ADVANCES IN ELECTRONICS AND ELECTRON PHYSICS VOLUME 6 5
EDITOR-IN-CHIEF
PETER W. HAWKES Laboratoire d'Optique Electronique du Centre National de la Recherche Scientijique Toulouse, France
ASSOCIATE EDITOR -IMAGE PICK-UP A N D DISPLAY
BENJAMIN KAZAN Xerox Corporation Palo Alto Research Center Palo Alto, California
Advances in
Electronics and Electron Physics EDITEDBY PETER W. HAWKES Laboratoire d’Optique Electronique du Centre National de la Recherche Scientijque Toulouse, France
VOLUME 65 1985
ACADEMIC PRESS, INC. (Harcourt Brace Jovanovich, Publishers)
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NUMBER:49-7504
CONTENTS CONTRIBUTORS TO VOLUME 65. . . PREFACE. . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . ...............
ix xi
Dyadic Green’s Functions and Their Use in the Analysis of Microstrip Antennas SBRGIOBARROSO DE ASSISFONSECA and ATTILIOJosB GIAROLA
1.Sumrnax-y . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 111. Properties and Applications of Microstrip Antennas . . . . . . . . IV. Definition and Properties of Dyadic Functions and of Free-Space Dyadic Green’s Functions. . . . . . . . . . . . . . . . . . . . . V. Expansion of the Dyadic Green’s Functions. Expressions of the Dyadic Green’s Functions for Media with Plane Parallel and Cylindrical Layers. Calculation of the Asymptotic Expressions Using the Saddle Point Method . . . . . . . . . . . . . . . . . . . . . VI. Analysis of the Effect of the Dielectric Substrate in the Radiation Pattern of Microstrip Disk Antennas. . . . . . . . . . . . . . . . VII. A Study of the Effects of a Dielectric Cover on the Radiation Characteristics of a Microstrip Ring Antenna. . . . . . . . . . . . VIII. Analysis of the Influence of the Dielectric Substrate on the Radiation Patterns of Microstrip Wraparound Antennas . . . . . . . . . . . IX. Study of the Influence of Excitation of Surface Waves on the Radiation Efficiency of the Space Waves and the Directivity of a Microstrip Disk Antenna . . . . . . . . . . . . . . . . . . . . . X. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 3 4 7
11 32 42 55 66 88 89
Ink-Jet Printing J. HEINZLand C. H. HERTZ I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Drop-on-Demand Methods . . . . . . . . . . . . . . . . . . . . 111. Methods Employing Continuous Jets . . . . . . . . . . . . . IV. Methods Employing Mechanical Valves . . . . . . . . . . . . V. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . V
91
..
. .
I11
132 16 1 163 166
vi
CONTENTS
Theory of Image Formation by Inelastically Scattered Electrons in the Electron Microscope H . KOHLand H. ROSE I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . I1. The Mixed Dynamic Form Factor . . . . . . . . . . . . 111. Theory of Image Formation . . . . . . . . . . . . . . . . . IV . Numerical Results . . . . . . . . . . . . . . . . . . . . . V . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . .
... 173 . . . . . 174 ... 185 ... 195 ... 213 ... 214 ... 224
Dimensional Terms for Energy Transport by Radiation and for Electromagnetic Quantities -Comments on the SI System BERTHOLD W . SCHUMACHER I . Why Another Discussion of Dimensional Terms? . . . . . . . . . . I1. Dimensional Terms and Physical Concepts. . . . . . . . . . . . . I11. Some Peculiar Aspects of the SI System . . . . . . . . . . . . . . IV . Electric and Magnetic Quantities . . . . . . . . . . . . . . . . . Appendix I . Magnetic Field Quantities and Their DimensionsEnergy Densities and Forces in the Fields . . . . . . . . . . . . Appendix I1. Numbers for the Characterization of Particle Beams . . Appendix 111. Numerical Values for Energy Densities . . . . . . . .
Image Calculations in High-Resolution Electron Microscopy: Problems. Progress. and Prospects D . VAN DYCK I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . I1. Principles of Image Formation . . . . . . . . . . . . . . . . . . 111. Present Status of Many-Beam Electron-Diffraction Calculations . . . IV . Systematic Approach to the Electron-Diffraction Problem . . . . . . V. Study of Existing Many-Beam Diffraction Formulations . . . . . . VI. A New Formulation: The Real-Space Method . . . . . . . . . . . VII. Recent Developments. Unsolved Problems. and Prospects for the Future . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .
229 230 235 242 247 278 292
296 297 299 308 316 324 337 353
vii
CONTENTS
Theory of Surface Electronic Structure E . WIMMER.H . KRAKAUER. and A. J . FREEMAN I. I1. 111. IV .
Introduction . . . . . . . . . . . . . . . . . . . Theoretical Framework . . . . . . . . . . . . . . Approach and Methodology . . . . . . . . . . . . Examples of Applications . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . .
........ ........ ........ ........ ........
358 360 363 367 429
INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
435
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CONTRIBUTORS TO VOLUME 65 Numbers in parentheses indicate the pages on which the authors’ contributions begin.
S ~ R G IBARROSO O DE ASSISFONSECA, Electrical Engineering Department, FundaC2o Universidade de Brasilia (UnB), 70.9 10-Brasilia, DF-Brazil(1) Department of Physics and Astronomy, Northwestern UniA. J. FREEMAN, versity, Evanston, Illinois 60201 (357) ATTILIOJosB GIAROLA,Coordenador Cursos Pbs-Graduacao, (UNICAMP)-Reitoria, 13100-Campinas, SP-Brazil(1)
J. HEINZL,Technical University of Munich, Munich, Federal Republic of Germany (9 1) C. H. HERTZ,Department of Electrical Measurements, Lund Institute of Technology, Lund, Sweden (91) H. KOHL,Institut fur Angewandte Physik, Technische Hochschule Darmstadt, Darmstadt, Federal Republic of Germany (1 73) H. KRAKAUER, Department of Physics, College of William and Mary, Williamsburg, Virginia 23185 (357)
H. ROSE,Institut fur Angewandte Physik, Technische Hochschule Darmstadt, Darmstadt, Federal Republic of Germany (1 73) BERTHOLDW. SCHUMACHER, Research Staff, Ford Motor Company, Detroit, Michigan 48239 (229) D. VAN DYCK,University of Antwerp, Rijksuniversitair Centrum, Antwerp, Belgium (295) E. WIMMER,* Department of Physics and Astronomy, Northwestern University, Evanston, Illinois 6020 1, and Institut fur Physikalische Chemie, Universitat Wien, Wahringerstrasse 42, A- 1090, Vienna, Austria (357)
* Present address: CRAY Research, Inc., 1440 Northland Drive, Mendota Heights, Minnesota 55 120. ix
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PREFACE
This volume of these Advances marks the introduction of a new theme, namely, image pick-up and display. Hitherto, a separate series of advances, edited by Dr. Benjamin Kazan, had been devoted to this subject. From now on, articles dealing with such topics will appear in Advances in Electronics and Electron Physics and Dr. Kazan will continue to be responsible for such chapters. We welcome the first of these, by J. Heinzl and C. H. Hertz, on ink-jet printing. The proliferation of electronic systems for processing and storage of information during the past few years has placed growing emphasis on the need for printing technologies that are capable of making this information available without the limitations associated with conventional impact-type printers such as bulk, noise, and speed. One ofthe most promising technologies now coming into commercial use is that of ink-jet printing, in which the deposition of fine droplets of ink on paper is electronically controlled. This has many advantages, which include silent operation, high speed, color capability, and the immediate creation of an image without further processing. The purpose of this chapter on ink-jet printing is, thus, to put the different systems currently being investigated into perspective and to provide a broad outline of the important principles involved for those readers not entirely familiar with the subject. The other five chapters range from the mathematical analysis of microstrip antennas to electron image formation, with chapters on units and radiation theory and on the theory of surface electronic structure. The first of these complements the contributions by F. E. Gardiol in earlier volumes of this series. The article by Berthold W. Schumacher is a personal, perhaps even polemical, view of the suitability of SI units in such fields as radiative transfer. The chapters on electron image formation are concerned with two aspects of this topic that have become of extreme importance. D. Van Dyck reconsiders the various methods employed for simulating high-resolution electron images, pointing out their strong points and their weaknesses, and shows how the latter can be circumvented. The slowly increasing availability of vector, parallel computers will certainly revolutionize this type of calculation, and I am sure that this survey will be appreciated by everyone who tries to employ these new machines to solve the problem of simulating images. In the other chapter on electron imagery, H. Kohl and H. Rose explore in depth the theory of image formation by inelasticallyscattered electrons. This is the first satisfactory treatment of this difficult subject, of which there is no xi
xii
PREFACE
proper study in even the most recent textbooks. I am extremelypleased to see such a full account appear in these Advances. The chapter by E. Wimmer, H. Krakauer, and A. J. Freeman on the theory of surface electronic structure is a major attempt to impose a pattern on a host of very complex material. I have no doubt that those working on this subject,not excludingrelative newcomers, will be grateful for this rich survey of, and guidance through, a difficult field.
ADVANCES IN ELECTRONICS AND ELECTRON PHYSICS, VOL. 65
Dyadic Green’s Functions and Their Use in the Analysis of Microstrip Antennas SERGIO BARROSO DE ASSIS FONSECA Electrical Engineering Department Fundapdo Universidade de Brasilia (UnB) Brasilia, B r a d
ATTILIO JOSE GIAROLA Electrical Engineering Department Unitlersidade Estadual de Campinas ( U N I C A M P ) Campinas, Brazil
I. Summary. . . . . . . . . . . . . . . . . . . . . . . . . . 11. Introduction
. . . . . . . . . . . . . . . . . . . . . . . . .
111. Properties and Applications of Microstrip Antennas . . . . . . . . . . .
2 3 4
A. Introduction. . . . . . . . . . . . . . . . . . . . . . . . 4 B. Description of Microstrip Antennas. . . . . . . . . . . . . . . . 4 C. Applications of Microstrip Antennas . . . . . . . . . . . . . . . 5 D. Cavity Model with Conducting Magnetic Side Walls . . . . . . . . . . 6 IV. Definition and Properties of Dyadic Functions and of Free-Space Dyadic Green’s 7 Functions. . . . . . . . . . . . . . . . . . . . . . . . . . 7 A. Introduction. . . . . . . . . . . . . . . . . . . . . . . . 7 B. Definition and Properties of Dyadic Functions . . . . . . . . . . . . C. Free-Space Dyadic Green’s Functions Associated with the Magnetic Field from 9 an Elementary Magnetic Current Source . . . . . . . . . . . . . . V. Expansion of the Dyadic Green’s Functions. Expressions of the Dyadic Green’s Functions for Media with Plane Parallel and Cylindrical Layers. Calculation of the Asymptotic Expressions Using the Saddle Point Method. . . . . . . . . . 11 11 A. Introduction. . . . . . . . . . . . . . . . . . . . . . . . B. Expansion of the Free-Space Dyadic Green’s Function for Media with Paral12 lel Plane Layers. . . . . . . . . . . . . . . . . . . . . . . C. Free-Space Dyadic Green’s Function Expansion for Cylindrical Concentric 13 Layered Media . . . . . . . . . . . . . . . . . . . . . . . D. Calculation of the Dyadic Green’s Function for Media with Three Plane and 14 Parallel Layers . . . . . . . . . . . . . . . . . . . . . . . E. Dyadic Green’s Function for Media with Four Plane and Parallel Layers. . . 18 F. Calculation of the Dyadic Green’s Function for Media with Three Cylindrical Concentric Layers. . . . . . . . . . . . . . . . . . . . . . 23 G. Calculation of the Asymptotic Expression of the Dyadic Green’s Function for Media with Three Plane and Parallel Layers. . . . . . . . . . . . . 28 1 Copyright 0 1985 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-014665-7
2
S. BARROSO DE ASSIS FONSECA AND A . J . GIAROLA
H . Calculation of the Asymptotic Expression of the Dyadic Green’s Function for Media with Four Plane and Parallel Layers . . . . . . . . . . . . . I . Calculation of the Asymptotic Expression of the Dyadic Green’s Function for Media with Three Cylindrical Concentric Layers . . . . . . . . . . . VI. Analysis of the Effect of the Dielectric Substrate in the Radiation Pattern of Microstrip Disk Antennas . . . . . . . . . . . . . . . . . . . . . . A . Introduction . . . . . . . . . . . . . . . . . . . . . . . . B. Calculation of the Equivalent Magnetic Current Density . . . . . . . . C. Calculation of the Radiated Fields . . . . . . . . . . . . . . . . D . Conclusion of the Section . . . . . . . . . . . . . . . . . . . VII. A Study of the Effects of a Dielectric Cover on the Radiation Characteristics of a Microstrip Ring Antenna . . . . . . . . . . . . . . . . . . . . . A . Introduction . . . . . . . . . . . . . . . . . . . . . . . . B. Calculation of the Equivalent Magnetic Current Density . . . . . . . . C. Calculation of the Radiated Fields . . . . . . . . . . . . . . . . D . Conclusion of the Section . . . . . . . . . . . . . . . . . . . VIII. Analysis of the Influence of the Dielectric Substrate on the Radiation Patterns of Microstrip Wraparound Antennas . . . . . . . . . . . . . . . . . A . Introduction . . . . . . . . . . . . . . . . . . . . . . . . B. Calculation of the Equivalent Magnetic Current Density . . . . . . . . C. Calculation of the Far Fields Radiated from the Element of Magnetic Current in a Form of a Ring . . . . . . . . . . . . . . . . . . . . . . D . Calculation of the Far Fields Radiated from the Microstrip Wraparound Antenna . . . . . . . . . . . . . . . . . . . . . . . . . E. Conclusion of the Section . . . . . . . . . . . . . . . . . . . IX . Study of the Influence of Excitation of Surface Waves on the Radiation Efficiency of the Space Waves and the Directivity of a Microstrip Disk Antenna . . . . . . A . Introduction . . . . . . . . . . . . . . . . . . . . . . . . B . Integral Representation of the Fields . . . . . . . . . . . . . . . C . Study of the Integration of H , . . . . . . . . . . . . . . . . . . D . Study of the Integral Representation of H , and H , . . . . . . . . . . . E. Study of the Poles in the Plane . Calculation of the Residues . . . . . . . F . Calculation of the Radiation Efficiency of Space Waves and of the Directivity of the Microstrip Antenna . . . . . . . . . . . . . . . . . . . . G . Conclusion of the Section . . . . . . . . . . . . . . . . . . . X . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . .
31 31 32 32 33 34 42 42 42 42 44 55 55
55 56 58 59 66 66 66 61 68 81 82 85
88 88 89
I . SUMMARY
This chapter is concerned with the influence of the dielectric substrate on some electrical properties of microstrip antennas and uses dyadic Green’s functions in media consisting of plane parallel or cylindrical concentric layers. The utilization of the cavity model with conducting magnetic side walls substantially simplifies the calculation of the radiation patterns . This model also allows a more detailed analysis of the excitation of surface waves along
DYADIC GREENS FUNCTIONS
3
the air-substrate interface. Analyses of the most commonly used microstrip antennas have indicated a pronounced influence of the characteristics of the substrate on the electrical properties of these antennas.
11. INTRODUCTION
The recent developments of the elements and arrays of microstrip antennas have been highly influenced by the large advance in the technology of microwave integrated circuits in the last few years. The basic concept of the microstrip antenna was first published by Deschamps (1953). The subject, however, did not receive much attention from the scientific community up to the end of the 1960s and the beginning of the 197Os, when Byron (1970) published a paper describing a new antenna consisting of a conducting patch separated from the ground plane by a dielectric layer. Not much after, a patent was granted to Munson (1973) for a microstrip antenna element, and Howell (1972) published studies on microstrip patch antenna elements with rectangular and circular geometries. From this time on, a large number of papers were published on microstrip antennas, with particular emphasis on the techniques for the implementation of these antennas and on the analyses of their electric behavior. Carver and Mink (1981) and Mailloux et al. (1981) published reviews that are useful for an immediate introduction to the subject. An analysis of the theoretical and experimental contributions published on the subject indicates the necessity of further studies concerning the influences of the dielectric substrate on the electrical properties of microstrip antennas. Thus, as one of the main objectives of this work, dyadic Green’s functions and the cavity model with conducting magnetic side walls are used to investigate the effect of the dielectric substrate on the far field and on the excitation of surface waves in microstrip antennas. In Section 111, the main properties and applications of microstrip antennas are discussed, and the cavity model with conducting magnetic side walls for the calculation of the radiated fields from the antenna is presented. In Section IV, the definition and basic properties of dyadic functions and of the free-space dyadic Green’s functions for use in the following sections are presented. In Section V, an introduction to the expansion of the free-space dyadic Green’s functions in vector wave functions in the rectangular and cylindrical coordinate systems is presented. Expressions of the dyadic Green’s functions for media with three and four layers, plane and parallel to each other, and with three concentric cylindrical layers are then obtained. Also obtained in
4
S. BARROSO DE ASSIS FONSECA AND A. J. GIAROLA
this section are the asymptotic expressions of these dyadic Green’s functions using the saddle point method or the method of steepest descent. In Section VI, the solution of problems of microstrip antennas using the mathematical formalism introduced in the previous sections is presented. The effect of the dielectric substrate on the radiation patterns of the microstrip disk antenna and for two different modes of operation of the antenna cavity is analyzed. In Section VII, the effect of a dielectric cover on the radiation patterns of microstrip ring antennas is studied. In Section VIII, the effect of the dielectric substrate on the radiation patterns of wraparound antennas is considered. In Section IX, and analysis of the problem of excitation of surface waves in the dielectric-air boundary of microstrip disk antennas is developed with the objective of obtaining the effect of the dielectric substrate on the directivity and efficiency of excitation of space waves. In Section X, concluding remarks are presented, based on the results obtained from the analyses developed in the previous sections. 111. PROPERTIES AND APPLICATIONSOF MICROSTRIP ANTENNAS A . Introduction A brief discussion about the definition and basic properties of microstrip antennas is presented in this section. The cavity model with conducting magnetic side walls, which will be used in subsequent sections for the analysis of these antennas, will also be discussed.
B. Description of Microstrip Antennas
The most simple configuration of a microstrip antenna consists of a conducting patch separated from the ground plane by a dielectric substrate, as shown in Fig. 1. The conducting patch may have various geometrical forms, such as rectangular, circular, triangular, and ring. There are also other types of microstrip antennas such as those that may be adapted to curved surfaces; these are discussed in Section VIII. The feed of these antennas may be accomplished by means of microstrip lines connected to the side of the patch or by means of a coaxial cable with its central conductor passing through the ground plane and the dielectric substrate and connected to an adequate position of the patch. The conducting
DYADIC GREEN’S FUNCTIONS
5
,conduct 1 ng patch
FIG.1. View of a microstrip patch antenna.
materials are usually copper or gold and the dielectric substrates are of various types with relative permittivities varying from 2 to 10. These antennas have been widely investigated (Carver and Mink, 1981; Bahl and Bhartia, 1980). The mechanism of radiation is discussed in some detail by Bahl and Bhartia (1980).
C . Applications of Microstrip Antennas
Although the technology of microstrip antennas is still in its infancy, these antennas are finding uses in a variety of applications, some of them include (Carver and Mink, 198 1 ; Munson, 1979) satellite communications, doppler radars and others, radio altimeters, command and control, high-speed vehicles (airplanes, missiles, and boosters), remote sensing, and radiators in biomedical applications. The main advantages and disadvantages of these antennas are discussed by various authors (Carver and Mink, 1981; Howell, 1975; Derneryd, 1976, 1978; Agrawal and Bailey, 1977; Wood, 1980; Bahl et al., 1980; Garvin and Munson, 1977; Munson, 1974; Collier, 1977; Bahl, 1979) and are summarized below: (a) Advantages Low weight and small volume Low profile and possibility of adapting on surfaces of varbus forms Low production cost Possibility of obtaining linear and circular polarizations by adequate choices of the feeding point Can be easily integrated with other devices Feeding lines and matching devices can be fabricated simultaneously with the antenna structure.
6
S. BARROSO D E ASSIS FONSECA A N D A. J. GIAROLA
(b) Disadvantages Narrow frequency band Low gain Radiation in one hemisphere only Low isolation between feeding and radiating elements Possibility of excitation of surface waves Low power capacity The effects of some of these disadvantages may be minimized with a careful antenna design. For example, the excitation of surface waves (as will be discussed in Section IX) may be decreased with the use of substrates with small thicknesses and low values of relative dielectric permittivities. D . Cavity Model with Conducting Magnetic Side Walls
In the analysis of microstrip antennas, a theoretical model is quite often used, based on the following observations (Lo et al., 1979): (a) The electric field in the region between the conducting patch and the ground plane has only the component normal to the plane of the patch, and the magnetic field has only the components parallel to this plane. This is a reasonable assumption for the usual cases since the thickness is commonly much smaller than the wavelength. (b) The electromagnetic fields in the region between the two conductors do not vary along the direction normal to the plane of the conducting patch. This is a reasonable approximation since the substrate thickness d is much smaller than the wavelength in the substrate I for the frequencies of interest. (c) The magnetic field component tangent to the edges of the patch is equal to zero. This results from the fact that there is no electric current component normal to the edges of the patch. Based on these observations, the region between the patch and the ground plane may be treated as a cavity having upper and lower electric conducting planes and conducting magnetic side walls. Thus, the fields in the antenna are considered as those existing inside the cavity. The knowledge of the electric field component tangent to the conducting magnetic side wall allows the definition of an equivalent magnetic current density in this wall based on the Huygen’s or equivalence principle (Harrington, 1961). The problem of obtaining the radiated fields from a microstrip antenna may thus be changed into a much more convenient problem of obtaining the radiated fields from a magnetic current source located in a stratified medium. The formalism often used to analyze microstrip antennas neglects the presence of the dielectric substrate that separates the conducting patch from
7
DYADIC GREEN’S FUNCTIONS
the ground plane. The main objective of this chapter consists of the study of the influence of this substrate on some properties of the antennas by using the cavity model and dyadic Green’s functions for stratified media. The presence of the dielectric layer is responsible for the excitation of surface waves. This subject is examined in Section IX.
Iv. DEFINITION AND PROPERTIES OF DYADIC FUNCTIONS AND O F FREE-SPACE DYADIC G R E E N S FUNCTIONS A . Introduction The applications of dyadic Green’s functions for the solution of a variety of boundary value problems in electromagnetics were significantly expanded after the publication of the book by Tai (1971). The combination of the existing work on the subject enriched by the valuable contributions of the author provided a book that served as the basis for a series of publications with a special emphasis in the areas of antennas and of electromagnetic wave propagation. This work is based on the mathematical development presented in Tai’s book. In this section, an introduction to the basic material is presented, starting with some definitions and basic properties of the dyadic functions and of the free-space dyadic Green’s functions. B. Definition and Properties of Dyadic Functions
A dyadic function D is defined as the association of two vector functions A and B as follows, (1)
D = AB
The vector functions A and B are known as the anterior and posterior elements, respectively, of the dyadic function D. If A
= A,,?
+ A y j + A,z^
and
B = B,A
+ B y j + B,i
then D = AB = (Ax,?
+ A y j + A,P)(B,A + By$ + BZP)
After expanding,
+
D = A,B,i,? A,ByAj AzBxz^A A,B,@
+
+
+ A,B,&I + AyB,jA + A y B y j j+ A y B , j 2
+ A,B,z*z^
(2)
8
S. BARROSO D E ASSIS FONSECA A N D A. J. GIAROLA
and with the definitions, D(,) = A,B,A
D(,) = A,B,A D(') = A,B,A
+ A,B,j + A,B,z^ = B,A + A,B,j + A,B,2 = B,A + A , B , j + A,B,2 = B,A
(34 (3b) (3c)
The anterior scalar product between the dyadic function D and the vector function C is defined by C * D = (C.A)B = B(C.A) = B(A.C)
(5)
From Eq. (5) we observe that the anterior scalar product provides a vector along the direction of the posterior element B of the dyadic function D. The posterior scalar product is defined as D - C = A(B.C) = (B.C)A = (C.B)A
(6)
As may be seen in Eq. (6), the posterior scalar product provides a vector in the direction of the anterior element A of the dyadic function D. The transposed dyadic function b of D is defined as
B = BA
(7)
such that D * C= C . 6 The anterior and posterior vector products between C and D provide dyadic functions and are defined as Cx D
= (C
x A)B
(9)
Dx C
= A(B
x C)
(10)
The application of the operators divergent and curl on the dyadic function D results in a vector and a dyadic function, respectively: V*D
= (V
.D'"')A + (V .D'y')j + (V *D'")h
V x D
= (V
x D(,))A
+ (V x D',')j + (V x D'"')P
The unit dyadic function I defined by I=A3
+ j j + 2.2
(11)
(12)
DYADIC GREENS FUNCTIONS
9
has the following main properties: A.1
V.(lY)
=
1.A
=
A
= VY
where A is a vector function, and Y is a scalar function of position.
C. Free-Space Dyadic Green’s Functions Associated with the Magnetic Field from an Elementary Magnetic Current Source The fields are assumed to have a harmonic time dependence expressed by exp( - iwt), where w is the angular frequency of the source. The sources are assumed to contain magnetic currents M only. Thus, the vector wave equations for the electric and magnetic fields in a homogeneous, linear, and isotropic dielectric having permittivity c0 and permeability po are given by
V x V x H - k:H
=
V x V x E-kfE=
iwcOM -V x M
where k, = w& is the wave number in the dielectric. The solutions for Eqs. (1 6) and (1 7) in terms of an electric vector potential F are E(R)
=(
~ / E ~ x) VF(R)
H(R) = iw[l
(18)
+ (l/k:)VV.]F(R)
(19)
where F(R)
= c0 j!JG,(R/R’)M(R’)
dv‘
(20)
and G,(R/R’) is the Green’s function given by (Tai, 1971) G,(R/R’)
= exp(ik, IR -
R‘1)/4nIR - R’I
(21)
For an infinitesimal magnetic current source with a moment l/iwEO and located in R’, along the x direction, as shown in Fig. 2, M ( R ) may be written as M(R’)
= (l/iwc,)G(R -
R’)A
where 6(R - R’) is the Dirac delta function. From the definition of the Dirac delta function, we obtain
(22)
10
S. BARROSO DE ASSIS FONSECA AND A. J. GIAROLA
FIG.2. Infinitesimal magnetic current source in free space.
With M(R’) given by Eq. (22), F(R) may be obtained from Eq. (20), with the result
F(R) = ( l/io)Go(R/R)2
(24)
Let G‘,“)(R/R’) be the magnetic field produced by the elementary source. Thus, from Eq. (1 9),
Gt)(R/R’) = [l
+ (l/k:)VV.]Go(R/R’)2
(25)
Also, from Eq. (16), with M(R’) given by Eq. (22), Gt)(R/R’) has to satisfy the wave equation expressed by
V x V x G‘,“’(R/R) - k:Gg’(R/R’)
= 6(R
-
R)A
In addition, G‘,“)(R/R’) satisfies the radiation condition for R lim R[V x Gg)(R/R’) - ik,R x G‘,“)(R/R‘)]
=0
(26) + co. Thus,
(27)
R+30
The functions Gdy)(R/R’) and G‘,“)(R/R’)may be obtained for infinitesimal magnetic current sources similar to that given by Eq. (22) but along the y and z directions, respectively. The dyadic Green’s function for the free space is defined as
Go(R/R’) = G t ) ( R / R ) 2 + Gby’(R/R’)j
+ Gt’(R/R’)L
(28)
This function satisfies the following wave equation:
V x V x Go(R/R’) - k:Go(R/R’)
=
16(R
-
R’)
(29)
11
DYADIC GREENS FUNCTIONS
and the radiation condition lim R[V x G,(R/R)
-
ik,l? x G,(R/R’)]
=0
(30)
R+CC
In addition, from Eqs. (25) and (28), we obtain
Go(R/R’) = [I
+ (l/k?)VV]G,(R/R’)
(31)
The knowledge of the free-space dyadic Green’s function G,(R/R’) allows the calculation of the magnetic field H(R) produced by an arbitrary magnetic current density M(R’). An expression for this calculation may be obtained using the vector Green’s theorem in a way similar to that shown by Tai (1971), with the result that
H(R)
=
iwEO
111
-
G,(R/R’) M(R) du‘
(32)
v‘
The integration is over the volume u’ where the sources are located. When M is a magnetic surface current density, the integration is over a surface s’ where M is located. Thus,
H(R)
=
iws,
ss
-
G,(R/R’) M(R’) ds‘
(33)
S‘
v. EXPANSIONO F THE DYADICG R E E N S FUNCTIONS. EXPRESSIONS OF THE DYADIC GREEN’S FUNCTIONS FOR MEDIA WITH PLANE PARALLEL AND CYLINDRICAL LAYERS. CALCULATION OF THE ASYMPTOTIC EXPRESSIONS USING THE SADDLE POINTMETHOD A . Introduction
In this section, an expression of the free-space dyadic Green’s function expanded in terms of its eigenfunctions by using the vector wave functions M and N introduced by Hansen (1935) will be presented. For media consisting of plane parallel layers, the boundary conditions are imposed on planes z = constant. Thus, for these media, it is convenient to use an expansion in n and I that contains the corresponding eigenvalues of the variables 4 and r of the circular cylindrical coordinate system, respectively. For the case of media consisting of cylindrical concentric layers with the axis along the z direction, it is convenient to use an expansion in n and h, where h is the eigenvalue
12
S. BARROSO DE ASSIS FONSECA AND A. J. GIAROLA
corresponding to the z variable of the same coordinate system. With these expansions and by using the method of superposition of scattered fields, dyadic Green's functions are obtained for the solution of a variety of problems in microstrip antennas. With the objective of obtaining the far fields, asymptotic expressions of the dyadic Green's functions are obtained by using the saddle point method. B. Expansion of the Free-Space Dyadic Green's Function for Media with Parallel Plane Layers
The vector wave functions M and N are solutions of the following homogeneous vector wave equation (Tai, 1971): V x V x F - k2F = 0
(34)
where k is known as the propagation constant. The functions M and N obey the following symmetric relations (Tai, 1971): x M
(35)
M = (l/k)V x N
(36)
N
= (l/k)V
In cylindrical coordinates, the functions M and N are given by (Tai, 1971):
where
k:
= A2
+ hZ
(39)
and J,(Ar) is the Bessel function of first kind, order n, and argument Ar. The subscripts e and o of the vector wave functions M and N are used to indicate even and odd functions, respectively. By using the method of Ohm-Rayleigh (Tai, 1971), the free-space dyadic Green's function is obtained as an eigenfunction expansion and is valid for observation points outside the source region. The interested reader is referred
13
DYADIC GREENS FUNCTIONS
to some of the materials concerned with the proper use of this expansion (Tai, 1973, 1980; Rahmat-Samii, 1975; Johnson et al., 1979). As a result, G,(R/R) is given by
+ N:ni(hl)N$ni(-hl), + Ngnl( -hl)N$ni(hl),
z
2 z’
z
Z’
(40)
where the following simplified notation is used:
In addition, 6, is the Kronecker delta (equal to 1 for n = 0 and equal to 0 for n f 01,
h,
=
Jm~
(42)
and
k: = C
O
~
~
~
E
~
(43)
The functions M‘ and N are the M and N functions defined with respect to the r’, @, and z’ coordinates that correspond to the location of the source R‘. The expansion [Eq. (40)] contains an integration in 1and a summation in n while it is closed in h.
C . Free-Space Dyadic Green’s Function Expansion for Cylindrical Concentric Layered Media
For a cylindrical layered medium concentric with the z axis, it is convenient to use an expansion in n and h that is valid for observation points outside the source region. Additional information concerning the proper use of this expansion is available (Tai, 1973,1980; Rahmat-Samii, 1975; Johnson et al., 1979) and is recommended to the interested reader. The expression for G,(R/R‘) is
14
S. BARROSO DE ASSIS FONSECA A N D A. J. GIAROLA
The functions M and N for this case are given by Eqs. (37), (38), and (39) by replacing 1 with a. In Eq. (44),M ‘ l ) and N(’)are defined in terms of the Hankel function of the first kind, such that J,(ur), which appears in M and N, is replaced by HL’)(ur).The prime sign is used to indicate that the vector wave functions are defined with respect to the source coordinates. In addition,
u=
JW
(45)
While Eq. (40) is an expansion closed in h, Eq. (44)is closed in 1. D. Calculation of the Dyadic Green’s Function for Media with Three Plane and Parallel Layers The geometry used to obtain the dyadic Green’s function for the present case is shown in Fig. 3. By using the method of superposition of scattered fields, we have the following: In region 3, the fields are equal to zero since the conductivity is infinite. In region 1, the fields are those transmitted from region 2 through the boundary in z = d. Thus,
G“”(R/R’) = GF2’(R/R’) (46) where G“’)(R/R’) is the dyadic Green’s function in region 1 due to sources located in region 2. The subscript T is used to indicate that Gk12)(R/R’) is the transmission function. In region 2, there are reflections that occur at the boundaries z = 0 and z = d, such that G””(R/R’)
=
G$’”(R/R’)
+ Gf:)(R/R’) + Gf:)(R/R’)
Region 1
______ -a
_-____ Region 2 b,€, 0.0
+a
7777-7T Region 3 O=CU
FIG.3. Geometry used to obtain the dyadic Green’s function.
(47)
15
DYADIC GREENS FUNCTIONS
where G\22'(R/R') has the same expression as that of the free-space dyadic Green's function, having the parameters c and p o of region 2 instead of E~ and p0, and ~~
h2 = Jk:
- A2
(48)
with
k:
(49)
= w2p0&
The dyadic Green's functions with the subscripts R , and R , are used to include the reflections from the boundaries z = 0 and z = d, respectively. In order to have satisfied the boundary conditions in z = 0, z = d, z = z', and the radiation condition, the functions appearing in Eqs. (46) and (47) are constructed as follows:
+ N(h,)[CN'(
-h2)
M(hJM'( - h2) M( -h2)M'(h2)
+ N( - hJ[KN'(
+ DN'(h,)]}, + N(h,)N'( - hz), + N( h2)N(h2),
- h2)
-
+ LN'(h,)]},
z
2d
d 2 z 2 Z' 0 5 z 5 Z'
0 =< z 5 d
In Eqs. (50)--(53),the notations G(R/R') and MgnA were replaced by G and M for convenience. After substitution of these functions into Eqs. (46) and (47), the general expressions of the dyadic Green's functions for regions 1 and 2 are obtained. The coefficients A , B, C,. . ., L, which appear in Eqs. (47)-(53), may be obtained from the boundary conditions at z = 0 and z = d. As seen in Section IV, the dyadic Green's function may be decomposed into the vector function components along the directions $ 9 , and 2: G = G(x)R+ G(Y)j)+ G(z)& (54)
16
S. BARROSO D E ASSIS FONSECA A N D A. J. GIAROLA
Each vector function component corresponds to the magnetic field produced by an elementary source as indicated by Eq. (25). Thus, the anterior element of G must satisfy the same boundary conditions as the magnetic field at the boundaries z = 0 and z = d:
2 x [V x G(22)] = 0, (1/~)2x [V x G(”)]
= (1/e0)
at z
=0
2 x [V x G“2)],
(55)
at z = d
(56)
As an example of application of the boundary conditions, consider Eq.
(54) applied to the M functions:
Having in mind Eqs. ( 3 5 ) and (37), each term of Eq. (58) may be written as
With these results in Eq. (58), we obtain A exp[i(h, - h,)d] = 1
+ E + I exp( - i2h,d)
F + J exp( - i2hzd) = B exp[i(h, - h,)d]
(63)
(64)
17
DYADIC GREENS FUNCTIONS
This procedure may be followed with the three boundary conditions [Eqs. (55), (56), and (57)] and with the functions M and N, resulting in -1+F-J=O
(65)
E-1=0
(66)
1 +H+L=O
(67)
G+K=O
(68)
Aexp[i(h, 1
-
h2)d] = 1
+ G - K exp(-i2h2d)
=
+ E + lexp(-i2h2d)
(69)
~ e x p [ i ( h ,- h 2 ) d l ( h l J F Z & 2 J m ) (70)
H - Lexp(-i2h2d)
=D
exp[i(h, - h , ) d ] ( h , J = / h , , / m ) (71)
1
+ E - 1 exp( -i2h2d) = A(eh,/e,h,)exp[i(h, F + J exp( -i2h2d) = B exp[i(h, h2)d]
- h2)d]
(73)
-
F - J exp( -i2h2d)
1
+ G + K exp(-i2h2d)
= B(ehl/eOh2) exp[i(h, =
(72)
-
h2)d]
(74)
C(~JFF@&~JFT@ exp[i(h, - h2)dl (75)
H
+ L exp(
-
i2h2d) = D
(
E
J
~
/
E
~ exp[i(h, J ~ -) h2)d] (76)
Since the dyadic Green’s function of interest is the one that is valid for region 1 G(12),where we wish to find the radiated fields, the only coefficients of importance are A , B, C, and D. After solving Eqs. (65)-(76), we obtain A=
t2hl[1
+
2h2 exp[ - i(h, h,)d] - h2[1 - exp(-i2h2d)]
+ exp(-i2h2d)]
B=A
where
<=&=A
(77) (78)
18
S. BARROSO DE ASSlS FONSECA AND A. J. GIAROLA
Region 4
FIG.4. Geometry used for calculation of the dyadic Green’s function.
With the knowledge of the coefficients A , B, C, and D, the function G‘12’, which is valid for a three-layered medium, is completely determined. E . Dyadic Green’s Function for Media with Four Plane and Parallel Layers
The geometry used for the solution of media with four plane and parallel layers is shown in Fig. 4. In this case, the source is assumed to be located in region 3 and the media are characterized as follows: Region 1 : po, E
~ CT, =
0,
k,
Region 2: po, E , ,
CT,,
Region 3: po, E , ,
CT,, k , =
Region4:
CT =
k,
=
= o,/poco
oJp0c2[1
wJ&~,[l
+ (ia,/wc,)] + Fo3/oE3)1
co
With the knowledge that the electromagnetic fields in region 4 are equal to zero and that the reflections occur at the boundaries z = 0, z = d , , and z = d , and by using the method of superposition of scattered fields, the following expressions may be written:
G‘I3’(R/R’)= Gy3’(R/R’) G‘23’(R/R’) = Gk2,’(R/R’) G‘,,’(R/R’)
=
Gk3;’(R/R‘)
+ Gk2,’(R/R’) + Gk32)(R/R‘) + GL3,’(R/R’)
(84)
The notation used here is similar to that for the three-layered medium problem.
