ) is replaced by a complex potential V(X,y ) i W ( x , y). If the coherence of the elastic wave is to be preserved after inelastic electron scattering, then the inelastic process must be essentially a coherent process. For the preservation of diffraction contrast, the inelastic wave in the forward direction Yl(q,) should be coherent with the inelastic-elastic wave Y,(ql + 2ng). If it is assumed that plasmon excitation is essentially a coherent process, it is possible to calculate the diffracted intensities including the effects of inelastic electron scattering (81, 82). The inelastic wave Y l ( q l ) corresponds to the modification of Yo by the angular distribution for plasmon excitation, that is,
+
90
D. L. MISELL
and the diffracted wave Y,(q, + 2ng) corresponds to the modification of YMby a similar angular dependent term, that is,
Iq, 1 = + 8E2)1'2 represents the angular deflection due to plasmon excitation. The phase relationship between Yi(q,) [or Yi(ql 2ng)l and Yo (or YJ is not clear from Eqs. (82) and (83); it is unlikely that the inelastic wave is coherent with either the unscattered wave or the elastic wave (83,84).Furthermore, the coherence of electrons which have lost different amounts of energy E is not evident; it would seem that the coherence assumption is valid for a given q1 and for a discrete energy loss, that is, S,(E) = 6(E - E J . A detailed discussion on the coherence of the inelastic scattering is deferred until Section
+
II,G. It is noted that if inelastic interactions are explicitly included in the calculation of diffracted intensities (e.g. Eqs. 82 and 83), then the Fourier coefficients Vji)of the complex potential (which represent absorption) should include only those inelastic processes that lead to scattering outside the objective aperture; thus Vii)(and t y ) )are dependent on the objective aperture size (81).
F. Numerical Results for the Angular-Energy Distributions In order to exemplify the effect of inelastic-elastic scattering on the angularenergy characteristics of the transmitted electron beam, numerical results will be given. Of particular relevance will be the mean free path values for inelastic and elastic electron scattering (determining the relative probabilities of the two types of scattering processes), the angular distributions and the distribution of energy loss. The scattering within the objective aperture will be a relevant quantity in relation to the contribution of inelastic and elastic scattering respectively to the electron microscope image. All calculations have been made using the " incoherent " analysis for elastic and inelastic electron scattering (Section 11,E).In particular, the electron scattering of " amorphous" carbon are examined, since this provides an approximation to the scattering by organic, polymeric, and unstained biological materials (618). 1. Mean Free Path Values
In Table 111 are listed mean free path values for elastic and inelastic electron scattering in evaporated (" amorphous ")carbon, aluminum, and gold. The elastic mean free path AE is calculated from the free atom formulation,
91
DEFECTS IN ELECTRON-OPTICAL IMAGES
TABLE I11 MEANFREEPATHVALUE(A&
FOR
ELECTRON SCATTERING"
Carbon ( Z = 6). p = 2000 kg m - 3 for evaporated carbon. 10 20 40 60 80 100
167 323 612 870 1103 1314
96 171 299 407 500 583
61 112 20 1 277 344
404
124 223 383 541 677 775
71 132 236 334 419 488
142 263 475 662 830 985
54 100 186 263 332 395
100 I82 318 434 538 629
14 26 37 46 54
Aluminum (Z = 13) 10 20
87 163 305 436 553 659
40 60 80 100
48 90 158 215 265 308
32 58 104
144 179 210
Gold (Z = 79) 10 20
8 15 28 40 50 59
40 60 80 100 ~~
~
36 68 120 161 200 232 ~
7 12 23 32 40 48 ~~
~
7
~~~
The elastic mean free path A, is calculated from the free atom theory. The inelastic mean free path values are from the free atom theory A',') and a composite theory hi2); the corresponding total mean free path values are AV) and AY).
Eq. (17), in the first Born approximation. The inelastic mean free path values A!') and Ai2) correspond respectively to the free atom theory and the composite model (see Section 11,C). Since " amorphous" carbon is used as a model for polymeric and biological materials, the mean free path values for this material are also presented in graph form (Fig. 2). For carbon it is seen that AE 1.5-2.0 A, and it is expected that inelastic electron scattering will represent a major contribution to the transmitted intensity in biological specimens (see also Section II,F,4). It is noted (in the absence of complex relativistic effects) that the ratio &/Al increases with E,; thus the relative probabilities of inelastic and elastic scattering, [l - exp( - t/A,)] and [ 1 - exp( - t/&)]eXp( - t/Al) respectively, do not alter significantly in favor of the elastic component as E, increases. In the extension of the mean free N
D. L. MISELL
92
I
0
25
50
75
100
Eo IkeV)
FIG.2. The dependence of the mean free path (A A) on the incident electron energy (Eo) for electron scattering in “amorphous” carbon. SuffixesE, I, and T refer respectively to elastic, inelastic, and total (elastic+inelastic) scattering. Superscripts (1) and (2) refer to calculations from the free atom formulation and the “composite” model respectively.
path values to organic and polymeric materials, the A values for carbon are scaled by p J p 0 (the respective densities of carbon and the “ carbonaceous” material) ;alternatively the scattering by the material can be considered in terms of the mass thickness p t (18). Experimental values for A, in evaporated films are in moderate agreement with the values for Ai2)(85, 86) and significantly lower than the free atom values A!’). The total mean free path values, AT, are calculated from A;’ = A-’ E + A;’. In the case of Al, A, 0.6-2.0 A,; the lower figure is more realistic for A1 where the dominant mechanism for electron energy loss is plasmon excitation. The ratio of inelastic to elastic electron scattering, I/E, for scattering into all angles, is greater than unity for t/A, 2 0.4. For Au the elastic mean free path is significantly shorter than AI (AE 0.1-0.3 A,); it is noted that Ai2) for Au is an overestimate since one-electron excitations are expected to predominate over plasmon excitations (65). N
-
DEFECTS IN ELECTRON-OPTICAL IMAGES
93
2. Angular Distributions The angular distributions for elastic, R(0), and inelastic, S(0), electron scattering may be measured, in both the conventional transmission microscope and the scanning transmission microscope, by scanning the scattering (diffraction) pattern across a fixed aperture. The measured intensity distribution I(0) then represents a differential cross section, where the element of solid angle ASZ is defined by the solid angle subtended by the fixed aperture at the specimen; the number of electrons scattered into 0 is defined by 21c8Z(8) for a cylindrically symmetric distribution Z(8). For a comparison with theory, it is usual to normalize the experimental distribution to represent the fraction of the incident electron beam scattered per unit solid angle at a scattering angle 0, that is, Z(0)/(Zo An), where I , is the incident beam intensity, and the theoretical distributions presented in this section have been calculated per unit solid angle to be directly comparable with experiment. In order to separate experimentally the elastic and inelastic components of the transmitted electron beam, it is necessary to scan the diffraction pattern across a fixed aperture placed above an electron energy analyzer or an electron spectrometer. For an energy loss near E = 0, the elastic distribution together with the unscattered component, &(8, E ) + Du(O, E ) , will be obtained; the thermal energy distribution of these electrons extends from - E l to + E l , where El is a rather arbitrary dividing line between elastic and inelastic scattering. Inelastically scattered electrons with an energy Eo - Eare characterized by the distribution D,(B, E ) . The angular distributions, obtained by an integration of the D(8, E ) over E, are for the elastic (+ unscattered) electrons and for the inelastic scattering given respectively by + EI - EI
[&(e, E ) 4- Du(8, E)1dE
R(0) f exP(- l/A~)Iid0)
(84)
and
for a specimen of thickness t ; for energy loss values greater than Em,,, D,(O, E ) is negligibly small. The angular distributions R(8) and S(0) are calculated from Eqs. (68) and (72) respectively; the component distributions R,(@ and S,(@ for n scattering events are calculated by a repetitive convolution procedure using the single-scattering distributions for elastic, Rl(8), and inelastic, Q,(8), electron scattering (87). For Q , ( Q the composite model (Bohm-Pines/ Ferrell/free atom formulation-see Section II,C) is used and for R,(8) the free atom formulation is used (see Section 11,A). Figures 3a and 3b show the
94
D. L. MISELL
.-
c)
--. ------------- ---. : 1 -
_------
C
0
::10'
----
0
-->a
---.
lL
\
lo-'
(a)
0.00
0.01
0.02
0.03 8 (rad)
FIG.3(a) FIG.3. The angular distributions for electron scattering in carbon. 0 is the angle of scattering and the incident electron energy is (a) Eo = 20 keV, (b) Eo = 100 keV. A-inelastic component; B-elastic component; dashed line, t = 100 A; solid line, f = 500 A.
theoretical angular distributions for elastic (B) and inelastic (A) electron scattering in carbon films for t = 100 A ( p t = 20 mg m-') and t = 500 A ( p t = 100 mg m-'); this thickness range is representative of normal sections used in the electron microscope. The most dominant feature is the sharply peaked small-angle behavior of the inelastic distributions; because of multiple elastic-inelastic interactions, the angular half-width of S(0) increases with specimen thickness and decreasing incident beam energy (compare Figs. 3a and 3b). It is evident that within the normal objective aperture, CI = 0.0050.02 rad, the inelastic scattering exceeds the elastic scattering, except for the thinnest film ( t = 100 A, E,, = 100 keV-Fig. 3b). In particular, for 0 = 0.005 rad (Eo = 100 keV), the inelastic-elastic ratio, Z/E, varies from 2-12 for a film thickness variation 50-1000 A ( p t = 10-200 mg m-'); equality
95
DEFECTS IN ELECTRON-OPTICAL IMAGES
I0 '
10' c
'L
vl
0
5
10'
c U
ul
.-zi
$
2
lo'
LL
10'
10'
0.00
0.01
elradl
FIG.3(b)
-
between the inelastic and elastic scattering occurs at 0 0.008-0.1 rad for this thickness range. For Eo = 20 keV (Fig. 3a), the inelastic intensity exceeds the elastic intensity for t9 < 0.02 rad, except for specimens less than 50 A in thickness. The inclusion of elastic-inelastic interactions in the formulation for the inelastic scattering significantly increases the angular broadening of these curves as compared with the corresponding results for plural inelastic scattering only; hence elastic and inelastic electron scattering cannot be considered independently. It is noted that the unscattered component has been omitted from the graphs of Fig. 3 because of the dependence of I,,(@ on the electron optics of the specific microscope; Z0(Q may be determined from a secondary experiment with no specimen in place and added to the theoretical elastic curves after appropriate normalization. The angular distributions for electron scattering in evaporated (" amorphous") carbon films ( t = 50-1500 A, Eo = 30-80 keV) are in moderate
96
D. L. MISELL
agreement with the theoretical results based on the free atom formulation (56); the composite theory for the inelastic electron scattering does not give quite such good agreement, This latter point is surprising since the mean free path A, agrees more closely with this composite theory than with the free atom formulation. It is most likely that this discrepancy is a result of the incorrect formulation for the elastic scattering [which is involved not only in the calculation of R(8) but also of S(8)l; the free atom formulation for elastic scattering in “ amorphous” carbon can only be an approximation (85). The extension of the theoretical calculations for carbon to organic materials, in the absence of fine structure effects in R,(8) and Q,(8), would simply involve an adjustment of the mass thickness of the carbonaceous specimen to an equivalent mass thickness of carbon (p = 2000 kg m-3); for example, a 200 A film of formvar (p = 1230 kg m-’) would be taken as equivalent to 123 A of “ amorphous” carbon. Experimental investigations on the scattering properties of organic and polymeric specimens are notable by their absence; the only detailed experimental investigations of R(8) and S(8) are those for polystyrene (88) and a biological specimen (89), but a theoretical interpretation has not been attempted. Experimental measurements on the angular distributions for electron scattering in evaporated films of A1 and Au show only moderate agreement with the free atom theory (90); however, the agreement is not improved by using the composite model for the inelastic scattering. 3. Angular Energy Distributions
The energy distribution of the unscattered and elastically scattered electrons is determined by the energy distribution N ( E ) (e.g. the thermal energy spread) ; N ( E ) is therefore instrument dependent and will be omitted from this section. The inelastic distribution of energy loss D,(8, E ) is dependent on the scattering within the specimen, and the results for the distribution D,(8, E ) may be subject to experimental verification. The theoretical results for D,(O, E ) are based on the analysis of Section II,E, Eq. (74), using the composite model for the calculation of the Sn(8), and the experimental energy loss distribution, F(E), is used to evaluate the f , ( E ) [after a calculation off,(E) from Eq. (62) (68)]. Implicit in the theoretical expression for D,(8, E ) , Eq. (74), is the assumption that D,(& E ) is a separable function of 8 and E ; as pointed out in Section II,C, the relationship between E and 8 (the dispersion relation in E ( 8 , E ) ) is not readily determined. The neglect of dispersion means that the position of the peak, E,,, and the energy half-width, E , , 2 , of the energy loss distribution (for single scattering conditions) will be invariant with 8. In the case of materials which give rise to electron energy loss distributions with discrete loss lines (half-width 1-5 eV, e.g. Al, Si), the dispersion effect is
DEFECTS IN ELECTRON-OPTICAL IMAGES
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marked (63); however, for carbon-like spectra, where the half-width of the main peak (centered at 20-25 eV) is 15-20 eV (6,91), it is not significant (18). D,(O, E ) may be measured as a differential quantity by the procedure outlined in the preceeding section. In Figs. 4 and 5 results are given for 2n8DI (8, E ) for E , = 20 keV (Fig. 4) and for E, = 100 keV (Fig. 5) and for “amorphous” carbon films of thickness t = 100 8, (pt = 20 mg m-’) and t = 500 A ( p t = 100 mg m-’). These results for D,(8, E ) are presented in the form of the energy loss distribution at an angle 8 (integrated over all azimuthal angle 4 ) for the convenience of placing all curves on a common intensity scale. The first maximum at 23 eV corresponds to the first “ plasmon” loss in amorphous carbon; the intensity of this energy loss (corresponding to single scattering) decreases with increasing 0. The intensities of the subsidiary maxima (at n x 23 eV), relative to the intensity of the 23 eV energy loss, increase with the angle of scattering, reflecting the increased broadening of the angular distributions S,,(8) with increasing n values. For t = 500 8, at 8 = 0.0015 rad ( E , = 100 keV, Fig. 5b, curve A) the single loss process is predominant, but as 8 increases to 0.005 rad (Fig. 5b, curve C) the two“ plasmon” loss is evident and the overall scattered intensity has decreased relative to that intensity at smaller 8. As the specimen thickness increases or equivalently the incident electron energy decreases, the probability of plural electron scattering increases and multiple “ plasmon ” losses become prominent in the D,(8, E ) curves; this is seen in the sequence t = 100 8, (E, = 100 keV, Fig. 5a), t = 100 8, ( E , = 20 keV, Fig. 4a), t = 500 8, ( E , = 100 keV, Fig. 5b), and t = 500 A ( E , = 20 keV, Fig. 4b). In the absence of detailed experimental results for carbon films, it can only be stated that the DI(8, E ) curves presented are physically acceptable. Experimental studies on the variation of the electron energy loss distribution with 8 for organic materials are sparse; the results for the nucleic acid DNA (92) and a biological specimen (89) are consistent with the theoretical D,(B, E ) curves both in profile and in the relative intensities between curves for different 8. Evidence for the correctness of the formulation for D,(8, E ) is indirectly provided by measurements on the variation of the carbon electron energy loss spectrum with the size of the objective aperture. This aperture function [ G ( E ) ] , , which represents the actual distribution of energy loss that contributes to the electron microscope image, is defined by (18)
-
[ G ( E ) ] ,= 2n /‘D1(8, E)8 do. 0
For objective aperture sizes (semiangle a) in the range 0.002-0.025 rad, the [G(E)],curves obtained for “amorphous” carbon films in the thickness range t = 300-800 8, ( p t = 60-160 mg m-2, E, = 50 keV) were in moderate agreement with the theoretical [G(E)], curves. Figure 6 shows such a
A
0
20
40
60
80 100 Energy l o s s (eV)
0.2
I
w. Q,
I
n
0, 0.1 E N
FIG.4. The variation of the electron energy distribution D(0, E ) of carbon with the angle of scattering -8; E is the energy loss. A--8 = 0.0025 rad, B-6 = 0.005 rad, C--8=0.01 rad. (a) t = 100 A, (b) t = 500 A. The incident electron energy Eo = 20 keV. 98
A
2
A
Energy loss (cV1
FIG.5. The variation of the electron energy distribution D(0, E ) of carbon with the angle of scattering; E is the energy loss. A--8 = 0.0015 rad, B--8 = 0.0025 rad, C-O= 0.005 rad. (a) t = 100 A, (b) r = 500 A. The incident electron energy is Eo = 100 keV. 99
100
D. L. MISELL
comparison for t = 410 8, (pt = 82 mg m-’, Figure 6a) and for t = 590 8, (pt = 118 mg m-2, Fig. 6b). The effect on the electron energy loss distribution of limiting the angular acceptance of the image forming system can be clearly seen; there is, with decreasing a, an overall decrease in the number of electrons collected, but the decrease in the intensity of the [G(E)], curves at higher energies ( E 50 eV, corresponding to plural inelastic scattering) is more marked than that at lower energy loss values ( E 25 eV, corresponding to single inelastic scattering). Thus there is a systematic decrease in the intensity of the 48 eV maximum as a varies from 0.025 rad to 0.002 rad, where the distribution exhibits a prominent maximum at 24 eV only. An experimental problem in the measurement of D,(& E ) for organic and polymeric materials is electron beam damage in the specimen (88, 93-952) ; the scanning transmission electron microscope would appear to be an ideal instrument for these measurements on D,(& E ) (96). Measurements on the electron energy loss distributions of several polymers, notably, formvar, collodion, polyethylene, and polystyrene, have been made (97, 6, 98, 99), but further information on the angular variation of these spectra is not readily available. The electron energy loss spectra of several condensed aromatic hydrocarbons (91) show fine structure in the 1-10 eV region, although the probability of these low energy loss processes appears to be significantly lower than that of the 23 eV “ plasmon ” loss. For materials of lower carbon content, notably the nucleic acid bases, the similarity with the carbon spectrum is no longer evident (96).
-
-
4. Scattering Within the Objective Aperture
The fraction h(a) of each component of the transmitted electron beam that is scattered within the objective aperture, semiangle a, is not directly related to the image formed in the electron microscope. However, h(a) summarizes the results contained in the angular distributions for elastic and inelastic electron scattering; further h(a) gives an indication of the relative contributions of the inelastic, I Y,( ’, and elastic, I YE1 ’, scattering to the final image. h(a) is defined respectively for the elastic and inelastic scattering by
for cylindrically symmetric scattering functions. hu(a), the corresponding fraction for the unscattered component is invariant with a, provided that the
20
40
6 0
Energy 0 l0o s s IeVI
I00
FIG.6. The variation of the electron energy loss distribution [G(E)J,of carbon with the objective aperture size a ; E is the energy loss. The dashed curves are the experimental results and the full line curves are theoretical results: A--or large, F3-a = 0.014 rad, C-a = 0.005 rad. (a) t = 410 A, (b) t = 590 A. The incident electron energy is 50 keV.
(bl 100
200
300
400 t(A)
500
DEFECTS IN ELECTRON-OPTICAL IMAGES
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FIG.7. The variation of the ratio of the inelastic to elastic electron scattering, Q(-a)/ &(a), with the specimen thickness t for (a) carbon, (b) aluminum, (c) gold. A--a = 0.01 rad, E-a = 0.02 rad. The incident electron energy is Eo = 20 keV. Dashed line, “com-
posite” theory; solid line, free atom theory.
angular half-width of I,,(@ (due to finite source size) is less than 0.001 rad (the smallest practical value for a). Tables IV and V list h,(a), hl(a), and hu(a) for the three components of the electron beam after transmission through “ amorphous ” carbon films in the thickness range 50-500 A ( p t = 10-100 mg m-’). For E, = 20 keV (Table IV) the two objective apertures are chosen to correspond to normal values used in the conventional electron microscope. It is noted that for the thinnest film given, hl/h, is significantly greater than unity, and increases further to a value -20 for t -,500A. In the case of an incident electron energy of 100 keV (Table V), h,/hE is also significantly greater than unity. The results for the ratio h,/hEare presented in Figs. 7 and 8 for “ amorphous ” carbon, Al, and Au. The A1 and Au results are presented in order to demonstrate that, even for materials where A, > &, h,/hEcan be greater than unity.
L
0 P
TABLE IV FRACTION OF INCIDENT ELECTRON INTENSITY SCATTERED WITHINTHE OBJECTIVE APERTURE. CARBON, 20 keV" CL = 0.01
a = 0.02 rad
rad
U
tch
E 50 100 150 200 250 300 350
400 450 500
6.37, -3 (5.95, -3) 9.01, -3 (7.86, -3) 9.58, -3 (7.81, -3) 9.06, -3 (6.90, -3) 8.05, -3 (5.72, -3) 6.88, -3 (4.57, -3) 5.73, -3 (3.55, -3) 4.68, -3 (2.71, -3) 3.77, -3 (2.04, -3) 3.01, -3 (1.52, -3)
I 1.15, -1 1.73, -1 1.96, -1 1.98, -1 1.89, -1 1.75, -1 1.58, -1 1.42, -1 1.26, -1 1.11, -1
(1.43, -1) (2.07, -1) (2.25, -1) (2.20, -1) (2.03, -1) (1.83, -1) (1.61, -1) (1.41, -1) (1.23, -1) (1.08, -1)
E 2.22, -2(2.08, -2) 3.16, -2(2.75, -2) 3.36, -2(2.74, -2) 3.19, -2(2.43, -2) 2.84, -2 (2.02, -2) 2.44, -2(1.62, -2) 2.03, -2(1.26, -2) 1.67, -2(9.66, -3) 1.35, -2(7.30, -3) 1.08, -2(5.46, -3)
I 1.45, -1 2.27, -1 2.68, -1 2.83, - 1 2.83, -1 2.72, -1 2.57, -1 2.40, -1 2.22, -1 2.05, -1
(1.82, -1) (2.75, -1) (3.14, -1) (3.21, -1) (3.12, -1) (2.94, -1) (2.72, -1) (2.49, -1) (2.27, -1) (2.07, -1)
6.84, -1 (6.39, 4.69, --I (4.09, 3.21, -1 (2.61, 2.20, -1 (1.67, 1.50, -1 (1.07, 1.03, -1 (6.84, 7.04, -2(4.37, 4.82, -2(2.79, 3.30, -2(1.79, 2.26, -2(1.14,
-1) -1) -1) -1) -1) -2) -2) -2) -2) -2)
a Semiangle a rad. E-elastic scattering, I-inelastic scattering, U-unscattered component. Main figures are calculated from the free atom/Bohm-Pines theory and figures in parentheses are calculated from the free atom theory. The figure after the comma represents the exponent.
a
r
$ 8
r
TABLE V FRACTION OF INCIDENT ELECTRON INTENSITY SCATTERED WITHIN THE OBJECTIVE APERTURE. a: = 0.005 rad
U
E 50
100 keV"
a: = 0.01 rad
4Q
100 150 200 250 300 350 400 450 500
CARBON,
2.69, -3 (2.64, 4.90, -3(4.70, 6.69, -3 (6.27, 8.11, -3(7.45, 9.22, -3 (8.29, 1.01, -2(8.87, 1.07, -2(9.22, 1.11, -2(9.38, 1.14, -2(9.40, 1.15, -2(9.31,
I -3) -3) -3) -3) -3) -3) -3) -3) -3) -3)
4.57, -2 (5.99, -2) 8.49, -2(1.10, -1) 1.18, - 1 (1.51, -1) 1.46, - 1 (1.85, -1) 1.70, - 1 (2.12, -1) 1.90, - 1 (2.34, -1) 2.06, -1 (2.51, -1) 2.19, - 1 (2.64, -1) 2.30, -1 (2.73, -1) 2.38, - 1 (2.79, -1)
E 8.97, -3 (8.78, 1.63, -2(1.56, 2.23, -2(2.09, 2.71, -2(2.49, 3.08, -2(2.77, 3.37, -2(2.97, 3-58, -2(3.08, 3.72, -2(3.14, 3.82, -2(3.15, 3.86, -2(3.13,
I -3) -2) -2) -2) -2) -2) -2) -2) -2) -2)
5.32, -2 (6.99, -2) 9.99, -2(1.30, -1) 1.41, -1 (1.81, -1) 1.76, -1 (2.24, -1) 2.07, -1 (2.60, -1) 2.34, - 1 (2.90, -1) 2.57, -1 (3.15, -1) 2.76, -1 (3.36, -1) 2.92, -1 (3.52, -1) 3.06, - 1 (3.65, -1)
9.02, 8.14, 7.35, 6.63, 5.99, 5.40, 4.88,
-1 (8.84, -1) - 1 (7.81. -1)
-1 (6.90, - 1 (6.09, -1 (5.38, - 1 (4.76, -1 (4.20, 4.40, -1 (3.71, 3.97, -1 (3.28, 3.59, -1 (2.90,
-1) -1) -1) -1) -1) -1) -1) -1)
Semiangle u rad. E--elastic scattering, I-inelastic scattering, U-unscattered component. Main figures are calculated from the free atom/Bohm-Pines theory and figures in parentheses are calculated from the free atom theory. The figure after the comma represents the exponent,
106
D. L. MISELL
FIG.8. The variation of the ratio of the inelastic to elastic electron scattering, h,(a)/ hE(a),with the specimen thickness t for (a) carbon, (b) aluminum, (c) gold. A--a: = 0.005 rad, B-cz = 0.01 rad. The incident electron energy is Eo = 100 keV. Dashed line, “composite” theory; solid line, free atom theory.
Because of the short mean free path value for elastic electron scattering in Au, multiple scattering effects are a dominant feature of the h,/h, curves (Figs. 7c and 8c). Table VI shows a comparison of the theoretical calculations and experimental results for h,(a); the experimental results were based on the [G(E)], curves presented in the preceeding section for evaporated carbon films. The theoretical model that gives the overall best agreement with experiment is the free atom theory, although for the thicker films and the smaller a values, the composite theory gives quite good agreement with experiment.
G. Localization and Coherence of Electron Scattering Of particular relevance to image formation in the electron microscope (conventional and scanning types) will be the localization of the scattering process ;the possibility of resolving atoms or, at least, the molecular structure
:)
100
200
30C
L -
400 t
(AI
500
108
D. L. MISELL
TABLE VI A COMPARISON OF THE EXPERIMENTAL VALUESFOR hl(u)AND THE THEORETICAL CALCULATIONS FOR CARBON"
Film thickness
585 A
405 A
675 A
tcA, u(rad)
Exp.
Theory
Exp.
Theory
Exp.
Theory
0.0016 0.0023 0.0038 0.0051 0.0085 0.0135
0.102 0.124 0. Y 66 0.194 0.256 0.324
0.124 0.149 0.194 0.223 0.287 0.359
0.084 0.104 0.145 0.173 0.240 0.321
0.099 0.121 0.1 63 0.192 0.260 0.344
0.060 0.076 0.109 0.134
0.086 0.106 0.145 0.172 0.240 0.327
0.198 0.282
a The incident electron energy is 50 keV and the scattering model used i s the free atom theory. Al = 355 A, A, = 745 A.
will be determined by the distance scale over which the electron interacts in transmission through the specimen. The coherence of the scattering, particularly in relation to the elastic and unscattered components, will be an important factor in the determination of phase effects (e.g. phase contrast) in the final electron microscope image. 1. Elastic Electron Scattering
In a single crystal the phase relation between electrons scattered by the different atoms in the unit cell appears to be well defined. Further, the phase relationship between the unscattered wave and the elastic wave is also defined ; in the first Born (or kinematic) approximation the elastic wave is n/2 out of phase with the unscattered component (see Section 11,A). Beyond the first Born approximation the phase delay due to elastic interactions is 4 2 plus an additional term (e.g. q(0));the deviation from n/2 introduces a certain degree of incoherence between the unscattered component and an elastic wave whose phase varies with the angle of scattering 0. The localization of elastic electron scattering is, on an atomic resolution scale, 1-2 A; the probability for elastic electron scattering is significant for 141 >, 3 A-1 (lgl 2 0.5 A-') which corresponds to a distance, J r l , in the g1-l or (q/2nl - I ) . This degree of specimen less than about 2 A ( 1 rl localization is affected adversely by the thermal vibration of the atoms in the lattice, the effect of which can be partially taken into account by the inclusion of the Debye-Waller factor exp( -BI gl ')(S4). N
I
DEFECTS IN ELECTRON-OPTICAL IMAGES
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2. Inelastic Electron Scattering It is most unlikely that the inelastic wave exhibits coherence with the unscattered and elastic waves (83, 84). In order to discuss image formation by the inelastic electron scattering, it is necessary to deride on the phase relation between electrons which have lost differing amounts of energy E and have been scattered through different angles of scattering (corresponding to the wave vector qnfor n inelastic interactions). Because of the essentially incoherent nature of inelastic processes, electrons which have made a different number of inelastic interactions are considered to be incoherent (83). The remaining problem is then the coherence of electrons with a given n but with differing E and 9,. In the case of a single crystal specimen there is a significant amount of experimental evidence for the preservation of diffraction contrast by inelastic electron scattering (100).Recent experimental work on the preservation of contrast has involved the use of energy selecting and energy analyzing electron microscopes; in the former instrument an image is formed from a narrow selected energy loss band E i-1 eV, whilst in the latter type of microscope the energy loss distribution from a small area of the normal electron microscope image is analyzed. Thus, within a narrow energy band k 1 eV, there is evidence for the preservation of contrast by plasmon scattering (101-104), one electron excitations (84), phonon excitations (quasielastic scattering, weak preservation) (103, 105, 106), and x-ray absorption processes (104). The requirement for contrast preservation (in a two-beam dynamical theory) is that the inelastically scattered wave (wave vector qn) should be coherent with the diffracted (inelastic-elastic) wave, qn + 2ng (83, 22, 51); deviations from q. N 0 lead to a loss in diffraction contrast (84). Similar conclusions to those obtained from the more complex theories of inelastic electron scattering in a single crystal (107, 108, 83, 109, 110) for the preservation of diffraction contrast by plasmon excitation have been made from the assumption of pre- and post-plasmon excitation outside the crystal (111); this possibility arises from the long-range behavior of small-angle inelastic scattering, and plasmon excitation within the crystal is treated as an essentially incoherent process. This latter explanation on the preservation of diffraction contrast by plasmon excitation omits the screening effect as determined by the dielectric constant E ( q , E ) (see below) and pre- and postexcitation regions 500 A in extent would appear rather large; further this model would not explain the preservation of contrast by multiple plasmon scattering (102, 84) and by one-electron excitations (84, 104). The coherence conditions given above are established on the basis of a discrete energy loss process 6(E - Ep);there is no evidence for electrons which have lost energy E being coherent with electrons which have lost energy E k 1 eV. The fact that diffraction contrast due to the inelastic scattering is weaker than that produced
-
D. L. MISELL
110
by the elastic scattering may be due to this incoherent superposition of electrons with different E. Also it is not often stated whether the objective lens was refocused for the recording of these selected energy micrographs; the effect of the chromatic aberration on the inelastic image could cause a loss in resolution (equivalent to a decrease in contrast, see Section IV,B). Experimental evidence indicates that, for a single crystal specimen, electrons with a given n and E exhibit coherence, that is, the inelastic wave !PI@, E ) may be calculated from a superposition of intensities for different n (see Section II,E)
where @,@,
E ) = P,"2[Sn(~)f(E)I"2 exp[k(% El1
with
P,
= (z/AJn exp( - z/Ar)/n!.
The exp[ip,,(8, E ) ] expresses the phase incoherence between electrons with different nand E. As a simplifying assumption, electrons with a given n and E are taken to be coherent over all 6 (or q,,); thus the 8 dependence of pn(8, E ) is represented by an arbitrary constant independent of 8 (or 4,). Theoretical work (83, 84) indicates a loss in diffraction contrast by the variation of qn but conclusive experimental evidence on this point is difficult to obtain. In the case of a single crystal the distribution Sn(0)= 1 d,,(fl)I should be calculated by an n-beam dynamical procedure (81, 82). Contrast preservation observed in the " inelastic " images of amorphous specimens and polycrystalline specimens (101, 112, 11, 104) and notably for a carbonaceous specimen (112) and for a biological specimen (113) does not necessarily represent strong evidence for the phase coherence of the inelastic component. The contrast results for amorphous specimens may be explained on the basis of scattering contrast (that is, the variation of h,(cl) with pt-see Sections II,F and V); because of the small-angle nature of the inelastic electron scattering, image contrast is expected to be inferior to the contrast obtained for the elastic component. For amorphous specimens the inelastic distribution I 'PI(& E ) I may be calculated from Eq. (88) and the Sn(8)are calculated by a convolution procedure (see Section 11,E). Before information on 1 Y,(8, E ) is available in a complete form, the localization of the inelastic electron scattering must be considered. The parameter which describes the screening of the electrons in the solid from the incident electron is 1 / E ( q , E)(60, 61, 63). The screening is a measure of the localization of the inelastic scattering. One-electron excitations, including
I
DEFECTS IN ELECTRON-OPTICAL IMAGES
111
core excitations, and thermal diffuse scattering (phonon excitations, which contribute only a small fraction to the scattering within the normal objective aperture) are localized phenomena (84,51).The plasmon excitation in metallic specimens is a delocalized phenomenon over a radius of about 20A(114,115). It would be expected that the “ plasmon” excitation at - 2 3 eV in organic and biological materials would be of a more localized character; in these materials the concept of a significant free electron distribution (as in metallic specimens) would seem unrealistic. For the “ plasmon ” excitation in organic and biological materials, the localization can be considered from the behavior of l/&(O, E ) in the region of the “plasmon” loss E E p ; a small value for this reciprocal implies a large screening effect and thus a localized excitation. Near E = E , , Re [1/&(O, E ) ] has a value nearly zero and Im [1/c(O, E ) ] is a direct measure of the energy loss profile obtained under single scattering conditions at 0 N 0 (see Section I1,C). The energy loss profile for the nontransition metals can bedescribed approximately by a Lorentzian profile(Drude free electron model) (64)and a similar profile accurately describes the 20-25 eV “ plasmon ” loss in “ amorphous ” carbon, organic materials (e.g. polystyrene, formvar, collodion, condensed aromatic hydrocarbons, the metal phthalocyanines) (86, 88, 91, Is), and materials of biological significance (6, 92). Thus
-
- E2)’.r2 -Im[l/e(O, E ) ] = Ep27E/[(EP2
+ E2].
(89)
-
The parameter t is related to the energy half-width of the energy loss profile by T N 1 / E 1 , 2 .In the region of the plasmon loss ( E E,,), -Im[l/c(O, E ) ]N E , / E , / , , that is, the screening is related directly to the half-width of the energy loss profile for plasmon excitation. In the case of Al, E, = 15 eV, E l , , = 0.8 eV (63), and -Im[l/&(O, E ) ] N 18; for “amorphous” carbon and organic materials, E , = 20-25 eV, E,,2 = 15-20 eV giving -Im[l/e(O, E ) ] N 1.5; both results are consistent with optical data. This relationship between the screening and the energy half-width E , j 2 is not a unique result obtained by the use of Eq. (89) but similar relations may be derived for other realistic line shapes (e.g. Gaussian and Maxwellian profiles), provided that the sum rule (61) ~omIm[l/&(O, E ) ] E dE = -rcE,2/2
(90)
is satisfied. The implication of these calculations on I/E(O, E ) is that the “ plasmon” excitation in organic materials is a localized phenomenon (certainly significantly more localized than the plasmon interaction in metallic specimens) and I Y,(O,E ) I is characteristic of the specimen structure on a short-range scale (- 5 A). A calculation of the localization of plasmon excitation using an
112
D. L. MISELL
uncertaintity principle, A4 * Ar 1, implies that Ar is identical for both metallic and nonmetallic specimens ( I 15). Experimental evidence on this localization problem is not evident; the selected energy images appear to contain detail on a 50 8, scale rather than a 5 A scale, but this may be the result for other factors, e.g. chromatic aberration and incoherence effects. N
111. IMAGEFORMATION BY THE ELASTIC COMPONENT
The theory of image formation in light-optical systems has been extensively developed (116); any corresponding development of a theory of image formation in the electron microscope is based on the light-optical formulation. In particular, there are two approaches to the problem of calculating the image that results after light transmission through an object and lens system: (i) transfer theory describes the relationship between the image wave and the scattered wave immediately behind the object, and (ii) diffraction theory describes the relationship between the image wave and the diffracted wave. The equivalence of the transfer theory and diffraction theory of image formation is evident (117, 118). The principle difficulties in developing the lightoptical theory to the electron-optical system are related to differences in terminology, notation, and the choice of coordinate systems; once these minor problems are resolved, the development of a theory of image formation in the electron microscope is almost a formality (119).In the application of the transfer theory to the electron microscope (conventional and scanning types), the illumination conditions (e.g. spatial and chromatic coherence of the incident electron beam) and lens aberrations (e.g. spherical aberration and chromatic aberration of the objective lens) may be included in the calculation of the image intensity. Here the transfer theory will be developed from the initial conditions of coherent illumination (applied particularly to the phase and amplitude objects-Section III,A) to a consideration of partially coherent illumination (Sections III,B, III,C, 111,D). In this section the theory of image formation by the elastic component (together with the unscattered component) of the transmitted electron beam will be considered; the theory of image formation by inelastic electron scattering, which is taken to be incoherent with the elastic and unscattered components, is given in Section IV.
A. Coherent Illurnination 1. Transfer Theory
Optical transfer theory (117, 116) will be applied to image formation in the conventional transmission electron microscope (1, 3 ) ; in particular, the integral diffraction equation, frequently used in the calculation of electron microscope images, is derived in its usual form (14, IS, 120).
DEFECTS IN ELECTRON-OPTICAL IMAGES
113
The incident electron wave is represented by a plane wave exp(iK, r) of unit amplitude; KO is the wave vector that describes the direction of the incident electron. The incident electron is scattered by the specimen and the scattered wave t j 0 immediately after the object (specimen) is dependent on the conditions of specimen illumination and the electron scattering properties of the specimen. In this section it is assumed that the incident radiation is monochromatic (KO= constant) and coherent in the z direction (KOis a constant along the optic axis). If the object is such that all electrons incident on the object are transmitted (that is, negligible backscattering), then the current density immediately after the object is
where ro = ( x o , yo) are coordinates in the object plane. The inequality in Eq. (91) represents the attenuation of the elastic scattering by inelastic electron scattering in the specimen; the effect of the inelastic scattering may be treated as an absorption effect, although the role of inelastic electron scattering in image formation must also be considered (see Section 11,E). The electron microscope transmits information on $,,(ro) to an image plane; the image wave $i(ri) is defined by image coordinates r i = (xi,yi). It is assumed that noise, for example, mechanical vibrations of the electron microscope column, granularity of photographic emulsions, and noise in electronic recording systems, may be neglected; then there is a unique relationship between $o and ll/i. This relationship,which must be linear, may be expressed by the following: if $o --* $ i and ll/o’+ $i’, then linear transfer + Bt,bi’. The nonlinearity of the electron theory implies that all/, + /h+b0’ microscope is evident if the relationship between the mass thickness of the object and the optical density of the developed photographic plate is considered. can be represented by (1) The linear relationship between ij0 and
G(ro’, Ti) is the image wavefunction corresponding to a point source at ro in the object, that is, $o(ro) = 6(ro - ro’). G(ro, Ti) is determined by the electron-optical image forming system, that is, G depends on the aberrations of the objective lens, on the position and shape of apertures, and on the defocusing. The transfer theory is simplified if it is assumed that all object points ro have the same image disk around their geometrical images at ri = Mr, for magnification M , that is, G depends on (ri - Mr,) and not on ro or ri separately; the isoplanatic approximation corresponds to (116, 1)
114
D. L. MISELL
The isoplanatic approximation is not satisfied if the optical system exhibits aberrations depending on ro (e.g. third-order astigmatism, field curvature, coma, and distortion). Aberrations depending only on the initial direction of the electron trajectory (scattering angle), such as spherical aberration, chromatic aberration, defocusing, and axial astigmatism, do not violate the isoplanatic approximation. If the field of view is small (large electron-optical magnification M ) , then the isoplanatic approximation can be considered satisfied ; considerable complexity arises in transfer theory in the absence of the isoplanatic approximation (121). In the isoplanatic approximation, Eq. (92) becomes
which represents a convolution in two dimensions; the application of a Fourier transformation gives (122)
x exp k n i v
- ($- ro)] dr, dri,
(95)
where Si(v), So(v), and T(v) are the respective Fourier transforms of $i(ri), +o(ro), and C(r); for example,
j
-
~ , ( v= ) + ~ r ~ e x p ( 2 n i ri/M) v dri
(97)
The inverse transform of Si(v) is defined by
5
$i(ri) = Si(v)exp(- 2dv rJM) dv, where v ri = v, xi
+ v,,yi ,
(98)
dv = dv, dv,,.
T(v) is termed the amplitude transfer function (or the pupil function) and if T(v) is known, the relationship between $o and $i is uniquely defined. T(v) can be derived for an electron-optical system from the properties of the objective lens (see below). The current density in the image plane is ji(ri)
that is, ji(ri) =
1j
=
I $i(ri) I 2 5
(99)
1 9
S,(v)T(v)exp( - 2niv ri/M)
1
2,
(loo)
DEFECTS IN ELECTRON-OPTICAL IMAGES
115
which relates the measured intensity distribution j i to the electron scattering properties of the specimen ($to) and the electron-optical system (G).The Kirchhoff diffraction integral (see below), as applied to the scattered wave (So) in the back focal plane (Fourier plane) of the objective lens, corresponds to the transform of Sito $ti. The “Kirchhoff” integration is over the transparent part of the objective aperture and takes account of the phase shift introduced into Soby lens aberrations and defocusing. Each point in the Fourier plane of the objective lens corresponds to a Fourier coordinate (spatial frequency) v, and So(v)represents the diffracted (scattered) wave in the Fourier plane. v corresponds to a point rB in the back focal plane in the relationship rB = fAov.
(101)
f is the focal length of the objective lens. In polar coordinates 8 = (0, 8 = A,v
or
8 = K’/Ko.
4)
( 102)
Thus v defines a reciprocal distance in the object ( l/ro); for a crystalline specimen v is equivalent to the reciprocal lattice vector g (see Section 11,A). K ’ = KO - K is the difference between the wave vectors of the incident electron and the scattered electron. It is seen from Eq. (96) that So(v)is modified by T(v)to give the wavefunction s,(~); T(v)can be expressed in terms of the wave aberration function W(v)(representing the deviation from a spherical wave front for a perfect lens system) by (116, I , 2) N
T(v)= (I/M)exp[ - iKoW(v)]B(v),
(103)
which describes the phase shift KO W introduced by the objective lens and the effect of an apertue by the aperture function B(v).B(v)= 1 for the transparent parts of the aperture and B(v)= 0 for the opaque parts. Both W(v)and B(v) are usually referred back to the coordinate system r B = (xB,yB),rather than v, using the relation (101). If the lens is subject to spherical aberration, defocusing, and axial astigmatism, then (116, 2)
Af e2 - - e 2 cos 24. w(e)= e4 + 4 2 2 cs
cA
116
D. L. MISELL
C, is the third-order spherical aberration coefficient, C, is the coefficient of astigmatism, and Af is the change in focal length of the objective lens (Af < 0 corresponds to underfocusing and Af > 0 to overfocusing). In Eq. (103) the negative sign in the argument implies a phase advance with respect to the electron wave traveling along the optic axis of the objective lens. The term Gaussian image plane refers to an aberration free lens system with Af = 0. Since W(v)and B(v) are respectively known from Eq. (104) and the geometry of the objective aperature, T(v) can be determined. If information on I)i is available then information on I)ocan be derived using Eqs. (96), (97), and (103) and a calculation of the inverse transform S o . As is known information on I)i is limited to intensity quantities such as current density, ji, contrast, C, or optical density of the electron micrograph. These latter quantities are not linearly related t o any property of the object; only in the case of the weak phase or weak amplitude objects is the relation between C and t+b0 linear (see Section 111,A,2) (123, 3, 2). The diffraction theory of image formation is derived from the fundamental equations (96), (97), and (98) of the transfer theory. In its most common form, the diffraction integral (98) is transformed to polar coordinates (0, (b) in the aperture plane using
for the small-angle approximation (sin 0 = 0) which is valid in the conventional transmission electron microscope (0 -40.1 rad). Equation (98) becomes
where H(0) = (l/Wexp[-iKo
x ( W No)
(107)
is the 0 equivalent of Eq. (103) and x(0) = W(0) (Eq. 104). In bright field electron microscopy with the normal circular aperture D(e)=I
=o
0.s;Bia a<e
OIC$S~R, 0 5 (b 5 2n.
(108)
a is the semiangle subtended by the objective aperture at the specimen.
"(0) represents the diffracted wave in the Fourier plane of the objective lens; Y(0) includes the unscattered component [represented by S(0)]. If
DEFECTS IN ELECTRON-OPTICAL IMAGES
117
“(0) is separated into the unscattered component /? S(0) (/?’ represents the fraction of the incident beam that is unscattered) and an elastic wave YE@), then Eq. (106) becomes
x exp
[- iKo (xi0 cos 4 + y i B sin 4)10 d0 d 4 .
(109)
’.
The image intensity is calculated from I Il/&, yi) I From Parseval’s theorem, derived from the properties of the Fourier transform (122),it may be shown that
(1 10)
Equation (1 10) expresses the physical situation that if the background intensity p’/M2 is subtracted from the image intensity, then the number of electrons ( j i - p’/M’) is identical to the number of electrons scattered elastically within the objective aperture, kE(a)(see Section II,F) (124). Equation (109) has been used to calculate the optimum condition of defocusing for the resolution of a single atom or an array of single atoms (125l.?O), with /? set equal to unity (that is, neglecting the attenuation due t o inelastic electron scattering and the effect of elastic scattering in reducing /?). In the case of an object with circular symmetry, such as obtains for a single atom, Eq. (109) becomes (14), neglecting axial astigmatism ( C , = 0 in Eq. 104),
B
Il/i(ri)= -
M
KO = +YE(B)exp[- iKo x ( B ) ] J ~ ( ri K 0/M)O ~ do. M 10
(1 11)
It is noted that in the derivation of Eqs. (109) and (1 1 I), the incident electron wave is assumed to be coherent; explicitly neglected is the effect of chromatic aberration [due to the energy distribution N(E)]on Il/i. The chromatic aberration term that occurs in the wave aberration function is of the same order of magnitude as the spherical aberration and defocusing terms (see Section II1,C). Thus a resolution criterion based on a consideration of spherical aberration, defocusing, and the diffraction limit (specified in Eq. (1 1 I ) by the Bessel function J,, and the value for a) is not strictly valid. f(O)exp{i[q(O) + 11/21) In the case of scattering by a single atom ‘uE(6) (see Section II,A), where the phase term exp{i[q(0) 7c/2]}expresses the phase difference (delay) between the elastically scattered wave and the unscattered
+
N
D. L. MISELL
118
wave, p S(O). The image intensity, j i ( i i ) ,is (neglecting second-order terms) (131,124)
pz
2KoP ];j(B)sin[-q(6) +M2
j , ( r i )= 7
M
+ K o ~ ( 0 ) ] J oKO( Tri6 )6
do, (112)
and the contrast C(r,) is given by C(ri) =
j i ( r i )- background intensity
background intensity
ZZ]
MZ =pz[j , ( r J - -
9
that is, C(r,)= 2K0P-’
f(O)sin[-q(8) 0
+ Ko~(B)]Jo(Kori6/M)6d0. (113)
On axis with ri = 0, the maximum contrast is obtained for (131) (i) overfocus, Af > 0
Af = (,lO/az)- C, a2/2,
(ii) underfocus, Af < 0
Af = -C,u2/2.
c1’
< (2A0/Cs)”2 (1 14)
-
The overfocus condition (i) is established on the basis of the oscillatory nature of the sin[Kox ( B ) ] term in Eq. (112) [q(O) constant]; the general condition is that the sin[Ko x(O)] term should be nonzero for 0 < 0 < c1 and none of the spatial frequencies transmitted by the objective aperture (v,, = ./,lo is)absent from the image. The underfocus condition is based on a partial cancellation of the spherical aberration term in x ( O ) by the defocus term. Maximum contrast is obtained for aoptN (4/20/Cs)”4 and Afop, N - (1, Cs)’’z which give the maximum value for the integral in Eq. (1 13) (125,127).The modification to this criterion, when chromaticaberration is included in the diffraction integral, is considered in Section II1,C. For a single crystal specimen YE(0)= Y, and the angular integral of Eq. (109) becomes a summation over the diffracted waves g that are transmitted through the objective aperture; for a single diffracted beam g C(rJ = 2Ko P - ‘(t/t,)sin[Ko x(2qJl
(1 15)
in the kinematic approximation and for a crystal in the exact Bragg position (see Section 11,A). Evidently maximum contrast is obtained for sin[Ko x (20,)I = +_ 1 (0, is the Bragg angle), that is, Af is the solution of K0[2Af Og2 + 4CsBe4] = (2n - 1)42 (132). This latter condition refers to the resolution of lattice planes in the electron microscope; evidently the observation of lattice planes in the electron microscope is a test of the performance of the instrument
DEFECTS IN ELECTRON-OPTICAL IMAGES
119
rather than a general condition for the high resolution microscopy of crystal specimens (131). In the application of transfer theory to image formation in the scanning transmission electron microscope, the amplitude transfer function T(v) refers to the lens system preceeding the specimen (16, 17, 2). The wave aberration function W(v) refers to the defects of the condenser lens (also referred to as an objective lens) which produces the scan spot and B(v) is the appropriate aperture function. The wavefunction immediately before the object G(r, , I,’ is) defined for a scan spot position r,’ (which varies with time) by G(r, ,r,’) = jexp[ -iK, W(v)]B(v)exp[2niv * (ro - r,’)] dv.
(1 16)
The isoplanatic approximation in Eq. (1 16) corresponds to the assumption that the scan spot geometry is independent of ro‘ and the scan spot position is defined by (r, - r,’). The specimen modifies the incident wave G ; the image wavefunction $i(ri, r,’) N 1)~(0, r,’) for a narrow selecting aperture in the image plane (ri N 0) is given by +i(o,
ref> = J ~ ( r 0 ro’)+o(ro) dro 9
9
or
(1 17)
-
x exp(2niv r,)dv dr,
that is,
tji(0, r,’)
=
,
I
-
T(v)S,(v)exp( - 2niv r,’) dv.
(1 18)
S,(v) is the diffracted wave; the similarity between Eq. (98) derived for the conventional electron microscope and Eq. (1 18) for the scanning electron microscope is remarkable (16). A Fourier transformation of Eq. (118) gives analogous results to Eqs. (94) and (96) obtained for the conventional microscope, that is, $i(o, rot)
=
/ ~ ( r o- ro’)$o(ro) dro,
(1 19)
and Si(0,v) = T(V)S,(V). The transformation of Eq. (1 18) to polar coordinates (where the angular coordinates now refer to the Fourier plane of the condenser lens) gives an
D. L. MISELL
120
analogous result to Eq. ( I 1 1); the conditions for maximum contrast also appear to be identical to those derived for the conventional electron microscope, Eq. (1 14) (16). However, for the scanning transmission electron microscope, the field emission source seems to have a high degree of chromatic coherence (11) and chromatic aberration is not expected significantly to affect the maximum contrast conditions. 2. The Weak Phase and Weak Amplitude Objects The preceeding analysis is illustrated by application to the weak phase object and weak amplitude object (133, 123,3);in particular, a linear relationship between j i and the electron scattering properties of the object (e.g. phase shift) may be derived. In the transmission electron microscopy of thin specimens ( t = 100-500 A, E , = 100 keV), the probability of elastic scattering is small and the interaction between the incident electron beam and the object can be considered as a phase shift in the electronwave introduced by variations in the potential distribution of the specimen (phase grating approximationsee Section 11,A). The object wavefunction IClo(ro)may be written as
or $o(ro) = exp[irl(ro) - Y(ro)l,
where q(ro) is the phase shift term and a(ro) [or y(ro)-both real quantities] represents an attenuation of the coherent incident wave by inelastic electron scattering (absorption). If a(ro) I [or y(ro) 01 then the specimen is referred to as a phase object, and for q(ro) 4 271 (19) N
-
for a specimen of thickness t and CT = -2~me&/h’; Eq. (121) represents the phase grating approximation for q(ro) (see Section 11,A). The amplitude object [a(ro)< 1 and q(ro) 01 corresponds to either a thick specimen (backscattering) or a specimen where inelastic electron scattering predominates (strong absorption). For the weak amplitude object, the equivalence of the two forms of Eq. (120) is evident, that is,
-
or
DEFECTS IN ELECTRON-OPTICAL IMAGES
121
with &(To) E y(ro). &(To) may be related to the imaginary part of the complex potential (representing the effect of absorption) by (see Section I1,E)
Both the weak phase object and the weak amplitude object represent approximations to an electron microscope specimen. Initially the weak phase object is considered, that is, e(ro) 0 and q(ro) 6 27t, with (123)
-
$o(ro)
= 1 + irl(r0) + o[rlZ(rO)l,
( 124)
and the scattered wave, So(v), is given by
-
~ , ( v )= J $o(ro)exp(2niv ro>dr, = 6(v)
+ i Jq(ro)exp(2niv - ro> dro.
(125)
Equation (125) describes the angular distribution of the transmitted electron beam; the delta function 6(v) represents the unscattered beam in the 0 = 0 direction and the second term in Eq. (125) represents the complex scattering amplitude of the object. Using Eq. (121) for q(ro) in Eq. (125) gives So(v) = 6(v)
+ ia jJJV ( x o ,y o , z)exp(2dv
ro) dxo dy, dz.
(126)
The integral in Eq. (126) represents the three-dimensional Fourier transform of the potential distribution in the object; in the particular case of a single atom, the integral represents the electron scattering amplitude gB(v)for an atom (in the first Born approximation-see Section I1,A). Equation (125) may be written as So@)= 6(v) with
I
+ iA(v),
-
~ ( v= ) q(ro)exp(2niv ro>dr, .
( 128)
From Eq. (96) for the conventional transmission electron microscope, S,(V)= T(v) 6(v) + iA(v)T(v) and the equation for the image wavefunction $i(ri) is
+
$,(ri) = T(0) i IA(v)T(v)exp( -27th
r i / M )dv.
( 129)
122
D. L. MISELL
In the derivation of Eq. (1 29), the " shift " property of the Dirac delta function
J T(v) 6(v - v,)exp( - 27civ
r i / M )dv = T(vo)exp(- 2niv0 r , / M )
has been used with vo = 0. In bright field microscopy B(0) = 1, and from Eq. (103), T(0)= 1/M. Hence the first term on the right-hand side of Eq. (129) describes a uniform background with density 1/M2 of the incident intensity. The second term describes a modulation of this background. In dark field microscopy T(0)= 0, and Eq. (129) shows that the dark field contrast exceeds that of the bright field image for a weak phase object. The contrast C(ri) in the bright field image is then ( 2 ) C(ri>= [I $i(ri)
I
-
1/M21/1/M2
= i~ JA(v)[T(~)- ~*(-v)lexp(-2niv
(130) *
r i / M )dv,
neglecting second-order terms. If the aperture has cylindrical symmetry about the optic axis [T(v) = T( -v)]
C(r,) =
Q(v)d(v)exp(-27civ
*
ri/M)dv,
(131)
where Q(v) = 2 sin[K, W(v)]B(v).
(132)
Equations (130) and (131) define a linear relationship between C and the phase shift r(ro) (both real quantities); the identity of Eqs. (1 13) and (131) is evident for the elastic scattering by a single atom (with p set equal to unity). Q(v) is referred to as a (phase) contrast transfer function and its behavior is relevant to the discussion of the effect of lens aberrations on the electron microscope image. Figure 9 shows graphs of the v dependence of Q for E , = 20 keV (Fig. 9a) and Eo = 100 keV (Figs. 9b and 9c) for several defocus settings (A-A f = 0 A, B-A f = - 1000 A, C-Af = 1000 A); the wave aberration function is calculated from Eq. (104) with C, = 2 mm and C, = 0. The maximum spatial frequency, vmax, transmitted by the objective aperture is determined by the semi angle a(v,,, = ./A,); hence for 0 > a no structural information on spatial frequencies v > vmax is present in the electron micrograph. The spacings, Y, ,resolved in the specimen are limited to ro > 1/vmax and structural features corresponding to ro .c l/vmaxare not present in the electron micrograph; evidently the objective aperture size cannot be increased to resolve smaller spacing because of the adverse effect of lens aberrations (see below on the convolution effects of lens aberrations). The effect of lens
DEFECTS IN ELECTRON-OPTICAL IMAGES
123
aberrations and defocusing on the image may be explained on the basis of the functional behavior of Q(v),although the effect of lens defects may be more evident from the behavior of a convolution function (see below). It is seen that I Q(v)l 5 2 and since Q(v) multiplies A(v) in the integral (131), all spatial frequencies transmitted by the objective aperture are attenuated by a factor 1 Q(v)1. In particular, when Q(v) = 0 for v = vo (zeros of Q ) the micrograph contains no information on the spacing ro = I/vo. The contrast C depends essentially on an integral of A(v)Q(v)over all v c v,, (ti N 0), and the oscillatory behavior of Q(v) will determine the maximum contrast (cancellation of negative and positive contributions to the integral). Defocusing of the objective lens can be used to enhance spatial frequencies that would otherwise be present only as small contributions in the final image (e.g. Fig, 9c-v = 0.21 A-'); this apparent improvement in resolution of certain spacings may also lead to spurious features in the micrograph (referred to as " phase noise "). The discussion given above applies only to the idealized weak phase object and the neglect of the second-order terms (particularly near v = vo) in the derivation of Eq. (131) should be noted. Subject to this latter provision, the effect of lens aberrations and defocusing on the image can be described by the convolution integral
1
C ( ~ J= q(ro)q[(ri/M) - r01 dro
9
(133)
where q(r) is the inverse Fourier transform of Q(v) or a Fourier-Bessel transform for the normal circular aperture, that is,
s
-
q(r) = 2 sin[Ko W(v)]B(v)exp(- 2niv r) dv
lo sin[Ko W(v)]J0(2nvr)v dv. V m U
= 47t
Evidently the equation (1 33) may be inverted to give an expression .x the phase shift q(ro) ~ ( r o =z )
1C(ri)M[(ri/M)
- r01 dri >
(134)
where M ( r ) is referred to as a deconvolution function, that is, the convolution of the contrast function C with M is an effective correction of the electron micrograph for lens defects and leads to structural information on the specimen; the behavior of the potential distribution V(ro) in the specimen may be inferred from q(ro). However, this information on q(ro) is not only limited to those spatial frequencies which are not significantly attenuated by Q(v) but
D. L. MISELL
124
2
I
-0a I
c
-
2
I
-
-2 c
0
-I
-2
,
(b)
of the *‘ phase contrast’’ transfer function Q(v) with the spatial - 1000 A, C-Af= 1‘000 A . values : A-Aj’= 0 A, B-Af= rad; (b) Eo = 100 keV,a = 0.005 rad.
DEFECTS IN ELECTRON-OPTICAL IMAGES
125
FIG.9. (c) Eo = 100 keV,a = 0.01 rad.
also to those which do not have a significant noise component (e.g. due to emulsion granularity). Reconstruction (deconvolution) techniques based on Eq. ( 133) and variations thus yield only limited information on the specimen structure. Evidently a thorough focal series of electron micrographs will yield more information on those spatial frequencies which were absent in a single electron micrograph (134, 1 3 4 ~ )The . uniqueness of the solution of Eq. (134) (and variations) for q(r,) must be established in order to have confidencein the reconstruction technique. Figures 10 and 1 1 show respectively the convolution functions, q(r), and the appropriate deconvolution functions, M(r), corresponding to the phase contrast functions of Fig. 9; a detailed analysis on the evaluation of M(r) is given in the Appendix. The slow convergence of the q(r) to zero is particularly noted; the implication is that the effect of lens aberrations on the image extends to large r values. It is evident that underfocus of the objective lens improves this convergence (compare curves A-A f = 0 A and B-Af’ = - 1000 A); the reduction in the convolution effect ofq(r) for underfocus of the objective lens is consistent with the partial cancellation of the spherical aberration term by the defocus term in Eq. (104). Evidently from the curves given, a resolution criterion based on the half-width of the function q(r) is not meaningful.
126
D. L. MISELL
I
-0.02
-
I II
:IB I
I
/
-0.04/
,’ (a)
FIG.10. The radial dependence ( r ) of the “phase contrast” convolution function 0 Ir(, B-Af= - 1000 A, C-Af= 1000 A . (a) Eo = 20 keV,a = 0.01 rad; (b) Eo = 100 keV,a = 0.005 rad.
q ( r ) for defocus values: A-1J.f-
127
DEFECTS IN ELECTRON-OPTICAL IMAGES
I
-L
,I
I
-0.1
d
- ; 1
I I
I
:I3
;
-0.2.
I I I
I
-0.3
I
- I; I I
I
I
-0.4. ( c i
FIG.10. (c) Eo = 100 keV,cr = 0.01 rad.
It is noted that for the scanning microscope, an identical result to Eqs. (131) and (133) may be derived for the weak phase object (16, 2); Q(v) and q(r) then refer to the condenser lens system which preceeds the specimen. The contrast function C for an amplitude object [q(ro) 0, &(To) < 1 in Eq. (1 22)] may be derived (2) : N
f
C(rJ = - Q,(v)E(v)exp( -27th
-
- r,/M) dv,
(135)
Evidently the (amplitude) contrast transfer function Q,(v) has a value -2 at the smaller spatial frequencies where Q(v) 0. Figure 12 shows the graphs of the amplitude transfer function for the same lens parameters as those used to calculate the phase contrast transfer function Q(v). Corresponding to Eqs.
128
D. L. MISELL
FIG.1 1 . The radial dependence ( r ) of the “phase contrast” deconvolution function M(r) for defocus values: A-Af = 0 A, B-Af = -lo00 A, C-Af = 1000 A. (a) Eo = 20 keV,o! = 0.01 rad; (b) Eo = 100 keV,cr = 0.005 rad.
DEFECTS IN ELECTRON-OPTICAL IMAGES
129
FIG.11. (c) E o = 100 keV,u=O.OI rad.
(133) and (134) for the phase object, the equations for a weak amplitude object are
and &(To)
= -
J C(r,)M,[(r,/M) - rol dr,.
( 139)
q , ( r ) is the inverse Fourier transform (or Fourier-Bessel transform) of Q,(v). Graphs of the convolution functions, ql(r), and deconvolution functions, M,(r), corresponding to the Ql(v) (for bright field microscopy) are shown respectively in Figs. 13 and 14. The convergence of q,(r) to zero is slow; in general, this improves with underfocus of the objective lens, corresponding to a partial cancellation of the spherical aberration term by the defocus term in the expression for the wave aberration function, Eq. (104). In the case of the weak phase and weak amplitude object, Il/o(ro)
N-
1 - dro)
+h(bl)9
( 140)
and the contrast function is given by (2) C(r,) =
/ Q(v)A(v)exp(- 2niv - r J M ) dv / Q,(v)E(v)exp( -
- 2niv * r J M ) dv;
(141)
130
D. L. MISELL
el(.)with
FIG.12. The variation of the “amplitude contrast” transfer function spatial frequency Y for defocus values: A-Af = 0 A, B-Af = -1000 A, C-Af= (a) Eo = 20 keV,a = 0.01 rad; (b) Eo = 100 keV,cr = 0.005 rad.
loo0 A.