19
DYADIC GREENS FUNCTIONS
The dyadic Green’s functions that appear in Eqs. (82), (83), and (84) are adequately chosen and with enough coefficients in order to satisfy the boundary and radiation conditions. The result is
+ N(h,)[CN’(
-
h3)
+ DN’(h,)]},
+ N(h,)[G’N(
-h3)
+ H”(h3)]},
z
+ N( -h,)[K’N’( - h3) + L’N(h3)]},
M(h,)M‘( - h3) M( - h3)M‘(h3)
+ N(h3)N( - h 3 ) , + N( hJ)N‘(h3), -
2 d1 + d2
d, 5 z 5 d,
(85)
+ d2
(87)
5 z 5 dl
(89)
5 z 5 dl 0 5 z 5 Z’
(90)
0
Z’
where
=Jm h2 Jm
h,
=
(92)
and the simplified notation for G, M, and N was used. By following a procedure similar to that for the case of a three-layered medium, we conclude that the anterior element of G must satisfy the same
20
S. BARROSO DE ASSIS FONSECA AND A. J. GIAROLA
boundary conditions as the magnetic field at the boundaries z = 0, z = d , , and z = d , + d , . Therefore, 2 x [V x G ( 3 3 ) = ] 0,
at z
=0
(94)
(l/&,)e x [V x G ' j 3 ) ] = (1/~,)2 x [V x G ( 2 3 ) ] , h x G ( 3 3 )= 2 x G ( 2 3 ) ,
at z
6'13)
=2
6'23)
,
at z
(95)
=d,
( 1 / ~ ~x) 2[V x G(13)]= (I/&,)& x [V x G ( 2 3 ) ] ,
2
at z = d ,
=d,
(96) =d,
at z
+ d,
+ d,
(97) (98)
By following a procedure similar to that of the previous subsection, and by using the boundary conditions [Eqs. (94)-(98)] to the terms in M and N of Eqs. (85)-(90), a system of equations for the calculation of the coefficients is obtained. 1-J+J'=O
-I+I'=O l+L+L'=O
K+K'=O 1
+ I - I' exp( - i2h3d,) = { - E exp[ - i(h3 + h,)d,] + E' exp[ - i(h3 - h,)d,]} hZ EZh, J - J' exp( - i2h3d1)= { - F exp[ - i(h3 + h2)dl] &3 ~
+ F' exp[ - i(h3 - h , ) d , ] } hZ &2 i2h3d1)= {C exp[ - i(h3 + h,)dl] &3
~
h3
+ K + K' exp(
-
(105) L
+ L' exp( - i2h3dl) = { H exp[ - i(h3 + h,)d,]
+ H' exp[ - i(h3 - h,)d,]} 1 + I + I' exp( - i 2 h 3 d 1 ) = E exp[ -i(h3 + h,)d,] + E' exp[ - i(h3 - h,)d,]
E
3
J
m
E
2
J
m
(106)
(107)
21
DYADIC GREENS FUNCTIONS
+ J’ exp( - i2h3d1)= F exp[ - i(h, + h2)dl] + F‘ exp[ -i(h3 - h2)dl] 1 + K - K’ exp( - i2h3d1)= { - G exp[ - i(h, + h2)dl] J
+ G exp[ -i(h3 L
- L’ exp( - i 2 , d l ) = { - H exp[
-
- h,)d,I}
(108)
h , J W h,JZZ+h:
i(h3 + h2)dl]
+ H’ exp[ -i(h3
-
h,)d,]}
h,JZZ+h: h3J22+hS (1 10)
A
=
{ - E exp[
- i(h,
+ E’ exp[-i(h, B
=
{ - F exp[
+ F’ exp[ C
=
+ h,)(d, + d,)] -
h2)(dl
+ d , ) ] } E&oh, Zhl
(1 11)
~
+ h,)(d, + d,)] &ohz i(h, - h2)(dl + d , ) ] } ___
- i(h, -
(1 12)
Ezh,
+ h2)(dl + d 2 ) ] + G‘ exp[ - i(h, h2)(dl + d , ) ] } { G exp[-i(h,
-
+ h,)(d, + d,)] + H’ exp[ - i(h, - h,)(d, + d , ) ] }
E
o
J
m
&JFTq
D = { H exp[ - i(h,
E
o
J
m
E
2
J
m
(1 14) A
B
=
E exp[ -i(hl
+ h,)(d, + d,)]
+ E‘ exp[ - i(h, - h,)(d, + d,)]
(1 15)
+ h,)(d, + d,)] + F’ exp[-i(h, h,)(d, + d,)]
(116)
=F
exp[ - i(h,
-
22
S. BARROSO DE ASSIS FONSECA AND A. J. GIAROLA
C = { - G exp[-i(h,
+ G exp[-i(h, D
=
{ - H exp[-i(h,
+ H exp[ -i(hl
+ h,)(d, + d,)] -
h,)(d,
+ d,)]}
h,,/ce
h l , / m
+ h2)(dl + d,)] h,Jm - h,)(dl + d,)]} h l , / m
(1 17)
('18)
From the solution of Eqs. (99)-(118), the coefficients A, B, C, and D are obtained, and the function G ( I 3 )is completely determined. The result is
+ h3)dl] + exp[i(h, - h3)dl] expC-i(h, - h,)d](a3 + 1) { - 1/[exp(-i2h3d,) + 1]}[a3 exp(i2h2d,) + 11 x expCi(h,
B=
x exp[i(h,
+ h3)dl] + exp[i(h,
-
h3)dl]
(119)
( 120)
DYADIC GREENS FUNCTIONS
23
F. Calculation of the Dyadic Green’s Function for Media with Three Cylindrical Concentric Layers The geometry used for the calculation of the dyadic Green’s function for media with three cylindrical concentric layers is shown in Fig. 5. The parameters of media 1,2, and 3 are = 0,
k,
= w&
IS = 0,
k,
= w&
Region 1: pa, ea, CJ Region 2: p o , E , , Region 3 :
CJ =
co
In addition, it is assumed that the sources are located in region 2 and that the perfect conducting cylinder and the dielectric layer extend to + co and - cc along the z direction. The method of superposition of scattered fields is again used, with the result
G“”(R/R’) = G$l2)(R/R’) G”’)(R/R’)
=
GiZz)(R/R’)+ Gf:)(R/R’)
(128)
+ Gk\’)(R/R’)
(129)
In order to have the boundary and the radiation conditions satisfied, the dyadic Green’s functions are conveniently expanded in terms of the vector wave functions M and N with a sufficient number of coefficients. The result is
24
S. BARROSO DE ASSIS FONSECA AND A. J. GIAROLA
FIG.5. Geometry used for calculation of the dyadic Green’s function.
25
DYADIC GREENS FUNCTIONS
+ [zg,Mg,,(h) + J ~ , N ~ n , ( h ) l M ~ , h, ()+ CK+g,,(h) + J+,M&)IN$?( -h) + [Ki,Ng,,(h) + ~ ~ , M ~ , , ( h ) l N ~ , , ( - h ) } , b 5 r 5 a
(133)
where a2 = k: - h2,
y2 = ki
-
hZ
(134)
From reasons similar to those in the previous cases, the anterior element of G must satisfy the same boundary conditions as the magnetic field at the boundaries r = a and r = b. Thus, F x [V x G‘22)]= 0
at r
=b
(135)
(1/c2)3 x [V x G ‘ z z ) ] = (1/eO)3 x [V x G ( ” ) ]
at r = a
F x G ( 2 2= ) f x G(”-), at r = a
(136)
(137)
From these conditions, the following relations between the coefficients of (1 30) to (1 33) are obtained:
+ I + E[Hp(yb)/J,(yb)] = 0
(138)
+ J[Ja(yb)/H!,”’(yb)]= 0
(139)
+ I’[J,(yb)/H“’(yb)] = 0 F‘ + J’[Ja(yb)/Hr”(yb)]= 0 1 + K + G[H!,”’(yb)/J:(yb)] = 0 H + L [ J , ( y b ) / H ~ ” ( y b ) ]= 0 G‘ + K‘[J:(yb)/H~”’(yb)]= 0 H + L‘[J,(yb)/H!,’’(yb)] = 0
(140)
1
F E‘
- EH!,”’(ya) _+ F(ihn/k2a ) H ~ ” ( y a )- IJ;(ya) = - AH!,”’(aa) &
F(y2/kZ)HI,”(yu) J(y2/k2)J,(ya) = B(a2/k1)H!l’(aa) -
(142)
(143) (144)
(145)
J(ihn/k2a)J,(ya)
B(ihn/k,a)H!,”(aa)
+
(141)
(146) (147)
H p ” ( y a ) - E’H!,”’(ya) k F‘(ihn/k,a)H!,’)(ya)- Z’Jr(ya) f J ’ ( i h n / k , ~ ) J , ( y a )= - A‘Hb“’(aa) f B’(ihn/k, a)H!,”(aa) (148)
26
S. BARROSO DE ASSIS FONSECA A N D A. J. GIAROLA
In these expressions, the following notation was used:
where = y, a and d cients were omitted.
= a, b. In
addition, the subscripts 8. and g,, of the coeffi-
DYADIC GREENS FUNCTIONS
21
After solving this system of equations, the coefficients An,Bgn,Agn, Bgn, CEn,Dgn,CEn,and Din, which appear in the dyadic Green's function of region 1, may be obtained as follows: (164)
28
S. BARROSO P E ASSIS FONSECA AND A. J. GIAROLA
where E, = & 2 / E O
G. Calculation of the Asymptotic Expression of the Dyadic Green’s Function for Media with Three Plane and Parallel Layers Before the dyadic Green’s functions are used to calculate the radiated far fields from microstrip antennas, it is convenient to obtain their asymptotic expressions. The saddle point method (Tai, 1971; Collin, 1960; Brekhovskikh, 1960; Felsen and Marcuvitz, 1973) may be used to obtain the contribution of the space wave to a first-order approximation. From this method, a complex integral of the type (178) -m
Also, h, is the saddle point obtained from @(h,) = 0, where +’(h,) is the first derivative of &h) with respect to h evaluated at h,. In addition, @’(/I,) is the second derivative of 4 ( h ) with respect to h evaluated at h,. The conditions that must be satisfied for the application of the saddle point method are (a) p has to be much greater than unity; (b) the magnitude of $(h) has to be on the order of unity and should present an extremum at h,, such that 4’(h,) = 0; (c) the variation off ( h ) with h has to be slow in the vicinity of h,.
DYADIC GREENS FUNCTIONS
29
Consider, as example, one of the terms of G“’(R/R’) given by Eq. (50):
The limits of integration of Eq. (181) may be changed using the operational method presented by Tai (1971) with the result
where M‘’) is defined with respect to the Hankel function of first kind and is obtained from M by replacing J,(Ar) by Hk’)(Ar). Following a similar procedure with the other terms of G(”’(R/R’), the result is
+ Ng’,:(hljCCN&nA(-hz) + DNgnA(h2)I)
(183)
Assuming that Ar is large in comparison with unity, the Hankel function of first kind may be approximated by its asymptotic expression (Tai, 1971): / ~i)(” f$,’)(lr) = ( 2 / 7 ~ A r ) ’ -
+
‘1’
exp(ilr)
(1 84)
With these expressions in M(’)and N(’)and neglecting the terms of order equal or higher than ( l ~ ) - results ~ ’ , , are
By substituting these expressions in Eq. (183), the results are
G“”(R/R’)
30
S. BARROSO DE ASSIS FONSECA AND A. J. GIAROLA
This expression may be written as
f ( I )exp[ip4(l)] d l where
+ hlz i "2-6, f ( I )= 1 _h,_ (- i)n+ /',( 47c
p+(l) = I r
-
n=O
-)
2 cos (n4){ -i$[AMgnA(-h,) n l r sin
Before using Eq. (179) to obtain an asymptotic expression for (154), observe that
h,
=J
k m -
or
+ B(O)Mg,,(t)] + ( - cos et + sin 02)[C(0)NgnS( t ) + ~ ( e ) ~ ~ , , ( t ) l } -
( 199)
31
DYADIC GREEN'S FUNCTIONS
where t
GG
= h2ILo= k,&
s = A, = k,
sin 8
As a result, from Eq. (179), we obtain
H . Calculation of the Asymptotic Expression of the Dyadic Green's Function for Media with Four Plane and Parallel Layers Following a procedure similar to that shown in the previous subsection, an asymptotic expression for G(13'(R/R'), given by Eqs. (82) and (85), may be obtained. The final result is identical to that given by Eq. (202). The only difference is that the coefficients A(@, B(B), C(8), and D(8) are now given by Eqs. (1 19)-( 122). I . Calculation of the Asymptotic Expression of the Dyadic Green's Function for Media with Three Cylindrical Concentric Layers
By using the saddle point method and a procedure similar to that shown in Subsection V.G, we obtain f(h,)
i " 2 9 ( - i)"' 8n ,=,
=-
x
1
Miid(
-
+[ +
[
h,)
+ [A:,(@
cos sin n$$
=
?Iho
-
:ts
-
cos iC,,(e) sin n@
+ D,(@
-
cos ic%,(e)sin n@
+ Dp,(O) cos
where
u
[A,(@
1/2f$)1'2{
= (ki
-
k:
C O S ~8)li2
'OS
sin
sin n 4 6 - iBg,(8) cos n@]
1
iBQ,(e)cos sin n@ ME,,( - h,)
n4e]N.$l(
-
h,)
32
S. BARROSO DE ASSIS FONSECA AND A. J. GIAROLA
As a result, the asymptotic expression for G‘”’(R/R) is given by
where AEn(8),Bg,,(8),. . .,DQ,(O) are the coefficients given by Eqs. (164)-(171) with Mlh=ho = U = Ylh=ho
k1 Sin 8
= U = Jk:
- k: COS’
(206)
8
(207)
The asymptotic expressions of the dyadic Green’s functions obtained in this section will be used in the following sections for the calculation of the far fields radiated from some of the most common microstrip antennas.
VI. ANALYSISO F THE EFFECTO F THE DIELECTRIC SUBSTRATE IN THE RADIATIONPATTERN OF MICROSTRIP DISKANTENNAS A . Introduction
In this section the dyadic Green’s functions for media consisting of planeparallel layers and the cavity model with conducting magnetic side walls are used for the analysis of the effect of the dielectric substrate in the radiation patterns of microstrip disk antennas. The radiation patterns of these antennas for various values of the dielectric constant and dielectric thickness were obtained. The influence of the dielectric constant on the radiation pattern was
DYADIC GREENS FUNCTIONS
33
observed to be of particular importance at lower radiation angles and for larger values of dielectric thicknesses. It has been observed that the results of the present analysis converge to those obtained using the standard procedure that neglects the existence of the dielectric substrate as the dielectric thickness approaches zero. Curves of the 3-dB beamwidths as functions of the ratios between the dielectric thickness and the disk radius agree with the available data.
B. Calculation of the Equivalent Magnetic Current Density A microstrip disk antenna with a radius a and constructed on a dielectric substrate with a thickness d and dielectric permittivity E is shown in Fig. 6. By using the cylindrical cavity model with conducting magnetic side walls and assuming that the fields inside the cavity do not vary along the z direction, such that only the TM,, modes are excited, the electric field inside the cavity is given by (Derneryd, 1979)
E
=
E,J,(k,r’) cos p4‘2
(208)
With the condition that the tangential component of the magnetic field be equal to zero at the conducting magnetic wall, we obtain,
where Jb(k,a) is the derivative of J,(k,r’) at r‘
= a.
FIG.6. Microstrip disk antenna.
34
S. BARROSO D E ASSIS FONSECA A N D A. J. GIAROLA TABLE I
SOME ROOTSOF EQ. (209) Mode
Root
TMll TM21 TMo2 TM3i
k 2 a = 1.841 k 2 a = 3.054 k 2 a = 3.832 k’a = 4.201
Given k,, the radius a of the disk may be obtained from Eq. (209) for each mode p . Values of k,a for some of the TM,, modes are given in Table I. From Eq. (208), the value of the electric field at r = a is given by
Ep,(r’ = a ) = E,, cos p 4 ‘ 2
(210)
An equivalent magnetic current is then obtained from the electromagnetic fields at the magnetic wall of the antenna, based on the equivalence principle (Harrington, 1961),
M,,
= E,,xt
M,,
=
at rr = a,
0 5 z’ 5 d
(2 12)
= a,
0 5 z’ 5 d
(213)
or E,, cos p4’&
at r’
C. Calculation of the Radiated Fields For the calculation of the far fields radiated from the microstrip disk antenna excited in the TM,, modes, Eq. (33) may be used. In this expression, G(”)(R/R’) given by Eq. (202) is used instead of G,(R/R’), and M,, given by Eq. (213) is used for M(R). The area S‘ is that of the side wall of the cavity defined at r’ = a and 0 5 zr 5 d. The result is
[ [j
Hp,(R) = ioe,
S‘
G(”)(R/R’) Mp,(R’)dS‘
1
r’=a
Osz’5d (214)
35
DYADIC GREENS FUNCTIONS
or
Hp@)
i m Oexp(ik,R) =
2ntRtan0
c (2 a
-
do)( - i>”
n=O
+ 1 ‘OS
sin
atr‘=a
(n#)E,, fo2’d@ fodr’ dz’
Osz’sd
(215)
After using the relation
and the orthogonality properties of the trigonometric functions, the following result is obtained:
HOp,(R)= ( E , , o E , ~ cos 0/tk2R sin 0) exp(iklR)(-i)P+lJp(sa) x sin p 4 [ - D(B)[exp(itd) - I] - C(B)[exp( - itd) -
111,
p >0 (2 17)
Hbp,(R)= (E,,oE,u cos 0/t2R sin 0) x exp(ik,R)( -i)P+l[~Jp-l(sa)- (p/a)J,(sa)] x cos p4{A(o)[exp( -itd) - I] - B(B)[exp(itd) - I]},
p >0
(218) It is important to observe that the apparent singularity that occurs in Eqs. (217) and (218) at 0 = 0 disappears owing to the presence of the Bessel function J,(sa), since, for small arguments (Abramowitz and Stegun, 1965), J , ( x ) x’o (l/v !)(x/2),
(2 19)
36
S. BARROSO D E ASSIS FONSECA A N D A. J. GIAROLA
With B(8) = A(8) and D(8) = -C(8) as given by Eqs. (78) and (80), respectively, Eqs. (217) and (218) may be written as - 2E,,ocOp cos
t k , R sin 8
Hop,(R) = x
H+,,(R) =
8
exp(ik,R)( - i)PJp(su)C(8)sin(td) sin(&),
~ E , , W Ecos ~ U8 exp(ik R)( - i)" t2R sin 8 P x [sJ,- ,(su) - J,(su)]A(B) sin(td) cos(p4),
p >0
(220)
,
~
U
p >0
(221) The electric field components in the far field are given by (Harrington, 1961)
where the intrinsic impedance of medium 1 is given by
The electric fields obtained from the standard method, which neglects the existence of the dielectric layer and which will be used for comparison with present method, are (Derneryd, 1979): Eo,,
=
-(#'k,[exp(
- j k , R ) / R ] ( ~ ~ u / 2 ) B n ( ksin , u 8) cos p 4
(225)
Egpq= (j)Pk,[exp( -jk,R)/R](Vou/2)Bp(k,usin 8) cos 8 sin p+ (226)
These expressions are obtained for a time dependence of expGot) instead of exp( - iot) used in the present chapter. With the results from Eqs. (220)-(223)and from Eqs. (225) and (226),the radiation patterns for the modes TM,, and TM,, were obtained and are shown in Figs. 7-14. The values for the Bessel functions were obtained by means of an appropriate digital computer subroutine.
37
DYADIC GREENS FUNCTIONS
7 \\\\
tQ2
c
FIG.7. Radiation pattern of the microstrip disk antenna.E,, E plane, TM
fp = 0".
,mode,
E, =
2.55,
FIG.8. Radiation pattern of the microstrip disk antenna. E,, H plane, TM,, mode, E, 2.55, = 90".
=
38
S . BARROSO D E ASSIS FONSECA A N D A. J. GIAROLA
= 0.526
$=0.158
\
FIG.9. Radiation pattern of the microstrip disk antenna. E,, E plane, TM,, mode, E,
= 2.55,
4 = 0".
FIG. 10. Radiation pattern of the microstrip disk antenna. E,, H plane, TM,, mode, E, = 90".
2.55, q5
=
39
DYADIC GREENS FUNCTIONS
# =0.526* # =0.368.
$ =0.263CO n:v
FIG. 1 1 . Radiation pattern of the microstrip disk antenna. E,, E plane, T M , , mode,
E,
=
9.60, q5 = 0".
FIG. 12. Radiation pattern of the microstrip disk antenna. E,, H plane, TM,, mode, E, = 9.60, q5 = 9 0 .
40
S. BARROSO DE ASSIS FONSECA AND A. J. GIAROLA
FIG.13. Radiation pattern of the microstrip disk antenna. E,, E plane, TM,, mode, 9.60, = 0".
FIG.14. Radiation pattern of the microstrip disk antenna. E,, H plane, TM,, mode, 9.60, 4 = 90".
E, =
E, =
41
DYADIC GREENS FUNCTIONS
After examining the radiation patterns, the following observations are made : (a) The effect of the dielectric layer is more pronounced for lower radiation angles, i.e., when 0 -+ 90". (b) The 0 component of the electric field is more affected than the 4 component by the presence of the dielectric layer. In particular, for the dominant mode ( p = q = l), the effect of the dielectric layer on the 4 component of the electric field is negligible. (c) The 3-dB beamwidth of the radiation pattern for the E , component increases with the increase of d/a for the values of E , considered here (2.55 and 9.6). This observation agrees with that made by Araki and Itoh (1981). (d) The results obtained by using the present method approach those obtained by using the standard method when the d/a ratio approaches zero. (e) The curves of the 3-dB beamwidths as functions of the d/a ratio agree with those obtained by Araki and Itoh (1981), as shown in Fig. 15. (f) It is important to note that the radiation pattern in the E plane is not nearly constant for high values of E , and d/a, as predicted by the standard method (Bahl and Bhartia, 1980).
I
0.005
I 0.1
I
-
0.2 d/a
I
0.5
FIG.15. Beamwidth of 3 dB as a function of the d/a ratio, for EdO) and Eo(x). Dotted line represents theory of Araki and Itoh (1981); solid line represents this theory.
42
S. BARROSO D E ASSIS FONSECA A N D A. J. GIAROLA
D. Conclusion of the Section
In this section, a method more rigorous than the conventional one for the calculation of the radiation pattern of microstrip disk antennas was presented. One advantage of the present method is that the asymptotic expression obtained for the dyadic Green’s function may be used in structures with different geometrical forms, such as rectangular, triangular, and ring, as long as the equivalent magnetic currents on the magnetic wall of the cavity defined by the conducting patch, dielectric substrate, and ground plane are known. Not enough experimental data are available for a complete comparison with the results obtained here for the microstrip disk antenna. The curves of the 3dB beamwidth as a function of the d/a ratio approach those shown by Araki and Itoh (1981) obtained by using the Hankel transform domain analysis with the method of moments. The differences observed in the radiation patterns at low radiation angles, when compared with other standard methods, were also observed in the results presented by Chew and Kong (1981), which used a more rigorous analysis based on the vector Hankel transforms and on the method of moments. VII. A STUDY OF THE EFFECTSO F A DIELECTRIC COVER O N THE RADIATIONCHARACTERISTICS O F A MICROSTRIP RING ANTENNA A . Introduction
In their natural environment, the microstrip antennas may occasionally or permanently be covered by a dielectric layer, such as a layer of snow or a dielectric protective layer. The presence of these layers alters some of the properties of the antenna, such as the resonance frequency and the Q factor (Bahl and Bhartia, 1980). The effect of the dielectric cover on these properties of a microstrip ring antenna was analyzed by using a variational technique (Bahl et al., 1980). In this section, the effect of a dielectric cover on the radiated far field from a microstrip ring antenna is analyzed. The cavity model with magnetic conducting side walls and the formalism of the dyadic Green’s functions for stratified media will be used in this analysis. B. Calculation of the Equivalent Magnetic Current Density
A microstrip ring antenna covered with a dielectric layer is shown in Fig. 16; also shown in this figure is the coordinate system used. The inner and outer radii of the ring are given by a and b, respectively. The electric field in
DYADIC GREENS FUNCTIONS
43
FIG.16. Microstrip ring antenna with a dielectric cover.
the region between the conducting ring and the ground plane at the resonance of the TM,,, modes is given by (Bahl et al., 1980).
E = E,[J,(kr’)Nb(ka) - Jk(ka)N,(kr’)] cos(p&)i
(230) where E , is the value of the electric field between the outer edge of the conducting ring (r‘ = b) and the ground plane at 4 = 0, p is an integer number identifying the mode TM,,,, k =2nfi/A0 (231) E, is the effective dielectric constant of the structure, A. is the free-space wavelength, and J , and N , are the Bessel functions of first and second kind, respectively. Here, J, and N , are the derivatives of J , and N , , respectively, with respect to the argument. With the condition that the radial current component at the edges of the conducting ring be equal to zero, the following relation is obtained: Jb(kb)Nb(ka) - Jb(ka)Nb(kb) = 0
(232)
From Eq. (232), k may be obtained for the various modes of the cavity with inner and outer radii a and b, respectively. An approximate value for k, when p 5 5 and ( h - a)/(b
+ a) 5 0.35
(233)
+ b)
(234)
is given by (Bahl et al., 1980):
k
= 2p/(a
The resonance frequency is given by
where c is the velocity of light in free space.
44
S. BARROSO DE ASSIS FONSECA AND A. J. GIAROLA
By using the theory developed by Bahl et al. (1980) and Bahl and Stuchly (1980) for the calculation of E,, the effect of the dielectric cover on the resonance frequency of the ring was observed to be small for the values of relative permittivities and loss tangents used in this work. As a result,f, was assumed to be given by
where E,, = &%/.sois the relative permittivity of medium 3. The electric fields at the inner and outer edges of the conducting ring may be obtained from Eq. (230) as E,
= Eo[J,(ka)Np(ka) - J,(ka)N,(ka)]
E,
=
cos pq5,
Eo[J,(kb)N’,(k~)- Jb(ka)N,,(kb)] cos pq5,
r‘
=a
(237)
r’ = 6
(238)
From the equivalence principle, the fields radiated from the antenna may be obtained using the equivalent magnetic currents on the magnetic side walls of the ring, given by
M(r’ = a ) = -P x EJr’ = a)2 = E , x [J,(ka)N,(ka) - Jb(ka)N,(ka)] cos pq5’,
r‘ = a (239)
M(r’ = b) = P x Ez(r’ = b)L = - E , x [J,(kb)N,(ka) - J6(ka)NP(kb)]cos p @ ,
r‘ = b (240)
C. Calculation of the Radiated Fields
The field radiated by the antenna in medium 1, which is the free space, is obtained by using the dyadic Green’s function G(13) (R/R), given by Eq. (202), and the equivalent magnetic current density given by Eqs. (239) and (240). The coefficients A ( @ , B(B), C(Q, and D(0) of the dyadic function are given by Eqs. (1 19)-(122). The integral in Eq. (33) is thus given by
ss
H(R) = imO G!?z;(R/R’)- M(r’ = a) dS’ Si
+ i m OjJG!?II(R/R’)
*
M(r’ = b) dS’
(241)
S5
where S; is the area of the inner apperture at r‘ = a and 0 5 z’ 5 d l , and S2is the area of the outer apperture at r‘ = b and 0 S z’ d , .
45
DYADIC GREENS FUNCTIONS
With the expressions for G“3’(R/R’) and M, the components of Eq. (241) are given by
c
cos
exp(ik,R)E, H,(R) = WE, _ _ _ ~(2 - do)( -i)”+’ 2rctRtan0 ,=,
x-
dJ,(sa) cos (n@) cos(p$’)[A(0) exp( - itz’) aa sin
- b[J,(kb)Nb(ka)
x
H,(R)
=
WE,
x -
sin ( n 4 )
-
+ B(8) exp(itz‘)]
J;(k~)N,(kb)]
ab
cos ( n @ )cos(p@)[A(8) exp( - itz’) + B(8)exp(itz’)] sin exp(ik,R)E, 2ntR tan 0
1 (2 - 6,)( - i y +
lcos
sin
n=O
I
(242)
(n4)
tn sin J,(su) ( n @ )cos(p+’)[ k C(0) exp( - itz’) & D(0) exp(itz’)] cos k3a
~
b [J,( kb)Nb(ka) - Jb(ka)N,( kb)] JO2’d4‘ J,*’dz‘ tn k3 b
x J,(sbfin (n4’) cos(p@)[ cos
C(8) exp( - itz’) T D(0) exp(itz’)]
I
(243) By using the orthogonality properties of the trigonometric functions and integrating, the result for p > 0 is
H,
=
exp(ik,R)E, im, 2( - i)p+ cos(p6)rc 2ntR tan 8 [exp( -itd,)
i
-
11 - B(8) [exp(itd,) ~
t
- Jb(k~)N,(kb)][sJ,- a[J,(ka)N,(k~)- J’,(k~)N,(ka)][sJ,-
-
I
13
P
‘(sb) - J,(sb)] l(sa) -
46
S. BARROSO DE ASSIS FONSECA AND A. J. GIAROLA
He
exp(ik,R)E, 2( - i)p+ sin(p4). 2ntR tan 8
= ioxO
[exp( - itd,)
i
-
11 - D(8) - [exp(itd,) t
- Jb(ka)N,(kb)]
tP ~
k3 b
-
11
J,(sb)
- ~[J,,(ka)Nb(ka) - Jb(ka)N,,(ka)] tp J,(su) k3 a ~
}
(245)
The components of the far electric field are obtained from H , and He by using Eqs. (222), (223), (244), and (245). For the case of the TM,,, mode, of particular interest, since the radiation pattern has a maximum along the normal to the plane of the ring, the result is
’ - t2R tan 8 exp(ik,R) cos 4
E -
‘lWEoE0
’
E -
-
‘ l W E 0 Eo exp(ik, R) sin 4 tRk3 tan 6
x { C(@[exp( - itd,) x KJl(ka)N;(ka)
-
11 - D(B)[exp(itd,)
-
l]}
J;(ka)N,(ka)lJ,(s4
- CJl(kb)N;(ka) - J;(ka)N,(kb)lJl(sb)l
(247)
where
J ; ( k a ) = kJo(ka) - ( l / a ) J , ( k a )
(248)
N;(ka) = kNO(ka) - (l/a)N,(ka)
(249)
After a numerical evaluation of Eqs. (246) and (247), the radiation patterns for the E, and E , components of the electric field were obtained for various values of the parameters of media 2 and 3. The values of the relative permittivity and loss tangent of medium 3 were those from two typical dielec~ and tan 6, = 0.0012) and the tric substrates, the R. T. Duroid 5870 ( E =~ 2.32 alumina (er3 = 9.8 and tan 6, = 0.004).
DYADIC GREENS FUNCTIONS
47
The results are shown in Figs. 17-24 and the following observations are made:
(a) The radiation patterns are practically insensitive to variations of the loss tangent tan 6, of the dielectric cover for the observed range of 0.001 5 tan 6, 2 0.01. For this reason, the value of tan 6, = 0.001 has been used for the calculation of the results. (b) The effect of the relative permittivities of media 2 and 3 is more pronounced for lower radiation angles as (3 approaches 90".
FIG.17. Radiation pattern of (a) JE,I at 9 = 0", (b) I E,I at 9 = 90". Parameters used: E,, 2.32, tan 6 , = 0.0012, d , = &in., frequency = 2.09 GHz, E , ~= 2.0, tan 6, = 0.001.
=
48
S. BARROSO DE ASSIS FONSECA A N D A. J. GIAROLA
d 2 - 0.1 crn 0.5crn 2.0crn
FIG.18. Radiation pattern of (a) lEsl at 4 = o",(b) IE,I at 4 = 90". Parameters used: E,, = 2.32, tan 6, = 0.0012, d , = &in., frequency = 2.09 GHz, E,, = 10.0,tan 6, = 0.001.
DYADIC GREEN'S FUNCTIONS
FIG.19. Radiation Dattern of (a), ,IE,I" , at dJ, = 0".(b), , IE,I,+,' at dJ = 90". Parameters used: €-3' 2.32, tan 6 , = 0.0012, d , = 4 in., frequency = 2.09 GHz, E,, = 2.0, tan 6 , = 0.001. ~
\
I
49
=
50
S . BARROSO DE ASSIS FONSECA A N D A. J. GIAROLA
A-
d 2 = 0.1 crn 0.5 cm 2.0 cm
FIG.20. Radiation pattern of (a) [Eel at 4 = 0",(b) IE,I at 4 = 90". Parameters used: E,, 2.32, tan 6 , = 0.0012, d , = Bin., frequency = 2.09 GHz, E,, = 10.0,tan 6 , = 0.001.
=
51
DYADIC GREENS FUNCTIONS
\
-e=
52
S. BARROSO DE ASSIS FONSECA A N D A. J. GIAROLA
\,&--FIG.22. Radiation pattern of (a) lEel at C#J = O",(b) IE41 at C#J = 90". Parameters used: E,, = 9.8, tan 6, = 0.004,d , = &in., frequency = 1.02 GHz, E,, = 10.0, tan 6 , = 0.001.
53
DYADIC GREENS FUNCTIONS
d 2 = 0.1 crn
0.5cm
\&/ FIG.23. Radiation pattern of (a) ]Eel at 4 = o",(b) IE+I at 4 = 90". Parameters used: E,, 9.8, tan 6, = 0.004, d , = Q in., frequency = 1.02 GHz, E,, = 2.0, tan 6, = 0.001.
=
54
S. BARROSO DE ASSIS FONSECA AND A. J. GIAROLA
FIG.24. Radiation pattern of (a) IE,( at $J = 0", (b) IE41at $J = 90". Parameters used: E,, 9.8, tan 6, = 0.004, d , = 4 in., frequency = 1.02 GHz, E,, = 10.0, tan 6, = 0.001.
=
DYADIC GREENS FUNCTIONS
55
(c) The 8 component of the electric field is much more sensitive to variations in the thicknesses and relative permittivities of the dielectrics than the $I component. The highest variation of this $I component with d , occurred for high relative permittivity values of both dielectrics (er3 = 9.8, E,, = 10.0). is (d) For the cases when E,, is low and E,, is high and vice versa, practically independent of d , in the range 0.1 cm 5 d , 5 2.0 cm. (e) For low values of d,, an increase in E,, causes a noticeable increase in the 3-dB bandwidth. (f) It is important to note the peculiar behavior of the 3-dB beamwidth with the various parameters considered. In some cases, an increase in d , causes a continuous decrease of the 3-dB beamwidth, while in other cases, just the opposite occurs. For E,, an increase of the 3-dB beamwidth for an increase in d , occurs when E,, is much larger than E , ~ ( E , , = 10.0, E,, = 2.32), while for E,, this occurs when E,, and E,, are nearly the same ( E , ~= 2.0, E,, = 2.32 or E,, = 10.0,er3 = 9.8). While this observation has not been fully investigated, it may be due to surface waves that are excited between the boundary of the dielectrics.
D. Conclusion of the Section In this section, the influence of a dielectric cover on the radiation pattern of a microstrip ring antenna using dyadic Green’s functions and the cavity model with conducting magnetic side walls was studied. Comparison of the results obtained here with other experimental or theoretical results was not possible since they were not readily available. It is important to note that the asymptotic expression of the dyadic Green’s function G(l3)(R/R’)is general and may be used in the study of similar antennas with other geometrical forms, such as rectangular, and disk.
VIII. ANALYSISOF THE INFLUENCE OF THE DIELECTRIC SUBSTRATE ON THE RADIATION PATTERNS OF MICROSTRIP WRAPAROUND ANTENNAS A . Introduction
High-speed vehicles, such as airplanes, rockets, and missiles have a serious aerodynamic problem and require that the antennas have a low profile and adapt well to their surfaces. These two requirements can usually be met well by microstrip antennas and they become very strong candidates for such
56
S. BARROSO DE ASSIS FONSECA AND A. J. GIAROLA
applications (Munson, 1974). In this section, the effects of the dielectric substrate on the radiation pattern and pattern coverage of microstrip wraparound antennas are analyzed by following a procedure similar to that shown in Sections VI and VII. B. Calculation of the Equivalent Magnetic Current Density
One of the microstrip antennas commonly used on the fuselage of missiles and rockets is of the wraparound type, which was described by Munson (1974) and is shown in Figs. 25 and 26. Beside the feeding structure, the antenna has a microstrip line section with a very low characteristic impedance. The width of this line is L and the length is W, as indicated in Fig. 25. Usually, W is nearly equal to one-half of a wavelength in the dielectric. This line section may take the shape of a ring, as shown in Fig. 25 and may be adjusted to the fuselage of a high-speed vehicle, as shown in Fig. 26. The antenna radiates through both of its apertures, separated by W = &/2, where is the wavelength in the dielectric. In order to have an omnidirectional radiation pattern in the plane normal to the axis of the vehicle’s fuselage (i.e., in the plane z = 0 of Fig. 26), the number of feeding points (as shown in Fig. 25) must be larger than the number of & contained in L (Munson, 1974). This feeding structure and the assumptions that the fields between the conducting strip and the cylindrical ground do not vary
FIG.25. Microstrip wraparound antenna.
57
DYADIC GREENS FUNCTIONS
T'
Dielectric
+al
L=
FIG.26. Microstrip wraparound antenna considered here.
with r' and that the electric field has only a component along the r' direction will be used to calculate the radiated fields from the apertures located at z' = - Ad/4 and z' = Ad/4, as shown in Fig. 26. By using the cavity model with conducting magnetic side walls, the fields radiated from the antenna are calculated as if the source consisted of magnetic currents uniformly distributed on both apertures located at z' = - A,/4 and z' = &/4. The magnetic current densities in these apertures are given by
M I = 2 x EoP = Eo$,
at z'
M,
at z ' =
=
-2 x (-,TOP)
= Eo&
=
Ad/4
(250) (251)
where E , is the value of the uniform electric field at the apertures. Equations (250) and (251) take into account the fact that W z Ad/2 and that the electric have equal magnitudes but opposite phases. field in z' = Ad/4 and z' = In order to simplify the present analysis, the method of multiplication of radiation patterns will be used. Thus, instead of obtaining the fields radiated from both apertures, it is necessary to calculate only the fields radiated from an aperture located at z' = 0, with a magnetic current density M = Eo& as shown in the next subsection.
58
S. BARROSO DE ASSIS FONSECA AND A. J. GIAROLA
C. Calculation of the Far Fields Radiated from the Element of Magnetic Current in a Form of a Ring By using Eq. (33), with G ( ” ) ( R / R ) given by Eq. (205) and with M(R) given by M = E , 3 at z’ = 0, H,,(R) is given by
-
dHb’)(ur’) cos cos {[A;,(8) sin n 4 6 - iB,(O) cos sin n@][ ar’ sin
(252) Since the integral of sin n4‘ in a complete period of 4‘ is always zero for any integer value n and the integral of cos n4’ in the same range of 4’ is only different from zero for n = 0, Eq. (252) is reduced to He@)
=
o s O E O k lsin 8 exp(ik R ) 2Ru2
From Eq. (253), it is observed that H,,(R) contains only the component along the 4 direction. The integrals of Eq. (253) may be obtained by means of an integration by parts dr’
=
[r‘Hb‘)(ur’)]; -
dr’
=
[r’J,(ur’)]:
Hb”(ur‘) d(ur’)
:c
--
J,(ur’) d(ur’)
(254)
(255)
DYADIC GREENS FUNCTIONS
59
In addition, the following relation may be used:
+
Hg’(ur’)= Jo(ur’) iNo(ur’)
(256)
where N,(ur’) is the Bessel function of second kind. By using Eqs. (254)-(256),Eq. (253) is simplified to
H&I
m 0 E O k lsin 6 exp(ik, R ) 2Ru2
=-
1
aJo(ua)- bJo(ub) aJ,(ua) - bJo(ub) en=O
aNo(ua) - bNo(ub) en=O
or H,d
=
f, ~
c
1
No(ur’) d(ur‘)
-oxOEOkl sin 6 2RuZ
where 1
Z1
= aJo(ua) -
bJ,(uh) -
Pun
J
Jo(ur’)d(ur’)
(259)
ub
:
2, = uNo(uu) - bNo(ub) - -
No(ur’)d(ur’)
Jub
The far electric field radiated by the magnetic current ring is obtained from Eee,= v , H ~ ~with , , H,,, given by Eq. (258) and q l r 377 R.