2
I
--a
-
r
DEFECTS IN ELECTRON-OPTICAL IMAGES
131
0
0
-I
-2
FIG.12. (c) EO= 100 keV,a = 0.01 rad.
the zero values of Q correspond to maxima in Q , . On the basis of Eq. (141) neither &(TO) nor q(ro) can be determined. If a semicircular objective aperture is used instead of the normal circular aperture, then two micrographs taken with complementary semicircular apertures contain sufficient information to calculate both &(To) and q(ro) (135,2); for this configuration, the contrast transfer function has an absolute value of lexp[iKo W ( v ) ] )= 1 and all spatial frequencies appear with unit weighting in the electron micrograph. Alternatively, two micrographs taken at different Afmay be used to determine both q and E (134u, 1357). The detail with which the weak phase and weak amplitude objects have been treated is not related to their practical importance. Of more relevance is the behavior of the transfer functions Q(v)and Q l ( v )and the corresponding resolution (convolution) functions q(r) and q , ( r ) ;these functions can be used in a general analysis of the effect of lens aberrations on the electron microscope image. The theoretical approach is then to calculate tl/i(ri) from the assumed object structure tl/o(ro)(and its electron scattering properties) from Eq. (94) or Eqs. (96) and (98) with
132
D. L. MISELL
FIG.13. The radial dependence ( r ) of the “amplitude contrast” convolution function A, C-Af= lo00 A. (a) Eo = 20
q l ( r ) for defocus values: A-Af= 0 A, B-Af= -1000 keV,a = 0.01 rad; (b) Eo = 100 keV,a = 0.005 rad.
DEFECTS IN ELECTRON-OPTICAL IMAGES
133
1( c ) FIG.13. (c) Eo = 100 keV,a = 0.01 rad.
and
= (271/2M)~ ' m a x [ Q , ( v ) iQ(v)]Jo(2nvu)v dv 0
for an aperture with cylindrical symmetry. The practical limitations of the linear contrast theory, as applied to the weak phase object or the weak amplitude object, are clearly related to the practical realization of such a specimen. This realization is particularly difficult for specimens containing low proton number atoms, notably organic, polymeric, and biological specimens, where inelastic electron scattering is a predominant factor (see Section 11,F). I f inelastic electron scattering is considered as an absorption process, then t,bo(ro) may be written i n terms of a(r,) and q(ro) (Eq. 120); if the inelastic scattering is significant a(r,) 6 1. Only if a(r,) 1 or a(r,) constant, invariant with r,, can linear contrast transfer theory be applied to image formation by the elastic component. a(r,) constant corresponds to the assumption that the inelastic scattering is not very dependent on the variations in the specimen structure; experimental evidence on " high resolution " electron microscopy using the inelastic
-
-
-
134
D. L. MISELL
FIG. 14. The radial dependence ( r ) of the “amplitude contrast” deconvolution function M , ( r ) for defocus values: A-Af = 0 A; B-Aj’= -1000 A, C-Af= lo00 A. (a) Eo = 20 keV,a = 0.01 rad; (b) Eo = 100 keV,a = 0.005 rad.
DEFECTS IN ELECTRON-OPTICAL IMAGES
135
FIG.14. (c) EO = 100 keV,cl:= 0.01 rad.
component of the transmitted electron beam would be of outstanding importance. In the current literature most electron micrographs of the “ inelastic” image (within an energy loss interval AE = f 1 eV) would seem to display image detail on a 50 A scale rather than a 5 A scale. The analysis of Section II,G indicates that the localization of the inelastic scattering in “carbonaceous ” specimens is 5 A but there is no experimental work available that supports or rejects this hypothesis. For completeness the case of a(ro) constant (a) is treated by linear contrast theory, that is,
-
-
+o(ro) = a eXP[i?(ro)l,
(14) with u2 1: exp( - t/AJ (see Section 11,E). For the weak phase object, the image intensity is calculated from Si(ri)= aT(v)6(v)
+ iaA(v)T(v),
and the contrast would appear to be identical to that given by Eq. (131); however, the background intensity is not a 2 / M 2 and the incoherent background intensity (1 - a 2 ) / M 2due to the inelastic scattering should be added to the a 2 / M 2term. Thus the contrast in Eq. (131) is reduced by a factor a2. Formally the analysis of the phase object can be extended to take account of the second-order terms, that is, +o(ro)
=4 + W o )
- tv2(ro)+ oIr13(ro)l>,
(145)
I36
D. L. MISELL
and the contrast in bright field microscopy is given by (neglecting third-order and higher order terms)
C(rJ = /A(v)Q(v)exp( - 2niv rJM) dv - 4 IA,(v)Q,(v)exp(-2niv
*
ri/M)u'v
IZ
- r i / M )dv A(v)Q,(v)exp( -2niv - rJM) A(v)Q(v)exp( -2niv
where A,(v) is the Fourier transform of q2(ro). Equation (146) may be rewritten as a sum of convolution integrals, that is,
Evidently neither Eq. (146) nor ( I 47) is readily inverted to give the phase term q(r,). Contrast that arises from the modification of the phase shift term q(r,) and the amplitude term &(To) may be referred to as phase or amplitude contrast; this contrast mechanism is a result of the introduction of certain phase relationships between the unscattered and elastically scattered electrons by, for example, underfocus of the objective lens. The terms " phase" and '' amplitude" should be reserved for the wave optical theory, and such terms are inappropriate to the description of scattering contrast (sometimes misleadingly referred to as amplitude contrast) which arises from the differential effect of the objective aperture on the electrons scattered from a mass thickness pr. Thus scattering contrast may be calculated from the variation of I S,(v) I * and h(a) (the fraction of the incident electron beam scattered within the objective aperture) with pr (see Section 11,F); in the literature scattering contrast is sometimes defined in terms of the scattering outside the objective aperture (34, 136). Since this type of contrast depends only on the intensity distribution in the Fourier plane, phase effects are irrelevant; thus it is not a contrast mechanism that one would expect to be included in the wave theory, where phase terms have a significant role. It would seem from the literature that phase/amplitude contrast and scattering contrast are to be treated as independent subjects (34, 136) giving the impression that electron microscope specimens are chameleon in character. It is evident that both contrast mechanisms are relevant to image formation in the electron microscope. At the higher incident electron energies ( E , 100 keV, r 200 A) phase effects can N
N
DEFECTS IN ELECTRON-OPTICAL IMAGES
137
be expected to dominate the differential behavior of h(a) (see Table V) whilst for the lower incident electron energies ( E , 5-20 keV, t 200-500 8 ) phase effects are minimal (e.g. due to multiple inelastic-elastic electron scattering) and scattering contrast is the main contrast mechanism (see Table IV). An analysis of image formation using the scattering contrast mechanism is deferred until Section V.
-
-
B. Spatially Incoherent Illumination
The incident electron wave is termed spatially incoherent if the wave vector KO varies, when the wave is then formed from electrons traveling in different directions defined by K O .The distribution of K O , F(Ko),is the angular distribution of the electrons incident on the specimen. KO may be considered as a wave vector with two components K, and K,, since ro = ( x o , yo) lies in the object plane. The scattered wave t,bo is dependent on ro and KO,that is, Go = I,ho(Ko, ro) and for monochromatic radiation it may be written as ( I ) G o W o ro) = I,ho(ro)exp(iKo* ro) 9
( 148)
for elastic electron scattering, provided that the thickness, t , of the specimen when multiplied by the angle between KOand the z axis is less than the smallest detail to be resolved in the specimen (lateral effect, see also Section V,F) (137). Initially a general distribution F(Ko) is considered (Section III,B, 1 ) and the transfer theory is applied to a calculation of the image contrast for a phase object. For a specific form of F(Ko), namely uniform illumination of the condenser lens aperture, an equation for the image intensity is derived (Section 111,B,2);the cases of spatial coherence and incoherence are considered.
I . General Distribution The image wave of Eq. (94) may be written, for a given KO,as
since G is not dependent on KO for monochromatic radiation ( I ) . The image intensities I t,bi(Ko, ri)l corresponding to different KO are superimposed incoherently, that is (138, 117),
138
D. L. MlSELL
or
x exp[iK,
*
(r, - r,')] dr, d r i dK,,
(151)
where F(K,) is normalized such that
1
+m
F(Ko) dK,
+m
F ( K , , K,,) d K , dK, = 1.
= -m
-m
The superscribed asterisk * indicates the complex conjugate of a function. The integration over KOin Eq. (151) may be performed by using the definition of the Fourier transform of &'(KO),that is, O(ro) = J F(K,)exp( - iKo * r,) dK, ,
(1 52)
where the function 4, (sometimes referred to as y ) defines the spatial coherence of the incident electron wave; 0 may be determined by electron interference microscopy for any particular gun geometry (139). A coherent source corresponds tor F(K,) = 6(K,), 0(r,) = 1 and complete spatial incoherence corresponds to F(K,) = constant, O(ro) = d(r,). The substitution of Eq. (1 52) for 0 in Eq. (151) gives
For a coherent source either Eq. (151) or Eq. (153) gives for the image intensity (138)
that is, the equation (94) for coherent illumination (KO= constant). In the case of incoherent illumination Eq. (1 53) becomes (138)
Evidently the integral (155) is an intensity convolution integral and I G(r) 1 is an intensity convolution function in the sense that G(r) is an amplitude without convolution function. The integral may be inverted to give I $o(ro)l any assumptions on the behavior of $,(r0) (e.g. the phase grating approxia exp[iq(r,)], j,(ri) is mation). It is noted that for a phase object, $Jr0) constant [C(ri)= 01 since 1 $o(ro) I = a'; hence the structural information carried by the phase term yl(r,) is irretrievably lost due to the incoherence of the electron source. Thus according to the wave optical formulation, strucN
139
DEFECTS IN ELECTRON-OPTICAL IMAGES
tural information will be transmitted to the image plane only if the amplitude (absorption, which may also include elastic scattering effects) term u varies with r,; in this case information on 1 a@,) I = a(ro)’ can be derived from Eq. (1 55). The case made for low voltage electron microscopy ( E , = 5-20 keV) relates to the absence of phase contrast effects in the low voltage electron microscope (140, 141) and the accentuation of absorption and scattering contrast effects. Phase contrast, that is, contrast arising from the phase relationships between electrons scattered from different object points r, , can lead to “spurious” structural features in the final image and to dificulties in the interpretation of “ phase ” contrast images (3). The essential incoherence of the electron beam may be assured by the use of an extended electron source (140). Thus the structural detail resolved in the low voltage microscope will be related to the dependence of u(ro)2and h(a) on the specimen structure; the resolution attained in low voltage microscopy is currently about 40 8, (142). It has been suggested that the use of a “filter” lens (energy selecting device) to remove the “ background ” inelastic scattering should improve image contrast (140, 142). In the case of incoherent illumination I G(r) I may be calculated from
I G(r) I ’= (1/4M2)[q1W2 + q W I ,
(156)
where q,(r) and q(r) may be calculated from the contrast transfer functions Q,(v) and Q(v) (see Eqs. 142 and 143). The Fourier transform of I C(r) 1 is given by a convolution integral, that is, g(v) = =
i I G(r) 1 exp(2niv r) dr 1T(v’)T*(v v’) dv’, -
(157)
which is termed the autocorrelation function of T (116). In the general case of partial coherence of the incident electron wave, the phase contrast will be inferior to that in the image for a coherent source a[l + ~(r,)] (e.g. Eq. 131) (117,3,143,144). For a weak phase object, $,(r0) the image intensity is derived from Eq. (1 53): N
ji(ri)= j ,
+ ia2 J’I q(ro)@(ro‘- r,)G
(L- ) (L 2
ro G* 2
rot) dr, dr,’
where j , = u2
Jj@(ro’ - ro)G(5M - r,)G*($
- r,’)
dr, dr,‘
(159)
140
D. L. MISELL
is the background current density; second-order terms in q(ro) have been neglected in Eq. (158). The contrast C(ri) is defined (in bright field microscopy) by = [ i d r i ) -jB]/?B
C(ri)
3
that is,
where
- @(ro- ror)G(;
- ro’)G*($
- ro)] dro’, (161)
and I-@,, ri) is real. Note that the addition of the incoherent background, due to inelastic electron scattering, reduces the contrast by a factor -a2 (see Section 111,A). For a coherent source, @(ro) = 1, the integral over ro’ in Eq. (161) may be evaluated to give
r ( r o , r i ) = i M [ G ( $ - ro) - G*($ =q ( 2
-ro)]
- ro),
since j , = a 2 / M 2and
1G(r) dr
=
jJ T(v)exp(- 2niv
= T(0).
*
r) dv dr ( 162)
-
Hence for coherent illumination the contrast is identical to C(ri)in Eq. (1 33). As expected, for incoherent illumination, @(ro) = h(r0), C(ri) 0 [T(r,, r i ) = 0, j , = constant, second-order terms are nonzero]. Thus the contrast decreases from a maximum for coherent illumination to near zero for the case of complete incoherence. It is noted that for the weak amplitude object, I - &(To), the contrast function C(r,) is not too adversely affected by the incoherence of the source. From Eq. (161) r is a function of [@,/&I) - ro] and not separately of ro and ri; thus Eq. (16 I ) is analogous to Eq. ( I 33) and the isoplanatic approxi-
-
DEFECTS IN ELECTRON-OPTICAL IMAGES
141
mation is not invalidated by spatial incoherence (116). Equation (161) may be rewritten as a convolution integral C(ri) = J” q(ro)r[(ri/M) - r01 dro *
( 1 63)
Since the function G(r) for coherent illumination is known from the inverse transform of T(v), T(r) may be calculated from Eq. (161) for any given distribution F(Ko) or its Fourier transform @(r). Both F(Ko) and @(r) are measurable functions, although information on F(Ko)would seem to be more accessible (see below). Using the definition, Eq. (97), for the Fourier transforms of I)i, I)o, and G, Eq. (151) may be rewritten in diffraction integral form, that is, (145)
- 2xi(v - v’)
-
]F(Ko)dv
Ti-
dv’ dKo , (1 64)
and the integration over KO represents the convolution of SOSO*with F. In the conventional transmission electron microscope F(Ko) represents the angular distribution of the incident electron beam after focusing (and collimation) by the condenser lens system. If the condenser aperture (semiangle a, subtended at the specimen) is small, then F(Ko)= &KO) and the integration over KO in Eq. (164) gives the coherent case, Eq. (100). Equation (164) is transformed to polar coordinates with
8 = A, v
and
8,
= ,lo K0/2n = Ko/Ko,
( 165)
where 8, defines the angular coordinate of the illumination. Hence Eq. (164) becomes
where d8 = 0 d0 d4, 8 * ri = x,O cos 4 $- y i O sin 4 in the small-angle approximation and H ( 8 ) is the amplitude transfer function, Eq. (107). l0(8,)[= &‘(KO)]represents the angular distribution of the incident electron beam; lO(Oc) may be measured using a small-angle diffraction technique (56) and for the normal double condenser lens system, the angular halfwidth of Io(ec)is significantly less than 0.001 rad.
142
D. L. MlSELL
2. Specijic Distribution In order to illustrate the analysis of the preceeding section, a specific form for F(Ko) is chosen, namely ZO(ec) = l/na,2 for a condenser lens aperture illuminated by an electron beam of uniform intensity. The integration over KOin Eq. (151)
W0’ - ro) = jF(K,)exp[
.
- iK, (r,’
- r,)]
dKo ,
represents the two-dimensional Fourier transform of F(K,). In polar coordinates and for the cylindrically symmetric Zo(Oc, 4) Eq. (167) becomes O(ro‘ - r,) =
% ~ ~ o ( X o O , / r-, ’rol) 8, do,, 7%
that is, @(ro’ - r,)
= 2J,(K0 a,
1 ro‘ - r,, /)/KO a, I ro’ - ro 1 .
(1 68)
J , is the Bessel function of order one. Two points ro and r,’ in the object are said to be coherently illuminated if KOa,( r,’ - ro 1 4 1 and @ 1 (138); incoherent illumination corresponds to a, a (objective semiangle) and @(ro‘ - r,) 4a@,’ - r,). Thus the preservation of the phase relationship between electrons scattered from different points depends on the semiangle of the cone of illumination. It is not evident that coherent illumination is preferable to incoherent illumination for the optimum “point to point” resolution; in fact the reverse is true ( I ) . In the case of incoherent illumination, the intensity convolution function 1 G(r) 1 for C, = 0, Af = 0, and in bright field microscopy, is calculated from
-
-
’,
that is, IG(r))’ = [v,,,J1(2nv,,,r)/r]’ ( M = 1) which is the intensity distribution due to Fraunhofer diffraction at the objective lens aperture. Thus for the incoherent case, two points in the object can be resolved (Rayleigh criterion) if the points are separated by 1 .22n/K0a. In the case of coherent illumination, the convolution to formji(ri)is that of tjo(ro)and the amplitude function G(r), and the “point to point” resolution is determined by the overlap of the Jl(Koar)/r, corresponding to a point separation of 1.64n/K0a. More generally, for a lens subject to spherical aberration and defocus, incoherent illumination gives an improved point resolution in the image as compared with coherent illumination. However, this does not mean that
143
DEFECTS IN ELECTRON-OPTICAL IMAGES
incoherent illumination is essential for “ high resolution electron microscopy; under conditions of spatial incoherence, the enhancement of image contrast by phase contrast will not be effective (see above analysis). Calculations for a phase ” object (e.g. single atoms, a thin crystal) demonstrate clearly that resolution is a secondary consideration to contrast (120, 132). This adverse effect of spatial incoherence on the image contrast for “biological specimens (139) demonstrates the subsidiary role of image resolution. In the case of coherent illumination, theoretical considerations indicate that, by the use of hollow cone illumination (annular condenser aperture), the point resolution criterion given above could be improved by a factor 2 (117,146).The use of hollow cone illumination with partial coherence appears to improve phase contrast effects, although this has been verified only in the light optical case (where all dimensions are scaled up by several orders of magnitude) (143). The transfer theory, given above for the conventional electron microscope, may be applied to the scanning transmission microscope, in which the condenser lens system is the equivalent of both the condenser and objective lenses of the conventional microscope. Thus the illumination angle a, is identical to the objective (condenser) aperture semiangle c1 but the point resolution (based on the Rayleigh criterion) is identical to that of the conventional microscope for a given coherence @ of the electron source (17). ”
“
”
-
C . Chromatically Incoherent Illumination
If the irradiating beam is not monochromatic or if energy losses have been produced by the interaction of beam and specimen, then the amplitude convolution function G depends on I K I = K, that is, G = G(K, ro , ri). If the incident electron beam has an energy distribution the illumination is principally incoherent. This distribution is represented by F(K,) or N(E), where E represents the energy spread about a most probable value E,; image formation by the energy loss electrons will be discussed in Section IV. &’(KO) and N ( E ) are normalized such that
j
+m
JbmF(Ko)d K , = 1
and
N(E) dE
= 1,
-m
where for a heated tungsten filament N ( E ) may be represented by an exponential distribution (see Section II,E, Eq. 66). F(K,) and N ( E ) are both known (or measurable) functions for a given electron gun configuration (69). Initially spatial coherence of the incident electron beam (KOconstant) will be assumed and the inclusion of the chromatic incoherence in the transfer theory will be considered for the case where electrons with different KO are incoherent (Section lII,C, 1); the possibility of coherence over a small interval
144
D. L. MISELL
in KO,AK,, will be included (Section 111,CJ). The chromatic aberration due to the energy distribution N ( E ) and the effect of chromatic aberration on the image intensity distribution j i will be discussed in Section III,C,2.
I . General Analysis In the case of elastic electron scattering $o may be considered independent of KO and the image wavefunction is given by ( I )
The image intensity j , is calculated from an incoherent superposition of monochromatic electron intensities, that is, m
ji(ri) =
joI+i(Ko
3
Ti)
I
d ~ o
or
Equation (171) may be rewritten in terms of the Fourier transforms of $o and G (Eq. 97) as (145) ji(ri) =
JmjJ S~(~)S,*(~’)T(K~,
V)T*(K,,
0
v’)
x exp[ - 2ni(v - v’) ri/M]F(Ko)dvdv’ dK, , (172)
or on replacing KO by E : j i ( r i )=
J+
m~~ i m
s,(~)s,*(~’)T(E, V ) T * ( E , v’) x exp[ - 2ni(v - v’) r , / M ] N ( E dv ) dv‘ dE. (173)
The amplitude transfer function T(E, v) now includes a term describing the effect of chromatic aberration, that is, for a lens subject to spherical aberration, chromatic aberration, and defocusing (J),
M with
(174)
W ( E ,V)
cs
= - I(E)‘v4
4
f + AA(E)’v’. + C , E1(E)’v2 2 2EO
145
DEFECTS IN ELECTRON-OPTICAL IMAGES
C, is the third-order chromatic aberration constant of the objective lens. Formally the incident electron wavelength 1 ( E )is dependent on E, that is, (34) A(E) =
12.26
( E , - E)1’2
[I
+ 0.978. IO-‘(E,
- E)]-”2A,
with E and E,, in eV. If E,, = 20 keV, then the thermal energy width, which is less than 2 eV, produces a deviation from A, of less than 5 parts in lo5. Since A(E) occurs only in the phase term of Eq. ( I 73), it may be replaced by A, without significant error. In Eq. (173) the integration over E is evaluated to give
ji(ri) =
-
S,,(V)S,*(V’)T(V)T*(~’)L(~, v’)exp[ - 2ni(v - v’) ri/M] dv dv‘,
where L(v, v’) is the Fourier transform of N(E), that is,
and T(v) is the transfer function including only spherical aberration and defocusing, Eq. (103) and (104). Consider now the application of Eq. (175) to a phase object, $,,(r,,) a [ I + ~(r,)];the image intensityji is then (in bright field microscopy): N
A(v)L(v, O)T(v)exp
-
2rciv ri
where second-order terms in q have been neglected. The contrast is then given by
C(rJ
=
IA(v)U,(v)exp(-2niv * ri/M)dv
+
I
A(v)U,(v)exp( -2niv
*
rJM) dv, (178)
where Ll,(v) = 2 Re[L(v, O)]sin[K, W(v)]B(v)
and U2(v) = 2 Im[L(v, O)]cos[K,, W(v)]B(v).
146
D. L. MISELL
Re and lm denote respectively the real and imaginary parts of L(v, 0 ) ; if N ( E ) is symmetrical about E = 0 then Im[L(v, O)] = 0. Thus the equation for C(r,) may be written as a convolution integral:
where for the normal circular aperture (with cylindrical symmetry) u(r) = ul(r)
+ u2(r)= 471
/Irnax
Re[L(v, O)]sin[Ko W(v)]J,(271vr)vdv
+ 471
J:max
Im[L(v, O)]cos[K, W(v)]JO(2x~r)v dv, (181)
and the isoplanatic condition is valid for chromatically incoherent illumination. Equation (180) may therefore be inverted to give an expression for q(r,) for the given u(r). Equation ( I 75) may be transformed to polar coordinates using Eq. (102), that is, (245) ji(ri) = ( ~ , / 2 . ) 2
1+
"J'Swe)y*(ew(E,e)H*(E,
el)
-
-m
x exp[ - iKo(O - 0') r i / M ] N ( E )d0 d0' dE, (1 82)
where
W E , 0) = (l/M)exp[-iK,y(E,
e)i w)
with D(0) defined by Eq. (108) in bright field microscopy and from Eq. (174):
? ( E ,0)
=
C, O4
C , E02
AfO
++2EO 2 . 4
The O equivalent of L(v, v'), Eq. (176), is
and
for an object with circular symmetry.
147
DEFECTS IN ELECTRON-OPTICAL IMAGES
2. Effect of Chroniatic Aberration Equations (174) and (183) indicate that the phase shift, introduced in the scattered electron wave by the chromatic aberration defect of the objective lens, is comparable with those introduced by spherical aberration and defocus. The phase shifts introduced by chromatic aberration (C.A.), spherical aberration (S.A.), and defocus (def.) are respectively - K O C, E02/2E0, -KO CsH4/4, and -K0AfO2/2. If the objective aperture semiangle u is taken to have the value 0.01 rad for Eo = 20 keV and Eo = 100 keV, then the following phase shifts may be calculated for O = a (C, = C, = 2 mm, Af = - l O O O & E I l 2= 1 eV): 20 keV C.A. -3.66 rad, S.A. -3.66 rad, def. +3.66 rad; 100 keV C.A. - 1.68 rad, S.A. -8.48 rad, def. 8.48 rad.
+
For 6, < CI the chromatic defect term becomes dominant because of the O2 dependence in contrast to the d4 dependence of the spherical aberration term. Since the angular dependence of the defocus and chromatic aberration terms are identical, it is possible that underfocus of the objective lens, in an attempt to cancel partially the effect of spherical aberration, may effectively cancel the chromatic defect of the lens. In order to assess quantitatively the effect of chromatic aberration on the image, the behavior of the transfer function U(v) = U,(v) + U2(v), Eq. (179), is considered ; in particular, its functional behavior is examined for a symmetric N ( E ) , namely a Gaussian distribution ( b / ~ ) exp( " ~ -bE2) (an approximation to Eq. 66 for p large) and the energy half-width E I l 2= 2(ln 2/b)'". Hence (147) U(v) = 2 sin[Ko W(v)]B(v)exp The behavior of the transfer function U(v) is shown graphically in Fig. 15 for Eo = 20 keV (Fig. 15a) and Eo = 100 keV (Fig. 1%) (C, = C, = 2 mm) and for several defocus values (A-Af = 0 A, B-Af = - 1000 I$, CAf = 1000 A). It is seen that the effect of chromatic aberration is to attenuate the amplitudes of the higher spatial frequencies v (= 6,/A0). Because of the strong functional dependence of U(v) on v (as v4 in the exponential term), the effect of the chromatic aberration is marked for O >, 0.01 rad and E,,2 1 eV. Evidently for the smaller angles of scattering and E I l 2< 1 eV, the effect of chromatic aberration on U(v) is minimal. Thus, although the above analysis applies to both the conventional transmission and scanning transmission electron microscopes, it is evident that the effect of chromatic incoherence on the scanning microscope image is negligible (E,,2 0.1 eV) except for large angles of scattering (6, 2 0.03 rad). It is also noted that the
-
-
148
D. L. MISELL
FIG.15. The variation of the “phase contrast” transfer function U ( v ) with the spatial frequency v when chromatic incoherence is included (energy distribution half-width E112). The defocus values are: A-Af = 0 A, B-Af= - 1000 A, C-Af= lo00 A. (a) Eo = 20 = 2 eV. keV, CL = 0.01 rad, E , , * = 1 eV; (b) E , = 100 keV, a = 0.01 rad,
DEFECTS IN ELECTRON-OPTICAL IMAGES
149
-
chromatic defect mainly affects the small-scale structure in the image (r, I/v) and it is expected that chromatic aberration will only lead to a loss in resolution on a 2-4 A scale (see below). As with spatial incoherence, the main effect of chromatic incoherence appears as a loss in image contrast (147). Corresponding to the maximum contrast criterion, Eq. (1 14), the revised clop, and Af,,, are for ri = 0:
Afopt N -
[L+ (-)* KO c c 20cs
Eo
cs
k-
- 112
.
The previous estimates for tlOpt(0.01 15 rad, Eo = 20 keV; 0.0093 rad, E, = 100 keV) based only on the spherical aberration defect (C, = 2 mm) should be decreased to -0.0067 rad ( E , = 2 0 keV) and -0.0081 rad ( E , = 100 keV); b has been taken as 2.77 eV-‘ ( E l , 2 = 1 eV) and C, = 2 mm. As expected the revised clopt for the lower incident energy (20 keV) is modified substantially by the inclusion of chromatic aberration; note that these estimates for tlopt and Af,,, are only approximations in the absence of derived analytic results. The effect of chromatic aberration on the image may be more evident from the radial dependence of the convolution function u(r); Fig. 16 shows the behavior of u(r) corresponding to U(v)in Fig. 15. In Fig. 17 a comparison of u(r) ( E , , 2 = 2 eV) and q(r) (chromatic coherence, E l j 2 = 0 eV, see Eq. 133) is shown. It is noted that u(r) converges to zero more rapidly than q ( r ) ; it is not obvious that chromatic incoherence leads to a deterioration in image resolution. Because of the oscillatory behavior of u(r) and q(r) it is difficult to assess the exact convolution effects produced by these functions. If it is noted that
from the functional behavior of Q(v) and U(v)respectively, then it is evident that the convolution effect of u(r) mainly affects the image resolution in the 2-5 A region (verified by numerical calculations). It is noted that underfocus of the objective lens (Fig. 17b) is effective in a partial cancellation of the chromatic aberration defect. The deconvolution functions M ( r ) , corresponding to the convolution functions u(r) in Fig. 16, are shown in Fig. 18 (see also Appendix). It is noted that in the analysis of chromatic incoherence, the E integration has been evaluated before the integrations over v and v’ (or 0 and 0’). Formally the integrations over v and v’ should be evaluated before the integration
D. L. MISELL
150
0.04
t-
-0.02
I
-
I
I
I I
I
I 9
.
/
/
- 0.04 -
I
I I
1
-
-0.1
;
*
I
I
L
I
I
s
:e
-0.2
-; I
I I II
-0.3
-
FIG.16. The radial dependence ( u ) of the “ phase contrast” convolution function u ( r ) including the effect of chromatic incoherence. The defocus values are: A-Af = 0 A, R-lf- IWOA, C-AfIOWA. (a) .Eo=20 keV, n=0.01 rad, E l i 2 = 1 eV; (b)
DEFECTS IN ELECTRON-OPTICAL IMAGES
151
0.10
0.00
FIG.17. A comparison of the radial dependence ( r ) of the “phase contrast” convolution functions q(r) (chromatic coherence, E l i z = 0 eV) and u(r) (chromatic incoherence, E,,,=2eV).(a) Af=OA,(b) A f = - l ~ A . E o = l O O k e V , a = O . O 1 r a d .
D. L. MISELL
152
I
I
L I
I
I
I
-0.1
I I I
. Irg I I I I I I I
-0.2
- (b)
FIG. 18. The radial dependence ( r ) of the “phase contrast” deconvolution function M(r) including the effect of chromatic incoherence (energy distribution half-width E1,2). The defocus values are: A-Af= 0 A, B-Af= -lo00 A, C-Af= loo0 A. (a) Eo = 20 keV, cc = 0.01 rad, E,,2= 1 eV; (b) Eo = 100 keV, o! = 0.01 rad, E,,Q = 2 eV.
153
DEFECTS IN ELEC,TRON-OPTICAL IMAGES
over E ; electrons with different E are then superimposed incoherently, that is, from Eq. (173) ji(ri) =
lyr 1 /So(v)T(E, v)exp( -2niv
*
12
rJM) d v N ( E ) dE.
(188)
Evidently for chromatic coherence, N ( E ) = 6 ( E ) , Eqs. (188) and (175) [L(v,v’) = I ] both lead to the coherent case, Eq. (100). However, for the other extreme of complete chromatic incoherence, N ( E ) constant, Eq. (1 75) becomes [with L(v, v’) = 6(v - v’)]
-
ji(ri)
=
11s o ( v ) ~ V1) dv ( 189)
for an object with circular symmetry. Equation (189) expresses the fact that the image is formed by a scattering contrast mechanism and that phase contrast effects are absent; the absence of any lens aberration terms in Eq. ( I 89) reflects some doubt on the general validity of Eq. ( I 89), derived under the assumption that the order of the v and v’, and E integrations may be reversed.
3. Partial Chromatic Coherence Although it is improbable that the incident electron beam will exhibit phase coherence over the enrgy distribution N ( E ) , it is possible that coherence exists within a small energy interval A E between E and E + AE. The analysis of Section III,C,I may be modified to include this coherence. In Eq. (171) the component wavefunctions $ i ( K o , ri) in the image plane have been multiplied by $i*(Ko, ri) before performing the integration over KO or E. If electrons within the energy interval ( E , E + A E ) exhibit coherence, then the tji(Ko, ri) are first summed over A E before multiplication by the complex conjugate. Provided that A E is small, Eq. (171) may be rewritten as
x N ’ ( p A E ) dr, dro’, (190)
where ( p t I)AE
N ( E ) dE,
N’(p AE) = P AE
154
D. L. MISELL
and G ( p A E , r) is approximately constant within a given A E (r constant). Alternatively Eq. (190) may be rewritten in terms of the Fourier transforms of t+bo and G:
-
x exp[-2rci(v - v‘) r i / M ] N ’ ( pA E ) dv dv’.
(191)
It is noted that for A E + 0, Eq. (191) reduces to the chromatic incoherence equation (1 7 3 ) . If AE becomes large, implying complete chromatic coherence, N’(E) N 1 and the transfer function T contains an energy term J?, which reflects the average effect of chromatic aberration on the image intensity j i . Equation ( I 9 I ) then becomes ji(ri) =
IJ
S,(v)T(E, v)exp( - 2riv * rJM) dv
Ij.
( 192)
In addition to the chromatic aberration effect due to the energy distribution N ( E ) of the incident electron beam, the effect of fluctuations in the accelerating voltage, AEo(t), and in the objective lens current, A l ( t ) , on the image intensity can be considered (147). In contrast to the analysis on the distribution N ( E ) , these time dependent variations can give rise only to an incoherent superposition of image intensities; the incoherence arises because the detecting system records, at a given instant of time t , intensity. Hence the image intensity j i is calculated from a superposition of image intensities over the period 1, of recording, that is,
The effect of the distribution N ( E ) on ji has been omitted for simplification. As in Section III,C, I , I)and , G may be replaced by their respective Fourier transforms, Eq. (97), to give ji(ri) =
JIl
]So(v)TIAEo(f),Al(z), v]exp(-2niv
*
r J M ) dv
Ij
dt.
(194)
The transfer function T is given by
T[AE,(t), AI(r), v]
= (I
/M)exp{ - iKo W’[AE,(t), AZ(t), v]}B(v) ( I 95)
with
cs
W’[AEo(t),A l ( f ) ,V ] = - AO4v4 4
Af cc +i O 2 v 2 + - AO2v2 2 2
--
155
DEFECTS IN ELECTRON-OPTICAL IMAGES
A I ( t ) > 0 corresponds to an increase in the objective lens current, whilst A E o ( t )> 0 represents a decrease in the accelerating voltage, which conforms with the definition of E given at the beginning of Section III,C. These variations in E, and l c a n be considered as effective changes in the focal length of the objective lens, that is, (3)
Af'
=C
, [ F -
I
D. Spatial and Chromatic Incoherence
The practical case in which the incident electron beam has both an angular and energy distribution is considered; the analysis is a formal extension of the content of Sections II1,B and III,C. The angular-energy distribution is defined by F(Ko, K O ) ;it is assumed that electrons with different KO or KO are incoherent. The image intensity ji(ri) is then calculated from the incoherent superposition of these electrons, that is,
F(Ko, KO)is normalized such that F(Ko, KO)dKo dKo lom
= 1.
For the incident electron beam F(Ko, KO) is a separable function of KO and K O ,that is, F(Ko)N(E). In terms of the Fourier transforms of i 0and G, Eq. ( 1 96) may be rewritten as
ji(ri) =
1I 1
SO(v -
2)
T(E, v)exp( - 27th * ri/A4) dv
Ij
x F(K,)N(E) dK, dE (197)
or, in polar coordinates,
I
2
iKoO * rJM) d0
x lo(O,)N(E) do, d E .
(198)
Formally the integrations over v (or 0) should beevaluated before performing the integrations over KO(or 0,) and E.
156
D. L. MISELL
In the application of transfer theory to image formation in the electron microscope, the Fourier transform, Eq. (97), was defined with v r = v,x vvy. This corresponds to the Fraunhofer diffraction integral for the object wavefunction t),(r0), with obliquity factors such as
-
+
exp( - iK, rO2/2z) and exp( - KOr o 2 / 2 f )
+
omitted; z is the distance between the electron source and object andf = f o A$ The inclusion of such obliquity factors corresponds to a Fresnel diffraction integral (116) and the relatively simple Fourier transform analysis presented in Section I11 is not strictly valid. It seems that there is only one particular type of object where Fresnel diffraction is an important feature of the electron microscope image, namely a specimen which exhibits a sharp discontinuity (e.g. an edge or a hole), where Fresnel fringes are a dominant feature (148, 149).The obliquity factors are then important since $,(ro) varies rapidly for a small change in ro . The Fraunhofer-Fresnel diffraction integral may be used to calculate the amplitude distribution due to an edge structure (150); the Fresnel diffraction contribution may be calculated from the integral (150)
j J/,(ro)exp(iK, Afro2/2f02>dro
( 199)
neglecting terms such as exp( - iKo rO2/2z)and exp( - iK, rO2/2f,) withf, % A.f. The change in the Fresnel fringe system as the objective lens current is adjusted from underfocus (Af < 0) to overfocus (Af> 0) may be explained on the basis of the integral (l99), and for Af = 0 the image is then the “true edge (148) is approxistructure. The Fresnel fringe spacing d N h.l(A,(Af mately I3 ( M = 1) for Af = & 5000 A ; this defocusing is about a factor of 5 larger than the Afvalues used to improve image contrast by phase effects (see Section III,A) (125, 127). The inclusion of the effects of lens aberrations, lens apertures, and the source incoherence in the Fresnel transform does not seem to have been examined in as much detail as the corresponding FourierFraunhofer transform. It is noted that Fraunhofer diffraction at an edge structure also gives rise to fringes in the image even for Af = 0 (118, 1 5 / ) , although these fringes are not as dominant as the Fresnel fringe system (150). In practice the specimen in question is not opaque but typically a thin carbon film; electron transmission through the film complicates the fringe system (150). The phase shift introduced in the transmitted electron beam is ascribed to a refractive index property ofthe film; the phase shift 6 introduced into both the unscattered and elastic components is given by (34) ”
a
6 = Ko(p - I ) c% K O (V0(1/2E0 where V , is referred to as the mean inner potential; V , defines a local average effect of the potential distribution V(r) on the incident electron beam. In
157
DEFECTS IN ELECTRON-OPTICAL IMAGES
the case of a crystalline specimen V , corresponds to the zero-order Fourier coefficient of V(r) (g = 0); more generally
The combination of the edge wave and the transmitted wave gives rise to a complex fringe system (128). This refractive index property has led to the suggestion of an electronoptical analog of the light microscopy phase plate for the enhancement of image contrast (128, 152). For an “amorphous” film of carbon ( Vo ‘u -7.5 eV) a phase shift of 4 2 may be introduced by using a film of thickness t = 220 60 A ( E , = 100 keV and depending on the experimental measurement or the theoretical model chosen). The phase plate is placed so as to intercept the zero-order beam ( v 0 , 0 0) and the scattered electron wave is unmodified. Thus in application to the phase object (Section III,A), $,(ro) u[l + ~ ( r , ) ] , the unscattered component is defined by u d(v)exp(irr/2), that is, from Eq. ( 129)
-
-
$;(ri) = uT(O)exp(irr/2)
-
+ iu S A(v)T(v)exp(-2rriv
- r i / M )dv,
(200)
and the image contrast is given by
where Q,(v) = 2 cos[K, W(v)]B(v). Evidently the image contrast is increased over the corresponding contrast in the absence of as phase plate, Eq. ( I3 I), provided that the same optimum tl and Af‘are used (Eq. 114) (128). Alternatively the phase plate may be used to introduce an additional rr/2 phase shift in the scattered wave and the zero-order beam is allowed to pass through a hole in the phase plate. In practice the design of such a phase plate is not easy, particularly “for amorphous carbon where the incoherent scattering will contribute to a loss in image resolution. The emphasis is on the improvement of image contrast rather than on image resolution, which may deteriorate as a result of inelastic scattering ( - 35 % inelastic, 54 7; unscattered in a carbon film of thickness 250 A) in the phase plate. I t is concluded that, provided $,(r0) does not exhibit rapidly variations with r , , the Fresnel transform may be omitted from the Fraunhofer-Fresnel transform. Since Fresnel diffraction gives very little information on the actual specimen structure, it is particularly useful to be able to omit this factor in the electron microscopy of a “normal” specimen. However, since Fresnel fringes are adversely affected by source incoherence, lens aberrations, ”
-
158
D. L. MISELL
and instabilities in the accelerating voltage and the objective lens current, Fresnel diffraction at an edge structure is a useful guide to these defects, particularly in the partial correction of axial astigmatism. IV. IMAGEFORMATION BY THE INELASTIC COMPONENT It is evident that inelastic electron scattering is relevant to image formation in the electron microscope; it does not just give a uniform incoherent background in the electron micrograph. The quite strong diffraction contrast effects exhibited in selected energy micrographs and the contrast in the selected energy images of amorphous specimens show clearly that structural information is available from the “inelastic” image. In consideration of the predominance of the inelastic electron scattering in “ carbonaceous ” specimens (e.g. organic, polymeric, and biological specimens), it is necessary to consider the image formed by the inelastic component in as much detail as given to the corresponding problem for elastic scattering. The formulation of a theory of image formation by the inelastic component is a more complex problem; this is evident from the discussion of Section II,G on the coherence and localization of the inelastic scattering processes. Characteristically the literature on image formation in the electron microscope dispenses with the contribution from the inelastic component either on the basis of the incoherent diffuse background or by the use of the “ theoretical energy selecting microscope; a notable exception is the work of Haine (120). The main calculations on image formation by the inelastic component, including the effect of chromaticaberration, have been made within the geometrical or “incoherent approximation (120, 153, 153a, 4, 5 , 89); the assumption on the complete incoherence of the inelastic component is the major simplifieation (see Section V). The electron-optical wave theory of image formation by the inelastic component does not appear to have for reference the corresponding lightoptical formulation. The analysis given in this section represents an extension of transfer theory, as applied to elastic electron scattering (Section 111), to a consideration of the inelastic electron scattering. In addition, information on the coherence and the localization of inelastic scattering is used (see Section ”
”
11,G).
A . Coherent Illumination
-
The incident electron wave is described by a plane wave exp(iK, r), where ( K O (= 2n/A0; the incident electron beam is spatially coherent (KOis constant) and monochromatic (KOis constant). The scattered wave is described by I + ~ ~ (KK, To), , where K and K refer to the spatial (angular) and energy characteristics of the inelastically scattered wave immediately after the object.
DEFECTS IN ELECTRON-OPTICAL IMAGES
159
The scattered wave includes electrons that have been inelastically scattered n times; thus formally the image wave Il/i may be written as a function of n, K, and K , that is, electrons with a given n, K, K are coherent:
where the amplitude convolution function G is dependent only on the modulus of K (representing the energy loss in transmission through the specimen) ( I ) and $, are the component wavefunctions of $o. Thus if electrons with a different n and K are incoherent, the image intensityj,(rJ is calculated by the incoherent superposition of the $Jn, K, K , T i ) , that is, (154)
ji(ri) =
2 SKoSJ’$,(K, ro)$:(K,
n=l
ro’)G
0
x
dr, drO’d K . (203)
The upper limit on the K integral is K O ,representing electrons which have lost no energy in transmission through the specimen; $,,(K, r,) = 0 for K 2 KO (corresponding to energy loss processes only). In order to simplify the analysis the assumption has been made that for a given n, electrons with different wave vectors K are coherent; this would appear valid only for K N KOor K’ N 0 (see Section U,G) (83, 84). Equation (203) for the image intensity may be rewritten in terms of the Fourier transforms of $, and G as j,(ri)
=
2 SEo//Sn(E, v)S,*(E, v‘)T(E, v)T*(E, v’)
n=l
0
x exp[ - 274v - v’) * r i / M ] dv dv’ dE,
(204)
where
-
s,,(E, v) = J $“(K, ro)exp(2niv ro>dr, and
(205)
1
.
T(E,v) = G(K,ro)exp(2rriv
To)
dr, .
The integral over K has been replaced by an integral over E ; energy loss processes correspond to E > 0 and for medium energy electrons (E, = 20100 keV) transmitted through thin film specimens ( t = 100-500 A), the probability of energy gain is considered negligible. The upper limit on the E
160
D. L. MISELL
integral, E , , may be replaced by an infinite limit, since S,(E, v ) is negligibly small for large arguments E. T(E, v ) is the amplitude transfer function including the spherical aberration, chromatic aberration, and defocusing terms, Eq. ( I 74). Sn(E, v ) represents the scattered wave in the Fourier plane of the objective lens for the n times inelastically scattered electron. Equation (204) is transformed from the variable v to the angular coordinate 8, where
e =2 r r v / ~ ,
(206)
to give the following equation for the image intensityj[:
x exp[ - iK(8 - 8’) ri/M] d8 do’ d E .
(207)
I@,,(O, E ) ] represents the intensity distribution for an electron scattered n times inelastically in the specimen (see Section Ir,E); H(E, 8) is the amplitude transfer function, Eq. (183). The variation of K with the energy loss, E, is not negligible as was the case in considering the effect of the energy distribution N ( E ) on the elastic image (see Section Il1,C). However, the variation of K with E, which is E/2E0, affects only the terms (K/277)’ and the exponential phase factors in Eq. (207), and the approximation K = KO (a constant) should not introduce any serious error into subsequent calculations. Strictly this approximation is not necessary since the integration over K can be evaluated numerically, but in order to derive analytic results, in relation to the effect of chromatic aberration on the inelastic image (see Section IV,B), this simplification is made. The replacement of K by KO in Eq. (207) gives N
x exp[iKo y ( E , 8‘)]D(8’)exp[- iKo(8 - 8’). rJM] d8 do’ d E . (208)
The assumption that electrons with the same E and n are coherent is necessarily an approximation; for a given resultant energy loss E, the inelastically scattered electrons may lose energy by different processes and by different amounts. E is then a sum of individual terms E l , E, , . . . , En; thus Eq. (208) shoud not only contain a summation over n but a further summation over all possible sets ( E l , E , , . . . , En) = E and the integration over E is to be omitted. Only for energy loss processes which give discrete loss lines nE, will the set (4, E 2 , . . . , En) be unique. Experimental evidence indicates that there is some preservation of diffraction contrast by electrons with a given n and E for multiple plasmon losses (102, 84).
DEFECTS IN ELECTRON-OPTICAL IMAGES
161
Following the analysis of Section H,E, the inelastic wave is calculated from an incoherent superposition of the @,(8, E ) for each E, that is,
n
P,,is the Poisson distribution (f/AJn exp( - ?/AJ/n!for independent scattering processes (83). The distributions +,,(8, E ) are calculated from, Eq. ( 7 9 , E)+n*(e, E ) = on(e, E ) N Sn(e)f,(E),
where Sn(8) is the angular distribution after n inelastic events with energy distributionfJE). In Eq. (75) for the +,,(8, E ) the phase term exp[ip,,(B, E ) ] is such that electrons with a different n and E are incoherent and that p,, is independent of 8. Thus in Eq. (208) the phase factor involved in the product On@,E)@,,*(8’,E ) is irrelevant since inelastic waves corresponding to different n and E are superimposed incoherently and the constant phase term (independent of 8) cancels in this product. Thus in Eq. (208) On(&E)@,,*(O‘,E ) = Pn +,,(W,,(O’)LW
(209)
and 4” is a function of 0 only, namely S,,(O)”’; the S,,(8) may be calculated either by the incoherent approximation (see Section II,E,I) for an amorphous specimen or by an n-beam calculation for a single crystal (Section II,E,3) (81, 82). The localization of the inelastic processes is relevant to the calculation of the $“(8) and the,f,(E), that is, the spatial distance over which +n and f,,are averaged. Since the phase factors in exp[ip,(8, E ) ] are not relevant to a calculation of j i , it is possible to determine experimentally both +,,(8) and fn(E)*
Thus the image intensity due to inelastic electron scattering may be calculated on the basis of the following assumptions that are not inconsistent with theoretical and experimental work: (i) the inelastic component is incoherent with the main beam and the elastic component; (this justifies the separate treatments of image formation by the inelastic and elastic ( + unscattered) components); (ii) the distribution Q,(8, E ) may be written as a separable function of 8 and E (see Section 11,D); (iii) electrons with differing n are incoherent even for the same E ; (iv) electrons with different E are incoherent even for a given n ; (v) inelastically scattered electrons with the same E and n are coherent for all 8; (this is valid for small-angle scattering with qn N 0). A further assumption which is unlikely to be valid but is made to simplify the preceding analysis: (vi) electrons inelastically scattered with the
162
D. L. MISELL
same resultant E and a given n are coherent; and lastly (vii) inelastic electron scattering is a localized phenomenon in organic, polymeric, and biological materials. In principle the distribution On(O, E) may be calculated by an average over the delocalized region (once specified). The validity of these assumptions may be subject to experimental investigation. Formally the possible coherence of electrons within an energy band between E and E + AE(K and K + AK) may be included in the calculation of the image intensity. The wavefunctions $,(K, K , ro) are superimposed coherently within the energy interval AE; electrons in different energy bands are superimposed incoherently. Thus
x exp[ - iKo(B - 0’) rJM] d0 de’,
(210)
where
B. EfSect of Chromatic Aberration
In order to investigate the effect of the energy loss distribution on the inelastic image, the integration over E in Eq. (208) is considered, that is,
with
iKox(0) is the phase shift term including only spherical aberration and defocusing, Eq. (104). The evaluation of the integral for the @, 0‘) requires a model for the distribution f , ( E ) . The f,(E) are calculated by a repetitive folding of f , ( E ) (see Section 11,D); the profile for f,(E) is represented by a Lorentzian, Eq. (89), for carbon, organic, and biological materials and for the nontransition metals (see Section 11,G). However, with a Lorentzian line shape for f , ( E ) , it is not possible to evaluate thef,(E) analytically and in general the integration (212) can only be evaluated numerically except for n = 1. As an approximation to f i ( E ) , the symmetrical Gaussian curve f , ( E ) = (b/n)’/’
DEFECTS IN ELECTRON-OPTICAL IMAGES
163
exp[-b(E - Ep)2]is used in Eq. (212); it is noted that although f , ( E ) is nonzero for E 5 0 , the value of the exponential term is negligibly small. Equation (21 2) becomes for the Gaussian profile in@,0’) = (b/nn)’I2
/om
exp[(iKoC, E/2Eo)(OZ- V2)]exp[- b ( E - nEP)’/n]dE,
or changing the variable of integration to E
=E
- nE,
/,,(0, 0’) = (b/nn)”’ exp[( - iKo C, nEp/2Eo)(02- S”)] m
.
1
- nE,
exp[( - iKo C, E)/2E0)(S2- W2)]exp(- bE2/n)dE (213)
Except for the lower limit on the variable E, - nEp,this integral is the Fourier transform of a Gaussian; for E < -nEp the exponential factor is vanishingly small and the lower limit on the integral may be replaced by a negative infinite limit, that is,
(214)
The optimum conditions for image formation by the inelastic component, neglecting for the present the elastic contribution to the image intensity, may be considered on the basis of Eq. (214). The angular dependence of the oscillatory term is the same as that of the defocusing term in x(0), that is, - iKo Af (0’ - W2)/2.With n = 1 , corresponding to a main peak at Ep in the energy loss distribution (and assuming multiple peaks at nEp are not significant), the maximum value of the exponent in the oscillatory term is -91.5 rad for E, = 20 keV (C, = 2 mm, Ep = 25 eV, ct = 0.01 rad) and -42.4 rad for E, = 100 keV (a = 0.01 rad). These phase shifts are an order of magnitude greater than the corresponding spherical aberration phase shifts (- 3.66 rad and - 8.48 rad respectively with C, = 2 mm), and may be cancelled by underfocus of the objective lens; for E, = 20 keV, A f = -2.5 pm and for E, = 100 keV, Af = -0.5 pm. This partial cancellation of the chromatic aberration defect is a direct result of displacing the Gaussian image plane to focus electrons with energy E, - E,. For a specific n, the corresponding defocus necessary to cancel that particular oscillatory term is Af = - C,nEp/Eo. Thus, depending on the specimen thickness, an optimum defocusing can be calculated in order to minimize the effect of chromatic aberration on the inelastic image. This underfocus will produce a large chromatic aberration effect on the elastic image and a decrease in image contrast as a result of a phase shift + iK, C, nEp(02- O”)/2E0.Assuming that the inelastic component carries useful information on the specimen, then the decision on whether to
164
D. L. MISELL
defocus the objective lens by the large amounts given above will depend on the magnitude of the elastic and inelastic contributions that are transmitted by the objective aperture, that is,
Numerical results for hE and h, are shown in Figs. 7a and 8a (also Tables IV and V) for carbon. The cancellation of the phase term in Eq. (214) leaves, as the main chromatic aberration effect, the exponential decay term
If n = 1, then the 8 dependence of the exponential factor with b = 0.007 e V 2 [corresponding tof,(E) for carbon] is:
Eo = 20 keV E,, = 100 keV
B = 0.002 rad 0.47 0.84
B
= 0.003 rad
-
0.02 0.28
8 = 0.004 rad 0.00 0.07.
Thus the 11(0,0‘) term is negligible for 0 a unless 8 N 8‘. For a in the range 0.0025-0.02 rad, the contribution from 11(0,0‘) to the 0 and 0’ integrands in Eq. (211) will only be significant for small arguments. Thus for a specimen which is characterized by a broad fi(E), it is evident that the chromatic aberration defect will not be very dependent on the objective aperture size. This is consistent with the results of calculations on the chromatic defect using the “incoherent” approximation (5) (see Section V). In the case of a specimen giving rise to a discretef,(E) (e.g. Al, b = 3.0 eV-2), the exponential attenuation of the phase term is not so marked. It is noted that the above conclusions are made on the basis of a Gaussian model for the energy loss distributions and in general it is not possible to cancel exactly the exponential phase term by a single value for the underfocus; the approximation K = KO used in order to separate the angular and energy integrations in Eq. (207) also affects these conclusions. The attenuation effect of the exponential term on the phase term of the ln(O, 0’) explains why, for even a uniform illumination of the objective aperture (1 q5,(0) 1 = constant for 0 -< 8 -< a), the chromatic defect is overestimated in the equation (6) rc
= CCS(E)~/EO
(216)
DEFECTS IN ELECTRON-OPTICAL IMAGES
165
for the radius of the “disk of confusion” r,; g ( E ) is a measure of the energy loss, for example; the most probable energy loss (7), or the energy halfwidth of the loss distribution (where a factor of 0.5 is introduced into Eq. (216) for focusing on the most probable loss) (9) or for materials exhibiting multiple plasmon losses, g ( E ) = (f/AJEp or 2.35Ep(t/AJ1”(155). Possibly Eq. (216) may be used to estimate r, if the maximum value for 8, u, is replaced by the 8 value for which the exponential decay factor decreases to (say) 0.5 of its value for 8 = 0; for carbon 8 = 0.002 rad at 20 keV and 8 N- 0.0028 rad at 100 keV. More usual for inelastic electron scattering in thin specimens is a nonuniform illumination of the objective aperture (see Section 11,F); in this case the replacement of u by the mean angle of scattering 8, also leads to an overestimate of r, (115). Formally the effects of the spatial incoherence and the chromatic incoherence of the incident electron beam may be included in the calculation of the inelastic image (154), but the appropriate equations are not amenable to numerical evaluation. The main effect of the angular distribution Zo(Oc) and energy distribution N ( E ) is to introduce further incoherence in the inelastic component. It is suggested that, under conditions of incoherent illumination, the analysis of image formation by the inelastic component may be adequately given by the “ incoherent ” theory (Section V). It is noted that, although the analysis of the inelastic electron scattering is equally valid for the conventional and scanning transmission electron microscopes, the chromatic defect of the inelastic image obtained in the scanning type microscope will be negligible. This fundamental difference between the inelastic images obtained in the two types of microscope arises from the absence of a lens system after the specimen in the scanning electron microscope. Thus the scanning electron microscope is most suited to an evaluation of inelastic scattering and its role in image formation since the energy losses of the scattered electron beam do not sensibly affect the resolution of the inelastic image. The inelastic image is not quite free from defects since the aberrations of the electron spectrometer of the scanning microscope will affect the image resolution.
v.
INCOHERENT THEORY OF
IMAGE
FORMATION
As a viable approach to the treatment of image formation by the inelastic component the incoherent or geometrical theory is presented in this section. Evidently the complete neglect of “phase” effects makes this treatment of limited use in the consideration of the elastic component. The main feature of the incoherent approximation in the facility with which the effect of lens aberrations on the imagecan be related to experimentally measurable quantities. In so far as inelastic scattering in “carbonaceous materials is considered, ”
166
D. L. MISELL
the dominance of inelastic scattering is evident (see Sectio.1 11,F). For these amorphous specimens there is no definitive experimental evidence for the coherence of the inelastic component. In the incoherent approximation the contribution from the components of the transmitted electron beam to the image may be calculated by scattering contrast (Section V,A) and the effect of lens aberrations on this image may be expressed by a convolution integral (Section V,B). The resolution functions appropriate to spherical and chromatic aberration may be related to the angular-energy characteristics of the transmitted electron beam (Sections V,C,D,E). Because intensity functions only are involved, the inversion of the convolution integral, to give an image corrected for lens aberrations, is possible. A . Calculation of Scattering Contrast Images
The contribution of each component of the transmitted electron beam to the image intensity at ri is calculated from the differential scattering outside the objective aperture, that is, by a scattering contrast mechanism. Two assumptions are made in relation to this incoherent theory and particularly to image formation by the inelastic component: (i) electron scattering gives an incoherent transmitted electron beam; (ii) electron scattering is a localized phenomenon and the transmitted electron carries information on the specimen structure on an atomic dimensions scale. Postulate (i) is invalid for the elastic and unscattered components of the transmitted electron beam. Postulate (ii) in relation to the inelastic component implies the absence of a delocalized electron density distribution within the specimen (see Section 11,G). For small-angle inelastic electron scattering the alternative models, relating to the partial coherence and delocalization effects of inelastic scattering are discarded ; if the inelastic component exhibits partial coherence, then the wave optical formulation is mandatory ; partial delocalization may be included in the following analysis once the degree of localization has been specified. The specimen structure is represented by a mass thickness variation Apt above a uniform level p o r. For the purposes of the present section the electron scattering in a " carbonaceous " specimen is identified with the scattering in an equivalent mass thickness of carbon. Thus the specimen structure is represented by
For a localized interaction, the image intensity at ri = Mr, may be calculated from the scattering within the objective aperture, h(u, p t ) . For each com-
DEFECTS IN ELECTRON-OPTICAL IMAGES
167
ponent of the transmitted electron beam h(cr, p t ) may be calculated directly from the angular-energy distribution D(0, E ) (see Sections II,E, II,F), that is, .+a2
h(a, p t ) = 27~J
-m
.a
J D(e, E)O dB d E 0
for cylindrically symmetric scattering functions. In the incoherent approximation, the image intensity, H(ri), is calculated from an intensity sum of the h(a, p t ) appropriate to each component of the transmitted electron beam, that is,
Hdri) = Hdri) + HE(ri) + HI(rJ9
(219)
where H(ri) = ( l / W h [ a ,~l(r0)1* Thus for a perfect (aberration free) lens system there is a 1 :1 correspondence between the object and image (ri = Afro).
B. Convolution Integral In this section is considered the effect of objective lens defects on the component image intensities, H(ri), in the conventional transmission microscope. Due to lens aberrations a point in the object will not correspond to a single point in the image plane. The effect of lens aberrations on H(ri) will be determined by the angular-energy characteristics of the electron scattered at ro in the specimen. The resolution function l ( r i , ro) is defined for each component of the transmitted electron beam; thus the image intensity at ri will be given by
with r’ = Mro. Formally I is dependent on both r’ and t i and not just on (Ti - r’) since the resolution function is dependent on the angular-energy characteristics of the transmitted electron beam at a particular object point. The approximation, that Zis independent of the variation in specimen structure (expressed by Apt) and is dependent only on pot, is made; the approximation I(ri, r’) = I(ri - r’) corresponds to the isoplanatic approximation in the waveoptical formulation (see Section 111). In the following sections expressions are derived for the resolution functions I(r) for an objective lens subject to the spherical and chromatic defects. C. The Spherical Aberration Function As a result of spherical aberration electrons scattered from a point object, through different angles of scattering 0, are brought to a focus in the Gaussian
I68
D. L. MISELL
image plane at a radial distance r measured from the ideal image point in this plane, where (34)
r = MC,03. (221) An estimate of the spherical aberration defect obtained by setting 8 = c( (the maximum value for 8) represents a pessimistic evaluation of r, particularly for the inelastic component where the intensity peaks very much in the smallangle region (see Section 11,F). In the case of the perfect objective lens the information from a point r,, in the object arrives a t a single image point ri = Mr, . However, for an objective lens subject to spherical aberration (assuming no chromatic defect) the information D(0, E) transmitted by the specimen is transferred from the coordinate system (8, E) to r using Eq. (221). In order to derive the spread function W(r) due to spherical aberration, the equality of the fraction h(a) and the number of electrons in the image plane is used, that is, h(a) =
/ d8 / a
+m
-m
D(0, E ) d E =
1
W(r) dr,
(222)
with rmax = MC, a3. W(r) dr represents the fraction of the transmitted electron beam which arrives in the image plane between r + dr and r. The left-hand side of Eq. (222) may be transformed from 8 space to r space using Eq. (221) to give for a cylindrically symmetric D(0, E) (4)
and the integration over E may be evaluated (see Section 11,E). The scattering models used for D(0, E) are such that the integrand in Eq. (223) is finite for all r. However, it is evident from Eq. (223) that W(r) is singular at r = 0; this is a direct result of the coordinate transformation, Eq. (221). The number of electrons at r is proportional to r-1’3 for a uniform illumination of the objective aperture. The effect of spherical aberration on the image formed by each component of the transmitted electron beam will be determined by the appropriate angular distribution and not by the energy distribution. Since the angular spread of the unscattered component is determined by the finite source size, the spherical aberration of this component will only be significant for small r values (r < 0.1 A). The close relationship between the distributions W(r) and the corresponding angular distributions for elastic and inelastic scattering is shown by calculations of W(r) for a “carbonaceous” specimen with thickness 100 A(pt = 20 mg m-’) and 500 A (pt = 100 mg m-’). In Fig. 19 a comparison is made of W Eand W , for E, = 20 keV (Fig. 19a) and for E, = 100 keV (Fig. 19b); C, = 2 mm, M = 1. Even in the case of the thinner specimen (dashed lines) the contribution to the spherical aberration from the inelastic
DEFECTS IN ELECTRON-OPTICAL IMAGES
169
component (A) far exceeds that from the elastic component (B). The ratio WJW, is not quite as large as the IIE ratio of the angular distributions; this is because r varies as O3 and W(r) for the larger r values (e.g. 8-20 A, Fig. 19a) is determined mainly by the large angle scattering (0.008-0.01 rad).