D. Calculation of the Far Fields Radiated ,from the Microstrip Wraparound Antenna
The method of multiplication of radiation patterns is used for the calculation of the far fields radiated from the microstrip wraparound antenna. Since the distance between the two apertures is equal to one-half dielectric wavelength &/2, the group pattern is given by (Jordan and Balmain, 1968): cos(xl, cos 0 / 2 l , )
60
S. BARROSO DE ASSIS FONSECA AND A. J. GIAROLA
The unit pattern is given by Eq. (258). Thus, the electric field radiated by the antenna is
I E , I K C O S cos ( ~ 0/2&)(sin 6/u2>I z , c A , , ( ~ ) + AB,(~)I+ iz, A,,(e) I (26 1) From Eqs. (164) and ( 1 66), we obtain
+
A,,(@
= ~ ~ ~ , ( U ~ ) / ~ U U [ Q H ~E’ ,”( (uU ~U / u)~ ) P H ~ ” ( u ~ ) ]
Aho(9)
= - i2Hb”(ub)/nua[QHb”‘(~a)
(262)
+ E , ( o ~ / u ~ ) P H ~ ” ( u u )(263) ]
with P and Q given by Eqs. (174) and (175), respectively, and knowing that Eq. (1 73) is reduced to
X I , = , = (ih,n/a)[l - (u2/u2>],=,
=0
(264)
From Eqs. (262) and (263), the following result is obtained:
Aeo(8)
+ A:,(@ = 2No(ub)/nua[QH~’”(ua)+ ~ , ( u ~ / u ~ ) P H ~ ’ ) (265) (ua)]
The fraction of the area of the radiation pattern with amplitude lower than - 8 dB with respect to its maximum is (Munson, 1974) F N = J:?”
Jg sin 9 d9 d 4 + J:.“””
sin 9 d9 d4/J:?O0
J;!”
sin 9 d9 d 4
(266) where /lo is the angle for which the amplitude of the radiation pattern is dB with respect to its maximum value. The pattern coverage is defined by Munson (1974) to be
C,
=
100 - FN
-
8
(267)
The radiation patterns obtained in the E plane, for any value of 4, of the wraparound antenna and of the ring of a magnetic current are shown in Figs. 27-34. In Fig. 35, curves of the pattern coverage as a function of the dielectric thickness, the radius b, and the relative permittivity of the dielectric for a frequency of 2.0 GHz of a microstrip wraparound antenna are shown. After examination of these figures, the following observations are made: (a) There is a strong influence of the relative permittivity E, in the radiation patterns. For higher values of E,, the radiation patterns are less dependent on the dielectric thickness within the range considered here. (b) For the same dielectric thickness, an increase in E, causes a noticeable reduction in the pattern coverage.
DYADIC GREEN’S FUNCTIONS
61
FIG.27. Radiation pattern of a microstrip wraparound antenna. [ E e l ,E plane, frequency = 2.0 GHz; E, = 2.55, b = 1.5 in.
FIG.28. Radiation pattern of a microstrip wraparound antenna. [ E e l ,E plane, frequency = 2.0 GHz, E. = 9.6, h = 1.5 in.
62
S. BARROSO DE ASSIS FONSECA AND A. J. GIAROLA
FIG.29. Radiation pattern of a microstrip wraparound antenna. [ E e l ,E plane, frequency = 2.0 GHz, E, = 2.55, h = 10 in.
FIG.30. Radiation pattern of a microstrip wraparound antenna. [.Eel,E plane, frequency = 2.0 GHz, E, = 9.60, b = 10 in.
DYADIC GREEN'S FUNCTIONS
63
FIG.31. Radiation pattern from a ring of magnetic current. IEel, E plane, frequency = 2.0 GHz, 6 , = 2.55, h = 1.5 in.
0.2 mm a -b
0.4
c I
FIG.32. Radiation pattern from a ring of magnetic current. IEel, E plane, frequency GHz, E, = 9.60, h = 1.5 in.
= 2.0
64
S. BARROSO DE ASSIS FONSECA AND A. J. GIAROLA
FIG.33. Radiation pattern from a ring of magnetic current. IEel, E plane, frequency = 2.0 GHz, E, = 2.55, b = 10 in.
FIG.34. Radiation pattern from a ring of magnetic current. / E e l , E plane, frequency = 2.0 GHz, E, = 9.60, b = 10 in.
65
DYADIC GREENS FUNCTIONS I
I
I
I
I
I 1.0
I 2 .o
I
I
I
3.0
4.0
5.0
0.2
1
I
D i e l e c t r i c thickness (mm) FIG.35. Pattern coverage as a function of the dielectric thickness for E, = 1.5 (-) and 60 (- - -) in., at a frequency of 2.0 GHz.
=
2.55 and 9.60 and
for b
(c) For the low value of E, = 2.55, the pattern coverage decreases as the dielectric thickness increases, contrary to the prediction made by Munson (1974) that the pattern coverage is only a function of the diameter of the cylinder and independent of the antenna thickness. (d) For both values of E, considered here, the pattern coverage did not vary appreciably when the diameter of the cylinder increased from 3 to 60 in. at the frequency of 2 GHz. (e) A comparison of the radiation patterns for a ring of magnetic current and for the wraparound antenna indicates that the interference between the two apertures of the wraparound antenna is more pronounced for lower values of E,. This is an expected result since the separation between apertures is
66
S. BARROSO DE ASSIS FONSECA AND A. J. GIAROLA
always equal to one-half of the wavelength in the dielectric such that the physical distance between apertures decreases as E, increases for the same frequency. (f) For the frequency of 2.0 GHz, considered here, the radiation pattern does not vary much as the diameter of the cylinder increases from 3 in.
E . Conclusion of the Section
In this section, the radiation patterns from a ring of magnetic current and from a wraparound antenna were obtained with the objective of examining the effects of the relative permittivity and thickness of the dielectric. No theoretical or experimental values from other sources were available for comparison with the present results. However, the results have indicated that, contrary to the prediction made by Munson (1 974), the presence of the dielectric layer causes an appreciable effect on the radiation pattern for radiation angles near the axis of the cylindrical surface. One of the advantages of the present method is that the asymptotic expression of the dyadic Green’s function used is also valid in the analysis of other types of antennas that may be adapted to the cylindrical surface.
Ix.
STUDY OF THE INFLUENCE OF EXCITATION OF SURFACE WAVES ON THE RADIATIONEFFICIENCY OF THE SPACE WAVES AND THE DIRECTIVITY OF A MICROSTRIP
DISKANTENNA A . Introduction
In this section, a study of the excitation of surface waves along the dielectric-air boundary of a microstrip disk antenna is presented. The use of the cavity model with conducting magnetic side walls is justified due to the adequate accuracy of the results (Carver and Mink, 1981) and because it allows the separate calculation of the space and surface waves excited by the antenna following a simple procedure. Following this model, the conducting patch is replaced by the equivalent magnetic current existing at the aperture of the antenna. As a result, the problem consists in the calculation of the fields in free space excited by a magnetic current source immersed in a dielectric layer placed over a perfect electric conducting plane. Thus, in addition to the fields refracted through the dielectric-air boundary, the fields excited in the form of surface waves along this
DYADIC GREENS FUNCTIONS
67
boundary should also exist. The main objective of the analysis presented here is to determine the influence of the surface waves on the directivity and the efficiency of space wave launching in microstrip disk antennas.
B. Integral Representation of the Fields Since the interest in this section includes not only the electromagnetic fields radiated in the form of space waves but also the surface waves, the asymptotic expression of the dyadic Green's function is not adequate for the present analysis. The original expression given by Eq. (187) should be used for G("'(R/R'). As discussed in Section VI, the equivalent magnetic current that produces a maximum in the radiation pattern along the direction normal to the plane of the disk (see Fig. 6) is given by
M(R')
= E,J,(k,
a) cos 4'6
(268)
The radiated magnetic field is obtained from Eq. (33) by using Eqs. (1 87) and (268). By using the asymptotic expressions of the Hankel functions with A = B and C = - D, H(R) is written as
x exp[i(llr
+ h,z)] cos 46 - c sin (h,d)
x exp[i(Ar
+ h,z)] sin 4( -h,F + ' 2 )
k l k , hz a
(269)
In Eq. (269) we observe that H, is proportional to cos 4, while H , and H , are proportional to sin 4. From Maxwell's equations, we verify that the field radiated by the antenna along the directions 4 = 0" and 4 = 180" is transverse magnetic with respect to z and TM,, while it is transverse electric with respect to z and TE, along the directions 4 = f90". Thus, for 4 = 0" and 4 = 180"
E = E,P
+ E,P,
H
a n d 4 = )90"
E = E,$,
H
= H,F
H,$
(270)
+ H,P
(27 1)
=
The integrations of H,, H,, and H , are studied separately as seen in the following subsections.
68
S. BARROSO D E ASSIS FONSECA A N D A. J. GIAROLA
C. Study of the Integration of H, By using Eq. (77) for A in Eq. (269), H, is given by
H, =K
aJ,(Aa) exp( -ih,d) sin(h,d) ~ 2 h , hcos(h,d) 2 - ihi sin(h,d) aa
S_dA
+ h,z)]
x J&exp[i(Ar
where
K
-i)5/2 cos 4
= wa~,E,J,(k,a)(
(273)
The integrand of Eq. (272) is an even function of h,. As a result, the branch cuts due to h , are eliminated. The function h, = has branch points at I pressed by
=
Jm
(274)
fk,. Consider A as a complex variable and exI=u+iv
To obtain the branch cuts for A
h: = k:
- (u
=
(275)
fk,, the following procedure is used:
+ iu)’
= k:
-
uz - i2uv
+ v2
(276)
Assuming that medium 1 has low losses, then, k, = k,,
+ ik,,,
(277)
k,, B k,,
From Eq. (276),
h:
= k:,
+ i2k1,k,,
- k:,
- u2 - i2uv
+ vz
(278)
or h: = [(u’ - uz) - (k:, - k:,)]
+ i2(kllk,,
- uv)
(279)
Consider now the curves Re(h:) = 0 and Im(h:) = 0, where Re and Im are the operators that take the real and imaginary parts from a complex quantity, respectively. In the complex plane A, these curves are curve Re(h:)
= 0: u2 -
u2 = k:,
curve Im(h:) = 0: uv and they are shown in Fig. 36.
= k,,k,,
- k:,
(280) (281)
69
DYADIC GREENS FUNCTIONS V
I
path of integration
\ \ \
FIG.36. Curves of Re(@
=0
and Im(h:)
=
0 in the plane 3, = u
+ iu.
The magnetic field given by Eq. (272) is that of a wave that propagates away from the source and attenuates with the distance, as long as the following conditions are satisfied: Re@,) > 0
(282)
Im(h,) > 0
(283)
From an examination of Eqs. (280) and (281), and having in mind the condition that k , , 9 k12, the regions in the complex 1plane for which Re(h:) and Im(h;) are larger or smaller than zero may be obtained and are shown in Fig. 36. The integrand of Eq. (272) is considered to be defined in a Riemann surface with two sheets. In the upper sheet, known as the proper sheet, Im(h,) > 0
(284)
70
S. BARROSO DE ASSIS FONSECA AND A. J. GIAROLA
In the lower sheet, known as the improper sheet, Im(h,) < 0. In order to have Im(h,) sheet,
0 in all upper sheet, it is necessary that, in this Arg(h:) > 0
(286)
where Arg is the operator that takes the argument from a complex quantity. Thus, the branch cut is defined as Arg(h:,) = 0
(287)
Re(h:) > 0
(288)
Im(h:)
(289)
or
=0
From a study of Fig. 36 and Eqs. (288) and (289), the branch cuts are those shown in Fig. 37. Also shown in this figure is the branch cut due to the branch point A = 0 introduced by the Hankel function.
FIG.37. Branch cuts in the complex 2 plane.
71
DYADIC GREENS FUNCTIONS
FIG.38. Upper sheet of the Riemann surface when k , ,
+
0'
With the assumption that the losses approach zero,
k , 2 -0'
(290)
the hyperbolas shown in Fig. 37 degenerate and approach the real and imaginary axes, as shown in Fig. 38. The complex plane shown in this figure is the upper sheet of the Riemann surface. In this figure, the regions for which Re(h,) > 0 and Re(h,) < 0 are also shown. The conditions for convergence of Eq. (272) [given by Eq. (282) and (283)] are satisfied if the path of integration from 1= - co to 1= +GO is that shown in Fig. 38 on the upper sheet of the 1plane. Thus, the radiation Iin this path and the integral representacondition is satisfied at any value of , tion of H , is completely specified. For convenience, the following complex variable is defined: p=D+iT
(29 1)
72
S. BARROSO DE ASSIS FONSECA AND A. J. GIAROLA
The following change of variable is made:
I = k , sin p r = R sin 0
z = R cos 8
(294)
where R and B are the variables in the spherical system of coordinates. Thus,
h , = (k:
I r + h,z
- k:
sin’
p)l12
= k , cos p
(295)
= k , R cos(B - p)
(298)
The sign of the square root of Eq. (295) was chosen such that for I + 0, h, + k , . This sign will be studied rigorously later in this section. As a result of Eq. (292), the branch points of h , are eliminated. The mapping of the I plane into the p plane is done by replacing B given by Eq. (291) in Eqs. (292) and (295) and with the assumption that k , is a real quantity. Thus, k , sin CJ cosh z
(299)
Im(I) = k , cos u sinh z
(300)
Re(h,) = k , cos u cosh z
(301)
Im(h,)
(302)
Re(I)
=
=
- k , sin u sinh z
The mapping of the first and third quadrants of the upper sheet of the I plane, for example, is accomplished by considering that in these quadrants, for k, real, Im(h,) > 0 and Re@,) < 0. Thus, the transformation of these quadrants to the p plane is obtained from Eqs. (301) and (302) as: sin u sinh z < 0
(303)
cos u cosh z < 0
(304)
For u varying in the interval - n 2 u 5 n, the solution of Eq. (303) is -n
for z > O
or
O
The solution of Eq. (304) for the same range of z is +n
or
-n
for z < O
73
DYADIC GREENS FUNCTIONS
for any value of t. The solution that satisfies Eqs. (303) and (304) simultaneously is --n O (305) for
+-n
t
(306)
To identify the first and third quadrants in the p plane, it is necessary to go back to Eqs. (299) and (300). In the first quadrant, Re@) > 0 and Im(A) > 0. Thus, from Eqs. (299) and (300): sin cr cosh z > 0
(307)
cos cr sinh z > 0
(308)
The solutions for --n 5 cr 5 -n are as follows: (a) From Eq. (307), 0 < cr < n for any value oft, (b) From Eq. (308), -3. < cr < 3-n for z > 0 or for z < 0,
+-n
or
--n
-3-n
Thus, Eqs. (307) and (308) are simultaneously satisfied for or A study of Eqs. (305), (306), (309), and (310) indicates that the first quadrant of the upper sheet of the 1 plane is transformed into the region 3-n < cr < n,t < 0, for - n 5 cr S -n of the plane. The mapping of the II plane in the plane is shown in Fig. 39. There are branch cuts in the II plane that no longer exist in the /? plane. The path of integration P in the II plane is transformed into the path P‘ in the fl plane. In Fig. 39, the letters S and I indicate the regions where the upper and lower sheets of the 1 plane were mapped. The subscripts indicate the respective quadrant. For example, S , is the region in the 8 plane that corresponds to the second quadrant of the upper sheet of the Riemann surface. With this transformation, the integral representation of H , is written as follows: r
1
where exp[ - ih,(fl)d] sin[h,(/?)d]k, cos 8
74
S. BARROSO DE ASSIS FONSECA A N D A. J. GIAROLA
I I 14
s3
I I
I I
I
Te I I\
51
___I)
rf
C
I
I2
'--
p2
I
FIG.39. Mapping of the 2 plane in the B plane.
The saddle point method is used to integrate Eq. (3 11) and an asymptotic expression for H , is obtained. Thus, the path of integration P' is deformed to the steepest descent path passing through the saddle point. The main contribution to this integral results from a small section located near the saddle point. The saddle point for Eq. (31 1) is obtained as follows: ( d / d p ) [ i cos(0 - p)]
Since the region of interest in the
= 0 -+
psp= 8 - nn
P plane extends from r~ =
-IT
to
r~ =
+z, n = 0. Thus,
Psp =
(313) The steepest descent path passes through the saddle point and is defined by ImCF(P>I= ImCF(Bsp)l
(314)
DYADIC GREENS FUNCTIONS
where
F ( p ) = i cos(8 - B) and, with Eq. ( 3 13),
Thus, the steepest descent path is defined as Im[F(B)]
= cos(8 - a) cosh z =
1
( 3 17)
This equation gives also the steepest ascent path. In order to identify the steepest descent path, a procedure presented by Collin (1960) may be used. In Fig. 39 the saddle point Bsp = 8 and the steepest descent path (SDP) are shown for a given value of 8. In the present work, 8 is restricted to vary in the range
The deformation of the integration path from P' to SDP is done using the Cauchy's theorem. Since the path is closed by the sections P , and P,, as shown in Fig. 39, it is verified that the contributions to the integral along these sections are zero, since they are located in the upper sheet of the Riemann surface (Collin, 1960). Thus, the integral along the path P' is equal to the integral along the steepest descent path added to the contributions due to the poles of the integrand that may eventually be enclosed by the path as the Cauchy's theorem is applied. The contributions due to these poles constitute the diffracted wave that propagates along the dielectric-air boundary. The contribution to the integral due to the steepest descent path constitutes the geometrical optics component, known as the space wave that is refracted through the dielectric-air boundary. Note that an adequate series expansion of the functionf(B), given by Eq. (312), may eliminate the poles in the complex p plane. However, branch points are introduced and the diffracted wave along the dielectric-air boundary is given by the sum of the lateral waves obtained from the integration along the various branch cuts (Tamir and Felsen, 1965). The result of the contour deformation may be summarized as follows:
J
p,
J
= SDP
+ 2ni 1residues
( 3 19)
76
S. BARROSO DE ASSIS FONSECA AND A. J. GIAROLA
The contribution to the space wave that results from the integration along the SDP was obtained in Section VI and is rewritten here, for convenience,
HF
=
-i2E0ws0a cos 0 sin[k,(t’ - sin’ B)’/’d] k:(t2 - sin’ O)R sin 8 x
H!P =
cos
1
1 k, sin 8Jo(k,a sin 8) - - J,(k,a sin 0) exp(ik,R) a
[
i2EOw&,cos 8 sin[k,((’ - sin’ @‘/’dl tk:(t’ - sin’ 0)”’R sin 0 x sin 4C(0)J,(k1a sin 8) exp(ik,R)
(320)
(321)
where the coefficients A(0) and C(0) are given by Eqs. (77) and (79), respectively, with h , and h, given by
hl = k, cos 0,
hz = kl(t2 - sin’
Q1/’
In order to obtain the diffracted wave, it is necessary to inspect the poles of the integrand of H,. This is more conveniently realized in the complex plane A = u + iu. After proceeding with their location and characterization, these poles are mapped into the plane. It is thus necessary to make a preliminary study of the sign of the root of h , = (k: having in mind the convergence of the integral. This study is based on Fig. 40. For A real (i.e., A = u) and varying from - co to + co,the following observations may be made, from Fig. 40: (a)
( k , - A)1’2(kl + A),’’
= (k, - A)1/2[kl -
(-A)]’/’
(322)
(b) The argument of (k, - A) varies within the following range: n > Arg(k, - A) > 0
(323)
+ A) varies within the following range: 0 < Arg(k, + A) < n
(324)
(c) The argument of (k, As a result,
)n > Arg(k, - A),/’ > 0
0 < Arg(k,
+
<.3
Another observation is that n > Arg(k, - A)
As a result,
+ Arg(k, + A) > 0
(327)
77
DYADIC GREENS FUNCTIONS
FIG.40. Geometry that allows us to obtain the sign of (k:
- d*)'''
Since the integral converges for Re&) > 0 and Im(h,) > 0, it is concluded that the positive sign should be adopted for (k: - ,l2)lI2. This will be done for the whole development and justifies the sign adopted for the square root in Eq. (295). The poles of the integrand are solutions of
t2hl cos(h,d)
= ih,
sin(h,d)
(329)
or
h,d tan(h,d)
=
(330)
-ie,h,d
where E,=
t2=
- El%
(331)
The surface waves that may be excited propagate in the r direction and attenuate in the z direction. Thus, from Eq. (272), the surface waves are obtained from the poles that satisfy
1= u(rea1) kl
(332)
< IuI
(333)
As a result,
h,
=
h:
= (ki
(k: - u2)l/, = i(u2 - k:)'', -
u2)
=
ia
(334) (335)
78
S. BARROSO DE ASSIS FONSECA A N D A. J. GIAROLA
where CI is a real and positive quantity. From Eqs. (330), (334), and ( 3 3 9 , the results are h,d tan(h,d) (CI@
= &,ad
+ (h2d)’ = (E,
-
(336) l)(kId)’
= 6’
(337)
It is important to note that Eqs. (336) and (337) are the same equations as those for the symmetric TM modes in dielectric slabs (Collin, 1960). The solutions of Eqs. (336) and (337) may be obtained graphically as shown in Fig. 41. Here, k , and k, are assumed to be real and a value of E, = 2.55 is taken, for example. Z not have a The first excited mode that occurs in the range 0 5 6 5 ~ I does S 6 5 .n, the solutions found for negative cutoff frequency. In the range
FIG.41. Example of a graphical solution of Eqs. (336) and (337).
79
DYADIC GREENS FUNCTIONS TABLE 11 SOLUTIONS OF EQS.(336) AND (337) FOR VARIOUS VALUESOF dla = 2.55
E,
E, =
9.60
dla
h,d
ad
h,d
ctd
0.050 0.158 0.263 0.368 0.526
0.072 0.226 0.373 0.587 0.715
0.002 0.021 0.057 0.153 0.243
0.087 0.275 0.457 0.639 0.906
0.00079 0.008 0.023 0.049 0.122
values of a do not correspond to surface waves. For the range 0 5 6 5 i x , the following range of d/a corresponds:
0 5 (d/a) 5 0.517
(338)
where d is the dielectric thickness, a is the disk radius, and E , is chosen equal to 9.6. For E, = 2.55, the range of d/a is 0
5 (d/a) 5 0.873
(339)
Practical values of d/a commonly used in the construction of microstrip antennas are within the ranges of Eqs. (338) and (339). Thus, only the first mode, corresponding to the solution obtained in the range 0 5 6 5 in,will exist in these structures for TM, modes. The values of ad and h,d for various values of d/a are shown in Table 11. The relation between 6 and d/a that is necessary for the construction of Table I1 is obtained from
d2 = ( E , - t ) ( k , d ) , = ( E , = C(E, -
-
l)[k2~(kJk2)(d/~)]~
1)/&,1Ck2a(d/a)12
(340)
At the resonance of the mode TM,,, the value of k2a is 1.841 and
6 = 1.841[(~,- l ) / ~ , ] ” ~ ( d / ~ )
(341)
For the solutions corresponding to the surface waves, ct must be a real and positive quantity. Complex solutions of Eqs. (336) and (337) may also exist. These solutions correspond to modes that are characterized by the existence of a flux of power along the direction z, perpendicular to the dielectric-air boundary. These modes are commonly known as the leaky waves or loosely coupled modes. These modes do not satisfy the radiation condition in the
80
S. BARROSO DE ASSIS FONSECA AND A. J. GIAROLA
complex A. plane. However, in the j? plane, after the transformation defined in Eq. (292), the deformation of the contour of integration for the application of the saddle point method may loop some of these complex poles that are located in the improper sheet of the Riemann surface. In these cases, the contributions due to these poles must be considered (Collin, 1960). Thus, the complex solutions of Eqs. (336) and (337) must be examined. This may be accomplished using the method presented by Collin (1960). With the definitions ad = T
+ iU
(342)
h,d = X
+ iY
(343)
Eq. (336) may be written as (X
+ i Y ) tan(X + i Y ) = E,(T + iU)
(344)
After expansion, two equations are obtained: 1 X sin 2X - Y sinh 2Y T=E, cos 2X cosh 2Y
(345)
+ 1 X sinh 2Y + Y sin 2X u = - cos 2X + cosh 2Y
(346)
E,
After eliminating ad in Eqs. (336) and (337), the result is X
+ iY =
k & , G / [ E , ~+ tan2(X + iy)]'I2
(347)
After expansion of Eq. (347), two equations are obtained: 2X + cosh 2Y ) x = f [Q' e,G(cos + (2 sin 2X sinh 2 Y ) ] 114cos -tan-'
[:
2 sin 2X sinh 2Y
Q
I
(348) Y= f
[Q'
~,G(cos2X + cosh 2 Y ) 2 sin 2X sinh 2Y + ( 2 sin 2X sinh 2 Y ) ],/,sin -tan-' Q
[:
1 (349)
where
Q
= .$(cos
2X
+ cosh 2Y)' + sin'
2X - sinh' 2Y
(350)
A study of Eqs. (348) and (349) shows that if the pair ( X , , Y,) is a solution, then the pairs ( X , , - Yl), ( - X , , Y,), and ( - X , , Y,) are also solutions. Therefore, only positive values of X and Y are necessary in the solution of these equations.
DYADIC GREENS FUNCTIONS
81
- -_--
12.0 ..
10.0-
8.0-
6.0-
4.0-
2.0-
Roots for reol ond p o s i t i v e v a l u e s of a
Eq. (7.86) 3.O 4.0 5:O FIG.42. Graphs of Eqs. (348) and (339) for E,
6.0
X
= 2.55.
By using the digital computer subroutine RTMI from the IBM Application Program, which provides the roots of an equation of the type F ( x ) = 0, the curves of Eqs. (348) and (349) for various values of 6 were constructed. For the values of E, and 6 considered here, the system of Eqs. (348) and (349) does not have complex roots. The only roots that were found are those that correspond to real and positive values of a. These values have already been obtained and shown in Table 11. It is thus concluded that leaky modes or leaky waves do not contribute for the field representation in the present case. For E , = 2.55, the graphs of Eqs. (348) and (349) are shown in Fig. 42. Similar curves are obtained for E, = 9.60.
D. Study of the Integral Representation of H , and H , After examining the poles of the integrand of Eq. (269) for the magnetic field components H , and H,, we observe that the surface waves for the TE, modes are not excited in the dielectric-air boundary for practical values of the d/a ratio used in microstrip disk antennas. By using Eq. (79) for the coefficient C, the poles are obtained from
t h , cos(h,d)
- ith,
sin(h,d) = 0
(351)
or h,d cot(h,d) = ih,d
(352)
82 For h ,
S. BARROSO DE ASSIS FONSECA AND A. J. GlAROLA =
ia, the result is h2d cot(h2d) = -ad
(353)
As expected, Eq. (353) is the same as that obtained for odd TE modes in dielectric slabs (Collin, 1960). For the first TE mode, the cutoff frequency is obtained from (Collin, 1960)
where Lo is the free-space wavelength. For the usual values of d/Ao and of E, and for the frequencies of interest, we conclude that the TE, modes are not excited. Thus, in practice, only the TM, modes are excited. E . Study of the Poles in the p Plane. Calculation of the Residues
For the calculation of the poles I , , it is necessary to select a value for the frequency. A value off = 3.0 GHz, which corresponds to ,lo = 10 cm, will be used as an example. Equation (334) will be used and is repeated here, for convenience. h 1 -- (k: - A')'/'
= ia
(355)
The calculated values of ,lp, based on the data of Table 11, are shown in Table 111. The values presented in this table are for those poles that are enclosed by the contour of integration in the /3 plane. Thus, they are the ones that contribute for the calculation of the fields. The mapping of the values of A,,to the corresponding values of p,, in the /?plane is done by means of Eqs. TABLE 111 VALUES OF 1, AND 8, OBTAINED WITH THE DATAOF TABLE 11 Poles associated with surface waves E,
= 2.55
dla
1,
PP
0.050 0.158 0.263 0.368 0.526
62.81 63.25 63.93 66.19 61.69
4 2 - i0.036 4 2 - i0.116 4 2 - i0.187 4 2 - i0.353 4 2 - i0.391
E, =
9.60
*,
8,
62.85 63.06 63.51 64.39 67.46
n/2 - i0.025 4 2 - i0.086 4 2 - i0.147 7112 - i0.222 n/2 - i0.382
83
DYADIC GREENS FUNCTIONS
c 0.4 0.3
0.2 0.1
(
8 , -2.55
S1 FIG.43. Location of the poles in the complex fi plane.
(299) and (300). The location of the poles in the /l plane corresponding to the surface waves for both values of E, used here are shown in Fig. 43. The critical values dc of the angle 6 that characterize the contribution of each pole may be obtained from these data. This is done by means of Eq. (317), which defines the steepest descent path and with the values of 8, from Table 111. The results are shown in Table IV. TABLE IV
VALUES OF THE CRITICAL ANGLE 0, A FUNCTION OF E, AND d/a E,
=
2.55
E, =
9.60
d/a
0,
dja
0,
0.050 0.158 0.263 0.368 0.526
87.9 83.4 79.4 70.2 68.2
0.050 0.158 0.263 0.368 0.526
88.6 85.1 81.6 77.4 68.6
AS
84
S. BARROSO DE ASSIS FONSECA AND A. J. GIAROLA
The regions of contribution of the surface waves corresponding to each pole are then defined as
0, < 0 < 3.
(356)
According to Felsen and Marcuvitz (1973), we observe that the absence of the contribution due to the surface waves, for 0 < Bc, does not imply that these waves do not exist. It simply means that the amplitude is so small that it is below the error resulting from the asymptotic approximation used. For a given value of I , and its corresponding Bp, the residue may be easily obtained. The expression of the integral representation of H , in the I plane is written as H,= K
sm
dI
-m
exp( - ih,d) sin(h,d) [’h,hz cos(h,d) - ihi sin(h,d)
The integrand of Eq. (357) may be written as the ratio of two functions of A, as follows:
where
f ( I ) = exp( - ih,d) ~in(h,d)[~~,(Iu)/da](2/~Ir)’~~ exp[i(Ir g(A) = t2h,h2 cos(h,d) - ihi sin(h,d)
+ h,z)]
(359) (360)
If A, are the zeros of g(I), and h , , and h Z pthe corresponding values of h , and h,, respectively, then, the residue is given by residue
= K[f(I,)/g’(I,)]
(36 1)
where g’(A,) is the derivative of g ( I ) with respect to I at I = I,. After completing the operations indicated in Eq. (361), an expression of the magnetic field, H Y , of the surface wave for each value of I , may be obtained as
1;
i2nK exp( - ihl,d) sin(h,,d) A aJ u)(&)l’zexpLi(Ipr su H4
- [2[I,hl,d sin(h,,d)
- I,h,,h,-b
cos(h,,d) - I,h,,h;,l + i[I,h,,d cos(h,,d) + 21, sin(h,,d)]
+ h,,z)] cos(h,,d)] (362)
DYADIC GREENS FUNCTIONS
85
A comparison of Eqs. (320) and (362) indicates that the space wave varies with distance in proportion to R - ' , while the surface wave varies as r-1'2. The surfaces of constant phase are a sphere and a cylinder for the space and surface waves, respectively. Thus, the space wave is a spherical wave, while the surface wave is a cylindrical wave.
F . Calculation of the Radiation EfJiciency of Space Waves and of the Directivity of the Microstrip Antenna In antennas designed to excite surface waves, the efficiency of surface wave excitation was defined to characterize these antennas (Collin and Zucker, 1969). Similarly, since the main interest here is to excite the space wave, a radiation efficiency of space waves is defined. For the lossless case, there is no cross coupling between the power contained in the surface wave P,, and that contained in the space wave P,, (Collin and Zucker, 1969). Thus, these two powers may be calculated separately. The radiation efficiency of space wave for a microstrip antenna is defined as Y = PSPAPSP + PS")
(363)
The power contained in the surface wave may be determined from the integration of the Poynting vector in a cylindrical surface with a large radius and infinite height. For this, it is necessary to obtain first the z component of the electric field E,. From Maxwell's equation,
V
X
H
=
-iWEOE
(364)
and neglecting the terms proportional to F3I2,the following relation is obtained :
The wave impedance in the radial direction r is then given by ZTM= - E,/H,
= Ap/WEO
(366)
This expression of the impedance for the TM mode may be compared with (21.6b) of Collin and Zucker (1969). Using the Poynting's theorem:
86
S. BARROSO DE ASSIS FONSECA A N D A. J. GIAROLA
With H F given by Eq. (362), the integral [Eq. (367)] is given by psu =
27c2we0a2E3~(k15a) exp(2aPd) sin2(hz,d)[A,Jo(A,a) - ( ~ / U ) J ~ ( A , U ) ] ~ ~,[((’a,d 2)1, sin(h,,d) (<’a, l h 2 , - <’a&; hz,d)A, cos(h,,d)12
+
+
+
(368) The power contained in the space wave is obtained from the integration of the Poynting vector in a hemispherical surface with a sufficiently large radius R, such that the far fields may be used in the integration. The result is
For far fields, EZP = q Hsp $,
(370)
E F = -qlHSP e
and Eq. (369) is written as
After introducing in Eq. (371) the expressions in Eqs. (320) and (321) for H F and HZP,respectively, and integrating in 4, the result is
x
i
cos2 6 sin’(k,dJ-)[k:
sin 8J;(kla sin 6 )
~~
+ (a’
(4’
-
sin’ 6)[5“ cos’ 8 cos’(k,d&’
- sin’ 8)
sin 9)-’J:(k1a sin 8) - (2k,/a)J0(k,asin 8 ) J l ( k , a sin 8)]
(5’
- sin2 8) s
i n ’ ( k , d J ~ ) ]
cos2 8 sin2(k1d,/5’ - sin’ 8)J:(k,a sin 9) + t4a2 sin 6[(t2 sin’ 8) c o s ’ ( k , d ~ -
+ cos’ 6 sin’(k,dJ=)]
I
~ ~ )
(372)
i
It should be noted that the apparent singularity of the integrand at 8 = 0” disappears if J , ( k , u sin 8) is replaced by its asymptotic expression for small arguments (Abramowitz and Stegun, 1965): J l ( k , a sin 9) 5 3kla sin 8
(373)
DYADIC GREENS FUNCTIONS
87
By noting that the total power radiated by the antenna is equal to the sum of the power contained in the space wave and the power contained in the surface wave, the directivity may be obtained from DIR
+ P,,)
= 4xR2S(8 = Oo)/(Psp
(374)
where S(0 = O o ) is the value of the power density along the direction 0 = 0" and corresponds to the integrand of Eq. (369) for 8 = 0". The integral in Eq. (372) was calculated by means of a digital computer program for E, = 2.55 and 9.60 and for various values of the d/a ratio. The results for the efficiency and directivity are shown in Fig. 44 and the following observations are made: (a) The directivity does not vary much with the d/a ratio for both values of E, used. (b) The powers P,, and P,, increase with the d/a ratio. For some values of E, and d/a, the increase in P,, is such that the directivity decreases with an increase in the d/a ratio. An increase of the antenna beamwidth with the dielectric thickness was reported by Araki and Itoh (1981) when the hypothesis was raised that this observation was due to the excitation of the surface waves in the dielectric-air boundary. (c) It is convenient to note that, for small values of the d/a ratio, the results for the directivity agree with those presented by Derneryd (1979) for both values of the E, used.
L 01 .
02
03
0.4
'0.5
d /o
FIG44. Efficiency ( 0 )of space wave launching and directivity (x) as functions of d/a for microstrip disk antennas with E, = 2.55 ( - - - -) and 9.60 (-).
88
S. BARROSO DE ASSIS FONSECA A N D A. J. GIAROLA
(d) The efficiency of the space wave excitation decreases rapidly with increases in the d/a ratio for high values of E,, reaching a low value of 69% when d/a = 0.526 and E, = 9.6. For low values of E,, the efficiency decreases slower with increases in the d/a ratio, reaching a low value of 85% when E, = 2.55 and d/a = 0.526.
G . Conclusion of the Section
In this section, the problem of excitation of surface waves on the dielectric substrate of microstrip disk antennas was analyzed. The results shown may be of relevance in the design of microstrip antennas, particularly with respect to the choice of the dielectric substrate. It is important to note that the antenna efficiency obtained here, based on the assumption that the power contained in the surface wave is lost, behaves just the opposite of the antenna efficiency considering as power loss dissipated in the metal surfaces and in the dielectric. In the latter case, the efficiency increases with d/a ratio. In addition, since the propagation of the TE modes in the dielectric surface can be practically neglected, the coupling between two elements of an array should be less when the perpendiculars to the center of each disk are located in the plane 4 = 90" than when these perpendiculars are in the plane 4 = 0". This fact may be observed from the experimental results presented by Jedlicka et al. (1981) for relatively large values of spacing between elements.
X. CONCLUSIONS The results of the analyses developed in this chapter show that the dielectric substrate used for the construction of microstrip antennas exerts a strong influence in the electric properties of these antennas. For some specific applications, such as in missiles and rockets, there is no doubt that the knowledge of how the radiation pattern of these antennas varies with the dielectric permittivity and thickness is of real importance. From Figs. 27-35, which show curves of the radiation patterns and of pattern coverage of microstrip wraparound antennas, we conclude that these antennas should use dielectrics with the lowest possible values of relative permittivities in order to allow the maximum possible pattern coverage. Also, keeping a low value of relative permittivity and using the smallest possible dielectric thickness will help in reducing the excitation of surface waves on the dielectric surface. It is important to note again, as was done in the beginning of this chapter, that not many experimental data were available for comparison with the
DYADIC GREEN’S FUNCTIONS
89
theoretical results obtained here. The development of these antennas occurred not too long ago so that there is not an abundance of material on this subject. In addition, these antennas are particularly important for applications in the military area, so that specific informations are not easily available. Before concluding this chapter, it is important that some observations be made concerning the models adopted for the microstrip antennas in the analyses developed in the previous sections. The cavity model with conducting magnetic side walls has some limitations, as discussed in Section V. However, the results using this model are reasonably accurate, particularly with respect to the calculation of the far fields (Carver and Mink, 1981). The great advantage of using this model is that it allows the calculation of the radiated fields in a simple and convenient manner. The other known theoretical methods, such as those discussed by Araki and Itoh (1981), and Chew and Kong (1981) and Newman and Tulyathan (1981), use the method of moments in order to obtain the current distribution in the antenna with a higher precision. These methods, however, do not allow a detailed analytical study of the dependence of the radiated fields with the various parameters of the antenna. A more careful study of the surface waves, such as that presented in Section IX, would have been more difficult if a more complex model had been used for the analysis of the antenna.
REFERENCES Abramowitz. M., and Stegun, I. A. (1965). “Handbook of Mathematical Functions.” Dover, New York. Agrawal, P. K., and Bailey, M. C. (1977). IEEE Trans. Antennas Propay. AP-25,756. Araki, K., and Itoh, T. (1981). IEEE Trans. Antennas Propay. AP-29,84. Bahl, I. J. (1979). Microwaues 18, 50. Bahl, I. J., and Bhartia, P. (1980). “Microstrip Antennas.” Artech House, Dedham, Massachusetts. Bahl, 1. J., and Stuchly, S. S. (1980).I E E E Trans. Microwaue Theory Tech. MTT-28, 104. Bahl, I. J., Stuchly, S. S., and Stuchly, M. A. (1980). IEEE Trans. Microwave Theory Tech. MTT28, 1464. Brekhovskikh, L. M. (1960). “Waves in Layered Media.” Academic Press, New York. Byron, E. V. (1970). Proc. Phased-Array Antenna Symp., 1970, p. 187. Artech House, Dedham, Massachusetts, 1972. Carver, K. R., and Mink, J. W. (1981). IEEE Trans. Antennas Propag. AP-29,2. Chew, W. C., and Kong, J. A. (1981). I E E E Trans. Antennas Propag. AP-29,68. Collier, M. (1977). Microwave J . 20,67. Collin, R. E. (1960). “Field Theory of Guided Waves.” McGraw-Hill, New York. Collin, R. E., and Zucker, F. I. (1969). “Antenna Theory,” Part 2. McGraw-Hill, New York. Derneryd, A. G. (1976). IEEE Trans. Antennas Propag. AP-24,846. Derneryd, A. G. (1978). Microwave J . 21, 77. Derneryd, A. G. (1979).IEEE Trans. Antennas Propag. AP-21,660.
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S. BARROSO DE ASSIS FONSECA AND A. J. GIAROLA
Deschamps, G. A. (1953). USAF Symp. Antennas, 3rd, 1953. Felsen, L. B., and Marcuvitz, N. (1973). “Radiation and Scattering of Waves.” Prentice-Hall, Englewood Cliffs, New Jersey. Garvin, C. W., and Munson, R. E. (1977). IEEE Trans. Antennas Propag. AP-25,604. Hansen, W. W. (1935). Phys. Rev. 47, 139. Harrington, R. F. (1961). “Time-Harmonic Electromagnetic Fields.” McGraw-Hill, New York. Howell, J. Q. (1972). Dig. Int. Symp. Antennas Propag. SOC.,1972, p. 177. Howell, J. Q. (1975). IEEE Trans. Antennas Propag. AP-23,90. Jedlicka, R. P., Poe, M. T., and Carver, K. R. (1981). IEEE Trans. Antennas Propag. AP-29, 147. Johnson, W. A., Howard, A. Q., and Dudley, D. G. (1979). Radio Sci. 14,961. Jordan, E. C., and Balmain, K. G. (1968). “Electromagnetic Waves and Radiating Systems.” Prentice-Hall International Editions, Englewood Cliffs, New Jersey. Lo, Y.T., Solomon, D., and Richards, W. F. (1979). IEEE Trans. Antennas Propag. AP-27, 137. Mailloux, R. J., McIlvenna, J. F., and Kernweis, N. P. (1981). IEEE Trans. Antennas Propag. AP29, 25. Munson, R. E. (1973). US. Patent 3713162. Munson, R. E. (1974). IEEE Trans. Antennas Propag. AP-22,74, Munson, R. E. (1979). IEEE Newsl. Antennas Propag. SOC.21, No. 3. Newman, E. H., and Tulyathan, P. (1981). IEEE Trans. Antennas Propag. AP-29,47. Rahmat-Samii, Y. (1975). IEEE Trans. Microwaoe Theory Tech. MlT-23,762. Tai, C. T. (1971). “Dyadic Green’s Functions in Electromagnetic Theory.” Intext Publishers, Scranton, Pennsylvania. Tai, C. T. (1973). Proc. I E E E 61, 480. Tai, C. T. (1980). Math Note 65. University of Michigan Radiation Laboratory, Ann Arbor. Tamir, T., and Felsen, L. B. (1965). IEEE Trans. Antennas Propag. AP-13,410. Wood, C. (1980). Proc. Inst. Electr. Eng. 127,231.
.
ADVANCES IN ELECTRONICS AND ELECTRON PHYSICS VOL. 65
Ink-Jet Printing J . HEINZL Technical University of Munich Munich. Federal Repuhlic of German);
C . H . HERTZ Lund Institute of Technology Lund. Sweden
I . Introduction . . . . . . . . . . . . . . . . . A . Ink Jets versus Conventional Printing Methods . . . B. Short Review of Ink-Jet Methods . . . . . . . . C . Present Fields of Application for Ink-Jet Printing . . . I1. Drop-on-Demand Methods . . . . . . . . . . . . A . Principle . . . . . . . . . . . . . . . . . . B. Transformation of Electrical Pulses into Pressure Waves C . Wave Propagation in the Ink Channel . . . . . . D . Transformation of Pressure Waves into Velocity . . . E. Drop Shaping, Refill, and Reverberations . . . . . F. Other Questions . . . . . . . . . . . . . . . I11. Methods Employing Continuous Jets . . . . . . . . A . The Break-Up of a Continuous Fluid Jet . . . . . B. Ink-Jet Oscillograph Recorders . . . . . . . . . C . The Sweet Method . . . . . . . . . . . . . . D . The Hertz Method . . . . . . . . . . . . . . IV . Methods Employing Mechanical Valves . . . . . . . A . Introduction . . . . . . . . . . . . . . . . B. Systems Employing Mechanical Valves . . . . . . V . Conclusions . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . .
91
. 91 . 92 . 103 . . . . . . .
.
.
111 111 113 115 122 127 131 132 132 141 142 152 161 161 162 163 166
I. INTRODUCTION A . Ink Jets versus Conventional Printing Methods
During the last few years. printing by ink jets has increasingly been used in different fields of graphics. since this type of printing has special advantages over conventional printing methods . Ink-jet printers are especially 91 Copyright 0 1985 by Academic Press. Inc. All rights of reproduction in any form reserved. ISBN 0-12-014665-7
92
J. HEINZL AND C. H. HERTZ
suitable for use in certain applications, such as computer output devices and word processors, as well as for marking of checks or packaged consumer products. The usefulness of ink jet systems stems from the fact that they are essentially nonimpact printers with the following attributes:
1. high speed of printing; 2. silent, nonimpact operation; 3. electrically controllable by computers; 4. paper requires no special aftertreatment; 5. color capability; 6. in some methods, printing is possible on curved surfaces situated a few centimeters from the printing device. In the future, the low cost and the small size of the printhead may provide further advantages. For these reasons, research efforts are today being conducted at many industrial organizations to develop different methods for controlling ink jets by electrical signals. In view of the complexity of the subject and the variety of ink-jet devices that have been developed, the remaining portion of Section I is devoted to a review of the most important ink-jet printing methods so far developed. It is hoped that this will provide an overview for those readers not fully acquainted with the subject. For readers interested in understanding the subject in greater depth, the two most important classes of ink-jet systems, namely, the drop-on-demand and continuous-jet types, are discussed in more detail in Sections I1 and 111, respectively. In Section IV a brief discussion is given of systems employing mechnical valves, and in Section V some comparisons are made between the different types of systems and comments made regarding the future of ink-jet systems. B. Short Review of Ink-Jet Methods
The use of ink jets in recording devices dates back to 1930 when research workers such as Ranger, Diekmann, Schroter, Hansell, and Richards used continuous jets of ink to record signals or pictures. A good review of these earlier methods is given by Schroter (1932). None of these devices, however, was useful enough to be produced in large quantities. The first successful product using ink jets was developed by Elmquist (1951) in Sweden. By replacing the pen in direct-writing oscillograph recorders by an ink jet, he was able to decrease the inertia and thereby increase the upper frequency limit of such recorders to 1000 Hz, a tenfold increase over the conventional recorders at that time. These ink-jet recorders are still in wide use, mainly in hospitals, for the recording of ECG and other physiological signals. The principle of these recorders will be described in more detail in Section 1II.B.
INK-JET PRINTING
93
Elmquist’s recorder can be used for the printing of alphanumeric characters in only a very limited way. In the beginning of the 1960s, however, Sweet (1964, 1965) developed another type of ink-jet recorder using a principle that proved to be of much more general use. Furthermore, during the same years, Winston (1962) developed an ink-jet typewriter based upon still another principle. Finally, around the same time, two Swedish groups, Stemme and Larsson (1973) in Gothenburg and Hertz et al. (1967) in Lund, devised two additional methods for controlling ink jets by electrical signals. Thus, by the end of the 1960s four fundamentally different principles for inkjet printing existed. These were improved and altered by other workers with most of the ink-jet systems in use today having evolved from these four principles, which were described by Kamphoefner in a review article in 1972. Further reviews were prepared by Carnahan (1975), Carnahan and Hou (1977), and Meyer and Hoffmann (1975). Generally, ink-jet printers can be divided into two main families: 1. drop-on-demand systems in which electrical signals are used to control the moment when an individual droplet is ejected; 2. continuous-jet systems in which an ink jet emerges continually from a nozzle under high pressure. This jet then disintegrates into a train of droplets. Electical signals are then used to control either the direction of the droplets (Sweet method) or to disperse the droplets into a mist (Hertz method).
There are several other ink-jet methods that do not fall into this classification. However, since they are of minor or no importance today, they will be treated very briefly in this paper, and only the principles of the two important families of ink jets will be discussed in this introductory section. 1. Drop-on-Demand Systems
The impulse or drop-on-demand ink-jet system (Kyser, 1981; Stemme and Larsson, 1973; Zoltan, 1972) is a system of great fundamental simplicity. As shown by the examples in Fig. 1, it consists of an ink chamber C that is connected to a nozzle N . Part of the chamber consists of a piezoelectric element P that can be excited by a short electrical signal pulse S . If the element P is suitably designed, this excitation causes a sudden change in ink pressure in the chamber that leads to the ejection of an ink drop from the nozzle N . The chamber is then refilled with ink from an ink reservoir through the conduit L. In all the systems, the ink pressure in the chamber at rest is maintained slightly below atmospheric pressure. In the place where the nozzle opens to the air, a concave meniscus forms whose slight curvature counterbalances the negative static pressure, giving the meniscus a stable position within the nozzle. Stemme (1973) was the first to demonstrate how this negative pressure is
94
J. HEINZL AND C. H. HERTZ
S
ISI
P
/ /
C
N
L
N
(el
FIG.1. Different types of drop-on-demand systems containing nozzle N , ink chamber or ink channel C , ink conduit L, and piezoelectric transducer P or heating element H , excited by short electrical signals S. (a) Round bimorph piezoelectric transducer and ink supply at the nozzle (Stemme, 1973); (b) strip-shaped piezoelectric transducer (Kyser and Sears, 1972); (c) thin tube of piezo-ceramic surrounding a glass tube (Zoltan, 1972); (d) thick piezo-tube cast in plastic unit (Heinzl, 1975); ( e ) bubble jet using a heating element (Hara and Endo, 1981).
INK-JET PRINTING
95
created by ink coming from a reservoir some centimeters below the nozzles. Because of the slight vacuum, the rest position of the meniscus is stable, regardless of whether the plate surrounding the nozzle is wettable or not. Any ink that may be on the nozzle plate, when it comes into direct contact with the meniscus, is sucked back into the nozzle by the low pressure, thus making it self-cleaning. The vacuum method allows a great number of nozzle openings to be placed close together on the surface of the output plate without the need to separate them by any grooves or ledges. This stable rest position of the meniscus and the fact that the nozzle is self-cleaning have accounted for the marked progress in the drop-on-demand systems in comparison to other systems (Winston, 1962; Ascoli, 1968). If the meniscus is displaced, it will tend to resume its stable position. However, the restoring force is small and cannot be increased much, since it depends on the surface tension of the ink, the limiting angle between ink meniscus and nozzle material, and the geometry of the nozzle. Some of the drop-on-demand systems use these capillary forces to refill the nozzle shaft after the ejection of a droplet; others use them only for long-term stabilization of the rest position of the meniscus. This can also be seen from the different descriptions of drop generation. Kyser et al. (1981) describe in a lumped-mass model how drops are formed in a drop-on-demand system. The transducer first squeezes the ink channel and ejects a drop from the nozzle. When the transducer resumes its initial position, the channel regains its former volume, and a volume corresponding to one drop is now missing in the nozzle shaft. This ink is then replenished from the reservoir by capillary forces. In this description, the ink is treated as an incompressible fluid, and the elasticity is concentrated in the transducer. The capillary refilling of the nozzle has been examined and described by Beasley (1977), taking into account inertial forces, surface tension forces, and viscosity. Beasley also considers ink as an incompressible fluid. Pressure waves and delay time are therefore neglected in his treatment. In contrast, Arndt (1974) treated the propagation of pressure waves in the ink channel as an acoustical problem as early as 1974. He focused on the propagation of energy and the disturbing reflections at the end of the channel. He therefore described nonreflecting terminations of the channel. Acoustical models that consider the nozzle as a closed end of the channel with a constant factor of reflection predict fairly well the changes in drop velocity dependent on the drop rate (Manini and Scardovi, 1982; Hofer and Talke, 1982). An acoustical model that not only takes into account the transport of energy but also the transport of fluid by pressure waves has been given by Heinzl and others (1982). This model is described in greater detail in Section 11. In this model the nozzle is not considered as a closed end of the channel, but as a transducer that transforms pressure into ink velocity, and vice versa, and even stores kinetic energy. The reflection factor of pressure
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waves at the nozzle, then, is not constant but depends on the ink velocity in the nozzle shaft. The consideration of fluid transport by pressure waves shows that it depends on the matching of the nozzle with the ink channel, on the pressure history, and on how much ink remains in the nozzle shaft after drop ejection. Good matching makes capillary refill unnecessary. All the fluid transportation is then done by pressure waves. In a separate work, Rosenstock (1982) shows which kind of pressure impulse is optimal for drop shaping. He describes the pressure history that makes drop shaping fairly independent of surface tension forces. According to Rosenstock, the optimal pressure history is a sequence of low pressure, high pressure, and low pressure. First, the low pressure draws the meniscus from its rest position back into the nozzle shaft. Second, the high pressure accelerates the ink still within the shaft to flight velocity and makes it emerge from the nozzle at nearly constant velocity. The subsequent low pressure causes a rapid constriction of the escaping drop and brings the meniscus back to its stable rest position. In this way fast-flying drops whose kinetic energy greatly exceeds their surface energy can be produced. However, the movement of the transducer must be well controlled; only then can a well-defined pressure history be produced. Apart from that, as stated before, the nozzle must be well matched to the channel. In general, drop-on-demand systems use nozzle openings ranging from 30 to 100 pm, and the upper limit of the drop rate ranges from 2 to 10 kHz. An upper limit of 10 kHz requires either an ink channel with a nonreflecting end (Arndt, 1974), a compensation impulse at the transducer (Kasahara et al., 1981), or an adaptation of the steering impulses to what happened before. The latter is especially necessary for the first drops of a series (Hofer and Talke, 1982).Without such measures, about 3 kHz can be achieved with fairly uniform drop velocity. If it is not required that the meniscus resume its rest position after ejection of every drop, drop rates of 25 kHz and even far more can be achieved (Hofer and Talke, 1982). In this case, however, the quality of the first and the last drops of a series differ from those of the others. Hence, a single droplet can no longer be expelled on demand. In spite of the limited drop rates associated with intermittent or discontinuous operation, impulse jets have proved to be very useful for the printing of alphanumeric characters. Since each jet system is very small and simple, a number of these systems can be combined into one printhead that is moved across the paper, replacing, for example, an electromechanical wire-matrix printhead (Stemme and Larsson, 1973; Heinzl and Rosenstock, 1977; Zschau, 1978; Doring, 1981,1982). Many printheads of this kind, containing between 7 and 40 nozzles, have been developed. So far only one of them has proved its reliability, even in heavy-duty applications and in large numbers, and is now well established on the market (Heinzl, 1982).
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The first applications of such systems to the production of color prints have recently appeared on the market (Duffield, 1982). However, high-resolution office quality printheads are still unavailable, although they are being developed in several laboratories and have been made public occasionally. The number of pattern lines and, accordingly, the number of nozzles necessary in one printhead, ranges between 20 and 40 for office quality. The qualities of paper and ink and the target precision of the nozzles must meet high standards. For many applications, it would be most desirable to have a large number of nozzles arranged in a linear array extending across the width of the paper similar to some thermal printhead. During printing, this array could then remain stationary (instead of traveling along the line like a conventional printhead), requiring only that the paper be moved. For this purpose 800-2500 nozzles would have to be arranged in one or several closely spaced rows. Up to now, however, no method has been found to produce narrowspaced piezoelectric transducers in great numbers economically. A possible solution recently described by Hara and Endo (1981) and Vaught et al. (1984) makes use of an array of ink jets in which electrically heated thermal elements produce vapor bubbles within the nozzle shaft. When a bubble expands, it ejects an ink drop, following which it condenses again. Here the requirements that the ink must meet are rather complex, and it remains to be seen whether sufficient lifetime of the nozzles can be achieved. Compared with the continuous-jet systems described later, the drop-ondemand principle is less useful for printing on curved surfaces or where the ink droplets must traverse large distances. The velocity of the ejected drops is limited typically to a range between 2 and 5 m/sec and the direction of drop travel is hard to confine within close limits. The distance between the recording head and the paper is therefore normally maintained between 1 and 2 mm in practical applications. In concluding this section, the reader should be warned against the apparent simplicity of the impulse-jet principle. In practice, reliable operation is achieved only if a large number of parameters are carefully chosen and the physics of the process is well understood. This will be discussed in more detail in Section 11. 2. Continuous-Jet Systems In the drop-on-demand systems described earlier, the ink drops are accelerated essentially by the electric input signal itself. Since the acceleration of a mass is involved, the rate of droplet generation and the printing speed of drop-on-demand systems are limited. To increase the speed of operation, continuous ink jets, generated by forcing ink through a small nozzle, must be used. As will be shown later, the direction of these ink jets can be controlled electrically by charging the ink drops of the jet. Since the drops can be charged at a much greater rate than they can be generated by acceleration,
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they can be controlled ten to a hundred times faster in continuous-jet systems than in drop-on-demand systems. However, continuous-jet systems consist of more parts and usually consume larger amounts of ink since the jet operates continuously even when ink is prevented from reaching the paper. a. The Sweet Method. It is well known that a fluid jet produced by forcing liquid through a nozzle breaks up into a train of drops shortly after leaving the nozzle. This is due to instabilities occurring on the surface of the jet (Rayleigh, 1878).This spontaneous drop-formation process can be influenced by inducing mechanical vibrations on the jet using, for example, a piezoelectric transducer. If the frequency of the mechanical vibrations is made approximately equal to the spontaneous drop-formation rate, the dropformation process is synchronized by the forced mechanical vibration and therefore produces ink drops of uniform mass (Savart, 1833; Magnus, 1859; Sweet, 1964; Schneider and Hendricks, 1964; Mansson, 1971). Furthermore, it is known that at the point of the formation the drops can be electrically charged. This can be accomplished by establishing an electrical field between this point and an external electrode, provided the liquid has sufficient conductivity so that it remains at the potential of the nozzle (Hendricks, 1962; Sweet, 1964). Sweet (1964, 1965, 1971) used this effect to develop a high-speed ink-jet recorder, and Lewis and Brown (1967) extended the use of Sweet’s concept to alphanumerical printing. The principle of this important development is shown in Fig. 2 (Stone, 1978). Here a high-speed, continuous jet is produced by forcing ink under high pressure from a pump through a nozzle. Shortly after leaving the nozzle, the jet breaks up into drops close to the control electrode E . A piezoelectric crystal P, attached to the ink conduit, induces mechanical vibrations on the continuous part of the jet, thereby ensuring that all drops formed by the jet have the same mass. Now, if an electrical voltage is applied to the control electrode E positioned at the point of drop formation while the ink flowing through the conductive conduit is kept at ground potential, a charge is induced on the drops. Provided the conductivity of the ink is large enough, this charge is roughly proportional (but of opposite sign) to the voltage applied to the control electrode E . After leaving the control electrode, the drops traverse an electrical field generated by applying a voltage H I / of several kilovolts between the two deflection electrodes D, and D , . In this field the drops are deflected in proportion to their charge. Thus, their leading position on the recording paper is displaced in the vertical direction by an amount proportional to the signal voltage applied to the control electrode E . If this signal voltage changes with time, it can be recorded as a trace on the continuously moving recording paper.
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FIG.2. Principle of a character printer using the Sweet method. A continuous-ink jet breaks up into droplets inside the control electrode E . Due to mechanical vibrations generated by the piezoelectric ceramic P, all droplets have equal mass. Each droplet can be charged individually by applying a suitable control voltage to the electrode E at the moment the droplet is formed at the point of drop formation. Depending on this charge, the droplet will be deflected in the transverse electric field between the electrodes D, and D , , thus allowing the formation of characters.
Sweet (1964) has shown that signals with frequencies up to 5 kHz can be recorded by this method. By adding a gutter G into which undesired drops can be deflected, characters can also be printed as indicated in Fig. 2 (Lewis and Brown, 1967). For low character quality (5 x 7 matrix), printing speeds up to 1500 ch/sec have been obtained. If higher character quality (more dots per character) is desired, the printing speed must be lowered accordingly for reasons given in Section 111. While Sweet’s system has not been commercialized for recording electrical signals, the character printing ability of this system has been used widely (Stone, 1978). In such systems, ink jets having a diameter of 30 to 80 pm and traveling with a speed of 10 to 20 m/sec are commonly used. The drop-formation rate, controlled by mechanical vibrations, is in the frequency range of 50 to 120 kHz. To avoid electrical breakdown, the deflection voltage cannot be made much higher than 5 kV, thus limiting the character size to 4-10 mm. The length of the deflecting electrodes is about 5 cm, and they are spaced at a distance of 5 mm from each other. Again, while the principle described here seems to be relatively simple, a large effort was required to develop a reliable apparatus. Difficulties such as nozzle clogging, problems with synchronizing the drop-formation process with the signal voltage, and the influence of previously charged drops on the charging of new drops at the drop-formation point were hard to overcome in the earlier years of ink-jet printing. However, by now, character printers
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using the Sweet principle are used in many applications and have proven quite reliable. A more detailed discussion of this method will be given in Subsection III.C.3. b. The Hertz Method. The ink-jet method developed at the Lund Institute of Technology (usually called the Hertz method) has many similarities with the Sweet method since it also uses a continuous jet, the drops of which are charged by a control electrode (Hertz et al., 1967; Hertz and Minsson, 1972, 1974). However, it provides intensity modulation of the jet instead of deflection. Furthermore, the diameter of the jet nozzle is smaller (10-20 pm) and the velocity of the droplets is higher (-40 m/sec). Figure 3 illustrates the mechanism used in this method. A fine jet of ink J is produced by forcing ink under high pressure through a nozzle N . Directly after leaving the nozzle, the ink jet traverses a ring-shaped control electrode E , which is connected to a voltage source V. The other side of the voltage source is in electrical contact with the electrically conductive ink in the tube supplying ink to the nozzle. If this voltage is zero (Fig. 3a), the jet travels in a straight line toward the moving recording paper where it produces a trace. However, if a voltage V of sufficient magpitude is applied between the electrode E and the conducting nozzle N (assuming the use of conductive ink), it is found that the ink jet is transformed into a spray of fine droplets as shown in Fig. 3b. When the ink jet is dispersed in this manner no recording trace is obtained on the paper. Thus, by controlling the voltage of the electrode E in a suitable way, the ink reaching the paper can be switched on and off, resulting in an on-off intensity modulation of the trace. The spray phenomenon described here results from the fact that a fluid jet spontaneously breaks up into a train of drops shortly after leaving the nozzle.
FIG.3. (a) In the Hertz method, an ink jet is formed by forcing ink under high pressure through the fine nozzle N . This jet can be transformed into a spray of small droplets by applying a suitable voltage V to the control electrode E as shown in (b). In that case no trace is obtained on the moving paper web. This allows an intensity modulation of the trace.
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If the electrode E is placed near this point of drop formation and a suitable voltage is applied to the electrode, the tip of the jet will be charged as well as the drops formed at the tip as in the case of Sweet’s arrangement. However, if the control voltage is high enough, these charges give rise to a strong repulsion between the drops, which transforms the jet into a spray a few millimeters from the point of drop formation as shown in Fig. 3b. The preceding phenomenon is best demonstrated with a very fine and high-speed jet, having a diameter of 10 to 20 pm and a velocity of 40 m/sec. Such a jet can travel about 6 cm through air without an appreciable increase in diameter. If a voltage of more than 1OOV is applied to the control electrode, the drops of the jet will depart from the jet axis increasingly with the voltage. At 500 V the jet is entirely dispersed into a well-defined spray. The solid angle formed by this spray is voltage-dependent and will increase to about i n s r at lOOOV. In most cases no disintegration of the droplets by internal electrical forces takes place (Fillmore et al., 1944). If short voltage pulses of 500 V are applied to the control electrode E , the jet will turn into a spray each time the voltage is switched from 0 to 500 V, thus disrupting the recording trace generated by the jet on the moving paper web. For a 10-pm jet traveling at 40 m/sec, the upper frequency limit for this on-off modulation lies well above 500 kHz. This surprisingly high frequency limit makes the method useful for many graphic applications. In this system shown in Fig. 3, the spray may still reach the paper and generate a colored background that seriously decreases the quality of the recording. Two simple ways to prevent this are indicated in Fig. 4. In Fig. 4a, an aperture is placed in front of the recording paper, which intercepts the fine
(a)
(bl
FIG.4. Suppression of background generated by spray. (a) Aperture in front of the recording paper intercepts most of the spray, but allows the undisturbed jet to pass. (b) Charged drops of spray are deflected by a transverse electric field, while uncharged drops of linear jet pass through the field undisturbed.
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droplets of the spray while allowing the undisturbed jet to pass through the small orifice in the aperture plate. This scheme actually allows an intensity modulation of the recording trace since the included angle of the spray will change with the voltage level applied, therefore controlling the number of spray droplets that pass through the orifice. The aperture in Fig. 4a, however, will always allow some drops to pass through and create a slight background. This can be avoided by using an electric deflection field as shown in Fig. 4b. Since the drops of the spray are electrically charged, they can all be deflected and prevented from reaching the recording paper. The uncharged drops of the undisturbed jet, on the other hand, can traverse the field without being affected. This method of background suppression is much more effective but allows only an on-off modulation of the trace. Before the systems shown in Fig. 4 can be applied to alphanumeric printing or facsimile devices, however, the geometry of their electrode systems must be simplified as described in Subsection III.C.4. As opposed to the Sweet method, the Hertz method allows only the intensity modulation of the recording trace. However, the upper frequency limit of the Hertz method is higher and no piezoelectric element is required to control the drop-formation process. Also, it is not necessary to synchronize the drop formation with the control signals since the drops of the jet are so small that single misplaced drops on the paper are not visible to the naked eye. In general, the Hertz method thus seems to be simpler than the Sweet method except for the fact that the smaller nozzle diameter increases nozzle clogging problems. 3. Electrostatic Ink-Jet Formation In the early years of ink development, it was shown that a continuous jet of ink can be produced by a strong electrostatic field acting on the fluid meniscus protruding from a relatively large (150-pm diameter) nozzle (Richards, 1952; Winston, 1962; Dunlavey, 1972; Ascoli, 1968; Ascoli et al., 1975; Mutoh et al., 1979). It could be shown that these jets consisted of electrically charged drops of approximately equal size and charge. Such a stream of drops could be deflected by electrical signals that were applied to deflection plates along the jet path in a manner very similar to the x-y deflection of an electron beam in a cathode ray tube (CRT). By using suitable electrical signals on the deflection plate, alphanumeric characters could be traced by the jet on a recording paper at a speed of up to 50 ch/sec (Kolb, 1966). However, this method proved to be insufficiently reliable, and only limited use has been made of it in typewriters or printing heads. The unsatisfactory reliability is caused by the difficulty in limiting the ink meniscus to the inside of the nozzle only. If other parts of the nozzle are also
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wetted by the ink, the size of the meniscus varies. This changes the drop generation process and thereby the drop size and the deflection of the drop stream in the deflection field, thus making the printhead nonoperative. The wetting of the outer parts of the nozzle is probably due to small satellite drops that are generated during the formation of the print drops (cf. Section III.A.3). These satellite drops tend to form a fine mist that is attracted to the nozzle due to the electric charge of the drops. Eventually this results in the wetting of the entire nozzle, thus causing breakdown of the printing operation. Work on this otherwise elegant method was, therefore, discontinued around 1975 and will not be described in more detail in this chapter. 4. Methods Employing Mechanical Valves A very simple control of a continuous ink jet can be obtained by the use of an electromechanical valve in the ink conduit to the nozzle. This approach has gained in importance since valves were designed that operate at frequencies of 100 Hz and above. In spite of its relatively low speed and large nozzles, this method has proved to be very useful for printing large characters on boxes, cartons, and similar packages as well as for producing designs on carpets.
C. Present Fields of Application f o r Ink-Jet Printing
Ink-jet printers have already become established in a number of commercial applications. Some of these application areas are briefly outlined together with the print samples produced. To allow the assessment of the print quality, each sample is magnified as indicated by the l-cm bar included in all the figures. For technical reasons, colored prints (Figs. 6-8) are reproduced in black and white only. a. Computer Terminals. Since ink jets are noiseless, fast, and directly controlled by electrical signals, they are ideally suited for computer output devices such as terminal printers and teletypes. Since the requirements here on character quality are relatively low and high speed is desirable, printing heads consisting of 7-24 closely packed drop-on-demand systems that are moved along the platen during printing are used. As an example, Fig. 5 shows the printhead of a successful commercial printer. The head contains nine nozzles of the cylindrical piezoelement type in Fig. Id (Heinzl and Rosenstock, 1977). The printhead moves in front of a platen similar to that of a conventional mechanical typewriter. A sample printed with a speed of 150 ch/sec is also shown in the figure. A similar teletype printer using 12 piezoelements printing at the rate of 270 ch/sec has been marketed for several years by the
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FIG.5. (a) Printhead containing nine drop-on-demand systems used in the Siemens PT88 low-cost alphanumeric computer terminal. (b) Magnified view of print sample produced at a speed of 150 ch/sec.
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same producer. Several other companies have introduced similar products on the market with somewhat varying characteristics and reliability. By increasing the number of drop-on-demand systems in the head, a character quality close to that of conventional typewriters seems to be obtainable if reliability problems can be overcome. During the last few years, color CRT displays have found increasing use as output devices for computer graphics. However, there are still no simple printers that generate a hard copy of the CRT image in color on paper. Ink jets should be an ideal answer to this need, and the very first color ink-jet printers appeared on the market during the latter part of 1982. They use a 12nozzle printhead similar to the one shown in Fig. 5 ; however, four of the 12 nozzles in the head are supplied by magenta, four by yellow, and four by cyan ink. The head is driven back and forth at right angles to a slowly moving continuous paper web (Duffield, 1982), and the nozzles are activated by electrical signals suitably delayed with respect to each other. Figure 6 shows the resolution obtained with such a color printer. In another color printer recently introduced for the reproduction of computer color displays, the paper is mounted on a rapidly rotating drum with four impulse jets moving slowly parallel to its surface.
FIG.6 . Magnified print sample obtained with a three-color plotter for computer graphics. The plotter uses a modified version of the drop-on-demand printhead shown in Fig. 5a and is produced by Advanced Color Technology, Inc.
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FIG.7. (a) High-resolution four-color ink-jet plotter using the Hertz method for computer graphics. During the plotting operation the drum rotates with a surface speed of 5 m/sec. A carriage containing four computer-controlled ink jets with colors magenta, yellow, cyan, and black, simultaneously moves parallel to the drum axis. In the figure the carriage is hidden by the drum (Photograph courtesy of Applicon, Inc.). (b) Print sample showing resolution obtainable.
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Large-size ink-jet color plotters (up to D-sized paper) for computer graphics have been on the market for several years. In these drum plotters the Hertz method is used because of its speed and high resolution. Figure 7a shows the printer itself while Fig. 7b is a highly magnified photograph of a print, which demonstrates the resolution obtainable. Due to their geometrical form, drop-on-demand systems can be mounted closely to each other. This is especially true for the bubble-jet system described in Fig. le, which can be produced on a silicon substrate by methods developed for integrated circuit production. With this technique, arrays of 128 bubble-jet nozzles have been produced on one chip with a nozzle density of 12 nozzles/mm. Using four such arrays with the colors magenta, yellow, cyan, and black, color prints could be produced at a rate of one letter-sized page in 90 sec. Gray scale or color shades were obtained by the digital halftone method using groups of four adjacent nozzles to print one pixel. In this way 64 shades could be produced in each color at every pixel of the image, the pixel density being 3 pixels/mm. A magnified view of such a picture is given in Fig. 8. b. Document Printers. In certain fields of office activities, large numbers of forms must be marked with serial information stored, for example, on a
FIG.8. Magnified sample produced by a bubble-jet color plotter for computer graphics. The plotter produces color shades by controlling the dot size in each pixel. One pixel is produced by four adjacent jets; jet density = 12 jets/mm.
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computer tape. Examples of this are the printing of names on tax forms, the completion of personalized advertisements, and the printing of numbers on lottery tickets. Since the information to be printed changes with each form, conventional printing methods cannot readily be used. For such applications, ink-jet systems of the Sweet type have proved to be very effective. An extreme example of this is an ink-jet printing press built with about 1000 closely spaced by individually controlled Sweet ink jets that can print entire documents at high speed on a paper web under computer control. The Sweet method has also been used in a fast word processor for the electronic office (Buehner et al., 1977), the very high print quality of which is demonstrated in Fig. 9.
c. Printing Codes and Routing Information. In many printing situations, printing on curved or rough surfaces is required. In addition, frequently personalized information must be printed in which the text of every printed unit changes. Typical examples are the printing of optically readable bar codes or alphanumeric text on checks prior to automatic processing, printing of addresses on letters or newspapers, and the marking of packaged industrial products with a production date or serial number (Fig. 10) (Stone, 1978; Stolzenburg, 1982). Here conventional impact printers obviously cannot be used. Drop-on-demand jets also cannot be employed since usually high printing speeds as well as printing distances, as large as 2-5 cm, are required. In view of this, only continuous-jet methods have been used for these tasks. A print sample is shown in Fig. 11. d. Facsimile Printers. Ink jets should be ideally suited for facsimile receivers because of their high printing speeds. Figure 12 shows a magnification of a print produced by such a machine. One page of A4 size can be printed in two minutes using a printhead with six nozzles such as shown in Fig. 5, which is somewhat modified for the purpose. However, due to the slow growth of
FIG.9. Print sample of IBM word processor 6640 using continuous-ink jet controlled by Sweet method. Print speed = 80 chisec.
INK-JET PRINTING
FIG.10. Beer can with ink-jet information printed on bottom by A. B. Dick printer.
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FIG.11. Sample of an address printer (A. B. Dick). Maximum printing speed
=
1500 ch/sec.
FIG.12. Copy produced by drop-on-demand ink-jet facsimile transceiver (Siemens HF 2050). Print-out time is 2 or 3 min per petter-sized page complying with CCITT recommendation (group 2).
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this field in Europe and the U.S., only limited commercial use of the drop-ondemand principle has been made for this application so far. However, because of the usefulness of facsimile transmission for Kanji character documents, the development of this area will probably be spearheaded by companies in Japan. e. Special Purpose Printers. Finally, some ink-jet processes have been developed for specialized applications such as the printing of patterns on carpets and textiles (Taylor, 1980), as well as the production of large posters in color (Suenaga, 1971). Most of these processes use mechanical valves as a means for jet control and are, therefore, relatively slow and have a resolution of one line pair/mm or less. However, since the ink flow in these systems is more than 100 times as large as in the electrically controlled jets described in Section I.B.1-3, areas as large as 4 x 3 m can be covered in about 1 h using a drum printer. In carpet printing, this time has been decreased appreciably by using an array of closely spaced, relatively large nozzles in each color, which decreases resolution. Mechanically controlled valves are also used in printheads for the marking of containers and other large goods with characters about 2 to 10 cm high. 11. DROP-ON-DEMAND METHODS A . Principle
Ink printers that eject drops on demand have much in common with mechanical wire-matrix printers, the so-called needle printers. In both devices, an electrical signal causes ink to be transferred from a supply or reservoir onto paper and to be fixed there. The ink is transferred over a distance of 1 to 2 mm to the paper and the printhead must be moved in relation to the paper so as to ensure the right spacing of the dots on the paper. In wire-matrix printing, the ink reservoir consists of a textile ribbon that holds ink on and between its fibers. This “reservoir” is not protected from the surrounding air. Accordingly, the ink must be nonvolatile to prevent it from drying. By means of a mechanical impact against the ribbon and paper, the ink is squeezed out of the ink ribbon, impressed, and rubbed into the paper. The energy needed by the impact mass to fix one dot on the paper amounts to W sec. This is the difference in kinetic energy of the moving mass before and after the impact. Ink printing with drops on demand is an effective method that reduces this energy drastically. In this case, the energy necessary to form an ink drop that produces a dot is about lo-* W sec. This energy is equal to the surface and kinetic energy of the drop.
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A drop of 100-pm diameter produces a dot of about 300 pm on the paper. g; to make it fly at a typical speed of 5 m/sec, 0.75 x Its mass is 0.6 x lo-* W sec of kinetic energy are necessary. Since the viscosity of the ink (lo-’ N sec/m2) is smaller than the viscosity of the ink in the ribbon by a factor of losses due to viscosity occur only within the nozzle and are of the order of lo-’ W sec for one drop. Since the paper absorbs the ink by capillary forces (Bechtel et al., 1981), the mechanical pressure used in wirematrix printing is not necessary. In wire-matrix printing, mechanical energy is transmitted from an electromagnetic transducer to the tip of a print needle by the motion of the needle. The electrical energy required to print a dot is about lo-’ W sec, approximately one-tenth of this energy being dissipated at the tip of the needle in striking the ribbon. The remainder is transformed into heat, together with a tiny, but especially annoying part that is transformed into noise. Since the wire is stiff and inelastic, as well as carefully guided to avoid loss of energy by buckling, only the kinetic energy of the wire is of importance. In ink-jet printing, the ink itself serves to transmit energy from the transducer to the nozzle. In most cases, piezo-ceramics are used for the transducer. To print one dot, the transducer needs an electrical input energy of 10-7-10-s W sec. The percentage of power not consumed for ink transfer is greater than in needle printing. However, since the total energy dissipated per dot is smaller than in needle printing by a factor of the heat generated and the noise emitted are of no importance. A comparison of the electrical and mechanical energies required to produce a single dot on paper is shown in Fig. 13. If we compare ink as a means of energy transmission to the wire in the needle printer, one essential difference becomes apparent. The energy transmission between transducer and nozzle orifice cannot be described adequately if the ink is considered to be an incompressible fluid. Owing to its compressibility, ink can store energy in the form of pressure waves. Therefore,
&jg$
Wire Printer
Drop on Demand I
;f9
I
I
1
II
1
W sec FIG.13. Electrical and mechanical energy used to make one dot on the paper. E,, stands for the electric energy supplied to the transducer. Em stands for the mechanical energy involved in the printing process, i s . , the kinetic energy of the flying needle consumed in the impact process (respectively, the kinetic energy and the surface energy of the flying droplet).
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113
if we want to describe energy transmission between the transducer and nozzle, it is necessary to consider the propagation of these waves within the ink. This is especially true if we want to understand the function of the nozzle. Before the entire system of drop generation and its limitation can be described, it is necessary to study the generation of pressure waves by the transducer, their propagation within the ink in the channel between the transducer and nozzle, and their transformation into velocity within the nozzle. In this connection, not only the transportation of energy but also the transportation of fluid by pressure waves will be focused on to make the mechanism of nozzle refill understandable.