D. The Chromatic Aberration Function Chromatic aberration in the final image arises from the energy and angular distributions of the transmitted electron beam. Due to inelastic electron scattering a fraction of the incident electron beam will emerge with an energy spread about E, (normally as an energy loss, E > 0) and the elastic component will have an energy distribution N ( E ) (see Section 11,E). As a result of chromatic aberration an image of a point object is formed at a radial distance r from the ideal image point, where (34)
r = MC, OE/Eo,
(224)
and E defines the deviation of the energy of the scattered electron from E, . Assuming that chromatic aberration only is operative, electrons scattered from a point object with a common value for the product BE arrive at the same radial distance r from the ideal image point. The calculation of the chromatic aberration by setting B = a and using a single value for E(e.g. energy half-width or most probable energy loss) may be viewed as a pessimistic estimate for r. In the expression for r the actual distribution of energy loss should be used (18). The chromatic defect equation (224) represents a transformation of the transmitted information D(0, E ) from coordinates (0, E ) to image coordinates r. The intensity distribution Z(r) in the image plane is defined in a similar manner to W(r), Eq. (222), with rmax= MC, aE,,,,,/E, . Slight differences occur in the derivation of Z(r) for the three components of the transmitted electron beam. In the case of inelastic scattering the limits on the E integral are 0 and infinity (or Em,,) and the expression for Z,(r) is (153, 4, 89) b " z,(r>= - J D,(B, br/e)do,
(225)
emin
with b = E,/MC, and Omin = br/E,,,,, . In the case of the elastic and unscattered components the energy distribution N ( E ) is nonzero for E < 0; electrons with E < 0 give rise to r values in the opposite sense to that for E > 0 and the expression for Z,(r) is (4)
D. L. MISELL
170 I
o-p
lo-'
- lo-+ L
L
3
10-1
lo-'
lo-'
FIG.19. The spherical aberration function W ( r )for carbon; r is the deviation from the ideal image point. (a) Eo = 20 keV, a = 0.01 rad; (b) Eo = 100 keV, a = 0.005 rad. Ainelastic component, ELelastic component; dashed line, f = 100 A; solid line, f = 500 A.
For a symmetric N ( E ) DE(6, br/O) = D,(B, - br/B).
It is evident from Eqs. (225) and (226) that the distributions Z(r) are singular at r = 0, although the number of electrons at r is finite. It is apparent from calculations on ZEand WEthat the chromatic defect of the elastic image is at least comparable with the spherical aberration defect; Figs. 20 and 21 show the comparison of ZEand W Efor Eo = 20 keV (Fig. 20) and for Eo = 100 keV (Fig. 21); C,= C,= 2 mm, M = I , and E,,2= 1 eV. It is clear that the chromatic aberration function for the inelastic component 2, exceeds ZEby an order of magnitude for r 6 A, Eo = 20 keV (Fig. 22a) and for r > 0.5 A, Eo = 100 keV (Fig. 22b). The comparison of Z, for two different objective aperture sizes, shown in Fig. 23, demonstrates that Z , varies only slowly with c1. Thus it would seem that, except for large radial
=-
DEFECTS IN ELECTRON-OPTICAL IMAGES
171
lo-'
I 0-3
10-
lo-.
I
Ib)
0
I
2
3
r [A)
FIG.19(b)
distances, the chromatic defect of the inelastic image will not significantly vary with a (see also Section V,F); this is a direct result of the predominantly smallangle nature of the inelastic component. E. The Combined Aberration Function For an objective lens subject to both spherical and chromatic defects the displacement from the ideal image point is given by an addition of Eqs. (221) and (224), that is, As was the case with the chromatic aberration function, it is necessary to
treat separately the three components of the transmitted electron beam. Following the analysis of Section V,D, the total aberration function for the inelastic component, TI@),is given by ( 4 )
1
b "
T(r) = -
DIIB, b(r - MC,03)/O]d0
emIn
D. L. MISELL
172
I 0-3
lo-*
-L
3-10-3 L
N
lo-'
lo-'
to-
FIG.
20. A comparison of the spherical aberration W(v) and chromatic aberration
Z(r) functions for the elastic component (for carbon). (a) a = 0.01 rad, (b) a = 0.02 rad.
The incident electron energy is Eo = 20 keV and the energy half-width of the distribution is = 1 eV. Dashed line, t = 100 A; solid line, t = 500 A.
for a cylindrically symmetric D1(fl, E ) ; Omin is the only positive real root of r = MC, O3 + MCcBE,,,,,/Eo. For the elastic and unscattered components the derivation of the equations for T, and Tu is complicated by the nonzero values of N ( E ) for E < 0 (4). The differences between TI and ZI for the inelastic component are not very marked ; this demonstrates the subordinate role of spherical aberration on the total aberration function for the inelastic component. F. Eflect of Chromatic and Spherical Aberration on the Image
In order to evaluate the effect of lens aberrations on image resolution, T(r) is applied as a convolution function to the ideal image H(r); I(r) is simply related to T(r) by a normalization factor, that is,
173
DEFECTS IN ELECTRON-OPTICAL IMAGES 10-
\ I
\
I
\
I
\
lo-'
-L
3 .
10-3
1
L
N
lo-'
lo-'
lo-'
b)
5
10
I5
r
(A)
FIG. 20(b)
with
I
T = T(r, po t ) dr In Eq. (229) it has been assumed that the convolution function I is dependent only on p o t and is independent of the mass thickness variation Apt. As mentioned in Section V,B, this approximation is the equivalent of the isoplanatic approximation and this simplification is made because of the difficulties involved in evaluating a convolution integral whose kernel Z(r, r') depends separately on r and r'. If only one component of the transmitted electron beam is considered, then Eq. (220) reduces to F(r) = IH(r')Z(r - r') dr'.
(230)
It is evident that the application of the convolution function Z(r) to H(r) is computationally difficult due to the singularity of Zat r = 0. In order to avoid this problem, Z is divided into two components: (i) a delta function pS(r)
174
D. L. MISELL
>
1
3
2
I
FIG.21. A comparison of the spherical aberration K(r)and chromatic aberration Z(r) functions for the elastic component (for carbon). (a) a = 0.005 rad, (b) o! = 0.01 rad. The incident electron energy is 100 keV and the energy half-width of the distribution is Ell2 = 1 eV. Dashed line, t = 100 A; solid line, t = 500 A.
for 0 I r 5 Ar and (ii) a well-behaved function T(r, p o t)/T for Ar 5 r 5 r,,,,,; the constant p is calculated from (5)
B = W ,p0 t ) -
Ar
T(r, p0
o drl
Equation (230) may then be written as
for, say, the inelastic image and I(r) is now defined as zero for 0 5 r I Ar; Ar is taken as approximately 0.2 A (p N 0.02). The first term on the righthand side of Eq. (231) then represents that part of the image which is unaffected by lens aberrations.
175
DEFECTS IN ELECTRON-OPTICAL IMAGES
loo .
lo-'
I o-'
--_
(b)
I0-5 0
2
I
3 r
(A)
FIG. 21(b)
In order to evaluate the effect of chromatic and spherical aberration on the inelastic image, H(r), of a " carbonaceous " specimen, a mass thickness structure S(ro) with a single peak superimposed on a uniform p o t is chosen. Figure 24 shows the results of the calculations for F for two specimen thicknesses: p o t = 40 mg m-' (Apt > 0) (Fig. 24a) and p o t = 200 mg m-' (Apt < 0) (Fig. 24b); Eo = 100 keV, C, = C, = 2 mm and M = 1. Curve A represents the ideal image H(r) as a function of the radial distance r; the background intensity has been omitted from these graphs. With increasing LY (curves B-D) there is a progressive decrease in the maximum value for F(r), corresponding to an increase in the chromatic defect of the image. However, the dependence of F(r) on c( is not very marked except at large radial distances r. It is also evident, from the long tail of the F(r) curves, that an estimate of the chromatic defect cannot be calculated on the basis of an " r " half-width. As expected from the comments of Section V,E, the removal of the spherical aberration contribution from I,(r) produces no significant effect on the F(r) curves of Fig. 24. Consistent with the above calculations on the convolution effect of I, are the following resolution figures for the chromatic
176
D. L. MISELL
lo-’
lo-+
-
10’
N
lo-‘ .
lo-’ -
(a) 10-8
0
10
20
30
r cR1
FIG.22. A comparison of the chromatic aberration functions Z(r) for the inelastic (A) and elastic (B) components (for carbon). (a) Eo = 20 keV, a = 0.01 rad; (b) Eo = 100 keV, CL = 0.005 rad. Dashed line, t = 100 A; solid line, t = 500 A.
defect: -4-6 8, for p o t = 40 mg m-’ and -6-8 1$ for po 1 = 200 mg m-’ for cx values in the range 0.0025-0.01 rad and Eo = 100 keV. Evidently these estimates for the loss in image resolution are almost an order of magnitude less than the estimates of chromatic aberration based on a uniform illumination of the objective aperture, for example, (9)
r = C,aE,,,/2Eo.
(232)
Equation (232) provides an upper limit for the image resolution as limited by chromatic aberration and the incoherent approximation a lower limit to this resolution; the latter result is consistent with the wave optical theory (see Section IV,B). It is apparent from recent experimental work (153, 89) that the chromatic defect is nearer to the lower limit, even for thick specimens [30008, of a biological section observed at 75 keV (89)l and that the main
DEFECTS IN ELECTRON-OPTICAL IMAGES
177
loo
lo-'
--
10"
L
N
lo-'
10-
I 0"
rfA)
FIG.22(b)
effect of chromatic aberration is as a decrease in image contrast (see Fig. 24) (89). The effect ofdefocusing the objective lens may be included in the incoherent theory; the defocus A f causes a displacement from the ideal image point given by r = MAfO, or, including lens aberrations in the expression for r :
r
= MC,
e3 + MC, OE/E, + M Af 8.
(233)
In contrast to the wave theory, Af appears only in the role of displacing the Gaussian image plane by underfocus of the objective lens in order to minimize the effects of lens aberrations; phase contrast effects are excluded. Clearly the advantages of the incoherent theory are due to the direct relationship between the image intensity H(r) and the scattering properties of the specimen D(0, E). Thus for an image where only one component of HT(r) is dominant, the convolution integral, Eq. (2301, may be used to correct the
D. L. MISELL
178
10’
.
lo-’
.
FIG.23. The variation of the inelastic chromatic aberration function Z ( r ) with the objective aperture size a.(a) Eo = 20 keV, A - a = 0.01 rad, B-a = 0.02 rad; (b) Eo = 100 keV, A--cc = 0.005 rad, B-a = 0.01 rad. Dashed line, I = 100 A; solid line, t = 500 A.
image for resolution effects; Z(r) may be calculated from an experimental B(0, E ) distribution averaged over a large area of specimen. In the scanning electron microscope the component images may be separated (11); further, because the “ objective ” lens is located in front of the specimen, the aberration functions appropriate to each component will be identical. In the scanning microscope the resolution function Z(r) will depend only on the angularenergy characteristics of the incident electron beam and will be independent of the electron scattering properties of the specimen; thus Z(r) may be measured for use in an incoherent theory. Two other defects of the electron microscope image, namely the lateral and depth effects, may be discussed using an incoherent approximation. The lateral effect corresponds to a loss in image resolution due to the lateral deflection of the scattered electron in transmission through the specimen (ZJZ, 137). An estimate of this effect may be given from the specimen thickness
DEFECTS IN ELECTRON-OPTICAL IMAGES
179
and the most probable value Om.*, of the angular deflection. In the case of inelastic electron scattering 0,,,+ < 0.001 rad ( E , = 20 -100 keV, t = 100500 A of carbon) and the maximum lateral effect is less than 1 A even for a thick section. In contrast the lateral effect for the elastic component is significant: 3-18 A for t = 100-500 A and E, = 20 keV; for E, = 100 keV the lateral effect is approximately 1-2 A. These are conservative estimates and significantly larger values may be obtained by, for example, using the values for the mean angle of scattering. The depth effect arises from the superposition of the electron scattering from successive " planes " in the specimen, particularly if the structural features of the section vary through the thickness (131, 137). Both the lateral and depth effects are relevant to image formation in a conventional transmission and a scanning transmission microscope.
-
VI. CONCLUSIONS In this review an attempt has been made to describe the effect of lens aberrations and of the electron source incoherence on the image formed by the elastic and inelastic components of the transmitted electron beam.
Image i n t e n s i t y
I
L
L
- I2
12 r (A,
image !ntcnrity
FIG.24. The effect of chromatic aberration on the “inelastic” image for a variation in thickness (mass thickness) about a uniform value (a) t = 200 k, ( p o t = 40 mg my’), (b) f = lo00 8, ( p o t = 200 mg m-2). The solid line (A) represents the “inelastic” image (background of uniform intensity omitted) in the absence of chromatic and spherical aberration. The discontinuous curves represent the “inelastic” image as modified by lens aberrations; B-a = 0.0025 rad, C-a = 0.005 rad, D-a = 0.01 rad. The incident electron energy is 100 keV.
DEFECTS IN ELECTRON-OPTICAL IMAGES
181
The main consideration appears to be the loss in image contrast rather than the deterioration in image resolution (sections IIJ and IV). For unstained biological materials the predominance of the inelastic scattering indicates that the main contrast mechanism is scattering contrast; phase contrast will have a subordinate role in image formation. Several schemes have been proposed for the enhancement of the scattering contrast; the technique of strioscopy (dark field technique), using an objective aperture with a centre stop to remove from the transmitted beam the zero-order beam and the small-angle inelastic scattering, leads to a substantial increase in image contrast for unstained biological and polymeric materials (156-158). In the scanning transmission electron microscope there are a number of procedures available for contrast enhancement (11-13); the versatility of the scanning microscope arises from the use of an electronic detection system, which enables, for example, signal differentiation to be accomplished (12). In the conventional microscope several procedures for contrast enhancement have been described (159, 160); notably the use of an annular objective aperture (zone plate) leads to an increase in image contrast; this is designed to intercept those spatial frequencies which give negative (positive) contributions to the phase contrast function Q(v) (see Section 111) and negative (positive) contributions to the contrast integral, giving C(r,), are removed. It is evident from current research that the design of the annular aperture is difficult; it must be constructed with a particular geometry depending on the lens aberration coefficients (e.g. C,) and on the degree of defocus, AA of the objective lens. It is also noted that not only does the theory of an effective zone plate explicitly assume a weak phase object, but also the removal of certain spatial frequencies by the opaque parts of the aperture leads to an image that contains no information on certain spacings in the object (3). Light-optical techniques for contrast enhancement have also been based on the use of an annular aperture with the optical diffraction pattern of the electron micrograph (161). Image reconstruction (deconvolution) techniques, either by light-optical or computational methods (159, 160, 134, 134a) are based on the practical realization of a weak phase object and neglect completely the contribution to the image from inelastic scattering. A deconvolution procedure will always lead to an apparent improvement in image resolution but the effect of “noise” must be carefully evaluated in order to have confidence in the final result. Evidently a through focal series of micrographs is mandatory; the reconstructed result, for example, q(r,,) in the case of the weak phase object, should be independent of Af(for given objective aperture size a, which limits vmaX). The application of holography to electron microscopy has been theoretically evaluated (162), and assuming incoherent illumination this optical technique has been applied to an image taken from the scanning transmission microscope ( 1 6 2 ~ ) .
D. L. MISELL
182
Of the less ambitious optical reconstruction techniques, the method for the removal of “noise” in the electron micrograph caused by, for example, photographic granularity is of general application (163). As stated by the authors ( 1 6 3 , the elimination of spatial noise from the electron micrograph leads to a deterioration in resolution due to the simultaneous removal of structural information. The resolution of the photographic emulsion is not a serious limitation provided that a moderately high electron-optical magnification, M , is used (~20,000)(131). The more general problem of evaluating the specimen structure from a “perfect image still represents a major project; the nonlinear relationship between the image intensity, ji(ri), even in the case of a simple scattering contrast model [as determined by h(a)],and the specimen structure (e.g. mass thickness p t ) is evident. Probjects involving the visualization of single atoms avoid the real problem; such projects give information only on the ultimate performance of the electron microscope. It is probable that a major factor which limits image resolution (and image contrast) is radiation damage in the conventional microscope (93-957). Radiation damage leading to bond rupture in organic and polymeric materials may be decreased by an increase in the incident electron energy, E,, for a given specimen thickness (94,95a);the gain with increasing Eo is limited to a threshold above which atom displacement becomes a major cause of radiation damage (8). Associated with the radiation damage is a corresponding loss in specimen mass, which may be as high as 80 % of the original mass for a polymer observed at E,, = 20-100 keV (93, 95, 164). Clearly the electron microscopy of unstained biological and polymeric specimens will effectively be an observation of a residual carbon skeleton; a preliminary evaluation of image contrast on the basis of the residual ”carbon ” structure has been made (164). Apparently the scanning transmission electron microscope may be used to examine organic and biological materials with a minimum amount of radiation damage to the specimen (11, 96); but the difficulty of attaining an ultrahigh vacuum (lo-’ torr) with the typical biological specimen may be a limitation. There is large scope for advancing the current knowledge on inelastic electron scattering and its role in image formation; the important factors which are relevant relate to the coherence and localization of the inelastic scattering, particularly in polymeric and biological materials. Both these quantities may be evaluated by attempting to take high resolution (5-10 A) electron micrographs using only the inelastic component. Here the scanning microscope would facilitate such a study, since the chromatic defect of the image is negligible; the use of the energy selecting microscope is also a viable approach. In the present work, it has been suggested that the inelastic processes in organic materials are localized and that the coherence of the inelastic beam is limited to electrons in a given small energy interval, AE (Sections I1 and ”
DEFECTS IN ELECTRON-OPTICAL IMAGES
183
IV). A modification to the electron interference microscope or the observation of Fresnel fringes with the inelastic component may provide definitive evidence on the coherence of the inelastic scattering. Measurements of the angularenergy distribution D,(8, E ) under high spatial resolution will give information on the sensitivity of the inelastic scattering to variations in the specimen structure. In the extreme assumption of the complete incoherence of the inelastic component, the incoherent theory leads to the possibility of correcting an electron micrograph for the chromatic defect (Section V). The neglected field of low voltage electron microscopy has much to commend it, particularly in application to the study of unstained biological materials. Although radiation damage, lens aberrations, and the lateral deflection effect (137) are severe disadvantages at low incident electron energies, the enhancement of image contrast is a major consideration (140142). Further, because image formation takes place under essentially incoherent conditions, possible artifacts due to wave interference effects are absent. The correction of the low voltage micrograph for lens aberrations could reduce the present resolution figure of 40 A to the “theoretical ” limit of 10 A. Despite the problems encountered in structure determination by electron microscopy, it is the only technique available for the analysis of complex structures, particularly amorphous specimens, below 100 A structural detail. The incentive to improve image resolution and image contrast either theoretically or by experiment remains strong. APPENDIX : DECONVOLUTION OF TWO-DIMENSIONAL DATA The solution of the convolution integral
to give information on i(r) is examined when the intensity distribution j(r) is affected by noise, n(r) (e.g. due to photographic granularity). The kernel s(r) represents the convolution function, which, for the wave optical theory, is related to the function G (Section 111) and for the incoherent theory is identified with the aberration function Z(r) (Section V). It is assumed that n(r) is introduced in the recording plane and not by the image forming system; thus it is unaffected by, for example, the source incoherence or the objective lens defects. The problems encountered in the solution of Eq. (A.l) for i(r) may be discussed in terms of its Fourier transform, that is,
+
J(v) = S(v)Z(v) N(v),
(A4
D. L. MISELL
184
where, for example, the Fourier transform of s(r), S(v), is defined by (A.3)
-
-
For certain values of the spatial frequency v, the distribution S(v) may have a zero or near zero values [e.g. S(v) 0 for v vo] and J(v) contains little information on Z(v) at these spatial frequencies; further for some frequency v > v, (e.g. due to the objective aperture limit) there is little or no information in J(v) on Z(v) (165). Thus the major contribution to J(v) at v vo or v vmaXwill be from the noise N(v),which is unaffected by S(v).An attempt to calculate Z(v) from Eq. (A.2), neglecting completely the effect of noise, will lead to a final solution for i(r) that contains large (and spurious) oscillations, arising from the division of J(v) by S(v) in regions v vo where S(v)-l = T(v)is large, and amplification of the noise component N(v) by the factor T(v).An inverse transform of Z(v), that is,
-
-
-
i(r) = /J(v)T(v)exp( - 2 ~ i *vr) dv,
64.4)
will then contain incorrect information on i(r). Thus a reconstruction procedure based on the complete neglect of the effect of noise can give information on spacings r which do not exist in the object. In particular, for v > vmaX it is possible to reconstruct spacings r < rmin(rmin l/vmax) in the object and an increase in image resolution is apparent. Although the effect of noise can be taken into account, it is not possible to eliminate the effect of noise on a deconvolution procedure, but the effect on the reconstructed result may be minimized. Below are given mathematical procedures for the deconvolution of data in the presence of noise; the mathematical solutions may be applied to optical reconstruction methods. (i) In the first case it is assumed that there is no information on n(r) beyond the general level of the noise. It is accepted that there is no information on Z(v)for v vo and for v > v, ,where the extent of the region around the vo depends on the noise level N(v). T(v)is set to zero for v v o , where S(v) is small; the regions of v for which S is small may be either calculated theoretically or determined from an optical diffraction pattern of j(r), J(v). Thus in the v regions where S is small ( T large), the effect of noise on the solution for i(r) is minimized. The reconstructed result is then given, from Eq. (A.4), by (166) N
-
-
i(r) = JJj(r’)T(v)exp[ -2niv = Jj(r’)M(r- r’) dr‘.
9
(r - r‘)] dv dr’
I85
DEFECTS IN ELECTRON-OPTICAL IMAGES
The function M(r) is referred to as a deconvolution function and M(r) is defined by M(r)
=
1
-
T(v)exp(- 2niv r) dv.
-
-
(A.6 )
Thus in the regions where J ( v ) N(v), all information on Z(v) is lost for v v,; the philosophy here is that no information on spacings ro l/v, is preferable to false information on these spacings. In the case of a cylindrically symmetric kernel s(r), Eq. (A.6) may be written as a Fourier-Bessel transform. (ii) If information on n(r) is available, then the reconstruction can be used, at least partially, to give reliable information on all spacings greater than rmin( 1/vmax); this does not overcome the problem of evaluating Z(v) at the actual zero values of S(v) but a more realistic reconstruction is possible. The “filter function” T(v) is defined by (167)
-
-
+ @n(v)I-’, T(v) = S*(v)QO(v)[l S(V)I 2@o(~)
(A.7)
where @,(v) is the Fourier transform of the autocorrelation function (bn(r): &(r)
=
Jn(r’)n(r - r’) dr’,
and @,(v) = I N(v) 1 ’.
(A.8)
@,(v) is the Fourier transform of (b,(r) with
4,(r)
=
Aj(r)
= j(r)
j A j ( r ’ ) Aj(r - r’) dr’,
- A - ’ sj(r’) dr’
for a small area A about the coordinate r. In the case of n(r) = 0, @, .+ 0 and T(v)3 S(v)-’. A review of deconvolution techniques is given in reference (168). In the calculations of Section 111, the deconvolution function M ( r ) has been calculated by method (i), which requires only an estimate of the noise level x (measured as a fraction of the maximum value ofj(r)); it is assumed that the error in J(v) is also x. In the regions where J(v) is small the division of J(v) by S(v) will amplify the noise component N(v), which has not been affected by convolution. Thus depending on the magnitude of x, there are regions of v where the value ofJ(v) is comparable with the error term N(v). We then omit from the function T(v) those contributions from S(v) that will amplify the noise (represented by x) by more than a specified factor. A suitable cutoff criterion, which may be modified in consideration of the actual data j(r), is that we omit from T(v) contributions from v values for which S(v) < l/e, corresponding to an amplification of J(v) by a factor e (3).Thus the effect of
D. L. MISELL
186
noise is not completely eliminated but the more serious effects are avoided. The omission of certain frequency regions from the deconvolution procedure leads to some loss of information on the specimen structure. Holographic deconvolution enables the multiplication J(v)T(v) to be performed optically (169). However, a suitable mask should be used to eliminate in the Fourier transform the spatial frequencies v vo . The holographic method is limited to functions that have positive values, since the photographic plate records intensity. Additional factors which must be carefully evaluated in an optical reconstruction technique relate to alignment errors and the defects of the light-optical system (e.g. spherical aberration, defocusing). The specific application of holographic techniques to electron microscopy has been evaluated theoretically (162) and experimentally for incoherent illumination (162a). (iii) A method which relies directly on the calculation of J(v)/S(v)from a through focal series of electron micrographs avoids the problem of noise amplification. For each electron micrograph in the series the noise component N(v) is the same, but the frequency gaps (v vo) in the appropriate S(v) vary with the defocus, Af (Section 111). Thus it is possible to construct an Z(v) by using only those J(v)/S(v)for which S(v) is significantly different from zero and the effect of noise on I(v) is minimal. The I(v) calculated from a set of micrographs comprises a complete frequency spectrum, except for frequenfor an objective aperture limit). The result for a given cies v v,, (-./Ao Z(v) may be confirmed by the evaluation of Z(v) for two different defocus values where S(v) % 0. The implication of the above analysis is that no information is available on I(v) for v = vo and v > vmaX;this corresponds to bandwidth limited dataj(r), where certain spatial frequencies are absent from J(v). The suggestion that these missing spatial frequencies may be reconstructed is based on the analytic continuation of J(v)/S(v)(170), that is, J(v)/S(v)may be used in a Taylor expansion about v(v # vo), near v = vo (or v > v,,) to give
-
-
-
However, analytic continuation is not a valid concept for data subject to error and Eq. (A.lO) must be viewed as a theoretical result of little practical value. The numerical differentiation of experimental data is notoriously inaccurate, particularly for the higher order differentials. ACKNOWLEDGMENTS The author is most grateful to Professor F. Lenz, whose lectures and unpublished notes on “Transfer of image information in the electron microscope” provided the basis for the sections on transfer theory. The author is grateful to his wife, Janet, for assistance in preparing the typescript and for proof reading.
DEFECTS IN ELECTRON-OPTICAL IMAGES
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141. 142. 143. 144. 145. 146. 147.
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162a. G . W. Stroke and M. Halioua, Optik 35, 50 (1972). 163. S. Boseck and H. Hager, Optik 28, 602 (1968). 164. P. Sadhukhan and D. G. Drummond, 7th Int. Con$ Electron Microsc., Grenoble 1, 37 (1970).
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ADDENDUM TO REFERENCES Due to an oversight by the author the following references by K. Kanaya and coworkers were omitted from the present review on image formation in the electron microscope. Although these papers appeared in print almost 20 years ago, acknowledgment of the major contribution of K. Kanaya to the subject is absent from the literature. As may be judged by the titles of the papers, the analysis of these papers considers a wave-optical analysis of image formation by the elastic and inelastic components of the transmitted electron beam and discusses the relevance of electron energy losses to image formation in the electron microscope. It should be noted that some of the approximations made by Kanaya in these papers would not now be necessary with the use of a computer in the evaluation of certain integrals.
K.Kanaya. Image formation in the electron microscope from the view-point of waveoptics. 1. Effect of spherical aberration, chromatic aberration and defocusing on elastic scattering. 11. Effects of chromatic aberration and defocusing on inelastic scattering. Bull. Electrotech. Lab. (Tokyo) 17, 679 and 756 (1953). K. Kanaya, Y.Inoue, and A. Ishikawa. Image formation in the electron microscope from the view-point of wave-optics. J. Electronmicrosc. 2, 1 (1954); Bull. Electrotech. Lab. (Tokyo) 18,517 (1954). K . Kanaya. Image formation by crystalline specimens in the electron microscope. I. Image Contrast. Bull. Electrotech. Lab. (Tokyo) 20, 610 (1956). 11. Image Contours. Bull. Electrotech. Lab. (Tokyo) 20, 801 (1956). 111. Fresnel diffraction fringes. Bull. EIectrotech. Lab. (Tokyo) 21,455 (1957).
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Recent Advances in Field Field Electron Electron Microscopy Microscopy of of Metals Metals Linfeld Research ResearchInstitute, Institute, Linfield McMinnville, Oregon McMinnvilfe, Oregon
..... .........................................,.................... ........................ .......... A. Band Band Structure StructureEffects. Effects ..................................................................... A. ....................................... .................................. B. Free Electron Metal .......
1. Introduction.. .................
194 194 i94 194 195 195 199 199 C. d-Band Emission . .......... ....,........' .................... 201 201 D. Many-Body Effects . ....................,.. ................................... 202 202 E. Potential Potential Barrier Barrier Corrections. Corrections.................................................................... ........ ..............,....... ..., 203 203 E. F. Surface States.......,........ 205 205 206 G . Adsorbate Effects .... .......... ........... .................................. ....... ........ 206 ,............ Techniques....._. .............................. *............................ 212 IIllll.. Techniques.. ............................................. .................. ..........,.... ..,..... ... 212 A. Coadsorption Coadsorption Experiments. Experiments........................,................................................................ ........ ..,.....,. ............. 212 212 A. 214 214 ents .... , ........, ............................... . C . Total Energy Distribution 21 21 55 D. Electron Impact Desorptio ................... 220 220 220 220 222 222 223 IV. Clean Surface Characteristics.......... ............. ..............................,........... 223 223 223 232 232 242 ............................................... C. Energy Exchange Effects ........ 242 .................. 252 .............................................................. D. Noise Studies ........... 252 256 256 ...................... 258 258 258 258 26I I B. Coadsorption.................................................. ............ ................. 26 262 262 265 265 273 273 275 275 277 VI. Emitter Su .............................................................. 277 277 ........................................... 277 .............. 278 278 ....................................................................... 283 ................ 283 ....................... 283 283 289 289 tions ..,.......... . 295 C. Instrument 295 296 296 304 References........................................................... ................................ ................... ......... 304 ,
11. Theoretical Considerations
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L. W. SWANSON AND A. E. BELL
1. INTRODUCTION During the past decade, field electron microscopy has become established as a powerful tool for elucidating a variety of phenomena occurring at metal and semiconductor surfaces. Examination of the total energy distribution of the field emitted electrons has recently brought to light certain inadequacies in the previously established Fowler-Nordheim theory and has opened the possibility of studying surface and bulk electronic structure. Important strides have also been made in gaining a more detailed and realistic theoretical description of the field emission process from both clean and adsorbate coated surfaces. More progress in the theoretical understanding of field electron emission is clearly necessary at the present time and will likely occur in the near future as theorists generally become more attentive to surface problems. Whereas considerable insight concerning the life and stability of field electron cathodes in practical environments was obtained during the early part of the last decade, significant technological exploitation of some of the unique properties of the field electron source has only recently been fully realized. For the most part technological uses have been confined to fine focus electron optical applications. The above mentioned subjects are reviewed in greater detail in the following sections of this paper. Since a number of excellent reviews of field electron emission cover the important developments prior to 1960 (1-4) this paper will deal primarily with subsequent progress. 11. THEORETICAL CONSIDERATIONS The field emission process at a metal-vacuum interface was given theoretical foundation in 1928 by Fowler and Nordheim (5, 6) who based their calculations on one-dimensional tunneling from a Sommerfeld metal at 0°K. Until recently, the resultant Fowler-Nordheim (FN) equation has been found adequate for describing field electron emission (FEE) from clean and adsorbate coated metals in the applicable temperature range. A modification of the FN equation t o include temperature effects was later treated quantitatively by several investigators (7-11). Although total emission current-voltage [Z( V ) ]measurements generally confirmed the expected temperature and field dependence, a more rigorous test of the theory was performed by Young and Miiller who determined theoretically (12) and measured experimentally (13) the total energy distribution (TED) of field emitted electrons from a tungsten substrate in the temperature range 77 to 300°K. Experimental attempts to measure the TED of field emitted electrons prior to the work of Young and
FIELD ELECTRON MICROSCOPY OF METALS
195
Muller were clouded by the erroneous belief that normal distributions were obtained from point-to-plane electrode geometries. In recent years the FEE theory has been expanded to include space charge effects (14), surface irregularities (15, 161, and quantum corrections (17) to the image force model, many-body effects (181, band structure effects (19-23) surface states (24),and inelastic (25) and elastic (26-28) scattering of tunneling electrons due to adsorbates. These theoretical advances have been in part inspired by experimental results (29-35) that could not be explained by the existing theoretical models. It will be instructive to review the relevant steps in the derivation of the TED equations pointing out the assumptions and approximations one must make in order to obtain the FN equation. A. Band Structure Efects
Common to most theoretical treatments of metal-vacuum tunneling is the assumption of specular reflection or transmission. That is, k , =f(k,, k J , the quasielectron momentum components perpendicular to the tunneling direction x are conserved during electron tunneling through a finite potential barrier; Fig. l a shows the usual potential energy diagram assumed for the one-dimensional tunneling problem from a metal into vacuum. Here it is assumed that the ionic potentials are small compared to the conduction electron energies so that the electrons see a constant potential well of depth V, = 4 + Ef, terminated at x = 0 by a surface. Here C#I and E, are the work function and Fermi energy respectively. The solutions to the Schroedinger equation inside this free electron metal are plane waves of the form I)I = exp (ik . x), where k is the electron quantum number. Since E K k2 is the general solution t o the Schroedinger equation, we note that the constant energy surfaces are spheres in k-space. The usual assumption depicts the surface potential barrier as depending on x only and the tunneling problem thereby becomes a soluble one-dimensional problem through use of the WKB approximation. The latter approximation assumes that the band structure is slowly varying in the barrier region and that far from the barrier the WKB solutions are eigenstates of a slowly varying potential. Later we will review the limitations of this model for tunneling from a tight binding d-band where. the crystalline potential varies rapidly with x and Bloch wavefunctions are more appropriately employed. The general formulation of the tunneling process from an independent particle viewpoint using the plane wave representation and the usual WKB approximation in the transition between region I and 11 has been accomplished by Harrison (36) for a metal of arbitrary band structure, i.e., the E(k) relationship is unspecified. The resultant expression for the current density
196
L. W. SWANSON AND A. E. BELL
x=o (a)
(b)
FIG. 1. (a) Diagram of a free electron metal of work function $ and Fermi energy EI . (b) Diagram of Bloch metal illustrating localized core potentials.
per unit total energy J ( E ) in the x direction is given by the expression
with
The integral in Eq. (1) is over the positive constant energy surface E as it projects on the y z plane perpendicular to the emission direction. As pointed out by Harrison, by using the independent particle approximation J(E) does not explicitly depend on the density of statesp(k). The functionf(E) is the electron distribution function in k-space and e - f ( k x )is the electron transmission probability which is frequently expressed by the symbol D(E - E,) where E, = f ( k , , k,) is the transverse energy component. The latter symbolism emphasizes the fact that in the case of specular tunneling the transmission probability of an electron of energy E can be formulated in terms of E and
FIELD ELECTRON MICROSCOPY OF METALS
197
k , , where k , is a constant of motion in the barrier region. The electron state in the metal can be expressed in terms of the two variables, k , and 4p, where 4, is the angle of k , in the plane perpendicular to the emission direction. Noting that k , = ( k t k;)’l2 and tan &, = k,/k,, the transformation between the two coordinate systems can be obtained from
+
dk, dk, = J, ddPdk, , where the Jacobian of the transformation J , = k , . Thus, Eq. (1) can be written in the following form for a band whose minimum is centered on
k=O:
According to the WKB approach,
so that D(E - E,) = D(E,). Since Ex is determined where the electron is free, i.e. E , = h2kt2/2mis a constant of motion outside the emitter, Ex is therefore given by
Ex = E - h2k12/2m
or dE, = dE, = k , dk,(h2/m). This allows Eq. (2) to be written in the form
where Elm= E,”(E, qbp), is the maximum value of El for a particular E and $p . Stratton (19) first pointed out that Eq. (3) could be written in the physically more suggestive form
The second integral was termed the “ band structure integral ” since a knowledge of the projection of the constant E surfaces on the k,, k, plane is required; it can be neglected only when Er El”, i.e., when one is dealing with nearly spherical energy surfaces in the y z plane. Gadzuk (21) criticized Eq. (4) as not being valid for a conductor of arbitrary band structure since “only in a free electron metal does E l = hk12/2rn.”Politzer and Cutler (U),
198
L. W. SWANSON AND A. E. BELL
on the other hand, dismiss Gadzuk's further assertion that Eq. (4) holds only for bands in which E cc k2 by noting that the functional dependence of E on k is unspecified in Eq. (4). They further note that for specular transmission through a one-dimensional barrier E , = h2kt2/2mis a constant of motion outside the emitter, but has no such physical significance inside the emitter. In order to develop Eq. (4) further the customary procedure is to perform a Taylor expansion off(Ex) about E, , f ( E x ) = HE,) - (4- Ef)c(Ef)+ (Ex- Er)2a(EA,
which allows Eq. (4) to be written in the form
provided that the quadratic and higher order terms of the Taylor expansion can be neglected. Murphy and Good (10)have derived the following expression for the coefficients b and c appropriate to the image force barrier:
b(Ef) =
4( 2m$3)'/2 6.83 x 10743/~u(y) 3AeF U(Y>= ,
F
1
c(E,) = - = d
2(2rn4)"2t(y)- 10.25 x 107c$1/2t(y) keF F 3
where the applied field F is in units of V/cm and the work function 4 in eV. The correction terms t ( y ) and v(y), which are due to the image potential, are tabulated (10) slowly varying functions of the auxiliary variable y = (e3F)1'2/4. The specific effect of band structure occurs in Eq. ( 5 ) only through the term in brackets and can be neglected when E,"/d % 1. Under usual fieldeIectron emission conditions, d 0.15 to 0.25 eV and for most degenerate metals exhibiting large Fermi energies E,"' is sufficiently large that the band structure term in Eq. ( 5 ) is negligible; however, for transition metals with partially filled narrow d-bands, the value of E," may be sufficiently small to be manifested through the band structure term. As shown by Fischer (37), in the case of spherical energy surfaces with a positive Fermi energy and an effective mass m, = r,m (so that E;"(E, 4,) = r,E), Eq. ( 5 ) becomes
=
J(E) = J,f(~)e"/"[l - e-'EE/d]/d,
(6)
where E = E - E, and where J , denotes the total current density (integrated over all E ) at O'K, which is related to F and 4 as follows:
FIELD ELECTRON MICROSCOPY OF METALS
199
- 1.54 x 10-6F2exp[ -6.8 x 107~3~2v(y)/l;l(A/cm2). 4t2(r>
(7)
Equation (7) is in essence the formulation of the FEE process as derived originally by Fowler and Nordheim (5, 6). B. Free Electron Metal
Neglecting the band structure term, Eq. (6) can be written J(E)= Jo eEld/d(1 + eE/pd)
(8)
where p = kT/d. Equation (8) is the form of the TED expression derived originally by Young (12) for a free electron metal where the Fermi-Dirac = [I ec/pd]-l. Because of the importance distribution is assumed, i.e. f(~) of higher order terms in the Taylor expansion of f(E,), the mathematical derivation of Eq. (8) breaks down for p 3 1 and becomes unreliable when p exceeds about 0.7. A graphical representation of J(E)vs. e/d yields a set of curves whose shape depends only on the dimensionless parameter p as shown in Fig. 2. At p = 0.5 an exactly symmetrical distribution of the emitted electrons occurs about Er . This condition can be written in terms of the so-called "inversion temperature ":
+
T* = d/2k = 5.67 x 10-5F/c#'/2t(~)(oK). I.Or
(9) Theoretical Curves P
'k\ \
0
0
,100
@
,200 .300 ,400
@ @ @ @
,500 ,600
-4.0
e/d
FIG.2. Theoretical total energy distribution plots based on the free electron model, Eq. (8) at various values of p ,
200
L. W.SWANSON AND A. E. BELL
If the average energy of the conduction carriers occurs at E,, no net energy exchange takes place and the significance of T* is that it separates emission heating (for T < T*) and cooling (for T > T*). The integration of Eq. (8) over E leads to the well known expression for the current density in the “ thermal-field ” (T-I;? region.
JTF= J,
Im eeld[l+
ee’pd]-’
dEjd
-03
(10)
= J, nplsin np.
Using the lower limit of - co in place of the correct value of zero in order to facilitate integration does not greatly alter the result because of the exponential decrease of J(E)for E < 0. For small values of p (i.e. low T o r high F)np sin np z 1 and the 0°K approximation of the FN equation J = J,, is obtained as given in Eq. (7) An analytical expression for the average energy of the emitted electrons relative to the Fermi level (E) = ( E ) - Ef can be obtained from Eq. (8) by performing the following integration: (E) =
jm EJ(E)d e / / - m / ( c ) -m
ds = -df(p).
(1 1)
The function f(p) is unity for p = 0 and zero for p = 0.5; thus, for field emitted electrons at T = 0, ( 8 ) = -d and at T = T*, ( E ) = 0. Levine (38) has shown thatf(p) E np cot np, so that Eq. (11) can be written in an alternative form : (E)
=
-np d cot np.
(12)
The form of Eq. (12) predicts ( E ) to be a function of p only, as shown graphically in Fig. 3. As mentioned previously the foregoing derivation is limited to p ? 0.7 (T Z 1700) because of the neglect of the quadratic and higher order terms in the Taylor expansion of the tunneling probability D(E - ET). Using computer techniques, this approximation can be avoided and more precise numerical values of the above equations can be obtained for values of p > 0.7. In the Appendix, a summary of the analytical expressions for (E) and J(E)is given for both the normal and transverse components of the emitted electrons extending from T = 0 and large F t o large Tand F = 0, i.e. from low temperature field emission to Schottky and thermionic emission. As shown by Christov (II ), approximate analytical expressions for the energy distributions and current can be obtained over most of the temperature-field range. Indeed, Figs. A1 and A2 show that only a small region between the T F and extended Schottky region cannot be described analytically. An interesting
FIELD ELECTRON MICROSCOPY OF METALS
20 1
D
FIG.3. A plot of (&>Idvs. p according to Eq. (12) where emitted electrons.
(F)
is the average energy of
result is the complete independence of the average transverse energy (El) on temperature in the field emission regime. C . d-Band Emission
Several authors (20-23) have pointed out that in the case of tunneling from metals with narrow d-bands, not only must one consider the shape of the energy surfaces through E,"(&, E ) but one must also evaluate the tunneling probability D ( E - E l ) using the more realistic Bloch wave functions as depicted schematically in Fig. lb. Itskovich (20) first considered the problem of solving D(E - E l ) in terms of a metal with Bloch states and arbitrary energy dispersion. Although Itskovich develops the formalism necessary to solve D(E - El), he contends it is impossible to solve the expression for D ( E - El) except in the case of free electrons. Gadzuk (21) also calculates D(E - E,) for a Bloch metal using tight binding wavefunctions and transfer Hamiltonian methods. Gadzuk further postulates a tight binding d-band that is described by a linear dispersion of the form E = b 1 k 1, where b is a parameter. The linear E(k) relationship does not greatly alter the functional form of the energy distribution expression from that of a quadratic E(k) relationship; however, the transmission coefficient of d-band tunneling appears to be reduced over s-band tunneling by a preexponential factor of to according to the recent theoretical works by Gadzuk (21)and Politzer and Cutler (22,23).Politzer and Cutler (22), using a one-dimensional model with a triangular surface barrier and 3d
202
L. W. SWANSON AND A. E. BELL
Bloch wavefunctions for fcc ferromagnetic nickel, find that the free electron transmission coefficient D(s) is 1 to 2 orders of magnitude greater than the 3d transmission coefficient D(3d). In disagreement with Gadzuk’s (22) similar treatment they find the ratio D(s)/D(3d) to be independent of F a t all electron energies. D. Many-Body Eflects Gadzuk (28) has used the many-body approach involving the thermodynamic Green’s function method to treat field emission from superconductors, and nonideal metals, i.e. those in which electrons collide with phonons, impurities, and lattice imperfections. The method was also applied to the effect of a finite analyzer resolution. In the simple case of a free, noninteracting electron gas, Gadzuk‘s treatment devolves to the Fowler-Nordheim result. Gadzuk’s expression for the TED is:
where q and A&, a)are electron momentum and spectral weight functions, respectively, for the right-hand side; k and A,(q, w) are the corresponding quantities for the left-hand side; Ek is the energy of an electron on the metallic side of the field deformed potential barrier and E, the energy on the right-hand side. Equation (13) may be simplified by noting that for field emission in a perfect energy analyzer the right-hand spectral function is sharp so that A,(q, o)+ 6(Eq - a),in which case we have (14)
The Fowler-Nordheim result is obtained by noting that the left-hand spectral function A,@, E,) is also sharp so that A&, E,) 46(Ek - Eq). Application to superconductors may be made by replacing A,@, E,) by
which represents the spectral function for an elementary excitation in a superconductor; Ek’ = (Ek2+ S2)”’ where S is the energy gap parameter and Ek the energy from the Fermi surface in the non superconducting phase. When J ( E ) is evaluated for typical values of d = 0.15 eV and S 1.5 x lod3 eV, a narrow peak of halfwidth of -3 mV is superimposed on the TED results from a normal metal. Gadzuk raised the question of whether the superconducting properties of the metal would be destroyed in the surface region; however, Applebaum and Brinkmann (39) have pointed out that in
-
FIELD ELECTRON MICROSCOPY OF METALS
203
the superconducting tunneling phenomenon the region of the metal being sampled is of the order of the coherence length of the electrons which for superconductors is of microscopic dimensions. Electrons scattering within the emitter can be accounted for in Gadzuk’s model by replacing A&, E,) in Eq. (14) by a spectral function which describes collision broadening. Gadzuk used a Gaussian line shape:
which yielded upon integration : J = J,, exp[r2(T)/2d2],
(16)
where r is a factor dependent on the magnitude of the broadening effects. Equation (16) of course, is the standard F N result multiplied by the broadening factor. Extending this approach to the case of 3d magnetic transitions metals, Osborne(40) has shown that effects on both theTED a n d J can be anticipated. Specifically, an antiferromagnetic metal such as chromium is predicted to exhibit an energy gap in the antiferromagnetic phase approximately two orders of magnitude larger than the bandgaps associated with most superconductors. Whether the strong electron correlation effects associated with narrow d-band metals which undergo magnetic transitions can be observed in the TED or total emission measurements remains to be experimentally justified. E. Potential Barrier Corrections
Other attempts to refine the FEE theory have included more realistic expressions for the potential energy barrier V ( x ) through which the electrons tunnel. Originally, only the image potential term was employed: V(x) = -e2/4x V(x) = 0
- eFx
for x 3 x,, for x e x s ,
(17)
where x, defines the “surface” of the metal. Cutler and Nagy (17) included an additional quantum correction term ye2/4x2,where q is a small positive number (approximately 0.068 A for tungsten) that depends on the electronic properties of the metal. Thus, the modified potential barrier becomes V(x) = -e2/4x
V(x)= 0
+ qe2/4x2- eFx
for x 3 x , , for x e x s .
(18)
204
L. W. SWANSON AND A. E. BELL
The important feature of the modified potential Eq. (18) is that it gives rise to a thicker potential barrier for the tunneling electrons than the image force potential Eq. (17). The effect of the quantum term is most manifest at high fields and was put forth as an explanation for the current deviation from the F N law at F 5 5 x lo7 V/cm. Figure 4 shows the deviation from the FN law using Eq. (18) for the potential barrier. Modinos (15, 16) suggested an additional correction to the classical image force law due to atomic size surface irregularities. He derived a term of the form-[e2/4x3. Including all correction terms the potential barrier becomes : V ( x ) z -e2/4x
+ qe2/4x2- 5e2/4x3- eFx.
(19)
loe/ F (VOLTS, CM" ) FIG.4. A Fowler Nordheim plot of log,,J vs. 1/F illustrating deviation (dotted line) from the FN law (full line) due to a modified potential barrier represented by Eq. (18) in which 77 = 0.679 A (dashed line) or 77 = 0 (solid line). In both cases 4 = 4.5 eV and V, = 10 eV; this latter quantity is the constant electronic potential in the interior of a free electron metal referred to an electron at rest at infinity. [From P. H. Cutler and D. Nagy, Surfuce Sci. 3, 71 (1964).]
FIELD ELECTRON MICROSCOPY OF METALS
205
The combined effect of the two correction terms is t o cancel one another depending on the relative values of q and l. Since experimental results indicate that the quantum correction should predominate, Modinos (16) suggests a larger value of q should be used. Originally, the experimentally observed deviations from the F N law at high field strength had been attributed to space charge reduction of the applied field at the cathode. More recently, Nowicki (14) calculated the effect of space charge for a spherical electrode by considering, not only the reduction in field, but also the increase in the image barrier due to electronic charge in the cathode-anode space. Nowicki determined that the influence of space charge on the applied field at the cathode is less important than its influence on the image potential barrier. Unfortunately, since both space charge and quantum corrections predict similar alterations to the F N law, it is difficult to verify their relative importance without further experimental results. F. Surface States Forstmann (41) has discussed the conditions under which surface states will exist in nearly free electron metals and semiconductors. In addition to surface states existing within gaps of type A (see Fig. 5 ) on a Brillouin
r
I K-
X
FIG.5. Nearly free electron band structure with gaps of type A and B. [From F. Forstmann, 2. Phys. 235, 69 235 (1970).]
206
L. W. SWANSON AND A. E. BELL
zone boundary, there is also the possibility of surface states existing in gaps of type B which result from the crossing of two bands inside the Brillouin zone. In gaps of type A, surface states exist only if the potential matrix element is greater than zero, whereas in gap B surface states are always found. In a following paper dealing with d-band metals, Forstmann and Pendry (42) also examined the possibility of surface states in bandgaps arising from the cross over of the s-band and a d-band of the same symmetry. They conclude that a surface state is localized in about the two top-most layers for a model which corresponds to copper and that the energy of the surface state was about 5-6 eV below the Fermi level. On the basis of their model, Forstmann and Pendry concluded that surface states can also be expected to be encountered in many transiton metals. Gadzuk (43) has calculated the effect of d-band surface states on field emission tunneling from W (100) by using Harrison’s (44) d-band pseudopotential theory. He concludes that because the existence of the surface states depends critically on the surface boundary conditions the surface states are strongly affected by the act of measurement. G . Adsorbate Effects
From the outset of the development of FEE theory and experiment the modulation of the emission current due to changes in Q, caused by adsorbed particles has been extensively investigated. Work function changes A 4 due to adsorption measured by FEE have generally agreed closely with measurements by other techniques. Gomer (45) has shown that due to the discreteness of the adsorbate its contribution to the surface dipole potential does not occur linearly with distance from the surface particularly at low coverage. Thus, as outlined by Gomer, a small correction to the field emission A$ must be made at low adsorbate coverage in order to compare results directly with contact potential measurements. Another effect on A 4 arises from the field-induced contributions due to the external applied field (4). The effective work function 4f including the fieldinduced contribution for an adsorbate of polarizability u and surface density of a. 0 can be written as
where E = F/F,, is the ratio of the externally applied field to the effective field at the adsorbate and g is either 2 (if the induced dipole moment pindiscentered on the plane of electric neutrality) or 4 (if pjnd is wholly contained in the adatom). Gomer (4) has shown that field induced work function changes that are small compared to 4 (i.e. 4 > 4, - 4) are manifested in the FN equation (7) through the preexponential, rather than the exponential term. The measur-
FIELD ELECTRON MICROSCOPY OF METALS
207
ment of the experimental intercept at 1 / V = 0 of a plot of log(Z/V2)vs. 1/V can thus be utilized as an independent method of evaluating a. This can readily be shown by first expanding @I2 as follows: $:I2
N
Cp3I2
+ 3 Cp112gnaFu,0 / 2 ~ .
Noting further that in the range of in Eq. (7) is approximately given by
4
(21) and F normally encountered, u(y)
u(4, F)N 0.943 - 0.146 x
10-6F/#2 (22) for F i n V/cm and Cp in eV, then combining Eqs. (7), (21), (22) leads to the following expression for the experimental intercept In A of the F N plot: In A = In[B/Cprt2(Cpf,F)]
+ 9.94/4~’/~ - 9.65 x 107Cp1/2gnaa0 0/e,
(23)
where B is proportional to emitting area and a is in units of cm3. Since for
e=o
In A, = In[B/4s t2(+, ,FJ]+ !1.94/4”~, (24) (where the subscript s refers to the clean substrate), it is possible by combining Eqs. (23) and (24) to obtain the following expression for U / E :
where Cpf has been replaced by its approximate value Cp in the first term of the numerator. In order to evaluate E , and hence, a, one may assume a Topping array of field-induced point dipoles centered on the image plane. The effective field at a specified adsorption site is therefore reduced by the accumulated effect of the field-induced dipoles in the adlayer and, for a square lattice site array, is given by Fo = F/E= F / [1 + 94u0 d)3’2], (26) where E = 1 + 9a(a00)3/2can be visualized as a two-dimensional dielectric constant of the adlayer. Sensible values of a have been obtained by application of Eqs. (25) and (26) for several adsorbates at various coverages; however, as will be discussed below, other factors influencing the preexponential factor In A must also be considered particularly at low coverages. Besides affecting FEE through dipolar induced work function change, it has recently been shown that an adsorbate can alter the electron emission in at least two other ways. These mechanisms can be categorized as follows: 1. Elastic Scattering (tunnel resonance (26-28)). 2. Inelastic Scattering (electron-phonon interaction (25); electronelectron interaction). Both of the above effects show up most distinctly in the TED and to a varying degree in the total current-voltage relationship.
208
L. W. SWANSON AND A. E. BELL
The mechanism of tunnel resonance set forth for the FEE case first by Duke and Alferieff (27) showed that monatomic adsorbates can cause observable structure in the TED and effect the preexponential term in the F N equation. The mechanism postulated was in essence a perturbation of the tunneling electrons through wave-mechanical interference effects due to the presence of discrete atomic potentials lying outside the main electronic charge cloud of the bulk metal. (See. Fig. 6). According to the one-dimensional pseudopotential model employed by Duke and Alferieff, atomic or molecular bound electronic levels of an adsorbate lying within the conduction band provide windows of enhanced electron tunneling which are manifested as subsidiary peaks in the TED. The atomic potential was represented both by a square well with attractive core parameterized by its depth and width and by a repulsive delta function potential which was equivalent to orthogonalization of the tunneling electron wavefunction to the occupied tightly bound adsorbate electron orbitals. Plausible forms for the pseudopotential for both metallic and neutral adsorbates were suggested and their effects on both the TED and FN equation were calculated. A treatment similar to that of Duke and Alferieff was carried out by Modinos (28) who developed a method for calculating the transmission coefficient through a potential barrier with a short range adsorbate potential superimposed on it, by utilizing the interaction formalism of the T scattering matrix. Both mono and multilayer cases were treated using a one-dimensional A
3 FIG.6. Electric potential energy diagram and TED for tunnel resonance enhanced field emission. W represents the width of the square well potential of the adsorbates and N(E)the expected TED curves. [See Swanson and Crouser (1641
FIELD ELECTRON MICROSCOPY OF METALS
209
6 function pseudopotential for the adsorbate atomic core. In general, the predicted effects on electron tunneling were similar to those outlined by Duke and Alferieff, particularly for the monolayer case. Using a perturbational approach, Gadzuk (26) also developed a theoretical framework for tunnel resonance in terms of the parameters relating to the position (relative to the Fermi level) and shape of the broadened adsorbate energy level. Although this approach lacks the mathematical exactness of that used by Duke and Alferieff, the influence of the adsorbate parameters is more clearly delineated. Based on the one-dimensional model of Fig. 6, in which a Lorentzian shaped virtual electronic level in the adsorbate is positioned at energy A (relative to the Fermi level), Gadzuk has shown that the TED including transmission resonance can be expressed as JJE)
Trz + [ + (E-A)2+r2
= J(E) 1
(27)
where T,, is the ratio of the adsorbate coated to clean tunneling probability and r is the halfwidth of the broadened adsorbate level which increases with decreasing metal adsorbate distance (46). The unperturbed TED, namely J(E)is given by Eq. (8). As pointed out previously (26). the first term in the right-hand side of Eq. (27) is the direct tunneling expression. The second term is the resonance tunneling factor which assumes a Lorentzian shaped broadened adsorbate level. The third term is an interference term between the direct and indirect channels. Thus the resonance peaks will exhibit a skewed Lorentzian shape centered near A. Clearly, the influence of tunnel resonance on the TED will be reduced as r increases and as A increases above E, . According to a highly simplified model (46) the virtual level A will shift downward with field F according to Fx,. A more complicated field shift may occur in practice upon inclusion of the image term, atomic or molecular polarization, and Stark effect. Assuming no temperature dependent change in x,, , no particular first-order effect of temperature on the shape or displacement of the tunnel resonance peak is expected. Now let us examine the effect of electron-phonon interaction on the tunneling electrons. According to Fig. 7, electrons tunneling from the metal at energy level E = 0 inelastically scattered by a phonon excitation ho will ultimately tunnel through a separate channel at E - ho. Thus, the TED shape at the Fermi level will be replayed at E - hw, reduced by a transition probability factor Tep.In this case the expression for the TED can be expressed as
where the second term represents the unperturbed TED and the third term is due to electrons which channel separately at E - hw.
L. W. SWANSON AND A. E. BELL
210
EMITTER
COLLECTOR
ANODE
FIG.7. Electric potential energy diagram for electron-phonon interaction during field emission where fiw is the interaction energy and N ( E )the expected TED curve.
A theoretical description of the electron-phonon interaction has been given elsewhere for the vacuum field emission case (25) and for the closely analogous tunnel diode configuration (47). If the transition is sharp (i.e. no lifetime broadening effects) the leading edge shape of the e-p transition will be due to the temperature broadening of the Fermi level electrons. Thus, the TED structure due to an e-p transition should exhibit a leading edge which broadens with temperature. It has been shown (25, 47,48) that Tep
K N C I(nIPxt0)12, n
where N is the adsorbate density, P, the surface normal component of the dipole moment, n is the nth vibronic level, and (n is the vibronic wavefunction upon which Px operates. For N N 5 x IOl4 atoms/cm2 Flood (25) shows that Tep a 0.01. Thus a very sensitive energy analyzer will be required to detect e-p transitions for small molecules. A Stark splitting of an electron-phonon transition can be envisioned in the case of degenerate vibrational, rotational, or bending modes. However, a qualitative prediction as to the magnitude of such field effects on the electronphonon interaction is not possible from the present theoretical status. At best the effect of field on e-p transitions is expected to be smaller than for tunnel resonance. This is borne out by the experimental results of Lambe and Jaklevic (48) which show that tunnel electron spectra of complex molecules is strikingly similar to the corresponding field free infrared spectra. Finally, let us consider the electron-electron (e-e) interaction. Our focus here is upon the possibility of the tunneling electron to excite the adsorbate electrons in the upper filled molecular orbitals to low lying excited states as depicted in Fig. 8. Since the tunneling electrons have several volts of energy
FIELD ELECTRON MICROSCOPY OF METALS
21 1
FIG.8. Electric potential energy diagram for electron-electron interaction during field emission where hv represents the electronic excitation energy of the adsorbed molecule and N ( E )the expected TED curve. [See Swanson and Crouser (164).]
relative to the uppermost filled state of the adsorbed molecules, excitation to levels a few volts above the ground state is possible. Cross sections for delocalized n orbital electrons should be of the order of the molecular dimensions for the highly conjugated molecules. Qualitatively speaking, the expression describing the TED structure due to e-e interactions should have a form similar to Eq. (28) where the transition probability Teewould be proportional to molecular orbital areas and the matrix element coupling the ground to the first excited electronic state of the adsorbate. As illustrated in Fig. 8, the transition may not be sharp due to the possible broadening of the electronic levels undergoing transition. However, we anticipate to be rather small for deep lying levels where the metal-adsorbate orbital overlap will be small. If r is sharp, i.e. I'< kT, the shape of the Fermi level TED is simply replayed at E - ho as in the case of electron-phonon interaction. We therefore expect temperature broadening to be manifest in the e-e transition where r < kT. The detailed effect of field on the e-e peak position cannot be ascertained from this cursory treatment; however, it will be proportional to the difference between the field shift in the ground and excited state levels. Stark shifts for highly polarizable molecules are likely to be important. From elementary considerations two Stark shifts can be identified for electronic transitions.
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L. W. SWANSON AND A. E. BELL
In the case of a nondegenerate state one may observe a quadratic shift in an energy level E, with field. E, = F2a/2.
(29)
It is actually the difference between the field shift of two electronic levels E, and E,' that is observed in the transition. For ct = 50 A and F = 0.3 VIA the value of AEJAF = 0.5 A. Thus, the quadratic Stark shift will be quite small for even highly polarizable molecules. A linear Stark shift arises from the action of an electric field on a degenerate energy level E,, . The energy level will then split into two levels given by En = Eo & eFlZl12
(30)
where Z , , = UI*Zl2U2dx. In essence Z1, is roughly the mean difference in atomic size in states 1 and 2, where U, and U2are the appropriate wavefunctions. For the hydrogen atom one can show that ZI2N 3a0, where a. is the Bohr radius. Thus, for the hydrogen atom, AE,/AF= 1.5 A. Clearly, from these elementary considerations the linear Stark is expected to be the predominant effect. By way of summary, a comparison between Eq. (27) depicting elastic scattering and Eq. (28) depicting inelastic scattering yields the following expectations: the shape of the subsidiary TED structure in the case of elastic scattering is Lorentzian and unrelated to the Fermi level emission peak; the opposite is generally expected to be the case for inelastic scattering, whether electron-electron or electron-phonon, provided the energy levels are sharp compared to kT. In addition, elastic tunnel resonance and electron-electron interaction should exhibit a larger AEIAF than an electron-phonon interaction. 111. TECHNIQUES General discussions on processing and construction details relating to field electron emission microscopy (FEEM) have been presented in the literature (1-4). Therefore this section is confined to a discussion of embodiment modifications which have extended the use and application of FEEM in recent years. A . Coadsorption Experiments
The investigation of adsorption on an atomically smooth and clean substrate (the emitter) has been one of the primary research applications of the FEEM. For the most part such investigations have been confined to a single component adsorbate until recently when two-component adsorbtion studies have been reported (49-53). Figure 9 shows a typical tube embodiment for the
FIELD ELECTRON MICROSCOPY OF METALS
213
A
FIG.9. Field electron emission tube containing both a chemical and a sublimation source. The cesium capsule F is broken and condensed in nozzle G where it is sublimed onto the platinum disk A. The latter is heated resistively so that controlled doses of cesium can be deposited on the emitter C; B is a heatable platinum bucket which can contain cesium fluoride for deposition of cesium fluoride on the emitter or copper oxide from which oxygen may be obtained by heating. E is an anode ring in front of the emitter while D is the pin connection to the conductive coating (stannous oxide) on the inside of the tube.
investigation of the coadsorption of cesium and oxygen on a tungsten field emitter (51). The sources are mounted at right angles to each other thereby reducing the cross-contamination. In an experiment of this type the FEEM is frequently immersed in liquid nitrogen or helium in order that volatile adsorbates will not exhibit spurious vapor pressures greater than lo-'' Torr. In the Fig. 9 rube both a chemical and sublimation source are depicted. The cesium capsule F is broken and condensed in nozzle G where it is sublimed onto the platinum disk A. The latter is heated resistively so that controlled doses of cesium can be deposited on the emitter C . The oxygen source B is a resistively heated platinum bucket containing in situ formed copper oxide. Heating the bucket to -700°K causes the chemical breakdown of the copper oxide so as to liberate oxygen which is deposited on the emitter in reasonably controllable doses. Using this basic design principle a variety of coadsorbate combinations can be conveniently investigated by this method. A review of the various
214
L. W. SWANSON AND A. E. BELL
coadsorbate systems which have been invesrigated by FEEM is given in a subsequent section.