B. Transformation of Electrical Pulses into Pressure Waves The transducer forms part of the wall of the ink cavity that leads to the nozzle as shown in Fig. 1. In most cases, piezo-ceramic is used as transducer. Of course, a number of other physical effects, such as the electrostatic or electrodynamic, electromagnetic or magnetostrictive effect, could be used as well. However, with the exception of the bubble jet (Hara and Endo, 1981), so far only drop-on-demand systems using piezo-ceramics for this purpose have been described. A piezo-ceramic plate alters its thickness up to 0.2% if an electric field is applied between its opposite surfaces. At the same time, transverse contraction or expansion alters its length and width up to 0.1 %. With the materials known today, electrical fields that might cause larger deformations lead sooner or later to a depolarization of the material, resulting in loss of piezoelectric effect. These small changes in dimension can be made to work directly on the fluid. Alternatively, they can first be magnified mechanically. In particular, if the piezo-ceramic plate is laminated on one side to a metal plate, the resulting compound plate bends under the influence of an electrical field, just as a bimetal bends under the influence of heat. Stemme (1973) was the first to use round plates made of piezo-ceramics and metal for producing drops on demand, while Kyser and Sears (1972) were the first to use rectangular plates. Other arrangements use a piezo-ceramic transducer in the form of a tube as first demonstrated by Zoltan (1972). In tubes, the inner and the outer surfaces are electroded, and the tube encloses the ink channel. Subsequently, Heinzl(l975) designed the tube so that only the inner-not the outer-diameter changed when an electric field was applied. This made it possible to fix such tubes in a rigid cast without affecting the changes of the inner diameter. Assuming that the wall is not too thick, this can be shown by the following considerations. If the tube is unrolled into a plane, a rectangular plate of thickness h, width b, and length 2rcrm,where rm is the mean radius, is obtained. The inner radius of the tube can then be described as ri = rm - (h/2),
114
J. HEINZL AND C. H. HERTZ
and the outer radius as ra = rm + (h/2). The piezo-effect affects the thickness h, and rmis deformed by the transverse contraction. Let the radial mechanical tension that occurs in the wall of the tube immediately after a step in the electric field be cr,the modulus of elasticity of the ceramics E and v, the Poisson number. We then have
Ara = (Ah/2) + Arm = [(h/2) - vr,]cr/E Now if h = 2vrm, there is no change in the outer radius. On the other hand, the change of the inner radius will be doubled. In Fig. 14 the various types of transducers are shown. The round piezoceramic as well as the strip-shaped transducer can be arranged so that the ink is separated from it by the metal layer. In Fig. 14c a glass tube separates the ceramic from the ink, the ceramic being on the outside. The outer radius of the ceramic can change without inhibition, while changes of the inner radius deform the glass tube, thus applying force to the ink. In Fig. 14d the outer surface of the tube is fixed. The four arrangements shown here differ primarily in efficiency of energy conversion. In Figs. 14a and 14b the piezo-ceramic can be well matched to the compressibility of the ink. Although the metal plate must be deformed along with it, thin foils of piezo-ceramics can be used. Hence the necessary field strengths can be obtained with low voltages. The electric power neces-
FIG.14. Different arrangements of piezo-ceramic used as transducer: (a) round disk, (b) strip, (c) thin tube sustained at the inner surface, and (d) thick tube sustained at the outer surface.
INK-JET PRINTING
115
sary to charge the load admittance of the transducer is transformed into hydraulic power with high efficiency. Without the ink acting as a load, the motion of the transducer increases remarkably. Thus, a force is impressed upon the ink. The efficiency of the squeeze-tubes is not as good. Compared with the enclosed ink, they are more rigid by approximately a factor of 10. If the tubes shown in Fig. 14c are thin-walled, the change in circumference due to transverse contraction dominates, whereas the direct expansion hardly contributes to the power transmission. The glass tube and not the ink is the main load; thus, the necessary voltage is higher than with well-designed plates. In Fig. 14d the walls of the transducer tube are especially thick; therefore, the required driving voltage is even higher. The mismatch here is particularly striking. Compared to the compliance of the tube, the compliance of the enclosed ink is negligibly small. The motion of the tube is not hindered by the ink and is nearly the same whether there is ink or not. Unlike the transducers shown in Figs. 14a and 14b, this transducer generates a change of volume in the ink cavity. Due to the mismatch between the transducer and the ink, transients in the ink are not fed back into the transducer. The various transducers shown in Fig. 14 differ greatly in their vibration modes. The round plates are very soft (Doring, 1982) and can be compared to jumping sheets as used by fire brigades to save people. They have relatively low resonant frequencies typically between 5 and 10 kHz. Thus, fast rises in the electrical pulses are not transformed. The ink damps the system well, and long rectangular plates are considerably stiffer. Apart from their thickness, their width is their characteristic dimension. The interaction between the plate and ink is described by Kyser (1981) by a coupled system of equations. The resonance frequencies of the oscillating tubes on the other hand have been calculated by Armenakas et a/. (1969). When oscillating longitudinally, the tube has its lowest resonance frequency. This mode is, however, strongly inhibited by the surrounding cast as seen in Fig. 14d. A typical resonance frequency in this mode is 0.2 MHz. The circumferential mode as well as the thickness mode is used to interact with the ink. These modes are not inhibited by the cast. Since the resonance frequencies of the transducer shown in Fig. 14d are, for example, 0.6 MHz and 3 MHz, respectively, it is able to transform steps with a rise time of 1 psec or less directly into changes in volume of the enclosed ink. These changes in volume are superimposed onto the transients previously caused by the transducer within the ink cavity. C. Wave Propagation in the Ink Channel
If the diameter of the ink channel is small compared with its length, the channel can be described as a one-dimensional waveguide. Unless the ink channel alters its cross section and unless the channel wall alters its stiffness,
116
J. HEINZL AND C. H. HERTZ
the pressure steps created by the transducer run along the channel undisturbed and at sound velocity. Such a channel enclosed in a transducer tube is shown in Fig. 15. Independent of channel shape, pressure differences within the channel disappear by transient processes. Rosenstock (1982) describes the pressure history at a cross-section D of the channel. In response to a voltage step at the transducer, a pressure wave of the length of the transducer tube passes cross section D. Since the wave propagates in two directions, its amplitude is one-half of that impressed onto the ink by the transducer at the moment of deformation. If we describe any given time dependence of the voltage applied to the transducer by infinitesimally small steps, we can calculate the pressure signal p e ( t ) at the position D as the sum of infinitesimally small step responses. where t , = t - (a/c) and t , = t - (a/c) - (b/c). Here a is the distance from the transducer to the position D,b the length of the transducer, c the velocity of pressure waves in the channel, U ( t ) the voltage signal applied to the transducer, and k a constant factor of the dimension N/m2 V, which describes the transduction of voltage into pressure. The movement of the pressure waves in the channel can be calculated by the methods of acoustics, for example, by using the concept of impedance (Kurtze, 1964) in which the acoustical impedance is given by 2 = pc/A, where A is the cross-sectional area, c the velocity of the pressure waves, and p the density of the ink. As long as the acoustic impedance along the channel does not change, the direction and amplitude of the wave motion remain undisturbed. However, changes of impedance generate reflections in which part of an incident wave is reflected and returns while the remainder continues to propagate in the forward direction as shown in Fig. 16. The pressure wave propagation in a onedimensional waveguide can be described well by the method of finite channel elements whereby the channel is divided into small sections, so that a pressure wave can pass one of them in the time interval At. These sections should be chosen so that steps of impedance coincide with the boundary between two sections. Again, the pressure wave introduced into the channel is assumed to be in the form of a sequence of steps. The intervals between these steps should be integer multiples of At. Thus, we can ensure that after time multiples of At, all pressure steps reach the boundaries between the sections. The pressure within every section is then described by two pressure waves moving in opposite directions. Figure 17 shows an example of a channel that contains one change in cross section and the description of the pressure waves in the finite channel sections. Figure 17c explains how Fig. 17d should be read. In Fig. 17c, the time course of an impulse of length At is represented. In order to obtain the response to a pressure step, a sequence of pulses is superimposed.
t=t,
t
rz
t
FIG.15. Propagation of a pressure wave in response to a step in voltage. (a) At the time t,, the transducer tube has narrowed down the ink channel and compressed the enclosed ink. (b) At the time t,, the transient process has already begun; two pressure waves of the length of the transducer run along the channel in opposite direction. (c) Time-path diagram of the pressure waves. (d) Pressure history at cross section D. After a delay time ajc, a rectangular impulse of duration b/c arrives at cross-section D a s a response to a step in voltage at the transducer.
118
J. HEINZL AND C. H. HERTZ
; r
z
\ t
47 12
FIG.16. A pressure wave is split up due to a change of impedance: p. is the amplitude of the arriving pressure wave; pr the amplitude of the reflected pressure wave; pd the amplitude of the passing pressure wave; r l , the reflection factor at the transition from section 1 to section 2 of the channel; and Zi the acoustic impedance of section i. R , , = p,/p, = ( Z , - Z , ) / Z , + Zl), pd =
P.
+
PC,
Z,
= ~ici/Ai.
By this method the response in Fig. 18a is obtained. This procedure is particularly suitable for computer calculations. Figure 18b shows the step response when the cross section is altered in two steps, and Fig. 18c shows the response when the number of impedance steps caused by equal changes in cross section is infinitely high. It can be seen that the step response is no longer linear as soon as there is more than one change in cross section. With increasing number of cross-sectional changes, the pressure propagation becomes more diffuse. Ink channels that have no changes in cross section between transducer and nozzle, and a significant change in cross section between transducer and ink reservoir can easily be described. Such a configuration is shown in Fig. 19. For this configuration, the pressure signal p , ( t ) at the position D is given by Pe(t> =
kU(t1) - KU(t,)
+ R,,KU(t,)
-
R,,KU(tJ
(2)
where t z = t - (a/c) - (b/c) t , = t - (a/c) t , = t - (a/c) - (b/c) - ( 2 d / ~ ) t , = t - (a/c) - (2b/c)- (2d/c) and R , , is the reflection coefficient at the entrance to the reservoir. The pressure signal at the position D that is initiated by the transducer in a previously quiescent ink channel can be described by Eq. (2) for a time interval of
119
INK-JET PRINTING
..
2 '
I
-2
t1
2 '
I
w
FIG.17. Method of finite channel elements used to determine the response to a pressure step occurring at a change in cross section: (a) the arriving pressure wave before it reaches the change in cross section, (b) the reflected and the passing pressure wave, (c) the time-path diagram of a pressure wave of the length Az and its split-up, (d) the finite channel elements and the pressure wave in simplified representation. Here A and B are two cross sections used for the observation of the pressure waves.
120
J. HEINZL AND C. H. HERTZ
I I
P, t-
I*
I
- t
FIG.18. Response to a step in pressure occurring at one or more changes in cross section: (a) single-stage change; (b) two-stage change; (c) continuous change. Here p . and pr are seen at cross section A , and pd is seen at cross section B.
+ +
2b 2d)/c without superimposition of reflections from the nozzle. If several steps of impedance occur, such a simple description is not possible. In those cases, the method of finite channel elements can be used. The transfer function of the configuration of Fig. 19 can also be described in terms of frequency. If a sinusoidal voltage is used as a test function, Eq. (2) leads to the transfer locus and to the transfer function shown in Fig. 20. Evidently, the transducer is not able to generate dc pressure components. Therefore, we would expect the need for some sort of valve somewhere in the ink channel-valves that open and close to allow a flow of ink only in the direction of the nozzle. For this purpose, Suga and Tsuzuki (1982) suggested fitting mechanical microvalves in the ink cavity in front of and behind the transducer. Since, however, drop-on-demand systems actually work without mechanical valves, we must look for some directional effect in or in front of
(3a
INK-JET PRINTING
D
R,,
121
pet
Z
FIG.19. (a) Ink channel with an open determination closely attached to the transducer. (b) Pressure history in place D as a response to a step in voltage at the transducer.
the nozzle. But where, then, is the equivalent of this rectifier valve? Does the nozzle function in a manner similar to a vibrating chute, which transports objects by moving quickly forward and then slowly backwards, providing nonlinearity as a result of the difference between static friction and sliding friction? Doubtless, capillary forces generate an effective nonlinearity at the nozzle. So we could think of breaking through the capillary forces by means of a short pressure pulse, thereby ejecting a drop, and thereafter by using a slight negative pressure for a long period, which does not exceed the capillary forces. One-directional effect can be found in the nozzle, if we consider the kinetic energy of the ink in the cylindrical element comprising the nozzle and especially the change of energy with time. With the ink mass m of this element, the velocity u, the radius of the nozzle rD, and the length of the ink mass within the nozzle I, the kinetic energy is &in
= $nU2
= ip7Crk/V2
Since the length 1 of the ink volume in the nozzle shaft may change with time and u = i, as long as no ink leaves the nozzle, the time derivative of the kinetic energy is Ekin= +p7~rhu(21d u 2 )
+
Here lies a possibility of using the nozzle as a valve with directive effect. There is a second, even stronger directional effect, which cannot be avoided even if we wanted to. Ink can be accelerated or slowed down by tractive forces only as long as it has not left the nozzle. Once the ink is outside the nozzle, its motion can only be influenced by surface tension or external flow resistance. A description of the directional effect of the nozzle based on the propagation of pressure waves is given by Heinzl et al. (1982) and will be illustrated in the following sections.
122
J. HEINZL AND C. H. HERTZ
0
30
60
90
120 f/kHz
(bl FIG.20. (a) Transfer locus and (b) transfer function of the arrangement according to Fig. 19 for specific data.
D . Transformation of Pressure Waves into Velocity In the following process, the nozzle diameter is assumed to be much smaller than that of the ink channel behind. For the sake of simplicity, the nozzle in Fig. 21 is represented as a cylindrical opening of length s with diameter 2r, in front of a cylindrical ink channel of diameter 2r,. The length of the ink volume as long as it is within the nozzle cylinder is 1. From the acoustical point of view, the nozzle is a configuration with two large steps of impedance
INK-JET PRINTING
123
FIG.21. Nozzle and wave propagation within the nozzle shaft. (a) Dimensions of the nozzle;
(b) response of the nozzle to a pressure wave coming from the channel.
located close to each other. The first step is the change in cross section between the channel and the nozzle. The second step, at the distance 1, results from the sudden change in density and sound velocity at the boundary between the ink and air. In operation, an incident pressure wave is reflected almost completely at the rear of the nozzle. The sum of the arriving wave and the reflected wave is then propagated into the narrow nozzle cylinder. At the ink-air phase boundary, for example, at the meniscus, the wave is again almost completely reflected, but with a change of sign. The negative phase of the pressure wave now runs back along the nozzle shaft until it meets the widening of the cross section where it is again reflected with a change of sign, etc.
124
J. HEINZL AND C. H. HERTZ
In the above process the arriving pressure wave transports fluid from the transducer to the nozzle. The pressure wave within the nozzle cylinder brings fluid to the meniscus and runs back as a negative pressure wave, thereby transporting additional fluid toward the meniscus. These reflections now follow each other at close intervals and superimpose on other parts of the pressure wave that arrive before or after. And since all the waves that run back and forth transport fluid in the same direction, a finite fluid velocity v occurs in the nozzle shaft. The increase of this velocity can be understood by using the method of finite channel elements. In general, the following equation applies for fluids: Ap
= (AV/V)k,
with
k = c2p
where k is the compressibility factor, V the volume, and AV the change in volume. Since the pressure wave Ap moves with the velocity c, the following equations apply:
( A V / V ) c = Av
and
Ap = Avcp
On the arrival of wave p e at the meniscus, its velocity u is 0. The velocity now increases in steps of
APlCP at intervals of At
=
= 2PJC
l/c. Assuming that u(0) = 0, we get fi
= Av/At = Ap/pc At = 2pe/pl
At the same time, the wave reflected from the back wall of the nozzle becomes weaker. Starting with v = 0, R , , = - 1 , and rK B rD, we get, according to Fig. 21,
we can see clearly that the nozzle cannot be described as a simple impedance step. Since it is complicated to consider the wave propagation for v = 0, it is useful to change the method of description immediately before the rear wall of the nozzle: Inside the nozzle, acoustical and fluid mechanical effects must be connected in the equations. In the following discussion, surface and viscous forces are neglected so that we can focus on elasticity and inertia only. If we assume a pressure difference p between the channel and the surrounding air, an outflowing velocity u establishes itself according to the equation of motion p
= p/2v2
125
INK-JET PRINTING
The transient time needed to establish the velocity depends on the mass of the ink volume within the nozzle cylinder that must be accelerated. Thus, we find for the transient period p = :pu2 lpd
+
As pressure waves run along the channel, the nozzle is exposed to a pressure difference that is made up by three partial pressures: the instantaneous pressure p , ( t ) of the pressure wave that arrives at the rear wall of the nozzle, the instantaneous pressure p r ( t ) of the pressure wave reflected by the nozzle, and that static difference po(t) between the pressure inside and outside the channel to which surface tension contributes as well. This is indicated by the following equation: ~ ( t =) Pe(t) + Pr(t)
+~
d t )
The equation of motion describing the transformation of the pressure into outflow velocity in the nozzle then is p,
+ pr + p o = f p u 2 + Zpd
(3)
The equation of continuity is obtained by comparing the mass flow within the nozzle with the mass flow in the channel that is caused by the pressure waves: u d t r k n p = ( p e / c 2 ) r i n c pd t
-
(pr/c,2)rincpd t
+ u K r i n pd t
or ur&’ri = (Pe/cP> - (Pr/cP>+
If all the transport of fluid is caused by sound waves, the flow velocity uK in the channel is 0. In this case, the following continuity equation must be fulfilled : Pe
- ~r = ucprk/ri
(4)
By using initial values for 1 and u and by neglecting po(t), these equations can be solved by numerical integration for a given excitation pe(t). Equations ( 3 ) and (4) together describe a remarkable feature of the nozzle. Within the ink channel it is a place of reflection with a changing reflection factor. The reflection factor depends on the velocity of the ink volume in the nozzle cylinder. This can be understood by considering the response of the nozzle to an arriving pressure step, Figure 22 shows the step response that depends on the initial values and on the size of the step. In this case, the initial values were v = 0 and 1 = s. This can best be seen if the steady state is considered. For p , = constant, = 0,l = s, and f = c r ; / r i , the steady-state velocity is, following Eqs. (3) and (4), ~
u=
Jrnf Jrn +f -
=
126
J. HEINZL AND C. H. HERTZ
"I
FIG.22. Step response at the nozzle. The nozzle acts like a channel termination that opens and closes. As long as the velocity within the nozzle shaft is low, the main part of the arriving pressure wave is reflected. With increasing velocity, the reflection diminishes and may even act like an open end and turn negative.
This relation is represented geometrically in Fig. 23. Factorfis the product of sound velocity and the ratio of the cross sections of nozzle and channel. It characterizes the interdependence between the amplitude of the arriving pressure wave on the one hand and the steady-state value of the reflected wave and of the velocity in the nozzle on the other hand. The change in velocity and the change of the reflection factor are characteristic for the step response according to Fig. 22. With p e = constant and 1 = s = constant, the equations change over to = CPe + P r + PO - ( P / ~ N ' I / ~ P and P r = -b
e
+ Pr +
PO
-
(~/2)u~I(c/O(r3ri>
With 21 = 0, and po = 0, we obtain d = 2pe/lp
and
pr = -pe(c/l)(rk/ri)
- 5-
FIG.23. Final steady-state velocity and amplitude of the arriving and reflected wave shown as a function of the nozzle data.
INK-JET PRINTING
127
as can be expected from acoustic considerations. Both equations describe also the transformation of kinetic energy back into pressure waves by the nozzle. Like an antenna, the nozzle sends the kinetic energy stored in the nozzle shaft back into the channel. In contrast, it can release energy in the outward direction only by ejecting drops. Equation (4) describes the emitted pressure wave, Eq. (3) the decrease of velocity in the nozzle shaft.
E . Drop Shaping, ReJill, and Reverberations It goes without saying that the shaping of drops can only be described accurately if pressure and mass forces as well as viscosity and surface forces are considered. Bogy (l978,1979a,b,c) demonstrated how the problem can be tackled as long as the changes in velocity caused by pressure forces are small in comparison with the outflow velocity. For the shaping of drops in dropon-demand systems, this assumption cannot apply, since it would mean the neglect of the mass forces. Fromm (1980, 1981) gives a detailed description of the treatment of surface forces without, however, taking into account the pressure pulse from the channel. Lee et al. (1982a,b) give an estimation of the time for the formation of a free drop. While taking into account viscosity and surface tension, they neglect, however, the mass forces caused by the pressure signal. The question of what can be neglected and simplified depends on the size of the shares that the different forms of energy contribute to the shaping of a drop. When slow drops are shaped, surface energy dominates (Doring, 1982); when fast drops are shaped, kinetic energy dominates. For this distinction the Weber number is useful. With one-sixth of the radius of a drop taken as the characteristic length, the Weber number expresses the ratio of the kinetic energy to the surface energy of a spheric, free-flying drop. Rosenstock (1982) describes the shaping of fast-flying drops at We = 4. Consequently, in a first approximation he neglects surface forces and viscous forces and takes only into account the mass forces in the axial direction that result from the pressure pulse at the nozzle. He demonstrates how the constriction of the liquid jet can be controlled by the time dependence of the pressure a t the nozzle. The essential relation is shown in Fig. 24 and leads to Eq. (5): - tfd)/u]; z = tfv (5) r 2 ( z )= $ [ ( I where rD is the radius of the nozzle, v and ij the velocity and acceleration at the nozzle orifice, respectively, r ( z ) the radius of a slice of the ink jet at position z , and t , the flight time after departing from the nozzle. If the slices of ink that have left the nozzle at different times reach position z simultaneously, the cross sections must be added: r$(z) = # ( z ) The radius in position z , then, is rT(z).
1
128
J. HEINZL A N D C. H. HERTZ
-1
t
I
V+dv vdt v
FIG.24. Deformation of an ink slice, with all particles keeping up their velocity in flight direction: (a) slice leaving the nozzle; (b) the same slice after a flight time t,.
This consideration leads Rosenstock (1982) to the optimal pressure pulse that generates fast-flying droplets at the nozzle. It starts with low pressure, followed by high pressure, and again low pressure. Initial high pressure, in contrast, causes an acceleration while ink is leaving the nozzle. This acceleration creates a slow-flying tail in front that must be overtaken by the following ink particles of the droplet. The ink particles that have left the nozzle with different speed can only be kept together by surface forces. If these forces are not strong enough, the droplet breaks up into partial droplets that continue their flight at different speeds. Figures 25-27 demonstrate the generation and the effect of an optimal pressure pulse in an especially simple channel. Figure 25a shows a channel
“t 2
(a)
(b)
FIG.25. Simplified ink channel with nonreflecting termination. (a) Dimensions; (b) driving pulse producing optimal pressure history.
129
INK-JET PRINTING
sec
FIG.26. Arriving pressure wave p e , reflected pressure wave p , at the nozzle, and the position l - s of the meniscus during the first 80 psec following the driving pulse according to Fig. 25. As soon as the ink leaves the nozzle, the time path of the ink slices is plotted in intervals of 1 p e c .
with a nonreflecting end. Figure 25b shows an electrical impulse that, according to Eq. (l), is transformed into a pressure pulse. Figure 26 shows the pressure pulse p e arriving at the nozzle and the reflected pressure wave pr according to Eqs. (3) and (4), as well as the position I of the meniscus calculated by integration of velocity u in the nozzle shaft. We can see that the meniscus first withdraws into the nozzle shaft and then is accelerated, so that it has already reached the full flying speed of the droplet to be formed when passing the nozzle opening. The time-path curves of single ink particles are indicated in intervals of 1 psec. The following low-pressure wave reinforced by the reflected wave from the rear opening of the nozzle prepares the breaking off of the droplet and constricts the droplet vigorously. Figure 27 shows the contours of the emerging droplet at different times according to Eq. (5), where surface forces and viscosity are neglected. The break-off point is distinctly marked. In this case the transport of fluid by the pressure waves has made the meniscus assume its former rest position without capillary refill being necessary.
60 50 60 70
I
100
200
t/rsec z/ym
FIG.27. Shaping of a drop according to Eq. ( 5 ) . The resulting contours are shown a t four different times.
130
J. HEINZL AND C. H. HERTZ
The results of calculations that take into account surface forces and viscosity as well as inertia forces differ only slightly from those shown here, as long as an optimal driving pulse is used. If surface forces are taken into account, a further time-dependent pressure component po(t) is obtained in addition to p e ( t ) and p , ( t ) in Eq. (3). However, only during the constriction of the drop and its break-off does the amplitude of this component show significant values. During the other time, this component is negligible. Outside the nozzle the drop-shaping is smoother but not principally different (Penningsfeld, 1984). Not even viscosity in the nozzle is of significant influence on the results of the calculations if taken into account. Its effect is quite similar to a narrowing of the nozzle and a slight deficit in energy. Only the free-flying drops do not take their final spherical shape if viscosity is neglected. If the channel is somewhat more complicated than the channel in Fig. 25, additional effects occur. The channel in Fig. 19, for example, is joined to the ink reservoir by a marked enlargement. The first reflection of the pressure wave from there can be used, together with the direct pressure wave from the transducer, to establish the desired pressure history. For this reason the transducer should be located close to the point of reflection. Equation (2) describes the pressure-dependence at the nozzle. Since the nozzle also reflects parts of the pressure waves, an additional term must be introduced into Eq. (2) for times t > (3a + 2b 2d)/c:
+
pe(t> = k U ( t l ) - K U ( t 2 ) where t , = t - (LZ/C) t2 = t t3
+ R12KU(t3) -
-
R12KU(t4)
+ R12GpI'(t5) ( 6 )
(u/c) - (b/c)
= t - (u/c) - (b/c) - (2d/c)
t , = t - ( ~ u / c )- (2b/c)
-
t4 =
t
-
(u/c)
-
(2b/c) - (2d/c)
(2d/c)
The dissipation factor 6 represents approximately the losses during the travel of the pressure wave along the channel and back. This dissipation factor can easily be found by experiment to a first approximation. However, the use of such a dissipation factor is not very accurate, since the attenuation is not linear. Actually, high-frequency components of the pressure waves are more strongly attenuated than the low-frequency components. This attenuation does not depend on fluid viscosity, but on the elasticity and damping of the channel wall and the created dispersion of propagation velocity in the channel (Wehl, 1984). Drops ejected during the reverberation period differ in flying speed. Their speed depends on which part of the echo is superimposed on the generating pressure pulse. This is shown by the fact that the speed of the second droplet plotted against the time delay of the first droplet shows maxima and minima in periodic intervals (Manini and Scardovi, 1982; Hofer and Talke, 1982). The period is 2(a - b - d)/c. Therefore, these reverberations limit the drop
INK-JET PRINTING
131
rate. Hofer and Talke (1982) have overcome this limitation by selecting a suitable sequence of electrical pulse amplitudes that are dependent on the case history. Kasahara et al. (1981) describe a compensation pulse applied after one period that extinguishes parts of the reverberating waves in the channel. Both procedures depend on the sound velocity in the channel. It is therefore much safer to make the echoes fade away by softening and damping at the channel walls. In this case, it is useful to increase the distance between transducer and nozzle, so that the generation of droplets is finished before the second echo arrives.
F . Other Questions Many other questions that arise in connection with drop-on-demand systems have not even been mentioned in the foregoing pages, for example, composition of ink, cavitation, air and gas bubbles in the channel, cleaning and wetting of the nozzle, compensation of temperature changes, the close arrangement of many nozzles as well as their adjustment, crosstalk between the nozzles, protection against shock and acceleration, and, last but not least, the long-term stability of the process. These issues have often been mentioned, but scarcely ever treated in the literature so far. On the other hand, it is quite understandable that those who have gathered experience in this field tend to keep it to themselves. We thus confine ourselves to pointing out some problems. The ink may serve as an example. It must convey the pressure energy from the transducer to the nozzle, serve as a valve and as a seal at the nozzle, and carry the coloring matter. It must not dry up in the nozzle opening, especially since the nozzle is not closed when unused. On the other hand, it should be instantly smearproof on the paper (Heinzl and Rosenstock, 1977). It should consist of a proper solution, so that the coloring matter does not slowly clog up the nozzle. O n the other hand, it should not be possible to dissolve the color in the paper, not even by its own solvent. It should be freeze-proof, so that it can be transported in an open freight compartment of a plane. The viscosity of the ink should be comparable to that of a multigrade oil. Thus velocity and size of the drops can be stabilized by merely heating the nozzle shaft or by adjusting the driving pulse over the full temperature range. In the bubble-jet system, at least, it is additionally necessary for the ink to be boilproof, so that-driven by a vapor bubble-it leaves the heating element and the nozzle shaft without leaving residue. It should keep well and be storable and, last but not least, should make clear and contrasting colored or black bleachproof dots on the paper. It is clear that the development of an ink meeting all of these requirements is a tedious and lengthy process, and it is not surprising that several patents have been issued describing ink formulations. The same is true for the other
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J. HEINZL AND C. H. HERTZ
problems mentioned, the solution of which often involves a large amount of unpublished know-how. Therefore, these subjects are not discussed in greater detail in this chapter. 111. METHODSEMPLOYINGCONTINUOUSJETS
A. The Break-Up of a Continuous Fluid Jet
As indicated in Section I, several methods are available for controlling continuous ink jets by electrical signals. To understand the mechanisms used in these methods, a certain knowledge of liquid jets is required. Therefore, this section will be devoted to the physics of such jets. 1. The Drop Formation Process Normally, liquid jets are generated by forcing a liquid at high pressure through a nozzle with a circular opening. After exiting from the nozzle, the jet usually travels through the air in the form of a continuous laminar flow cylinder, the diameter of which may contract slightly if the velocity distribution inside the jet changes due to viscous forces (McCarthy and Molloy, 1974).This continuous part of the jet disintegrates spontaneously into a train of droplets at a well-defined point as will be discussed later. At very high speeds, the jet, instead of being continuous, assumes a turbulent-like flow pattern (Batchelor and Gill, 1962; Viilu, 1962; Phinney, 1973a,b; McCarthy and Molloy, 1974). The transformation from laminar to turbulent-like jet flow depends on the aspect ratio L/d of the nozzle (McCarthy and Molloy, 1974), where L is the length and d the diameter. However, since only laminar flow jets can be used in ink-jet printing, we will turn our attention to those jets. Already in 1833 Savart showed that laminar-flow jets breaks up into a train of drops at some point due to surface tension. This is due to the fact that the surface energy of a liquid sphere is smaller than that of a cylinder containing the same volume. Therefore, a cylindrical fluid column, for example, a jet, is inherently unstable and will eventually transform itself into spherical drops (cf. Fig. 28a). Rayleigh (1878) was the first to treat this drop formation theoretically for a nonviscous liquid jet. He assumed a disturbance a. of the jet radius created at the nozzle exit that produces spatial oscillations of the jet radius r (cf. Fig. 29). These oscillations increase in amplitude exponentially while traveling down the jet, so that the radius of the jet r can be expressed as r = ( d / 2 ) + aOeqrcos(2nx/i) (7) where t is the time after leaving the nozzle, d the nozzle diameter, q the growth rate factor, and i the wavelength of the disturbance along the jet axis x. By
INK-JET PRINTING
133
FIG.28. (a) Spontaneous drop formation of a 10-pm diameter jet with drops traveling at a speed of 20 m/sec. Mottled background is due to giant pulse laser illumination. (b) Same as in (a), but drop-formation process is controlled by mechanical vibrations with a frequency of 1.08 MHz. Jet velocity = 39 mjsec.
calculating the increase in surface energy caused by the oscillations as well as their kinetic energy and by using Lagrange’s method, Rayleigh was able to show the existence of a relation between the growth factor q and the wavelength I. of the disturbance. From this relation he could deduce that the growth factor had a maximum q, = 0.97Ja/Pd3
(8)
at the wavelength Am
= 4.51d
(9)
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J. HEINZL AND C. H. HERTZ
r A
\
(b)
DROP FORMATION POINT
FIG.29. Drop-formation process. (a) Normal jet profile; (b) jet profile with magnified radial scale.
where CJ is the surface tension and p the density of the liquid. He further showed that disturbances with a wavelength of 1 5 nd could not appear on the jet since their growth factor was 5 0 (Rayleigh, 1878; Sweet, 1964; Bruce, 1976). Since of all possible disturbances with different wavelengths the disturbance with the largest growth factor predominates, the disturbance with the wavelengh I , will develop most rapidly and therefore become visible on the jet (cf. Figs. 28 and 29). Weber (1931) treated the same problem but included viscosity in his calculations. By considering the forces acting within the jet, he arrived at a result very similar to that of Rayleigh, but the maximum of the growth rate was shown to increase markedly with viscosity at viscosities above 10 CP(Bruce, 1976). From Eqs. (7) and (8), the distance from the nozzle to the point of drop formation L can be calculated. If we let T be the time necessary for the amplitude of the disturbance to become equal to the jet radius d/2, we have d/2
= aOeqT
(10)
Then, if u is the speed of the jet, we find for the length L
L
= uT = ( u / q ) In(d/2ao)
( 1 1)
By using Eq. (8), we then obtain
L
=
1.03 l n ( d / 2 a 0 ) u ~
The factor 1.03 ln(d/2a0) is strongly dependent on the amplitude a, of the initial disturbance generated at the nozzle exit. Weber (193 1) determined this
INK-JET PRINTING
135
factor from Haenleins measurements (1931) to be about 12, indicating a value of .ao as low as 4 x 10-(jd. In this case, the distance to the point of drop formation is given by
L
= 12vJpd3/o
(12b)
Others (Minsson, 1972; Bruce, 1976; Richardson, 1954) arrived at slightly different values, probably due to differences in nozzle smoothness that determine the initial amplitude a. of the disturbance. It is obvious from the preceding material that a drop is formed each time a disturbance wave approaches the point of drop formation. Thus, the frequencyf, of spontaneous drop formation is equal to f , = v/A,
= vf4.51d
(13)
and the maximum drop-formation frequency f,,, attainable is fm,,
= v/nd
(14)
In ink-jet printing, jets with a diameter of 50 pm traveling at 20 m/sec are frequently used. According to Eq. (1 3), their drop-formation frequency would be about 90 kHz, which is in good agreement with experiment. 2. Control qf Drop Formation Process
Since the drop formation process is caused by a disturbance initiated at the nozzle, it can be controlled by generating mechanical vibrations a t the nozzle, which are transferred onto the jet. This can be achieved by attaching an ultrasonic generator to the nozzle, for example, a piezoelectric crystal, the resonance frequency of which is close to that given by Eq. (13). In that case, the drop-formation rate is synchronized to the frequency of the mechanical vibrations resulting in a very regular train of drops, each with exactly the same size (Sweet, 1965; Schneider and Hendricks, 1964; Lindblad and Schneider, 1966; Magarvey and Taylor, 1956; Mgnsson, 1971) as is shown in Fig. 28b. This possibility of generating drops of exactly the same size is of crucial importance for the printing process based on the Sweet method described. To ensure that the mechanical vibration generated by the piezoelectric crystal actually controls the drop formation, the disturbance amplitude a0 caused by this vibration of the jet surface must be appreciably larger than the disturbances generated spontaneously at the nozzle exit. According to Eq. (12a), this results in a decrease of the length L from the nozzle to the point of drop formation. In practice it is found that the position of this point can be determined within certain limits by changing the amplitude of the mechanical
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J. HEINZL A N D C. H. HERTZ
vibrations through amplitude control of the excitation voltage of the piezoelectric crystal (Bruce, 1976): Hermanrud et al. (1982) used this effect in a novel method to print characters. For proper control of the drop-formation process, it should be observed that the mechanical vibrations of the nozzle do not impart transversal oscillations to the jet. Otherwise, the jet may split up at the point of drop formation into two or more diverging drop trains that cannot be used in ink-jet printing (Hermanrud and Hertz, 1979; M b s o n and Welinder, 1976). Finally, it should be mentioned that periodic disturbances can be caused on the jet even by electrostatic means. Thus, Sweet (1964) and Gunn (1975) described methods in which one or more electrodes are situated along the continuous part of the jet. If an ac signal of suitable frequency is applied to these electrodes, electrostatic forces will cause disturbances on the jet of sufficiently large amplitude to control the drop formation. However, this method has not been of practical importance in ink-jet printing.
3. Satellite Drops A more detailed study of the drop-formation process reveals that it is more complex than described so far. Already in Fig. 28a we see that small socalled satellite drops are generated between the large drops. This phenomenon is shown in more detail in Fig. 30, which indicates that shortly before drop formation ligaments are formed between the drops (Keur and Stone, 1976). Given suitable conditions, these ligaments detach themselves from the adjacent drops and form one or more small “satellite” drops that normally travel with a slightly different speed compared to that of the main drops. As a result of this they merge with the preceding or trailing drop on their flight through the air. Satellite drops are generated with and without ultrasonic drop-formation control and can arise in many different modes as shown by Chaudhary and Maxworthy (1980a,b) and Curry and Portig (1977). Since Fillmore et al. (1977) found that the occurrence of satellites in the ink stream was very detrimental in ink-jet printing using the Sweet method, a large effort has been made to avoid satellite formation. Curry and Portig (1977) reported the existence of a region of certain frequencies of the ultrasonic generator and disturbance amplitudes where no satellites were formed and which could be used for ink-jet printing. Alternatively, reasonably good results can be obtained with forward-merging satellites (Keur and Stone, 1976). Pimbley and Lee (1977) showed that the satellite formation was due to second-order effects
-*-.
4
a
FIG.30. Drop formation with forward merging satellites.
0
INK-JET PRINTING
137
of the jet instability. In addition to the work of Pimbley and Lee, the mechanism leading to different modes of satellite formation was studied theoretically in several papers (Lee, 1974; Chaudhary and Redekopp, 1980; Bogy, 1979a,b), and reasonably good agreement with experimental results was achieved. Bruce (1976) investigated the behavior of fluids consisting of water with different additives. He found that the drop-formation frequency and length of the continuous part of the jet were in agreement with the equations derived by Rayleigh (1878) and Weber (1931) as long as the jet fluid had Newtonian properties. The drop formation was not influenced by the addition of detergents, which lowers the static surface tension of the liquid since new surface is continuously created at the jet nozzle. However, the behavior of viscoelastic fluids obtained by small additions of high molecular weight polyethylene oxide (Polyox) differed appreciably. While stream stability was found to be excellent over a wide range of viscosity and surface tension with Newtonian fluids, it was appreciably decreased by the addition of Polyox. Similar behavior was found with fluids containing a dispersion of fine particles. The addition of Polyox appreciably increased the length to the point of drop formation (Goldin et al., 1969; Hermanrud, 1981). Actually, with viscoelastic liquids the drop formation becomes less distinct since long, thin ligaments are formed between the drops, which do not break for an appreciable time after the drops have formed. Therefore, relatively large additions of Polyox to the ink should be avoided in ink-jet printing applications since this might prevent the proper charging of the drops. 4. Ink Properties It is obvious that the ink used in continuous-jet printing is of crucial importance to the functioning and reliability of the process. Therefore, the ink properties required for ink-jet printing will be discussed briefly. The diameter of the ink jets used in continuous-jet printing is usually in the range of 10 to 60 pm. Thus, the jet nozzles themselves must have the same small size. This generates problems, owing to nozzle clogging. Therefore, fine filters must be used upstream from the nozzle to avoid dirt particles in the ink from reaching the nozzle. Furthermore, during the time the printing apparatus is not operated, the ink should not dry in the nozzle since a solid deposit will also result in clogging. This difficulty can be avoided by using a humectant in the ink so that it never dries except by absorption on the paper. Finally, the ink must have good electrical conductivity, usually less than 500Rcm, since the time constant of the charging of the drop must be appreciably shorter than the time required to form one drop. Aside from these main properties the ink must meet several other requirements. First of all, biological growth must be prevented in the ink by addition
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J. HEINZL AND C. H. HERTZ
of fungicides to avoid nozzle clogging by fungi or bacteria. To facilitate the generation of the ink jet, a lower viscosity is chosen than that of ink in dropon-demand systems. Normally, viscosities lie in the range of 1 to 3 cP. To avoid toxicity, inflammability, and corrosion of the ink systems, most of the inks used are water-based except in special applications, for example, the printing of identifying information on glass or plastic containers where fastdrying inks must be used. Thus, a normal solvent for an ink intended for printing on paper would be 80 % water 20 % humectant (for example, glycerin or polyglycol) to which a suitable fungicide is added. Electrical conductivity can be adjusted either by the addition of suitable inorganic salts or some acid. To obtain the required color, a suitable dye must be added to this solvent. Color pigments cannot be used since they clog the nozzle or, if this does not occur the filter in the ink system. The choice of the dye is critical since it should not form a solid deposit with the humectant if the ink is allowed to dry in the nozzle. Also, it should meet other requirements such as light fastness, nontoxicity, good optical density, and good resistance against water and mechanical abrasion after being deposited on the paper. Usually, it is impossible to meet all the requirements at the same time. This has resulted in a large amount of work on the development of suitable inks for ink-jet printing, reflected by the large number of patents related to this subject. More detailed accounts of the properties required for ink-jet printing have been given by Jaffe and Mills (1982), Ashley et al. (1977), and Beach et al. (1977). It is obvious that these requirements limit the choice of inks severely. Therefore, it is difficult to obtain an ink with an optical density and light fastness similar, for example, to pigmented inks, which, as mentioned, clog the filter and nozzle in most ink-jet systems. For similar reasons, chemicals used in conventional textile printing cannot be used.
+
5. The Compound Jet To ease these restrictions on the ink, Hermanrud and Hertz (1979) developed an alternative way to generate an ink jet. This jet emerges from an orifice at least 1-2 m in diameter but is still small enough to be used in electrostatic ink-jet printing devices (Hertz, 1975; Hermanrud, 1981;Hermanrud et al., 1982; Hertz and Hermanrud, 1983). The principle of the method is shown in Fig. 31. Here a 10-20-pm diameter fluid jet emerges under high pressure from a nozzle that is submerged in a second fluid contained in a suitable vessel of arbitrarily large diameter. To distinguish between these two fluids, the fluid of the jet is called the primary fluid, and the surrounding liquid is called the secondary fluid. Since the direction of the primary fluid jet is normal to the surface of the secondary fluid and its speed is large enough, it can emerge through the sur-
INK-JET PRINTING
139
0 0
t FIG. 31. Compound jet. A high-speed jet emerges from the nozzle N slightly below the surface of a stationary fluid (ink). After breaking through the surface, the primary jet is enclosed by a concentric layer of ink that has been dragged along by viscous forces between the jet and the ink.
face into the air. On its way through the secondary fluid, the primary jet entrains layers of the secondary fluid close to the jet surface due to viscous forces (Fig. 32). The jet emerging into air therefore consists of an inner core of primary fluid with a concentric layer of secondary fluid. The thickness of this layer depends on the distance between the nozzle and the surface of the secondary fluid as well as on jet speed and fluid viscosities. Since the two fluids are usually chemically different, such a jet is called a compound jet. The compound jet breaks up into a train of drops due to Rayleigh’s instabfiity in the same way as conventional jets. Therefore, the compound jet can
140
J. HEINZL AND C. H. HERTZ
FIG.32. Microphotograph of a compound jet. A primary jet (20% ethanol and 80% water) emerges from a 75-pm nozzle into pure water and breaks the surface, thereby creating a compound jet. The black line originating at the top is generated by a colored tracer fluid injected there to show stream lines in the water.
be used in ink-jet printing devices employing most of the methods developed for the electrical control of liquid jets. Furthermore, since the diameter of the vessel containing the secondary fluid can be arbitrarily large, the properties of the secondary fluid are of minor consequence since drying and clogging of this large surface can be avoided without difficulty. Hence most inks can be used as secondary fluids. On the other hand, the primary fluid should be chosen so that it will not clog the nozzle. Suitable liquids for the primary fluid are water, water-glycerin, or water-ethanol mixtures. As mentioned earlier, the amount of secondary fluid entrained by the primary jet depends on a variety of factors. Therefore, we would assume that it might be difficult to control the total amount of fluid in the compound jet. However, if the secondary fluid is supplied to the vessel shown in Fig. 31 at a constant rate, for example, by a peristaltic pump, the surface of the secondary fluid will immediately attain an equilibrium position such that the secondary fluid flow supplied will equal the amount of secondary fluid carried away by the primary jet. Thus, by controlling the flow of secondary fluid supplied to the vessel, the composition of the compound jet can be controlled at will. Compound jets have been used successfully in different ink-jet printing applications using various types of inks, including india ink, as secondary fluids (Hermanrud, 1981).
INK-JET PRINTING
141
B. Ink-Jet Oscillograph Recorders Direct-writing oscillographs using moving-coil systems driving a mechanical arm with a stylus are very popular since they create a directly visible trace on untreated recording paper. However, due to the inertia of the arm and stylus, their upper frequency limit is only about 100Hz. To improve this, Elmqvist (1951) substituted for the mechanical arm a small glass nozzle that was fastened to the moving system of the galvanometer. By forcing ink under high pressure through that nozzle, he produced a continuous ink jet, the direction of which could be controlled by the current through the galvanometer coil. The ink jet easily traveled 6- 10 cm through the air and generated a recording trace on a moving paper web. Since the inertia of the nozzle is much smaller than that of the previous mechanical arm, the upper frequency limit of such recorders is more than 10 times higher than that of conventional recorders (Elmqvist, 1951). Figure 33 shows the principle of a more recent version of this type of recorder (Elmqvist and Almgren, 1977; Kaiser, 1959; Sima, 1966). It consists of a cylindrical permanent magnet that is magnetized perpendicular to its axis. The magnet is attached to a 0.1-mm glass capillary, which passes through a central hole along the magnet's axis. After emerging from the magnet the capillary is bent 90" and drawn out into a 10-pm nozzle. The capillary serves as an ink conduit and also provides the restoring torque that returns
FIG.33. Ink-jet oscillograph recorder. A small permanent magnet M is supported by a fine glass tube G leading ink to the nozzle N. If a signal current is applied to the electromagnet, a magnetic force will turn magnet M and nozzle N, thereby changing the direction of the ink jet emerging from nozzle N in accordance with the strength of the signal current. The necessary restoring torque is provided by tube G .
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J. HEINZL AND C. H. HERTZ
the nozzle to its resting position. If ink is forced through the nozzle, an ink jet with a diameter of 10pm will travel through the air toward the recording paper. As shown in Figure 33, the magnet is placed in the field of an electromagnet that is energized by the signal current. This causes the magnet to turn around its axis by an amount roughly proportional to the current, thereby changing the direction of the nozzle. This deflection is immediately recorded on the paper. By suitable design of the magnet system, the deflection of the trace can be made proportional to the signal current, allowing this instrument to be used as a direct-recording oscillograph. Due to the low moment of inertia of the magnet-nozzle structure, the upper frequency limit of this oscillograph is as high as 1000 Hz. Recent improvements of the design (Elmqvist and Almgren, 1977) have increased this to more than 2.5 kHz. Since the magnet’s inertia and the restoring torque of the capillary constitute an oscillating system, the magnet is placed in a damping field of oil to suppress resonance peaks. This type of oscillograph is being produced by Siemens-Elema in Sweden and is widely used, predominantly for the recording of electrocardiograms (ECG) and the like in hospitals. Because of the very high upper-frequency limit of the more recent versions of the instrument, it can also be used to print alphanumeric characters at speeds up to 40 ch/sec (Lloyd et al., 1972). This is accomplished by modulating the signal current in such a way that the ink jet is pointed at the different dot positions of a 5 x 7 matrix for about 2 msec/ dot, while switching between the dots is done at the maximum speed of the oscillograph system. Thereby visible dots are generated on the paper at desired points with the trace between them barely visible because of the high switching speed. Character printers of this type are used to add text onto the recording paper of ECG recorders, for example, patient identification. C . The Sweet Method
It has already been mentioned in Subsection I.B.3 that Sweet (1964) was the first to demonstrate the use of a continuous-ink jet for the recording of electrical signals by electrically charging its drops at the point of drop formation. While Sweet’s oscillograph was not turned into a commercial product, it showed the feasibility of the method and paved the way for further developments. Lewis and Brown (1967) proposed to use Sweet’s method for the printing of characters, and several companies (Stanford Research Institute, Recognition Equipment Corp., and A. B. Dick) began to develop this idea into commercial printing devices for different fields of application. The success of these developments then triggered many other companies to develop and, in some cases, to market products using the Sweet method.
143
INK-JET PRINTING
m
m
a
NOZZLE V I B R ATOR
CHARGING ELECTRODE
*a*
RECORD RECEJVI N G SURFACE
CONTROL SIGNAL FROM ELECTRONICS
FIG.34. Sweet-type printhead.
A simplified diagram of a printing device of the Sweet type developed by the A. B. Dick Company is shown in Fig. 34 (Stone, 1978). Here a nozzle with a diameter of 65 pm is placed about 6-8 cm from the recording or receiving surface. The nozzle is vibrated mechanically by an ultrasonic transducer with a frequency of 66 kHz and the ink jet emerging from the nozzle at a speed of about 15 m/sec is transformed into a train of drops of exactly equal size at a point about 8 mm in front of the nozzle. Since the point of drop formation is situated inside the charging electrode to which a signal of up to lOOV is applied, the tip of the continuous part of the jet will acquire a charge approximately proportional to the amplitude of the charging signal. When the drops are formed, they carry this charge and are, therefore, deflected in the strong transverse deflection field on their way to the recording paper (Lindblad and Schneider, 1967). Since the droplets all have the same mass, their landing position on the paper can be determined at least theoretically with great accuracy by the charging voltage. In this way, up to 1300 characters per second can be printed with the device shown in Fig. 34. Each character consists of a 5 x 7 matrix in which one drop of ink is placed in each matrix position. Drops that are not desired in the matrix are not charged at all and, therefore, are caught by the gutter. From there they are recirculated to the ink reservoir. Unfortunately, the actual printing process is much more complicated than described here since the charging and deflection of the drops depend on
144
J. HEINZL A N D C. H. HERTZ
many factors (Kuhn and Myers, 1979; Fillmore et al., 1977). We now discuss these factors in more detail. 1. Drop Charging
To achieve maximum deflection in a given deflection field, the drops should acquire as much charge as possible. However, two fundamental processes limit the amount of charge that can be placed on a drop. Since the electric forces generated by the charge on the surface of the drop counteract surface tension, the drop will disintegrate if the charge becomes too large. This effect was originally treated by Rayleigh (1882) who found the limiting charge Q to be
Q=JW
(15)
where r is the drop radius and CT the surface tension. Rayleigh’s calculations were refined by Hendricks and Schneider (1962) and confirmed by experiments (Hendricks, 1962; Taylor, 1964). It should also be observed that a continuous liquid jet becomes unstable in a high electric field (Magarvey and Outhouse, 1962; Huebner, 1969; Mutoh et al., 1979), and a corona discharge may develop at the sharp tip formed by such a jet (Mhnsson, 1972). In all ink-jet printing devices, the charge Q applied to the drops is lower than that given by Eq. (1 5), thus usually no disintegration of drops is observed in either the Sweet or the Hertz methods. The second charging limitation is caused by mutual electrostatic repulsion among the charged drops. This force disturbs the linear train of drops and turns them into spray at charging voltages usually much lower than those necessary to cause drop disintegration (Fillmore et al., 1977; Hertz and Mhnsson, 1972; Sweet, 1971). Because of these factors, the maximum charging voltage is limited and usually kept below 150 V. To control accurately the charge produced on the drops by the charging voltage, it is necessary that the point of drop formation be always kept at a constant potential, for example, at ground potential. This can be achieved only if the conductivity of the ink is high enough so that the voltage drop along the continuous part of the jet can be neglected compared with the charging voltage. Thus if R is the resistance of the continuous part of the jet, q the charge on one drop, C , the capacity of a drop relative to the charging electrode, and V , the voltage applied to this electrode, we should have for the potential V of the point of drop formation
V
= R I = Rqf
4 V , = q/C,
(16)
wherefis the number of drops formed per second. From this, we find RC,
< l/f
(17)
INK-JET PRINTING
145
i.e., the time constant of the drop-charging process should be smaller than the drop-formation time. This condition limits the electrical resistivity of the ink to less than 500 R cm, a value that can be shown to be relatively independent of the jet diameter. 2. Historical Efect Unfortunately the charge induced on a drop at the point of its formation is determined not only by the voltage of the charging electrode but also by the charge on the drops formed earlier. This is due to the capacitances C,,, C,,, C , , , . . . that exist between the drops as indicated in Fig. 35. Fillmore et al. (1979) showed that charge errors of 20% and more might arise as a result of the charge of previous drops, and Minsson (1971, 1972) demonstrated that the charge on the drops may even change sign following a rapid change of the control voltage. In the latter case the two drops of opposite charge usually merge. Since the actual charge acquired by a newly formed drop is partially determined by the charge on the drops formed earlier, this effect is often called the historical effect. The historical effect often appreciably distorts the drop charge resulting in large errors in drop positioning on the recording paper. Different methods to cope with this problem have been devised as indicated later. However, it should be mentioned here that the historical effect can be lowered by decreasing the inside diameter of the charging tunnel of the control electrode as much as conveniently possible and by placing the point of drop formation inside the tunnel. If this diameter is small enough, the walls of the charging tunnel act as an electrostatic screen and markedly decrease the capacitance between the point of drop formation and all but the first of the previous drops (Fillmore et al., 1977). The influence of the other drops on the charging process of a new drop is, thus, greatly reduced.
I '
cc=+ I
'
-I tc12
/
#'
FIG.35. Simplified equivalent charging circuit with three previously charged drops.
146
J. HEINZL AND C. H. HERTZ
3. Satellite Drops It is essential for the Sweet method that all drops have the same mass. As mentioned before, this can be achieved by mechanically vibrating the nozzle with a frequency similar to that of natural drop formation. However, even when controlling the drop-formation process by such vibrations, satellite droplets are often formed. Since these satellites have very different masses compared to the other drops and usually merge eventually with the preceding or following drop, they severely disturb the printing process and thus cannot be tolerated (Fillmore et al., 1977). The jet parameters, therefore, must be adjusted carefully so that no satellites are formed. Satellite formation seems to be least likely when the main drop spacing is five to seven times the jet diameter (Curry and Portig, 1977). By using this print window, high-quality printers for alphanumeric characters can be built (Buehner et al., 1977). Since satellite drops are smaller than the normal drops, Yamada et al. (1982) have used satellite drops alone to obtain high-resolution printing from relatively large nozzles (Cohen, 1983). In his printer, Yamada used the Sweet method and intercepted the large drops by a gutter.
4. Phase Control It is important to keep in mind that in the Sweet method every drop leaves a visible mark on the recording paper and, therefore, must be placed correctly. This assumes that a correct charge is induced on each drop at the moment when the drop is formed and separates from the continuous part of the jet. Thus, the charging signal must attain the correct value at exactly this moment, i.e., the charging signal and the drop-formation process must be synchronized. Since slight changes in jet velocity, surface tension, viscosity, or temperature influence the drop-formation process, the phase difference between the charging signal and the mechanical nozzle vibrations controlling the drop formation must be adjusted (Pullen, 1982). Two ways have been used to achieve this adjustment. Carmichael (1977) describes a printing device that measures the position of strongly deflected drops close to the paper by detecting their charge. Each time the printer is at rest, some drops are deflected into this detector, which generates a signal proportional to the deviation of these drops from the correct position. This signal is then used to adjust the phase as well as other device parameters (IBM 6630, A. B. Dick printer). Alternatively, in the Domino printer half-width charging pulses are applied, the length of which is incremented in time until a charge detector placed along the jet stream shortly after the charge electrode no longer detects a signal (Pullen, 1982).The necessity of this phase adjustment, of course, adds complexity to the Sweet printers since the sensing of the small drop charges requires sensitive electronic circuitry.
INK-JET PRINTING
147
5 . Drop Dejection After emerging from the charge electrode, the drops are deflected in a transverse electric field (cf. Fig. 34). If L is the length of the deflection plates and D the distance from the deflection plate’s entry to the recording paper, we have for the deflection x on the paper x
= (qE/rnuZ)L[D - (L/2)]
(18)
where q, m, and u are the charge, mass, and the velocity, respectively, of the drops. If appropriate values for the drop parameters (about l00pm) and length of flight D (about 5-10cm) are introduced in this equation, it is seen that the field strength E must be relatively close to the electrical breakdown of air to effect deflections of about 5-10 mm. This cannot be improved much by increasing L or D since the jet loses speed as it travels through the air due to air friction eventually resulting in a scattering of the drops. For the same reason, the deflection is given only approximately by Eq. (18). Also, drop-todrop interactions due to air drag and electrostatic repulsion are not taken into account by this equation.
6. Air Drag In his early work, Sweet (1964) found that drops moving through air at high speeds close to each other interact because of aerodynamic forces. If, for example, a linear train of 10 drops following each other moves through the air, the leading drop is retarded most by air friction and therefore tends to merge with the drop behind it after a short time. This process repeats itself until the entire train of drops has merged into a single drop (Minsson, 1971). The situation becomes even more complex if closely spaced drops follow slightly different trajectories as shown in Fig. 36 (Fillmore et al., 1977; Twardeck, 1977). Since the merging of drops is very undesirable in high-quality printing, adequate spacing must be provided between the drops. This can be achieved either by the guard drop technique (Fig. 37) or, alternatively, a suitable drop sequence can be selected for each type of character so that two drops following each other travel to character matrix positions spaced far from each other (Fillmore et al., 1977). Hendricks (1980) investigated the air drag problem in detail and proposed the use of an aspirator creating partial vacuum to eliminate the influence of air drag. 7. Electrostatic Interaction
If several drops travel through the air close to each other, even electrostatic forces will tend to deflect them from their course (Fillmore et al., 1977; Minsson, 1971, 1972). This interaction also can be avoided by increasing the
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FIG.36. Electrostatic repulsion and aerodynamic drop interaction effects. Thin arrows represent electrostatic repulsion, and heavy arrows represent aerodynamic forces. These forces may lead to the merger of adjacent drops (drop 1 + 2).
droplet spacing by using the methods indicated earlier. Therefore, for highquality character printing special drop sequences have been worked out for each character to minimize both air drag and electrostatic interaction effects (Fillmore et al., 1977). 8. Correction Procedures
Three effects, i.e., historical effect, air drag, and electrostatic repulsion, are inherent to the Sweet method and must be compensated for to minimize their influence on print quality. This is achieved essentially in two ways. Since all of the effects mentioned earlier decrease strongly with drop spacing, one way to eliminate these problems is not to use every drop in the jet stream for printing. As shown in Fig. 37, one or several uncharged “guard” drops can be
+
-
GUARD DROPS
FIG.37. Uncharged guard drops increase distance between print drops. In-flight interaction is thereby diminished.
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INK-JET PRINTING
placed between the printing drops, thereby separating the charged printing drops from each other (Fillmore et al., 1977). Obviously this will decrease print speed since a large part of the ink issuing from the nozzle is returned to the ink reservoir, but very high print quality can be achieved as demonstrated by the IBM 6640 document printer (Buehner et al., 1977). However, if high printing speeds and low character quality can be accepted (A. B. Dick printer, Domino printer), these aberration effects can be compensated for electronically as indicated in Fig. 38. Here, the signal from the digital data source indicating the character to be printed is converted into a column-by-column description of the dot positions of that character in a ROM circuit. From this ROM the main signal to the charging electrode is derived via a D/A converter. To this signal two correcting voltages are added to correct for the historical effect as well as for in-flight interaction. These voltages are obtained from a look-up table to which the value of the two previous drop charges are supplied. Other schemes using more sophisticated look-up tables are also used. 9. Printer Hardware While printers based on the Sweet method are usually more complicated than drop-on-demand devices, they can be used in many instances where the latter cannot meet the requirements. Compared to drop-on-demand devices, Sweet printers are superior in speed and can print at printhead-to-paper distances of as much as 1 to 2 in. Therefore, they can print on curved or uneven surfaces that occur commonly in product marking (e.g., beer cans or plastic bottles), or they can be used to apply bar codes on letters and checks at high
k CONVERTER
PRECED I MG DROP LOOK-UP TABLE
F
'
D/A HISTORICAL DROP CORRECTION
I f
:
1,
IN
D/A - FLIGHT
ELECTRODE
-
FIG.38. Electronic circuitry for the correction of historical and drop interaction effects.
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speed. Furthermore, since they use a continuous jet, fast-drying ink can be used, which is necessary when printing on nonadsorbing surfaces such as glass or plastics. Finally, the quality and resolution obtained with Sweet printers is presently still superior to that of commercial drop-on-demand printers. In view of this, many industries have developed and are marketing Sweettype printers. Recognition Equipment, Inc., was probably the first one to market a reliable printer for applying bar codes on checks for automatic sorting. The printer used fluorescent ink and operated at a maximum speed of 2000 ft/min. A. B. Dick (Keur and Stone, 1973) introduced the method for the marking of products such as beer cans, plastic bottles containing hair shampoo, pharmaceuticals, etc. This printer, as well as the Domino printer for the same application, prints at speeds up to 1500ch/sec. The Moore Business Form Company used the method for address printing on business forms. IBM in their word processor 6640 made use of the high print quality obtainable with the Sweet method. In their printer, a 24 x 22(40 x 28) matrix was used to form characters, the quality of which cannot be distinguished from conventional typewriting with the naked eye (Kuhn and Myers, 1979; Buehner et al., 1977). A similar printer was later built by the Sharp Corporation. The Sweet method has also been used to produce color copies of high quality. IBM demonstrated very impressive high-quality color prints made on an experimental drum copier (Crooks et al., 1982; Jaffe et al., 1981), and Kobayashi et al. (1982) built a color ink-jet printer, printing on a continuous web, to be used as computer output peripheral. Both systems used four ink jets to provide colors of magenta, yellow, cyan, and black. Even facsimile devices were developed (Knuth et a/., 1978). To increase speed and enable printing on paper as well as textile webs, Mead Digital Systems developed a highly sophisticated printing bar (cf. Fig. 39) in which a linear array of up to 1280 parallel jets emerged at a speed of 3 m/sec from a 26-cm-long orifice plate. The jets have a diameter of 75 pm and are spaced about 0.2mm from each other. After emerging from their nozzles, each jet transverses an individual charge electrode. All the jets then travel through a common deflection field toward the recording paper web. Only those jets that are uncharged reach the paper, the charged droplets being deflected and intercepted. This is a purely on-off system capable of printing at web speeds of up to 200 m/min. Because of the high complexity of the system, only single-color printers have been commercially produced. By using fewer nozzles and printing at lower web speeds, Cambridge Consultants Ltd. in England have developed a web printer using a slightly different approach (Keeling, 1981;Pullen, 1982; Stolzenburg, 1982). As illustrated in Fig. 40, the jets are spaced at about 1 cm from each other and each jet
INK-JET PRINTING
FIG.39. Mead printing bar.
FIG.40. Multijet web printer (Courtesy of Cambridge Consultants, Ltd).
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J. HEINZL AND C. H. HERTZ
prints a strip of the same width. Printing speeds in the range of 0.1 to 1 m/sec have been demonstrated in the laboratory at medium to high resolution. D. The Hertz Method
The physical process used by the Hertz method has already been illustrated in Fig. 4. Similar to the Sweet method, the drops of the jet are electrically charged by a control electrode placed close to the point of drop formation. However, by applying a higher voltage to this electrode, the electric charge induced on the drops is so high that the drops repel each other on their way to the recording paper (the individual drops do not break up, cf. Section III.C.l). In this way the linear jet consisting of a train of drops is transformed into a spray. This phenomenon was observed by AbbC Nollet as early as 1749. Fillmore et al. (1977) calculated the distance z at which the average displacement of a drop from the jet axis is one drop radius. The relation they obtained is z
=
1.05 x 10-3(ud12/q)
m
(19)
where z is measured from the point of drop formation, u is the jet velocity, d the jet diameter, I the distance between two drops, and q the charge of the drops. Since I d, and q is approximately proportional to dV, we find N
z (ud2/V) (20) where V is the control voltage. Thus z is strongly dependent on jet diameter. From this we can deduce that the Lund method can be realized only with jet diameters less than 50 pm since, otherwise, unreasonably high control voltages must be used. It has been mentioned earlier that the upper frequency limit for the on-off modulation of the jet lies above 100 kHz, i.e., more than lo5separate dots per second can be printed on a moving recording paper. This is due to the fact that the spontaneous drop-formation frequency of a 10-pm jet traveling at 40 m/sec lies somewhere around 800 kHz. Since the charge on the drops determines their interaction and thereby their trajectory when traveling toward the paper, the upper frequency limit for the jet switching is somewhat lower than the number of drops formed per second. This is demonstrated in Fig. 41, which shows a jet about 4 mm from the nozzle consisting of lo6 drops/sec (MBnsson, 1971). The drops were charged by supplying 500-V pulses 10 p s long to the electrode E at a rate of 4 x lo4 pulses/sec. Thus, a train of 10 charged drops is followed by 15 uncharged drops. Farther down the jet path, the charged drops will have left he jet axis and can be prevented from reaching the recording paper as described later. N
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INK-JET PRINTING
1 mm
.
%A,>..
~
..................I ......,.,".......... I
1 L . . . . . . . . . . . . . . . . .
..
.L...D.O.*.
DIRECTION OF JET FIG.41. Train of drops charged by 500-V pulses 3 mm from the point of drop formation. Pulse length = 10 / I S ; pulse repetition rate = 40 kHz; jet parameters are the same as in Fig. 28b. Repulsion can be observed between the charged drops, which show a tendency to leave the jet axis on the right side of the photograph.
Thus, only the uncharged drops will generate dots on the paper, a visible dot usually consisting of 5 to 10 uncharged drops. The repulsion between the charged drops can easily be observed, while the uncharged drops tend to approach each other and merge. This is especially pronounced at both ends of the uncharged drop trains, which is due to air resistance and electrical forces between charged drops. Also, the historical effect mentioned earlier plays an important role in the drop merging, which therefore increases with the magnitude of the control voltage. Since the merged drops often have intermediary charges and are relatively large, they can affect the print quality (Miinsson, 1972). Therefore, the control voltage should be kept as low as possible. Since single drops from a 10 to 15-pm jet are very small, they cannot be observed on the recording paper by the naked eye. To generate a visible dot, at least 5-10 drops must be placed onto the same spot of the paper. The exact placement of each and every drop is, thus, not critical: A single misplaced drop does not impair the image quality except for a slight increase in background. Because of this, the mass and charge of the drops do not have to be controlled as exactly as in the Sweet method, and no special care is required to synchronize the control signal and the phase of the drop-formation process. Even the historical effect is only of minor importance. Therefore, in all practical applications only spontaneous drop formation is used, thereby eliminating the need for an ultrasonic transducer to generate mechanical vibrations on the jet. On the other hand, the print quality obtained with the Hertz method increases with the number of drops formed per second, making advisable the use of ultrasonic agitation of the jet for maximum print quality. 1. The Electrode System
In Fig. 4 a device was shown that allowed the intensity modulation of a trace on a recording paper by an electrical signal controlling the ink jet. However, as mentioned, in this case the drops of the spray still reach the paper and generate a general background that seriously decreases the quality of the recording. A simple way used to prevent this is shown in Fig. 42, in which the
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J. HEINZL AND C. H. HERTZ
T
C
m 5
0
mm
fo pump
FIG.42. (a) Simplified control electrode structure. A tube T of electrically conductive, porous material forms the electrode controlling the jet J on its way to the recording paper on the rotating drum D. Ink collected on the inside of the tube T is sucked away by a vacuum maintained in the casing C. (b) Expanded view of electrode.
electrode E is essentially in the form of a tube 7', which is made of electrically conductive material. When a suitable voltage is applied to this tube, the jet is broken up into a spray as described earlier, the drops of which are deposited on the inside of the tube. To remove these drops, the tube is made from porous material and enclosed in a casing, which allows the ink to be sucked away through the walls of the tube by a suction pump. For better reduction of the background generated by the spray, the tube is often tapered down to a small hole at its far end as shown in the figure. While the undisturbed jet can pass through this hole to the recording paper, nearly all the drops of the spray will be intercepted. In an actual arrangement, the nozzle is made of a glass capillary tapering down at the far end to form an orifice with an internal diameter of 10 pm from
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INK-JET PRINTING
which the jet is ejected. The other end is connected to a suitable filter that prevents small particles in the ink from entering the capillary and clogging the nozzle. As shown in Fig. 42, the nozzle end of the capillary is placed about 1-2 mm from the control electrode tube. In this way the point of drop formation of the jet lies approximately at the entrance of the tube. This device was used for the reproduction of high-quality images on a facsimile recorder. The recorder consisted of two drums D, and D, mounted on the same axis and rotated with a surface speed of 3 m/sec (Fig. 43). On one of the drums an original was placed and scanned by an electrooptical device 0 with a scanning line density of 10 lines/mm. After amplification the signal obtained from the photosensor was used to control the ink jet that reproduced the original on the second drum. Ultrasonic excitation of the jet was used to optimize resolution and the maximum control voltage applied was 500 V. Black and white, as well as color pictures (using multiple jets), were reproduced with good resolution (Minsson, 1972). Figure 44 shows a magnified view of some black and white resolution test patterns. Since the amount of ink reaching the drum through the orifice in the control electrode of Fig. 42 depends on the angle of the spray and thus on the voltage applied to the control electrode, the intensity of the trace can be varied continuously from nearly zero to the maximum density of about 0.8 (Mhnsson, 1972). Thus levels of gray or shades of color can be readily produced with the electrode device shown in Fig. 42. Minsson (1972) measured the modulation transfer function of the facsimile recorder shown in Fig. 43 and found its limiting resolution to be 5 line pairs/mm. Higher recording speeds reduce resolution because of need for a larger jet. However, experiments with a 20-pm ink jet and a drum surface speed of 25 m/sec still gave acceptable results, the resolution having decreased to about 2 line pairs/mm.
1
I /
Dl FIG.43. Facsimile system using ink-jet method of the Hertz type
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J. HEINZL A N D C. H. HERTZ
FIG.44. Ink-jet reproduction of resolution test pattern. The numbers indicate the line pairs per millimeter in each pattern.
Despite the improvement obtained, the control electrode shown in Fig. 42 produces a certain amount of background even in the off-position of the jet, since a few ink drops of the spray always find their way through the electrode aperture to the paper independent of the spray angle. Since this was considered to be unacceptable for computer output printing, another method to prevent the spray drops from reaching the paper was developed, as previously mentioned (cf. Fig. 4b), by using a transverse electric field to deflect the charged drops into a catcher while allowing the uncharged drops of the linear jet to pass between the plates to the paper. Unlike Sweet’s method, the exact positioning of the deflected drops is of no importance. This allows the simplified electrode system shown in Fig. 45a to be used. Here the control electrode and the lower deflection electrode are combined into one piece, which is made of conductive porous material. Ink deposited on this material is removed by a suction pump. Close to the drum a razor blade edge fastened to the lower electrode acts as an interceptor for the deflected ink drops. In this system, the control voltage can be applied either to the lower electrode or to the conduit electrode as shown in Fig. 45a. The deflection field is established by applying a dc voltage of 1.5 to 2 kV to the high-voltage electrode. This system allows only on-off control of the recording trace and is therefore not useful for continuous gray scale generation. A suitable signal voltage is a digital signal of 200-V amplitude.
INK-JET PRINTING
157
1500 V
T signal in
(a) 1.5 kV
FIG.45. (a) Simplified electrode system. The control electrode and lower deflection electrode are combined into one structure. (b) Perspective view of system with three parallel ink jets as used in the color plotter of Fig. 46. Here an electric shield is inserted between high-voltage electrode and drum (not shown) to avoid ink on electrode from being attracted t o the drum by electric forces.
This system has the advantage that it allows the placement of several parallel ink jets close to each other, since the two deflection electrodes of the system can be made common to all of them as shown in Fig. 45b. Individual control signals for the jets can then be applied to their respective ink conduits. Crosstalk between the jets is avoided by making these conduits from plastic tubes with small cross section so that the electric resistance of the ink column in the tubes is > 1 M a , thereby essentially insulating the electrodes in the conduits from each other and the ground.
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2. Color Plotter This electrode system has been utilized in a new color plotter to be used as an output device for computer graphics (Smeds, 1973; Bladh, 1982; Bladh and Jern, 1980; Bladh et al., 1979). As shown in Fig. 46, the three ink jets of Fig. 45b are mounted on a carriage that is slowly moved by a screw drive parallel to the surface of a drum rotating with a constant surface speed of 5 m/sec. The ink jets have the subtractive primary colors magenta, yellow, and cyan and are aligned along the axis of the drum. When plotting an image, the carriage with the ink jets is slowly moved by a stepper motor parallel to the axis of the drum carrying the recording paper. Since the motion of the rapidly rotating drum is monitored by a shaft encoder, the exact position of each ink jet relative to the drum surface is known at each moment. Thus, by on-off modulation of the three colored jets an image can be printed on the recording paper. The on-off control of the jets during the plotting operation is effected by signals from a tape station. The image to be plotted is previously prepared on this tape in suitable format by a computer that has been fed by the user’s input data. Special software has been developed for this purpose (Jern, 1978). The plotter is produced on a commercial basis by Applicon, Inc. (Fig. 7) and was also made for the plotting of medical scintigrams by Siemens AG, (Kraus and Wiesmuller, 1975). In the present plotters, a recording line density of 5 lines/mm is used. To ensure the same resolution in the x and y directions, the size of a picture
n
Motor
FIG.46. Basic units of the color plotter. The information to be plotted is computed in a suitable format by a computer and stored on a magnetic tape. This tape is then read by a magnetic tape unit into a buffer memory. The read-in and read-out periods of the memory are governed by control circuits, and the read-out signals are used to modulate the intensity of the three ink jets (magenta, yellow, and cyan) mounted in front of the rotating drum carrying the recording paper.
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INK-JET PRINTING
element (pixel) was chosen to be 0.2 x 0.2 mm. At a drum surface velocity of 5 m/sec, such a pixel is printed in 40 psec. During that time the pixel receives about 20-30 drops of ink. The diameter of the ink jets is 15 pm, their velocity is about 40 m/sec, and the distance from the nozzle to the paper is 30 mm. Using these parameters, the plotter can print a letter-sized document in three colors in 1 min. Compared to the example shown in Fig. 44, the resolution of the plotter is appreciably lower. This reduced resolution was chosen for the following reason. The number of pixels in a letter-sized image having a pixel size of 0.2 x 0.2 mm is about 1.4 x lo6. Since for each pixel three colors have to be on-off controlled, the information content of the image is 3 x 1.4 x lo6 4.106 bits. This large amount of information must be handled by the computer when preparing the picture, which is reflected in the cost of computer time. An increase in resolution would have increased this cost above practical limits when the first plotter was planned in 1971. However, plotters with a pixel size of 0.1 x 0.1 mm have been constructed (cf. Fig. 47). By superposition of all three ink jets, a black color is obtained. Since every pixel can be on-off modulated, only a maximum of eight color shades can be obtained in each pixel by combining the three colors. However, since the pixel size is quite small, an ordered dither technique using a matrix consisting of 16 pixels was employed to create color shades (Jarvis et al., 1976; Judice et al., 1974; Nilsson and Nygren, 1981). By filling only certain pixels in each matrix with color, more than 4000 shades of color can be created. A magnified view of such a plot is shown in Fig. 47.
-
3. Printing of Alphanumeric Characters
Several different approaches using the Lund method for printing of alphanumeric characters have been investigated (Ernbo and Hertz, 1969; Ernbo, 1972), the most successful of which was proposed by Hertz in 1973 (Erikson, 1975). Here the jet direction is oscillated mechanically as indicated in Fig. 48 with a frequency of 1 kHz or higher. This can easily be achieved with an ink-jet oscillograph system of the Elmqvist type (cf. Fig. 33). In this way a sinusoidal trace is generated on the moving recording paper. On its way to the paper, the jet transverses an electrode system of the type described in Fig. 45b. By applying a suitable control signal to this electrode system, the ink jet can be on-off modulated in such a way that it can reach the recording paper only in those positions where ink is required'to mark the desired characters (cf. Fig. 48). Using a 5 x 7 matrix, quite high character printing speeds should be obtainable with this device. However, due to difficulties in synchronizing the character matrix control signal with the mechanical movement of the jet, reliable printing can be achieved only by printing on the upstroke of the sine
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J. HEINZL AND C. H. HERTZ
FIG.47. Generation of color shades by dither technique. Only on-off control of the ink-jet trace is used.
INK-JET PRINTING
161
t
FIG.48. Characters can be printed by intensity modulation of a single jet J, which is mechanically oscillated horizontally, generating a sine wave on the recording paper R . The paper is moved vertically at constant speed and the jet is normally held in the off mode. When the jet is switched to the on mode in a suitable time sequence by applying voltage pulses to the system electrode E, characters are generated as shown.
wave (Erikson, 1975). This, together with the limited amount of ink that can be deposited by so small an ink jet, allows for printing rates of not more than 500ch/sec at the most. The speed of this system is thus about three times lower than that of a Sweet printer. However, because of its greater simplicity, it has been used in addressing machines (Bell & Howell). Furthermore, a single jet can be used to write several lines of text on top of each other simultaneously, which proved useful in writing text on ECG records in the hospital (Jansson et al., 1976) and recording echocardiograms (Lindstrom et a/., 1973). 4. System Considerations
Contrary to the Sweet-A. B. Dick method, the Lund method is very insensitive to change in system parameters such as pressure, ink viscosity, and temperature. Also, control of the drop-formation process by ultrasonic vibrations is not required. This is partly due to the fact that the ink drops are so small that a single misplaced drop is not visible to the naked eye. However, since the Lund method requires a much smaller nozzle than all other ink-jet methods, problems of nozzle clogging and ink composition are aggravated. The production of the nozzles is also difficult, and they must be handled with great care. IV. METHODSEMPLOYINGMECHANICAL VALVES
A . Introduction
The methods described earlier comprise the most successful applications of ink jets in the graphic arts. Aside from these, several other concepts have been proposed. A typical example is the use of continuous jets using magnetic
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ink, which can be deflected by inhomogeneous magnetic fields (Kazan, 1966; Kuhn and Myers, 1979). Since most of these methods have not been realized outside the laboratory, they will not be discussed here. There are, however, some tasks in graphic arts that require the advantages of ink-jet printing stated at the beginning of this chapter but which cannot be realized with the methods described so far. These tasks require large amounts of ink, like the printing of large characters on boxes or packages, the printing of very large posters, and the printing of carpets. In all of these applications, the ink flow through the nozzles must be large and cannot be controlled by any of the mechanisms described earlier. Fortunately, however, at the same time the upper frequency limit of the control mechanism required in these applications can be as low as 500-1000 Hz without seriously impairing the quality and readability of the print. This makes it possible for electromagnetic valves to be used to control the ink stream.
B. Systems Employing Mechanical Valves The most direct of these methods has been developed by the Swedot company in Sweden and Printos Marsh company in the United States for marking boxes and packages with alphanumeric characters 3- 10 cm high. To this end seven to nine nozzles are mounted in a row on a printhead. The nozzles throw large-diameter continuous ink jets at relatively low speed against the surface to be printed. In the conduits that lead the ink from a common pump and ink reservoir to the nozzles, electromagnetic valves are installed, which allow on-off control of the ink flow to each of the nozzles separately. The upper frequency of this control is limited by the inertia of the valve-conduit system to about 500 Hz. If the printhead is moved at a suitable speed relative to the surface to be printed and the valves are actuated by electric signals for, say, a computer, alphanumeric characters can be printed at a speed of about 20-50ch/s. While the quality of the characters is not very high, it is good enough for most applications where marking by stencils and similar handheld devices have been used previously. A slightly different system for the control of ink jets has been developed by Suenaga (1971) for the printing of very large color pictures intended to be used as posters or life-size background in TV studios. In his NECO printing systems, Suenaga used four jets with the colors magenta, yellow, cyan, and black to print on a recording paper fastened to a large drum with a diameter of up to 2 m. The voltages to control the jets were picked up as optical signals from an original mounted on a small drum fixed on the same axis, thereby providing a magnification of the original equal to the ratio of the diameters of the drums.