B. Single Plane Techniques The FEEM projects the image of the hemispherical emitter onto the screen with an overall magnification M of M
N
xll S r ,
(3 1)
where x is the emitter to screen distance and r the emitter radius. For typical values of x and r, M 3 lo5;thus, extremely small areas of the emitter surface can be examined visually or by probe current measurements. The use of Z(V) data to arrive at work function change involves Eq. (7) which may be rewritten in terms of the directly measurable field emission current I and applied voltage V as = AfV2e-mf/v.
(32)
In view of this relation, it follows that a “Fowler-Nordheim plot” of the current-voltage relationship (In Z/V2 vs. 1/V) yields a straight line having a slope m f and an intercept A , with the vertical axis at 1/V = 0. The linearity of a F N plot is normally regarded as adequate proof that the emission is due to stable field emission. When the FN law is satisfied, it can be shown that 4f is related to the slope m, by the expression m, = 2.96 x 107(4,3/2//?f)s(y) (volts),
(33)
where s(y) is a tabulated function which is equal to 0.951 & 0.009 over the range of current densities normally encountered. Using the average work function ($) for a clean surface as a reference, the work function $, of a surface when coated with an adsorbate or of a particular crystal plane can then be determined from
4f = (4>[mrPrs(Y)lm
s(Y)12/3,
(34)
where m , and m are the slopes of the corresponding FN plots. Absolute determination of the work function requires a knowledge of the geometric factor /?(where F = PV), which can be determined from an electron micrograph of the emitter profile with an accuracy of about 15 %; however, when the work function of the uncoated surface is well known, both fi and the work function of a coated surface or a single crystal plane can be determined with good accuracy. Since a variety of crystal planes develop on the single crystal emitter surface during cleaning procedures the total current is necessarily an average weighted heavily toward the highly emitting (low work function)
FIELD ELECTRON MICROSCOPY OF METALS
215
planes. It can be shown ( 4 ) that the FN plot of I/V’ vs. I/Vyields an average value of (43’2//3)which is related to the individual regions as follows
(43’21P>= 1 f i 4;”IPi i
9
(35)
where fi is the fraction of the total current from the ith emitting region It should be emphasized that when adsorption greatly alters the emission distribution from that of the clean surface the value of (4) cannot always be related directly to contact potential (51). The pitfalls inherent in the determination of average work function can be overcome by measuring the electron current from individual crystal planes by suitably designed current probes (51,5456). A tube designed for this purpose is shown in Fig. 10 which allows measurement of the total energy distribution as well as emission from a single crystal plane. Magnetic deflection of the electron beam by external coils (B) allow measurements to be extended over all regions of the emitting surface. More recent designs have used electrostatic deflection of the beam rather than magnetic. The total area A , of the emitter of radius r “seen” by a small aperture probe which subtends a halfangle u at the emitter is A , r n(ru)2 .
(36)
Typical values of r = lo-’ cm and u = 1” give A , = 2 to 10 x cm2. For the well developed low index planes this value of A I is well within the size of the single plane facet. A FN plot of the probe current in conjunction with Eq. (34) allows clean single plane work function values or adsorption induced change in 4 to be measured. Thus, adsorption and electron emission processes can be conveniently studied on several different, well defined crystal planes of a given material by this technique. C . Total Energy Distribution Measurements
The measurement of the total energy distribution (TED) of field emitted electrons has, mostly, been carried out by a retarding potential technique. The diagram in Fig. 11 shows the relevant potential energy diagram for a typical retarding potential energy analyzer. The original design of the field electron emission retarding potential analyzer by Miiller has been improved by van Oostrom (56) whose modified tube design (32)is depicted in Fig. 10. Briefly, the tube is designed in such a way that electrons passing through a lens system are focused near the center of the spherical collector F. The electrode system of the analyzer consists of an anode D, a focusing electrode E, and a Faraday cage G . The latter electrode is operated near ground potential and
216
L. W. SWANSON AND A. E. BELL
FIG.10. Cross section of a field electron, emission retarding, potential energy analyzer tube employing magnetic deflection of the electron beam by placement of an electromagnet at B; C is a rotatable magnetic field concentrator;D is the anode; E is a focusing electrode; F is the hemispherical collector while G is a ground potential shield.
acts as a shielding electrode for the hemispherical collector (also operated near ground potential) and accordingly reduces the effect of undesirable reflection inherent in most retarding potential analyzers. In this particular design the anode potential Vf is constant and the focal length adjusted by varying the potential V , of the lens electrode E. The TED curves were taken for fixed values of the ratio V , /V , = 0.003 by sweeping the
FIELD ELECTRON MICROSCOPY OF METALS
217
v c s #c
FIG.11. Electronic potential energy diagram depicting position of emitter Fermi level with respect to collector Ferrni level; $E and +c are the emitter and collector work functions, respectively. Electrons begin to be accepted by the collector only when the collector work function is lowered by an external potential V, to below the level of the emitter Ferrni level.
cathode potential approximately 1.5 V. The energy distribution curves are obtained by graphical or electronic differentiation of the collector current Zp vs. tip bias voltage V , curves. A typical such curve is shown in Fig. 12. A small, rotatable external magnet B and internal concentrator C shown in Fig. 10 allows the beam to be deflected in order to position the desired crystallographic plane onto the aperture probe. More recent designs have replaced the dipole magnetic deflection with a quadrupole electrostatic deflector (30). The latter is more convenient in that the deflection angle can be made independent of anode voltage and the elimination of disturbing stray magnetic fields from the collector region is more easily accomplished. Analyzer resolution is found to deteriorate at large deflection angles for the above design. It was therefore useful to employ oriented wire so that emitters of a desired crystallographic direction along the axis could be fabricated. Such wire can be made by zone melting and electrochemical etching. Since resolution is critically dependent on emitter alignment, particularly with analyzer designs with large magnification of the source (30),it is useful to mount the emitter assembly on a bellows assembly in order to allow it to be positioned externally.
L. W. SWANSON AND A. E. BELL
>
V,
(VOLTS)
FIG.12. Plot of integral current curve, I , , obtained from a retarding potential electron energy analyzer, and corresponding differential curve J ( E ) both plotted as a function of retarding voltage, V,
.
It is difficult to reduce the energy resolution of the retarding potential energy analyzer below 20 to 30 mV and the sensitivity is limited to a signal-tonoise ratio of 1 to 10 %, i.e. AZ/Io 5 0.01 which in turn limits the depth into the conduction band for which meaningful TED measurements are possible. As first shown by Shepherd (57)an approach which circumvents this problem is the use of an electrostatic differential analyzer of the type described by Purcell (58). The tube shown in Fig. 13 is similar to the electron monochromator designed by Simpson (59, 60) which contains virtual entrance and exit apertures in order to eliminate spurious secondary electrons. By employing a virtual entrance aperture to the spherical deflectors and using a computer optimized decelerating lens, Kuyatt and Plummer (61) were able to develop
-
FIELD ELECTRON MICROSCOPY OF METALS
219
FIG.13. Spherical electrostatic deflection electron energy analyzer. Voltages shown are typical, though not unique, operating values. [After C. E. Kuyatt and E. W. Plurnmer, Rev. Sci. Instrum. 43, 108 (1972).]
the Purcell type analyzer into a form capable of energy analyzing field emitted electrons over a current range of up to lo8 with a resolution as low as 10 mV. Basically the electrons enter the hemispherical electrostatic analyzer after being decelerated to approximately 1 to 2 eV. The electron beam passing through the aperture is refocused at the entrance of the analyzer spheres and only electrons in a particular energy range AE are transmitted to the electron multiplier. The resolution AE/Eo (where Eo is the energy of the electrons entering the spherical section is approximately
AE/Eo N w/r, + a',
(37)
where w is the diameter of the entrance and exit apertures, ra is the radius of the electron trajectory. and u is the half-angle of the diverging beam entering the spherical section. Since E = I eV, w/r, = 0.016, and in practice u2 4 w/r,, one can easily calculate a resolution of 16 mV which can be further reduced without loss in current by increasing r, . By virture of the fact that only electrons in the energy increment AE enter the collector, the signal to noise ratio
220
L. W. SWANSON AND A. E. BELL
AI/Zo can be made sufficiently small to allow sampling of several orders of magnitude of current range. D . Electron Impact Desorption
The desorption of a chemisorbed layer by low energy electrons is an important phenomenon in surface physics which has undergone considerable study in recent years by a variety of techniques. The FEEM has been utilized to study electron impact desorption (EID) of a variety of adsorbate-substrate combinations. Whereas most techniques of EID measure the mass-to-change ratio of the desorbing species, the use of FEEM is confined to the measurement of the amount of adsorbate remaining after EID through changes in work function. By use of probe methods one may investigate EID on several crystal faces at once provided that the electron beam impinges uniformly over the surface. If the electron beam flux and work function-coverage relationships are known one may evaluate total cross sections for EID by this method. Because of the extreme sensitivity of FEE to adsorbate coverage, EID cross sections as low as lo-’’ cm2 can be detected and measured. Several FEEM tube configurations have been employed to investigate EID. Figure 14 shows one such embodiment whch combines single crystal face probe techniques with EID (62).A thermionic emitter B in the Faraday collector C can be used along with the suppressor electrode elements as a part of a lens system to focus electrons onto the emitter. In essence, the tube is designed so that the same set of electrodes are used, at different times, as the beam-forming electrodes for the bombardment beam and as Faraday collection electrodes for emission from a small portion of the emitter; this insures that the bombarding electrons impinge nearly normal to the same plane from which the measured field emission current is obtained. The tube has side arms containing adsorbate sources and a phosphor screen to monitor the field electron pattern.
E. Sputtering Measurements Figure 15 shows a diagram of a tube employed for the investigation of sputtering of a field emitter tip by a cesium ion beam of known energy and flux (63). The basic elements of the tube, in addition to the emitter and field electron microscope embodiment, include a cesium ion beamforming gun and Faraday cage collector in line with the emitter and gun axis for the purpose of measuring the cesium ion flux impinging on the emitter surface. The cesium ion beam is formed by vaporizing neutral cesium on a heated platinum ribbon which forms a beam of cesium ions through surface ionization.
FIELD ELECTRON MICROSCOPY OF METALS
22 I
FIG.14. Electron-desorption probe tube in which A is a field emitter, B is a tungsten filament used as an electron source, C is a lens arrangement that can be used either to focus the electron beam or as a Faraday collector, D is a phosphor screen, and E is an electron collector plate. [SeeBennette and Swanson (62).]
FIG.15. Cross section of a field electron emission tube used for the investigation of sputtering of a field emitter tip by a cesium ion beam of known energy and flux. The cesium ion beam is formed through surface ionization. [See Strayer et al. (63).]
222
L. W. SWANSON AND A. E. BELL
The field emitter can be viewed through the microscope embodiment, which is perpendicular to the ion beam. In this way, visual assessment of the sputter damage can be conveniently made. Most of the quantitative information is obtained through measurement of the onset and degree of microscopic surface roughness induced through the sputtering events. The roughness calculation is obtained from F N plots where the change in slope is due to a change in fl. This, of course, implies the removal of surface cesium so that the surface work function is unchanged. When studying inert gas the removal of adsorbate can be accomplished thermally in the temperature range 20 to 100°K depending on the inert gas employed. For cesium, one may employ field desorption at 77°K as a means of removing adsorbed cesium; this can be accomplished without perturbing the sputter damage because of the low field strength (- 3 x lo7 V/cm) at which cesium is removed. The FE technique is primarily used to examine threshold energies for sputtering because of the sensitivity of emission current on p. Sputtering yield vs ion energy relationships can be obtained on a relative basis only so long as surface roughness is proportional to sputtering. The latter is generally found to be the case for slightly sputtered surfaces.
F. Field Emission Retarding Potenrial Measurements FEE techniques can also be employed to measure the "true" work function of a macro-monocrystal surface by a special electron gun design which is essentially a retarding potential analyzer. The basic difference in requirements from the van Oostrom analyzer discussed in Section III,C is a removable planar collector surface and associated alteration of the electron optical system. The unique capability of this technique for work function measurement is discussed more fully in Section IV,B,2. The basic requirement of the electron optical system for this application is to transform the highly divergent electron beam into a parallel beam normal to the collector substrate surface and to simultaneously decelerate it to zero volts. In order to maximize the analyzer energy resolution the electron source must be highly apertured (64) which in turn causes a very low beam transto mission coefficient of the order of However, if the emitter is to be operated at room temperature the resolution of the analyzer need only be 100 mV (65); therefore, an electron optical system which sacrificed unnecessary resolution has been designed (66) for this application in order t o obtain a larger collector current to speed the data acquisition. Rather than aperture the primary beam to the usual 1" half-angle 8, for this application 6' g 8"; depending on the orientation ofthe emitter this aperture angle allowed a beam transmission of the order of 10%. Currents in the A range were
-
N
FIELD ELECTRON MICROSCOPY OF METALS
223
easily obtained in the focused spot, thereby allowing the gun to be used as an electron source for other applications as well. The electrostatic focusing system used in the analyzer shown in Fig. 16 consists of an anode, two Einzel lenses, and a 500 line/in. decelerating mesh electrode which established parallel equipotential surfaces in front of the collector. A two-stage electrostatic focusing system with a virtual cross-over in front of the first lens was chosen over a single stage because of its greater optical efficiency. As the beam enters the Einzel lens it is partially decelerated and forms a virtual image of the source -2 mm behind the emitter tip. The second Einzel lens focuses the virtual tip image into a -0.5 mm spot size at the mesh electrode where further deceleration occurs. The lens system was aligned and mounted securely on four longitudinal glass rods. Both the emitter and anode could be removed as a unit from the tubular anode holder. In this way the emitter, which was held in place by a Corning 1720 glass bead in a molybdenum tube, could be easily replaced and prealigned in the center of the 10 mil anode aperture prior to insertion into the anode holder. The angular convergence of the beam at the collector was fixed by geometry to be < 1.4" for a well focused spot. Hence, negligible loss in resolution resulted from the angular deviation of the beam from perpendicularity at the collector. The single crystal collector substrates of this study were shaped and mounted in the holder as shown in Fig. 16. The face of the collector crystal was circular with a diameter of 200 mils. This was sufficiently large compared to the 20 to 40 mil beam size to eliminate edge effects. Thermal and electron induced desorption cleaning of impurities at the collector crystal was accomplished through electron bombardment. Collector crystals could be easily replaced by removing the glass seal which holds the collector support rod. IV. CLEANSURFACE CHARACTERISTICS A . Total Energy Distribution
In Section I1 a review of recent advances in the theoretical description of the TED of field emitted electrons was given; in this section we shall review recent results of TED measurements. Initially, TED measurements (13) were directed at providing a more rigorous proof of the Sommerfeld freeelectron model assumed by Fowler and Nordheim in their original derivation of the field electron emission process. The early results (13) confirmed the theoretical expectations and no serious experimental challenge of the adequacy of the original FN model was made until energy exchange results (29) and subsequent TED results (31-33) from
BOMBARDMENT FILAMENT
? P
3r
r
FIG.16. Field electron emission retarding tube consisting of a field electron emitter, anode, two Emzel lenses, and a 500 linelin. decelerating mesh electrode which is designed to ensure parallel equipotential surfaces in front of the collector. [See Strayer er al. (66).1
225
FlELD ELECTRON MICROSCOPY OF METALS
certain crystal directions of tungsten and molybdenum could not be explained in terms of the strict free-electron model. The TED results for W(100) and Mo(100) shown in Figs. 17 and 18 exhibit significant departures from the theoretical expectations given in Fig. 2. Additional peaks in the TED were observed at E = -0.37 and -0.15 eV for W(100) and Mo(100) directions respectively. TED measurements along other crystallographic directions of tungsten and molybdenum generally agree with Eq. (8) (32). However, more recent studies (24) using a differential energy analyzer have allowed the extension of TED measurements further below the Fermi level; these results show fine structure in the TED along most crystal directions of tungsten as well as additional structure on the W(100) results 0.78 to 1.5 eV below the Fermi level. I.0r
a9
/\a 1
-
-
0.8
E r .2
0.7
a
E .-
(100) Totol Energy Distribution
L
-
0.6-
2' 0.4 ? 9
-7
0.5
0
0.3-
0.20.1 -
"
-410
t
I
I
-3.0
-2.0
-1.0
0
I.o
2.0
3.0
cld
FIG.17. Experimental total energy distribution along the (100) direction of a W fieM emitter as a function of p , where d = 0.174 eV and F = 4.08 x lo7 V/cm. [See Swanson and Crouser (32).]
An explanation of these anomalous results was first attempted in terms of bulk band structure effects (32,67)along the lines of Stratton. Further studies (24) have shown that the peak at E = 0.37 eV in the W(100) TED disappears with gas adsorption while the peak at E = 0.78 eV is unaffected. On the basis of theoretical advances discussed in Section II,F these results appear to be more reasonably explained by the influence of both surface states and band structure effects on the TED. Figure 19 shows the band structure calculation for W(100) and the corresponding TED results plotted in terms of an enhancement factor N E )
=J'(E))/JW,
(38)
V,
(VOLTS)
FIG.18. Experimental total energy distribution along the (100) direction of a M O field emitter as a function of field. [See Swanson and Crouser (33).1
FIG. 19. (a) Relativistic energy bands for the <100) direction of W calculated by Loucks [Phys. Reo. Lett. 14,212 (1965)] showing the two gaps creating the splitting of the three A, bands. (b) The experimental enhancement factor R(E)for the (100) plane of W (defined by Eq. 38) [After E. W. Plummer and J. W. Gadzuk, Phys. Rev. Lett. 25, 1495 (19701.1.
FIELD ELECTRON MICROSCOPY OF METALS
227
where J’(E)is the experimental value and J(E)the Eq. (8) value. The three A7 bands result in two bandgaps coincident with the E = -0.37 and - 1.5 peaks in R. These peaks also coincide with the energy range of the predicted surface states. The peak at E = -0.78 eV, because of its smaller enhancement and insensitivity to adsorption, is believed to be due to a ramification of the d-band structure originating at r7+. The TED from Cu (I 11) would also be expected to display a deviation from the FN model because of the well known necks in the Fermi surface along the (111) directions. In effect, the absence of electronic states at E, should cause a slight broadening and shift of the leading edge of the TED toward lower E . Measurements by Whitcutt and Blott (34) appear to confirm these general expectations for the TED from Cu (1 11). Recent careful measurements of the high energy “ Boltzmann tail ” have shown a current dependent tail in excess of the Boltzmann tail extending far above the Fermi energy (35, 68). This anomalous result, independent of crystal direction, is believed to be due to electron-hole interactions at the surface. Figure 20 shows the unexpected current enhancement above the
6 (eV)
FIG.20. TED from W(111) at 78 K. The fields and current densitiescalculated from the slope of the Fowler-Nordheim (FN) plot are 0.326, 0.355, and 0.37 V/A and 4.7 x lo3, 2.5 x lo3, and 5.2 x lo3 A/cmZ.The curves are normalized properly from FN plots. Note the anomalous leading edge which increases with current density. [From J. W. Gadzuk and E. W. Plummer, Phys. Rev. Lett. 26, 92 (1971).].
228
L. W. SWANSON AND A. E. BELL
Fermi level obtained by Gadzuk and Plummer (68) for W(111). Lea and Gomer (35)extimated the unexpected current i* (i.e. that produced above the Fermi level in excess of that predicted by Fermi statistics) by noting that the probability of observing an electron of energy Ef + A’ depends on a factor f(A)governing scattering in the metal, multiplied by the probability that both electrons have tunneled; A‘ is an energy above E, measured with respect to it. The tunneling probability, P,in the WKB approximation is
where c is a constant. To first order in A‘, P is given by
P 5: exp( - 2c@I2)/F. Since the total current i is proportional to exp( -cCp3”/F), we see that i* cc i 2 . Gadzuk and Plummer (68) have carried out a more detailed analysis by considering the quasiparticle decay modes in the interacting electron gas. In the present scheme the quasiparticles are represented by the hot holes; then using “ low-energy analysis ” and the random-phase approximation, Gadzuk and Plummer were able to calculate the hole scattering probability per unit length. Next the supply function or secondary electron flux at energy Ek > E, resulting from the hole source at Ep < Ef was calculated from transport equations by Ritchie (69) for the resulting electron and hole fluxes in a homogeneous electron gas containing a steady uniform source emitting one electron per unit volume per unit time per unit energy interval. The volume density of holes created is not uniform throughout the bulk because they are confined by the image potential attraction to within a few screening lengths of the surface. Finally the high energy TED tail,jo‘,was obtained by multiplying the supply function by the WKB tunneling probability. The resulting distribution for Ek > E, yields the correct energy dependence of jc’,the correct field dependence, and correct order of magnitude for the anomalous tail. The effect of the image potential localization was to enhance the magnitude of the anomalous tail. Measurement of the TED for several major crystal directions of tungsten in the temperature range 77 to 900°K gave good agreement with Eq. (8) (32). As the parameter p exceeds 0.7 the usual WKB approximation together with the Taylor series expansion of the WKB exponent about the Fermi energy does not provide an adequate description of the tunneling process. Various other approximations are discussed in the Appendix. Gadzuk and Plummer (70) measured the TED from a tungsten “built-up” emitter at 1570°K and found that numerical evaluation of the full analytical form of the WKB
229
FIELD ELECTRON MICROSCOPY OF METALS
approximation adequately described their results even when emission was partly over the work function potential barrier. Their results also provided direct confirmation of the adequacy of the image potential model to distances 3 to 4 A from the metal surface. Before the realization that surface and bulk electronic effects were able to dramatically alter TED shapes it was hoped tht TED measurements combined with FN plots would provide a method of calculating the true work function value of a single crystal face (71, 72). For E < 0 Eq. (8) becomes J(E)= Jo e"'/d.
(41)
Thus a plot of log J(E)vs. E gives a straight line whose slope me is given by
me = l/d. (42) As pointed out by Young ( 7 4 , Eqs. (33) and (41) can be combined to give the following expression for the emitter work function Q e : 9 e
=
- 3mf t(y)/2me V ~ Y ) ,
(43)
where V is the anode voltage at which the TED was taken. Accordingly, a value of emitter work function can be ascertained from combined TED and FN measurements via Eq. (43) which eliminates assumptions concerning ($) and (P>. Figures 21 and 22 show integral collector current-voltage data obtained from various crystal faces of W and Mo plotted according to the integrated form of Eq. (41), namely,
+
10g[(Zo - Zc)/Zo] = - Qc/2.3d Vc/2.3d,
(44)
where Zo is the maximum collected current level, V, is the emitter-to-collector bias voltage, and Q, is the collector work function. The inclusion of surface patch field effects leads t o a correction term which multiplies the right-hand side of Eqs. (33) and (44) by factors ( -+Fp/pV and (+F,/flV+ I), respectively, where Fp, the net strength of the patch field, is positive for a high 9 plane and negative for a low 9 plane relative to the surroundings. It was shown by Young and Clark (71) that for crystal planes greater than 100 A in diameter a work function difference of at least 1 eV is necessary to cause an appreciable patch field correction. A more rigorous analysis of patch field corrections taking into account the spherical nature of the field emitter was carried out by Politzer and Feuchtwang (7.3). Their results suggest a patch field correction which is at least eight times larger than that of Young and Clark (71). In spite of various degrees of fine structure noted in the TED curves from W and Mo, excluding (100) direction, it was possible to obtain reasonable
+
L. W. SWANSON AND A. E. BELL
230
I-, 1.0
.
plone
- 112
+. x A
o
-
Ill
116 310
I10
-
d (ev) 0.135 0.154
'
0.141
- 0.149 - 0.177
-
I 5.6 -5.8 -6.0 FIG.21. Typical results of the integral field-emission current plotted according to Eq. (44) from various directions of a clean W emitter at 77°K. The calues of d are obtained from the slopes of the plots. The collector work function $c is obtained from the intercept on the abscissa axis at AZp/Io= 1 . [See Swanson and Crouser (32).]
23 1
FIELD ELECTRON MICROSCOPY OF METALS I.o
0.90.8 0.7 0.6
I
-
-
.-
0.4 0.5
0.3
-
0.2
-
0-
110 112
-
0.1 a09 H0 0.08n 0.07-
-
2
0060.0 5
-
0.0 3 0.04
0.02
Ef
0.0 I - 46
I
-4.7
-4.8
,
,
-4.9 -50 Vt (VOLTS)
-5.1
-5.2
-5.3
FIG.22. Typical results of the integral field-emission current plotted according to Eq. (44)from various directions of a clean Mo emitter at 77°K. The values of d are obtained from the slopes of the plots. Collector work function rj5 is obtained from the intercept on the abscissa axis at AZp/Zo = I .
agreement with Eq. (44). Table I gives values of q!Je and 4' for comparison from Mo and W substrates. Values of q!Jc were obtained from Eq. (33) using (4) values of 4.52 and 4.20 eV for W and Mo, respectively. Here we see that agreement between q5e and q!Jf values is good for the (112) directions, but relatively poor for most other directions, particularly the (1 11) direction. Some of the disagreement may be due to local faceting of the thermally annealed end-forms; this is unlikely to be the only cause of the disagreement in the (1 11) results. Thus, at this juncture, with the effects of surface states, band structure, and many body interactions on the TED results, it can be concluded that evaluation of the work function using Eq. (44) cannot be a reliable method until further understanding of the deviations from the simple
232
L. W. SWANSON AND A. E. BELL
TABLE I
WORKFUNCTION VALUES FOR W AND Mo (32, 76)
Mo
W Plane 110 112 112 (flashed) 100 111
310 310 (flashed) 43 1
MeV)
$&V)
+,(eV)
MeV)
6.40f0.09 5.05 f0.05 4.84 & 0.06
5.79k0.04 5.00 f0.02 4.93 5 0.01 4.59 f0.02 4.49rt0.02 4.28 =t0.01 4.21 f0.01
5.12&0.16 4.45 & 0.07
4.81 zkO.09 4.51 ='c 0.07
-
4.82i0.10
4.35 f0.02 4.00It0.08
4.32 f 0.14
4.02 f0.06
-
4.80+.0.03 4.16 0.03 4.34 f0.02
FN model is obtained. In spite of this disappointing result, the unveiling of a host of previously unappreciated surface effects through TED measurements portends an interesting future for TED measurements.
B. Work Function Measurements 1. Emitter Values
The use of the probe FEEM to measure relative work function of various crystal faces of an emitter, introduced many years ago by Mijller (54), has received considerable attention in the past decade. The convenience of this technique used in conjunction with a quadrupole electrostatic deflection electrode makes it extremely attractive for quick measurement of 4f for the major crystal faces. Due to the high surface to volume ratio at the emitter tip, surface tension forces dictate a near atomically smooth surface upon thermal heating. Field evaporation cleaning at low temperature yields a surface free of quenched-in atomic disorder and can be carried out on most refractory metal emitters. An advantage of employing field evaporation cleaning occurs when faced with bulk contaminates which concentrate at the surface in thermal treatment. Also, the large surface-to-volume ratio of the emitter does offer the possibility of exhausting the surface region from bulk impurities by flash heating. Figure 23 shows the variation of the relative work function on short heating cycles at the indicated temperatures for a Re substrate. The lowering of d, in the mid-temperature range arises from an accumulation of low work function bulk impurities at the surface. Higher temperature cycling increases 4 to its clean value due to the rapid thermal desorption of the impurities to relative to bulk diffusion.
233
FIELD ELECTRON MICROSCOPY OF METALS
Ice
1
I.='\,
98-
1
1
1
1
1
, , , , , , , , , , , , .
1
VALUE AFTER FLASHING AT- 2900°K O /O -:
'\\
.-
8.%-
'O\
8OJ \O
94-
9 2 . . . . . . . . . . . . . . . 1
4)
loo0
1500
2000
TEMPERATURE
2500
3000
OK
FIG.23. Variation of work function # of a rhenium emitter initially flashed at 2900°K and heated for 75 sec intervals at indicated temperatures. Results are normalized to the work function #, obtained after initial high temperature flashing.
The variation of /? (i.e. of local electric field at a given applied voltage) with angular separation 8 from the emitter apex is one of the primary difficulties in utilizing Eq. (34) to obtain meaningful results for the variation of relative work function for off-axis crystal planes. This variation in #I is primarily due to the increased shielding of the tip by the emitter shank which causes a monotonic decrease in /? with increasing 8. In addition, another variation in p occurs due to differing amounts of thermodynamically motivated local faceting of certain crystal faces (74). This local faceting varies in magnitude with the temperature at which the emitter is annealed prior to thermal quenching and is largest for the low index planes because of their lower surface energy. The slight effect of annealing temperature on the value of 4f derived from Eq. (34) for a given plane, first noticed by Muller (54), is caused at least in part by temperature-dependent local faceting. To allow correction of the first problem the variation in with 8 has been established experimentally (32) by measuring the relative variation in the FN slope m, for various (310) planes along the [loo] zone line of a (310) oriented emitter. If the reasonable assumption that each (310) plane exhibits identical values of 4 is accepted, then the relative variation of p with 8 can be obtained from mF.The relative value of #I measured for the (310), (130), (310>, and (T30) directions are shown in Fig. 24, where Po refers to the on-axis value. The data points fall within the limits established by an analytical calculation (75, 76) of p/p0 for two sphere-on-orthogonal cone models imitating two emitter profiles with a slight and a pronounced constriction. The deviation of the data from the solid curve representing an average emitter
234
L. W. SWANSON AND A. E. BELL
8 (degrees)
FIG.24. p(e)/,!?, is the relative variation of electric field with angular distance from emitter apex, 8.Dashed curves: (1) emitter with pronounced constriction; (2) emitter with slight constriction. Solid line: for average emitter shape. Experimental data for ,8(6)/& given by circles. Lower curve gives the relative variation of linear magnification with 8;data points indicated by crosses. [See Swanson and Crouser (32).]
shape does not exceed 2%, which is within the accuracy of the F N plots. It is therefore concluded that the corrections to the probe FN work function calculation, based on the average emitter shape curve of Fig. 24, are reasonably accurate. Thus, one of the major sources of error in the FN slope method of determining & for various crystal faces is removed. Also plotted in Fig. 24 is the variation of linear magnification M vs. 8 which is related to the probe hole area A, and the emitter area A , “seen” by the probe hole as follows: M = (Ap/AO)1’2. (45) The emitting area can be obtained directly from the experimental intercepts A f of the FN plots by noting
4= CA,B2/9,
(46)
where C is a constant. Whence, according to Eqs. (35) and (46) the value of M at an angle 8 from the emitter apex relative to the magnification M , along the emitter axis is given by M ( w f , = L ~ , I A , ( ’ ~ ) I ~ ’ ~ P ( .~ ) / P ,
(47)
FIELD ELECTRON MICROSCOPY OF METALS
235
The values of M(8)/Moin Fig. 24 decrease more rapidly with 0 than fl(8)/flo, as expected from the enhanced beamcompression caused by the emitter shank; the exact functional form of the relation M = f ( B ) cannot be established because of possible distortion of the beam by the deflecting magnetic field, particularly at large 8. The values of the work function 4, given in Table I for the various crystal faces were obtained from Eq. (34) by making the appropriate corrections for variations in B(8) as given in Fig. 24. Prior to each measurement the emitter was either annealed for 300 sec at 1OOO"K or flashed to 1800°K and quenched to 77°K at which temperature each of the measurements was made. The results show that the annealed values of dr for the (21 1) and (310) planes of W are slightly larger than those obtained by flashing and quenching. Measurements by Young (77) on W(110) using thermally and field evaporated end forms show no difference in df due to atomic disorder; hence the variations with annealing history is likely due to an alteration of the macroin geometric end form with temperature. In general the W results in Table I for #f agree closely with those of many other investigators (54, 56, 78, 79) using the FEE probe method. Another useful feature of the FEE probe method is the ability to accurately measure the work function temperature coefficient d$/dT. A thorough review of the theoretically expected contributions to d4/dT has been given by Herring and Nichols (80) and experimentally confirmed by both contact-potential (81) and FEE methods (82). Using FEE methods, van Oostrom (82) measured values of dd/dT by monitoring the change in current at constant voltage between 78 and 293°K. A more accurate and less restrictive method consists of obtaining FN plots at each temperature. Using the more general TF formulation given in Eq. (10) the 0°K expression for mf given in Eq. (33) becomes +f
m, = V(1 - np cot np)/2.3 - 2.96 x 107$3/2s(y)/pr,
(48)
When the correction term 6 = V(l - np cot np)/2.3 cannot be neglected, the work function at temperature T becomes
d(T> = 4 0 [ h + s)/mfo12/3[D(~)/So12/3,
(49) where the subscript 0 refers to the low temperature value and p(T) corrects for the thermal expansion of the emitter of radius r and is approximately given by B(T) = P O ( 1 - w.1.
(so)
Thus, field electron Z(V) data can be utilized to evaluate the temperature coefficient of the work function over the temperature range of validity for Eq. (10). In practice, the upper limit of temperature is determined by the threshold of the field-induced geometric rearrangement of the emitter
L. W. SWANSON AND A. E. BELL
236
surface due to surface migration which, in most cases, occurs below the temperature where Eq. (10) ceases to be valid. The work function temperature coefficients for various planes of W and Mo computed according to Eqs. (49) and (50) are given in Figs. 25 and 26 (32, 76). Perhaps the most interesting feature of the temperature coefficients of 4 for W and Mo are their variation in both sign and magnitude with crystal direction. The Mo results agree in sign with the W data, but not in magnitude for the (112) and (116) planes. Further, the (116) results of Mo exhibit a relationship of the form
4 = 3.95 + 4.5 x 10-"T3(eV) rather than the linear (i.e. 4 = 4o aT) form observed for other directions of W and Mo. Interestingly,the Mo result seems to be in approximate accord with contact potential difference results by Gel'berg (83),which were observed by Young (77) to have the form
4 = $o
+ 0.44 x 10-'oT3(eV).
Both results are observed to exhibit a cubic dependence on T over the temperature range 77 to 1000°K. 6.0
1
I
I
I
I
1
( I101
O-O-
5.8
I
-
0 0
56
-
SA
-
52
-
4.8
-
-
-
(100)
O O --o --
4.6
>
0-0
--
4A=-0-----0 0 -
o(130)
-0 0-0
I
4.2L
Id0
I
300
(111)
I
500
(1161 I
roo
900
FIG.25. Temperature dependence of work function for various planes of clean W substrate as determined from field electron emission data. [See Swanson and Crouser (33.1
237
FIELD ELECTRON MICROSCOPY OF METALS I
I
I
4.!
4c
4.z
0
1
->a ~
4.2 I
41
4.0
39
1
100
I
200
I
300
I
I
400 500
I
600
I
I
700 000
TC" K)
FIG.26. Temperature dependence of work function for various planes of clean Mo substrate as determined from field electron emission data. [See (Swanson et at. (76).]
Herring (80) pointed out four major contributions to d4/dT that must be considered. They are (1) thermal expansion of the lattice, (2) effect of atomic vibrations on the internal electrostaticpotential, (3) chemical potential, and finally (4) the effect of electronic specific heat. Inasmuch as effects (1) through (3) are not expected to lead to an alteration in sign with crystal direction, one may interpret the appearance of both positive and negative values of d4ldT as evidence for the importance of band structure effects [i.e. item (4)above] which may vary in sign. For example, it is interesting to speculate that large negative d+/dT values observed along the (loo), (1 12), and (1 10) directions may arise from nearly filled narrow d-bands along these directions (84). Examination of the postulated electronic band structure of tungsten (8.5) shows such possibilities exist along the (110) and (100) directions where nearly filled d-bands occur. Further theoretical study of
238
L.
W. SWANSON AND A. E. BELL
these and other temperature effects on 4 particularly as they relate to transition metals will be needed in order to fully understand their relative importance. 2. Collector Values
Field emission retarding potential (FERP) techniques described in Section II1,F offers one of the few electron emission methods of measuring the true work function of a macroscopic collector surface. This method circumvents many of the model limitations encountered in measuring 4 for electron emitters. A review of these limitations has been described for thermionic, field, and photoelectric emission by Itskovich (20, 86). The FERP approach to work function studies, introduced many years ago by Henderson (87), has largely been neglected with the exception of recent studies of polycrystal surfaces by Holscher (88) and Kleint(89). As will be shown below, the success of the FERP method rests on the theoretical and experimentally verifiable fact that the voltage threshold for collection of field emitted electrons occurs at E, at 0% or can be described by a Boltzman distribution, i.e. exp(E, - E)/kT, a temperature T . For the retarding potential method diagramed in Fig. 11, the emitted electrons can be collected at a metal surface of work function 4, only if their total energy E meets the condition: E > 4, + E, - V, . The condition V, = 4, represents the current cutoff since electronic states above E, are not populated, and the relationship between collected current Z, and V , is given by Eq. (44). It is clear that the values of $o and d can be obtained from the intercept and slope respectively of a plot of loglo(Zo- Zc)Z/ vs. V, as shown in Figs. 22 and 23. The principal source of uncertainty in this method is due to the occurrence of electron reflection which alters the slope and intercept of Eq. (44) plots. Alternatively Eq. (8) may be differentiated with respect to E in order to obtain the difference in energy E~ between the peak of the TED and E,: E~
= kT ln[kTj(d - kT)].
(51)
This equation, plotted in Fig. 27 at several values of d, may be used to obtain the theoretical value of ep which is equal to 4, - Vp. Since Vp (the position of the maximum dZc/dVcon the energy axis) can be obtained experimentally, the value of 4, can be obtained directly from the TED curves as depicted in Fig. 28. It can easily be shown that the analyzer resolution and electron reflection even at Vp have a minor effect on the accuracy of Vp particularly when the emitter is at room temperature. Values of 4, obtained by the FERP method for several single crystal surfaces are given in Table I1 (66). One can observe that the values compare favorably with those obtained by other methods.
1
I
I
70 60
1
I
d * 0.25 cv
i
I
/ -
I
\O.lO
FIG.27. The difference in energy E, between the peak of the total energy distribution curve and the Fermi energy level as a function of temperature T and energy parameter d.
45
47
4s 61 COLLECTOR VOLTAGE (VOI
FIG.28. TED plots for collector crystals of W (IlO), W(111), and W(100)obtained in a retarding potential tube.
L. W. SWANSON AND A. E. BELL
240
TABLE 11 SUMMARY OF WORK FUNCTION VALUES OBTAINED BY VARIOUS TECHNIQUES
Material W(110) W(100)
W(111)
Ir(l1 I) Ir(l10)
Nb(100) Ni(100) Cu(100)
Field emission (eV) (collector values)
5.25 f 0.02 4.63 f0.02 4.47 f 0.02 5.76 & 0.04 5.42 i~0.02 4.18 f0.02 5.53 0.05 5.10 f 0.05
Thermionic emission ( W
Photoelectric emission (eV)
5.35 z t 0.05 (90) 4.60 0.05 (90) 4.40 0.02 (90) 5.79 f 0.03 (91)
Field emission (eV) (emitter values) 5.9 0.01 (32) 4.7 & 0.05 (32) 4.45 f 0.03 (32)
**
3.95 z t 0.03 (92)
3.87 & 0.01
5.22 & 0.04 (93) 4.9 (94)
Another interesting feature of this method is the ability to measure the electron reflection coefficient of both elastically Re and inelastically Ri, scattered electrons. Since the reflected electrons are collected at the mesh electrode (see Fig. 16), the sum of the collector current Z, and mesh current Z, is given by lp= I,
+ I,,
(52)
where Ipis the emitter current arriving at the mesh. Noting that the mesh transmission is given by Zp’/Zp = T , where Zp‘ is the current impinging on the one obtains collector, and that ( I - R) = Zc/Zp’,
- R) = Zc/T(Ic+ Is).
(53) Since T, Z, and I, are measurable quantities R may be determined as a function of V, as shown in Fig. 29 for an Ir (1 11) substrate. Note that the abscissa is given in terms of the primary beam energy Ep + V, - $ c . When V, - 4, > V, - $s (s refers to the screen-mesh electrode) those reflected electrons which l~oseenergy through inelastic processes will be returned to the collector. Thus, by fixing V, 5.0 V only specularly reflected elastically scattered electrons will escape from the collector thereby allowing Re to be measured. By setting V, >E,(max) all reflected electrons return to the mesh and the total reflection coefficient R, is measured. Hence, it is possible to measure the inelastically reflected electron coefficient Ri, by noting that Ri, = R,- R, . Both Ri,and Re are given in Fig. 29 for Ir (111). It can be noted that the threshold value of Ep is -8 V larger than for Re and that Ri, > R, for Ep> 20 V. An unusual feature for this result is the unusually large value of (1
COLLECTOR
:
CURRENT
,----_-_--_-___________________
-
It
I I I
I
4
c
Y
+ z
" 0o
(theoretical )
-----------
oooo%oap~~
0--00-
0
0
0O0
0
"-Jo.tl
0 00
-oo 0
--
0
0 0
0"
-"pomooo~---
"0 0
10.7
0-
0
w
a a
2 la a
2
0
w
J
J
0 0
5
0' ELECTRON ENERGY E,, ( o V )
- - - - - - O v
r
FIG.29. Experimental and theoretical Z(YJ curves obtained from Ir(l11). Solid line curves show inelastic R,. and elastic R, reflection coefficients. [See Strayer et al. (66).J
242
L. W. SWANSON AND A. E. BELL
Re as E, 40. In spite of the large value of Re near Ep= 0, the value of 6, can be accurately determined and the value of R , can be measured within a few tenths volt of Ep= 0. In summary the FERP method is uniquely suited to conveniently and accurately measure true work functions of single or polycrystalline surfaces. In addition, reflection coefficients for both elastically and inelastically scattered electrons can be measured down to Ep 0. C . Energy Exchange Effects
Electron emission is accompanied by energy exchanges between the conduction electrons and lattice, which becomes particularly important at the very high emission densities feasible with field and thermal field (T-F) emission cathodes. Their study is of basic interest as it provides a complementary check, through a direct measurement of the average energy of emitted electrons, of the theory of field and T-F emission; it is also ofpractical importance because these energy exchanges control the cathode emitter tip temperature and set and upper limit on the feasible emission density. Most work on this subject has been directed toward attempts to verify the relationships for T* and (&} expressed in Eqs. (9) and (11). There are two main emission induced energy exchange phenomena. The familiar resistive Joule heating effect was studied in the case of field emission by Dyke el al. (95) and Dolan, Dyke, and Trolan (96). In the usual case where resistivity increases rapidly with temperature, resistive heating by itself leads to an inherently unstable situation at high emission densities. Since stable high density emission is observed (97), it was concluded that another factor must exist having a strong and stabilizing influence on the cathode tip temperature. Such a stabilizing factor is provided by the energy exchange resulting from the difference between the average energy of the emitted electrons, ( E ) , and that of the replacement electron supplied from the Fermi sea, (E’). In the case of thermionic emission this phenomenon, discussed by Richardson (98) and later by Nottingham (99),is well known and produces cooling of a cathode with a work function 4 by an average energy amount e+ + 2kT per emitted electron. The corresponding effect in field and T-F emission was first discussed by Henderson and Fleming (100) who were unable to detect it experimentally, and was a subject of controversy (99, 100) with respect to the correct value of ( E ’ ) and hence the direction of the effect (cathode cooling occurs when ( E ) > ( E ’ ) and heating when ( E ) < (E’)).Earlyresults (101) tended to support the view of Nottingham who took ( E ‘ } to be the Fermi energy and, on that basis, predicted heating of the cathode in the case of field emission. Thus, the energy exchange corresponding to the replacement electron
FIELD ELECTRON MICROSCOPY OF METALS
243
at energy E, was referred to as the “Nottingham Effect.” More recent experimental results (29) fortungsten indicated that ( E ’ ) # E, ,which, at first glance seemed to support the view of Henderson and Fleming. However, this agreement was found to be fortuitous and the observed difference between ( E ‘ ) and E, was attributed to the fact that conduction processes in tungsten are not well described by the free electron model. The combined effect of resistive and Nottingham phenomena has been treated in the special case of a tungsten field emitter initially at room temperature (102). Levine (38) gave a theoretical analysis of a similar problem. Dreschsler (103) has reported both departure and agreement with theoretical predictions for the temperature dependence of the Nottingham Effect and for the value of the inversion temperature for tungsten. Consider a cathode field emitting electrons with an average electron energy ( E ) , which are collected at the anode and subsequently conducted back to the cathode through a conductor at temperature T. Letting ( E ’ ) be the average energy of the conduction electrons in the emitter, then a net energy exchange of an amount EN
=(E)
- (El)
(54)
is released to the lattice at the cathode. As pointed out originally by Nottingham (99), for the case T = 0 conduction in the external circuit occurs at the Fermi level (except through the batteries) and EN
= < E ) - Ef’,
(55)
where El” is the 0°K Fermi energy. The functional dependence of ( E ) on and F is given by Eq. (11); however the assumption that ( E ‘ ) = Ef must be examined further. For a Fermi gas at temperature T, conduction involves only those electron states for which the derivative of the Fermi function df((E)/dEis finite. Since the derivative of the Fermi function is appreciable only in a range of a few kTabout E , , i t is clear that conduction is limited to states near the Fermi surface at low temperatures. In most experiments, the values of kTranged from 0.02 to 0.07eV, while experimental values of eN ranged from -0.60 to +0.30 eV; hence, near the inversion condition (eN = 0) a variation of (E’) by a few hundredths of an eV may cause the inversion temperature to depart significantly from the predicted value based on Nottingham’s assumption that ( E ’ ) = E , . It is therefore instructive to obtain the expression for the average energy of the charge carriers in a conductor possessing a temperature gradient and internal electric field. An expression for ( E l ) has been given by Mott and Jones (104) for a conductor in which the electron energy E is an arbitrary function of the wave vector k . A slightly different expression for ( E ‘ ) has been derived by Seitz
244
L. W. SWANSON AND A. E. BELL
(105) by considering the Sommerfeld-Lorentz solution of the Boltzmann transport equation. In the latter case the interaction of the electrons (viewed as a degenerate Fermi gas) with the lattice is contained in a function A(E, T ) representing the mean free path of the conducting electrons. The final form of the expression to the first order in kTIE, is
where A'(Ef, T ) is the value of the first derivative of 1 with respect to E at the Fermi level. The function )I(E,, T ) is undetermined by the SommerfeldLorentz theory and must be calculated from more detailed quantum-mechanical approaches. The direction and magnitude of the variation of ( E ' ) with T is ultimately governed by the function )I(JZf, T ) . Upon examining the expectation regarding the direction of the variation of ( E ' ) with T for a more general model, Mott and Jones (104) concluded that d ( E ) / d T should be positive for metals possessing partially filled bands and negative for those possessing nearly filled bands. They further concluded that transition metals with overlapping s- and d-bands, in which the main resistence producing factor is s d transitions, should exhibit a positive d(E')/dT for a nearly filled d-band, whereas for a partially filled d-band the sign of d(E')/dTwill depend on the E(k) and A(E, 7') relationships near the top of the Fermi distribution. As expected, both positive and negative d(E')/dT have been measured for the transition metals and, for many, both the magnitude and sign of d(E')/dT vary with temperature. In view of the limited knowledge of the E(k) and R(E, T ) relationships for most transition metals, it is more fruitful to estimate the variation of ( E ' ) with T from experimental values of ( E l ) vs. T rather than from first principles through Eq. (56). This can be accomplished as pointed out by Seitz (105), by expressing ( E l ) in terms of the thermoelectric power S as follows :
( E ' ) = El - eTS,
(57)
where
and p is the Thompson coefficient of the metal. The quantity eTS represents the reversible heat carried by the current whose direction of flow, relative t o the current flow and temperature gradient, depends on the sign of p. In keeping with the usual convention a positive p signifies the evolution of heat as electrons go to places of higher temperature and according to Eq. (57), a lowering of the average energy of the charge carriers.
FIELD ELECTRON MICROSCOPY OF METALS
245
Based on Potter’s (106) values of S for tungsten, one can expect a small decrease in ( E ’ ) - E, with increasing temperature (e.g. 0.015 eV at 900°K). The main difficulty in measuring energy exchange phenomena in field and T-F emission is the strong localization of these phenomena and of the associated temperature changes; this localization results from the cathode geometry (very sharp needles with a conical shank and a tip radius well below one micron) with which controlled field emission is most reliably obtained. A determination of both the magnitude and the location of the energy transfer requires measurement of the temperature at the emitting area itself, which is of the order of cm’. For this purpose, temperaturesensitive coatings of materials which alter the cathode work function have been used to sense the local tip temperature. Measurements of this type conclusively established (29) the existence of emission heating and cooling domains; within the limit of experimental accuracy, they also confirmed the magnitude of the energy exchange and its localization within a few tip radii of the cathode tip. However, the complex experimental conditions (pulsed emission, large field, and temperature gradients near the tip, etc.) limited the accuracy of this approach; therefore, a more precise method was used to measure the magnitude (but not the location) of the energy exchange and the inversion temperature. A method (29) employed to obtain quantitative measurements of the energy exchange accompanying field emission was a refinement of that used by Dreschsler (103) who obtained field and T-F emission from random protrusions on the surface of very thin wires. Briefly stated, the method was to provide an emitter-support filament of sufficient thermal impedance that the small heat input resulting from a low field electron emission current may be detected sensitively through the associated change in temperature and resistance of the filament. Reliance on emission from several random protrusions of unknown number, geometry, and location creates uncertainties in the interpretation of the data which was avoided by confining emission to a single-field emission needle (whose precise geometry could be determined in an electron microscope and from field electron emission I- V characteristics) mounted at the center of a smooth wire an inch long and approximately 1.1 mil in diameter. The thermal impedance of the structure was sufficiently large that emission-induced power inputs as low as 10 pW could be detected and measured. This had the advantage of permitting good measurements to be made at low dc current levels where the emission is highly stable and where the Nottingham Effect strongly predominates, and resistive heating (which can be calculated only approximately) has only a relatively small effect. The emission-induced power input H at the emitter was derived from the change AR in support filament resistance caused by the associated change AT in support filament temperature.
246
L. W. SWANSON AND A. E. BELL
Assuming for the remaining part of this discussion that the Notingham energy exchange with the lattice is cN = ( E ) , i.e. that ( E ' ) = E , , then one obtains for the total power input H to the lattice
+
H = ( & ) I , H,,
(59)
where Z, is the total emitted current. H , is the power input due to Joule heating, and positive values of H refer to a net power input to the lattice. An approximate expression for the resistive power exchange for a conical emitter of radius r and cone half-angle u is given by the following expression: H,
p(T)Ie2/nur,
(60)
where p ( T ) is the bulk resistivity at the temperature T . The complete expression for H obtained from Eqs. (12), (59), and (60) is
H = nkTI, cot np + p(T>Io2/nur.
(6 1)
An analytical expression for the temperature Toat the emitter apex compared to the temperature TI at the emitter base (assumed fixed) can be obtained (102)for a truncated conical shaped emitter of radius ro at the point of truncation; neglecting radiation losses and assuming ro = r the expression takes the form :
where K is the thermal conductivity. The first term in Eq. (62) is due to the Nottingham Effect whereas the second term is caused by Joule heating. As mentioned previously, the observed temperature stability of field emitters at high current density levels, where Joule heating by itself should cause strong instability, provided initial evidence of the Nottingham Effect. It was subsequently observed that strongly bound adsorbed layers on tungsten, such as zirconium-oxygen layers, which lowered the work function, also reduced the emitter temperature and allowed even greater emitted current densities before the onset of excessive tip heating. In one experiment (29) illustrated in the photos of Fig. 30 a peak pulse current (duty factor approximately of 57 mA was obtained from a clean (110)-oriented tungsten emitter before the onset of instabilities due to Joule heating. This current caused sufficient heating to activate diffusion phenomena, producing slip planes and roughening of the tungsten surface. After annealing the emitter to restore a smooth surface and applying zirconium coating to the emitter a pulse emission current of greater than 102mA was obtained. This results from the increased emission cooling characteristics of the lower work function (4 = 2.8 eV) surface. Thus, the tip temperature was substantially less than for the clean emitter at half the current. As shown in Fig. 29c. the
FIELD ELECTRON MICROSCOPY OF METALS
247
FIG.30. (a) Typical field electron pattern of clean and smooth (1 10)-oriented tungsten emitter. (b) Pattern of the same emitter, showing cumulative disruption of the clean tungsten surface (just prior to vacuum arc) resulting from emission of a peak pulse current of 57 mA at 15.3 kV (4 = 4.52 eV) with a duty factor of (c) Pattern of the same emitter, with an adsorbed zirconium-oxygen layer and emitting stably a peak pulse current of 102 mA at 14.8 kV (I$ = 2.8 eV for the brightly emitting areas). (See Ref. 29.)
lowering of the work function due to zirconium adsorption occurs selectively in the (100) regions, while the remaining surface exhibits nearly clean tungsten characteristics. Thus the emitted current density in the zirconiumcoated regions of Fig. 29c is approximately four times that of the clean tungsten surface in Fig. 29b. These observations are consistent with the foregoing theory of the Nottingham Effect. Equation (7) shows that a constant current density J
248
L. W. SWANSON AND A. E. BELL
requires ~'I'/F
z const. = C.
(63) Combining Eqs. (9) and (63) gives the following expression for the inversion temperature at a constant J : T* = 5.67 x 10-5 4/t(y)c (64)
for 4 in eV. Thus, a lower T* (or larger cooling effect) is obtained at a constant J as 4 decreases. Alternatively, for a given allowed tip temperature a larger J can be obtained as 4 decreases. Figure 31 shows the measured power exchange H at a clean tungsten emitter as a function of I, for various temperatures (29). Three general observations can be obtained from the data given in Fig. 31 : 160
-
0
140
-
@ T = 792'K
120
-
100
-
T -297.K
Q T'562'K
-P -
% 80 c
-
i
I 60-
40
-
20
-
0
100
200
300
400
500
FIG.31. Experimentally determined power exchange H a t the emitter as a function of field emitted current I. at the indicated temperatures for clean tungsten where 4 = 4.52 eV. Negative values of ff indicate emission cooling. [See Swanson e l al. (291.1
FIELD ELECTRON MICROSCOPY OF METALS
249
1. H increases nearly linearly with I, at low temperatures. 2. It appears that the amount of heating at a specified I, decreases with increasing temperature. 3. The Nottingham inversion (i.e. H = 0) occurs at increasingly higher values of I, (or F) as the emitter temperature increases. The first two observations are in qualitative agreement with the predictions of Eq. (61) since the Joule heating term is negligibly small in the present experiments. Recalling that the inversion temperature T* corresponds to p = 1/2, it follows thatp = T/2T* and Eq. (61) can be rewritten (neglecting the Joule heating term) : H E HN = ZkTI, cot(zT/2T*).
(65)
At low temperatures or large electric fields the term cot(nT/2T*) is insensitive to F,thereby causing H to vary in a near linear fashion with I,; also, as T increases the T cot (nT/2T*) term decreases, thereby causing H to decrease. As T approaches T*, H is no longer proportional to I , , but instead depends more sensitively on the variation of T* with F and, hence, I , . When the condition T = T* is attained, then H = 0. At higher temperatures Nottingham cooling is observed at low currents. However, as the emitted current is increased (at fixed emitter temperature) T* increases according to Eq. (9) and the transition from Nottingham cooling to heating occurs at a current which increases with emitter temperature. The results of Fig. 31 are in qualitative agreement with these predictions of Eqs. (9) and (65). The experimental results can be compared quantitatively with theory in two ways. First, the inversion temperatures given theoretically by Eq. (9) can be compared with those obtained experimentally for clean and zirconiumoxygen coated tungsten. The experimental and calculated values of the inversion temperature are compared in Fig. 32 for the clean and low work function zirconium-oxygen coated tungsten surfaces. In both cases the experimental inversion temperatures vary linearly with field, but are substantially below the predictions of Eq. (9). An earlier study of the Nottingham Effect by Drechsler (103) also revealed an anomalously low value of T* for tungsten. The above mentioned discrepancies are significant since emission cooling of tungsten field emitters appears to occur at lower temperatures and to be much more important than predicted by the Sommerfeld model. This is further illustrated by the second method of comparing the experimental results with theory, as shown in Fig. 33 where the average energy exchange ~ H/Zc), as obtained experimentally and also theoretically per electron ( E = from Eq. (12), are plotted as a function of F. The experimental values of E~ given in Fig. 33 for clean tungsten at 297°K are larger than theory, while at higher temperatures the experimental values of cN are considerably less than
L. W. SWANSON AND A. E. BELL
250
Clean W (4 = 4.52 eW (a) experimental (a') calculated
Zr on OW (4 = 2.67eV) (b) experimental (b') calculated
t Y
/
2I)
3.0
/
/
I y
/
/
/
/
/
/
/'
/
4.0 F (10' V/cm)
5.0
6.0
FIG.32. Experimentally determined inversion temperatures T for a clean and zirconium-coatedtungsten emitter as a function of applied electric field. Dashed curves are the respectivecalculated inversion temperatures according to Eq. (9). [See Swanson et al. (29.).]
expected from theory particularly at low values of F. Hence, at a fixed field .zN decreases with emitter temperature much more rapidly than predicted by theory, particularly at low values of F. Thus, both the clean tungsten and zirconium-oxygen coated tungsten results provide clear evidence of a significant departure of the Nottingham Effect from the theory based on the assumption (E') =" Ef. In an attempt to explain the observed anomalies in the Nottingham Effect, one is led to examine the two basic premises of the theory. First, does Eq. (8) adequately describe the total energy distribution over the temperature and field range investigated ? Excluding the W(100) results, measurements (32) of the TED performed in the temperature range 77-900°K on tungsten have generally supported Eq. (8), and, therefore, the model upon which Eqs. (8) and (12) are based. Although measurements of the TED exhibit minor discrepancies from FN
25 1
FIELD ELECTRON MICROSCOPY OF METALS
/ i
-0.3-
-0.4
-
-0.5
-
(0)
T
(b) T
(d)
46
I
I J 1
40
I
50
I
I
297' 562.
K K
(c) T m792.K (d) T
A
-0.J
-
I
I
52 54 F (MV/cm)
I
I
56
961.
'
50
K
I
I
60
FIG.33. Solid curves represent the experimentally determined energy exchange per electron eN with the tungsten lattice as a function of applied electric field Fat the indicated temperatures for clean tungsten. Dashed curves are correspondingones calculated according to Eq. (12). [See Swanson et al. (29).]
theory as discussed in Section IV,A they are not sufficient to account for the observed anomalies in the Nottingham Effect. This suggests an examination of the second basic premise of the Nottingham Effect, namely that the average energy of the replacement electrons is equal to the Fermi energy. Investigations of the magneto resistance (107, 108) and surface conductance (109) of tungsten have provided evidence that electrical conduction in tungsten involves nearly equal numbers of positive and negative charged carriers. If the holes of a nearly filled d-band take part in the conduction, one may expect, according to the discussion following Eq. (56), a negative value of d(E')/dT. Furtheremore, if the hole band possesses a low degeneracy temperature Td, the contribution to Eq. (56) of the neglected higher order terms becomes significant at T + Td;a conduction mechanism of this nature,
252
L. W. SWANSON AND A. E. BELL
which causes ( E ’ ) t o decrease with increasing temperature, may partially account for the anomalously low observed inversion temperatures. A variation of ( E ’ ) with temperature should result in a uniform displacement of the experimental curves of eN(F),at various temperatures, from their theoretical values. As shown in Fig. 33, this is confirmed for the 297°K results, but the direction of the shift requires a positive ( E ’ ) - E,. In contrast, the higher temperature results of Fig. 33 exhibit a deviation from theory which requires a value of ( E ’ ) - E, whose magnitude and sign varies with field. Engle and Cutler (110) considered the effect of different surface barriers on the energy exchange process and concluded that such corrections could not account for the experimentally observed discrepancy. The same authors (111) later considered the effect of a nonequilibrium distribution of conduction electrons on the replacement electron and found ( E ’ ) several hundredths of an eV less than Ef.This lowered the inversion temperatures appreciably but not enough to agree with experimental observations of Drechsler (103) and Swanson et al. (29). The anomalous enhancement of the Boltzman tail of the TED described earlier (35,68),and believed to be caused by Auger type processes involved in the replacement of holes left by the emitted electrons, can account for a slight additional lowering of the apparent energy level of the replacement electrons. It appears at present that a combination of electron-electron interactions and bulk conduction complexities characteristic of nonfree electron metals account for the observed discrepancies.
D. Noise Studies Extensive theoretical and experimental studies of statistical fluctuation in the current from a FEE cathode have been carried out by Kleint and Gasse. (112-218). Both shot (119) and flicker (120,121) noise, which have been examined in connection with thermionic emission, have been observed from clean (112) and adsorbate (114) covered emitters respectively. More interesting of the two types is flicker noise which can be related to stochastic processes due to surface diffusion and desorption of an adsorbate. In noise measurements one usually obtains the mean square noise power (usually reduced to a one ohm resistance) (fir2)which is related to the well known spectral density function W ( f )as follows:
Wf2)f,...f2 = J,k) ds.
(66)
According to early work by Schottky (119) with regard to shot noise in thermionic emission, W ( f )is frequency independent and linearly dependent on current I , , that is W ( f ) = 2eZ0. (67)
FIELD ELECTRON MICROSCOPY OF METALS
253
Both of these expectations have also been observed (112,115) for FEE from clean surfaces over the frequency lo2 to lo5 Hz and a factor of ten change in Zo. With the presence of adsorbed layers flicker noise becomes dominant and can generally be related tofas follows:
Wf)
lif”,
where 0.95 < E < 1.2 for residual gas adsorbates (115) and 0.8 < E < 1.3 for adsorbed layers of potassium (118). Figure 34 shows a typical variation of W(f) with f obtained by Kleint (115) for a field emitter coated with residual gas at room temperature. The shape of the curve in Fig. 34 has been explained
FREQUENCY
FIG.34. Plot of spectral function W ( f )vs. frequency for a W field electron emitter covered with a residual gas layer. (1) is the experimental curve, (2) the theoretical curve calculated according to Eq. (68), and (3) is the shot noise level; curve (4) is the sum of the curves (2) and (3). [After C. Kleint and M. Gasse, Forsch. Phys. 13, 499 (1%5).]
by a model in which the adsorbed molecule can exist in two states of lifetime T~ and T ~ Using . earlier calculations for semiconductor noise (114, Kleint (115) has shown that such a model leads to the following expression for W(f):
W(f)= 4(AZ)*
T o f1
(To
+ 7,)2
.-
1
0
+ w202
where l/a = I/?, + I/?,, w = 2nf, and AZ is the change in current when the adsorbate switches from state 0 to 1. Figure 34 shows a plot of Eq. (68) where z0 = 2.59 msec, ti = 1.4 msec, and A I = 5.81 x lo-’’ A. A reasonably close fit is obtained provided that shot noise is taken into account at high frequency. The spectral density function for shot noise includes the two terms 2eZ0
254
L. W. SWANSON AND A. E. BELL
+ 4kT/R,, where Re is the equivalent input resistance to the spectrum analyzer. The temperature dependence of flicker noise for the above model can be obtained by postulating an activated process for the transitions between the With the further assumption that N two adsorbed states, i.e. T~ = AieELIkT. adsorbate molecules are undergoing uncorrelated transitions the spectral distribution becomes (115): W ( j )= [AZ2NkT/w(E,- E,)] (arctan w z1 - arctan w zo).