INK-JET PRINTING
163
Suenaga used jets very similar to conventional paint spray guns, where paint is sucked out of a nozzle and converted into a spray by a rapidly moving stream of air. By controlling the air stream with mechanical valves, he was able to regulate the amount of ink deposited on the drum with an upper frequency limit of 500 Hz. Since the ink is deposited in the form of a spray, the width of the ink trace on the paper is about 2 mm or more; however, due to the size of the pictures this low resolution does not matter when viewed from some distance. A drum surface speed of up to 2 m/sec can be used. The printing of carpets had been a very time-consuming task until recently when the Deering Milliken Research Corporation developed a carpet printing machine by using a large number of fluid jets carrying dye that are mounted on a printing bar positioned across the moving carpet web. The internozzle distance is 2.5 mm and the nozzle diameter about 100pm (Klein and King, 1975). Each jet is individually on-off controlled by an air stream, which is blown at right angles to the jet stream (Taylor, 1980). In the off position, the jet is deflected into a gutter. The air streams themselves are controlled by the electromagnetic valves energized by signals from a signal source, determining the pattern to be printed. If multicolored patterns are desired, several printing bars are positioned one after another along the direction of web movement, each bar applying one color.
V. CONCLUSIONS From the literature on ink jets, it is evident that extensive research has been conducted during the last two decades to make use of electrically controlled ink jets for the printing of characters and other tasks in graphic arts. In spite of this, the new technology has come into more widespread use only recently. This is due to several circumstances, for example, the presence of competitive technologies, such as different types of electrostatic printing and the demands of the market. However, the most important reason for this long delay was probably the need for fundamental research and the collecting of know-how, which proved to be of decisive importance to the development of reliable ink-jet systems. Some of the more important results of this research effort have been discussed in Sections I1 and 111. From the present use of ink-jet methods outside the laboratory, it is obvious that two fundamentally different technologies have proved their usefulness and reliability: drop-on-demand and continuous-jet methods. From the experience we have today, these two technologies seem to fit into different parts of graphic arts because of their properties. Drop-on-demand methods are relatively slow and do not allow printing distances of more than a few millimeters between the printhead and the record receiving surface, but they
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are comparatively inexpensive to produce. Furthermore, because of their simplicity, drop-on-demand devices can be mounted close to each other to form a multinozzle array, whereby their speed becomes comparable to that of continuous-jet systems in many applications, for example, teletype and facsimile printers. Continuous-jet systems, on the other hand, excel by their speed and their ability to print on surfaces as far as 2-3 cm from the printhead. Therefore, they are mainly used today to print short alphanumeric information or bar codes on irregular surfaces, such as checks, envelopes, beer cans, and industrial packages. Since these jets have small diameters and high dropformation frequencies, they are also used for the printing of high-resolution color images on computer output devices. Attempts to use continuous jets as printers in word processing equipment (Buehner et al., 1977) apparently have not met with great success. Furthermore, while an array of closely stacked continuous-jet devices promises remarkable printing speeds (Mead Digital Systems), such systems become very complex and consequently less reliable. Therefore, the dream of replacing the conventional rotary newspaper press by a sophisticated ink-jet array with electronic input only seems to be fairly remote today. Generally, it can be said that continuous-jet systems because of their greater complexity and consequently higher cost are presently used in more specialized applications only. Therefore, the number of drop-ondemand systems in actual use probably vastly exceeds the number of continuous-jet systems. It has already been mentioned that the electrostatic laser printer is considered to be a serious competitor to ink-jet printers. This is especially true since the price of laser printers has been reduced appreciably in the last few years as a result of a large research and development effort. Therefore, at the present time it may be hard to judge which of the two systems will prevail in the future, or if each of them will find special niches of its own, for example, drop-on-demand devices in teletypes and small computer output printers. However, the color capability of ink-jet devices probably gives this technology an edge over electrostatic methods for some time to come. This is especially true since color is being used increasingly in computer CRT displays. Therefore, ink-jet printers can be expected to gain importance in the fields of computer hard copy output devices and color facsimile machines. Even in the field of medical imagery, such as computer tomography, ultrasonic imaging, and nuclear magnetic resonance (NMR), ink-jet printers may become of importance. Today it does not seem to be impossible that highresolution color ink jets might replace the conventional photographic color printing process used to make enlarged prints from amateur color films. Hence ink jets might be the natural printout device for the newly introduced electronic camera. However, this will be a question of cost and of output velocity. Up to now, ink printing is still a slow method compared with laser
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printing; ink printing is fast only in comparison with needle printing. Printing velocity strongly depends on resolution. To obtain a resolution of 4 dots/mm or 100 dots/in., a continuous jet with 1 nozzle per color needs approximately 1 min/sheet (Am’) whether the Sweet method or the Hertz method is used. At this degree of resolution one continuous jet can be replaced by six drop-on-demand nozzles. At a drop rate of 3 kHz, the six drop-on-demand nozzles achieve the same printing velocity, even if the entire sheet is to be covered. At a resolution of 12 dots/mm or 300 dots/in., one continuous-jet nozzle is equivalent to about 15 drop-on-demand nozzles. Consequently, if more than 16 drop-on-demand nozzles per color are provided, the printing velocity can be increased in comparison with that obtained by one continuous jet with one nozzle per color. We could also think of increasing the comparatively small drop rate of drop-on-demand nozzles. However, a significant increase in this drop rate is difficult to bring about. It makes the nozzles expensive, less simple, and less robust. Higher drop rates lead only to a higher printing velocity if the velocity of the nozzles with respect to the paper is increased as well. As a consequence, irregularities in the velocity of the flying drops would lead to greater inaccuracies in dot positioning on the paper. Therefore, the better solution is to increase the number of nozzles and to use simpler and smaller transducers even if the drop rate of the single nozzle is reduced thereby. Of special interest are nozzle arrays with transducers and nozzles spaced in coincidence with their resolution, for example, 12 nozzles/mm for a resolution of 12 dots/mm, and which are extended over the entire width of the paper. A resolution of 4 dots/mm requires 800 nozzles per color for a sheet of 200-mm width. The output of one sheet takes only 1.2 sec even if the drop rate does not exceed 1 kHz. A resolution of 12 dots/mm requires 2500 nozzles per color, and the output of one sheet takes 3.6 sec at a drop rate of 1 kHz. The bubble jet shows that such arrangements are feasible, and printed samples (Fig. 5) show the high standard that has been reached, if only under laboratory conditions so far. It remains to be seen whether the electric heating element producing bubbles within the ink is the best transducer possible. It is doubtful whether an array can be developed which can be produced economically and which has high reliability and long lifetime. This reasoning is true provided that the drop-on-demand and continuous-jet systems can print in the on-off mode only, i.e., that each pixel is formed in one of two possible states: ink or no ink. This is probably correct for drop-on-demand systems for some time to come. Small continuous jets, however, have very high drop-generation rates and often use a large number of drops to print one pixel. Thus by varying this number, different shades of a primary color might be generated in each pixel, which should improve image resolution greatly compared with ink-jet systems which reproduce shades of
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color by matrix methods. However, at present the latter method is still generally in use. Finally, we might consider if there are fields in graphic arts where jets are not used today but may be applied in the future. For high-volume production of books, newspapers, wallpapers, and textiles, ink jets will probably not attain the speed and print quality of conventional printing presses. However, these machines are clumsy and not suitable for the production of small numbers of prints, especially samples of images and textiles prior to their large-scale production on a printing press. Here ink jets may find interesting applications provided that the same ink can be used as in the conventional printing process. A similar situation exists in the ceramics industry where decorations have to be applied to pottery before it is sent to the kiln for firing. In many of these applications, pigmented inks or colors are used, which presently cannot be used in ink jets. However, if this difficulty can be circumvented, for example, by the utilization of the compound jet, several new fields of use for ink jets should develop.
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ADVANCES IN ELECTRONICS A N D ELECTRON PHYSICS. VOL. 65
Theory of Image Formation by Inelastically Scattered Electrons in the Electron Microscope H. KOHL AND H. ROSE lnstitut fur Anyewandte Physik Darmstadt. Federal Republic of Germany
I. Introduction . . . . . . . . . . . . . 11. The Mixed Dynamic Form Factor . . . . . . A. The Phase Problem in Electron Scattering . . B. Properties of the Mixed Dynamic Form Factor C. The Generalized Dielectric Function . . . . 111. Theory of Image Formation . . . . . . . . A. Image Formation in STEM . . , . . . . B. Image Formation in FBEM . . . . . . . C . The Reciprocity Theorem. . . . . . . . D. Phasecontrast . . . . , , . , . . . IV. Numerical Results. . . . . . . . , . . . A. Image of a single Atom in an FBEM. . . . B. Image of a Single Atom in STEM , . . . . C. Images of Assemblies of Atoms . , . . . . D. Image of a Surface Plasmon . . . . . . . V. Conclusion . . . . . . , . , , . . . . Appendix. . . . . . . . . . . . . . . References . . . . . . . . . . . . . .
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I. INTRODUCTION The main objective of electron microscopy is to obtain high-resolution images of thin specimens. Modern instruments yield point-to-point resolutions of about 3 A. Thus, it is possible to elucidate the properties of the object on an ultramicroscopic scale (Bethge and Heydenreich, 1982; Reimer, 1984). In such an instrument, the image is formed by electrons, which are scattered in various directions within the objective aperture. In electron energy loss spectroscopy (EELS), we measure the intensity of inelastically scattered electrons as a function of energy loss and scattering I73 Copyright 8 1985 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-014665-7
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angle. This technique is widely used in solid-state physics to obtain information about the dynamic properties of the specimen as a whole. For recent reviews see Raether (1980), Schnatterly (1979), Silcox (1979), and Colliex (1984). Recently, both techniques were combined under the name of analytical electron microscopy. Thus, it is now possible to obtain energy-loss information from a small volume of a few cubic nanometers. By registering only those electrons that have suffered a characteristic energy loss (say a K-loss corresponding to a particular element), we can quantitatively determine the elemental distribution in an object (Isaacson and Johnson, 1975; Egerton, 1979, 1984; Joy, 1979; Maher, 1979; Colliex, 1984; Colliex and Mory, 1984). An extensive bibliography has been given by Egerton and Egerton (1983). Experimental (Isaacson et al., 1974) and theoretical (Rose, 1976a,b) investigations show that images taken with inelastically scattered electrons are blurry as compared to an elastic image of the same object. This “delocalization of the interaction” makes the interpretation of inelastic images more difficult. Despite the striking progress in experimental abilities, a detailed theory of the process of image formation is still lacking. Most reviews on image formation treat the contribution of the inelastically scattered electrons as a deleterious side effect (Misell, 1973; Humphreys, 1979). In view of the recent progress in analytical electron microscopy, it seems worthwhile to examine image formation by inelastically scattered electrons in more detail-first, the scattering properties of the object (Section 11) followed by electron-optical considerations (Section 111) and image calculations (Section IV). 11. THE MIXEDDYNAMIC FORMFACTOR
A . The Phase Problem in Electron Scattering
The standard geometry for a scattering experiment is shown schematically in Fig. 1. The object is illuminated by an incident plane wave with wave vector k,. A far-field detector registers all electrons that have been scattered in the direction k, and determines their energy and angle of scatter. For simplicity, we shall restrict our discussion to the case of purely elastic scattering. The differential cross section dg/dQ gives the probability that an electron is scattered into the solid angle dQ. It is given by where f ( 0 , 4 ) denotes the scattering amplitude for the direction determined by the polar angle 0 and azimuth 4. The differential cross section is equal to
THEORY OF IMAGE FORMATION
i
175
.c ko
object
detector FIG.1. Conventional arrangement for diffraction measurements; K = k, - k, is the scattering vector, and k, and k, are the wave vectors of the incident and the emergent electron, respectively.
the square of the modulus of the scattering amplitude. For fast incident electrons with wave vector k,, scattered by a thin object, we can use the firstorder Born approximation to calculate the scattering amplitude. For scattering in the direction k,, we then find f(K)
=
-
m &
1
V(r) exp(iKr) d3r
where K = k, - k, denotes the scattering vector, V(r) the object potential, m, the electron mass, and 12 Planck’s constant. Thus, to first order, the scattering amplitude is proportional to the Fourier transform of the object potential. As can be seen from Eq. (l), we can extract only the modulus, not the phase, of the scattering amplitude from a diffraction experiment. Thus it is impossible to determine unambiguously the object potential from experimental diffraction data. This so-called phase problem has grave consequences for diffraction experiments, because there are an infinite number of functions with a prescribed modulus of the Fourier transform. (For example, a translation leads to a shift in phase of the scattering amplitude.) Even with additional information, it is often impossible to determine the object potential unambiguously. For example, it is impossible to distinguish between the potentials V(r) and P(r) = V ( - r), because T(K) = f’*(K), and
176
H. KOHL AND H. ROSE
thus lY(K)I2 = If(K)12. This fact has serious consequences for crystallography, where a noncentrosymmetric structure cannot be distinguished from its inverse, even though the types of atoms in the unit cell and their scattering properties may be known. Thus, even with considerable amount of a priori information, we cannot determine the object structure by a scattering experiment as shown in Fig. 1. If we consider inelastic scattering, the situation is even more complicated. It is not possible to determine the spatial structure of the transition density of the excitation by examining the scattering data. All we can do is compare the measured data with theoretical calculations. However, as we have seen in the case of elastic scattering, agreement between theory and experiment does not necessarily mean that the theoretical model is correct. Thus, a method to measure the phase of the scattering amplitude is of great experimental interest. Let us now discuss a setup that allows us to determine the phase of the scattering amplitude (Fig. 2). A biprism splits the incident wave into two
K' K
FIG.2. Arrangement for determining phase effects in elastic and inelastic scattering. The object is illuminated by a coherent superposition of two plane waves with wave vectors k and k . The detector registers the intensity in the direction of k,. [From Kohl (1983a). Copyright 1983 North-Holland Publ. Co., Amsterdam.]
THEORY OF IMAGE FORMATION
177
FIG.3. Intensity distribution in the specimen plane when two inclined plane waves with relative phases 4 = 0 or 4 = n are impinging on the object. Each dot represents an atom, and a is the lattice constant. The origin is located on an atom. The directions of incidence of the two waves enclose an angle i j a . [From Kohl (1983a). Copyright 1983 North-Holland Publ. Co., Amsterdam.]
coherent plane waves, which form interference fringes in the object plane (Mollenstedt and Diiker, 1956). The initial state of the electron is then given by (1/42)(exp(ikr)
+ exp(i4) exp(ik’r))
(3)
where k and k‘ are the wave vectors of the two incident plane waves impinging on the object. The factor ei4 determines the phase relation between the two waves. To understand its meaning a little better, let us first calculate the intensity distribution in the object plane. It is given by I(r)
=
1 + cos[(k - k ) r - 41
(4)
The vector r denotes the position on the object. Suppose the object is periodic with a lattice vector ae,. We adjust k and k’ so that k - k = (2n/a)e, is a reciprocal lattice vector.’ Then the interference fringes have the same periodicity as the object. The phase 4 determines the position of the fringes on the lattice (Fig. 3). Depending on 4, either the atoms will be illuminated predominantly (4 = 0) or else the space in between ( 4 = n). The probability of registering an electron in the detector will depend on the position of the interference fringes on the object. The count rate of the detector is proportional to
1.m)+ exP(i4>.f(K’)l2
(5)
where K = k - k, and K’ = k’ - k, denote the two scattering vectors. By measuring the intensity as a function of K and K‘, we can obtain the phase of the scattering amplitude (Gabor, 1957; Hoppe, 1969a,b; Hoppe and Strube,
’ We use the convention a;aj = 2nsij, where a; is a reciprocal lattice vector and aj a lattice vector in real space. This definition, which is common in solid state physics, differs from the one used in crystallography (a;aj = S i j ) by a factor of 2n.
178
H. KOHL AND H. ROSE
1969; Berndt and Doll, 1976, 1978, 1983). Formula (5) shows that an experiment as depicted in Fig. 2 cannot be completely described by calculating cross sections. The phases play an important role in this type of experiment. Let us now consider inelastic scattering events. The detector is assumed to register the electrons as a function of energy and angle of scatter. T o describe the scattering process, we must use the product states of the incident electrons and the object state. (The influence of exchange is neglected here.) The initial state is given by
+ exp(i4) exp(ik’r)] Im)
(l/J2)[exp(ikr)
(6)
where Im) denotes the initial state of the object. Here the object is assumed to be initially in a pure state. The final state is equal to eikf‘In). For fast electrons and thin objects, we can use the first-order Born approximation to determine the count rate. Since the detector registers only the direction and energy of the scattered electron but not the final state of the object, we must sum over all possible final object states with a given energy. Using Dirac notation
I kfn)
= eikf’In>
we obtain for the transition rate from the initial to the final state
+ exp(i4)( km I V I kf n ) (kfn I V I k’m) + exp( i&)(kml Vlkfn)(kfnl Vlkm))6(wm -
-
+
o, o)
(7)
Here V denotes the interaction potential between the incident electron and the object, while hw, and hw, are the energies of the initial and final state of the specimen, respectively. The first two terms stem from the scattering of the partial wave from k (k’) to k,. They appear likewise in a calculation of a cross section. The following two interference terms contain all spatial information about the object. For scattering experiments as shown in Fig. 1, the object properties can be described conveniently by means of a density-density correlation function (Van Hove, 1954; Platzman and Wolff, 1973). We shall generalize Van Hove’s result to the case of two (or more) interfering incident waves. With V
=
C V(r i
-
Ti)
THEORY OF IMAGE FORMATION
179
where V(r - ri) describes the interaction of the incident electron with the ith particle, we obtain
where V(K) =
s
V(r) exp( - iKr) d3r
is the Fourier transform of the interaction potential V(r). For Coulomb interactions, we obtain V(K)
= e;/EoK2
where - e , is the charge of the electron and c0 the dielectric constant of the vacuum. In Eq. (8), S(K, K’, 0)=
J 2n
m
(pK(t)P-K,(o))T
-m
exp(iot) d t
(9)
is the mixed dynamic form factor (Rose, 1976a,b), where PK(~) =
C~xPC-
iKrj(t)I
j
is the Fourier-transformed density operator in the Heisenberg representation and ( )T denotes the thermal average. In Eq. (8), p m gives the probability that the object was in the state Im) before the scattering took place. By inserting Eq. (8) into Eq. (7), we obtain
+ I V(K’)I2S(K‘,K’, o) + exp(i+)V(K)V*(K’)S(K, K’, o)+ exp( -$)V*(K)V(K’)S(K’,
w = (n/h’)[I V(K)I2S(K,K, o)
K, o)]
(10) The calculations are outlined in Appendix A. Here we have included a thermal average with respect to the initial states of the object. The transition rate w is given by a sum of four terms, each of which is a product of the Fouriertransformed density-density correlation function and two conjugate Fourier transforms of the interaction potential. Thus, Eq. (10) can be viewed as a generalization of the relation given by Van Hove (1954). The last two terms of Eq. (10) contain the function S(K, K’, w ) with K # K’, whereas in a conventional scattering experiment only terms with S(K, o)= S(K, K, o)occur. Our apparatus in Fig. 2, thus, enables us to obtain additional information about the object.
180
H. KOHL AND H. ROSE
The first two terms in Eq. (10) can be obtained from a conventional scattering experiment. If we now measure the intensity with k and k’ fixed for two different phases 4, we can solve the two resulting equations for the two unThis procedure can be repeated for known values S(K, K’, w ) and S(K’, K, o). all values of K and K by varying the direction of the incident electrons and the detector. Thus, the function S(K, K’, o)is a quantity that can be measured experimentally (at least in principle). It contains information about the spatial distribution of the excitation within the object. (Subsections I1.B and 1I.C will deal with the properties of the mixed dynamic form factor. The more experimentally oriented reader can bypass them without loss of comprehension.) B. Properties of the Mixed Dynamic Form Factor
In Subsection II.A, we found that the mixed dynamic form factor S(K, K’, o)can be viewed as a generalization of the conventional form factor S(K, o).We therefore ask Which properties of S(K, o) are also true for S(K, K’, w)? First of all, let us consider two sum rules. The integral
j
m
S(K, w> do
=
(PK(o)P-K(o))T
= S(K)
(1 1)
-m
yields the static form factor S(K) (Kittel, 1964). By using Eq. (9) and interchanging the order of integration, we find
j
m
=
-m
=
(12)
Accordingly the right-hand side is called the mixed static form factor. Using the operator identity = po the number operawhere K = I K 1, H is the Hamilton operator, and tor, we obtain the generalized Thomas-Reiche-Kuhn sum rule Jm -00
w{S(K, w )
+ S(-K,
o)}dw
=
hKZ ~
m0
N
181
THEORY OF IMAGE FORMATION
Here N denotes the number of particles in the object (Kittel, 1964). By employing the double commutator [ [ H , &], P-K’], we obtain [[& PKI? P-K’I = -(hZKK’/mO)pK-K’
(15)
Taking the thermal average, we find
1 =
-
Z
1 exp(-pEm)(Em
-
En)((mlPKIn)(nlp-K’lm)
m,n
+ (ml P -K’I n>( n I PK Im)) cu
=
-A]
w{S(K, K’, 0)
+ S( - K’,
-
K, w ) } dw
-m
where p m = e-DEm/Z denotes the occupation probability, Z the partition sum,
p = l/k, T, k , Boltzmann’s constant, and T the temperature. Thus,
The magnitude of the mixed dynamic form factor is related to the expectation value of the Fourier transform of the electron density for K - K’. The larger the density variation in the object, the more pronounced are the terms for K # K’ in the form factor. In contrast to the conventional dynamic form factor S(K, w), the mixed dynamic form factor is, in general, not a real quantity. Therefore, we cannot simply carry over the proofs for a certain property from S ( K , o ) to S(K, K’, w). Rather, we must check very carefully, if the proof for S(K, w ) makes use of the fact that the form factor is a real quantity. Symmetries of the object lead to further restrictions on the form factor. Suppose the specimen has a center of inversion at the origin. Then,
(dr,t)P(r’))T
=
(p(-r? L)d-r’))T
where p(r, t ) =
1
-
ri(t))
i
is the density operator in the Heisenberg representation and p(r) = p(r, 0). By using pK(t) = p(r, t)exp( -iKr) d3r, we find
S
S(K, K’, W ) = S( - K, - K’, W )
(17)
182
H. KOHL AND H. ROSE
For a periodic object with lattice vectors a,, a 2 , a3, the relation ( ~ ( r+ ai, L > P ( ~+’ ai))T
=
S(K, K’, w ) = exp[i(K
K)a,]S(K, K‘, w )
( d r , t)dr’))T
leads to
for i
=
-
(18)
1,2, 3. This means that either S(K, K’, w ) = 0, or else exp[i(K
-
K)ai]
= 1
The latter condition is equivalent to the condition K
-
K’ = g, where
3
g = Ch,a,
hiEH
i=l
is an arbitrary reciprocal lattice vector. In short, the mixed dynamic form factor is nonvanishing only if K - K’ = g. This can be understood by looking at Fig. 3. If K - K = g, the fringes have the same periodicity as the object. In that case, we illuminate predominantly one part of the unit cell, e.g., the atoms ( 4 = 0) or the space in between ( 4 = n). Classically speaking, the scattering is then characteristic for that part of the unit cell. If the difference K - K’ is unequal to any reciprocal lattice vector, the fringes on the object have a periodicity different from the lattice. The scattering is then given by an average over the elementary cell. By using a setup as shown in Fig. 2, we can determine the spatial structure of an excitation. If the object is invariant with respect to arbitrary translations, Eq. (18) is valid for any vector a. Then S(K, K’, w ) is nonzero only if the two scattering vectors are equal (K = K ) . In that case the scattered intensity no longer depends on the position of the fringes on the object. For such a free electron gas, the conventional form factor S(K, w ) gives a complete description of the scattering properties. Let us compare the conditions K = g and K‘ = g’ for elastic scattering with the corresponding condition K - K’ = g for inelastic scattering. Electrons scattered elastically can be found only if both scattering vectors coincide with reciprocal lattice vectors (K = g and K = g’). For inelastic scattering, the weaker rule K - K’ = g applies, because for such processes a quasi-momentum hq can be transferred to the crystal. Since we must sum over all possible final states with a given energy, many excitations contribute to S(K, K’, w ) for fixed o. Finally, we shall consider the influence of time-reversal symmetry on the mixed dynamic form factor. The relation S(K, K’, 0) = S( - K’,
-
K, 0)
(19)
THEORY OF IMAGE FORMATION
183
is a generalization of the well-known formula S(K, w ) = S( - K, o)(Kittel, 1964). To apply the time-reversal symmetry, all magnetic fields must be reversed (Landau and Liftshitz, 1965). If such magnetic fields are negligible, Eqs. ( 1 6) and (1 9) yield
For objects that are invariant under time reversal and inversion, the combination of Eqs. (1 7) and (19) yields
S(K, K‘, w ) = S(K’, K, W )
(21)
Since S(K, K’, w ) = S*(K’, K, w ) , this means that in these cases the mixed dynamic form factor is real. We could also exploit the implications of point symmetries of the object. A specific example (complete rotational symmetry) will be given in Appendix C. C. The Generalized Dielectric Function
The dissipation-fluctuation theorem yields a relation between the dissipative part of a linear response function and the fluctuations in thermodynamic equilibrium. The latter are described by correlation functions. Applying this theorem to electronic excitations, we find for the conventional form factor (Pines, 1962, 1964; Kittel, 1964; Geiger, 1968; Platzman and Wolff, 1973; Schnatterly, 1979; Raether, 1980)
S(K, to)
=
-{EOhK2V/nei[1 - exp(-phw)]} Im E-’(K, w )
(22)
where E denotes the dielectric function of the medium, V its volume, and Im the imaginary part. Hence, the dynamic form factor is proportional to the imaginary part of the dielectric response function E - which describes absorption effects in the object. Reading Eq. (22) from right to left, we find that we can calculate Im E - ‘(K, w ) from measured values of S(K, w). The real part can be obtained by use of the appropriate Kramers-Kronig relation. We now try to generalize Eq. (22) to include the mixed dynamic form factor S(K, K’, w). For the derivation, we must keep in mind that S(K, K’, w ) is a complex quantity due to the spatial Fourier transforms. Thus, appropriate care must be taken when generalizing properties from the conventional dynamic form factor to the mixed dynamic form factor. The corresponding linear response function is the generalized inverse dielectric function E&(o), which was introduced into solid-state physics by Adler (1962) and Wiser (1963). This function includes the so-called local field effects, which stem from the atomic structure of the specimen.
’,
184
H. KOHL AND H. ROSE
FIG.^. Spring model to visualize the polarization of a diatomic chain induced by a homogeneous external electric field.
The dielectric function relates the induced polarization to the applied external field. To understand its meaning, we consider a one-dimensional diatomic lattice (Fig. 4). This spring model describes the effect of the different polarizabilities of the two types of atoms. Suppose we apply a homogeneous external field to the object. The polarization itself will not be homogeneous; rather it will be the sum of Fourier components that have the periodicity of the lattice. It is convenient to use the external potential Oext,where E = grad Oexc and the induced charge eOpind = div P for the calculations. For an external potential Oext,the induced charge is given by (Adler, 1962)
+
Here g and g‘ are reciprocal lattice vectors; the indexes q g and q + g’ denote the respective Fourier components. The vector q is restricted to the first Brillouin zone. Calculation of such a dielectric function is so onerous a task that only few numerical results have been reported (e.g., Van Vechten and Martin, 1972; Hanke and Sham, 1975). Nevertheless, this dielectric function is of great interest in solid-state physics. Sturm (1982) has used the nearly free electron approximation to calculate it in the region of the plasmon frequency wp. He then discusses the influence of the band structure on plasmon dispersion and linewidth. For real solids, the charge density is nonuniformly distributed within the unit cell. Consequently the plasmon excitation is not a simple plane wave. Applying Bloch’s theorem, we find that it will be modulated by a lattice periodic function. Thus, there might even be plasmon bands. The conditions under which these could be detected are discussed in Sturm’s review (1982).
THEORY OF IMAGE FORMATION
185
The excited states of an insulator can also be calculated via the dielectric formulation. The lowest excitation energy is not given by the band gap; the electron-hole correlations must also be taken into account. Such excitonic effects have been discussed by Hanke (1978). For nonperiodic objects, the sum in Eq. (23) must be extended over all vectors K’. All other properties remain unaffected. The relationship between the mixed dynamic form factor and the dielectric function is given by S(K, K’, w ) = isOhV/2ne;[1
-
exp( -fiho)](K2~&w) - K’2~,&’(w)} (24)
The derivation is outlined in Appendix B. Since S(K, K’, w ) is a complex function, Eq. (24) is more complicated than the corresponding relation [Eq. (22)]. Also, the dissipative part of the response function is no longer given by its imaginary part (Johnson, 1974). The generalized dielectric response function can be obtained from the mixed dynamic form factor by the relation s&(w)
=
hKK,+ -in
a, ~
0 - 0’
4 ~
[1 -- exp( - Bhw‘)]S(K, K’, 0’) dw’
s,hK2V
[1 - exp( - phw)]S(K, K’, w )
Here # denotes the principal value; the integral describes the dispersion, and the following term is the absorption in the object. 111. THEORY OF IMAGE FORMATION
A . Image Formation in S T E M To calculate the intensity distribution in the image, we must first determine the scattering properties of the object. We restrict our investigations here to objects thinner than the depth of field. The first-order Born and the small-angle approximation can then be used to describe the elastically scattered electrons (Cowley, 1975; Colliex and Mory, 1984). To explain image formation, a knowledge of the differential cross section is not sufficient, because it does not contain spatial information. In a semiclassical picture, we can consider the electron as a classical particle with impact parameter b and calculate the probability for the excitation of a particular object state (Fliigge, 1971). This procedure is equivalent to assuming infinite resolution (Ritchie, 1981).
186
H. KOHL AND H. ROSE
In the more general case both the incident electron and the object are treated quantum mechanically (Rose 1976a,b). The degree of coherence between incident plane waves coming from different directions must then be taken into account. Thus, image formation in a scanning transmission electron microscope (STEM) (Fig. 5) can be interpreted as a more complicated version of the biprism experiment depicted in Fig. 2. To calculate the incident wave packet at the object plane, we assume a plane wave in front of the objective lens. The object is located in its back-focal plane. There the wave function of the incident electron with wavenumber k , is given by n
J ~ ( eexp~-iy,(e)i ) exp[-ik,(p
exmoz0)
- po)ei d2e
(26)
where A ( @ )denotes the aperture function. For a circular aperture subtending an angle €lo, we find A ( @ )=
1 0
for 101 SOo otherwise
The phase shift Yo(8) = k0{(C3/4)8~- (Af/2>02)
(28) introduced by the lens depends on both the coefficient of the spherical aberration C3 and the defocus Af. The vectors po and p denote the position of an
object
aperture spectrometer FIG.5 . Schematic setup in STEM for obtaining images with inelastically scattered electrons. Here 0, is the limiting objective aperture angle, and 0,defines the spectrometer acceptance angle. [From Kohl (1983a). Copyright 1983 North-Holland Publ. Co., Amsterdam.]
THEORY OF IMAGE FORMATION
187
object point and the position of the center of the spot, respectively. The latter can be influenced by the deflection coils of the instrument. We now generalize Eq. (10) to the case where the incident electron wave packet is given by Eq. (26). The measured quantity is the current per unit energy dIldE, which is given by (Rose, 1976a,b)
I , kk:E, dI d E n30;h E,
where D(O) =
1 0
for 10150, otherwise
is the detector function, I, the incident beam current, 0,the spectrometer acceptance angle (at the specimen), E , = 13.6eV is Rydberg's energy, and E , the energy of the incident electrons. To derive Eq. (29), we have employed the small-angle approximation. For small energy losses AE 4 E,, we then find the following expressions for the scattering vectors:
+ (0 - a)] K' = k0[0,e, + (0' O)] K
=
k,[O,e,
-
where 0, = AE/2E, is the characteristic scattering angle for the energy loss AE. The two integrations over 0 and 8' reflect the coherent summation of the plane wave components of the incident wave packet, which enters quadratically in the current. The integration over the angle O must be performed over the entire solid angle accepted by the spectrometer. The factor S(K, K', C O ) / K ~ K describes '~ the scattering properties of the object in the firstorder Born approximation. [For simplicity we have restricted our discussion to electronic excitations. Phonon excitations are discussed by Kohl (1983b).] Although a certain relationship exists between our theoretical approach of inelastic image formation and the light-optical concept of object transparency, the latter is in general not sufficient to describe correctly the coherence properties of inelastically scattered radiation. Hawkes (1978) tried to describe image formation by the inelastically scattered electrons by means of an inelastic specimen transparency function. Following a recent paper by Rose (1984), let us reformulate Eq. (29) in such a way that it can be compared more
188
H. KOHL AND H. ROSE
directly with Hawkes’ results. To do so we Fourier-transform all relevant quantities exp[i(Kp - K’p’)] d28d2W and
s
a(p) = A(8) exp[ - iy,(8)] exp[ - ik08p] d28 Conversely,
j
s ( ~ ~ = :w(p, ~p’, )w ) exp[ - i(Kp - K’p’)] d2p d2p‘
(32)
(33)
(34)
and A ( @ )exp[ -iyo(8)]
=
(qj
~ ( p exp(ik,Op) ) d2p
-
(35)
The function a( - p) is the amplitude of the wave function in the object planeup to the phase exp(ikz)-where the Gaussian focus is centered at the origin. Inserting these expressions into Eq. (29), we obtain dl dE
-
I , kk:E, 7c3Bgh E, x exp[ik,@(p’
-
p”)]D(@) d2p’d2p”d 2 0
(36)
The quantity w(p, p’, w ) has been called the “cross-spectral object transparency” by Rose (1984). For purely elastic scattering by a static potential in the dark-field mode, we obtain lo dE-rc30ih
E,
6(w)
jI
ja(p
-
p‘)T(p‘) exp( - ik,@p’) d2p‘
- lo k:EH 6(w) j u * ( p - p’)u(p - p”)T*(p’)T(p”) 7c38gh E,
x exp[ik,O(p’ - p”)]D(@) d2p‘ d2p“d 2 0
(37)
To include inelastic scattering, Hawkes (1978) has generalized Eq. (37) by replacing the object transparency T(p) by a phenomenologic complex transparency function T(p, w), where w determines the energy loss. Comparing the resulting expression with Eq. (36), we find that w(p’, p”, w ) should have the form NP’, P”, 0)= T*(P’, w)T(p”, w )
(38)
THEORY OF IMAGE FORMATION
189
However, w(p’, p”, (0)factors in this way only in the special case of transitions between pure nondegenerate states. In practice dipole excitations play a dominant role. For free atoms, then, at least one of the two object states must be degenerate.Due to theorthogonalityoftheobject states((n1m) = Oforn # m), the partial waves of the scattered electrons leading to different object states must be added incoherently, even though they might have suffered the same energy loss. The factorization, however, implies that for coherent illumination, those partial waves that have suffered the same energy loss remain coherent. Therefore, the object is not described accurately by an inelastic object transparency function. Rose’s (1984) cross-spectral object transparency must be used for a correct description of the inelastic image. Equation (29) is based on an assumption of fully coherent illumination. The importance of partial coherent illumination has already been emphasized by Hawkes (1978), who also worked out the implications of the inelastic transparency of the restricted form [Eq. (38)]. Rose (1984) has given a unified treatment of elastic and inelastic scattering, including partial coherence, by use of a modified Glauber approximation.
B. Image Formation in FBEM Let us consider a fixed-beam electron microscope (FBEM) as shown schematically in Fig. 6. The object is illuminated by a plane wave with wave
Kohler illumination
object
4%
imaging energy filter
image FIG.6. Image formation in an FBEM equipped with an imaging energy filter. The angles 0, and 8, are the illumination cone angle and the objective aperture angle, respectively. [From Kohl (1983a). Copyright 1983 North-Holland Publ. Co., Amsterdam.]
190
H. KOHL AND H. ROSE
vector k,. An imaging energy filter is incorporated in the imaging system below the objective lens. The latter can be either a Castaing filter (Castaing and Henry, 1962; Ottensmeyer and Andrew, 1980) or an omega filter (Rose and Plies, 1974; Krahl 1982). Since its detailed properties are of no concern for our considerations, we have drawn it as a black box. We assume that it removes all electrons, except those whose energy lies within a prespecified range, without introducing an aberration into the image. After the scattering process, the state of the system composed of the object and the incident electron is given by
The expansion coefficients A,(r) denote the electron wave function if the object has been excited from Im) to In). We shall restrict our discussion to thin objects so that the first-order Born approximation can be used. If we had no objective lens, the coefficients A,@) would be given by A,(r)
= exp(ik0r) 6,,
+ f,(K)Cexp(ik,r>/rl
(40)
for r + co,wheref,(K) is the scattering amplitude for the excitation of the nth state of the object and
k,
= ,/kg
- 2m0AElh2
(41)
is the wave number after the scattering has taken place. In an electron microscope, however, the electrons are registered in the image plane below the objective lens. Thus, the influence of the lens on the partial waves of the scattered electron must be taken into account. Next, let us calculate the value of A,@) in the image plane. In free space every single function A,(r) obeys the wave equation AA,(r)
+ k;A,(r)
=0
(42)
Also, the value of A,(r) just below the object plane is known. Therefore, we can use the Kirchhoff integral to calculate A,@) in any other plane, just as is done for elastically scattered electrons (Lenz, 1965, 1971; Hanszen, 1971; Hawkes, 1973,1984; Howie, 1984). For each A,(r), we must use the appropriate wave number k,. Propagation of electron waves in electric and magnetic fields (electron lenses) has been thoroughly discussed by Glaser (1943, 1952, 1956) and Glaser and Schiske (1953a,b), who have shown that the effect of the lens can be treated as a phase shift. The total current density per unit energy dj/dE is given by a sum over the partial current densities obtained from the amplitudes A,. Thus, for coherent illumination, contributions from partial waves resulting in the same final state of the object must be added coherently,
THEORY OF IMAGE FORMATION
191
whereas contributions leading to different final states must be added incoherently. In this way we find for axial illumination
x exp{ - i[y(0, AE) - y(W, AE)]} exp[ik(0
- el)p] d20 d2el (43)
where
y(0, AE)
=
k[(C3/4)e4 - (Af/2)e2
+ C,(AE/2E,)e2]
(44)
denotes the phase shift due to the defocus Af, the spherical aberration C,, and the chromatic aberration C,. The two vectors K and K’ are given by
K = k,[B,eZ
+ 01,
K’ = k,[O,e,
+ el]
(45)
and j , is the current density in the image plane if no object is present. The vector p denotes the position in the image plane but is referred back to the object. Equation (43) can be reformulated into a real-space description, similar to Eq. (36). The corresponding expression has been discussed by Rose (1984). Quite often the object in FBEM is illuminated incoherently from all directions, forming an angle less than 0,with respect to the optic axis (Kohler illumination). We must then average the current density per unit energy over all directions of the incident electrons, thus yielding
x exp{ - i[y(0, AE) - ?(el, AE)]} expCikp(0 - el)] d20 d20‘ d2@# (46)
The similarity between this formula and the corresponding formula for the STEM [Eq. (29)] will be discussed in the following subsection.