(69)
Thus if l/zo > o > I/z, the above equation predicts a linear dependence of W ( ( f )on T and llf. From the basic FN equation (see Eq. 7) one can obtain the following relation between AZand the work function change A 4 caused by the transitions between the states 0 and 1: A I = -$bq5‘”IA4/F,
(70)
where b is a constant. Depending on the details of the adsorbate states Aq5 can be related to the number of adsorbed molecules per unit area N i in a particular state and the dipole moment pi by
Aq5 = 2npi N i . (71) Assuming a two-state system as before and combining Eqs. (69 to (71) gives the following expression for W(f): W ( f )=
- arctan o z O ] [ p o N+O 4% - Eo)
Z2NkTBZ[arctanw q
(72)
where B = 37~bq5’/~/F and N = N o = N , . Thus, from Eqs. (72) it is clear that the flicker noise amplitude is quadratically related to I , increases with temperature, decreases as Ilfonly under certain conditions (as pointed out above), and is proportional to the adsorbate coverage and magnitude of the dipole moments of the different states of adsorption. Using a statistical thermodynamical approach Kleint (118) has extended the two-site model further to include diffusion, desorption, and the explicit dependence of noise on adsorbate coverage. From this approach one obtains a low noise factor at 0 = 0 and 6 = 1 with a general maximum in the midcoverage range-the exact coverage dependence being dependent on the magnitude of the mutual interactions. In the temperature range where only diffusion occurs the noise factor increases with temperature. The predicted frequency dependence of W(f)varies from llfz at high frequencies to independence off at low frequencies. This frequency dependence appears to have greater support from the existing experimental data than does a patch model of the flicker noise put forth by Timm and van der Ziel (122) which predicts a llf3I2 dependence.
255
FIELD ELECTRON MICROSCOPY OF METALS
Figures 35 and 36 show the mean square current deviation obtained by Kleint (118) for potassium and nitrogen on tungsten with increasing temperature. The occurrence in the nitrogen results of several peaks rather than one is believed to be associated with the desorption from various states of adsorption. Interestingly, the sharp minimum occurring in the potassium results at the minimum work function coverage also coincides with the minimum observed elsewhere (52) in the surface diffusion coefficient. This probably represents a two-dimensional phase change due to mutual interactions. Similar structure in the desorption spectra of hydrogen (123) and carbon monoxide from tungsten confirms the association of noise peaks with specific adsorption states. In summary, it appears that noise measurements can provide considerable information regarding the energetics and kinetics of adsorbed layers on a field emitter. As pointed out by Kleint one may extract activation energies of desorption and diffusion of surface layers from the details of the noise spectrum. More precise relationships between experiment and theory will be possible by using probe methods in connection with noise measurements. Y
r
=
3 N ' e
19 z
0
.' W 0 IW L
3
:
W
5: 2 W
E
I
1000 I E M I T T E R TEMPERATURE
I
-.-
FIG.35. Noise and voltage vs. temperature curves for potassium adsorbed on a tungsten field emitter. The voltage V,.,,, is that required to draw a total current of 0.6 PA from the emitter. [From C. H. Kleint, Surface Sci. 25, 411 (1971).]
256
L. W. SWANSON AND A. E. BELL
FIG.36. Noise and voltage vs. temperature curves of a high coverage (1) and a lower coverage (2) nitrogen layer; VZpArepresents the field emission voltage required to draw 2 pA current from the emitter. [From C. H. Kleint, Surface Sci. 25, 411 (1971).1
E . Magnetic Field Effects The application of a high magnetic field directed along the axis of a field emitter can effect the total emission current in the following two ways: first, a degenerate electron gas exhibits an oscillatory dependence of density of states and chemical potential on magnetic field, and second, spin polarization of the conduction electrons can be effected through external magnetic field application. Both of these magnetic effects on the metallic electrons have been detected experimentally and predicted theoretically as a perturbation in the emitted electron current. The quantizing effect of a magnetic field on a degenerate electron gas adds a term ko,(l + 3) to the energy of a free electron. Thus, the energy E, of an electron in a magnetic field H along the z direction is given by
E, = hw,(l+
3) + hZk2/2m,*
(73) where w, = eH/m*c is the cyclotron frequency of electrons of effective mass m* and 1 is a positive integer or zero. Several authors have derived the expression for the current density J in the presence of a longitudinal quantizing
FIELD ELECTRON MICROSCOPY OF METALS
257
magnetic field (124-126). By restricting consideration to the limit T-t 0 Blatt (124) was able to show
where A = 2e2He-[h(Er)+Er/dl /h Zc and
+
El = horn(( 4).
Carrying out the summation indicated in Eq. (74) it can be shown that J decreases monotonically with increasing H . In addition, discontinuities in the density of quantum states at the bottom of each Landau zone leads to an oscillatory component in J (124, 126). Experimental measurement (127) of J . vs. H for tungsten in the range H = 0 to 15 k C and for T = 77°K showed a distinct oscillatory component and an overall decrease in J with increasing H , thus verifying the above the-, oretical expectations. Although the possibility of obtaining spin polarized electrons from a field emitter was considered some time ago (128), only recently has it been verified experimentally for gadolinium (129, 130). The spin polarized electrons in the case of Gd at T = 85°K reached a maximum level of 8 % at H = 10 kG and decreased to zero at H = 30 kG. A detailed theoretical treatment of spin polarization of field emitted electrons from ferromagnetic 3d elements has been given by Obermair (131). In the usual theoretical approach a model is considered in which a split or magnetization of the conduction band electrons occurs through an “exchange field” BE = S/Jp, (where S is the spin magnitude and J the exchange coupling integral) which is parallel to the magnetization of the localized spins. For ferromagnetic 3d metals it has been assumed that 4s conduction electrons are polarized by a 3d-4s exchange interaction. This leads to two spin subbands and the existence of two different Fermi surfaces for spin states s = +,that is
+
E, = Ef(kf1) S J = E , ( k f f )- SJ,
(75)
where k,T and k f l are the k-vectors whose endpoints describe the two spinsplit Fermi surfaces. Since only states near the Fermi level contribute to the should be given by field emission current, the degree of polarization P,,,
258
L. W. SWANSON AND A. E. BELL
where N(E,) is the density of states at E, . Obermair (131) has considered these implications in regard to the FN equation and concluded that the degree of spin polarization in the bulk should be transmitted to the tunnel current provided that external local transverse fields and macroscopic field inhomogeneities do not cause excessive depolarization. Obermair (131) also concludes that the 3 d - 4 ~exchange coupling model has features which tend to cancel the net polarization of conduction electrons near E,; however, the 3d bands themselves which also have Fermi surfaces, depending on crystal direction, may contribute to the spin polarization of the tunnel current. He further points out that the rare earth elements with their localized 4f moments are indeed capable of interband exchange. Very recent experimental results tend to confirm the above expectations. Electron spin polarization (ESP) values up to 50 % at T = 4°K have been measured for field emission from EuS (132). The latter is a ferromagnetic insulator with a localized and polarized 4f electron band. In addition, Ni and Fe have exhibited degrees of spin polarization of 13% and a few percent respectively at T = 80°K (133). Interestingly, W has exhibited a degree of spin polarization of 20 % at 80°K and 20 kG (133). These results indicate the need for further understanding of the detailed role of band structure and orientational effects on spin polarization in field emitters. The low value of ESP for ferromagnetic Fe and high value for nonferromagnetic W remains to be fully understood. Clearly, the effect of high magnetic fields on the field emitted current remains a fruitful area for further research. In addition, FEE appears to be a useful high current density source of spin polarized electrons. V. SURFACE ADSORPTION A . Work Function-Coverage
Measurements
The probe technique has been used in the last decade to study a variety of adsorbates on various single crystal planes of tungsten. Of particular interest is the work of Gomer (55,234, 235) and co-workers in using work function measurements as a probe of surface coverage in equilibrated and unequilibrated adsorbate layers in order to determine differences in heats of adsorption between different planes of a field emitter. The procedure used was as follows: 1. Adsorbate from a heatable platinum platform was deposited onto the field emitter in reproducible doses. After deposition of each dose, the emitter was heated to thermally equilibrate the layer with the substrate. Next, the average work function, (+), was obtained from the total emission current measurements using Eq. (33).
FIELD ELECTRON MICROSCOPY OF METALS
259
2. In the case of the alkali metals, the flux of adsorbate corresponding to each dose was calibrated with the aid of a surface ionization detector. 3. From 1 and 2 above, a plot of (4) vs. average surface coverage, E, was constructed, assuming a sticking coefficient of unity of the adsorbate on the emitter. A graph of <4) vs. iiobtained by Schmidt and Gomer (136) is shown in Fig. 37 for K on W.
1 ,
I
1.0
I
1
1
2.0,
n XI6
I
3.0
I
I
&
4.0
I
,
d
50
(ATOMS CM2)
FIG.37. Plot of 4 vs. K atom density ii, and average coverage 8. [From L. D. Schmidt and R. Gomer, J. Chem. Phys. 42, 3573 (1965).]
4. Single plane work functions,
4j, were next measured as a function
of
( 4 ) and hence, from 3, of Fi for layers which were thermally equilibrated with the substrate after each dose. 5 . In order to determine 4j as a function of local adsorbate density nj , bj was measured for different dose sizes for immobile, unequilibrated layers. After all 4j values for a particular dose had been measured, the layer was thermally equilibrated, ( 4 ) was measured, and the corresponding value of Fi was determined from the information gained in step 3 above. Finally nj was determined by noting that
nj = nFi(cos &'/siny),
(77)
where E' is the angle between the source and the normal to the regionj being probed, and y is the angle between the emitter axis and the source. Swanson and Strayer (51) obtained + j vs. nj for Cs on the (1 10) and (100) planes of W by assuming a linear relationship between 4j and nj at low coverages. Graphs (55) of 4j vs. fi and nj for K adsorbed on the (1 lo), (21 1) and (1 11) planes of W are shown in Fig. 38.
260
L. W. SWANSON AND A. E. BELL
,
6.0
UNEPUILIBRATED
6-5.1
e 1
.
1
ID
1
(MONOLAYERS) 1
20
,
1
3.0
1
1
40
1
1
5.0
I
J
60
n 10" (ATOMS C M ~ ) FIG.38. Summary of equilibrated and unequilibrated work function results of K on the W(110)plane. The abscissa refers in the first case to 0 and ii and in the second to 0 and n. The 4 vs. ii curve obtained from total emission is also shown in the dashed heavy line. [From L. D. Schmidt and R. Gomer, J. Chem. Phys. 45, 1605 (1966).]
6. From each of the 4j vs. ii and nj plots, a plot of nj/n vs. ii was obtained. Differences in heats of adsorption H between planes i and j were then obtained from
nj/E = nj = exp( ++). ni/ii
ni
H.-H.
The nil@ratios obtained for K (55) and Cs (1.34) and for adsorption of Ar, Kr, and Xe (135) indicate surprisingly small differences in H for different planes. (AH < 0.4 eV for K on W at ii = 0.) The difference is especially small when compared to the large variation of H with ii. A sensitive field emission-flash filament technique has been developed by Bell and Gomer (137) in which the field emitter, used as a detector, was placed very close to a heatable W ribbon. The ribbon was dosed with CO from a heatable platinum platform cooled to 20°K; CO which did not stick on the ribbon was reflected back on to the field emitter, also at 20°K. This method is very sensitive to deviations of the sticking coefficient from unity. The CO initial sticking coefficients were found to lie very close to unity for all substrate temperatures up to 300°K. Continued dosing was accompanied by a constant high sticking coefficient followed by a sharp drop-off
FIELD ELECTRON MICROSCOPY OF METALS
26 1
to zero. Even with the substrate at 900"K,the initial sticking coefficient was greater than 0.7. Apart from this small drop-off in the initial sticking coefficient with temperature, the onset of the sticking coefficient drop-off declined to lower values of delivered adsorbate flux as the substrate temperature increased. The above technique has also been applied by Kohrt and Gomer (138) to a study of 0, adsoption on W(l lo). While the initial sticking coefficients were very high (20.9for substrate temperature I 100"K), a sharp decrease occurred when the substrate temperature was raised to 200°K. At 500"K,it had declined to -0.01. Desorption processes may also be followed by this method. Thus when the ribbon is covered with adsorbate, desorption can be followed by monitoring changes occurring on the field emission detector. In this way flash filament desorption spectra may be constructed which are similar to those obtained by other methods (139).The above experiments were carried out in a cryostat at 20°K so that the low temperature features of the flash-filament desorption process could be conveniently studied. Swanson and Strayer (240) have also adapted this technique to the study of Cs desorption from W. B. Coadsorption
Several coadsorbed systems have been examined in the field emission microscope for W emitters. Mostly the studies have involved adsorption of alkali metals and Zr (141) with the strongly electronegative adsorbates 0 (51, I42), F (52, 143), and Cl (144); H was also studied as a coadsorbate with K (145). The effect of the coadsorbed gas generally is to decrease the emitter work function below that due to adsorption of the metallic adsorbate alone. A low work function surface is of technological interest; of additional interest is the fact that bonding of the metallic adsorbate is increased by the presence of underlying 0 and F. While deposition of 0, and H, can be readily accomplished by heating suitable chemical sources (Cu,O and ZrH,) or by passing H, and 0, through suitable diffusion thimbles, deposition of C12 and the highly reactive F, present greater technical difficulties. By depositing an alkali halide on the emitter and heating to remove the alkali metal a reasonable amount of halogen may be left on the substrate. This method however does not result in a complete monolayer converage of the halogen because some is desorbed during the heating procedure designed to remove the alkali metal. In the case of adsorbed CsF a temperature of 1010°K is required to completely thermally desorb Cs, with the emitter biased at a positive potential to facilitate desorption of Cs as ions in the high work function, Cs terminal coverage desorption region. A maximum F/W work function of 5.14 eV was attained by Evans, Swanson, and Bell (52) by
-
262
L. W. SWANSON AND A. E. BELL
dosing CsF onto an emitter, and then equilibrating it at 1000°Kwith the emitter held at + 200 V with respect to the anode. Wolf (143), by a similar technique, reported the much higher value of 5.8 V. The minimum work function, attained by dosing the fluorinated emitter with Cs, was 1.1 eV for a W/F work function of 5.0eV. A similar result was also obtained for Cs adsorbed on O/W. Both the Cs/O/W (52) and the Cs/F/W (52,143) systems exhibited strong electron emission from the (21 1) planes at coverages approaching that corresponding to &,in. Schmidt and Gomer (49) have studied the system K/H/W. Little or no change was observed when the K covered emitter was exposed to H, , even for K coverages as low as 0.5. Atomic H generated in a heated W spiral, however, was found to adsorb; for 0.5 I OK I 1.0, the work function increased after H-atom exposure while for & > I, the work function decreased. On heating K-H layers for OK > 1, the normal W-K pattern was reattained at 140°K while for OK < 1, heating to 200-250°K resulted in patterns and work functions corresponding to K-H coadsorption which was otherwise accomplished by adsorbing Hz first, equilibrating, and then dosing with K. The minimum work function attained by this technique decreased below the corresponding K/W value by 0.2 eV. An interesting feature of the three coadsorbed systems Cs-0-W, Cs-F-W, and Cs-H-W became manifest during diffusion studies ; in all three cases, coadsorption of the gaseous adsorbate led to marked increases in the Cs and K equilibration temperature, especially near monolayer coverages. A brief study of the effect of 0 on Zr adsorption on W has been made by Swanson and Crouser (141) who suggested that preferential binding of Zr occurs on the (100) plane due to the increased strength of the Zr-0-W bond compared to the Zr-W bond. Small amounts of adsorbed oxygen are essential for the preferential emission of electrons from the (100) plane of a Zr-coated emitter. C . Nucleation Phenomena
The FEEM technique is well suited to the study of nucleation and epitaxy, especially the early steps involved in these processes. In addition, the technique can be used to study the important contributing processes of surface migration and thermal desorption. The advantages of the FEEM are: operation in UHV, high magnification (> lo5 x), and resolution (= 20 A). In addition, the emitter, which serves as the substrate, is a single crystal and, in the case of tungsten or other refractory metals, can be flashed free of adsorbed gases. Disadvantages, inherent in the field emission technique, are due to the difficulty of making reliable impingement and nucleation measurements (145).
FIELD ELECTRON MICROSCOPY OF METALS
263
Systems which have been studied to date, by nonequilibrium vapor deposition techniques using the FEEM, include Cu-W (145), Au-W (146), Ag-W (147), Fe-W (148),Zr-W (149), Ni-W (150), and Fe-Ir (151). In some cases the FEEM studies have been augmented by the complementary, powerful technique of field ion microscopy (150). A photo showing Cu epitaxed on W is shown in Fig. 39. The optimum growth conditions for obtaining single crystals have been established by Melmed (143, and involve the use of relatively high substrate temperatures (0.3-0.5T, , where T,,,is the melting point of the adsorbate
FIG.39. Photos depict (a) clean (110) oriented W field emission pattern, tip diameter approximately 5000 A. (b) Field electron emission pattern of partially grown Cu layer on W point. (c)Completed Cu growth. [From A. J. Melmed, J . Chern. Phys. 38, 1444 (1963).] Photo by courtesy of the author.
264
L. W. SWANSON AND A. E. BELL
material) and rapid deposition of adsorbate (10-100 atom layers per minute). At low temperatures, in the case of Cu deposition on W,granular deposits were obtained up to 300“K,beyond which nucleation began to occur at the (011) plane edges and on the (001) and (111) planes, but only at high Cu coverages (5 < 0 c 8).At temperatures above 575”C, nucleation and growth originated only around the (1 10) planes ; and crystalline (often single crystal) growths were obtained. Single crystals covering the entire emitting area were readily grown from 625°K to the highest temperature involved in Melmed’s studies, -1050°K.Polycrystalfine growth was also found to occur in this temperature range. Single crystal growth was favored by high temperatures and low deposition rates and may also be facilitated (152) by dosing perpendicular to the emitter, in which case only one side of the (110) planes edges is likely to provide a nucleation site for crystal growth ; whereas “head-on ” dosing from a ring, or other kind of source, in front of the emitter increases the possibility that nucleation will occur simultaneously on two or more (1 10) plane edges with resultant growth of a a polycrystalline layer. Smith and Anderson (150) have discussed the orientation relationships which occur in a number of systems: a close-packed plane of the adsorbate aligns parallel to the closest packed plane of the substrate, e.g. the ( I 11) planes of fcc adsorbates grow on the (1 10) planes of bcc W. No such general rule applies to the relationship between the directions of close-packed atom chains in the substrate and adsorbate. Hardy (153) and Paunov (154) have studied the nucleation of Hg on W which, in contrast to the systems alluded to above, is conveniently studied under equilibrium conditions. At an emitter temperature of 77”K, the critical coverage (about 4 monolayers) for nucleation was independent of impingent flux. At higher temperatures (154), the nucleation time t* was found to be constant for a given impingent flux up to a certain temperature T,, beyond which t* increased sharply. T, was found to increase with impingent flux. The sharp rise in nucleation time with temperature was attributed to the onset of equilibration between impingent and desorption fluxes and was interpreted as defining a critical supersaturation for the formation of nuclei. Gomer (155) and Melmed and Gomer (156) have developed techniques for the growth of a variety of materials in whisker form suitable for study in the FEEM. The whiskers consist of single crystals with an axial screw dislocation and were grown without thickening of the radius ( 100 A) up to a cm. Hg whiskers were grown at Hg pressures of lo-’ length of about 5 x to 10-6Torr on a flat electropolished W substrate cooled to -78°C.The growth could be followed by measuring the voltage required for a whisker field emission pattern. This declined with time as the whisker grew due to an increase in the field factor /?(h), where /?(A) is given by F = P(h)V. Gomer showed that the field at an emitter consisting of a hemisphere of radius r N
FIELD ELECTRON MICROSCOPY OF METALS
265
mounted on a cylindrical shank of length h is given by J'/F(free)
= h/(h
+ r),
(79)
where E;,,, is the field at the surface at the end of a whisker emitter of infinite length and, as noted earlier, is given by Ffree = PV where /3 z 0.2 cni-', hence F = 0.2 . [h/(h
+ r ) ] . V.
(80) Hence, the voltage V required for drawing a constant emission current, which corresponds to a constant field, is dependent on h. By measuring V as a function of time Gomer was able to determine growth rates for the whiskers. Growth rates were found to be exponentially related to growth time up to a certain time, beyond which growth rates leveled off. Gomer showed that if desorption from the whisker is ignored, growth of the whisker will continue exponentially as long as Hg atoms landing on the whisker sides are able to catch up with the advancing front. When this no longer occurs, the whisker is expected to lengthen linearly with time. Melmed and Gomer (156) were also able to grow whiskers of Al, Cu, Fe, Ni, Si, Mo, and C on W substrates with various impingement rates and substrate temperatures. From the above discussion it can be inferred that the field emission method provides a novel approach to the study of whisker growth. Before the advent of reliable techniques of epitaxial crystal growth on field emitters, the whisker technique constituted an important potential tool for the field emission study of substrates not easily obtainable in clean form. Since then, however, the previously discussed technique of epitaxial growth has superceded the whisker technique in this area of study because it appears to be easier to carry out. Recently, Melmed (156) has used the epitaxial technique to obtain clean Fe field emitter surfaces for the study of adsorption of 0,.
D. Total Energy Distribution* The field emission TED technique has been developed recently into a very powerful tool for the study of the effect of adsorption on adsorbate electronic structure (30, 157-159). Also, the TED technique appears to be a very sensitive probe for the identification of clean metal surface states (24, 41, 42, 160, 161). In a very recent (162)experiment, employing a high signal-to-noise ratio energy analyzer, energy losses due to excitation of adsorbed H and H, molecules were detected. A more detailed discussion of these results is given below. * A comprehensive review of Field Emission Energy Distributions will be published in Reo. Mod. Phys. by J. W. Gadzuk and E. W. Plummer, N.B.S., Washington, D.C. 20234.
L. W. SWANSON AND A. E. BELL
266
It is instructive to examine the effect of adsorbates on both R ( E ) (see Eq. 38) and on the Fowler Nordheim preexponential ratios, log A / A , , which can be expected to vary with coverage because of depolarization of the adsorbate according to Eq. (23). For Cs adsorption on Mo and Wo, Swanson and Crouser (159) found that for f3 < 0.1, the variation of log A / A , was significantly greater than unity for Cs on Mo(ll0) and for Cs on W(100) and on W(110). These high values of A / A , were attributed by Swanson and Crouser to the resonance tunneling enhancement of emission predicted by Duke and Alferieff (27). In an investigation of x nitrogen on W(100) at 78°K Ermrich (157) noted an anomalous behavior of the F-N plot, which he and van Oostrom interpreted in terms of the tunnel resonance model. In a subsequent TED study of x nitrogen adsorbed on (310) W at 78"K, Plummer, Gadzuk, and Young (163) found an additional peak (see Fig. 40) in the TED 0.17 eV below the Fermi level which they attributed to tunnel resonance through a narrow energy level of width -0.1-0.2 eV. In a series of experiments these authors and Clark (30,158)have also examined the affect on the TED of single adsorbed atoms of Sr, Zr, Ca, and Ba which give rise to enhanced electron emission because of tunnel resonance. Enhancement factors R ( E ) for
NITROGEN (310)W
v)
k
z
4.
FLASHED 800°K
3
>.
n a a z! m n a >
u . H
0
-5.0
- 4.5
E (ev)
x
FIG.40. TED of field emitted electrons for ,8 and y nitrogen (curves 1 and 4) and nitrogen (curves 2 and 3) on W(300). [From E. W. Plurnmer, J. W. Gadzuk, and R. D. Young, Solid State Commun. 7, 487 (1 969).J
FIELD ELECTRON MICROSCOPY OF METALS
267
FIG.41. Experimental TED enhancement factor for single Ba atoms on W(111).The three lower curves represent cases where adsorption of electronegative gases was observed before the Ba atom arrived. The shift in the narrow peaks with field was AE/eAF= 1.3 f0.3 A, while the peak separation was 0.29 f 0.01 eV. [From E. W. Plumrner and R. D. Young, Phys. Rev. B 1,2088 (1970).]
Ba atoms on (1 11) W, obtained by Plummer and Young (30), are shown in Fig. 41. The ground state valence shell configuration of Ba (6s') is at - 5.2 eV relative to the vacuum level, while the first and second excited states of Ba are a triplet 'D 6s 5d at about -4.05 eV and a singlet 'D 6s 5d state at -3.80 eV. Interaction with the surface is expected to lead to shifting and broadening of these levels, though the 6s 5d levels are not shifted and broadened as much as the 6s' level because of the contracted nature of the 5d orbital relative to the 6s orbital. A diagram illustrating the shifting and broadening as a function of Ba atom distance from the metal surface is shown in Fig. 42. On the low work function (3 10) and (1 1 1) faces three peaks were observed: a broad level, lower in energy, which originated from the ground state 6s' level and two sharp levels at higher energies, which can be related to the triplet 3D and singlet 'D 6s 5d excited states of Ba. The separation in energy between 3Dand 'D peaks was observed to be 0.29 +_ 0.01 eV which is exactly the separation between the two m = 0 states of the singlet and triplet states in the isolated
268
L. W. SWANSON AND A. E. BELL
L
/
-4.5 t
m
K W
z W
- 5.0
-5.0
-5.5
-5.5
r-
rDISTANCE FROM SURFACE
FIG.42. Pictorial representation of the broadening and shifting of the energy levels of Ba and Ca as they interact with the surface. The shape and position of the virtual levels at the surface are taken from the data on the low work function planes of tungsten. [From E. W. Plummer and R. D. Young, Phys. Reo. B 1, 2088 (1970).]
atom. As the substrate work function was increased, the narrow 6s 5d bands began to disappear above the Fermi surface and on the (110) plane the only visible state is the 6s’ virtual level. The general characteristics of width and relative position of the three “ atomic ” bands for Ba can be determined from the enhancement factor curves. Lea and Gomer (160) investigated the TED of Kr on various planes of W and found that the R(E)values for Kr adsorbed on W(111) and (211) were constant over the range investigated in accord with free electron-like behavior for the adsorbate/substrate. Moreover the log(A/A,) variation with coverage followed the depolarizatjon model. On the W (110) plane, however, these authors found log A / A , values which were much smaller than the values predicted by the depolarization model and attributed the low values to a “ decrease in transmission probability for weakly bound adsorbates, presenting a repulsive pseudopotential to metal electrons.” On the W (100) plane, increasing adsorption of Kr caused a progressive diminution of the TED fine structure which vanished almost completely at 8 = 1. Similar effects on the (100) planes were also noted by Swanson and Crouser (32)during coadsorption of Zr and 0,by Plummer and Young (30) as a result of adsorption of residual gas (thought to be CO), and by Plummer and
269
FIELD ELECTRON MICROSCOPY OF METALS
Bell (161)who examined the effect of Xe adsorption on the TED from the (1 1 l), (310), (loo), and (21 1) planes of W. Because of the great effect of even weakly bound adsorbates on the W(lO0) plane TED structure, its origin is now thought to be due to surface states rather than band structure effects as was originally postulated. The effect of diatomic and polyatomic adsorbates on the TED has been investigated by Plummer and Bell (162)and by Swanson and Crouser (164) respectively. The latter authors examined the large aromatic molecules, phthalocyanine and pentacene, and confined their interest to an investigation of the TED obtained from “ molecular patterns,” which in the case of phthalocyanine consisted of large quadruplet, doublet, and singlet spot shapes; only singlet shapes were observed and examined for pentacene. The TED curves for these large molecules exhibited marked structure down to as much as 3 eV below the Fermi level. Occasionally peaks above the Fermi were seen, as in Fig. 43,in which a large Fermi level-like peak at 950 mV above the Fermi level is shown. A possible explanation of these results is an Auger type mechanism in which the excited electronic state of the molecules is sufficiently long-lived (compared to the interelectron tunneling time) that a subsequent tunneling electron stimulates the de-excitation energy in the process and appears above the Fermi level. That this result was not frequently observed is indicative of the fact that for the most part de-excitation occurs faster than interelectron tunneling time. The wide variety of types of TED curves obtained for pentacene on W is similar to that obtained by Clark (165)by deposition of heavy (238) layers of Ge on W. The striking result depicted in Fig. 44 greatly resembles the TED calculated by Nicolaou and Modinos (166)
I 1.0
-
1
1
-;
T = 77’K
0.8-
C
a
?
e e
-
0.4
-1200 mV
+ 0.6
.-
-> .P . 0
0.20
Bias Voltogc
FIG.43. TED spectra of phthalocyanine on W(110). Note the “superelastic” peak 950 mV above the Fermi level.
270
L. W. SWANSON AND A. E. BELL
Bias
Voltage
FIG.44. TED spectra of pentacene on (310) tungsten; At$
= - 1.7 eV,
log A / A , = 2.0.
for a W surface covered by three monolayers of Ge (see Fig. 45). Other pentacene TED curves also show a resemblance to high coverage Ge on W TED curves. These results strongly suggest that tunnel resonance phenomena plays a strong part in the pentacene data. The adsorbate ground state level will lie approximately I - I$ + Fex, below the Fermi level, where x,, is the distance of the adsorbed molecule from the surface. The ground state level is then found to be -2.6 eV below the Fermi level. From data on the optical spectra, several excited state levels should be expected just below the Fermi level so that adsorption of pentacene in trimeric sandwich assemblies would then be capable of modulating the TED as in the case of triple layers of Ge on W. Plummer and Bell (162) have examined the TED of hydrogen and deuterium adsorbed on W using the high performance energy analyzer described in Section II1,C and found that a continuous change in TED shape took place as increasing amounts of hydrogen was adsorbed on W (100). This was interpreted in terms of a single adsorption state rather than the two-state model invoked by earlier workers (167)to explain the second-order p, and firstorder PI peaks observed in flash filament desorption spectra. In the light of the TED data, the Dl and p2 peaks may be explained in terms of atomically adsorbed H atoms recombining by means of a second-order mechanism (yielding the P, peak) at coverages below a certain critical value, and in terms of H, recombination from nearby pairs leading to pseudo-first-order kinetics for the PI peak at higher coverages.
FIELD ELECTRON MICROSCOPY OF METALS
271
The high signal-to-noise ratio of the differential energy analyzer/multichannel analyzer combination led to the discovery of a set of very weak energy losses in the range 0-0.550 eV which were found to be due to field emission electron excitation of W-H and W-H, vibrational modes (162). These results are shown in Fig. 46. That the energy losses were due to adsorbed hydrogen was established by measuring energy losses due to adsorbed deuterium. These were smaller by a factor of I/$ as expected for the vibrational model. Mileshkina and Sokol’skaya (168), and more recently Clark (165), have examined the effect on the TED of Ge adsorbed on W. Both groups found that for 0 I 1 adsorption of Ge lowers the field emitted current. Clark found that adsorption of single atoms of Ge has little effect on the TED shape other than that attributable to a 0.5 eV increase in the work function which was measured from FN plots. Clark’s TED results at monolayer coverage did not produce the two-peak structure found earlier (168) at the same coverage; however, Clark did observe similar structure in the TED at higher Ge coverages. When plotted in the form R(E)vs. E , the TED data indicates that tunnel resonance occurs even for the first layer. Calculations indicate that atoms in the first layer have a very broad resonance energy level centered more than 1 eV below the Fermi energy of the W substrate. This level is thought to be derived from the 4p2 Is level while the additional TED peak, first observed by Mileshkina and Sokolskaya, has been shown to result from resonant energy
FIG.45. Calculated TED curve from 3 layers of germanium adsorbed on tungsten. [From N. Nicolaou and A. Modinos to be published.]
272
L. W. SWANSON AND A. E. BELL
FIG.46. Experimental TED enhancement factors R(E)for hydrogen (curves A and C) and deuterium (curve B) on (1 11) W at 78 K. The arrows below the deuterium curve point to steps that are believed to be due to excitation of W-H bonds while the arrows 4, 5, 6 above curve A point to the expected positions for hydrogen based on a 42 isotopic shift; the unmarked arrow below the deuterium curve points to a step thought to be due to molecularly adsorbed deuterium which disappears when the emitter is warmed to room temperature. Curve C is the corresponding curve for hydrogen. [From E. W. Plummer and A. E. Bell, Proc. 1971 Int. Conf. Solid Surfaces, Boston, Massachusetts.]
levels (believed to be the 5s'P and/or 'P) of atoms in the second layer. Clark's TED curves obtained for triple layers (Fig. 47) agree well with the results of calculations by Nicolaou and Modinos (166) shown in Fig. 45. Clark was unable to reproduce the TED experiments of Mileshkina and Sokolskaya in which a large number of electrons originate above the Fermi level. Clark reports that a Ge/W alloy is formed when a Ge covered W
FIELD ELECTRON MICROSCOPY OF METALS
273
6 (ev)
FIG.47. Experimental TED of germanium on tungsten. [From H. E. Clark, Ph.D. Thesis American Univ., Washington, D.C. (1971).]
emitter is “cleaned” by flash heating. Although the field emission pattern is identical to the clean pattern, the TED for the Ge/W “alloy” is markedly different. It may be concluded that the field emission TED technique may be used to investigate: (1) the effect of adsorption on surface states; (2) the broadening and shifting of adsorbate energy levels due to adsorption; (3) vibrational energy levels of the adsorbate substrate and of the adsorbate; and (4)possible energy losses due to excitation of electronic energy levels of the adsorbate. E. Field Efects
The FEM technique is well suited to carry out investigations of the effect of high fields on the properties of adsorbed layers because of the ease of generation of high fields of both polarities. Negative field strengths are limited by the need to avoid excessive field emission while positive fields are limited by field desorption. In a study of the effect of electric field on the equilibrium concentration of Cs adsorbed on W, Swanson, Strayer, and Charbonnier (169) found that positive fields cause an increase in emitter concentration of Cs while negative fields bring about a decrease in emitter concentration. The field-induced change in emitter concentration is brought about by heating the adsorbate covered emitter and shank to initiate surface mobility of the adsorbate. The electric field is then turned on and heating is
274
L. W. SWANSON AND A. E. BELL
continued until no further change in adsorbate concentration occurs. At this point, with the field still on, the emitter is rapidly cooled in order to “freeze in ” the field-induced change in adsorbate concentration, and the new adsorbate concentration is determined by FN work function measurements. The results of the field effect on the equilibrium Cs coverage indicated that the process is thermodynamically induced and kinetically controlled. At equilibrium, the chemical potential on the shank (for which the electric field is zero) is equal to the chemical potential on the high field portion of the emitter. An increase in emitter coverage does not affect the shank adsorbate coverage because the shank area is about lo5 times as large as the emitter area. Swanson et al. (269), showed that the relation between coverage and field is given by
where H(0) is the coverage dependent zero field heat of the adsorption, where 0, is the initial adsorbate coverage for F = 0 and OF is the coverage after equilibration in the presence of a field; a and pi are, respectively, polarizabilities and dipole moments of the adsorbed species. A plot of
is shown in Fig. 48. The plots are very close to linear through F = 0 so that the aF2/2 term may be neglected. Hence, from Eq. (81) pi may be obtained from the slope of these curves. These values of pi are within a factor of two of those obtained from the Helmholtz equation. Other workers have studied Li (170), Ba (170), Na (272), and Yt (172) on W using the above technique. Swanson, Crouser, and Charbonnier (169) have also studied the effect of electric field on the diffusion rates of Cs on W. More recently, Vladimirov, Medvedev, and Sokolskaya (173) studied the effect of field on the diffusion of Ti on W. Utsugi and Gomer (174) have also studied Ba on W from the same viewpoint. Swanson et al. (269) interpreted the effect of field on diffusion in terms of a model based on a higher effective field in the saddle than in the trough position and in terms of pi * F and uF2/2field interactions. This led to the following dependence of Ed on P: EdF= Edo - (ko2 - k,Z)aF2/24-(ko - k,)pi * F where the electric field at the saddle and trough is, respectively,
F, = k,F, Fo=koF,
k, < 1, ko>1,
(82)
FIELD ELECTRON MICROSCOPY OF METALS
T=301°K
1
275
/* /*'
?'-
8,; 0.12 Electric Field 10 20 30
FIG.48. A plot of Eq. (81) for two initial coverages of cesium on a tungsten field emitter. The slopes of the curves are equal to the dipole moment of the W-Cs surface complex. [See Swanson et al. (169). J
and where Edo is the activation energy at zero applied field. For Cs on W, the aF2 term appears to predominate since the curve of EdFvs. F is parabolic and concave to the Faxis for Cs coverages in the range 8 0.2. For Ti on W, on the other hand, EdFvs. F curves are linear with negative slopes for all coverages up to 8 = 1 where the parabolic shape was obtained. Apparently, the p i . F term predominates for Ti on W, except at 8 = 1. From the slope of the EdF vs. Fplot, the dipole moment of Ti on W was found to vary from 3.4 D at 8 = 0.2 to -0.2 D at 8 = 1.
-
F. Electron Impact Desorption
The FEEM has been exploited to a limited extent as a tool for the study of the effect of electron bombardment on adsorbed layers. It suffers from the
L. W. SWANSON AND A. E. BELL
276
disadvantage that the desorbing species cannot be investigated if desorption occurs. Information about the residual layer is gained by studying the field emission patterns and variation of work function and preexponential as a function of electron bombardment, heating, redosing, etc. Menzel and Gomer (175,176) have investigated the effect of low energy (15-200 eV) electrons on Hz , CO, 0 2 ,and Ba adsorbed on W while Ermrich (278) has examined the effect of slow-electron impact on N z , CO, , CH4, and Xe adsorbed on W. Bennette and Swanson (62) also examined the coadsorbed systems Hg/O/W and Cs/O/W. Menzel and Gomer showed that cross sections can be determined (175) from emission changes as follows: For a given adsorption state j
-dNj/dt = n,’ajN j ,
(83)
where n,’ is the electron flux in electrons per square cm per sec, oj the desorption cross section in cm’, and N j the converge of state j in adparticles per cm’. For small coverage changes,
4 = 4,
+ CjNj,
(84)
where Cj is a constant. From Eqs. (83) and (84) we have
[ ( 4 o - 4,)/(4t - 4,)1, (85) where i is the current density in A/cm’ and 4o and 4, are, respectively, the cj
= (3.68 x 10-”/it) log
work functions at times 0 and r. In the case of electronegative adsorbates of small polarizability, the Fowler Nordheim preexponential factor A is related to coverate N , in which case Menzel and Gomer (175) showed that c$~ may also be related to measurements of A :
4j = (3.68
x 10-191it) log [iog(Ai/A,)/iog(A,/A,)
(86)
where A , , A , , and A , are, respectively, the FN preexponentials corresponding to times 0, t and co.The minimum cross sections which can be measured are limited by the detectable limit of work function change (-0.1 eV), maximum practical electron bombardment time ( lo4 sec), and the maximum bombardment current which, in order to limit heating of the emitter below 1-10°K for 100 eV electrons, is limited to about 1 mA/cm’. Under these conditions urnin lo-’’ cm’. For Cs (62),Ba (275), and also Hg (62)adsorbed on clean W, u for desorption was found to be below this limit. Preadsorption of oxygen however led to electron-induced desorption of Cs with cross sections which increased with Cs coverage to a maximum of 8 x lo-’’ cm’. Cross sections for electronegative adsorbates were found (175-178) to range from lo-’’ cm’ to 5 x cm’. Using the electron impact FEEM technique, one is able to distinguish between different binding states and is also able to bring about interconversion of one binding state to another (276). N
-
277
FIELD ELECTRON MICROSCOPY OF METALS
VI . EMITTER SURFACE REARRANGEMENT A . Sputtering
A useful but little used technique involves the FEEM to investigate sputtering phenomena. Strayer et al. (63, 179) have studied the effect of Cs ion sputtering of W emitters. As shown in Fig. 15, the Cs ions were emitted from a zeolite source and focused into a parallel beam of ions which impinged on one side of the emitter in a direction perpendicular to the emitter axis. A full description of the tube and techniques employed are given in Section III,E. Measurement of the electron emission current ratio Z3/Zo, which is exponentially related t o the sputter-induced change in p, provides a sensitive indicator of the degree of surface roughening, where Z, and loare, respectively, emission currents corresponding to the sputtered and smooth surface, each obtained at the same voltage. Figure 49 contains a plot of ZJZ0 vs. sputtering energy. Extrapolation of Zs/Z, to zero gave a threshold energy of approximately 30V in agreement with other work. The leveling off of the Zs/Zo vs. ion energy curve is probably indicative of a saturation of the degree of surface roughness. By plotting &/lo vs. T, the annealing of the surface damage can also be studied as may be seen From Fig. 50 in which the effects of sputtering begin to disappear at 100°K and do so completely at -750°K.It is necessary to carry out annealing experiments in a very good vacuum (< lo-'' Torr) in order to prevent the complicating effects of strong electronegative gas adsorption. Vernickel (180,182) has studied Ar ion bombardment of W and Mo field emitters and was able to estimate activation energies for the annealing process. 1.8,
I
1
1
I
I
1
0
IOU
0
I
500
1
1000
I
1500
I
2000
,
2500
1
3000
3500
1
4OOO
CESIUM ION ENERGY ( e V )
FIG.49. A plot of Is/Io vs. Cs+ ion bombardment energy, where lo and I, are the total emitter currents at a fixed voltage before and after ion bombardment.
278
L. W. SWANSON AND A, E. BELL
3.5* 3.0-
0
c
CSt'W
E = 3.8 kV
9
,4
2.5-
\v)
20
\ 90'6.
-
0 '
\"\
1.5 -
O\ O\
I0
I
I
I
I
,O
\ " b n
I
)O
FIG.50. A plot of Zs/Io vs. annealing temperature where loand I. are the total emitter currents at fixed voltages before and after ion bombardment of W.
These limited results are sufficient to illustrate the usefulness of this technique in studying surface roughening due to particle bombardment. A combination of this technique with field ion microscopy provides a powerful method of investigating surface sputtering and roughening on a near atomic scale. B. Surface SeK- Dzyusion
Diffusion in pure metals may involve either surface or volume diffusion. Herring (182) has developed criteria for distinguishing between these cases and has shown that the times, t , and tl, required for two geometrically similar structures to undergo proportional changes are related according to t , = IZ"t1,
(87)
where 1 is the scale factor relating the two; n has the value of 3 for volume and 4 for surface diffusion. Boling and Dolan (183) used this relationship to de-
monstrate that blunting of W field emitters of radii in the range 300 A < r < 3000 A takes place by a surface rather than a volume diffusion mechanism a t temperatures in the range 2600°K < T < 2900°K. They did this by measuring emitter radius as a function of time using an electron microscope; in this case A is given by IZ = r 2 / r , , where rl and r2 are initial and final radii, respectively. Use of field-electron emitters to measure surface migration constants has been made by several authors (184-190) who have mostly made use of the field electron emission process itself to monitor change in shape of the emitter.
FIELD ELECTRON MICROSCOPY OF METALS
279
Under these conditions the FEEM is a convenient tool for the measurement of various surface diffusion processes in UHV. Barbour et a1 (184) have applied Herring's (182) treatment of the kinetic change of the surface to the case of surface diffusion on FEE cathodes in the presence of an electric field F. They were able to show that the volume of material JMwhich flows per unit time across a line of unit length perpendicular to the direction of the migration at an arbitrary point M on the emitter surface is given by
where 0, is the volume per atom, y is the surface tension, Dois the diffusivity constant for surface migration, Q is the activation energy, A , is the surface area per atom, r is the radius of curvature of the emitter, and aMand C, are dimensionless parameters which depend only on the details of the emitter geometry. The quantity J M may be related to the rate of change in length dz/dz of the FEE cathode which occurs as a result of surface diffusion if we make note of the experimental fact that the rate of change of emitter length dzldr is much larger than the corresponding rate of change of emitter radius dr/dt. Under these conditions it can be shown that
dzldt = (2/rM) * JM,
(89)
where rMis the radius of a plane perpendicular to the emitter axis as illustrated in Fig. 51. For the zero field case, Eqs. (88) and (89) yield:
($),=
-yR,2D,e-Q1kT -1 2a, AokT r 2 r,
(90)
Barbour el al. (184) found that kM/rM N- 1.25/r so that
- yRoZDoe-Q1kT1.25 A , kT r3
The effect of electric field on (dzldt)may be obtained from Eqs. (88), (89), and (91) which yield the following expression for rate of change of emitter length in the presence of a field
where F, is the electric field and c, is the value of cW at the emitter apex. Hence the theory predicts that the emitter shortening rate will decrease with dc applied fields of either polarity. If the field is high enough so that electrostatic forces exceed surface tension forces, the net surface migration direction
280
L. W. SWANSON AND A. E. BELL
FIG.51. Successive positions ( I ) and (2) of the field emitter surface during thermal rearrangement. [From J. P. Barbour et al., Phys. Rev. 117,1452 (1960).]
is reversed and emitter material flows towards the emitter apex. However, uniform increase in emitter radius does not occur because of the difficulty of nucleating new atom layers in certain crystallographic directions. Consequently a more complex process called “field buildup” occurs in which the emitter gradually assumes a polyhedral shape. This process has important technological applications which are discussed in Section VII. Barbour el al. (184) monitored the FEEM pattern during diffusion in order to measure (dzldf). FEE currents increase exponentially with F while field effects on diffusion increase only as F2,so that by applying the screen voltage in short pulses of a suitable length and repetition rates they were able to minimize unwanted field effects on diffusion during pattern observation. Under these conditions electron emission from lattice steps of atomic height could be detected by a corresponding ring in the FEEM patterns. The dissolution of a ring during surface migration corresponds to the removal of a single atom layer from the emitter apex so that the change in emitter length
28 1
FIELD ELECTRON MICROSCOPY OF METALS
TABLE 111 SURFACE
MIGRATION CONSTANTS
Activation energy Metal
w (190) Ta (189) Mo (189)
QW 3.14 f0.08 2.61 & 0.10 2.28 f0.06
FOR VARIOUS
METALS
Surface tension y(dynlcm)
Diffusivity Do(cm2/sec)
Temperature range ("K)
2900 i300
4
1900-2800
2200 i200
0.8
1600-1900
can be measured with high accuracy by counting the rate at which rings collapse. Values of Q were obtained by measuring (dzldr), as a function of temperature and plotting Eq. (91) in Arrhenius form. To obtain D o ,the intercept of the above plot is required together with a value for y. This latter quantity was obtained from Eq. (92) by plotting (dz/dt),, vs. Fo2in which case the slope M of the resulting straight line is given by
M = r/(16nc0y).
(93)
Barbour et al. estimated co from electrostatic theory and obtained an average value of co = 0.5. By this method these authors and others were able to obtain values of y, Q,and Doas summarized in Table Ill for W, Mo, and Ta. Bettler and Barnes (187) have refined the Barbour (184) technique to measure Q,y, and Do values on individual crystal faces of various substrates, the results of which are summarized in Table 1V. Bettler and Charbonnier (190) have obtained activation energies QF of the surface diffusion of W in the presence of electric fields sufficiently high TABLE IV SURFACE MIGRATION CONSTANTS FOR VARIOUS METALS(187)
Metal
Activation energy QW)
Surface tension y(dyn/cm)
W(I 10)
2.95 & 0.10
w (100)
2800 & 10% (2100°K)
2.7 f0.3 2.3 & 0.2 2.1 f0.3 2.3 f 0.2 1.8 f0.2
Re (1010) Re (0001) Ir (I 1 1 ) Rh(ll1)
2200rt 15%
Diffusivity Do(cmz/sec) 0.5
2000-2600
0.3 0.9
1500-2300
4 x 10-2
I 700-2 I00 1200-1 500
-
Temperature range (OK)
282
L. W. SWANSON AND A. E. BELL
to cause field buildup. They measured electron emission current as a function of time at a constant temperature and electric field. A series of similar curves was obtained for other temperatures at the same field from which Q F was obtained via the following relationship: t = cTexp(Q,/kT),
(94)
where c is a constant for a given emitter at a give value of F and where t was chosen to represent the time required for the emission current vs. time curves to progress from one feature to another during the buildup process. The value of Q F , obtained by plotting log(t/T) vs. 1/T, was then found to be 2.44 f 0.05 eV/atom. This value is smaller than the zero field value of Q by an amount aFZ where a is a constant which is proportional to the polarizability of W atoms. The term aF2 arises because the energy of the migrating atom is field stabilized more in the saddle position than in the trough position as discussed in Section V,E with respect to the field effect on surface diffusion of adsorbed monolayers. Bettler and Charbonnier (190) showed how to obtain a by measuring t as a function of F at constant temperature. Q was then obtained from QF by noting that Q=
eF+ d 2 .
(95)
The value of Q obtained this way was found to be 2.79 f 0.08 eV/atom which compares with a value of 3.14 f 0.08 eV/atom obtained by Barbour et al. (184) by the ring counting techniques. Bettler and Charbonnier (190) believe that the difference between these results is statistically significant and indicates that buildup is not simply a reverse of the dulling process. In the buildup process, W atoms migrate out from the center of the (loo), (110), and (211) regions of the W emitter, producing an extension of these low index planes and a deposition of atoms at the intermediate regions; the emitter dullingprocess employed by Barbour et al., however, involves movement of W atoms over larger distances, in which case the migrating W atoms would have to traverse the atomically rough regions of the emitter to a greater extent than in the buildup process. This could account for the smaller value of Q observed in the buildup process. Melmed (186) has measured Q for the fcc metals Ni and Pt. by measuring the annealing rate of emitters which had been subjected to a small amount of field buildup before each diffusion run was begun at zero field. Melmed (186) also measured the activation energy for field buildup Q F for Ni and Pt. By using Bettler and Charbonnier’s (190) method of data analysis he was able t o obtain alternative values of Q. In contrast to Bettler and Charbonnier, Melmed was unable to detect any difference between Q for the zero-field anneal process and Q obtained from field buildup measurements. Melmed
FIELD ELECTRON MICROSCOPY OF METALS
283
attributes this to the likelihood that these processes are the converse of each other. From the above discussion it can be concluded that FEEM techniques are very useful for obtaining self-diffusion information about metals. Two zerofield techniques are available, that of Barbour e l al. (184) involving the sensitive ring counting method and that involving measurements related to the field emission current (286). Two substrate configurations are also available for the zero-field methods: the emitter dulling method in which diffusion occurs over large distances and the emitter annealing process which Melmed ( i S 6 ) believes is the reverse of the emitter buildup process. In addition to the zero-field techniques, a high-field technique (190) is available for obtaining QF and Q. The diffusion constants, obtainable from the various field emission measurements above are “average” values obtained for diffusion over a variety of crystallographic planes. Measurement of diffusion constants on single planes is perhaps more appropriately performed in the Field Ion Microscope as shown by Ehrlich (292)for the diffusion of W atoms on the low index planes of W. VII. TECHNOLOGICAL ADVANCES Use of FEE as a practical source of electrons for a variety of technological applications received considerable attention by the Dyke group in the 1950s and early 1960s at the Linfield Research Institute (2, 192, 293). The interest in FEE as an electron source stems from its following well known properties: (1) a cold source of electrons (as low as T = 0°K theoretical); (2) large current density (J r 1 x lo8 A/cmZ); (3) small optical source size (20 to 30 A); (4) highly nonlinear current-voltage characteristics ; ( 5 ) relatively narrow energy distribution. The exploitation of some of these properties for recent practical applications is described below in Section VI1,C. Inhibiting the full realization of FEE as an electron source for commercial application has been the stringent vacuum and environmental requirements necessary for stable, long-lived cathode performance. In the following section we review the current status and understanding of these problems. A . Cathode Stability and Life According to Eq. (7) the exponential factor (P3”/fl contains the sensitive parameters which control both short term and long term current stability. The cathode temperature and tube pressure will determine the relative importance of (p and /3 on current stability, although in many instances they
284
L. W. SWANSON AND A. E. BELL
are interrelated. Factors which influence current stability through j? may also lead to irreversible emitter destruction through a vacuum arc mechanism described in earlier work by Dyke and co-workers (95,96). Neglecting Nottingham heating and cooling and radiation heat loss, Dyke et al. (96) derived an expression relating the emitter temperature T,,,, emitter radius r, and current density J, for a conically shaped emitter. For an interior emitter cone angle of 11" they obtain for W T,,, = 9.5 x 10-4 Jc2r*Cc).
(96)
Assuming one desires T, I1000°C and r = cm the maximum to J, values correspond to lo7 to 10' A/cm2 for steady-state operation. The temperature rise time is such that the steady state is approached in 10 psec; thus pulse operation shorter than 1 p e c offers the possibility of attaining still higher current density levels without excessive heating. These conclusions were verified in more recent studies by Gor'kov et al. (194),who included radiation heat loss and a more explicit effect of the emitter cone angle. Their studies indicate a dramatic increase in T,,, for a > 50". Charbonnier et al. (195) utilized Eq. (62) and included both Joule and Nottingham energy exchange mechanisms in calculating the maximum J, required to heat the emitter to a Torr. This value of J, temperature corresponding to a vapor pressure of is believed to be the maximum allowable prior to a cathode initiated vacuum arc. Table V gives such values of J, for several cathode materials. On this basis we note that W, Mo, and Ir cathodes are the most durable and yield J, > 6 cm. x lo7 A/cm2 for a = 0.1 and r = 2 x TABLE V ULTIMATE TENSILE STRENGTH AND CORRESPONDING F.O ELECTROSTATIC STRESS
Material
W Mo Ir Pt Ni
cu Cr
Tensile strength ( 1 0 ' O dyn/cmz)
Jdo!
F.(MV/cm)
(104A/cm rad)
4.10 2.74
300 249
0.241 1.17-0.52 0.245 0.482
80.4 162-108 74.0 104
1.68 2.24 2.50 I .01 0.48 1.65 0.40
-
-
Also the cathode current density J, required according to Eq. (62) to heat the material to a temperature corresponding to a vapor pressure of lo-" Tom.
FIELD ELECTRON MICROSCOPY OF METALS
285
Another common failure mechanism for emitters of low tensile strength material is irreversible yield to electrostatic forces F,; the tensile stress due to the applied field is
f = Fs2/8n. For F = 3 x lo7 V/cm the tensile stress is 4.1 x lo8 dyn/cm2. Fortunately for most refractory metals the tensile strength exceeds this value. For example, see Table V for values of the tensile strength for several possible emitter materials. A geometric change of an emitter at constant voltage which increases p or the sudden appearance of a low work function impurity pocket at the emitter surface are the usual precursor mechanisms to a destructive vacuum arc. A regenerative increase in p can arise from two basic interrelated mechanisms : (1) cathode ion bombardment and (2) thermal-field buildup as described in Section VI,B. It is observed that an extrusion on the emitter surface caused by ion bombardment usually increases p and hence J at a constant anode voltage (196); this in turn may further increase the emitter temperature via Eq. (96) and lead to further extrusion due to field buildup via Eq. (92) where the term in brackets becomes positive. This process is regenerative leading to further increases of both the extrusion rate and current density and ultimately to a low impedance vacuum arc. Because of the extremely detrimental effect of ion bombardment on FEE cathodes it is instructive to consider in greater detail its fundamental mechanism. Martin et al. (196) found that a simple improvement of the vacuum environment did not eliminate emission current induced changes in 4 and 8. Through careful studies they were able to conclude that both ionic and neutral molecules were released from the anode surface due to electron impact. This mechanism (i.e. electron induced desorption) discussed briefly in Section III,D is able to cause the release of both ionic and neutral molecules from a gas covered anode at an approximate yield of lod3molecules/electron and lo-' ions/electron (197).Vernickel and Welter (198)have computed the trajectories of ions formed by electron impact ionization in the cathode/anode region of a field emitter and conclude that only those ions formed in a pencil-like volume whose radius is approximately 3r (r is the emitter radius) are able to impinge on the emitting area of the cathode. For the most part ions formed in the interelectrode region impinge on the emitter shank where, at least initially, they have no effect on 8. Using the above yields for ion and neutral electron impact desorption it can be shown that under normal operating conditions (i.e. P rz lo-'* Torr, I = lOpA, and a pumping speed of 1 liter/sec in the emitter region) the chief source of ions striking the emitting area of the cathode arise from neutrals which have been electron desorbed at the anode and subsequently ionized by the primary electron beam in the space close to the emitter (198). By rigorous outgassing of the anode surface stable
286
L. W. SWANSON AND A. E. BELL
field electron currents of several milliamperes (J = lo7 A/cm2) have been obtained for over 1000 hr at room temperature (196).Periodic temperature flashing has extended the emitter life in one case to 12,000 hr (196).On the basis of these observations it is important in the design of a stable field cathode to carefully consider the placement, configuration, and cleanliness of the anode surface on which the major portion of the electron current impinges. Another means of reducing undesirable desorption from the anode electrode is to limit the angular extent of the beam at the cathode surface by use of a (100)-oriented emitter and an adsorbate such as zirconium which selectively reduces the work function of the (100) plane of W on Mo. Similarly, field evaporation and field buildup of the (100) plane dramatically reduce the angular extent of the field electron beam. Studies (141)have shown that techniques of this sort can reduce the angular beam spread from approximately 35 to 10”. Residual gas adsorption on the emitter is not as serious a problem since it causes an initial current reduction followed by a region of stable current operation (199).As discussed in Section IV,D flicker noise increases with residual gas coverage and current level; thus, periodic temperature cycling to 2000°K can restore the cathode to its original clean and smooth surface condition. According to Fig. 50 surface damage due to ion bombardment can be rapidly annealed by operating the cathode above 800°K. In an attempt to relax the stringent vacuum and environmental requirements associated with stable field emission a detailed study of both dc and pulsed T F emission was performed by Dyke et a/. (200). Here the object was to cause instantaneous thermal desorption of residual gases and smoothing of microscopic surface roughness. Pulse field emission, consisting of applying the voltage in microand peak current levels up to second pulses at a duty factor of 1 to 2 x -50 mA, was successful in extending the average cathode life to 250 hr at T = 2040°K and 487 hr at T = 1970°K. Stable emission was obtained for tube pressures as high as Torr. For the 487 hr life studies a dc bias voltage sufficient to reduce the rate of emitter dulling to near zero was applied. According to Eq. (92) the field strength Fol required to balance the chemical potential gradient at the emitter apex causing dulling is N
F,, = (87~y/r)”~.
(97)
For W (y = 2900 dyn/cm), Fol = 8.1 x lO4r-lI2 (V/cm). The useful range of emitted current densities (J = lo4 to 108)A/cm2)corresponds to fields in the range 4 x lo7 to 8 x lo7 V/cm for materials such as clean W. In view of Eq. (97) the necessity to avoid buildup precludes dc operation of heated cathodes with radii in excess of 8 x cm, since such radii correspond to F,, values lower than 4 x 107/V/cm. Thus, for emitter radii in the lo-’ to
FIELD ELECTRON MICROSCOPY OF METALS
287
cm range F,, is considerably below that required for normal emission levels thereby eliminating the possibility of stabilizing emitter shape during dc TF operating of a smooth and clean W emitter. Use of thermally stable low work function adsorbates such as Zr offer the possibility of dc T F operating in a more useful range of emitter radii (141) because of the reduced field required for emission. Stable pulse or dc T F emission operation is also possible for extremely dull emitters where, according to Eq. (91) and Fig. 52, the emitter dulling (and buildup) rate is substantially reduced because of the l / r 3 dulling rate dependence. Recent studies in this laboratory indicate that after appropriate conditioning of the emitter, stable, long life (1000 hours) T F dc emission currents of 100 to 200 pA in the temperature range 1200 to 1800°Kare possible in the much relaxed vacuum environment of lo-* to lo-' Torr.
FIG.52. Plot of emitter dulling rate dr/rdt vs. emitter radius for various emitter temperatures for an emitter half-angle a = 0.10 rad.
288
L. W. SWANSON AND A. E. BELL
As pointed out by Dyke et al. (200),an important life terminating mechanism during TF operation is the migration and uncovering due to emitter dulling of low work function impurity clusters at the surface. Thus, floating zone melting techniques used to control crystallographic orientation of the emitter wire can also be helpful in reducing the impurity content of the emitter. Another means of enhancing the stability of a field cathode consists of the deposition of multilayer coatings of germanium on tungsten. Mileshkina and Sokol’skaya (201) noted that Ge coated W field emitters operated stably under conditions where bare W emitters showed considerable change in current. Stable emission was reported with residual pressures as high as loq7Torr. Two factors appear to be operating which account for the enhanced stability of such adsorbate coated emitters. First, the high degree of mobility of Ge at room temperature allows continuous annealing of sputtering damage of the thick adsorbate coating. Second, the low sticking coefficient of residual gases on Ge reduces the rate of change of current due to work function change. These factors, together with the fact that the emitter shank contains a nearly inexhaustable supply of Ge which can replenish Ge sputtered from the emitting region, accounts for the observed stability of coated emitters. Some effort has been directed at evaluating refractory metal carbides, borides, and nitrides with regard to providing a stable FEE source. The relatively high melting point and low work function are the attractive features of several of these metaloid compounds. Table VI lists the physical and electrical properties of a few of these compounds. Investigation of field emission from ZrC and LaB, by Elinson and Kudintseva (207) indicated that both materials were more stable with respect to ion bombardment than W and that operation in pressures up to Torr is feasible. The difficulty of fabricating homogeneous emitters of these materials has been the primary factor impeding progress in assessing the practical value of these cathode materials. On the basis of physical properties both TaC and HfC should be good emitter materials. In summary it is clear that the rapid advancement in vacuum technology and the understanding of mechanisms affecting emitter stability brings stable, high current level field electron emission into the realm of practical instrument application. Room temperature dc operation requires the most rigorous vacuum and environmental conditions. Both pulse and dc TF operation greatly relax environmental and vacuum requirements. The development of means of contending with field-thermal motivated emitter shape changes for dc TF emission has led to practical utilization of field cathodes. The primary disadvantage of TF operation is the increased width of the energy distribution; however for most applications this disadvantage,
FIELD ELECTRON MICROSCOPY OF METALS
289
TABLE V1 MELTINGPOINTS,
ELECTRICAL NITRIDES,
WORK FUNCnONS, AND
PROPERTIES OF SEVERAL CARBIDES,
AND BORIDES
Melting point (“C)
Material
#J
(eV)
Resistivity at 300°K ( p ohm-crn) ~~~
Hf HfC HfN Nb NbB2 NbC Ta TaC W W&
wc
2020 3930 3350 2450 3030 3530 3050 3900 3430 2350 2660
Zr ZrC ~ B
3590 2200
-
s
3.53 (202) 4.00 (205) 4.04 (202)
3.65 (202)
-
4.08 (202) 3.14 (202) 4.50 (202) 2.62 (202) 3.28 (202) 3.62 2.80(204)
30 109 (203) 49 (206) 16 (206) 65.6(206) 74 (206) 14.7 (206) 20 (206) 5.5 (203) 43 (203) 19.6 (203) 41 (206) 63 (206) 27
which primarily increases the chromatic aberration term, is outweighed by the useful life and stability with lower vacuum requirements. Protective adsorbate coatings and emitter materials of refractory metal carbides, borides, and nitrides present additional possibilities of practical field cathodes. B. Source Optics
Two features of the field cathode, namely its large current density (i.e. brightness) and small virtual source size make it an attractive electron source for fine focus applications. In order to illustrate the source optical properties it is instructive to compare the field cathode with a thermionic or Schottky electron source. The use of a heated tungsten point cathode was first proposed by Hibi (208) as an electron source of smaller size and brightness than the conventional cathode. The size of the emitter radii and subsequent field strength suggests emission enhancement due to the Schottky effect was the primary mechanism in Hibi’s early work. Cosslett and Haine (209)compared the tungsten hairpin thermionic and field cathode and concluded that the FEE cathode was superior for image spot diameters less than -1OOOA. Later Drechsler, Cosslett, and Nixon (210)compared pointed cathodes under conditions of cold field emission, TF and Schottky emission and concluded
290
L. W. SWANSON AND A. E. BELL
that aberrations in the lens systems used to image the source were the primary limit on image spot size. A rigorous analysis of the optical properties of a cold field and Schottky point source emitter was performed recently by Everhart (64). The model employed consisted of a cathode idealized as concentric spherical equipotential surfaces followed by a converging field of hyperbolic equipotentials which comprised most of the cathode-anode space. Everhart’s analysis of this model gave the Gaussian source size, axial position of the source, and the spherical and chromatic aberration constants. After including diffraction effects the minimum apparent source size papand maximum current density emitted M2Japfrom the apparent source was calculated as a function of angular aperture angle a(M is the lateral magnification). Table VII summarizes the results for optimum source size and maximum current for both Schottky and field emission. The superiority of the FEE cathode is clearly demonstrated by its smaller pap and larger M2JaP.The value of cathode current density Jo = lo4 A/cm2 for the field electron cathode could, of course, be increased to lo6 A/cm2 without great difficulty and thereby even more dramatically surpass the capabilities of the Schottky cathode. A further result noted from this study was that increasing r in the FEE mode at constant Presults in an increase in papand a relatively larger increase in Japand I. A more realistic analysis of the source optics of a pointed cathode has recently been carried out by Wiesner (211) using a model of a field emitter consisting of a sphere-on-orthogonal-cone (SOC). As shown earlier by Dyke and co-workers (75) the SOC model most closely approximates the shape and field distribution ofthe typical field emitter. The model is an exterior equipotential surface of a SOC which can be described analytically by three parameters Oo’, the exterior half-angle of the cone, poo, the radius of curvature of the apex, and y, the ratio of poo to the radius of the core sphere ro TABLE VII A COMPARISON OF THE MAXIMUM VALUEOF M ‘JaP,pap,AND USABLE BEAMCURRENT Z FOR AN IDEALIZED SPHERICAL POINTED CATHODE OPERATING UNDER FIELD AND SCHOTTKY EMISSION CONDITIONS (64)
Field emission F(V/cm) (A/cm2)
JO
r(t4 PaP(4
MZJap(A/cm’) I(nA)
-
4 x 107 104 0.25 50 -2
16
lo4
-.o
4x
106
100
1 .o
60 X
Schottky emission
-2 X 10’ 230
1
SO0 -160
-
13
-
4 x 105 10 1.o 2000 N
2.1
3.4
FIELD ELECTRON MICROSCOPY OF METALS
29 1
(see Fig. 53). Electron trajectory calculations for three separate initial conditions were carried out to an arbitrary stopping plane (anode) by the use of computer techniques. Trajectory tangents (caustics) in the stopping plane were extrapolated back to the (virtual) axis-crossing region as shown in Fig. 54 to form a virtual image. By examining the effect of various initial conditions on the virtual image Wiesner was able to sort out the contributions of the Gaussian, spherical, and chromatic aberration contributions to the
Fro. 53. The sphere at the end of the orthogonal cone model depicted in this figure is capable of generating equipotentials which closely approximate those generated at the surface of a typical field emitter (shown in figure as the profile surrounding the model). p r o m W. P.Dyke et a/., J. Appl. P h p . 24, 570 (1953).]
FIG.54. Sketch showing definitions of various symbols and coordinates used in the trajectory calculations of field emitted electrons. [From J. C. Weisner; see (211).]
292
L. W. SWANSON AND A. E. BELL
cathode image size for various cathode shapes, sizes, and applied field strengths. In calculating the virtual source radius pepthe terms of the various contributions including the diffraction limit on source size were added in quadrature. Figure 55 shows the variation of papand apparent current density lap normalized to the cathode current density Jo as a function of the total acceptance angle a. The Fig. 55 results are given for two field strengths and emitter sizes depicting optimum conditions for field and Schottky emission modes. In contrast to the spherical model each field strength resulted in an optimum source size which minimized pap. In fact Wiesner observes that blunter cathode shapes (i.e. larger cone angle and poo) result in smaller
FIG.55. Plots of virtual source radius p., and apparent current density J., normalized to the cathode current density Jo as a function of total acceptance angle a. The plots are shown at two fields: the greater field of 4 x lo' V/cm correspondsto field emission and the smaller one of 4 x lo6 V/cm corresponds to Schottky emission. [Plotted from data of J. C. Weisner; see GVI).]
FIELD ELECTRON MICROSCOPY OF METALS
293
pap due to reduction in the sizes of the disks of diffraction and chromatic aberration. Comparing Table VII and Fig. 55 we note that both the spherical and SOC models predict a smaller papfor field emission than Schottky emission by a factor of 10 to 50. More important is the comparison of Jap/Jowhich is proportional to source brightness. The Fig. 55 results for the SOC model show that Jap/Jofor field emission is roughly a factor of 60 times larger than I , for the field emission mode is easily lo4 times for Schottky emission; since . larger than the Schottky mode we conclude that source brightness for the former is lo5 times larger than the latter. Wiesner points out that for the SOC model the virtual source is 0.3 to 0.6 mm behind the cathode apex. He further notes that the initial transverse energy of the electrons E, is by far the largest of the effects causing an increase in source size over that given by the Gaussian image for small beam angles. In contrast to Schottky emission we show in the Appendix that (8,) for cold and TF emission is equal to d and independent of T. (See Eq. A.ll.) Wiesner uses a value for (E,) of 0.7 eV which is appropriate for Schottky emission but much larger than dfor cold or TF emission (in practice d z 0.1 to 0.2 eV). Thus Wiesner's results for the field emission value of pap are larger than necessary due to the unrealistically large value of E , employed. The important practical result is that for TF mode operation papis relatively unaffected by thermal broadening of the electron energy distribution. Another important factor contributed by the emitter cone is an overall compression of the beam. For a perfect sphere the linear magnification M is given by
-
M = R/spoo
9
(98)
where R is the emitter to anode distance and s = 1. According to Dyke and Dolan (2) the influence of the emitter cone causes s = 1.5 to 2.0 for real emitter shapes. According to Fig. 24, M also decreases with emission angle 8,. Wiesner's (211) theoretical analysis of the SOC model gives values for the angular demagnification K = a/O0. In addition, it can be shown that s = R / ( R - Zoo)K
(99)
for the SOC model. Using Wiesner's empirical values of K for various shaped SOC emitters, one finds s z 1.6 to 2.4 in good agreement with experimental observation. For small angular apertures the area of the SOC emitter A , seen by a probe of area A, is A , = A,(p,o/R)21/KZ.
(100)
For poo = 0.1 to 1 .O p, y = 2, and the emitter cone angle in the range 8 to lo", one obtains values of K between 0.40 and 0.50 (211). Thus for a given value of
L. W. SWANSON AND
294
A.
E. BELL
A, and R the value of the emitter area A , for the SOC is 4 to 6 times larger than A, for a concentric spherical system due to beam compression. Finally in Fig. 56, based on Wiesner’s results, we plot, papvs. I(a)/Jo,i.e. the total current contained in the optimum beam angle a normalized to J, , for both field and Schottky emission. The important feature in Fig. 56 is that point of crossing of the two curves which gives a region of papbelow which the field cathode is clearly superior from the standpoint of current for a given virtual source size. For equal values of J, the value of pap at the crossing point is -40,000 A; however, since J, for the field cathode is lo4 times larger, the crossing point for the non-normalized curves occurs well above papr 10 p. For large values of pap(i.e. pap5 lo3 A) the dependence ofJ(a) on papbecomes determined primarily by the spherical aberration coefficient c, , that is
-
3
(101) k a* Using Eq. (101) and the following approximate relation for I(a) when initial transverse velocities can be neglected: Pap
Z(a) E J , r 2 m 2 ,
(102)
On the other hand, for the thermionic it can easily be shown that I(a) K case the well-known relationship Z(a) oc p:L3 is obtained from Eq. (101) and the Langmuir equation. These trends are manifested in the Fig. 56 curves for
10’lo6
I
16~
I
16~
I
I
16’
lo2
I1
I
I (d)/J,
FIG.56. Plots of virtual source radius pal,vs. I(a)/Jowhere I ( = ) is the total current in the optimum beam angle a normalized to Jo , for both Schottky emission (F= 4 x lo5 V/cm) and field emission (F= 4 x 10’ V/cm). plotted from data of J. C. Weisner; see (211).]
FIELD ELECTRON MICROSCOPY OF METALS
295
the respective emission modes and explain why the field cathode provides a higher current than the Schottky cathode below a specified value of pap. These same considerations apply in the case of an additional strong magnetic or electrostatic lens which forms a real image of pap. Hence, for applications requiring micron or submicron electron beams the field cathode properly apertured is clearly superior to Schottky or thermionic cathodes.
C . Instrument Applications The use of the field electron cathode has been considered and examined for a variety of instrument applications including microwave amplifiers, cathode ray tubes, oscillography, switch tubes, transducers, and flash x-ray devices (192, 193). The most successful of these applications has been the flash x-ray tube which employs a field emission initiated vacuum arc which generates a submicrosecondlarge electron current and severely disrupts the emitter tip surface.* For this application multiple emitter combs containing several hundred emitters in a small area are employed. Instrument applications involving controlled dc field emission which appear most fruitful at the present time consist of micron and submicron electron beam devices such as scanning electron microscopes, integrated circuit fabricators, Auger, and x-ray microprobe analyzers and electron beam computer memory devices. Hibi (212)has promoted the use of point cathodes for the electron interference microscope because of the high degree of spatial coherence associated with the small virtual source size. The most significant instrument use of a FEE source has been in the scanning electron microscope (SEM) developed by Crewe and co-workers (199, 223, 224). Both a high resolution transmission SEM (213), with a resolution in the range of 5 A, and a 100 A resolution reflection SEM (214) have been successfully developed by Crewe using a cold (310) oriented W field cathode. In the SEM the substantial advantage of the field cathode over the heated cathode is dramatically demonstrated by the need for only one electrostaticimmersion lens to form a 100 A beam. Normally, heated cathodes require two to three lenses to form such a beam size. In addition, the larger beam current capabilities to A for field cathodes vs. lo-" to 10-l' A for heated cathodes) allows the field emission SEM to scan (and photograph) specimens in a few seconds as opposed to a few minutes for a SEM employing a heated cathode. Using differential pumping techniques to maintain a high vacuum in the cathode region and the specially shaped electrostatic lens designed by Crewe to reduce spherical aberration a SEM
* Available from Field Emission Corp., McMinnville, Oregon.
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L. W. SWANSON AND A. E. BELL
employing an oriented field cathode has recently been developed for commercial use.* It is apparent that in the coming decade substantial commercial and practical applications of FEE cold and heated cathodes will be made in devices employing scanning or fixed fine focus electron beams. This significant technological advancement will be made possible by the extensive basic and applied research on field electron emission during the past two decades. APPENDIX
Following the approach of Young (12) for FEE from a “free electron’’ metal, one may readily obtain the following expressions for the energy distribution functions: 4nmeD(E,) dE, dE, J(E, , El) dE, dE, = (A.1) hJ[exp(Exi-El - E,)/kT 11’ 4nmeD(Ex)dE, dE J(E, Ex)dE, dE = h3[exp(E - E,)/kT + 11 ’
+
In the one-dimensional model considered here the total energy distribution J(E) (i.e., Eq. 8) may be obtained by integrating Eq. (A.2) over all possible values of Ex; the transverse energy distribution J(E,) is obtained by integrating Eq. (A.1) over all possible values of Ex; the normal energy distribution J(E,) is obtained by integrating either Eq. (A.l) over El or Eq. (A.2) over E. Finally, the total emitted current J is obtained by double integration of either Eq. (A.1) or Eq. (A.2) over the appropriate range of the energy components. Equations (A.l) and (A.2) apply in the general case of FEE from a heated metal which includes thermionic (P= 0) and cold field emission (T = 0) as particular cases. For each case, the equations can be integrated analytically after the Fermi-Dirac distribution function {exp[(E- E,)/kT] - I}-’ and/or the transmission coefficient D(E,) are replaced by simpler approximations, which are sufficiently correct over the range of energies where electrons are emitted in appreciable quantity. Various cases for which approximate analytical expressions can be obtained are given below. A. Thermionic Emission
This case is very straightforward and the results are well known. The approximations used for the transmission coefficient are : D(E,) = 0 for Ex < El + 4, D(Ex) = 1 for Ex > Ef + 4, Available from Coates and Welter Instruments Corp., Sunnyvale, Calif.
297
FIELD ELECTRON MICROSCOPY OF METALS
where the zero of potential energy is taken at the bottom of the conduction band. Noting also that since E - Ef % kT in all cases of practical interest, one may neglect the factor 1 in the denominator of Eqs. (A.1) and (A.2).Integration over the appropriate range of the energy components yields: JOT
= [4nme(kT)'/h3]exp( - 4 / k T ) ,
(A.3a)
J(EJ dE, = JOT exp(- EJkT) d(E,/kT) for Ex 2 E,
+ 4,
(A.3b)
J(E,)dE, = JOT exp( -E,/kT)d(E,/kT) for E, 2 0,
(A.3c)
J ( E ) dE = J O T ( E / k T ) exp(-E/kT) d(E/kT) for E 2 E, + 4, (A.3d) where JOT is the total thermionic emitted current. The normal and transverse energy distribution functions are exponential for Ej > Ef + 4, and the total energy distribution function has the form indicated by Eq. (A.3d). The average value of the energy components for the emitted electrons are obtained from Eq. (11) which yields for the thermionic case: (E,)=kT+E,++,
(E,)=kT,
(E)=2kT+Ef+4. (A.4)
B. Schottky Emission
Schottky emission is defined as electron emission from a heated metal in the presence of an electric field which lowers the top of the potential barrier but which is not sufficient to cause appreciable emission by tunnel effect through the barrier. The approximation used for the transmission coefficient is : D(E,) = 0 for Ex < Ef 4 - Eo , D(E,) = 1 for Ex > E, + 4 - Eo , where E, is the Schottky term reduction of the potential barrier. As before, the factor unity in the denominator of Eqs. ( A .1) and (A.2)can be neglected over the range of electron energies of interest, and integration over the appropriate range of the energy components yields:
+
(A.5a)
J(E,) d E , = Jos exp
(-9;")
d(2)
(-$) -d(2)
J(E,) dE, = Josexp
for
for E,20,
( A .5 b) (A&)
(ASd)
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L. W. SWANSON AND A. E. BELL
where Jos is the total emitted current for Schottky emission. It is to be noted that the energy distribution functions can be made identical (except for an amplitude factor equal to the Schottky factor Jos/JoT)to those obtained for thermionic emission, simply by shifting the zero reference by Eo = (e3F)1/2for the normal and total components of the electron energy. Thus application of a low electric field at the metal surface does not change the width of the energy distribution functions, but simply displaces the total normal energy distribution by the amount J& by which the top of the potential barrier is lowered. Thus, the average values of the energy components for the emitted electrons are :
(Ex> = kT -+ Ef i4 - Eo ,
( E , ) = kT,
( E ) = 2kT -I-E f -+ 4 - Eo . (A.6)
C . Cold Field Emission
In cold field emission electrons are drawn from the cold metal by application a t the surface of an electric field large enough to yield an appreciable probability that an electron will escape by the tunnel effect through the narrow potential barrier. A generalization of the WKB approximalion (10) yields the following expression for the transmission coefficient :
D(Ex)= { 1 + exp[4(2m IEx1 3)’/2v(y)/3heF])-’. 64.7) Equation (A.7) remains valid even if 1 Ex1 is appreciably greater than the top of the potential barrier. This form of the transmission coefficient is too complex to allow analytical integration of Eqs. (A.1) and (A.2). However, using the Taylor expansion of D(Ex)about Ef (see Eq. 5), and noting that in the case of cold field emission, the Fermi-Dirac distribution function is unity for E c E, and zero for E > E , , one readily obtains the cold field emission current JOF and the energy distribution functions : JOF = (4n: me/h3)Dod2,
(2)
(A.8a)
( 7 ( ) 7 ) for Ex 5 E,,
dEx = JOF Ex - E, exp Ex - E, d
(A.8b)
J(EJ dE, = JOFexp (- E J d )d(E,/d) for E , 2 0, J(E) dE = JOFexp[(E - Ef)/d‘jd(E/d) for E < Ef,
(A.8c) (A.8d)
where Do = e-b(Er). The distribution functions reduce to zero when Ex and E exceed E,. As first noted by Young (12), there is an interesting “mirror” symmetry between the energy distribution functions for thermionic emission and cold field emission :the transverse energy distributions are both exponen-
FIELD ELECTRON MICROSCOPY OF METALS
299
tial, and the normal energy distribution in one case has the same form as the total energy distribution in the other case, with the energy parameter d (proportional to F ) playing for cold field emission the same role as the energy parameter kT for thermionic emission. Finally, the average values of the energy components for the field emitted electrons are readily obtained :
( E x ) = E f - 2d,
( E l ) = d,
( E ) = E, - d.
(A.9)
The range of validity of the field emission theory (Eqs. A.7-A.9) is limited by the condition that the top of the potential barrier remain above the Fermi level, which requires that the applied field F be smaller than qi2/e3.
D. T-F Emission T-F emission is defined as the electron emission from a heated metal under the influence of a strong applied electric field, when the relation between T and F is such that the major fraction of the emitted electrons escape by tunnel effect rather than over the barrier. T-F emission is related to cold field emission in a similar manner as Schottky emission is to thermionic emission. The addition of temperature in the former case and of an applied electric field in the latter case causes some enhancement of the emission, but does not fundamentally alter the emission characteristics. Thus, in T-F emission the emitted electrons originate predominantly near the Fermi level, and the approximation of Eq. ( 5 ) is still used for the transmission coefficient. The main difference in the cold field emission case is that the Fermi-Dirac distribution function is now a continuous function, instead of the discontinuous function (with values 0 to 1) corresponding to cold field emission. The total emitted current J T F and the energy distribution functions can be expressed in terms of the basic parameter p = kT/d, and integration of Eqs. (A.l) and (A.2) yields the expressions: JTF
=
=P 4nme D o d 2 - nP - JOF 7 h3 sin np sin np ’
(-2) ):(-
J(El) d E , = J T F exp - d
(A.lOb) for E , ~ o ,
sin np exp[(E -E,)/d] J(E) d E = JTF 7rp 1 exp[(E - E,)/kT]
+
(A.lOa)
(A.10~) (A.lOd)
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L. W. SWANSON AND A. E. BELL
These expressions of course reduce to the field emission expressions (Eq. A.8) for p = 0. As p increases, an increasing number of electrons are emitted above the Fermi level, resulting in a broadening of the total and normal energy distributions, as illustrated in Fig. 2 for the total energy. However, as indicated by Eq. (A.lOc), the transverse energy distribution remains unchanged. Comparison of the predictions of Eq. (A.lOa) with measured values of the emitted current indicates that Eqs. (A.lO) remain satisfactory approximations as long as p does not exceed about 0.7. Finally, the following expressions may be derived from Eqs. (A. 10) for the mean values of the energy components for T-F emitted electrons: (Ex) = Ef - 4
1 +f(P)I,
( E J = d’
(E) = Ef - &-(PI,
(A. 1 1)
with f ( p ) = np cot np.
The range of validity of Eqs. (A.11) is the same as that of Eqs. (A.10), i.e. approximately 0 Ip 5 0.7. E. Transition Region
There lies, between Schottky and T-F emission, a transition region in which electrons both above and below the top of the potential barrier contribute significantly to the total emitted current. If one approaches this region from the Schottky side, and gradually increases the applied field F, an increasingly large fraction of the emitted current arises from electrons escaping through the barrier by the tunnel effect. As long as F is not too large the normal energy distribution has a peak at the top of the potential barrier, which suggeststhat the transmission coefficient D(Ex)of Eq. (A.7) be expanded about its value at the top of the potential barrier E, = E, + 4 - E, , which corresponds t o y = 1, u(y) = 0, and D(E,) = 0.5 in view of Eq. (A.7). A somewhat lengthy derivation yields a satisfactory approximation for D(E,) near the top of the barrier : 1
D(Ex) = 1 + exp[(E, - E,)/c] ’
(A. 12)
with = h e t / 4 ~ 3 1I4,,,1/2.
In the present case most of the emitted electrons have energies well above the Fermi energy, and one may again neglect the factor 1 in the denominator of
301
FIELD ELECTRON MICROSCOPY OF METALS
the Fermi-Dirac distribution function. The total emitted current Jo and the energy distribution functions can be expressed in terms of the basic parameter q = c/kT,
and integration of Eqs. (A.l) and (A.2) yields the following expressions:
sin nq exp[(Es - Ex)/kTl J(Ex) dEx = Jo nq 1 exp[(Es - Ex)/c]
+
J(E,) dE, = Joe x p ( 3 )
d(2)
(A.13b)
for E, 2 0,
(A. 1312)
These expressions exhibit a striking similarity to those obtained for T-F emission and given by Eq. (A.10), and the same “mirror symmetry” is found which was noted by Young (12) in the more limited case of thermionic and cold field emission, i.e. temperature and electric field play symmetrical roles, the transverse energy distribution is exponential in both cases, and the normal energy distribution in one case has the same form as the total energy distribution in the other. As in the case of T-F emission, Eqs. (A.13) have a limited range of validity and obviously breakdown when q > 1, i.e. c > kT, in which case the applied field is so large that (near the top of the potential barrier) the supply function increases more rapidly than the transmission coefficient decreases with decreasing E x , and the use of a limited expansion of D(Ex)about its value at the top of the potential barrier is no longer justified since the energy distribution peak occurs below the top of potential barrier. The normal energy distribution is symmetrical with respect to the top of the potential barrier when q = 1/2, i.e. in the case exactly one-half of the total emitted current is contributed by electrons escaping through the potential barrier by tunnel effect. Finally, the average values of the energy components for emission in the transition region may be derived from Eqs. (A.13): <EX>
= Es
+ kTf(q),
(Et) = kT,
< E ) = Es + kT[1 +.f(q)I,
(A. 14)
where the function f ( q ) is as given before with p replaced by q. Again, Eqs. (A.14) are valid only when q is appreciably smaller than unity (i.e. q I 3/4 approximately).
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L. W. SWANSON AND A. E. BELL
F. Range of Validity of the Various Emission Theories The expressions given in the preceding sections provide an almost complete theoretical determination of the emitted current density and of the energy distribution functions for arbitrary values of cathode temperature and applied electric field. The diagram of Fig. A . l illustrates schematically the range of validity of the various approximations given above, in the case of a tungsten cathode. As shown, there are three major boundaries. The first boundary CC' corresponds to the condition q I 1, i.e.
F I Fl(T) = (nm112kT/he114)4/3.
(A.15)
Fl depends on the temperature but not on the work function of the cathode; in practical units,
Fl
1100 T4l3(V/cm),
where F and Tare in V/cm and O K , respectively. As long as the applied field is low enough so that condition (A.15) is satisfied, i.e. below the boundary line CC', the emission is predominantly thermionic in character. Below the boundary AA' (i.e. for F < 0.15 Fl) Eqs. (AS) and (A.6) based on the
:3655'K
FIG.A1 . Temperature-field domains for various electron emission mechanisms.
FIELD ELECTRON MICROSCOPY OF METALS
303
simple Schottky theory apply to a good approximation, e.g. to within 10% for the total emitted current density. Between the boundaries AA‘ and CC’, the emission will be referred to as the “extended Schottky emission,” and the more general expressions (A.12) to (A.14) must be used; these expressions break down completely for F 2 Fl, but appear fairly accurate for F I 0.75 Fl. The boundary BB’, corresponding to q = 0.5 or F = 0.4 F,, is of interest as it separates the region (below) where the larger fraction of the emitted current is contributed by electrons emitted over the top of the barrier ( E > E,) from the region where the majority of emitted electrons escape through the potential barrier by the tunnel effect. The secondary important boundary DD’ corresponds to the condition p 5 1, i.e.
F 2 F2 z 3(n1+)’/~kT/he.
(A.16)
F2 depends on both temperature and work function; in practical units,
F2 z 9.4 x 103@/2T(V/cm). Above the boundary DD’ the emission is of a field emission rather than thermionic character. Above boundary GG’, i.e. for F 4.2 F2 , Eqs. (A.7) to (A.9) apply to a good approximation (e.g. to within 10% for J ) , whereas the T-F emission theory, corresponding to Eqs. (A.lO) and (A.ll), would be used between boundaries GG’ and DD‘; the latter expressions break down completely when F I F2 ,but are fairly accurate for F 2 1.3 F2 . The boundary EE’ (corresponding to p = lj2 or F = 2F2) marks the separation between regions where the major fraction of the emitted electrons have initial total energies either above or below the Fermi energy. Finally, the upper boundary HH’ corresponds to an applied field:
=-
F3 = +2/e3 = 7 x lo6 $ 2 (Vlcm).
(A.17)
Above this boundary the field emission or T-F emission expressions do not apply because the top of the potential barrier is reduced below the Fermi energy. This region corresponds to emitted current densities of the order of 10” A/cmZ, and is well beyond the range which can be investigated experimentally at present. There unfortunately exists a gap between the regions of validity of the T-F emission theory and the extended Schottky emission theory; this gap corresponds in Fig. A.l to the shaded area between boundaries CC’ and DD’. In fact these analytical expressions become inaccurate near these boundaries, and the actual region where an analytical expression has not yet been developed is somewhat wider than the shaded area, extending approximately from 0.75 Fl up to 1.3 F2 as indicated earlier. To illustrate these considerations, Fig. A.2 shows the emitted current density J ( F ) for 4 = 4.5 eV and 4 values
304
L. W. SWANSON AND A. E. BELL
F ( V/CM 1
FIG.A2. Emitted current density vs. electric field for four values of cathode temperature; the solid curves JEsand JTFare derived from the extended Schottky and T-F theories.
of cathode temperature. The solid curves JEs and JTF are derived from the extended Schottky and T-F theories, which appear accurate respectively t o the left of points AA'A" and to the right of points BB'B". Since the actual emitted current density must be a smoothly varying function of F, it is estimated by interpolation in the intervals AB, leading to the dotted portions of the complete J ( F ) curves. For each cathode temperature, there is a region where the average total energy of the emitted electrons varies rapidly with the applied field, from a value near the top of the barrier to a value near the Fermi energy.
ACKNOWLEDGMENTS The authors are grateful to Mr. R. W.Strayer and Dr. F. M. Charbonnier for important contributions to this work and to Mrs. M. L.Schroeder and Mrs. M. L. Plagmann for coordination and typing of the manuscript. The authors also wish to thank the National Science Foundation for support during this work.
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Multiple Scattering and Transport of Microwaves in Turbulent Plasmas V. L. GRANATSTEIN* Bell Laboratories. Murray Hill. New Jersey AND
DAVID L. FEINSTEINf Cornell Aeronautical Laboratory. Inc., Buffalo. New York
I . Introduction ........................................................................................ A . Review of Single Scatter Analysis ...................................................... B . Analyses of Multiple Scattering ......................................................... C. Experiments in Controlled Thermonuclear Research (CTR) .................... D. Experiments in Reentry Physics......................................................... I1. Derivation of the Radiative Transport Equation ......................................... A . Heuristic Derivation ........................................................................ B. Polarization and the Vector Radiative Transport Equation ..................... C.3 Rigorous Derivation-Multiple Scatter Equations ................................. D.3 Other Methods of Derivation ............................................................ E.S Extensions and Special Cases ............................................................ I11. Applications and Model Calculations ....................................................... A . Iterative Solutions .......................................................................... B . Exact Solutions for Idealized Models ................................................. C . Diffusion Approximation ................................................................. D . Monte Carlo Solutions ......................................... IV . Comparison of Experimental Results and Model Calculatio A . The Regime of Large Scale Fluctuations (a A,, ) .................................. B . The Regime a A0 ......................... ............................................ V . Brief Summary .................................................................................... Glossary ............................................................................................. References ..........................................................................................
+
-
312 312 316 318 319 320 320 323 324 342 343 346 347 350 355 358 359 359 363 372 373 376
* Present address: Division of Plasma Physics. Naval Research Laboratory. Washington. D.C. 20390. t Present address: Department of Mathematics. University of Wisconsin. River Falls. Wisconsin 54022. 3 Sections IIC. IID. and IIE present a rigorous derivation of the radiative transport equation. and may be omitted without loss of continuity. 311
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V. L. GRANATSTEIN AND DAVID L. EINSTEIN
I. INTRODUCTION The interaction of electromagnetic waves with plasma density fluctuations first received attention in connection with radiowave propagation in the ionosphere. An excellent review of this work was presented by Bowles (I) in 1964, and contained a lucid analysis of the wave-plasma interaction based on the first-order Born approximation (i.e. the electromagnetic wave impinging on each electron in the plasma medium was assumed to be equal to the unperturbed incident wave). Bowles was, of course, fully aware that under appropriate circumstances the incident wave could be significantly modified both by refraction and by strong scattering, but he noted that “A comprehensive theory accounting for the effects of refraction and multiple scattering has yet to be published.” Since 1964, there has been considerable progress in formulating a theory of scattering by plasma density fluctuations which allows for a continuous modification of the wave as it propagates through the plasma medium. It is the purpose of the present paper to review some of this work. We will also discuss two classes of laboratory investigations of microwave scattering which have complemented the analytical development and have served as a test for the theoretical predictions. The first class of experiments has been carried out in certain high density plasma devices used in “ controlled thermonuclear research ” (CTR) ; while the second class of experiments has been motivated by interest in detecting and identifying objects (e.g. ballistic missiles) entering the earth’s atmosphere by scattering a radar signal from their ionized wakes. A . Review of Single Scatter Analysis 1. First-Order Born Approximation
To define the problem of interest, consider an incident plane electromagnetic wave with wave vector k, and frequency coo interacting with a plasma density distribution N(r, t ) which is a random function of space and time. The plasma density has a mean and fluctuating part, i.e. N(r, t ) = F(r) + AN@, 2) where the overbar denotes averaging over a time interval that is large compared with the characteristic correlation time of the fluctuations. The scattering may be thought of as consisting of two parts, viz, (1) coherent scattering from m(r) which will have the same frequency as the incident wave, and (2) incoherent scattering from AN@, t ) which will be randomly shifted in phase and frequency due to the time dependence of AN. In this paper, con-
MULTIPLE SCATTERING OF MICROWAVES IN PLASMAS
313
sideration will be limited to the incoherent scattering from the density fluctuations. If the wave is assumed to undergo a negligible perturbation as it propagates through the plasma medium, then the incoherent scattering is related in a rather simple manner to the space-time correlation function of the plasma density juctuations, C(r, R, 2’) = (AN(r
- R/2, t - 2‘/2)AN(r + R/2, t + 2‘/2)).*
Specifically, for an incident wave that is linearly polarized, the differential scattering cross section (per unit volume, per unit solid angle, per unit frequency interval) is
where the subscript B on crg indicates the first-order Born approximation. Also, the following conservation rules are satisfied w, = 0
k,
0
= k,
k w,
(W
+ K.
(2b)
The frequency and wave vector of the scattered wave are denoted respectively by o,and k, , while W and K are the frequency and wave vector of the plasma density fluctuations. The angle between k, and the incident electric field Eo is denoted by y ; the factor sin’y appears in Eq. (1) because the scattered wave is linearly polarized with its electric field being equal to the component of E that is normal to k,. The scattering volume is denoted by V , and re is the classical electron radius, i.e. re = e2/(471mec2eo) = 2.82 x m. Rationalized mks units are used. According to Eq. (2a), when the wave is scattered, it undergoes a frequency shift equal to the frequency of the plasma density fluctuations. If the time variations in plasma density are due to convection because of motion of the entire plasma at some random velocity v, then one may relate C(r, R, 2’) to the spatial correlation function C(r, R, 0) by C(r, R, 7 ’ ) =jdvC(r, R - vt’, o)P(r, v),
(3)
where P(r, v) is the probability density function of the convection velocity. Using this expression for C(r, R, z’) in Eq. (1) will yield daB -dw,
“ V
Jdr dR’ dvC(r, R’, O)P(r, v) 6(W
-v
K)eiK’R’, (4)
* The triangular brackets denotes the same averaging process as an overbar.
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V. L. GRANATSTEIN AND DAVID L. FEINSTEIN
where we have made the change of variables R' = R - vz'. Thus each value of v gives a contribution to the frequency spectrum of the scattered wave at the doppler shifted frequency ws=w0+ W = W ~ + V * K .
(5)
Generally, the frequency of the scattered wave will be both shifted from ooand broadened, corresponding respectively to a mean convection velocity and a distribution of velocity fluctuations around this mean. To obtain the total scattering cross section (per unit solid angle, per unit volume), one may integrate da,/dw, in Eq. (4) over frequency yielding bg
s mS(r, K) dr,
= re2 -
V
where S(r, K) is the power spectral density of the spatial pattern of fluctuations defined by S(r, I() = I d R eiK"C(r, R, O)/C(r, 0, 0),
(7)
where C(r, 0,O)= AN2@). For example, for the case of a homogeneous the power spectral density is Gaussian correlation, C(r, R,0) = AN2e-R2/a2, Then, the scattering cross section is S(r, K) = n3/2a3e-K2a2/4.
In general, the scattered power at a particular wave vector k, depends on the component of the spectral function at wavenumber K = k, - ko . Usually, the frequency shift in the scattered waves is small (i.e. I WI 4 oo)and, in that case, k, x ko = wo/c. Then the wavenumber of the plasma fluctuations, which contribute to the scattering depends on the scattering angle 8, and is given by K = 2k0 sin(BJ2). (9) It is clear from Eq. (9) that, if the plasma fluctuations are of much larger scale size than the wavelength of the electromagnetic wave (i.e. K 4 ko),then 8, 4 1 and the scattering will be peaked sharply around the forward direction. On the other hand, in cases where the spectral power extends out to smallscale fluctuations with K >, 2ko,the scattering will be more evenly distributed in angle. 2. Limits of Applicability of Firsst-Order Born Analysis
The preceding expressions are for the case where the first-order Born assumption is applicable. They will not be applicable unless the plasma density everywhere is small compared with the critical density N,, = (oO2/c2)(4nr,)-'. This may be understood by considering the relative dielectric
MULTIPLE SCATTERING OF MICROWAVES IN PLASMAS
315
constant of a plasma, 1- N / N , , . Unless N @ N,, ,refractive effects can strongly modify the incident wave. Another condition for the applicability of the first-order Born expressions is that only a small fraction of the power in the incident wave be scattered by the fluctuations. This is equivalent to the requirement
lo ds 4 D
CL,
1
where the variable s is the distance along the incident wave path and the plasma medium extends from s = 0 to s = D. CL, is the extinction coefficient due to scattering and equals the integral of the differential scattering cross section, o, over all solid angles, R, a, = J4= odQ. Inequality (lo), together with the requirement (1 1)
N @ N,,,
is sufficient to ensure applicability of single scatter theory. Some relaxation of these conditions is possible when one is interested only in scattering at certain angles (e.g. backscatter). This is discussed by Salpeter and Treiman (2). To obtain some physical insight into the meaning of inequality (10) in terms of the turbulence parameters, consider again the case of isotropic, homogeneous plasma density fluctuations with a Gaussian correlation function having correlation length a. This has already been used to find oB as given in Eq. (8); oB may be then integrated over solid angle to find a,; and then inequality (10) becomes a, D = n1/2(koa)2 g(D/a)[l
- e-lrozPz- F(ko a)] < 1,
(12)
is the mean square fluctuation where - in refractive index and, for the case of a plasma with N 4 N,, , An2 = )AN’/N:,. The factor F = (2/ko2a2)(1 + e-lroZaz ) - (4/ko4a4)(1- e-lroZaZ ), and it arises due to the presence of the polarization term sin2yin Eq. (8). For small-scale fluctuations (k, a @ l), the above condition for the applicability of single scatter analysis reduces to a Rayleigh-like expression
-
AN2 D a,D = - ( l ~ , a ) ~ -6 N:r a &2
< 1,
while, for large-scale fluctuations (koa % l), the reduction is n1/2
a,D = - ( k o a ) 2 4
AN2 D -- < 1. N,Z, a
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V. L. GRANATSTEIN A N D DAVID L. FEINSTEIN
-
This last expression limits one to very weak fluctuations A N 2 4 N:r since typically D > a . If inequalities such as (lo), (12), (13), and (14) are not satisfied, multiple scattering can be important, and then Eqs. (1) and (2) will not hold. The scattered wave can acquire a frequency shift much in excess of the frequency of the plasma fluctuations and can be scattered at large angles by fluctuations with K Q ko .
3. The Distorted Wave Born Approximation One straight-forward method of generalizing the first-order Born approximation is called the " distorted wave Born approximation " (DWBA) (3). In contrast to first-order Born, it includes the attenuation of the incident wave due to scattering and absorption in the plasma. In place of Eq. (6), one obtains for the cross section
where u, the total extinction coefficient, has contributions from both scattering and absorption, i.e. a = us
+ u,,,
The scattering coefficient, u s ,has been defined in the previous subsection. The absorption coefficient, aA, will be discussed in Section 11. When attenuation of the incident wave is primarily due to absorption, Eq. (15) is a useful improvement over the first-order Born theory. However, if the incident wave is significantly attenuated by scattering, then, in general, multiple scattering cannot be ignored and one must seek a more complete theory. B. Analyses of Multiple Scattering
Recently, some progress has been made in examining the general problem of applying Maxwell's equations to a wave which is scattered by random fluctuations of refractive index. As previously mentioned, when the fluctuations are weak, the situation is adequately described by single scatter theory. Extension of the theory to strong fluctuations, beyond the single scatter approximations, has proven to be a very complex problem (4-9). There have been analyses of the double scatter approximation (secondorder Born approximation) (10, ZZ). However, experience with this approximation indicates that it usually has too limited a range of validity to be of much practical use.
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In addition to second-order Born, there have been other attempts to carry through analyses valid under various different limiting conditions. First, several papers have appeared which consider the coherent propagation and the mean properties of the field (12-17). Secondly, there have appeared analyses of the effect of multiple scattering on both the coherent and incoherent radiation when k , a % 1* (diffusion approximation) (18-23); when this condition is satisfied, the scattering is strongly peaked in the forward direction, and it is possible to sum the significant part of the terms in a multiple scatter expansion. A third type of limited analyses has been carried out in which numerical methods are used to compute the scattering from random plasma slabs (24-29). Each of the above analyses with their three different limiting conditions (viz. calculations for coherent radiation only; large scale fluctuations, ko a B 1 ; and one-dimensional or slablike fluctuations) has furnished important insights. Nevertheless, it would be very desirable to have more general solutions. Unfortunately, as is well known, a general solution to Maxwell’s equations is, in practice, very difficult to obtain. Considerable simplification can be achieved if interest is confined to the intensity of the wave. The price paid for simplification in that case is the loss of information about the phase of the wave. The behavior of wave intensity in a random medium can be described by the equations of radiative transfer (27-29). DeWolf (30) has discussed the equivalence of Maxwell’s equations with the simpler equations of radiative transfer for purposes of describing the wave intensity. It may be shown that the two sets of equations are indeed equivalent, provided that the following conditions are satisfied: 1. Aoct, 4 1, where A, is the wavelength of the incident electromagnetic wave. 2. act, 4 1, where a is a correlation length of the plasma density fluctuations. 3. lo34 V . 4. I Vlnn I 4 k, ,where n is the refractive index of the plasma. If AN 4 Ncr, then condition 4 is equivalent to ak, % ANIN,, . 5. N(r, t ) = + AN@, t ) c N,, . Condition 5 states that the plasma density must be less than critical but not necessarily much less as in the requirements for single scatter analysis. In general, the five conditions above are considerably less stringent than the
m
* In this paper, we consider plasma fluctuations to be characterized by a single scale size or correlation length, a. In more sophisticated treatments of turbulence two scale sizes are often mentioned (i.e. outer and inner scales). However, in interpreting microwave scattering measurements in laboratory plasmas, it is usually not necessary to consider the concept of an inner scale size.
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single scatter conditions in inequalities (10) and (11). In addition, these five conditions for applying radiative transport analysis do not limit one to the k,a 9 1 regime. In recent years, there have appeared a number of important papers establishing the applicability of radiative transport analysis to electromagnetic scattering by plasma density fluctuations (31-37) while other investigators of plasma turbulence obtained various approximate solutions to the radiative transport equation (38-44). One reason for the popularity of the radiative transport equation with researchers in the field of plasma turbulence is that much valuable information about the solutions to this equation is obtainable from the literature on astrophysics and nuclear reactor technology (28,29,45, 46). The present paper will concentrate on the radiative transport approach to analyzing electromagnetic scattering by plasma fluctuations. This concentration was chosen not only to keep the size of this survey within reasonable limits but also because radiative transport analysis has wide applicability to multiple scattering situations encountered in the field and in the laboratory. Derivation of the radiative transport equations for the case of electromagnetic wave propagation in a turbulent plasma is presented in Section 11. Solutions of the equation are discussed and compared with appropriate laboratory experiments in Sections I11 and IV. C . Experiments in Controlled Thermonuclear Research (CTR)
In CTR, the central problem to be solved is to heat the plasma ions to a point where thermal energy overcomes Coulomb repulsion (making fusion of two ions possible) and simultaneously to keep a large concentration of plasma confined for a sufficiently long time so that encounters between ions are likely. The typical plasmas used in CTR are highly ionized and magnetized. Plasma instabilities are driven by electric and magnetic forces and the fluctuations which arise are of considerable concern. The ffuctuations can limit the confinement time of the plasma in the magnetic " traps"; on the other hand, they can influence the energy distribution of the plasma particles and can thus be useful in schemes for plasma heating. Microwave scattering has the potential of identifying the type of fluctuation, thus aiding in establishing its causes and in evaluating its influence on macroscopic plasma properties. In most studies dealing with the scattering of microwaves by plasma oscillations, inequality (10) holds and single scattering analysis is valid. These studies were recently reviewed by Marshall (47). An outstanding example of scattering from a monochromatic plasma oscillation is to be found in the work of Wharton and Malmberg (48). They scattered 8 mm microwaves from electron plasma oscillations, and studied the dependence of the scattering angle and the frequency spectrum of the scattered signal on the
319
MULTIPLE SCATTERING OF MICROWAVES IN PLASMAS
excitation frequency. For oscillations excited at a frequency W , the scattered power was peaked both at the particular frequency w, + W (as predicted by Eq. 2a), and also at one particular scattering angle. This peak scattering angle gave a value for the wavenumber of the oscillations (from Eq. 9) which was in good agreement with the theoretical dispersion relation for electron plasma waves. Increased frequency broadening of the scattered wave was observed when the plasma oscillations became turbulent under conditions of strong excitation. The power of the scattered - signal was proportional to the energy of the plasma oscillation (i.e. to AN2) as expected in the single scatter limit. High frequency oscillations like those examined by Malmberg and Wharton are connected with charge separation and do not cause large fluctuations of electron density. However, some low frequency plasma instabilities (convective or drift instabilities) can be accompanied by large electron density fluctuations. For example, in certain large CTR devices with diameter of Im,low frequency oscillations have been observed with A N 2 1024cm-6 and scale size a 5 cm. Then, for a microwave wavelength of 2 mm, it may be determined from Eq. (14) that u,D 10. Thus, single scatter analysis will be invalid. Extensive investigations of the multiple scattering of microwaves from low frequency plasma oscillations have been carried out in the ZETA device at Culham Laboratory (U.K.) (49-53) and in the ALPHA device at the A. F. Institute (Leningrad, U.S.S.R.) (54-60). This work was done with millimeter waves and with fluctuations of size a 10 cm; thus, theoretical consideration could be specialized to the diffusion regime I , -ga. Solutions of the Radiative Transport equation in this limit are considered in Section 111, and are compared with the experimental results at Culham and Ioffe in Section IV.
-
-
-
-
-
D. Experiments in Reentry Physics Extraterrestrial bodies usually develop a wake of weakly ionized gas upon entering the earth’s atmosphere. In the past, there has been interest in developing communication links based on radiowave scattering from meteor tails (1). More recently, interest has focused on man-made objects like spacecraft and ballistic missiles. In Anti Ballistic Missile Defense, there is considerable interest in being able to identify the characteristics (size, weight, etc.) of an object entering the atmosphere by radar scattering from plasma fluctuations in its wake. These plasma fluctuations are driven by the gas turbulence generated by the shear flow of air around the missile. Typically, the plasma density fluctuations in such a situation will be very strong, i.e. AN,,,/R >,10%. The mean plasma density N depends on many factors (body composition, altitude, distance from body, etc.) and varies over many orders of magnitude as the body descends; typically, N will change from negligible values to
320
V. L. GRANATSTEIN AND DAVID L. FEINSTEIN
values greater than N,, of a probing radar signal. Thus, multiple scattering will often need to be considered. Furthermore, since the radar probing is remote, high power microwave sources and large scattering cross sections are required (especially in backscatter), and this dictates using wavelengths which are approximately the same magnitude as the turbulence scale size. The condition k, a B 1 would result in very weak backscatter, and would be unsatisfactory. To aid in the understanding of radar scattering in the regime 1, a, several controlled laboratory experiments have been undertaken. We will review specifically the work on plasma jets at the Stanford Research Institute (61-65), and at the RCA Ltd. Research Laboratories (Canada) (66,67),as well as the experiments on an electrical discharge in a turbulent gas flow at Bell Telephone Laboratories (39, 68-71). In all these laboratory studies, the plasma density has been variable and both single and multiple scattering were investigated. Typically, the microwave wavelength was comparable to the scale size of the turbulence (i.e. l , / a l), and analysis of the multiple scattering results was very difficult. In Sections I11 and IVYthe experimental measurements will be reviewed together with the progress which has been made in obtaining applicable solutions of the radiative transport equations.
-
-
11. DERIVATION OF THE RADIATIVE TRANSPORT EQUATION In this section, we first discuss the heuristic derivation of the transport equation based on energy conservation and including polarization effects. We then describe a rigorous derivation starting from the multiple Scattering of the electromagnetic wave from individual electrons. The importance of going beyond a phenomenological or semiphenomenological derivation is that no explicit microscopic interpretation of the scattering extinction coefficient in terms of the turbulent plasma parameters is possible from a heuristic derivation. Hence, one must make some sort of arbitrary assumption about the form of this coefficient. Furthermore, an explicit interpretation of the coefficient is necessary if one wants to investigate the range of applicability of the transport equation. Finally, a rigorous derivation from first principles provides a logical framework within which to extend the theory. Before concluding the section, we will briefly consider several of these extensions. A . Heuristic Derivation
We first consider the scalar radiative transport equation and define the intensity of radiation Z(r ; Ti) as the rate at which energy is transported across a unit area at position r into a differeqtial solid angle di2 about Ti. We assume
32 1
MULTIPLE SCATTERING OF MICROWAVES IN PLASMAS
Z(r; A) represents intensity at a given frequency, and that energy exchange between frequencies can be neglected. The frequency dependent transport equation has been discussed in the literature (33),and will be briefly considered in Section II,E,2. The transport equation simply balances the losses and gains of Z in a volume element. A simple pictorial representation is given in Fig. 1.
ENERGY OUT
-I
ENERGY I!
( x t dx,, Y,
2;Rldy
dz
dn
I ( x , y , r ; n ) d y dz
x tdx
f
ENERGY GAINS J
Cl;
A1 dV d O
FIG.1 . Energy gain and loss in an elemental volume.
To consider the loss terms, we define the scattering luwp(r; A, A’) such that p ( r ; A, A’)I(r; A) dVdQ(A)
(16)
is the energy scattered from the direction A into the direction A’ by the scattering volume dV. p is analogous to the cross section, CT,which was presented in the discussion of single scattering in Section I. dQ(fi) is the differential solid angle about direction A. Define the scattering extinction coeflcient us@,A) as the integral of p ( r ; A, A‘) over all directions A’
is the energy loss from the beam travelling in direction A due to scattering in the volume dV. In general, aside from scattering, there may be energy lost in volume dV due to absorption. We will include the possibility of absorptive loss in our equation. Define the absorption extinction coeficient uA(r;A) such that
aA(r; 12)Z(r;12) dVdQ(A)
(19)
is the energy loss from the beam traveling in direction A due to absorption in the volume dV. Hence, the total energy loss in volume dV is:
a(r; fi)Z(r; A) dVdQ(li),
(20)
322
V. L. GRANATSTEIN AND DAVID L. FEINSTEIN
where a(r, A) is the extinction coeficient given by
a(r; A) = ctA(r; A)
+ as@;A).
(21)
In addition to energy lost from a beam traveling in direction A in dV, energy can also be gained by scattering into direction A from beams traveling in other directions. We now consider the gain emission term and define
such that J(r; A) dVdQ(A) is the energy gain in the beam traveling in direction A due to scatterings from A’ in the volume dV, With the above definitions, the transport equation is a statement of conservation of energy in the volume element dV. The change in energy being transported in elemental solid angle &(A) due to scattering in the volume element is Z(X
+ dx, y , Z;A) - Z(X, y , Z ;A) dy dz dR(A) = -tl(r; A)Z(r; A)
dVdR(A) + J(r; A) dVdR(A),
(24)
or, passing to the limit dV+ 0,
.
A VZ(r ; A)
+ cl(r ;A)Z(r ;A) = J(r ;A).
(25)
Sources in the volume dV may be easily included by adding a source term. Equation (25) is an integro-differential equation for the scalar electromagnetic field intensity Z(r; A). A common notation is to write tl = 1/1 where 1 is called the transport mean free path. To obtain an expression for the scattering lawp and extinction coefficients tl, and a A , it is necessary to investigate the more rigorous derivation of the transport equation to be discussed later. In the context of this heuristic development we may take as a simple first approximation the single scatter approximation discussed earlier (cf. Section 1,A) for the scattering law p(r; A, A’) = r e 2 A ~ S ( k, , ks).
(26) The scattering extinction coefficient is then obtained from Eq. (17). The absorption extinction coefficient may be obtained from the complex index of refraction as given by the Appleton-Hartree equation (72). For values of the electron cyclotron frequency and electron collision frequency which are small compared with the signal frequency, this is aA
= vz/(cNrx),
where Y is the electron collision frequency for momentum transfer.
(27)
MULTIPLE SCATTERING OF MICROWAVES IN PLASMAS
323
B. Polarization and the Vector Radiative Transport Equation
The derivation in Section II,A is readily generalized to consider the polarization of the electromagnetic field. This is conveniently done in terms of the Stokes parameters (28, 73). We write the electric field E, as
E, = [E,(l)l?,(I) + E,(2)&,(2)le-iW',
(28)
where &(i) are unit vectors defined in terms of a fixed unit vector f (usually, we take fi = lo= ko/ I k, I) and the direction of propagation of the scattered wave, fi, . We define &,(2) = k, x
Lo/ 1 k, x Lo I
orthogonal to scattering plane
and
(29)
2,(1)
=~
~ ( x2 fi, )
in scattering plane.
For backscatter when fi, = - f , , we relate the polarization vector of the scattered wave C,(i) to the polarization vector of the incident wave ?,(i) as follows : U I )=~ o ( 0 ,
U 2 ) = 60(2).
(30)
The vectors (,C 1) and 6,(2) are arbitrarily chosen to define a set of orthogonal rectangular coordinates with Lo. Note that, except for the case of backscatter, the unit vectors &(i) and e,(i) are not, in general, in the same direction. The Stokes parameters are then defined as
and the Stokes vector is
I=
f).
(35)
14
It is important to remember that I is not a vector in Cartesian coordinates, but rather a four-component vector in the mathematical sense. The vector radiative transport equations may be written by analogy with the scalar equation as A * VI(r; A )
+ a(r; A)I(r; A) = J(r; A),
(36)
324
V. L. GRANATSTEIN AND DAVID L. FEINSTEIN
where the emission vector is given by
J(r; A) =
4n
P(r; A, A’)I(r; A’) dQ(A’).
(37)
The scattering phase matrix P describes how the Stokes vector I is transformed in scattering from direction A’ to direction A. An explicit form for P will be derived from the multiple scatter equations in the next section. (See Table I in Section II,C,7.) Equation (36) may be formally solved in integral form as
-I-jr:exp
[- j;u(rff; A) dr”1J(r’; A) dr’,
where ro is some reference position. If u is not a function of position (the medium is homogeneous) Eq. (38) takes the simple form
I(r; A) = exp[- ( r - rolu(A)]I(rO;n)
+
exp[- Ir - r’la(A)]J(r’; A ) dr’. (39)
The physical meaning of Eq. (38) or Eq. (39) is clear. They state that, in propagating from a point ro to a point r, wave intensity is attenuated by a factor exp[ - a 1 r - ro 1 1; on the other hand, the intensity is augmented by contributions from the emission vector at all points along the ray path. A difficulty encountered in the above derivation based upon energy conservation is that wave interference effects are not considered. This yields an error in the backscatter direction which must be corrected. The nature of the correction and an explicit expression for it will be given in Section I1,E. C. Rigorous Derivation-Multiple Scatter Equations
To calculate the interaction of an electromagnetic wave with an underdense turbulent plasma, one should, in principle, be able to start with the wave scattering from individual electrons. The fundamental multiple scatter equations were first, formulated for the scattering of scalar waves by isotropic point scatterers (27). More recently, the analysis has been extended to quantum-mechanical systems (74, 75) and, of particular interest to us, to microwave scattering from turbulent plasma (31-33,38). In this section, we discuss the rigorous derivation of the transport equation starting with individual electron scattering. The development in this section is not essential to an understanding of the remaining sections of this paper; however, it does provide a logical framework within which one can extend the present analysis.
MULTIPLE SCATTERING OF MICROWAVES IN PLASMAS
325
1. Scattering by a Single Electron
Let us begin with the scattering by a single electron. Using classical considerations, the electromagnetic field at ra produced by an electron at r, = ra - R,, under an acceleration is (in mks units)
+
where e is the electron charge and ra is in the Fraunhofer zone. The vector quantities are shown in Fig. 2.
TRAJECTORY If ELECTRON
FIG.2. Electric field due to an accelerating electron.
Consider an incident plane wave
E,(r) = eo(l)E,(r) = C,(l)E,(l)exp[i(k,
-r
- ot)],
(41)
where k, is the incident wave vector. The wave causes an acceleration of the electron 0
= (elm,)&,(l)E,(l)exp[ik,
- r, - iot],
(42)
where me is the mass of the electron. This results in a scattered field E,(ra) = reEo(l) -&, x [fsx C,(l)]exp(ik, r, + ik, R,,), (43) R,, where the time dependence exp( -jot) has been suppressed. In terms of the unit vectors e,(l), &,(2) introduced in Eqs. (29) and (30), the component of the scattered field parallel and perpendicular to the scattering plane may be found by expanding the vector triple product in Eq. (43) and noting from the definitions that both a( 1) and C(2) are perpendicular to f , . The parallel component of the scattered field is
326
V. L. GRANATSTEIN AND DAVID L. FElNSTElN
and the component of the scattered field perpendicular to the scattering plane is
2. Multiple Scattering Scattering from the electrons can now be expressed in terms of the coupled multiple scattering equations. The electric field vector E(ru)for a wave arriving at a point ru is the sum of the incident field Eo plus that for the waves scattered from all M electrons
W , ) = Eo(ru) +
M
2
C C 2u8(.i)Eu8(.i). p(+a)=l
(46)
j=1
The vectors 2 4 j ) are unit vectors used to describe the state of polarization in propagation from rs to ra (see Figs. 3 and 4). The term EaB(j)represents the component of the electric field along fius(j)of the wave scattered from point rs to ra. The quantity in the sum of Eq. (46) represents the contribution to the field at ra due to scattering. It satisfies the equation
FIG.3. Vector quantities describing polarization and propagation.
FIG.4. Components of the multiple scattered field.
MULTIPLE SCATTERING OF MICROWAVES IN PLASMAS
327
where Eo(rp)is the incident field at rpand Eppt is the field at rp due to scattering at rs,. In Eq. (47)' A4 is the number of electrons in the system, Go(@) is the free space, scalar Green's function for the wave equation satisfied by the field given in Eq. (40). The Green's function is (48) -re exp(iko &p)lRap , with Rap = r. - rp. Equation (47) states that the component of the electric field with polarization i scattered from an electron at rp to r. is obtained by scattering the total field at rp to ra and taking the component in the ith polarization direction (the total field at rBis the sum of the incident field at rp and the vector sum of fields scattered to rp from all other points). Use of the scalar Green's function Go assumes that each scattering occurs in the wave zone of a previous scattering, i.e. Ik, $ 1 (see condition 1 in Section 1,B) and loI > re2 where 1 is the transport mean free path defined after Eq. (25) and lois the wavelength of the electromagnetic wave. The classical electron radius is very small so the condition l o l > re2 is well satisfied for all cases of interest. However, the first condition Zk 9 1 is not satisfied for all electron coordinates in the sum of Eq. (47). The tensor or dyadic Green's function which includes the near field effects should, in fact, be used for these electrons. However, it can be shown that this does not significantly affect the results since it is precisely those terms in the sum which required use of the dyadic Green's function that do not contribute substantially to the sum in Eq. (47) (76). The function Eap(i)satisfies a multiple scattering equation which may be seen as follows: The field Epp,is given by Go(aP)=
Epp, = i?sp,(l)Esp,(l)
+ i?pp,(2)Epp,(2).
(49)
Using Eq. (49) in Eq. (48) yields
-(
Eap(i)= G O ( ~ P ) C , , ( ~ )g o ( ~ ) ~ o ( r+s )
f
8 ' ( # 8 ) =1
P ~ ~ ~ ( w+(g~p e>s ( 2 ) ~ p p 0 1 ] . (50)
We define the nondimensional Thomson scattering amplitude fij(aP, PP') = Ga,Ai) * g p p , ( j )
describing scattering from direction
(51)
kss, to k,, and
P o ) = ~ap(i) gO(0
(52) describing the first-order scatterings of the incident wave. Using these definitions in Eq. (50), we obtain the desired multiple scattering equation fil(NP,
328
V. L. GRANATSTEIN AND DAVID L. FEINSTEIN
Equations (46) and (53) are the multiple scattering equations. Their structure has the following meaning; the field at ra is composed of three terms, the incident field, the once scattered waves, and finally the multiply scattered waves. Ignoring polarization bookkeeping, the multiple scatter equations (46) and (53) may be written as M
E(ra)= ~ ~ ( +r ~C) EaB
(54)
#(#a)= 1
M
The structure of these equations can be understood more clearly if we sequentially iterate Eq. (55) into Eq. (54).
+
f
M
B(#a)=l b’(#B)=l
+ .
(56)
Go(a/3)Go(PP’)Eo(rB~) *
Now consider the case where ru is an observation point and not an electron coordinate. Then we can rewrite Eq. (56) as
E(r,) = Eo(ra)
+
M
M
&9= 1
M
+2
Go(aP)Eo(ra)
c
/?’=1 /?(#B’)=l
+ .
Go(a/3’)Go(P’P)Eo(rs)
Next, in the terms involving multiple summations, the order of summation may be changed giving
Thus, the total field at r, may be written concisely as
J W u ) = Eo(ra) +
M
c l-(~P)E,@B),
j9=
where the operator
r(a/?) is
1
(57)
329
MULTIPLE SCATTERING OF MICROWAVES IN PLASMAS
3. Statistical Averages The operator r(ap)includes the effect of multiple scattering from all M electrons in the system. However, since their positions are random variables, it is necessary to find the ensemble average of r, i.e.
where PM ( 1 )- l ( r l Y r 2 , * . * Y r p - l , r p + l ,
...,r,;’p).
c M
dry
y(+B)=l
is the probability that M - 1 electrons are at positions rl, rz , ..., rp-l, rp+l, rM given that there is an electron at rp. Recall that r, is not an electron coordinate. The form of the conditional probability density function PgL depends on the correlation between the positions of the particles. In the simplest case, the particle positions are uncorrelated and Pgl may be written as a product of the individual probability density functions for the positions of M - 1 particles, i.e.
M
NOW,using the expansion for
r given in Eq. (58),
330
V. L. GRANATSTEIN AND DAVID L. FEINSTEIN
and the fact that Pl(ry)= N(r,)/M yields
(63)
We observe the following: the first term in the square brackets [ ] in Eq. (63) (the linear term in Go) is independent of the M - 1 integration variables ry (y # j?). Hence, the M - 1 integrations over the densities N(rJ may be performed yielding a term Go(cr/3).The second term (the term with the single sum) in the square brackets may be integrated over all but one of the integration variables. There are M - 1 terms in the sum; consequently, this yields a term
or, in the limit of many particles or large M ,
Similar arguments for the higher order terms lead to the equation
+ !dry dry.W(r,.)~(ry)Go(a~')Go(~~')Go(~j?) + - , (64) * *
or the equivalent integral equation,
(r(aj?>>, = Go(aP) + J dr,N(r,)GO(clj?)(r(rj?)>,
(65)
In this approximation to (T(aj?))s, correlations between electron positions are ignored but multiple scattering to all orders is considered. Hence, Eq. (65) should be a valid approximation for the uncorrelated high density case. Of course, correlations between electrons increase with the electron density and care must be taken when using results based on Eq. (65). Before closing this subsection, let us briefly consider the procedure for
33 1
MULTIPLE SCATTERING OF MICROWAVES IN PLASMAS
obtaining an improved form of r(a/?) by explicitly accounting for pair correlations. In place of Eq. (60), we then have
,- ...,
p(1)
rp-1,
rp+l,
...,rM;rp) = p d r 1 , - - -,r e , . ..,rp, ...,rM)/P1(rp) (66)
and pM(rl,
*.
* 9
rM)
= pl(r1)p1(r2)>
* * *
9
pl(rM)
- [1 + g h , rz) + d r 1 , r3) + - + g(rl, * *
+g(r2
Y
r3) f ' ' '
f g ( r M - 1 , rM)],
rM)
(67)
where PEA is a joint probability density function and g is the pair correlation function. In terms of the correlation function of plasma density fluctuations which was introduced in Section I,A,l Then, proceeding as in the uncorrelated case, we obtain as integral equation in (r(a/.?)), similar in structure to Eq. (65),
where h(r,) is found by keeping pair correlations in Eq. (64) (see ref. 25). Equation (68) is an improvement over Eq. (65) in that pair correlations are considered. Of course, triple and higher order correlations are still neglected. This neglect is justified when a 4 I (see condition 2, in Section 1,B). Polarization effects do not change the methods of analysis and merely amount to additional bookkeeping. Before concluding this section we present the following general definitions for ensemble averages of a function A .
(A)
= lAPMdr,, ..., dr,
332
V. L. GRANATSTEIN AND DAVID L. FEINSTEIN
The ensemble average of the form ( A ) p has of course already been used in evaluating (r(ap))p. The other averages, (A),,, and (A), will be used in subsequent sections. 4. Solutionfor the Averaged Green's Function
To solve Eq. (65) (or Eq. 68 if correlations are considered) we note that the Green's function Go (ap) satisfies the wave equation
(Va2+ ko2)G0(ap)= 4nre 6(ra - r,),
(69)
where 6(r) is the Dirac delta function. It is straightforward to find the equation for (T(a, p)). Simply operate on Eq. (65) with (Va2+ ko2)to obtain Pa2
+ ko2n12(ra)I(r(ap)>p = 4nre(ra - rp),
(70)
where we have defined the refractive index nl as
n12(ra) = 1 - N(ra)4nre/ko2 = 1 - N(ra)/Ncr.
(71)
If pair correlations are accounted for, then one must solve Eq. (68) for Eq. (65). In that case, it is found that all the above equations, (69) to (73), with the exception of Eq. (71) for the refractive index, are valid. The improved expression for refractive index including pair correlations is
(r(ap)), rather than
*
ex~[ik,nl(ra)(Rap - Rap * Rap)I*
(72)
Now (r(ap)), satisfies the wave equation in a medium of refractive index n. If we have 1 Vln n I 6 k (see condition 4 in Section 1,B) then the geometric optics or eikonal approximation is valid and the waves propagate along ray paths (77). We then have
(r(@)>, = -re
ex~[ikoSapllRap
9
(73)
where the eikonal is SUB=
J':" n(r) ds.
(74)
MULTIPLE SCATTERING OF MICROWAVES IN PLASMAS
333
5. Renormalized Multiple Scattering Equations We now rewrite the multiple scattering equations (46) and (53) using our . In Eq. (46) the first term was simply the averaged Green's Function (r(aB)), incident field at the observation point. In the renormalized equations the first term will be the coherent propagation to the observation point. One consequence of this renormalization will show up in the single scatter or first-order solution. The single scatter approximation to our original equations (46) and (53) was the first Born approximation while the single scatter approximation to our new equations will be the distorted wave Born approximation. First, for convenience, let us rewrite Eq. (46)
We define the coherent wave as
Ec(ru1 = Wru)),
(75)
and also from Fermat's principle
Ec(ra)= expW, &)E0 W ) ,
(76)
where the path integral in the eikonal S, (cf. Eq. 74) is taken from --co to ra. Using Eq. (46) with Eq. (75) we have
In terms of E, we may rewrite Eq. (46) as
Define
so that
The quantity I?,&) scatter equation
is the incoherent field and satisfies the new multiple
Eap(j)= (r(aB))p -Ij;.l(aB,BO)EJ.rg)
334
V. L. GRANATSTEIN AND DAVID L. EINSTEIN
Equations (80) and (81) are our renormalized multiple scatter equations and replace Eqs. (46) and (53)’ respectively. 6. The Transport Equation
We now introduce the energy density in order to derive a transport equation for the power flux. The energy density is
t&o< I 2
u(ra; 2)
*
E(ra) I
’>
(82)
and the coherent density
uc(ra;2) = u,(ra)[& . 20(1)12 E &o 12 * Ec(ra) I
’9
(83)
where we consider ra to be a point in space rather than an electron coordinate. The flux for a point far from the plasma for a wave with linear polarization 2 is cu(ra; 2). Now from Eq. (80)
so that Eq. (82) may be written as
-
bc0( 12 E(r)I2)
E u(ra; 2)
By virtue of the definition of Eap(j)in Eq. (79) the averages in the second two sums in Eq. (85) are zero. Also from Eq. (83) the first term is u,; hence
x (6
*
gapW2 * e a p . ( i ) [ E a p ( j ) ~ ~ p , ( i ) I ) .
(86)
Now we make use of the arguments used in Section II,C,3 in developing the expressions for the averaged Green’s function. The second term in Eq. (86) is the incoherent energy density, u i;using the definitions of ensemble averages
MULTIPLE SCATTERING OF MICROWAVES IN PLASMAS
335
at the end of Section 111,C,3, u ican be written as
x 2 .a
.2up,(i)($p(j)~.a‘(i))pp’
(87) The contribution of the “one” term has been discarded since it represents coherent energy density which already has been included in u, (i.e. the ‘‘one ” term is the limit lim A N + o u iwhich , is zero by the definition of ui). We now define uij(a,
u p m
B) = +o Jdrp, N(rp)N(rp,)g(rp
9
rp,)($p(j)~~a,(i>>,g’
(8ga)
The integrand represents the energy density at r. with polarization 2 arriving from a unit volume at ra . Equation (89) reminds one of the integral form of the transport equation (38). The next step is to convert this into a flux. To this end we define the following: The flux of coherent power Ic(ra)
(90)
= cuc(ra)
and, also by definition Iij(ra,11)
L(ra)dildji&o,/i
+ c J- a
R $dRap uij(a, /I),
(91)
where d i j is the Kronecker delta function and dL, is the Dirac delta function. The integral in Eq. (78) is to be taken along the straight line - A = Rpu/l&pl.
We have used the unit vector A to be consistent with common transport theory usage. It is equivalent to the k, introduced earlier. If we consider
336
V. L. GRANATSTEIN AND DAVID L. FEINSTEIN
I,, I, and u to be vectors composed of four elements (i.e. ij = [ l l , 12,21,22]) we may write Eq. (91) as
a,U(@,
+ c j- a R:,
I(ru > A ) = 1, 8fo,A
PI.
(92)
Comparing the definition of Zij in Eq. (91) and u i j in Eq. (89) we see 1 u(%; a) = - jdn(il) C
2
i. j = 1
8 * &,(i)&
- 4,(j)Zij(ruy A )
(93)
where we have used the fact that
Hence, Iii(ra
9
A) dWA)
is the power per unit area of radiation with polarization i propagating in a cone dn(A). We now proceed to obtain an expression for uij(a, p) in terms of
I ( W Y PI), I 2First we use Eq. (73) to approximate the Green’s function as follows: for
IR a , u I
= O(a>
e W) = I R,, I
(94a) (see Fig. 5a), where a is the correlation length and I is the extinction length (see condition 2 in Section 1,B)
(n’, PI), where k,, = k,
La,.
exp[iM,)k,,
’R,,,I(m
m,
Y
(94b)
On the other hand, when IR@@,l = O ( 4 @ O(0 =
I &,I
(953)
(see Fig. 5b), then
(mP’)),.
= exP[- in,(r,)ku,
R@J,l(r(aYPI>, *
(95b)
We thus have expressions for (r(a’P)), and ( T ( a , P’)),, in terms of (r(ap)), . The next step is to use Eqs. (94) and (95) with Eq. (81) to find the appropriate form of E,,(i) to substitute in Eq. (88b). Using the fact that for the conditions shown in Fig. 5a we have fil(a’PY PO) %L(aP, PO)
and fij(a‘P9
PP’) ghj(aP, PP’)
MULTIPLE SCATTERING OF MICROWAVES IN PLASMAS
(b)
FIG.5. (a) Vectors in Eq. (94). (b) Vectors in Eq. (95).
and using the definition of EUB(i)from Eq. (81)
Similarly for the conditions in Fig. 5b we have
331
338
V. L. GRANATSTEIN AND DAVID L. FEINSTEIN
and again from Eq. (81) for R,,,
< R,,
Using the results for Ea,,(j) and E,,,(j) we can obtain from Eq. (88b)
+ cross terms.
(102)
In deriving Eq. (102), we have used the fact that in Eq. (88b) the presence of g(r, r,,) in the integrand causes uij to vanish for R,,, 2 a. The " cross terms " in Eq. (102) are of the form
p r yp r y *6
;[ (r, +
ryJ - r,
1
~ ~ ~ , ','I ~ ~ ~ ~ , M Y
M Z
x x
cc a'
fil(M8, PO)
~c*(r,>(r*(~Y)>,(r(cly')>,.
f
(fjt(aP3
Pa')Ey,a,(t)>yy'
9
which vanishes since the average of the incoherent field vanishes (31).
(103)
MULTIPLE SCATTERING OF MICROWAVES IN PLASMAS
339
To simplify the notation we define the 4 x 4 matrix. (This is the scattering phase matrix introduced in Eq. 37.)
To convert Eq. (106) to the transport equation for the flux, we introduce the unit vectors. fi =Rap/lRaply
fi‘=Rpe/IRpeI*
(107)
Let us now multiply Eq. (106) by cR$ dRa, and integrate along the line parallel to fi to obtain
340
V. L. GRANATSTEIN AND DAVID L. FEINSTEIN
or using Eq. (92)
Note that the P matrix may be written as P(& PO) = P(h, k), P(a/3, P&) = P(A, fi’).
Also P(A, A’) depends only on the unit vectors fi, fi’, hence only on the angles defining these vectors. Now, we consider the integral in Eq. (109) SdreP(ap, BE)CU(P,
= Sdre ~ ( f iA’)cu(B&), ,
(111)
changing variables re,= re - r,,
dr,, = dr, = dR,, Rie dQ(A’),
(112)
and using Eq. (92) we have for Eq. (1 11)
= fdn(fi’M”fi, fi’)I(rp,hi) - P(fi, ~ d I , ( ~ p ) I .
(1 13)
Substitution of Eq. (113) in Eq. (109) gives the integral form of the vector radiative transport equation
fdQ(fi’)P(A, A’)I(r,, fi‘).
(114)
Note that the term PI, cancels. To summarize what has been done to derive Eq. (1 14) rigorously: we first wrote down the multiple scatter equations for the electric field in terms of scattering from single electrons. These equations were then modified to include the attenuation of the coherent wave within the first term. Using these modified equations for the field, the energy density was found, and finally the equations for this quantity was converted to an integral equation (radiative transport equation) for the flux.
MULTIPLE SCATTERING OF MICROWAVES IN PLASMAS
34 1
It would be more convenient if we express Eq. (114) in terms of the more familiar attenuation length 1. Using Eq. (73)
I (r(@>>plZ = (r(4>>p(r(% B>S* = (reZ/R,28)exp{ikoCSap- S,*,II = (rez/R$)exp[ -2ko Im Sap] = (r.”IR$)exp[
1.
-2k01m lr>(r’) ds’
Now the reciprocal attenuation length is I/&)
= 2k0 Im
n(r)
as may be seen by considering the example of a plane wave in a medium of complex refractive index. Thus Eq. (1 15) becomes
The transport equation ( 1 14) may then be rewritten as
*
/dQ(A’)P(A, A’)I(rp, A‘).
(118)
Differentiation of Eq. (1 18) along a path element 6s parallel to A leads to the familiar form of the radiative transport equation
which is identical in form to Eq. (36) derived heuristically earlier. The advantage in this present derivation is the specific form given to P and I and the ease with which Eq. (1 19) can be generalized. 7. Stokes Parameter Representation In the Stokes parameter representation P is a 4 x 4 matrix Pij. Using the definition of P, Eq. (104), and the Stoke’s vector Eqs. (31)-(34) we can write the matrix P in the Stokes parameter representation. The elements are shown in Table I where the polar coordinates (6, 4) and (Of7 4’) have been introduced for the respective unit vectors A and A’. The effect of collisions has been included (38).
342
V. L. GRANATSTEIN AND DAVID L. FEINSTEIN
TABLE I
MATRIXIN THE STOKES PARAMETER REPRESENTATION
SCATTERING PHASE
+
p1 = u,[sin 6' sin 8' cos 6' cos 8'cos($' - +)I2 pIz= u,cos26' sin2(+ - +') pI3= u,cos 6' sin(+ - +')[sin 0 sin 8' cos 0 cos 8'cos(+' pZl= uscos28'sin2(+ - +)
+
P22 = U,COS2(+'
- $11
- I$)
pt3= uscos8'sin[2($' - +)]/2 p J 1= 20,cos 8' sin(+' - +)[sin 6' sin 8' cos 6' cos 8'cos(+' - +)] P32 = - +)I P~~= u,{cos 6' cos 8'cos[2(& - +)] sin 6' sin 8'cos(+' - $)} P~~= u,{cos 6' cos 8' sin 6' sin 8' cod$' - +)}
+ e sm(+ + + Pi4 = = = = 0 us = re2[1 + ( ~ ~ / w ~ ~ ) ]J -dRg(r; ~ ~ ~R)exp[inlko(A' ( r ) A) Rl = re2[(1 + (vz/wo2)]-' J dRC(r, R, O)exp[i(l - N(r)/Nc,)'/2ko(ir' -A) P24
P34
P41
P42 = P43 =
-
R]
D . Other Methods of Derivation Several other methods of deriving the transport equation have been proposed. These deal with the scalar equation and we shall not consider them in detail. They can be, however, generalized to the vector case. I n this section, we describe the procedures used in the derivations. The close parallel between the radiative transport equation and particle kinetic theory is demonstrated by the phase space expansion derivation of the radiative transport equation (37) and Boltzmann equation (78, 79). Following the derivation of the Fokker-Planck equation of Brownian motion, one writes the equation of motion for the Lagrangian density of the electromagnetic fieId (80). This is done by introducing the probability function for the vector potential. The " Louisville Equation " for this probability is then Fourier transformed to obtain a " Fokker-Planck " equation for the equilibrium distribution. The operator in the Fokker-Planck equation is Hermite's operator; hence an expansion is made in terms of the Hermite polynomials. A first-order equation is derived which describes the propagation through the plasma of the mean electromagnetic correlation. To sum the higher order contributions, a diagramatic technique is used. Essentially what is done is that the short-range effects of the scattering are treated by wave optics and the long-range effects by the transport equation. Another method used to derive the transport equation which also considers the effects of correlated scatterers (35,36,81) is to formally Fourier transform the Bethe-Salpeter equation (4). This method is an extension of the formal perturbation methods applied to the random scalar wave equation
MULTIPLE SCATTERING OF MICROWAVES IN PLASMAS
343
(4,6,17).The inclusion of the random part of the operator leads to attenuation due to multiple scattering. One formally considers the Dyson and Bethe-Salpeter equations for the ensemble average of the scalar Green’s function and the double Green’s function for the random scalar wave equation (35,36). These quantities are then Fourier transformed to arrive at a generalized transport equation.
E. Extensions and Special Cases
As should be obvious from the development in Sections I1,A-D and in our discussion of the approximations used in the derivation of the transport equation there are many restrictions on the use of the transport equation. What one hopes to find through the formal procedures is a systematic manner through which the range of applicability of the analysis can be extended. One would ideally like to include these effects and still obtain a tractable equation. When the plasma density increases, the far field approximation (condition 1 in Section I,B) as well as neglect of triple and higher order correlations (condition 2 in Section 1,B) must be modified. In addition, the eikonal ray paths may have to be used in place of the straight line paths if the condition In - 1 I g 1 is not satisfied. Finally, if one wishes to investigate, through the use of the transport equation, the frequency dependence of the electromagnetic field, it is necessary to include an explicit description of the frequency in the formulation of the equation. In this section we will discuss what might be involved in including these additional effects, as well as the specialization to direct backscatter.
1. High Density As we mentioned when the plasma density increases, the distance between the scattering centers decreases and one must modify the transport equation. The general transport equation derived by the transformation of the BetheSalpeter equation (35,36) formally contains these effects, but it is not explicitly clear what must be modified in the reduction to the phenomenological transport equation. In the use of the phase space expansion (37), it is somewhat easier to see what should be done. The kernel formally contains these higher density corrections. In fact, an iterative procedure has been proposed to account for these effects (37). As we stated earlier, there is a close parallel with the kinetic theory of fluids and the problems are similar to those encountered in attempting to develop corrections to the Boltzmann equation. One starts with the N-particle distribution function which satisfies the Liouville Equation in phase space (82) and, by integrating over subsets of particle coordinates, obtains a set of coupled integrodifferential equations for reduced
344
V. L. GRANATSTEIN AND DAVID L. FEINSTEIN
distribution functions known as the BBGKY heirarchy (Bogolubov, Born, Green, Kirkwood, and Yvon) (82). Under certain approximations the first equation is the Boltzmann equation. This development is analogous to the phase space expansion derivation of the radiative transport equation (37). The problem in kinetic theory has been to develop a physically meaningful method to generalize the Boltzmann equation and still obtain a tractable formulation. In the formulation of the radiative transport equation, it is usually assumed that the ray paths between scatterings are straight lines. This condition was expressed as In - 1 I < 1 where n is the refractive index. If this condition is not satisfied and n varies in the medium, one must consider curved ray paths. This is the case in a refracting medium and is studied by Lau and Watson (32).In the initial development by Watson (31),the restriction of straight-line ray paths was made for convenience. However, this restriction is not necessary and relaxation of the restriction results in a modified transport equation with a rotation operator acting on the polarization indices. In notation consistent with our earlier development,
where the scattering strength or emission vector, J, is given by Eq. (37). There are two significant differences between Eq. (120) and Eq. (119). First, the attenuation length is modified
where n, is the real part of the refractive index. The second term represents the rotation of the polarization vectors as the radiation moves along a ray path. The components are given by [RI(r, A)lij = (- l ) j @ Z i j + l(rr A )
+ (- 1)%Zi+
lj(r, A )
( 122)
with @ = (R,
*
R,/I R,
x
R, 1)?,(2) v In n,.
(123)
2. Frequency Effects The generalization of the transport equation to include frequency effects is quite straightforward (33).The derivation follows closely that given earlier. Formally, the generalized equation may be written as
d ds
- I(r,
1
A, w ) + -Z(r, A, w ) =
10)
W
0
dw‘ jdR(A’)P(A, f i r ; w - wr)I(r, A’, or).
345
MULTIPLE SCATTERING OF MICROWAVES IN PLASMAS
Equation (124) is similar to Eq. (1 19) except that explicit dependence on the frequency is contained in I and in the scattering kernel P. We approximate the refractive index of the plasma by nI2(r)= I - wP2[w(w
+ iv)]-’,
(125)
where the effect of collisions has been included. Then (ijlP(fi,A’; w)lsr) = o,(fi, A’; w)(ijlplsr),
( 126)
where p is given by Eq. (104) or Table I and a,(A, A’; w ) is a*(& A‘;
0 )=
[1 + r 2
1
1“
,]Z2(r) x 2n - -“dz’eior’jdRg(r, R ; 7’)
(Vbo)
x exp[in,(r)k,(A‘
*
A) * R].
(127)
The frequency dependence is manifest in a,(& A’; o)through the first factor on the right-hand side of Eq. (127) and through the time transform of the time dependent correlation function (8.3).This function is defined in the same manner as the time independent correlation function except that the particle is specified to be at position rs at time tS . With the approximation of Eq. (127) we see that P will vanish for w < t,-’ where t, is a collision time. If I is confined to a narrow band of radiation, Eq. (124) may be integrated to yield the conventional transport equation.
3. Direct Backscatter
As discussed by Watson (31) there is a backscatter cone in which the intensity obtained from Eq. (119) is not correct; the cone angle is of order ( k , D)-’, where D is a characteristic size of the plasma medium. This arises only near backscatter because one can reverse the propagation vectors and obtain the same scattering diagram. These pairs of paths can interfere coherently and this interference is not included in the regular transport equation. However, this effect can be accounted for by taking a suitable linear combination of the solutions to the transport equation. This has been explicitly done by Watson (31,38) and in a slightly different context by deWolf (18, 19, 30), but has not been included in the original models of Feinstein (39,40) or Shkarofsky (41). To be more explicit, we define the transfer matrix T such that I(r, A) = TI,,
(128)
where I, is the intensity of the incident wave. The matrix T formally is the resolvent kernel of the integral transport equation (1 18). For the remainder
346
V. L. GRANATSTEIN AND DAVID L. FEINSTEIN
of this subsection we will only deal with backscatter A is incorrect. The correct equation we will write as
=
-Lo. Then Eq. (128)
I(r, A) = F I 0 .
(129)
Now since the correction only occurs for the multiple scattering terms, it is convenient to express T as the first-order term B plus a correction AT due to multiple scattering T=B+AT,
(130)
where B(r, - k o ) =
ds(r’)exp .Go
On making use of certain symmetry properties of the two multiple backscatter paths Watson (31) derives the relation between T and Y. These are shown in Table I1 (38). TABLE I1 THEBACKSCATTER TRANSFER MATRIX5- IN THE STOKES PARAMETER REPRESENTATION
111. APPLICATIONS AND MODEL CALCULATIONS
In order to obtain numerical results using the above development, it is necessary to formulate a model for the geometry and scattering properties of the turbulent plasma. One may divide the models that have been considered into four categories ; the first are iterative solutions to the transport equation (39, 41, 63, 64). These have the advantage that fairly complicated geometric and scattering functions can be easily handled; however, they suffer from the disadvantage that the range of plasma parameters to which they are applicable
MULTIPLE SCATTERING OF MICROWAVES IN PLASMAS
.
347
is quite restricted. The second are " exact " solutions for idealized models (38,40). The motivation for developing these solutions is that while they cannot usually be compared directly with laboratory results, they can provide a " benchmark " or standard with which one can compare various approximate schemes. The third category are solutions to approximations to the transport equation, in particular, the diffusion approximation (40). We consider this as a separate category since it is particularly well suited to the CTR work where k, a %- 1 and the scattering is strongly peaked in the forward direction. Finally, we have the purely numerical solution of the Monte Carlo approach (42,84,85).We will consider each of these categories in turn. Before proceeding we want to emphasize the difference between three properties : homogeneity of the plasma turbulence, isotropy of the turbulence, and isotropy of the scattering function. The first, homogeneity of the plasma turbulence means that the statistical averages are uniform in space. The second, isotropy of the turbulence means that the turbulent spectrum is not a function of the wave vector K but rather just of the magnitude I KI . The third, isotropy of the scattering function means that scattering has equal probability in all directions. The absence of any one of these properties poses very different and often difficult theoretical problems. A . Iterative Solutions We consider either the scalar transport equation (25) or the vector transport equation (36). These may be converted to an integral equation of the form of Eq. (38)
+ SWUI,
(132)
WI,+IWU[WI,+~WUI],
(133)
I
= WI,
and iterating once
I=
and with successive iterations I = WI,
+jwUw~+ , /Jw~w~wI,
+ - a - .
(134)
The operators in Eq. (134) are defined in Section II,B. The terms in Eq. (134) may be interpreted as follows: The intensity I (either scalar or vector) is found from the incident intensity I, [the I@,) in Eq. (38)] attenuated by scattering and absorption which occur along the path to the observation point. This is described by the operator W. Each successive integral represents an additional multiple scatter contribution described by the operator u. Each
348
V. L. GRANATSTEIN AND DAVID L. FEINSTEIN
successive multiple integral represents a higher order scattering contribution to the intensity. The propagation between scatterings is described by the operator W. On the right-hand side of Eq. (134) we explicitly show the terms for the incident attenuated beam, single scatter, and double scatter.
1. Convergence of the Iterative Solutions The series solution in Eq. (141) is called the Neumann series. Symbolically we may write it as m
I=
1G0"P,
(135)
n=O
where Go is called the single scatter albedo given by the ratio of the scattering extinction coefficient us to total extinction coefficient u, i.e. 80
= u,/a = us/(aA
+ us).
(136)
--
We thus see that for uS/uA4 1 or weak scattering compared to absorption, Go 0; while, for uA/tls 4 1 or weak absorption compared to scattering, Go 1 (in fact Go is restricted to the range 0 < Go I 1). The I" represents the nth order scattering contribution and depends on the geometry of the scattering system as well as the normalized scattering phase function. Convergence for the Neumann series is assured if I""/I" < l/Go.
(137)
This condition is well satisfied for Go 4 1 (weak scattering compared to absorption) and the Neumann series is rapidly convergent. On the other hand, when absorption is negligible (Go l), the convergence is determined by I"+'/In< 1. When the scattering is also not strong, I"" 4 I", so convergence I", and is still rapid. However, when the scattering becomes strong, 1"" the convergence is slow.
-
-
2. First-Order Models
First-order models are essentially the single scatter models (i.e. first Born and distorted wave Born approximations) discussed in Section I, so they will not be considered in detail here. However, the model developed by Guthart and Graf (63,64), although a single scatter model, attempts to account for multiple scatter effects in an empirical manner. Scattering and absorption attenuation are included by defining an extinction coefficient u as in Eq. (21). It was found, however, that in this single scatter model, the integral defining us in Eq. (17) overcompensates for the attenuation of the forward propagating wave. This is so
MULTIPLE SCATTERING OF MICROWAVES IN PLASMAS
349
because small-angle multiple scatterings convert energy from a coherent form to an incoherent form but do not constitute an energy loss. To correct this phenomenon, an effective scattering coefficient, cleff, was proposed, viz.
aetf= [a, - (as/b)]e-(es'ec)z + (cl,/b), where 8, is the scattering angle, and the constants b and 8, are to be determined empirically from experimental data. It is seen that for forward scattering (8, -,0), ueff = a,,while for 8, % O , , ueff x aJb. Another feature of the model is modification of the refractive index, essentially as was suggested in Eq. (125) or more specifically as the scattering phase matrix was written in Table I. The effect of a finite volume on the spectral function is also taken into account. Finally, we point out that this is a scalar model and does not consider polarization. Comparisons of the model with experiment will be made in Section IV.
3. Second-Order Models Now, to develop a model for the vector intensity we observe the following: For direct backscatter from the plasma there is no contribution from the first term on the right-hand side of Eq. ( I 34). Also, in backscatter the single scatter term has the same polarization as the incident wave and does not contribute to the cross-polarized component. We note, however, that Halseth and Sivaprosad (86) have shown that for angles of incidence greater than the critical angle for inhomogeneous slabs of plasma there can be a contribution to the crosspolarized component from this term due to total internal reflection. We will restrict our attention to angles outside this range. The second-order term contributes to both the direct and cross-polarized components. Both the models of Feinstein and Granatstein (39) and Shkarofsky (41) are second-order models in that they consider only terms to double scattering. In the former, which is a scalar model, some estimate of the depolarization was obtained by assuming that the double scatter term contributed to the direct and cross polarized components in the ratio of 3: 1. This ratio was found on the assumption that the contribution to the double scatter term could be estimated by two 90" scatterings. The plasma turbulence spectrum used in the calculations was based on experimental measurements, and calculations were performed for a slab and cylinder of homogeneous plasma. The special correction for backscatter discussed earlier (Section II,E) was not considered. Backscatter calculations without the special correction are compared, in Section IV of this paper, with experimental results for scattering at 8, = 160". This value of 8, is outside the cone where the backscatter correction is required.
350
V. L. GRANATSTEIN AND DAVID L. FEINSTEIN
For Shkarofsky’s model (41) which is essentially a vector model, considerable attention is given to analytic forms of the turbulent spectrum (87,88). Calculations of the scattering cross section were carried out using an analytic expression for the turbulent spectrum derived for the case of k o a b 1 and a homogeneous slab of plasma. Again, the specialization for backscatter is not included. B. Exact Solutionsfor Idealized Models
The advantage of an exact formulation is that, unlike the iterative solutions discussed above, they are not dependent on the rapid convergence of the series solution through the size of the expansion parameter; furthermore, they may give more insight into the physics since the mathematical analysis has been carried further than with a purely numerical solution. The disadvantage is that to obtain the solution many simplifying assumptions about the medium must be made. Recently, there have been considerable advances made in the mathematical treatment of the transport equation [see, for example, References (89-91)] and it does not seem unreasonable that in the near future more powerful mathematical techniques will be available that will permit a better representation of the actual experimental configurations. The first model we will consider was developed by Feinstein, Leonard, Butler, and Piech (40). It is for the scalar equation, and is based on the singular eigenfunction technique (45). To utilize existing solutions (92) it was assumed that the turbulent plasma was homogeneous and isotropic. Under these conditions the transport equation Eq. (25) is separable in 4, the azimuthal angle, yielding:
with the superscripts m denoting the mth component in the azimuthal expressions :
m
where p is a dimensionless scattering phase function defined by In Eqs. (139) and (140) t is the optical depth
MULTIPLE SCATTERING OF MICROWAVES IN PLASMAS
35 1
the simple scatter albedo Go is
+ %),
Go = a,/(%
(144)
and p is the cosine of the polar angle, i.e. = cos
e.
(145)
The scattering phase function, under the homogeneity and isotropy assumptions, is azimuthally symmetric. The medium is now assumed to fill the half-space y > 0 with a plane wave incident at y = 0 at angle ( p o , 40).The method is now to find the normal modes or eigensolutions to Eq. (145) and write the answer in terms of an expansion over this set of solutions: “(Y,
=
w 4,(K d>, v
( 146)
where the 4, are the eigensolutions and the A , are the expansion coefficients. The S indicates the sum is to be taken over the discrete modes and an integral over the continuum modes. With this model it is possible to study the angular distribution of the intensity as a function of turbulent plasma and incident wave parameters. The scattered intensity is governed by two nondimensional quantities. The first, ak, , where k , is the incident wavenumber, determines the anisotropy or degree of forward peaking of the scattering phase function. Here it is necessary to choose a form for the isotropic turbulent spectrum. For the purpose of illustration, although any analytic or experimental spectrum could be used, Feinstein et al. (40) chose the von Karman interpolation formula (23) S ( K ) = C’a3/(1 + U 2 K * ) ” ’ 6 .
(1 47)
In Eq. (147) C‘ is the normalization constant such that J-n
p(A, a’) dQ(fi’) = 4n.
Feinstein et al. (40) have investigated a range of ak, from 0 (isotropic phase function) to 7. The upper limit, ak, = 7, corresponds approximately to the values of k , and the axial turbulence spectrum reported by Feinstein and Granatstein (39). Watson (38) has investigated the case ak, = 0 extensively with a more general transport model which will be discussed later. The second nondimensional parameter is the single scatter albedo Go defined in Eq. (144) as the ratio of single scattering attenuation to total attenuation. This parameter ranges from 0 for a purely absorbing medium to 1 for a purely scattering medium. For the half-space, which corresponds physically to an optically thick medium, the scattered intensity depends on only ak, and G o . For a medium which is not optically thick the scattered intensity, as is
352
V. L. GRANATSTElN AND DAVID L. FEINSTEIN
physically obvious, also depends on the width of the medium. It is important to realize that 63, depends on the turbulence parameters as can be seen in Eqs. (18), (26), and (144) so that in practice one cannot vary ak, and 0, independently. Representative calculations with this model are shown in Figs. 6 and 7. The ordinate in these figures is Z/Io,radiance normalized to the incident intensity I,. Figure 6 shows the total radiance within the medium as a function of polar angle in the plane of the zenith for ak, = 5.5 and 0,= 0.7. The incident direction has a polar angle of 60". We observe that for small optical depths there is a maximum about the incident direction. As the optical depth increases this maximum shifts toward the zenith and the I
OPTICAL DEPTH
I I €
FIG.6. Radiance distribution within a semiinfinite medium as a function of polar angle in plane of the zenith (for von Karman spectral function given in Eq. 147). From Feinstein et al. (40).
353
MULTIPLE SCATTERING OF MICROWAVES IN PLASMAS 0.9-
u
WO
- P a PO'
0
I
2
3
6
4
I x I0
'
f
1
I
9
10
10-2
FIG.7. Single scatter albedo as a function of normal backscattered intensity from a semiinfinite medium (for von Karman spectral function given in Eq. 147; special backscatter effect discussed in Section II,E,3 not included). From Feinstein et al. (40).
effects of the incident direction diminishes. The persistence of a maximum about the incident direction depends on the degree of forward peaking or value of ak, . For larger values of ak, or more strongly peaked phase functions the persistence is greater. In Fig. 7, the albedo is plotted as a function of the normal backscattered intensity for several values of ak, . For the isotropic case, the first-order or single scatter approximation is also shown. Considering the albedo as a function of intensity is the classical inverse problem of transport theory where one tries to deduce the properties of the medium by measuring the scattered intensity. We note strong deviations from single scatter theory. We also observe that the same intensity yields a much larger albedo for the more strongly peaked functions. There are several shortcomings of this approach. Although the eigenfunction solution to the transport equation is formally exact, certain numerical calculations must be performed. These calculations become more difficult for large ak, . Work is progressing in this area (93) and one should be able to extend the range of ak, . However, for large ak, it might be more advantageous to look at the diffusion approximation (Section II1,C) which is particularly applicable in this range.
354
V. L. GRANATSTEIN AND DAVID L. FEINSTEIN
In addition to the obvious extension to consideration of polarization effects, any model calculation, if it is to be compared to experiments should include the effects of inhomogeneities and anisotropy in the spectral function as well as more realistic geometries. The above calculations also did not consider the special backscatter effect discussed in Section II,E,3. The second set of model calculations we consider in this section were carried out by Watson (38), and are based on the discrete ordinate solution to the vector transport Eq. (36). For a homogeneous slab of plasma, in the limit of ak, -+ 0 (og isotropic), the vector transport equation can be. written as
where
and we have used the normalization
as is appropriate for the vector transport formalism (38). Equation (149)is appropriate for boundary conditions which are azimuthally symmetric (this physically only occurs for normal incidence). Besides the vector form of Eq. (149), the differences between Eqs. (139)and (149) are due to: (1) the azimuthal symmetry of the boundary condition in Eq. (149)so no azimuthal expansion is necessary, and (2) the assumption of the isotropic limit so the scattering phase function, og, is a constant. The discrete ordinate technique consists of selecting a set of discrete pi and approximating the integral by a sum :
This yields a coupled set of differential equations which may be solved by standard techniques. Some representative calculations are shown in Figs. 8 and 9. In Fig. 8 normal backscatter, with the specialization for backscatter included, is considered as a function of slab thickness z and a, . In Fig. 9,the fraction of cross-polarized component to total scattered intensity is shown as a function of z and Go. These calculations were performed for small values of m( 3) in Eq. (150)so that the results are approximate. This may indeed be the reason the fraction of cross polarized component is sometimes greater than 0.5 (94). The necessary extensions of this model to more realistic systems are similar to the extensions discussed for the previous model except polarization has already been included while ak, > 0 has not been considered. N
MULTIPLE SCATTERING OF MICROWAVES IN PLASMAS
355
rFIG.8. Backscatter from a slab
ako
[T
is optical thickness of the s1a.b; do= a,/(&
+ 1 ; normal incidence]. From Watson (38).
+ orA);
FIG.9. Fraction of cross-polarization in backscatter from a slab (ako-4 1 ; normal incidence). From Watson (38).
C. Diflusion Approximation
In many cases it would be desirable to have an analytic formulation for the case of strong fluctuations and large ak, .This region is of particular interest in the CTR work. The iterative solutions already discussed are not applicable when the fluctuations are strong; while the exact solutions, at this stage of development, still have unsolved numerical problems for large uk, , For these reasons it is advantageous to investigate the diffusion approximation
356
V. L. GRANATSTEIN AND DAVID L. FEINSTEIN
(95,31-33) to the transport equation which is appropriate for the case of large ak, while still being applicable for strong fluctuations. Essentially, the diffusion approximation converts the integro-differential transport equation into a differential equation by expanding the scattering function in a Taylor series about the forward direction and keeping only the lowest order terms (31). This is a valid approximation for ak, % 1 since in this case the scattering is highly peaked in the forward direction. Consider the transport equation given in Eqs. (36) and (37) and supplemented by Table I. Consistent with ak, % 1, assume that ng given in Table I is og(ks- Lo) = og(cos 0,) = 0 except when the scattering angle 0, < Oe where Oe is a characteristic parameter small compared to unity. Observe that this is similar to the model discussed in Section III,A,2. We write
fi, where
=
&, + K
(151)
I K I < 1. Expanding the intensity gives I(r, R,) = I(r, Lo) + K VkoI(r, Lo)
Recall that I is a vector in the Stokes space while the other vectors are in coordinate space, and the differential operator operates on all the Stokes parameters. Now express the scattering phase matrix P as
where
's
D' = - 6- ~Q(EJK~~J,(cos 0,)
(1 57)
357
MULTIPLE SCATTERING OF MICROWAVES IN PLASMAS
or, equivalently,
D,
'I
=-
3
dn(L,)(i - cos e,)a,(cos
e,).
(158)
In Eq. (158) we have used the fact that K~ = 4 sin2t6, = 2(1 - cos 6,). Keeping terms to order K~ we have for the transport equation in the diffusion approximation
LJ
*
VrI(r, L o )
+ aAI(r, Lo) = ~D,[V.C~P(JO, LS)Iks=~J
*
[ V . C ~ I Ro)] (~,
(159)
+ DtV& I(r, Lo).
For the slab geometry with azimuthal symmetry this reduces to
where p is the cosine of the scattering angle and 5 is the optical distance along the slab normal. As an example we consider the ratio of scattered radiance coming back out of the slab (n/2 < 8, < n) to the incident intensity
+
where I is the radiance (in terms of the Stokes parameters I = Il 12). In Fig. 10, A' is plotted vs. Qo as calculated by Watson (38).Note that for all but Qo 40, most of the energy diffuses into the medium, and is not scattered out. This would be expected since most of the scattering is in the forward direction. In Section IV, further applications of the diffusion approximation will be considered. For these purposes, it is useful to express the mean spread of the
FIG.10. The radiation returning from a slab in the diffusion approximation (ako normal incidence; D, = a,/20). From Watson (38).
+1;
358
V. L. GRANATSTEIN AND DAVID L. FEINSTEIN
scattered beam (0,’) in terms of the diffusion coefficient D , . Neglecting polarization, the probability for scattering into solid angle dC2 centered on scattering angle 0, is given by a,(cos &)LodC2(Ls),where Lo is the path length in the plasma (2). Thus, for the mean square scattering angle we obtain
and using Eq. (1 58) in the small 8, limit
(0,’) = 6Lo D , .
( 163)
For the specific case of a Gaussian correlation function and negligible collisions D , = (&/6a)(m/n14NZr)
as may be verified by substituting the Gaussian form of a, into Eq. (158) and integrating. Then the mean square scattering angle is
D. Monte Carlo Solutions
The Monte Carlo solution (42,84,85) is perhaps the most direct means of solving the radiative transport equation for the complicated scattering functions and boundary conditions that occur in many experimental configurations. The method simply consists of tracing a large number of photons through a random distribution of scattering centers with a known scattering law. The probability of a photon traveling a distance I before scattering through an angle 0, is computed by selecting random numbers normalized to the known scattering properties of the medium. The photon distribution function is then calculated by keeping track of the individual photons. In this manner any physical medium can be simulated. Comparison with experiments will be discussed in Section IV. There are, however, several shortcomings to this approach. In particular, the probability of a particular event such as backscatter for ako 9 1 might be extremely small. Hence, if one was interested in backscatter an excessively large number of photons must be studied requiring a prohibitively large amount of computation time. In addition, the theoretical purists object on the basis that the method is not more than a numerical experiment and little physical insight can be gained. However, this is a moot question, and there is a growing community who feel that this kind of analysis is “ more real ” than much sophisticated mathematics where the true physics is lost.
MULTIPLE SCATTERING OF MICROWAVES IN PLASMAS
359
IV. COMPARISON OF EXPERIMENTAL RESULTS AND MODELCALCULATIONS A . The Regime of Large Scale Fluctuations (a 9 1,) As mentioned in the introduction, investigations of the multiple scatter of millimeter waves from plasma fluctuations with a 5-10 cm scale size have been carried out in the ZETA device in England and in the ALPHA device in the Soviet Union. Both these devices were toroidal vacuum vessels (with minor diameter -1m) filled with low pressure (- 1 mtorr) deuterium or hydrogen gas in which high-current pulsed discharges were induced. The mean plasma density during the current pulse was between 1013 cm-3 and 10'' ~ m - and ~ , the electron temperature was between 10 and 50 eV ( Rand T, were measured by Langmuir probe, microwave emission, infrared transmission, and neutral beam methods). The magnetic field due to the induced plasma current is self-constricting and tends to pull the plasma away from the walls; however, the configuration is hydromagnetically unstable. Partial stabilization is achieved by externally applying an axial magnetic field of some hundreds of gauss, but strong oscilIations of plasma density are still observed during the current pulse. The probing of this turbulent plasma with millimeter waves is somewhat simplified by the fact that the magnetic field is small enough so that its effect on the plasma dielectric constant can be ignored ; furthermore, the electron temperature is sufficiently high that collisional absorption is negligible. Thus the plasma dielectric constant is simply 1 - N/N,, . Initially, attempts were made to use millimeter waves in measuring the mean plasma density by standard interferometry techniques (51,57). However, the strong influence of the turbulence on the microwave signal made such measurements impossible. For plasma densities appreciably less than the critical density of the microwaves, the microwave signal was highly attenuated when propagated across the minor axis of the toroid (e.g. attenuation in ALPHA was 10-20 dB with 0.2 N,, < < 0.5 N,, ,and even stronger attenuation was observed in ZETA). Also very strong amplitude and phase fluctuations were found in the transmitted signal (59). The turbulence in these toroidal devices probably develops as a result of MHD convective instabilities, and its features are reminiscent of ordinary hydrodynamic turbulence. For example, the spectrum of the fluctuations measured with electric probes in the frequency interval 10 kHz to 2 MHz (59) is observed to be peaked at low frequencies and to fall off monotonically as frequency increases. No sharp peaks were found in the spectrum. Larionov and Rozhdestvenskii (59) have suggested that the plasma density fluctuations
m
360
V. L. GRANATSTEIN AND DAVID L. FEINSTEIN
are caused by collective motions in the presence of a mean plasma density gradient from the discharge axis to the walls of the discharge chamber. They estimated the size of the plasma density fluctuations by
AN,,,/N z aVN/N z alR' z 0.1,
(165)
as would be appropriate for ordinary hydrodynamic turbulence. (The scale size was taken as a = 5 cm as determined from correlation measurements with electric and magnetic probes, and the chamber minor radius, R', is 50 cm.) The importance of multiple scattering can now be gauged by roughly evaluating the magnitude of a,D, the number of scatterings a wave experiences in traversing a minor diameter of the toroidal plasma. Using Eq. (14) we have N
where the parameters of the plasma devices described in the preceding paragraphs have been used, and we have made the calculation for a 4 mm microwave wavelength. Thus a wave will undergo many scatterings from largescale fluctuations in traversing the plasma. Because k , a % 1, the scatterings will be at small angle (forward scattering) and will have the effect of converting the incident coherent wave into an incoherent beam with mean square angular width BO2 given roughly by Eq. (164). Outside this main beam of incoherent radiation there will be a much weaker signal arising from single large angle scattering out of the main beam. Now, if the time variation of the plasma fluctuations is primarily due to a random convection velocity v, then each scattering will result in a doppler shift of the wave frequency by an amount wok, sin(OJ2) (see Eqs. 5 and 9). Thus, the multiple small angle scatterings (0, 4 1) might produce little frequency broadening compared with the effect of a single large angle scattering. Wort and Heald (49) observed the frequency spectra of both the forward propagating beam and a signal scattered through 90". In line with the above discussion, the 90" frequency broadening was found to be substantially greater than in the forward direction, and was Gaussian in shape with a width of 4 MHz. Assuming this broadening to arise from doppler shifts in single scattering events, Wort and Heald deduced that the random convection velocity was also Gaussian with a width of 1 x lo5-7 x lo5 cmlsec depending on time during the current pulse. These measurements are consistent with estimates of the random velocity obtained from electric field, energy balance, and spectroscopic evidence. Measurements on the multiply scattered main beam have also been carried out to get a measure of the plasma density fluctuation. The angular spread of a beam of 2 mm microwaves transmitted through the ZETA plasma
361
MULTIPLE SCATTERING OF MICROWAVES IN PLASMAS
was measured (52,53) with the experimental arrangement shown in Fig. 11. The angular profile of the intensity in the transmitted beam was determined by using a movable receiving antenna, and was found to have a Gaussian shape, i.e.
qe,) = I(0)e-8s2/802
(167)
as shown in Fig. 12. The angular width 8, rms ranged from 20" to 60" during the current pulse. If one takes the correlation length of the plasma fluctuations to be a = 5 cm (from correlation measurements with electric and magnetic probes in ZETA), then Eq. (1 64) can be used to relate the measured values of 8, rmsto LOCAL OSCl LLATOR CARCINOTRON BALANCED MIXER
LF. AMPLIFIER RECEIVING
osc I LLOSCOPE
01ELECTRIC SHEET POWER SPLITTER
CARCINOTRON OSCl L L ATOR
FIG. 11, Arrangement for millimeter wave scattering measurements on ZETA. From Stott (42).
t
e: FIG. 12. Typical logarithmic plot of scattered radiance vs. scattering angle in ZETA measurements. From Stott (42).
362
V. L. GRANATSTEIN AND DAVID L. FEINSTEIN
AN,,, . We calculate that 20" < O0 , < 60" corresponds to 1.4 x loi3 cm-3
- I5
5 x -5
0
I -0
2.0
3-0
a
4.0
TIME (meact
FIG.13. Time dependence of AN,,, in ZETA, calculated from the scattering measurements, showing a quiescent period of reduced fluctuations.The lower trace is plasma current (peak value 350 kA). From Stott (42).
MULTIPLE SCATTERING OF MICROWAVES IN PLASMAS
363
multiple scattering [i.e. they neglected the source term, J(r; A) in the radiative transport equation, Eq. (25)].Thus, they postulated that the beam attenuation caused by the plasma fluctuations was simply due to single scattering of power out of the coherent wave; this was described by the equation
PIP, = X(N)e-’sD, (168) where Po is the power transmitted without plasma, P i s the power transmitted with plasma present in the path, and the factor X(N)accounts for absorption and coupling between the antennae. The values of a, which were deduced by using Eq. (168) must be regarded with suspicion, since in the experiment, u,D 2 10 and k o a w 75, and thus the contribution of multiple scattering to the forward propagating wave is expected to be so great as to completely obscure the attenuated coherent wave. Larionov et al. (55) were not completely unaware of this difficulty. In extracting a value of AN,,, from their estimate of a, by using Eq. (14), they attempted to compensate for the underestimation of a, by purposely underestimating the scale size a. They realized that small angle scattering by the larger inhomogeneities would not contribute to the signal attenuation, and therefore they somewhat arbitrarily chose a = A/bA= 2 cm, where 4Ais the angular width of the receiving antenna directivity pattern. However, even with this adjustment the calculated values of AN,,, were rather low (they calculated AN,,, w 4 x 10” cm-3 at IV = 1 x I O l 3 ~ m - ~ judging ), by subsequent more reliable determinations. In a later paper, Larionov and Rozhdestvenskii (59) obtained a more reliable and larger estimate of AN,,, by analyzing the amplitude fluctuations (scintillations) of the transmitted wave. They employed the expression -
x2 = [In (E/E0’)l2 = IN, DB arctan B,
where E,,’ is the amplitude of the applied wave with no plasma, E is the wave amplitude in the presence of turbulent plasma, and B = 4D/(k0a2).Equation (I 69) is derived by a perturbation method in References (22) and (24, and (2n)-’(D/a)1’3] and for is appropriate for large values of k,a [viz. k,a x2 4 1 (59). Using Eq. (169) only as a lower bound, since ;;” w 1 in the ALPHA experiments, it was calculated (59) that AN,,, = 1.1 x 10” cm-3 at N = lOI3 ~ m - ~ .
+
B. The Regime a
In contrast to the k,a
-
A.
+ I regime of Section IV,A, we will now discuss
a class of experiments in which the microwave wavelength was roughly equal
to the turbulence scale size. In this new regime, multiple scattering of
364
V. L. GRANATSTEIN AND DAVID L. FEINSTEIN
electromagnetic waves by plasma fluctuations is in general considerably more difficult to analyze; however, experiments have been carried out at several laboratories (39,61-71) specifically to gain insights which would lead to useful theoretical models. The experimental devices used were steady state rather than pulsed, and allowed for controlled variation of plasma density. A large range of plasma densities from much less than N,, to larger than N,, was investigated. Ordinary hydrodynamic turbulence was generated by subsonic shear flow of a weakly ionized gas. The plasma fluctuations which resulted from this tur1. bulence were used in the microwave scattering studies; typically, AN,,,/N No external magnetic fields were applied, and in general, the electron collision frequency was moderately small compared with the microwave frequency (v/wo5 0.1) so that attenuation of the microwaves was dominated by scattering rather than absorption. One of the main diagnostics used in all the experiments was a set of electrostatic probes which were biased to draw ion saturation current. These probes yielded measures of N(r) and ANrm&). Wavenumber spectra of the plasma fluctuations were in part deduced by measuring a frequency spectrum of the probe ion current, S,(W). This spectrum was interpreted as a onedimensional wavenumber spectrum along the gas flow axis by using Taylor’s hypothesis W = K f. Taylor’s hypothesis is a good assumption in turbulence diagnostics when the fluctuations in gas velocity are small ( A U , , ~4 17). This was satisfied to varying degrees of approximation in the different experiments. In one set of experiments at Bell Telephone Laboratories (BTL) (39,6871),the plasma fluctuations were produced by forcing Ar gas in turbulent flow through a glass tube. Electrodes were located 1 m apart on the tube axis. A dc arc discharge was then struck between the electrodes. The relative fluctuation of the convection speed was small. This is required for validity of Taylor’s hypothesis (Au,,,/ij x 4 %). The nature of the plasma fluctuations in such a device has been described in the literature (97-99).* Electron density in the turbulent flow tube was easily varied by changing the discharge current, a, and to first order, this did not affect the stochastic parameters (ANrmS/N, etc.) Furthermore, the microwave antennae could be located far from the plasma so that Fraunhofer zone (far field) analysis of the scattering geometry was valid. In most of the BTL microwave investigations, the diameter of the glass tube was 2.5 cm, while the diameter at which AN,,, had fallen to l/e times its axial value was D,,,w 1 cm. The axial correlation length was a, = 0.6 cm, and strong anisotropy was found to exist with the transverse scale size a, r a,/10. The microwave wavelengths used were lo= 0.88 cm
-
* Recent insights (99) indicate that the experimental values of Nreported in the microwave scattering experiments (39,68-71) should be increased by 60%, and this correction will be made when presenting results in the present paper.
MULTIPLE SCATTERING OF MICROWAVES IN PLASMAS
365
(giving kD,,, x 7) and lo= 0.44 cm (giving ko D,,, x 14). In other experiments in the A, a regime at the Stanford Research Institute, and at the RCA Victor Research Laboratories, the extent of the plasma was larger, and 14 6 ko D,,, 6 50. At the Stanford Research Institute (SRI) (61-65), microwave scattering experiments were performed using a plasma jet. The plasma was produced in a relatively high pressure chamber by thermal ionization in an oxygen-ethylene flame, and was then expanded through a nozzle (diameter = 2.5 cm) to form the jet in a lower pressure chamber with diameter = 1.2 m), The relative gas velocity fluctuation was Au,,,/~ 20% so that validity of Taylor’s hypothesis was marginal. Details of the device may be found in References (100) and (101). The plasma density was varied by seeding the flame with KCI; this produced some narrowing of the jet diameter (e.g., at a typical location for the microwave measurements, 50 cm downstream from the nozzle exit, D,,,x 12 cm with the value of AN,,, on the jet axis being ANo x 10” ~ m - and ~ , D,,, FZ 7 cm with AN, x 3 x 10” cm-j). The microwave antennae were located inside the jet chamber, and the chamber walls were coated with microwave absorber. In contrast to the measurements at BTL, both the near field radiation pattern of the antennae and the variation of plasma parameters along the jet axis had to be considered in making calculations of scattering from the plasma jet. The axial correlation length was a, FZ 2.5 cm and the transverse scale was a, x 1.25 cm. The microwave wavelengths used were lo= 3.2 cm (giving 14 6 k , D,,, 5 24), and I , = 0.97 cm (giving 45 5 k , D I l e5 78). At RCA Victor Research Laboratories (RCA) (66,67), a plasma jet device was also employed with approximately the same dimensions as the device at SRI. However, the plasma was produced in the high pressure chamber with an argon arc rather than a flame. This device is described in reference (102). Variation in N was achieved only by moving the position of the microwave illumination relative to the nozzle. The axial position of the microwave illumination varied from 30 cm to 70 cm downstream from the nozzle exit; however, over this range the correlation scale of the turbulence was apparently constant at a x 2 cm, and isotropy was found to exist (a, x a,). The transverse extent of the plasma fluctuations varied from D,,, x 8 cm at the 30 cm location to D,,, = 16 cm at the 70 cm location. The microwave wavelength used for all the RCA experiments was I , = 1.9 cm (giving 26 < k , D,,, < 53).
-
-
1. Experiments with N 4 N,,
When N < N,,, multiple scattering effects are weak and one can reasonably expect to predict the initial manifestations of multiple scattering with perturbation analyses. Preparatory to examining such weak multiple scattering, the experimenters checked predictions of single scatter theory.
366
V. L. GRANATSTEIN AND DAVID L. FEINSTEIN
-
The increase of a with A N Z as predicted by first-order Born analysis (see Eq. 6 ) was first observed in the flame jet device and reported by Guthart et al. (61). In that same paper, it was also erroneously reported that the measured absolute magnitude of the scattering cross section, a, was 6 to 11 dB larger than predicted by first-order Born analysis, and that a increased as only the first power of AN,,, for AN, > 0.003 N,, (at 1, = 3.2 cm and 8, = 160"). Later measurements in the flamejet (62,63) showed in fact that the measured magnitude of a agreed with Eq. (6) to within 2 dB for values of AN, up to N,, for large angle scattering (e.g. 0, = 160"). Weissman et al. (62) also described interferometer measurements involving signals to two receiving antennas with variable separation ; results of these measurements at low AN,,, also agreed well with predictions of first-order Born analysis, and demonstrated the potential of radar interferometry in determining the size of a remote turbulent plasma. Another set of measurements, which were interpreted using first-order Born analysis, involved the frequency spectrum of the scattered microwaves. In the turbulent jets (63,66), frequency broadening was primarily due to the spatial gradient of the mean gas velocity, and only a rough measure of the velocity fluctuations (factor of 2) could be obtained from frequency broadening measurements. However, in the pipe flow experiment at BTL (71), the largest contribution to frequency broadening was shown to arise from the velocity fluctuations; thus, a more precise measure of fluctuations in the convection velocity of the background gas was obtained. A frequency spectrum of the scattered microwaves measured by Granatstein and Philips is shown in Fig. 14. The frequency spectrum is peaked at o,- w,, = K I= 2ko ij cos [ (for backscatter at aspect angle c) as would be expected from Eq. (5). The frequency broadening was not found to increase substantially over the prediction of single scatter analysis until AN, > N,, . This observed failure of multiple scattering to strongly affect the frequency spectrum agrees with theoretical predictions (103). On the other hand, other effects of multiple scattering do appear at considerably lower values of ANrms.In a single scatter process, with incident electric field Eo perpendicular to the scattering plane formed by k, and k,, one would expect the scattered wave to be polarized parallel to Eo . However, a substantial cross-polarized component has been observed with AN, as low as 0.04 N,, (70,39). Granatstein and Buchsbaum (70) resolved the component of the radiance with polarization parallel to E,, I,,, and the crosspolarized component, Z, . The dependence of both I , , and Zl on aspect angle [ was determined. Their experimental arrangement is shown in Fig. 15. Results of the measurement are presented in Fig. 16. They are compared with a second-order solution of the transport equation obtained by Feinstein and Granatstein (39) and described in Section III,A,3.
MULTIPLE SCATTERING OF MICROWAVES IN PLASMAS
367
There was a degree of arbitrariness in the modeling; for example, the plasma was considered to be a statistically homogeneous cylinder with radius equal falls to A N o , and with the homoto the point at which the measured ANrms geneous AN,,, of the model taken as $ A N o . Nevertheless, it may be noted from Fig. 16 that the calculations do correctly predict the relative levels of I , , and ZL as well as the shape of the I , variation with aspect angle, including the broad maximum in ZL around 5 = 15".
+
r
7-
$a
6-
3
t
m
3
5-
d
g
4-
J
3-
2 4
a I-
Y
w
w (I
4
2I-
3 0
> o
I
0
2
I 4
I
I
I
I
6
8
10
1 2 1 4
J
FIG.14. Frequency spectrum of backscattered microwave signal (scattering from fluctuations in plasma column of discharge in turbulent gas; aspect angle 5 = 60";ho = 0.88 cm; ANo = 0.3Nc,; mean gas flow speed fi = 80 mlsec). VN is the spectrum of the background noise in the absence of plasma. From Granatstein and Philips (71). ELECTROSTATIC PROBE
1
m
F%XI:Up
CATHOOE
""
POLARIZATION DIPLEXER
::
11
FROM KLYSTRON
Ill
FIG.IS. Experimental arrangement for measuring fIIand f I as a function of aspect angle 6. From Feinstein and Granatstein (39).
368
V. L. GRANATSTEIN AND DAVID L. FEINSTEIN
-
t
-
0
-.
c FIG.16. Direct and cross-polarized radiance as a function of aspect angle 5. A comFrom Feinstein and parison of calculations with experimental results ( A N o = 0.04NC,). Granatstein (39).
Finally, in closing this subsection on the parameter regime a 1, and N 4 N,, , we take note of a recent experiment carried out by Shkarofsky and Ghosh (67). A monopole antenna was inserted into the plasma, and conversion of coherent radiation into incoherent radiation was studied as the wave propagated into the plasma. Their experimental arrangement is shown schematically in Fig. 17. The microwave circuitry was capable of distinguishing between the coherent wave intensity, I, , and the intensity of the incoherent radiation, Ii.The measurements of I, and Ii are plotted as a function of position in Fig. 18. One can clearly see the conversion of coherent energy to incoherent energy as the wave penetrates more deeply into the plasma. N
I
PLASMA JET
I
I
TRANSMITTING ANTENNA
+r
XI4 MONOPOLE RECE I V I NO ANTENNA
------W
X
FIG.17. Experimental arrangement for measuring conversion of coherent to incoherent radiation in a turbulent arc jet.
MULTIPLE SCATTERING OF MICROWAVES IN PLASMAS
‘
‘ 1
369
(EXPERIMENT)
10 I
0.8
I
I
I
I
I
I
I
I
* x
I
/’
fi,
/’
c
0.01 -10
-
1
-8
1
-6
, 4
1
-4
-2
0
(EXPERIMENT)
2
l
I 6
l 8
C
X
10
FIG. 18. Intensities of coherent and incoherent radiation vs. position. From Shkarofsky and Ghosh (67).
Figure 18 also shows theoretical curves which may be compared with the measurements. The theoretical curve for I, was calculated from 1, = 1, exp(
-
cis d x ) , -03
where a, was taken to have the same parameter dependence as in Eq. (14) but with a constant different from &/2 to account for the nongaussianity of S@). The theoretical curve for intensity of the forward-propagating incoherent wave was calculated from Ii = Z,,
j:
a,dx, m
(171)
3 70
V. L. GRANATSTEIN AND DAVID L. FEINSTEIN
which is applicable in the optical limit (A, 4 a) where the scattering is strongly peaked in the forward direction. The comparison of theory and experiment are shown in Fig. 18. For Z, the discrepancy is less than lo", while for Zi the discrepancy is as great as 30 % with experimental Zifalling below the predictions of Eq. (171). This is not surprising since an optical-limit expression like Eq. (171) would be expected to overestimate Zi in the forward direction. 2. Experiments as N Approaches N,, Measurements of the variation of a with N , as N approached the critical density, were reported by Granatstein and Buchsbaum (68, 70). At A, = 0.88 cm, with transmitting and receiving antennae on opposite sides of the discharge tube, they found a rose as (AN,)' until A N , x 0.2NC,;at that point a sharp saturation was observed with D almost constant until No x 2Nc,; then, 0 began to fall with further increase in AN, .* In backscatter, the saturation was less sharp, and was sometimes preceded by a region where a increased more rapidly than (AN,)'. This different behavior for backscatter may be due to the extra scattering term in backscatter as discussed in Section II,E,3. Of course, the observed saturation at AN, = O.2Nc, should not be expected to be characteristic of a wide class of tubulent plasmas. Such a phenomenon would depend on such factors as the relative scale size, all,, and the relative extent of the plasma, D/A, Indeed, when Granatstein and Buchsbaum performed a backscatter experiment with a smaller wavelength (A, = 0.44 cm), they found that saturation did not occur until AN, = 0.35 N,, , Feinstein and Granatstein (39) attempted a theoretical calculation of the behavior of 0 vs. AN, as AN, --f N,, . They used the first- and second-order terms of their Neumann series solution of the radiative transport equation. This effort must be regarded as ill-advised since there was no justification for ignoring higher order scatterings for AN, 2 0.07 N,, . (See Section III,A,I for a discussion of convergence of the Neumann Series.) A more intensive study of the variation of a with AN, was carried out at SRI on the flame jet plasma (63,65). The experimental arrangement is shown schematically in Fig. 19, and some of the results are presented in Fig. 20. Note that in this experiment, saturation occurs at larger AN, when 8, is larger; the curve for 8, = 150" does not saturate until AN, is larger than N,, . The theoretical curves in Fig. 20 were calculated by Guthart and Graf (63) using their semiempirical approach described in Section II,A,2. Recall that this calculation took into account the change in refractive index of the plasma as R approached N,, . That is, the wave vector of the plasma fluctuations from which scattering occurs was taken as K = (1 - H/Nc,)(ks - k,). I
*The dependence of u on ANo has recently been studied in the regime NS- N,, by Attwood (104).
MULTIPLE SCATTERING OF MICROWAVES IN PLASMAS
/
k
'TURBULENT FLAME JET WITH KCI
37 1
l N G ANTENNA
ANTENNA
FIG. 19. Experimental arrangement for measuring u vs. ANo in turbulent flame jet.
--A
EXPERIMENTAL CALCULATED
No (cm-3)
FIG.20. u vs. ANo as ANo + N,, (Ao = 3.2 cm, N,, and Graf (63).
= 1.1
x loL2cm-9. From Guthart
The decreasing magnitude of this wave vector as N approaches N,, will result in increasing the scattering cross section because of the nature of the spectral function S(K). Thus, as AN,, -+ N,, , the strong scattering attenuates the incident wave and tends to cause a saturation in oy but this effect may be cancelled out by the effect of an increasing S(K). Consequently, o apparently can continue to rise approximately as ANo2 long after strong multiple scattering has become significant (a, > 1/D) as shown both by the calculated and the experimental curves at 8, = 150" in Fig. 20. Another feature of the calculations of Guthart and Graf (63) was to compute the scatter attenuation of the forward propagating wave from an effective scattering coefficient as given in Eq. (138), viz.
+
aeE= [a, - ( ~ ( ~ / b ) ] e - ( ~(a,/b). "~~)~ The factors b and 8, were chosen empirically to fit the experimental data. It was found that the calculations were much more sensitive to the choice of b
372
V. L. GRANATSTEIN AND DAVID L. FEINSTEIN
than to the choice of 8,. In the calculations shown in Fig. 20, b was chosen to be equal to 6 . Encouragingly, b = 6 also fit some experimental data taken by Graf et al. (65) at 1, = 0.97 cm as well as results of a computer experiment by Hochstim and Martens (25,26,64). However, in matching this semiempirical model to measurements in the arc jet at RCA, b had to be chosen as 100 (105); such a large variation in the empirical factor b between two experiments with essentially similar parameters suggests that further study is warranted. V. BRIEFSUMMARY In conclusion, satisfactory solutions of the multiple scattering equations have been found in the optical limit @,a % l), and weak multiple scattering effects have been successfully described by perturbation analysis even when a A,, . However, in the case of strong multiple scattering with a A,, no satisfactory solutions of the multiple scattering equations are as yet available. This situation is depicted in the mapping of Fig. 21. The range of available experimental data is also indicated in that figure. Note especially that experimental data does now exist in the theoretically difficult region where k, a 1 and ANrms N,, . N
N
N
N
hss %//////////
%s;L%E:s
Eu::y::;Ly WITH
;LgTE0 RE-ENTRY m y s i c s
Id' 10-3
10''
FIG.21. Regions of parameters space where various approximations for electromagnetic scattering from turbulent plasma are applicable (asD evaluated from Eq. 12).
MULTIPLE SCATTERING OF MICROWAVES IN PLASMAS
373
GLOSSARY Correlation length of plasma density fluctuations Correlation length along mean flow axis Correlation length transverse to flow axis Ql A‘ Total radiance scattered through angles fom 90” to 180” normalized to incident intensity, Eq. (161) Empirically chosen parameter, Eq. (1 38)’ First-order part of T, Eq. (131) Speed of light in vacuum Two-point space-time correlation function of plasma density fluctuations D Spatial extent of turbulent plasma, diameter of plasma column, or minor diameter of plasma torus Diameter at which AN,,, = AN& Diffusion coefficient, Eqs. (I 57) and (I 58) Electron charge Unit vectors indicating direction of polarization of incident field Unit vectors indicating direction of polarization of scattered field Unit vectors indicating direction of polarization of field scattered by electron at rp to observation point at r. Electric field Incident electric field Amplitude of applied wave with no plasma present Scattered electric field Component of E, with polarization parallel to $ ( j ) Electric field scattered by electron at re to observation point at ra Component of EaBwith polarization parallel to e^a.p(j) Coherent electric field Incoherent electric field Incoherent part of Epla Nondimensional Thompson scattering amplitudes. See Eqs. (51) and (52) Pair correlation function. Defined after Eq. (67) Far field, scalar Green’s function for the wave equation. See Eq. U
Qz
(48)
Weighting function in integral equation for
+
3 74
V. L. GRANATSTEIN AND DAVID L. FEINSTEIN
.I
Column matrix specifying intensity and polarization of incident wave I,, Intensity of incident wave I, Part of I arising from cohexent wave I, Intensity of coherent wave I” nth term in Neumann series expansion of I 1”’ Azimuthal mode of I, Eq. (140) I , Intensity of incoherent wave Im Imaginary part o f . . . J Emission. See Eq. (22) J Emission vector. See Eq. (37) ko Incident wave vector k, Scattered wave vector k., Unit vector describing direction of wave vector on propagating from rDto r. K Wave vector of plasma density fluctuations 1 Transport mean free path L Modified 1. See Eq. (121) Lo Path length in plasma me Electron mass M Number of electrons in system n Refractive index of plasma nl Mean refractive index of plasma n, Real part of n R, R’ Unit vectors indicating direction of wave propagation N Plasma density R Mean plasma density AN Fluctuating part of plasma density AN,,, Root mean square value of AN ANo Value of AN,,, on axis of axisymmetric plasma N,, = (w2/c2>(4nr,)-’ Critical plasma density p(r ;R, ti’) Scattering law describing probability of scattering electromagnetic wave from direction R,’ to direction R . Analogous to differential scattering cross section a P(r; 4,a’) Scattering phase matrix (4 x 4). See Eq. (104) for components in “scattering theory” formalism. See Table I for components in Stokes formalism (gIP(cc8, 8 ~ ) I t t ’ ) Component of P in “scattering theory” formalism, Eq. (104) P I , Component of P in Stokes formalism. See Table I P(r, v) Probability density function of the convection velocity PM(rl,r 2 , . .. rM) Joint probability of electron 1 being at rl, electron 2 being at r 2 , . . , and electron M being at rM PGI ,(rl, . . r,Conditional probability function for electron positions. See Eq. (66) rDfl. . . . r, ; r d r Spatial coordinate r. A field point ro Reference position rs, re; r,,, r,; Electron positions r l , r 2 , .. r M ,etc. re Classical electron radius. 2.82 x 10-’5m
.
.
.
MULTIPLE SCATTERING OF MICROWAVES IN PLASMAS
r. - ro Vector denoting difference in position Matrix describing rotation of polarization vectors along curved ray paths. See Eq. (122) Distance along ray path Eikonal. See Eq. (73) and discussion following Eq. (76) Power spectral density of the pattern of plasma density fluctuations Frequency spectrum of electrostatic probe ion saturation current Time Retarded time Electron collision time for momentum transfer Transfer matrix which is the resolvent kernel of the integral transport equation. See Eq. (128) Part of T due to multiple scattering Corrected form of T including special backscatter effect. See Table I1 Electron temperature Operator used in iterative solution of radiactive transport equation. See Eq. (I 32) Energy density with polarization 6 at point r.. See Eq. (82) Coherent part of u. See Eq. (83) Four-element column matrix whose volume integral times c is the incoherent part of I. See Eq. (92) Component of U ( U , 8). See Eq. (88a) Convection velocity Scattering volume Frequency of plasma density fluctuations Operator used in iterative solution of radiative transport equation. See Eq. (132) Depth. Coordinate of symmetry, orthogonal to plane of the azimuth Coordinate in direction o f f Attenuation coefficient. u = 1 / 1 Single scattering attenuation coefficient Absorption coefficient Effective scattering coefficient, empirically determined, Eq. (138) Angle between k, and Eo Green's function operator. See Eq. (58) Permittivity of free space Aspect angle Polar angle Empirically chosen parameter, Eq. ( I 38) Scattering angle Angular width of transmitted beam; especially with Gaussian dependence, exp (- Q2/OOz) Root mean square value of 8,
R, - L o /L
Wavelength Wavelength of incident electromagnetic wave cos 0
376
V. L. GRANATSTEIN AND DAVID L. FEINSTEIN
Electron collision frequency for momentum transfer Optical depth, Eq. (143) 4?rp(fi,fi’)/a,(fi). Dimensionless scattering law Azimuthal mode of p , Eq. (141) plus. Dimensionless scattering phase matrix, Eq. (126) Differential scattering cross section (cross section per unit volume per unit solid angle) First-order Born approximation for u, Eq. (6) Improved version of un including collisional effects and change in n as N + N,, , but omitting polarization effects, Eq. (127) and Table I Distorted wave Born approximation for u, Eq. (15) Time change Optical thickness of slab Azimuthal angle Parameter used in defining R, Eq. (1 23) In (E/Eo’),Eq. (169) Frequency Frequency of scattered wave Frequency of incident wave Single scatter albedo, Eq. (1 36) Solid angle Ensemble averages. See Section 11,C,3 is equivalent to an overbar
ACKNOWLEDGMENTS The authors are indebted to the following individuals for their helpful comments and suggestions: S . J. Buchsbaum, M. A. Heald, A. Leonard, A. M. Levine, J. Lotsof, T. 0. Philips, S.N. Samaddar, I. P. Shkarofsky, F. D. Tappert, K. M. Watson, and D. J. H. Wort. A helpful literature search was carried out by G. G. Harris.
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90. Second Conference on Neutron Transport Theory, Los Alamos National Laboratory USAEC, CONF-710107 (April 1971). 91. Proceedings of the Atlas Symposium, No. 3, Inter-disciplinary Symposium on the Applications of Transport Theory. J. Quant. Spectrosc. Radiant Transfer 11, (1971). 92. N. J. McCormick and I. Kuscer, Bi-orthogonality relations for solving half space transport problems. J. Math. Phys. 7 , 2036 (1966). 93. H. G. Kaper, J. K. Shultis, and J. G. Veninga, Numerical evaluation of the slab albedo problem solution in one-speed anisotropic transport theory. J. Comp. Phys. 6, 288 (1970). 94. K. M. Watson, Private conversation, 1971. 95. B. Davison, “Neutron Transport Theory.” Oxford Univ. Press, London, 1958. 96. M. G . Rusbridge, A numerical experiment on the scattering of microwave radiation by a turbulent plasma. Plasma Phys. 10, 95 (1968). 97. G. A. Garosi, G . Bekefi, and M. Schultz, Response of a weakly ionized plasma to turbulent gas flow. Phys. Fluids 13,2795 (1970). 98. V. L. Granatstein, Structure of wind-driven plasma turbulence as resolved by continuum ion probes.” Phys. Fluids 10, 1236 (1967).
MULTIPLE SCATTERING OF MICROWAVES IN PLASMAS
99.
38 1
V. L. Granatstein and A. M. Levine, Comments on response of a weakly ionized plasma to turbulent gas flow. Phys. Fluids 14, 2247 (1971).
100. H. S. Rothman, H. Guthart, and T. Morita, Spectral distribution ofcharged particles in
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Author Index Numbers in parentheses are reference numbers and indicate that an author’s work is referred t o although his name is not cited in the text. Numbers in italics show the page on which the complete reference is listed.
A
Adams, R. N., 317(24), 377 Alexander, J. K., 21(22), 60 Alferieff, M. E., 125(27), 207(27), 208, 266, 305 Altenhoff, W. J., 31(31), 60 Amatuni, A. Ts., 257(128), 307 Andersen, W. H. J., 85(69), 143(69), 188 Anderson, J. S., 263(150), 264, 308 Appelbaum, J. A., 202,305 Arlinghaus, F. J., 69(28), 187 Armstrong, R. A,, 235(81), 306 Arp, H. C . , 54,61 Ashworth, F., 194(1), 212(1), 304 Attwood, D., 370, 381
B Badde, H. G., 157(152), 190 Badgley, R. E., 238(87), 306 Bahr, G. F., 100(95), 182(95), 189 Baker, B. G . , 240(93), 306 Band, W., 237(84), 306 Barabanenkov, Yu. N., 318(34, 35, 36), 342(35, 36), 343(35, 36), 378 Barakat, R., 112(118), 156(118, 151), 189, 190 Barbour, J. P., 242(95), 278(184), 279, 280, 281, 282, 283, 284(95), 286(200), 288(200), 306, 309 Bare, C., 17(18), 19(18), 60 Barnes, G., 233(75), 278(187), 281(187), 290(75), 306,309 Barrett, A. H., 40(47, 491, 61 Bekefi, G., 364(97), 380 Bell, A. E., 212(52), 260, 261(52), 262(52),
265(161, 162), 269, 270, 271(162), 306,307,308 Bell, S. J., 46(58), 61 Benesch, R.,80(55), 188 Bennett, A. J., 209(46), 305 Bennette, C. J., 220(62), 221, 276(62, 177), 277(179), 284(195), 306,308,309 Beth, H., 194(7), 305 Bethe, H., 76(41), 77(41), 187 Bettler, P. C . , 278(187, 190), 281(187), 282, 283(190), 309 Blackman, M., 76(45), 77(45), 188 Blatt, F. J., 257(124), 307 Blevis, E. H., 240(94), 306 Blott, B. H., 195(34), 212(53), 227, 263(149), 305, 308 Boersch, H., 156(148), 190 Boling, J. L., 278, 309 Bonham, R. A., 68(26), 187 Born, M., 10(6), 60, 112(116), 113(116), 115(116), 139(116), 141(1 16), 156(116), 189, 332(77), 380 Boseck, S., 182(163), 191 Bowles, K. L., 312, 319(1), 376 Bracewell, R. N., 23(25), 60 Branson, N. J. B. A., 26(27), 27(29), 60 Brinkmann, W. F., 202,305 Broderick, J. J., 17(19), 60 Broten, N. W., 17(17), 60 Brouw, W. N., 50(64), 61 Brown, W. P., Jr., 317(20), 377 Briinger, W., 92(85), 96(85), 188 Buchsbaum, S. J., 320(68, 70), 364(68, 70), 366(70), 370(68, 70), 379 Bugnolo, D . S., 318(43, 44), 378 Buhl, D., 40(52), 61 Burbidge, E. M., 58(76), 61 Burbidge, G. R., 55(74), 56(74), 58(76), 61 Burge, R. E., 80(56), 83(56), 92(86).
384
AUTHOR INDEX
93(87), 96(56, 901, 111(86), 188, 189 Buribaev, I., 257(127), 307 Burke, B. F., 25, 60 Bussler, P., 181(160), 190 Butler, F. E., 318(40), 345(40), 347(40), 350, 351(40), 352(40), 353(40), 378
C
Cameron, A. G. W., 46(59), 61 Carswell, A. I., 365(102), 381 Case, K. M., 318(45), 350(45), 378 Castaing, R., 109(102, 103, 105), 110(112), 160(102), 189 Chandrasekhar, S., 317(28), 318(28), 323(78), 377 Charbonnier, F. M., 195(29), 223(29), 242(101), 243(29), 245(29), 246(29), 247(29), 248(29), 250(29), 251(29), 252(29), 273, 274(169), 275(169), 278(184, 190), 279(184), 280(184), 281(184), 282(184), 283(184, 190, 193), 286(200), 288(200), 305,307, 309 Chernov, L. A., 317(22), 377 Cheung, A. C., 40(50), 40(51), 61 Childs, P. A., 68(27), 187 Chisholm, R. M., 17(17), 60 Chow, P., 317(16), 377 Christiansen, W. N., 10, 23(24), 60 Christov, S. G., 194(11), 200, 305 Chrobok, G., 257(130), 307 Clark, B. G., 17(18), 19(18), 60 Clark, H. E., 229, 265(158), 266, 269, 271,306,308 Clark, T.A., 17(20), 60 Cohen, M. H., 16(13), 16(14), 17(18), 19(18), 60 Colliex, C., 109(104), 110(104), 189 Collins, R. A., 46(58), 61, 212(53), 263(149), 306, 308 Conklin, E. K., 58(79), 61 Cooper, E. C., 220(63), 221(63), 277(63), 306 Cosslett, V. E., 65(7, 8, 91, 165(7), 176(9), 182(8), 187, 289, 309 Cowley, J. M., 67(19), 70(19, 31-33), 76(44), 78(19, 31, 33, 48), 79(53), 88(78), 89(19), 112(119), 187, 188, 189
Cox, H. L.,68(26), 187 Crewe, A. V., 65(11-131, 100(96), llO(11, 113), 120(1l), 178(1l), 181(11-1 3), 182(11, 96), 187, 189, 286(199), 295(199, 213,214), 309 Crick, R. A., 64(4, 5), 66(4, 18), 85(4), 86(4), 90(18), 92(18), 97(18), 111(18), 158(4, 5), 164(5), 169(4), 171(4), 172(4), 174(5), 187 Cronyn, W. M., 16(15), 60 Crouser, L. C., 195(29, 31-33), 208.211, 215(32), 223(29-32) 225(32), 226, 228(32), 230, 232(32), 233(32), 234(32), 235(78), 236, 240(32), 243(29), 245(29), 246(29), 247(29), 248(29), 250(29), 251(29), 252(29), 261, 262, 265(159), 266, 268(32), 269, 286(141), 287(141), 305, 307,308 Crowell, C. R., 235(81), 240(94), 306 Cundy, S. L., 90(84), 109(84), 110(84), 111(84, 114, 115), 112(115), 159(84), 160(84), 165(115), 188, 189 Cutler, P. H., 195(17, 22, 23), 197, 201(22, 23), 203, 225(67), 252(111), 305, 306, 307
D Daniels, J., 82(64), 83(64), 111(64), 188 Davidson, B., 318(46), 356(95), 378, 380 Denman, E. D., 317(24), 377 deWolf, D. A., 316(11), 317(18, 19), 345, 377 Ditchfield, R. W., 100(99), 189 Dolan, W. W., 194(2, 91, 212(2), 233(75), 278(184), 279(184), 280(184), 281(184), 282(184), 283(2, 1841, 284(96), 290(75), 293,304,305,306 Douglas, D. C., 320(65), 364(65), 365(65), 370(65), 372(65), 379 Downes, D., 31(31), 32(36), 60 Downs, G. S., 47(61), 61 Doyle, P. A., 89(82), 110(82), 161(82), 188
Drechsler, M., 233(74), 243, 245, 249, 252,289,306,307,309 Drummond, D. G., 182(164), 191 Dudenhausen, W. D., 75(39), 187 Duke, C. B., 195(27), 207(27), 208, 266, 305
385
AUTHOR INDEX
Dupouy, G., 65(10), 181(156, 157), 187,190 Dupree, A. K.,38(42), 61 Dyke, W. P., 194(2, 9), 212(2), 233(75), 242(96, 97), 278(184), 279(184), 280(184), 281(184), 282(184), 283(2, 184), 284, 285(196), 286(196), 288(200), 290, 295(192, 193), 304, 305,306, 309
E Eckstein, W., 258(132), 307 Edwards, S . F., 319(52), 342(79), 361(52), 378, 380 Eggenberger, D. N., 286(199), 295(199), 309 Ehrenstein, G., 40, 61 Ehrlich, G., 283, 309 Eisenhandler, C. B., 117(127), 118(127), 156(127), 190 El Hili, A,, 109(102, 103). 110(112), 160(102), 189 Elinson, M. I., 284(194), 288(207), 309 Elsmore, B., 12(9), 60 Engel, T., 258(135), 260(135), 307 Engle, I., 252(111), 307 Enjalbert, L., 181(157), 190 Erickson, H. P., 125(134), 181(134), 190 Ermrich, W., 265(157), 266(157), 276(178), 308,309 Ess, H. E., 347(85), 358(85), 380 Evans, B. E., 212(52), 261(52), 262(52), 306 Everhart, T. E., 222(64), 290(64), 306 Ewen, H. 1.. 33(37), 60
F Falicov, L. M., 209(46), 305 Fawcett, E., 251(107, 108, 109), 307 Feinstein, D. L., 318(39, 40), 320(39), 345, 346(39), 347(40), 349, 350, 351, 352, 353, 364(39), 366(39), 367(39), 368, 370, 378 Felsen, L. B., 317(16, 21). 377 Ferrell, R. A., 81(59), 188 Festenberg, C. V., 82(64), 83(64), 111(64), 188
Feuchtwang, T. E., 229.306 Findlay, J. W., 5, 11(3), 60 Finkel'berg, V. N., 318(35), 342(35, 81), 343(35), 378, 380 Fischer, R., 198,305 Fisher, P. M. J., 78(49), 188 Fisher, R. M., 88(79), 188 Fleming, G . M., 242, 307 Flood, D. J., 195(25), 207(25), 210,305 Floyd, R. L., 286(200), 288(200), 309 Foldy, L. L., 317(27), 324(27), 377 Ford, G. W., 343(82), 344(82), 380 Forstmann, F., 205, 206, 265(41, 42), 305 Fowler, R. H., 194(6), 199, 305 Frank, J., 125(134a), 131(134a), 181(134a, 160), 190
Franklin, K. L., 25, 60 Freeman, A. J., 80(54), 188 Frisch, U., 316(4), 343(4), 376 Fujimoto, F., 109(108), 189 Fujiwara, K., 78(52), 188 Fukuhara, A., 109(107), 189 G Gadzuk, J. W., 195(18, 21, 24, 26), 197, 201(21), 202,206,207(26), 209, 225(24), 227(68), 228(68, 70), 252(68), 265(24), 266,305, 306,308 Galt, J. A,, 17(17), 60 Garosi, G. A., 364(97), 380 Gasse, H., 252(112, 114), 253(112, 1151, 254( 1 15), 307 Gavrilyuk, V. M., 212(50), 261(142), 305,308 Gel'berg, A., 236, 306 Gevers, R., 73(35), 187 Ghosh, A. K., 320(66, 67), 364(66, 67), 365(66, 67, 102), 366(66), 368, 369, 372(105), 379, 381 Gjennes, J., 78(47), 88(77), 188 Glaeser, R. M., 100(95a), 182(95a), 189 Glauber, R., 67(20), 69(20), 187 Gleich, W., 258(133), 307 Glick, A. J., 82(62), 188 Goldberg, L., 38(42), 61 Goldstein, R. M., 17(20), 60 Gomer, R., 194(4), 195(35), 206, 212(4, 49), 215(4, 5 9 , 227(35), 228, 258,
386
AUTHOR INDEX
259(55), 260(55, 134,135), 261, 262, 264, 265(156, 160), 268, 274, 276(175, 176), 305, 306, 307, 308 Good, L., 31(31), 60 Good, R. H., Jr., 194(3, lo), 198, 212(3), 298(10), 304,305 Goradze, G. A., 257(125), 307 Gordon, K. J., 51, 61 Gorkov, V. A., 284,309 Graf, K. A., 320(63-65), 346(63, 64), 348(63, 64), 364(63-65), 365(63-65, 102), 366(63), 370(63, 65), 371, 372(64, 65), 379, 381 Granatstein, V. L., 318(39), 320(39, 68-71), 345(39), 346(39), 349, 351, 364(39, 68-71), 364(98, 99), 366(39, 70), 367(39), 368, 370(68, 70), 378, 379, 380,381
Gretz, R. D., 263(146), 308 Griffiths, D., 251(109), 307 Grinton, G. R., 79(53), 188 Grishina, T. A., 158(153a), 190 Grubb, D. T., 100(99), 189 Gundermann, E. J., 16(13), 60 Gush, H. P., 17(17), 60 Guth, E., 194(8), 305 Guthart, H., 320(61-65), 348(63, 64), 364(61-65), 365(61-65, 100, 101), 366(62, 63), 370(63, 65), 371, 372(64), 379, 381
H Habeck, D. A,, 110(113), 189 Hager, H., 182(163), 191 Haine, M. E., 112(120), 143(120), 158, 189, 289,309 Halioua, M , 181(162a), 186(162a), 19/ Hall, C. R., 117(130), 190 Hall, R. W., 26(27), 60 Halseth, M. W., 349, 380 Hama, K., 96(89), 97(89), 158(89), 169(89), 176(89), 177(89), 189 Hamming, R. W., 117(126), 189 Hanszen, K.-J., 64(3), 65, 112(3), I16(3, 123), 120(3, 123, 133), 121(23), 139(3, 143, 144), 143(143), 144(3), 147(147), 149(147), 154(147), 1 5 33),
181(3), 185(3), 187, 189, 190 Hardy, S. C., 264,308 Harris, J. L., 184(165), 191 Harrison, W. A., 195, 206, 305 Hashimoto, H., 88(80), 188 Haskell, R.E., 322(72), 379 Hawkes, P. W., 114(121), 189 Hayashi, M., 75(38), 78(38), 79(38), 187 Heald, M. A., 319(49), 360, 378 Heidenreich, R. D., 73(34), 108(34), 109(111), 117(126), 118(132), 136(34), 143(132), 145(34), 156(34), 168(34), 169(34), 187, 189, 190 Heiland, W., 258(132), 307 Helstrom, C. W., 185(167), 191 Helwig, R., 258(133), 307 Henderson, J. E., 238, 242, 306, 307 Henoc, P., 109(105), 189 Henry, J. C., 40(49), 61 Henry, L., 109(102, 103, 105), 110(112), 160(102), 189 Herring, C., 235, 237, 278, 279, 306, 309 Hewish, A., 11(7), 16, 46(58), 47(60), 60,61
Hey, J. S., 15(11), 60 Hibi, T., 138(139), 143(139), 190, 289, 295, 309 Hillier, J., 156(149), 190 Hines, R. L., 117(130), 190 Hirsch, P. B., 68(22), 72(22), 73(22), 74(22), 76(22), 87(73), 88(22), 109(22), 165(55), 187, 188, 190 Hobson, J. P., 285(197), 309 Hochstim, A. R., 317(25, 26), 372, 377 Hoernders, B. J., 131(135a), 190 Hoerni, J. A., 68(23), 76(40), 187 Hofmann, M., 257( 129, 130), 307 Hogbom, J. A,, 10,60 Holscher, A. A., 238, 306 Holt, E. H., 322(72), 379 Hopkins, H. H., 112(117), 137(117, 138), 138(138), 139(117), 142(138), 143(117), 189, 190
Hoppe, W., 131(135), 181(159, 160). 190 Hornby, J . M., 32, 60 Howie, A., 68(22), 72(22), 73(22), 74(22), 76(22), 78(49), 88(22, 80), 90(83, 841, 109(22, 51, 83, 84), 110(83, 84), 111(51, 841, 159(83, 84). 160(84), 161(83), 187, 188
387
AUTHOR INDEX
Huang, T. S., 185(168), 191 Hubbard, J., 82(60), 110(60), 188 Hudda, F. G., 283(191), 309 Humphreys, C. J., 87(73), 165(155), 188, 190
1
Ibers, J. A., 68(23), 187 Ingold, J. H., 289(205), 309 Inoue, Y., 191 losfisku, B., 236(83), 306 Ipavich, F. M., 59(78), 61 Isaacson, M., 100(96), 182(96), 189, 295(214), 309 Ishida, K., 109(110), 189 Ishikawa, A., 191 Itskovitch, F. I., 195(20), 201(20), 238, 257(125), 305, 306, 307
J
Jager, J., 100(91), 1 1 1(91), 189 Jaklevic, R. C., 210(48), 305 James, R. W., 74(37), 79(37), 187 Jauncey, D. L., 17(18), 19(18), 60 Jefferts, K. B., 41(54), 61 Johnson, B. B., 240(93), 306 Johnson, C. D., 97(92), I 1 1(92), 189 Johnson, D., 100(96), 182(96), 189, 295(214), 309 Johnson, F. B., 100(95), 182(95), 189 Johnson, H. M., 181(158), 190 Johnson, J. B., 252(120), 307 Johnston, T. W., 320(66), 364(66), 365(66, 102), 366(66), 379, 381 Jones, A. F. 84(68), 96(68), 184(166), 188, 191
Jones, E. H., 243, 244, 307 Jouffrey, B., 109(104), 110(104), 189
K Kainuma, Y., 109(108, log), 189 Kamiya, Y., 109(100), 189 Kanaya, K., 191 Kaper, H. G., 353(93), 380
Karal, F. C., Jr., 317(12, 13), 377 Keil, P., 83(66), 188 Keller, J. B., 316(5, 6), 317(12, 13, 16), 343(6), 376, 377 Kellermann, K . I., 17(18), 19(18), 52(69), 56(69), 57, 60, 61 Kenderdine, S., 12(9), 60 Kennedy, P., 257(129), 307 Kerr, F. J., 33(38), 36, 36(40), 61 Kislyakov, A. l., 319(54, 57), 359(57), 379 Kleint, C., 238, 252(112), 253(112, 115, I I8), 254(115), 255(123), 306, 307 Klimenko, E. V., 274(170, 171), 308 Klug, A,, 125(134), 181(134), 190 Knight, C. A,, 17(20), 60 Kobayashi, K., 100(94), 182(94), 189 Kogan, Sh. M., 257(126), 307 Kohrt, C., 261,307 Konsha, G., 236(83), 306 Kornelsen, E. V., 285(197), 309 Krzywoblocki, M. Z., 316(8), 377 Kudintseva, G. A,, 288(207), 309 Kulik, I. O., 257(125), 307 Kundu, M . R., 23(23), 60 Kupiec, I . , 317(16, 211, 377 Kuscer, I., 350(92), 380 Kuyatt, C. E., 218, 222(65), 306 Kvapil, J., 186(170), 191
L Lambe, J., 210(48), 305 Landau, L. D., 342(80), 380 Landecker, T. L., 28, 29(30), 60 Langer, R., 131(135), 181(160), 190 Lapchine, L., 181(157), 190 Larionov, M. M., 319(5460), 359(59), 362, 363, 379 Lau, Chau-Wa, 318(32), 324(32), 327(76), 344, 356(32), 378, 380 Lea, C., 195(35), 227(35), 228, 252(35), 265(160), 268, 305, 308 Leder, L. B., 65(6), 66(6), 90(6), 100(6), 111(6), 164(6), 187 Legg, T. H., 17(17), 60 Leonard, A,, 318(40), 345(40), 347(40), 350, 351(40), 352(40), 353(40), 378 Lenchek, A. M., 59(78), 61 Levine, A. M., 364(99), 381
388
AUTHOR INDEX
Levine, P. H., 200, 243, 305 Lenz, F., 64(1, 2), 112(1), 113(1), 115(1, 2). 116(2), 119(2), 122(2), 127(2), 129(2), 131(2), 137(1), 142(1), 143(146), 144(1), 159(1), 187, 190 Liepack, H., 233(74), 306 Lilley, A. E., 40(47), 61 Lipshitz, E. M., 342(80), 380 Litvak, M. M., 45, 61 Locke, J. L., 17(17), 60 L’vov, S . N., 289(203), 309
M McCormick, N. J., 350(92), 380 MacGillavry, C. H., 76(42), 188 Mackie, W., 222(66), 224(66), 238(66), 241(66), 306 McLeish, C. W., 17(17), 60 Madey, T. E., 270(167), 308 Maire, G. L. C., 240(93), 306 Malli, G., 80(55), 188 Malmberg, J. H., 318, 378 Manchester, R. N., 47(62), 61 Maran, S . P., 46(59). 61 Marandino, G. E., 17(20), 60 Marcus, S . M., 210(47), 305 Marshall, T. C., 318, 378 Martens, C. P., 317(25, 26), 372, 377 Martin, E. E., 242(95,97, 101),243(102), 246(102), 278(184), 279(184), 280(184), 28 1(184), 282(184), 283(184), 284(95), 285(196), 286(196), 306, 307, 309 Marton, C., 65(6), 66(6), 90(6), 100(6), 111(6), 164(6), 187 Marton, L., 65(6), 66(6), 90(6), 100(6), 111(6), 164(6), 187 Mathewson, D. S . , 23(24), 32, 50(64), 60,61
Mattheiss, L. F., 237(85), 306 Maxwell, A., 31(31), 60 Meclewski, R., 252(117), 307 Medvedev, B. K., 274,308 Medvedev, V. K.,212(50), 305 Meeks, M. L., 40(49), 61 Melmed, A. J., 261(145), 262(145), 263(145, 148), 264(152), 265, 278(186), 282,283(186), 308, 309 Mendlowitz, H., 65(6), 66(6), 90(6),
100(6), 111(6), 164(6), 187 Menz, W., 92(85), 96(85), 188 Menzel, D., 276(175), 276(176), 308 Menzel, E., 181(162), 186(162), 190 Metherell, A. J. F., 88(79), 89(81), 90(81), 109(106), 110(81), 111(115), 112(115), 161(81), 165(115), 188, 189 Mikheeva, E. V., 240(90), 306 Mileshkina, N. V., 271(168), 288, 308, 309 Mirandb, W., 181(162), 186(162), 190 Miroschnickenko, L. S., 289(204), 309 Misell, D. L., 64(4, 5), 66(4, 18), 68(27), 80(56), 83(56), 84(67, 68), 85(4), 86(4), 90(18), 92(18, 86), 93(87), 96(56,68), 97(18), lll(18, 86), 141(56, 145), 144(145), 146(145), 158(4, 5), 159(154), 164(5), 165(154), 169(4, IS), 171(4), 172(4), 174(5), 184(166), 187, 188, 190 Mitton, S., 53, 61 Miyake, S., 78(46), 188 Modinos, A., 125(28), 195(15, 16), 204, 205, 207(28), 208, 269, 272,305,308 Mol, A., 85(69), 143(69), 188 MSllenstedt, G., 139(142), 183(142), 190 Montagu-Pollack, H. M., 263(151), 308 Moodie, A. F., 67(19), 70(19, 31), 76(44), 78(19, 311, 88(78), 89(19), 112(119), 187, 188, 189 Morgenstern, B., 120(133), 190 Morita, T., 320(61, 62), 364(61, 62), 365(61, 62, 100, 101), 366(61, 621, 379,381
Mott, N. F., 243,244,307 Miiller, E. W., 194(3, 13), 212(3), 215(54), 223(13), 232, 233, 235(54, 77), 236(77), 304, 305,306
Mullin, C.J., 194(8), 305 Murphy, E. L., 194(10), 198, 298(10), 305 Mussa, G., 236(83), 306
N Nagata, F., 96(88), 97(89), 158(89), 169(89), 176(89), 177(89), 189 Nagy, D., 195(17), 203, 225(67), 305, 306 Natta, M., 109(105), 189 Naumovets, A. G.. 261(142). 274(170, 171), 308
389
AUTHOR INDEX
Nemchenko, V. F., 289(203), 309 Newton, R. G . , 316(3), 323(73), 376, 380 Nichols, M. H., 235, 237, 306 Nicholson, R. B., 68(22), 72(22), 73(22), 74(22), 76(22), 88(22), 109(22), 111(115), 112(115), 165(115), 187, 189 Nicolaou, N., 269, 272, 308 Niehrs, H., 117(129), 190 Nixon, W. C., 289,309 Nordheim, L. W., 194(5, 6), 199, 305 Nottingharn, W. B.. 242, 243, 307 Nowicki, R., 195(14), 205, 305 Nozikres, P., 82(61), 110(61), 111(61), 188
0 Obermair, G., 257, 258, 307 Odishariya, G. A., 278(188), 309 Olsen, H.,69(30), 187 Osborne, C. F., 203,305
P Palmer, P., 39(43), 40(44, 52), 44(55), 45(44), 61 Palyukh, B. M., 274(172), 308 Parsons, D. F., 181(158), 190 Parsons, S. J., 15(11), 60 Paschley, D. W., 68(22), 72(22), 73(22), 74(22), 76(22), 88(22), 109(22), 187 Pauliny-Toth, I. I. K., 52(69), 56(69), 57,61 Paunov, M., 264(154), 308 Peacher, J. L., 68(25), 187, 318(33), 321(33), 324(33), 344(33), 356(33), 378 Pelly, I., 258(134), 260(134), 307 Pendry, J. B., 206,265(42), 305 Penzias, A. A., 41(54), 58, 61 Pefina, J., 186(170), 191 Perrier, F., 181(157), 190 Pettengill, G. H., 2(1), 60 Philips, T. O., 320(71), 364(71), 367, 379 Phillips, J. W., 15(11), 60 Piech, K. E., 318(40), 345(40), 347(40), 350, 351(40), 352(40), 353(40), 378 Piliya, A. D., 319(55), 363(55), 379 Pilkington, J. D. H., 46(58), 61
Pines, D., 81(57), 82(61), 110(61), 111(61), 188 Plummer, E. W., 195(24, 30), 217(30) 218, 225(24), 227(68), 228(68, 70), 252(68), 265(30, 161, 162), 265(24), 266(30), 267, 268, 269, 270, 271(162), 305,306,308 Politzer, B. A., 195(22, 23), 197, 201(22, 23), 229, 305, 306 Potter, H. H., 245, 307 Powell, C. J., 96(88), 100(88), 111(88), 189 Protopopov, 0. D., 240(90,92), 306 Purcell, E. M., 33(37), 60, 218, 306
R Radhakrishnan, V., 47(62), 61 Radi, G., 87(74, 76), 188 Raether, H., 82(63, 64),83(64), 97(63), 110(63), 111(63), 111(64), 188 Raimes, S . , 81(58), 188 Raith, H., 68(24), 187 Ramberg, E. G., 156(149, 150), 190 Rank, D. M . , 40(50, 51), 61 Rauth, A. M . , 100(98), 189 Redhead, P. A., 261(139), 285(197), 307, 309 Reed, W. A., 251(107), 307 Rees, A, L. G., 70(32), 187 Regenfus, G., 257(129, 130), 258(133), 307 Reichley, P. E., 47(61), 61 Reimer, L., 100(93), 117(124, 128), 118(124), 136(136), 157(128, 1521, 182(93), 189, 190 Resh, D. A., 110(113), 189 Rhodin, T. N., 263(151), 308 Richard, C., 320(66), 364(66), 365(66, 102), 366(66), 379, 381 Richards, R. S . , 17(17), 60 Richardson, 0. A,, 242(98), 306 Rickett, B. J., 16(16), 60 Riecke, W. D., 137(137), 178(137), 179(137), 183(137), 190 Rinehart, R., 31(31), 60 Ritchie, R. H . , 228, 306 Roberts, M. S . , 49(65), 61 Robertson, D. S . , 17(20), 60 Robinson, B. J., 20(21), 60 Rodichkin, V. A., 319(58), 379
390
AUTHOR INDEX
Rogers, A. E. E., 17(20), 60 Roman, P., 67(21), 187 Rosenbaum, S., 317(14, 15, 16, 17, 21), 342(17), 377 Rosenbruch, K. J., 120(133), 190 Rothman, H. S . , 365(100), 381 Rozhdestvenskii, V. V., 319(54-59), 359(59), 362, 363, 379 Ruffine, R. S . , 316(11), 377 Rusbridge, M. G . , 362, 380 Ruthemann, G., 100(97), 189 Ryle, M., 11(7), 12(8, 9), 53, 56(72), 60, 61 Rymer, T. B., 97(92), 111(92), 189
S Sadhukhan, P., 182(164), 191 Sahashi, T., 158(153), 169(153), 176(153), 190
Sakaoku, K., 100(94), 182(94), 189 Salpeter, E. E., 315, 358(2), 376 Samsonov, G . V., 289(202, 203), 309 Sandomirskii, V. B., 257(126), 307 Sandqvist, A,, 41(53), 61 Scalapino, D. J., 210(47), 305 Scharpf, D., 257(129), 307 Scheinberg, B. N., 240(90), 306 Schemer, O., 65(14), 112(14), 117(14), 187 Schmidt, L. D., 212(49), 215(55), 235(79), 258(55), 259(55), 260(55), 305, 306, 307 Schmidt, M., 54(70), 61 Schomaker, V., 67(20), 69(20), 187 Schottky, W., 252(119, 121), 307 Schultz, M., 364(97), 380 Scott, P. F., 46(58), 61 Seitz, F., 244(105), 307 Shaffer, P., 289(206), 309 Shapiro, I. I., 2(1), 17(20), 60 Shepherd, W. B., 218,306 Shiskin, B. B., 257(127), 307 Shkarofsky, 1. P., 318(41), 320(66, 67), 345, 346(41), 349, 350(87, 88), 364(66, 67), 365(66, 67), 366(66), 368, 369, 372(105), 378, 379, 380, 381 Shreider, Yu. A., 347(84), 358(84), 380 Shultis, J. K., 353(93), 380 Shuppe, G. N., 240(90), 306 Sidorski, Z . , 258(134), 260(134), 307
Siegel, B. M., 117(127), 118(127), 156(127), 181(161), 190 Silver, M., 261(144), 308 Simpson, J. A., 65(6), 66(6), 90(6), lOO(6, 98), 111(6), 164(6), 187, 189, 218,306 Sivaprosad, K., 349,380 Sivers, I. L., 274(172), 308 Slater, J. C., 87(70), 188 Smart, J. W., 80(56), 83(56), 96(56), 141(56), 188 Smith, G. D. W., 263(150), 264, 308 Sneddon, I. N., 114(122), 117(122), 189 Snider, R. F., 342(78), 380 Snyder, L. E., 40(45, 52), 42(45), 61 Sobolev, V. V., 317(29), 318(29), 377 Sokol’skaya, 1. L., 271(168), 274, 278(185), 288,308,309 Sommerfeld, A., 194(7), 305 Southon, M. J., 263(151), 308 Stevenson, M. J., 40(48), 61 Stolz, H., 139(142), 183(142), 190 Stott, P. E., 318(37, 42), 319(52, 53), 342(37), 343(37), 344(37), 347(42), 358(42), 361(52, 53), 362, 378 Stoyanova, I. G., 158(153a), 190 Stratton, R., 195(19), 197, 305 Strayer, R. W., 212(51), 213(51), 215(51), 220(63), 221, 222(66), 238(66), 241, 242(101), 255(51), 259, 261(51), 273, 274(169), 275(169), 277, 278(189), 281(189), 286(200), 288(200), 305, 306, 307,308,309 Stringushchenko, 1. V., 240(92), 306 Stroke, G. W., 181(162a), 186(162a, 169), 191
Stroud, A. N., 110(113), 189 Sturkey, L., 78(49), 188 Sugata, E., 263(147), 308 Sullivan, W. T., 45, 61 Sun, N., 237(84), 306 Swanson, L. W., 195(29, 31-33), 208, 211, 212(51, 52), 213(51), 215(32, 51), 220(62, 63), 221, 222(66), 223(29, 31-33), 224(66), 225(32), 226, 228(32), 230, 232(32), 233(32, 76). 234(32), 235(78), 236(76), 237,240(32), 242(101), 243(29), 245(29), 246(29), 247(29), 248(29), 250, 251,252,255(51), 259, 261(51), 262(52), 265(159),
391
AUTHOR INDEX
266, 268(32), 269, 273, 274, 275, 276(177), 289(154), 286(141), 287( 141), 305,306, 307, 308,309
Swanson, N., 96(88), 100(88), 11 1(88), 189
Swarup, G., 23(25), 60 Swensen, G. W., Jr., 13(10), 60 T
Takahashi. S., 138(139), 143(139), 190, 295(212), 309 Takeda, K . , 263(147), 308 Tatarski, V. I., 317(23), 351(23), 377 Taylor, L. S., 316(9, lo), 377 Terzian, Y., 49(63), 61 Thomson, M. G . R.,65(16, 17), 119(16, 17), 120(16), 127(16), 143(17), 187 Thon, F., 131(135), 181(161), 190 Thorton, D. D., 40(50, 51), 61 Timni, G. W., 254,307 Timonin, A . M., 319(58), 379 Tantegode, A. Ya., 240(91), 306 Townes, C. H., 40(48, 50, 511, 61 Treiman, S. B., 315, 358(2), 376 Trepte, L., 139(143, 144), 143(143), 147(147), 149(147), 154(147), 190 Trolan, J . K., 233(75), 242(95-97), 278(184), 279(184), 280(184), 281(184), 282(184), 283(184), 284(95, 96), 285 (196), 286(196, 200). 288(200), 290(75), 306 307,309 Turner, P. S., 78(48), 188 U Uhlenbeck, G. E., 343(82), 344(82), 380 Unwin, P. N. T., 111(115), 112(115), 165(115), 189 Usnicki, B. J., 316(7), 377 Utsugi, H., 274, 308 Uyeda, R., 65(15), 109(100, 101), 110(101), 112(15), 187, 189 V
Van der Kruit, P. C., 50(64), 61 Van der Zeil, A,, 254, 307 Van Dorsten, A. C., 139(141), 183(141), 190
VanHove, L., 345(83), 380 Van Oostrom, A., 215(56), 235(56), 265(157), 266(157), 306, 308 Veninga, J. G., 353(93), 380 Verdier, P., 181(157), 190 Vernickel, H., 277, 285(198), 308, 309 Verschuur, G . L., 31(32), 38(41), 60, 61 Vladimirov, G . G . , 274, 308 Viatskin, A. Ia., 82(65), 92(65), 188 von Hoerner, S., 5(2), 60 Vorobev, Yu. V., 117(125), 118(125), 156(125), 189 Vyazigin, A. A,, 117(125), 118(125), 156(125), 189 W
Wall, J., 65(12, 13), 110(113), 181(12, 13), 187, 189, 286(199), 295(199, 213), 309
Watanabe, H., 109(101), 110(101), 189 Watson, K. M., 318(31-33, 38), 321(33), 324(31-33, 38, 74, 75), 338(31), 341(38), 344(33), 345, 346(38), 347(38), 351, 354(94), 355, 356(31-33, 38), 357, 366(103), 378, 380, 381 Weingartner, I., 181(162), 186(162), 190 Weinreb, S., 40, 61 Weissman, D. E., 320(61, 62), 364(61, 62), 365(61, 62, I O l ) , 366(61, 62), 379 Welch, W. .I. 40(50, , 51), 61 Weliachew, L., 52(67), 61 Welter, H., 285(198), 286(199), 295(199, 213), 309 Welter, L. M., 110(113), 189 Westerhout, G., 36(39), 61 Wharton, C. B., 318,378 Whelan, M. J., 68(22), 72(22), 73(22), 74(22), 76(22, 43), 87(72, 75), 88(22, 43, 75, 80), 100(99), 109(22, 106), 111(115), 112(115), 165(115), 187, 188,189
Vainshtein, B. K., 74(36), 187 Valdrk, U., 90(84), 109(84), 110(84), 111(84), 159(84), 160(84), 188 Vandenberg, N. R., I7(20), 60
Whitcutt, R. D., 195(34), 227, 305 White, R. S., 261(144), 308 Whitney, A. R., 17(20), 60 Wielebinski, R., 28, 29(30), 60
AUTHOR INDEX
392
Wiesner, J. C., 290, 291, 292, 293(211), 294,309 Wild, J. P., 24, 60 Wills, J. G., 68(25), 187 Wilska, A. P., 139(140), 143(146), 183(140), 190 Wilson, R. W., 41(54), 58, 61 Windram, M. D., 56(72), 61 Wolf, E.,10(6), 60, 112(116), 113(116), 115(116), 139(1la), 141(116), 156(116), 189, 332(77), 380 Wolf, E. D., 261(143), 262(143), 308 Wort, D. J. H., 319(49, 50, 51), 359(51), 360, 362, 378 Wyndham, J. D., 55(73), 61 Wynn-Williams, C . G., 32, 34(35), 60
Y Yakovleva, G. D., 284(194), 309
Yates, J. T., 270(167), 308 Yen, J. L., 17(17), 60 Yoshioka, H., 87(71), 88(71), I88 Young, R. D., 194(12, 13), 195(30), 199, 217(30), 222(65), 223(13), 229, 235, 236, 265(30, 158), 266(30, 158), 267,268, 296,298, 301,305,306,308 Z
Zanberg, E. Ya., 240(91), 306 Zeitler, E., 65(16, 17), 69(29, 30), 100(95), 119(16, 17, 131), 120(16), 127(16), 143(17), 178(131), 179(13l), 182(95, 131), 187, 189, 190 Zeppenfeld, K., 82(64), 83(64), 111(64), 188 Zuckerman, B., 39(43), 40(44,52), 44(55). 45(44), 61 Zweifel, P. E., 318(45), 350(45), 378
Subject Index A Abberation functions, in electron microscopy, 167-1 79 Absdrption extinction coefficient, 321 A. F. Institute (Leningrad), 319 ALPHA device (U.S.S.R.), 319, 359, 362-369 Amorphous specimen, scattering by, 79 Amplitude contrast convolution function, in electron microscopy, 132-1 33 Amplitude contrast deconvolution function, 134-135 Amplitude contrast transfer function, in electron microscopy, 130(-131 Angular-energy distributions, of electron scattering, 90-100 Antenna temperature, 3 Antiballistic missile defense, plasma fluctuations in, 319 Aperture synthesis method, in radio astronomy, 11-13,26 Atomic species, scattering by, 75
B Backscatter, radiative transport and, 345, 355, 367 Bell Telephone Laboratories, 320, 364 Born analysis, 366 Born approximation, 73, 312-317 Brightness temperature, of sky, 3
Chromatic aberration in electron microscopy, 147-153, 169-171 on inelastic image, 180 Chromatically incoherent illumination, 143-155 Chromatic coherence, partial, 153-1 55 Chromatic incoherence, defined, 155-158 Clean surface characteristics, in field electron microscopy, 223-258 Coadsorption, in field electron microscopy, 261-262 Coherent illumination, in electron microscope, 158-165 Coherent radiation, intensity of, 368-369 Cold field emission, 298-299 Continuum emission, from normal galaxies, 49 Continuum systems, in radio astronomy, 18-19 Controlled thermonuclear research (CTR), 318-319, 355-356 Convolution, holographic, 186 Convolution integral. in electron microscopy, 167 Copper, work function values for, 240
as 339 47 Crimean Astrophysical Observatory, 17 CTR, see Controlled thermonuclear research Culham Laboratory (U.K.), 319 Cygnus, ionized gas region in, 31
C
California Institute of Technology, 13 Carbon electron scattering in, 94 variation of electron energy distribution in, 98-99 Cathode stability, in field electron emission, 283-289
D Data processing, in radio astronomy, 18-22 Decametric bursts, in radio astronomy. 25-26 Density fluctuations, in turbulent plasma, 359-360 393
394
SUBJECT INDEX
E Earth galactic noise, and, 27-28 Jupiter and, 26 Elastic mean free path, in electron scattering, 91 Elastic scattering, field electron emission and, 207 Electron, field emitted, see Field emitted electron Electron desorption probe tube, 221 Electron-electron interaction, in field electron emission, 21 1 Electron energy analyzer, 219 Electron energy distribution, variation of in carbon, 98-99 Electron energy loss distribution, in carbon, 101 Electron impact desorption, 220, 275-276 Electron microscope and microscopy see also Electron scattering amplitude contrast convolution function in, 132-133 amplitude contrast deconvolution function in, 134-135 amplitude contrast transfer function in, 130-1 3 1 chromatic aberration in, 147-153, 169-1 7 1 chromatically incoherent illumination in, 143-155 coherent illumination in, 158-165 combined aberration function in, 171-1 72 convolution integral in, 167 deconvolution of two-dimensional data in, 183-186 electron energy loss distribution and, 101 electron scattering and, 66-1 12 field, see Field electron microscopy image formation in, 63-186 image formation by elastic component in, 112-1 58 image formation by inelastic component in, 158-165 incoherent theory of image formation in, 165-183 “inelastic” image in, 135
lens aberrations in, 167-179 linear contrast theory in, 133-134 objective aperture scattering in, 100-106 operation of, 64-65 partial chromatic coherence in, 153-155 phase contrast convolution function in, 126, 150-151 phase contrast deconvolution function in, 128-129, 152 phase contrast transfer function in, 148 phase contrast vs. frequency in, 1241 25 scanning transmission in, 119-120 scattering contrast images in, 166-167 spatial and chromatic incoherence in, 155-1 58 spatially incoherent illumination in, 137-143 specific distribution in, 142-143 spherical aberration function in, 167-169 transfer theory in, 112-120 weak phase and weak amplitude objects in, 120-137 Electron-optical images, defects in, 63-186 see also Electron microscope Electron-phonon interaction, in field electron emission, 210 Electric potential energy diagram, 217 Electron scattering see also Microwave scattering; Scattering absorption in, 87-88 by amorphous specimen, 74, 79 angular and energy distributions for, 66-1 12 by atomic species, 75 Born approximation in, 73, 312-317 combined inelastic-elastic, 85-90 elastic, 108, 207 in electron microscope, 66-1 12 free-atom factors in, 69 incoherent approximation in, 85-86 inelastic, 109-1 12, 207 inelastic-to-elastic, 103, 106 interband transmission theory and, 82 localization and coherence of, 106-1 12 mean free path value for, 91-92 multiple elastic, 74-79, 86
SUBJECT INDEX
multiple elastic and inelastic, 86 multiple inelastic, 83-84, 86 with objective aperture, 100-106 plasma excitation and, 109-1 I 1 by single atom, 67-72, 80-81, 117-118 by single crystal, 72-74, 76-79, 88-90 single inelastic, 79-83 in solids, 81-83 transfer theory in, 112-120 Electron scattering contrast images, calculation of, 166-167 Electron wave, spatially incoherent, 137-143 Electrostatic deflection electron energy analyzer, 219 Elliptical galaxies, 49 Emitter surface rearrangement, in field electron microscopy, 277-283 Energy density, in radiative transport equation, 334 Energy exchange effects, in field electron emission, 242-252 Extinction coefficient, 322 Extragalactic radiation, 48-59 Extragalactic radio sources, properties of, 5458
F Faraday rotation, 31 FEEM, see Field electron emission microscopy Fermi-Dirac distribution, free electron metal and, 199-200 Fermi level electron potential energy diagram of, 217 energy exchange effects in, 242-243 total energy distribution and, 239 Field cathode, at electron source, 289-290 Field effects, in field electron microscopy techniques, 273-275 Field electron cross section of, 216 trajectory calculations for, 291 Field electron distribution, electronphonon transition in, 210 Field electron emission see also field electron microscopy adsorbate effects in, 206212
395
average energy in, 200 cathode stability and life in, 283-289 collector values for, 238-242 d-band emission in, 201-202 electric field variation in, 234 electron impact desorption and, 275-277 Fowler Nordheim plot in, 204, 214, 222, 229 magnetic field effects of, 256258 many-body effects in, 202-203 noise studies and, 252-256 Nottingham Effect and, 245-251 nucleation in, 263-265 potential barrier corrections in, 203-205 retarding tube for, 224 source optics in, 289-295 surface adsorption and, 258-276 surface migration constants in, 281 surface self-diffusion in, 278-283 surface states in, 205-206 technological advances in, 283-296 theories of, 194-212, 302-304 total energy distribution (TED) technique in, 208-209, 265-273 tunnel enhancement of, 266-267 tunnel resonance in, 207-208 Field electron emission microscopy (FEFM), 212-223 see also Field electron microscopy Field electron microscopy see also Field electron emission basic design in, 213-214 clean surface characteristics in, 223-258 coadsorption in, 261-262 cold field emission and, 298-299 electron impact desorption in, 220 emission theories in, 302-304 emitter surface rearrangements in, 277-283 energy exchange effects in, 242-252 field effects in, 273-275 field emission retarding potential measurements in, 222-223 Fowler Nordheim plots in, 204, 214, 222, 229 instrument applications in, 295-296 of metals, 193-304 nucleation phenomena in, 262-265 single plane techniques in, 214-21 5 sputtering measurements in, 220-223
396
SUBJECT INDEX
surface adsorption in, 258-276 techniques of, 212-223 T-F emission and, 299-300, 303 theory in, 194-212, 302-304 thermionic emission and, 296-298 total energy distribution measurements in, 215-220, 223-232 transition region in, 300-301 Field emission current, plot of, 230-231 Field emission process band structure effects in, 195-199 free electron metal in, 199-201 at metal-vacuum interface, 195-196 sputtering in, 220-223, 277-278 Field emission retarding potential, 238 Field emitted electrons, 194-195 trajectory calculations for, 291 Flux density, of radio source Flux unit, 3 FN plots, see Fowler Nordheim plot Fowler Nordheim (FN) plot, in field electron emission, 204, 214, 222, 229 Fowler Nordheim preexponential ratios, 266 “Free electron” metal energy distribution functions for, 296-297 total energy distribution in, 199-201 Fresnel length, in radio astronomy, 15-16
H Hartree-Ryatt potential, 70 Haystack Observatory, 17 HELIOS project, 22 Holographic deconvolution, 186 Hydrodynamic turbulence, in plasma, 359-360 Hydrogen clouds, 36 Hydrogen distribution, declination and, 35 Hydrogen spiral structure, 36
I Image formation (in electron microscope) aberration in, 167-174 incoherent theory and, 165-183 transfer theory in, 119 Incoherent illumination, in electron microscope, 137-155 Incoherent radiation, intensity of, 368-369 Inelastic image, 135 chromatic aberration in, 180 Inelastic scattering, 207 Integral field emission current, 230-231 Interband transition theory, 82 Interferometry, very long baseline, 16-19 Interplanetary scintillations, 16 Interstellar scintillations, 16 Iridium, work function values for, 240 Isotropic background radiation, 58
C
J Galactic noise, 27-28 Galaxies radio, 52 Seyfert, 55 spiral and elliptical, 49 Galaxy (Milky Way) Faraday rotation and, 31 hydrogen clouds in, 36-37 molecular lines in, 40-44 pulsars in, 46-47 Green Bank, W. Va., radio telescope and interferometer in, 5, 7-8, 33 Green’s function multiple scattering and, 327 radiative transport equation and, 327, 333 Gum nebula, 31
Jodrell Bank radio telescope, 5 Jupiter emissions from, 26 magnetostructure of, 27 map of, 27 radio waves from, 25-26
K Kitt Peak, Ariz., radio telescope at, 7-9
L Linear contrast theory, in electron microscope, 133-134 Linfield Research Institute, 283
397
SUBJECT INDEX
M Magellanic Clouds, emission from, 49, 51 Magnetic field effects, in field electron emission, 256258 Max Planck Institute, Bonn, 6 Mercury, radio emission from, 25 Metals, field electron microscopy of, 193-304 Metal-vacuum interface, field emission process at, 194195 M51 galaxy, 50 Microwaves, multiple scattering and transport in, 311-376 Microwave scattering applications and model calculations in, 346-358 backscatter in, 355, 367 con trolled thermonuclear research and, 318-319, 355-356 diffusion approximations in, 355 exact solutions for idealized models in, 350-355 experimental results vs. model calculations in, 359-372 first- and second-order models in, 348-350 iterative solutions in, 347-350 model calculations vs. experimental results in, 359-372 Monte Carlo solutions in, 358 plasma density fluctuations and, 359-360 radiative transport equation in, 320-346 in reentry physics, 319-320 Molecules, radio interstellar lines from, 40-44 Molybdenum, work function values for, 232, 231 Monte Carlo solutions, in microwave scattering, 358 Multiple electron scattering, analyses of, 83-84, 316-318, 326-332 N
National Radio Astronomy Observatory, 1, 14 Neptune, radio emission from, 25 Neumann series, 348
Neutron star, pulsar at, 4 6 4 7 NGC5195 galaxy, 50 Nickel, work function values for, 240 Niobium, work function values for, 240 Noise studies, field electron emission and, 252-256 Nottingham Effect, 245-251 Nottingham inversion or cooling, 249 Nucleation phenomena, in field electron emission technique, 262-265
0 Objective aperture, electron scattering in, 101-106 One Mile Telescope, Cambridge, England, 12, 14, 34, 56 Ootacamund telescope, 15
P Parkes, Australia, radio telescope at, 5 Parsec, defined, 32 n. Phase-amplitude relationships in wave optical theory, 136 Phase contrast convolution function, in electron microscopy, 126-127, 150-151 Phase contrast deconvolution function, in electron microscope, 128-129, 152 Phase contrast transfer function, vs. frequency, 124, 148 Photography, radio analog of, 3 Physics, radio astronomy and, 1-2 Planets, radio observation of, 25-27 Plasma multiple scattering in, 316-318 single scattering in, 312-316 turbulent, 311-376 Plasma density fluctuations in, 319-320, 359-360 microwave signals and, 359 Plasmon excitation, in electron scattering, 109-1 11 Potential barrier corrections, in field electron emission, 203-205 Pulsars, 5,46-49
3Y8
SUBJECT lNDEX
u Quasars (quasi-stellar sources), 17-1 8, 54
R Radiation, extragalactic, 48-59 Radiative transport equation derivation of, 320-346 direct backscatter and, 345-346 energy density in, 334 extensions and special cases in, 343-346 frequency effects in, 344-345 heuristics of, 320-322 multiple scatter equations and, 324-342 polarization and, 323-324 statistical averages in, 329-332 Stokes parameter representation in, 341-342 Radio astronomy see also Radio telescope amplifiers in, 6-7 antennas in, 7 antenna temperature in, 3 aperture synthesis method in, 11-13 background radiation and, 58-59 continuum systems in, 18-19 data processing in, 18-22 defined, 1 extragalactic radiation and, 48-59 galaxy and, 27-33 high-resolution techniques in, 10-18 interplanetary medium in, 16 observations in, 2-4, 22-60 planets and, 25-27 radio galaxies in, 52-53 radio observations in, 22-60 rapidly varying phenomena in, 20 receivers in, 6 solar, 23-25 space, 21-22 spectral line systems in, 19-20 techniques in, 2-22 technology and observations in, 1-60 Radio Astronomy Explorer Satellite, 21 Radio galaxies, 52-55 Radio sources double, 52 flux density and, 3
Radio sun, as data processing problem, 21 Radio telescope see also Radio astronomy aperture synthesis method and, 11-13 earth rotation synthesis in, 13-14 Fresnel length in, 15-16 Green Bank, W. Va., instrument, 7-8 high resolution techniques in, 9-18 improvements in, 4 interferometer and, 13 single, 4-5 spectroscopy and, 4-9 types of, 5-6, 14-15 zenith attenuation in, 11 RCA Research Laboratories, 320, 365 Recombination lines, 38-39 Reentry physics, experiments in, 319-320 Rhenium, work function values of, 233
S
Scanning electron microscope (SEM), 295-296 Scanning transmission electron microscope, 119 see also Electron microscope Scattering see also Electron scattering of microwaves, see Microwave scattering multiple, 83-84, 316-318, 326-332 by single crystal, 76, 88-90, 117-118 by single electron, 325-326 see also Single scatter analysis Scattering extinction coefficient, 321 Scattering law, 321 Scattering phase matrix, in Stokes parameter representation, 342 SEM (scanning electron microscope), 295-296 Seyfert galaxies, 55 Single crystal, electron scattering by, 76 88-90, 117-1 18 Single electron, scattering by, 325-326 Single inelastic electron scattering, 79-83 Single plane work function, 259 Single scatter albedo, 348 Single scatter analysis, 312-316 Born approximation in, 312-317 distorted wave in, 316
399
SUBJECT INDEX
Single telescopes, 4-6, 10 Sky, brightness temperature of, 3 Solar corona, radio waves from, 23 Solar radio astronomy, 23-25 Solid, electron scattering in, 81-83 Somerfeld-Lorentz theory, 244 Space radio astronomy, 21-22 Spatial incoherence, electron in, 155-1 58 Spectral line systems, in radio astronomy, 19-20 Spectroscopy, radio telescopy and, 7 Spherical aberration function, electron microscope and, 167-169 Spiral galaxies, 49 Sputtering, in field electron emission, 277-218 Stanford Research Institute, 365 Stark-broadening effects, 39 Stokes parameter representation, 341-342 Sun proximity of, 33 radio waves from, 23-24 Supernova remnants, 47 Surface migration constants, 281 Surface self-diffusion, in field electron emission, 278-283
T Tata Institute, Bombay, 15 TED, see Total energy distribution Telescope, radio, see Radio telescope T-F emission, in field electron microscopy, 299-300, 303 Thermionic emission, in field electron microscopy, 296-298 Thomas-Fermi potential, 70 Total energy distribution clean surface characteristics and, 223-232 for copper and tungsten, 226-228 curves for, 271-273 Fermi level and, 239 field emitted electrons and, 194-195, 208-209, 265-273 for free electron metal, 199 measurements of, 21 5-220 Nottingham Effect and, 251 spectra of, 269-270
Transfer theory, in electron scattering, 112-120 Transmission electron microscope, 119-122 sec a/so Electron microscope Tungsten field emission total energy distribution for, 226-228 work function values for, 232, 236-237, 240 Tunnel resonance, in field electron emission, 207-208, 266-267 Turbulent flame jet, backscatter and radiation in, 367-371 Turbulent plasma magnetic field due to, 359 microwave scattering for, 372 multiple scattering and transport in, 311-376
V Van Allen belts, 27 Van Oostrom analyzer, 222 Vela X supernova, 47 Venus, aperture synthesis map of, 26 Very Long Baseline Interferometry (VLBI), 16-1 9 von KBrnxh spectral function, 351-352
W Wave optical theory, phase amplitude in. 136 Weak phase object, in electron microscope, 120-1 30 Westerbork Synthesis Telescope, 13, 15, 49 Work function value for rhenium, 233 single plane, 259 surface adsorption and, 258-261 temperature dependence in, 236-237 for tungsten and molybdenum, 232 for various techniques, 240
z Zeeman splitting, 32, 38 ZETA device (U.K.), 319, 359, 361-362
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