C. The Reciprocity Theorem In light optics the reciprocity theorem states that if the positions of the source and detector are interchanged, the light amplitude at the detector remains the same (Sommerfeld, 1978; Born and Wolf, 1965). Suppose the amplitude is u(S) at the source point S and u(P) at the image point P . If we put a point source with strength u(S) at P, the reciprocity theorem tells us that the amplitude at S will be u(P). Applying this theorem to a setup as shown in Fig. 7, we find that the amplitude at the detector is the same, regardless of whether we have the
192
H. KOHL AND H. ROSE
bS
source
I
xp Pi
objective lens
I
FIG.7. Simple arrangement of an imaging system that illustrates the reciprocity theorem.
source u(S) at S and measure at various image points P and P' or whether we move the source u(S) over the image plane and fix the detector at S . The theorem of reciprocity also holds for electron scattering off a static potential (Landau and Lifshitz, 1965). Thus, the image in a STEM with an axial detector and in an FBEM with axial illumination are the same for equivalent conditions (Cowley, 1969; Zeitler and Thomson, 1970a,b). The movement of the (virtual) source in a STEM is generated by a double deflection element. Before discussing inelastic scattering processes, we must define the meaning of reciprocity in these more general cases. For elastic scattering, reciprocity is due to time-reversal symmetry (Landau and Lifshitz, 1965) and leads to an important property of the elastic scattering amplitude for scattering of a wave ki to k,: f ( k i , k,) = f ( - k , , -ki) (47) Generalizing this result to inelastic processes, we find
f A k i k,) = f * m 4 - k,, - ki) The indexes rn and n denote the initial and final object states Irn) and In), while Im*) and In*) are the time-reversed states. The reciprocity theorem states that the amplitude for a transition from Im) to In) is equal to the timereversed transition amplitude from In*) to I m*). If inelastic scattering is taken into account, the reciprocity theorem cannot be used to prove the equivalence of images taken in FBEM and STEM, because the state reciprocal to the final state in FBEM is given by 3
THEORY OF IMAGE FORMATION
193
It consists of a coherent superposition of excited object states, which is clearly unequal to the initial state in a STEM, where the object is in its ground state. Nevertheless, Eqs. (29) and (46) look very similar. We shall, therefore, compare the images in STEM and FBEM. Let us assume that the instruments operate under equivalent conditions, namely, that the defocus, aberration coefficients, and objective aperture angles are equal. Let us further assume that the illumination angle in FBEM is equal to the spectrometer acceptance angle in STEM. If we then consider electrons that have suffered one particular energy loss AE, we find that, apart from the different forefactors, two further differences occur. First, in STEM the incident plane wave k , is focused, whereas in an FBEM the objective lens operates on the electrons k,, which have lost an energy A E in the object. Thus, in FBEM the chromatic aberration leads to an additional term in the phase shift y, which can be compensated by refocusing properly for the particular energy E , - AE. Second, the factor exp[ik,p(O - e’)] in STEM corresponds to exp[ikp(O - e’)] in FBEM. Since the energy loss is generally very small, this difference is usually too small to be detectable. Pogany and Turner (1968) used a different definition of reciprocity. In their famous paper, they name the approximate equivalence of the intensities an approximate “reciprocity of intensities.” We do not follow their definition, because in optics and quantum mechanics the term reciprocity is used to denote an exact property of a quantity (an amplitude) that obeys a particular differential equation (Sommerfeld, 1978). The approximate equivalence in intensities has a different origin and should be named differently.
D. Phase Contrast Thin phase objects are visualized in light microscopy by the phasecontrast method (Zernike, 1935). By introducing a proper phase plate in the back-focal plane of the objective lens, we obtain an intensity distribution in the image that, in the case of weak-phase objects, is proportional to the local phase shift (Goodman, 1968). Scherzer (1949) showed that in electron microscopy the phase plate can be obtained approximately by proper choice of defocus (Scherzer focus). This phase-contrast method can be used for thin objects and simplifies image interpretation considerably. For describing the phase contrast, it is sufficient to assume that the electrons are scattered by a static potential. This leads to an interesting question. Suppose the electron is scattered off a single atom; the transfer of momentum to the atom then results in a change of energy of the incident electron. Why is it then possible to see phase contrast or any interference between the unscattered and scattered electron waves? (We are grateful to several colleagues for raising this question.)
194
H. KOHL AND H. ROSE
Obviously the point is rather subtle. Phase-contrast images stem from the interference between the unscattered and the elastically scattered waves. In this case the object remains in its initial state. We shall show that there is always a finite probability amplitude that the internal state of the object does not change. For an FBEM with axial illumination, we obtain (Eusemann and Rose, 1982) A(8) exp[-iy(O)] exp(ik,pe)(f*(K)),
1
d28
(50)
where (f(K))T denotes the expectation value (thermal average) of the elastic scattering amplitude and all other quantities are as defined in Subsection 1II.B. The first term in brackets gives the constant current density if no object is present; the integral denotes the current density variations, which depend linearly on the scattering amplitude of the object. Evaluation of (f(K))T yields (Kohl, 1983b)
where VJK) is the Fourier transform of the interaction potential between the incident electron and the pth atom, and (plr,JT is the expectation value of the (Fourier-transformed) particle density of the pth atom. For a thin crystal plate, we find (f(K))T
= -
2c
exp(iKR,) exPC- ~ ( K ) I F ( K ) V 3 K )
(52)
Ir
where R, is the position of the pth atom within the elementary cell,
is the Debye-Waller exponent, and F(K)
=
N 0
for K equal to a reciprocal lattice vector otherwise
(54)
where N is the number of elementary cells in the object, M , the mass of a p atom, and s the number of atoms in the unit cell. The indexes q and o indicate the wave vector and polarization of a phonon, respectively, and fiqa
= (exp(Phmqa) -
1I-l
(55)
is the mean occupation number of the phonon mode qo. The calculation is given by Kohl (1983b) and need not be repeated here. The contrast decreases
THEORY OF IMAGE FORMATION
195
by a factor exp( - W ) . [The factor is not exp( - 2 W ) because the phase contrast depends on the amplitude, not the intensity, of the scattered electron.] Thus, the reason for the occurrence of phase contrast in spite of phonon vibrations is the same as the reason for the occurrence of distinct Bragg reflections off a crystal.
IV. NUMERICAL RESULTS A . Image of a Single Atom in an FBEM
To obtain quantitative results, we shall proceed in two steps. First, we shall calculate the mixed dynamic form factor S(K, K’, 0). For an arbitrary object, this is a rather hopeless task. Therefore, we must settle for simple examples. In this section we consider imaging of a single atom in an FBEM (Fig. 6) with axial illumination and an imaging energy filter. This filter removes all electrons unless they have suffered a distinct energy loss AE corresponding to a particular (dipole-allowed) transition. The objective aperture angle is assumed to be sufficiently small so that we can use the dipole approximation. Figuratively speaking, this means that the resolution limit of the microscope is larger than the diameter of the atom. In this case the mixed dynamic form factor for this particular transition is given by S(K, K’, 0) =
1 ~
25
+1 (nJMI exp(-iKrj)In’J’M‘)
x
M ,M ’ j, k
x (n’J’M’I exp(iK’rk)InJM)G(o - Am)
j, k
=
CKK’G(o - Am)
The only assumption used in the last step is rotational symmetry of the object. The proof is outlined in Appendix C . Since the quantum states with various M for a given 5 are degenerate, we must average over the 25 + 1 initial states ( n J M ) and sum over all 25’ + 1 final states In’J’M’). Note carefully that in this approximation the mixed dynamic form factor S(K, K’, o)factors into a strength factor CG(m - Am) and a shape factor KK’. Thus, the shape of the intensity distribution is the same for all atoms and depends only on 8, and on the microscope setting. The particular transition enters only via the strength factor CG(o - Am).
196
H. KOHL AND H. ROSE
Inserting Eq. (56) into Eq. (43), we obtain
with
Here J,(x) and J , ( x ) denote Bessel functions of the first kind. The total current density is given by a sum of two contributions. The first part, l A l i12, stems from the excitation of a dipole parallel to the incident electron, while IA,I2 describes the excitations perpendicular to the optic axis. For our numerical calculations, we assume that the image is recorded in the Gaussian image plane. Furthermore, we neglect spherical and chromatical aberrations. The results obtained with these assumptions are displayed in Figs. 8-1 1. The intensity distribution in the image is shown in Fig. 8 as a function of the (normalized) distance kp8, = 3.83p/d from the atom; d = 0.612/8, is the instrumental resolution limit (for an ideal lens). The dashed curve denotes the contribution of the parallel excitations IA II 1'; the solid curve shows the total intensity. The ratio 8J8, of the characteristic scattering angle eE = AE/2E0 to the objective aperture angle B0 is a measure of the energy loss. For large energy losses the aperture is almost uniformly illuminated. The intensity distribution in the image is then almost proportional to the Airy distribution [J,(kp8,)/(kp8,)]2. At large losses the perpendicular excitations contribute very little to the total intensity. With decreasing energy loss, the image spot starts to broaden as a result of the delocalization of the inelastic interaction. Its influence on the spot shape begins to show up for 8J8, = 0.5. For lower losses ( O E / 8 0 = 0.1 and 0.02), we find a ring of high intensity around the atom. This ring is due to the perpendicular excitations, which dominate low-loss images. The spot size increases even further. These effects of decreasing energy loss can be seen even more clearly in Fig. 9. Here we have drawn the image intensity within a thin ring p dp around 1 A , 12) ; the the atom. The solid curve shows the total intensity kp8,( [ A l2 dashed curve represents the contribution of the parallel excitations kp8,I A II 1.' For relatively large energy losses ( 8 J 8 , = 2.0 or OS), the intensity decreases rapidly within a short distance from the object. For lower-energy losses the image spot is much broader. Comparing the curves for 8J80 = 0.1 and those for 0.02, we find that the height of the first maximum hardly increases. Thus, the increase of the cross section is due to an increase of the
+
197
THEORY OF IMAGE FORMATION (b)
-
003
001
0
0 20 018
0.18
0 16
0.16
0 14
0.14
-6
012
-
012
-2
010
a?
-2 010
w
U
008
008
0 06
006
0 04
004
002
002
0
2
4
6
8
1
0
0
FIG.8. Radial intensity distribution in the FBEM image of an atom formed by inelastically scattered electrons with different relative energy losses OJB,. (a) @JOo = 2.0, (b) @,/O, = 0.5, (c) OJO, = 0.1, and (d) t),/O, = 0.02. The normalized radius kpB, = 3.83p/d is inversely proportional to the radius d of the Airy disk. The dashed curve represents the contribution of the parallel excitations ; the solid curve describes the total intensity.
intensity at a far distance from the center of the atom. The images of neighboring atoms will then overlap considerably, thus creating an undesirably high background intensity. Comparing the images at a given energy loss for various objective apertures, we must keep in mind that the scale at the axis changes also. To be specific, let us assume an energy loss AE = 120 eV and an incident energy of 60 keV. Then 0, = 1 x and the four energy-loss parameters correspond
198
H. KOHL AND H. ROSE 0301
lb)
025-
-z'
020015
I
I
I
m" 0 10
~
1 a
005-
07 0
5
10
15 k Pea
20
25
30
kpe,
FIG.9. Intensity contained within a ring of radius p and width dp in the inelastic atom images shown in Fig. 8 as a function of the normalized distance from the atom center. (a) OE/OO = 2.0, (b) O J O , = 0.5, (c) HE/@, = 0.1, and (d) OE/Oo = 0.02.
2 x lop3, 1 x and 5 x to objective aperture angles 8, = 5 x 10- '. The corresponding instrumental resolution limits are then given by d = 60,15,3, and 0.6 A, respectively. Thus kp8, = 1 corresponds to p % 15,4, 0.75, and 0.15 A in these images. We find that for increasing objective aperture angle the spot size decreases considerably. The instrumental resolution limit, however, can only be reached for a large energy-loss parameter (e,/e, = 2.0). We are thus led to the problem of determining the specimen resolution obtainable with inelastically scattered electrons. Since most objects consist of many atoms, rather than just two, use of the Rayleigh criterion would be inappropriate. For such problems other definitions, which relate the resolution to the radius of a circle containing a given percentage of the total intensity, have proven fruitful. The two most common definitions for the effective specimen resolution limit deff use either the radius of a circle that contains 84 % of all scattered electrons or else twice the 59 % radius. Both definitions are chosen so as to yield the same result as the Rayleigh criterion for a point scatterer. The fraction P(kp8,) of the total intensity falling into a disk of radius p is shown in Fig. 10. We find that the effective resolution limit increases with
199
THEORY OF IMAGE FORMATION
$IL a
IO-
04
02 00
5
10
15 kPeo
(b)
(a)
20
25
30
0
5
10
15 kpe,
20
25
30
10,
(C)
25
30
kpe,
kpe,
FIG.10. Fraction P ( k p 0 , ) of the total intensity contained within a disk of radius p in the images shown in Fig. 8. (a) OJO, = 2.0, (b) O,/O, = 0.5, (c) @,/On = 0.1, and (d) OE/OO = 0.02.
decreasing energy loss. The two definitions for deffyield different numerical values, but the general trend remains unaffected. Some authors have suggested use of the equation
as a rule of thumb for the effective resolution limit (Howie, 1981). This formula (or similar ones) can be inferred either from the uncertainty relation or else by replacing the aperture angle B0 by the characteristic scattering angle 0, in the Abbe formula
d
= O.6A/tI0
(60)
To justify this procedure, we presuppose that most of the electrons are scattered within a cone of OE around the optic axis. This implies that replacing the angular shape of the scattered intensity l/(0; + d2) with 1 for 0 S 8, and with zero otherwise will not affect the result too much. Looking at Fig. 10, we find that this crude rule of thumb does not describe reality very well. In particular, for low energy losses (0,/0, e l.O), the effective specimen resolution limit deffis much smaller than Eq. (59) indicates. The most important reason is the fact that the total intensity is the sum of two distinct parts. For low energy losses, the perpendicular excitations lead to the
200
H. KOHL AND H. ROSE
characteristic donut-shaped structure, which means that a large fraction of all electrons in the image are close to the atomic nucleus. These results are in qualitative agreement with recent experimental findings. Colliex et al. (1981), and Colliex (1982) have imaged small uranium clusters and found that the effective resolution limit, although worse than the point-to-point resolution of the instrument, is definitely better than we would expect from the rule of thumb [Eq. (59)]. This fact will be seen more clearly in the discussion of the image of a surface plasmon (Subsection 1V.D). Due to the poor signal-to-noise ratio in inelastic images, it is extremely difficult to measure intensity profiles. Thus, unfortunately, a detailed quantitative comparison is not yet possible. Another means for describing the properties of optical systems is to consider the diffractograms H(h/B,). They are the Fourier transforms of the image intensity and determine how well a periodicity in the object is transferred to the image. Some mathematical details are given in Appendix D. In Fig. 11 we have drawn the diffractograms for the images shown in Fig. 8. With decreasing energy loss, the decrease of H(6/8,) becomes steeper. The transfer of high-spatial frequencies in the object is then increasingly suppressed. For OE/O, 5 0.5, we find that the diffractograms even have a zero, meaning that this spatial frequency will not be found in the image. For 6 larger than this value, H(h/8,) is negative, thus resulting in a reversal of contrast. All results given in this chapter are also valid for a STEM with a small axial detector (0,4 6,) as can be seen from our discussion in Subsection 1II.C. The influence of the detector angle will be discussed in the following subsection. B. Image of a Single Atom in S T E M
To calculate the intensity distribution in a STEM image, we insert Eq. (56) into Eq. (29). We assume zero defocus (Af = 0) and neglect spherical aberration (C, = 0). In this case four of the six integrations can be performed analytically (Kohl, 1983b). The remaining integrals were evaluated numerically; some results are shown in Figs. 12-14. We have drawn the intensities for the energy loss parameters 8J8, = 2.0,0.5,0.1, and 0.02 for a fixed detector angle 0,.The dashed curve denotes the contribution of the excitations parallel to the optic axis. We see in all three figures that the spot size increases with decreasing energy loss. Figure 12 shows the intensity distribution for a comparatively = 0.28,). Here we note again the donut-shaped structure small detector (0, for low energy losses (OE/6, = 0.1 or 0.02). This is comparable to Fig. 8, but the effect is not as pronounced. Increasing the spectrometer acceptance angle
20 1
THEORY OF IMAGE FORMATION 601
"1
- 1 04 0
(b)
05
1
15
2
1.5
1 2
6/80
16 0 14 0
12 0 10 0
-? 9 -
r
80 60 40 20
00 -2 0
05
1
a / 00
15
2
-
0
5 05
1
0
618,
FIG.11. Diffractogram H(6/Bo) of the inelastic FBEM image of a single atom as a function of the normalized spatial frequency 6/B, in the case of axial illumination for four values of the characteristic energy loss parameter BJB,; (a) BJB0 = 2.0, (b) B J B , = 0.5, (c) BJB0 = 0.1, and (d) Os/Oo = 0.02. The dashed curve represents the contribution of the parallel excitations.
(0,= e,), we find only a very weak donut structure for 8J0, = 0.02. For an even larger detector angle (0,= XI,), the spot looks more and more like an Airy disk. At first sight it seems surprising that the spot shape is so strongly dependent on the detector angle. In imaging with elastically scattered electrons this dependence is very weak, so, in general, it can be neglected. Why is this different for inelastic scattering?
202
H. KOHL AND H. ROSE 0.25
(a)
1.0
0.20
d -
0.15
Y
Y
0.10
0.05
C
FIG.12. Radial intensity distribution of inelastic images obtained with a spectrometer acceptance angle (at the specimen) 0,= 0.28, for four values of the energy-loss parameter 8J8,; (a) OJ8, = 2.0, (b) OJO, = 0.5, (c) OE/Bo = 0.1, and (d) B J B , = 0.02. The dashed curve represents the contribution of the parallel dipole excitations.
Due to the small extension of the atomic charge distribution (RA < 0.5 A), an atom can be considered as a point scatterer for elastically scattered electrons at present resolution limits (d 2 2 8).Therefore, the electrons are scattered almost uniformly in all directions accepted by the detector. The detector angle, which determines the total current measured, then has no significant influence on the spot shape. With respect to inelastic scattering the atom cannot be considered a point scatterer, as is apparent from the spot shapes in Figs. 8 and 12-14. The strong
203
THEORY OF IMAGE FORMATION (b)
2c 18 \
16
-
z
-
14 12
10
b-.
8
E
\
4
' 0
FIG.13. Radial intensity distribution in the inelastic image of an atom where the spectrometer acceptance angle @, is equal to the objective aperture angle Oo. (a) OJOo = 2.0, (b) O,/H, = 0.5, (c) O,/O, = 0.1, and (d) OJO0 = 0.02.
dependence of the spot shape can be explained by the fact that electrons passing close to the nucleus have a large excitation probability, but due to their large deflection they contribute to the image only if the detector angle is sufficiently large. Therefore, the spot size decreases with increasing spectrometer acceptance angle. The perpendicular excitations play an important role for this effect. In their channeling experiments, Tafto and Krivanek (1982) observed the strong dependence of localization on scattering angle. They find that as the angle of scatter increases, the excitation is more localized.
H. KOHL AND H. ROSE
204
120-
(d)
100 -
100-
-
$-
80-
60-
3
80 -
60-
40-
40-
20-
20.
07 0
2
4
6 kpe,
8
1
0
0 ....... 0 2
4
6 kPe,
8
1
0
FIG.14. Radial intensity distribution in the inelastic STEM image of an atom obtained with a large detector 0,= 50, for four values of the energy-loss parameter BE/@,; (a) 8,/0, = 2.0, (b) Oe/Bo = 0.5, (c) 0,/0, = 0.1, and (d) 0,/0, = 0.02.
Maslen and Rossouw (1984) and Rossouw and Maslen (1984) evaluated the differential cross section for inelastic scattering in crystals. They used a hydrogenic model to calculate the mixed dynamic form factor for K-excitations (Maslen, 1983). Their results confirm the increase of localization for increasing scattering angle. Knowledge of the spot shape may be of special interest for high-resolution microanalysis. The current approach is to deduce the elemental concentration in a pixel from the number of counts for a characteristic energy loss (after subtraction of the background) by using a differential cross section (Isaacson
THEORY OF IMAGE FORMATION
205
and Johnson, 1975; Egerton, 1981a,b). However, this is true only if the spot is smaller than a pixel. The differential cross section only gives the total number of scattered electrons, regardless of where they appear in the image. If the resolution is so high that the image spot extends over several pixels, the spot shape must be considered in analyzing the data. As can be seen from Figs. 8 and 12- 14, this will be particularly important for high-resolution low-loss images (small 8,/0,). The Z-contrast method (Crewe et al., 1975; Carlemalm and Kellenberger, 1982; Colliex et al., 1984) is frequently used for obtaining high-contrast images of biological specimens. This method makes use of the fact that the ratio of the elastic to the total inelastic cross section is proportional to the atomic number Z (Lenz, 1954). Dividing the elastic by the inelastic image, we thus obtain an image whose intensity is proportional to the mean atomic number of the corresponding object point. A detailed theory of contrast, however, must take the different localizations of the elastic and inelastic images into account. As we divide the dark-field signal by the inelastic signal [lDF(p)/linc,(p)],the delocalization of the inelastic image is expected to lead to an increased contrast as compared to the ratio of the respective cross sections. Calculations of total inelastic images (summed over all possible energy losses) have been performed by Rose (1976a,b). Since the details of his work are too lengthy to be repeated here, we refer the interested reader to his original papers. C . Images of Assemblies of Atoms
So far we have been dealing with images of a single atom. In practice, however, any specimen consists of a large number of different atoms. Unfortunately, it is not possible to treat such an object in the same general manner as the case of a single atom. We shall discuss one approximation that gives some insight into the image formation process. The excitation spectrum can be divided into several parts. Quasi-elastic processes lead to energy losses smaller than the energy resolution of about 0.5-2 eV. Such losses are due to the excitation or annihilation of phonons. The low-loss region extends out to energy losses of about 50 eV. The dominant processes in this regime are plasmon excitations and transitions of valence electrons. The plasmon energies are not very sensitive to elemental composition, so their use for microanalysis is very difficult (Williams and Edington, 1976). The energies of valence excitations are strongly dependent on the chemical state of an element, so they can be used to identify molecular species (Isaacson, 1972; Johnson, 1979). We shall focus our attention on inner-shell losses whose energies are larger than 50 eV. The energies of the edges are characteristic for a particular
206
H. KOHL AND H. ROSE
element. The fine structure of the energy-loss spectrum yields valuable information on their local environment [extended energy-loss fine structure (EXELFS)] and on their chemical state [energy-loss near edge structure (ELNES)] (Stern, 1982). By registering the edge as such, we can obtain purely elemental information. In this latter case the natural approximation is to consider the atoms as single, independent entities. We thereby neglect the influence of the potential of the neighboring atoms on the secondary electron of an ionized atom. Since this EXELFS effect produces only small intensity oscillations in the energy-loss spectrum (typically less than 5 %), this approximation will be satisfactory. [Very close to the edge the error will be larger, as can be seen from ELNES and x-ray absorption near-edge structure (XANES) experiments (Pendry, 1983)l. By assuming the atoms to be independent from one another and using the form factor [Eq. (56)] for each, we obtain for an assembly of N atoms of one element in the object plane the total form factor N
S(K, K’, w ) = Ch(w - Aw)KK’
1exp[ - i(K - K’)pj]
(61)
j=1
where pj denotes the position of the jth atom. The excited states of the N atoms are degenerate, so we sum over all of them. Inserting this expression into Eq. (29) or Eq. (43), we find that the total intensity I , in the image is just the sum of the intensities due to the excitation of a single atom I , N
This situation differs sharply from that of an elastic dark-field image. There we must consider the interference terms between different atoms, because there is only one final state (which is equal to the initial state). Here the different final states can be classified by the number j of the atom that has been excited. All the contributions of the different atoms must be added incoherently, because the final object states are distinct. One possible way to determine the delocalization of the inelastic interaction experimentally is to measure the intensity distribution in the image of a straight edge of a thin foil (Isaacson et al., 1974). Suppose we take an amorphous object, whose density (per unit area) is given by
If the scatterers are independent, we obtain the intensity distribution by a convolution of the intensity distribution of a single atom with the particle
207
THEORY OF IMAGE FORMATION
density distribution. This is equivalent to multiplying the Fourier transform of the density times H ( 6 ) and using the inverse Fourier transform (Rose, 1976a,b). Thus
In Fig. 15 we have drawn the (normalized) intensity distribution in the inelastic image of the edge of a thin foil. We can see again that the image becomes blurrier as the energy loss decreases.
kxe,
kx0,
(a1
(b)
, -100
-50
-50
kx0,
00 kxe,
(Ci
(d I
00
50
100
- 100
50
FIG.15. Normalized intensity distribution in the inelastic FBEM image of a straight edge for the case of axial illumination as a function of the distance x from the physical edge for four values of the energy-loss parameter 0JBo; (a) BJV, = 2.0, (b) B J B o = 0.5, (c) OE/BO = 0.1, and (d) f~,/O, = 0.02.
100
208
H. KOHL AND H. ROSE
Similarly we can calculate the intensity I, in the image of an amorphous sphere as
Some results for OJO, = 0.1 are shown in Fig. 16. To obtain dimensionless quantities, we have displayed 1J(8n2R,”).For resolution limits larger than the radius of the sphere ( d = 2R,), we find a donut shape again. For smaller resolution limits (d = R,), this structure disappears. The intensity distribution is then more similar to the projected density; still, there is some delocalization left. Images of spheres have been taken by Colliex et al. (1981) and Colliex (1982). These authors found that the spot is broader than the sphere when inelastically scattered electrons are used to form the image. Let us now consider the inelastic image of a thin crystal. We assume the crystal to be so thin that the first-order Born approximation still holds. In Subsection 1I.B we found that for periodic objects S(K, K’, w ) is nonvanishing only if K - K’ = g. We use only this property, which holds for the two dimensions of the object plane, and the small-angle approximation [Eq. (3 l)]. Then g is a two-dimensional, reciprocal lattice vector. We make no assumption about the particular excitation; i.e., the atoms may be strongly interacting. When we take a lattice image in STEM, we are free to choose the position of our detector with respect to the optic axis. The importance of the detector position for elastic lattice images has been thoroughly discussed by Cowley and Spence (1981) and Spence and Cowley (1978). To establish the main point, let us consider the biprism experiment as shown in Fig. 2. We have already found (Subsection 1I.B) that to see the crystal structure we must choose k and k’ so that their difference is equal to a reciprocal lattice vector g. The lattice fringes then have the same periodicity as the crystal. To detect any 018.
-a“ -
I L q 010-
008006-
o i
-* a
(bl
010
006
004-
002-
018
(a)
004 ....,. .-.. ,
\,
-..
002
’... .
0
0 PlR,
1
2
3
4
p/R,
FIG.16. Radial intensity distribution in the inelastic image of a sphere for two values of the ratio d / R , between the resolution limit d and the radius of the sphere R,; (a) d / R , = 2 and (b) d/R, = 1. A STEM with a point detector or an FBEM with axial illumination has been assumed.
THEORY OF IMAGE FORMATION
209
elastic signal, we must place our detector in a Bragg position with respect to the incident beams. In STEM, the object is illuminated by a coherent superposition of all plane waves within the illumination cone, which is limited by the objective aperture. This wave packet is diffracted and the crystal is resolved only if the detector is located in an area where (at least) two diffracted disks overlap (Fig. 17). If we want to register inelastically scattered electrons, we must still fix k and k so that k - k = g, but now we detect electrons in all directions. Any excitation in a crystal can be classified by its wave vector q (or its crystal
FIG. 17. Scheme demonstrating the formation of an elastic lattice image in STEM. The incident cone is Bragg reflected. The lattice fringes appear in the image only if the detector is placed at a position where two cones overlap (hatched area).
210
H. KOHL AND H. ROSE
momentum hq), which lies in the first Brillouin zone. The energy-loss spectrum in a given direction k, reflects only those excitations whose wave vector q differs from the scattering vectors K and K' by arbitrary reciprocal lattice vectors. If, on the other hand, we look at a specific energy loss AE, we register electrons only in those directions corresponding to an excitation with energy hw, = AE. Since in any real experiment we have a finite energy resolution, the directions will not be so well defined. To detect a lattice image with energy loss AE in a STEM, we must fulfill a condition for the detector similar to the one for the elastic case. The disk representing the incident wave packet must be tilted by q (with hw, = A E ) and then Bragg-reflected (Fig. 18). The detector must be placed so that at least two disks overlap. If the spectrometer resolution is worse than the bandwidth of the excitation, the position of spectrometer is irrelevant. Note carefully that for thin crystals such images are due to single-scattering inelastic processes only. The image then reflects only the spatial structure of the excitation. It should not be confused with inelastic images of thicker crystals, in which multiple elastic and inelastic scattering processes are dominant (Fujimoto and Kainuma, 1963; Howie, 1963, 1984). The latter images look similar to elastic images; the contrast, however, is reduced. (a)
crystal
ax
Fia. 18. Comparison of the formation of elastic and inelastic lattice images in STEM. In (a) and (b), the incident and Bragg reflected cones are represented by their axes; (c) and (d) show the cross section through the cones at the detector plane. Lattice fringes can only be observed if the detector is placed in one of the hatched areas. The deflection 0, = q/k, resulting from the excitation of the crystal must be considered when describing inelastic image formation.
THEORY OF IMAGE FORMATION
21 1
In an FBEM with an imaging energy filter the image is formed with all electrons of a given energy that pass through the objective aperture. To form an elastic lattice image in an FBEM, at least two diffracted beams or one diffracted beam together with the unscattered beam must pass through the objective aperture. For inelastic lattice images, we must take the additional tilt by the angle O4 = q / k into account. Then if two diffracted and tilted beams pass through the objective aperture, we can see lattice fringes. If the energy filter transmits a whole energy interval simultaneously, the total image is given by an incoherent superposition of all inelastic lattice images resulting from excitations whose energies are in the chosen interval. In practice it is very difficult to prepare a crystal thin enough so that the first-order Born approximation holds. The argument can be carried over to thicker crystals, if we take hq to be the difference of the crystal momenta between the final and initial states of the object. Inelastic lattice images of crystals have been obtained by Craven and Colliex (1977). A quantitative description of such images is rather complicated. Thus, it is not yet possible to compare their results with theoretical calculations. In the case of thin crystals and independent atoms, the Fourier transform of an image is related to the diffractrogram of a single atom H ( 6 ) via r
r
x exp[ - i k ~ ( p- pj)] d2(p - p j ) exp( - ik6pj)
For a periodic object, the last sum is nonvanishing only if k6 is equal to a reciprocal lattice vector. If the atomic positions are known, it is then possible to measure H ( 6 ) at points that correspond to reciprocal lattice vectors. D. Image of a Surface Plasmon
As a further example of an inelastic image, let us consider the image of a plasmon on the surface of a metallic sphere or, similarly, on the surface of a spherical void in a metal. The properties of such objects have gained increasing attention over recent years (Fujimoto and Kumaki, 1968; Natta, 1969; Ritchie, 1981; Schmeits, 1981; Marks, 1982; Manzke et al. 1983; Penn and Apell, 1983; Lundqvist, 1983; Ekardt, 1984). The procedure to determine the intensity distribution is similar to the one described in Subsection 1V.A. To calculate the mixed dynamic form factor, we use a plasmon model given by Ashley et al. (1974, 1976). We consider only dipole plasmons that have an
212
H. KOHL AND H. ROSE
excitation energy of AE = EP/$ for metallic spheres and AE = E p Z for voids, where E , is the energy of a bulk plasmon. For the calculations (Kohl, 1983a,b), we assume an FBEM with axial illumination or a STEM with a small axial detector and neglect all aberrations (y = 0). For finite resolution, we use d / R , as a parameter. The resolution limit d and the radius from the center of the sphere p are both normalized to the radius of the sphere R , . Figure 19 shows the intensity distribution for d / R , = 0.2. The total intensity (solid curves) is again a sum of two parts. One can be viewed as resulting from the parallel excitations (dashed curves), the other as due to the perpendicular excitations. For a rather large energy loss (O,/O, = O.l), we obtain a full spot, whereas for lower energy losses the image is donutshaped. This ring of high intensity at the surface of the sphere is due to the perpendicular excitations. Images of surface plasmons were already reported 15 years ago by Henoc and Henry (1970). They used a neutron-irradiated foil that they imaged in an FBEM with a Castaing-Henry filter (1962) by selecting the plasmon loss electrons. In this case the voids appear as bright spots on a dark background, whereas they can hardly be recognized in the elastic image. Some experimental findings have been obtained that confirm our theoretical predictions. During the past years, Batson (1982a,b, 1985) and Batson and Treacy (1980, 1982) imaged small aluminium spheres in a STEM by using only those electrons that had excited a surface plasmon. In the course of specimen preparation, their spheres had been covered with a thin oxide layer. In their experiments both the instrumental resolution limit ( d / R , 4 1 ) and the energy loss (OJO, 4 1) were very small. More recently Ach6che et al. (1985) experimented with metallic spheres of tin, gallium, and uranium. In these images they find bright rings at the surfaces. Unfortunately, we cannot yet compare their experimental results quantitatively with our predictions from theory, because they used all surface plasmons in their experiments so far, whereas we have considered only dipole plasmons. Furthermore, the finite spectrometer acceptance angle influences the spot shape as well.
"6
05
PI R,
1
15
FIG.19. Radial intensity distribution in the image of a sphere, obtained by using only those electrons, which have excited a dipole surface plasmon, (a) B,/O, = 0.1 and (b) &/B0 = 0.005.
THEORY OF IMAGE FORMATION
213
Nevertheless the qualitative agreement between theoretical and experimental results is encouraging. Batson and Treacy (1982) and Colliex and Mory (1984) state that the excitation process is amazingly localized, in contrast to expectations from the rule of thumb [Eq. (59)]. The intensity distribution in their images indicates a specimen resolution limit of less than 40 8, whereas Eq. (59) would indicate approximately 800 A. These experiments show that, in spite of the delocalization of the interaction, we can obtain images with fairly high resolution even at rather low energy losses (AE z 5-15 eV). V. CONCLUSION
We have outlined a quantum-mechanical theory of image formation in the electron microscope, which considers both elastically and inelastically scattered electrons. The calculation of the inelastic image necessitates knowledge of the local scattering and excitation properties of the object. This information is contained entirely in the mixed dynamic form factor S(K, K’, w), which allows us to determine the shape of the inelastic image of thin objects within the first-order Born approximation. The mixed dynamic form factor can be viewed as a generalization of the ordinary dynamic form factor S(K,w). We have investigated the properties of the mixed dynamic form factor.and its interrelation with the generalized dielectric function including local field effects. The function S(K, K’, w ) can be determined experimentally by using an interference experiment as shown in Fig. 2. Recently, Schulke (1982) has shown that the mixed dynamic form factor of crystalline objects can also be measured by Compton scattering of x rays if the crystal is in Bragg position. The theory yields information about the coherence properties of the scattered radiation. In the case of coherent illumination, all partial waves leading to the same final object state are coherent with each other. Partial waves, however, that result in different final object states are incoherent with each other. By considering these coherence properties we can explain the occurrence of phase contrast in spite of the fact that the object atoms can absorb recoil energy, which is connected with an energy loss of the incident electrons. In Section IV we have calculated the intensity distribution in the image of two model objects. The image of an atom formed by the inelastically scattered electrons (in dipole approximation) is much more blurred than that formed by the elastically scattered electrons. This delocalization, however, is not as pronounced as has been assumed previously by Howie (1981). For small energy losses we found a donut-shaped intensity distribution. Contrary to elastic images, inelastic images in STEM exhibit a pronounced dependence
214
H. KOHL AND H. ROSE
on the spectrometer acceptance angle. The spot size decreases with increasing detector angle. Our calculations of images formed by dipole-surface-plasmon-loss electrons show that for low energy losses and high resolution the image of a sphere consists of a ring of high intensity about the projected surface. The delocalization is again less pronounced than has been assumed previously by Howie (1981). Our predictions agree reasonably well with recent experimental findings (Batson, 1982a; Colliex, 1982). Summarizing, we can state that the large progress in analytical electron microscopy has enabled the experimentalists to obtain inelastic images with high spatial resolution. Although the interpretation of the inelastic images is more difficult than the interpretation of elastic images, they contain information which is not accessible otherwise. For example, inelastic images can serve as a means for elucidating the local electronic structure at boundaries and surfaces. Therefore, these techniques will undoubtedly gain increasing importance for solid-state research.
APPENDIX A . Calculation of the Transition Rate
The interaction potential of the incident electron with the object is given by the operator
b' =
1 V(r - rj) j
where r and rj denote the positions of the incident and thejth object electron, respectively. The matrix element can be written as (kml Vlkfn)
=
1 (ml .i
=
Similarly, we find
7
(ml
s
exp[i(kf
-
k)r] V(r - rj) d3rln)
I-
J exp[i(kf
- k)(r
-
rj)] v(r
-
rj) d3(r - rj)
THEORY OF IMAGE FORMATION
215
The two scattering vectors K and K’ are defined as
K=k-kJ,
K’=k’-k
f
The operator
is the Fourier transform of the particle density operator. With the procedure introduced by Van Hove (1954), the product of matrix elements appearing in Eq. (8) can be transformed into a density-density correlation function. Taking into account the probability p m that the system is initially in state Im), we obtain for the sum
where ( )T denotes the thermal average, H , is the Hamilton operator of the object, and pK(t)is the density operator in the Heisenberg representation. By introducing the mixed dynamic form factor (Rose, 1976a,b) as
we eventually arrive at Eq. (8). The use of correlation functions such as (A7) is often advantageous, because it is then not necessary to calculate the corresponding wave functions explicitly. For collective excitations, it is in many cases possible to determine equations of motion for the density operator, which in turn lead directly to the correlation function (Pines, 1962, 1964).
216
H. KOHL AND H. ROSE
B. Interrelation of the Mixed Dynamic Form Factor and the Generalized Dielectric Function
To calculate the reaction of a solid when an external electric potential cDext(r,t ) is applied, we generalize the treatment given by Platzman and Wolff (1973). The Hamilton operator H can be written as the sum of an unperturbed operator H , and a perturbation H,(t) = -e,
s
p(r)cDexc(r,t) d3r
(B1)
where -eo denotes the charge of the electron and p(r) the density operator. By utilizing the Fourier expansions
mext(r,t) =
1 7 1cDf'(t) exp(iKr) K
we obtain
where V is the volume of the solid. By employing methods based on the linear response theory (Platzman and Wolff, 1973), we derive the following expression for the expectation value of the Fourier-transformed density operator
By inserting the Fourier transforms with respect to time,
into Eq. (B5), we obtain
217
THEORY O F IMAGE FORMATION
where the function
is the response function and 6 > 0 is a small quantity tending to zero. The response function describes the reaction of pKwhen a periodic potential @Ft of frequency OI is applied to the object. We now show the relationship existing between this function and the mixed dynamic form factor. To avoid unnecessary terms, we consider only the case o # 0. (For o = 0, the term ( p K ) T d ( o ) and similar ones occur in the formulas.) By using I
Zlr,
r
U,
J-u.
(p-K,pK(t))T
exp(iwt) dt
= exp( -Bhw)
S(K, K’, w )
(B10)
we obtain at the limit 6 .+ 0
+
x exp[i(o - o’ i6)tl O0
a,
do’d t
S(K, K’, o’)[ 1 - exp( - Pho’)] do’ w - w’ + i6 S(K, K’, 0’) [1 - exp( - Phw’)] d o ’
ine; hV
-~ [1 - exp( - Phw)]S(K, K’, w )
where # denotes the principal value. It should be noted that taking the imaginary part of (B11) yields a relationship between the linear response function F(K, K‘, w ) and the mixed dynamic form factor only if S(K, K’, w ) is real (e.g., for K = K’ or when the object is centrosymmetric and invariant with respect to time reversal). This function, however, is generally complex and fulfills the relation S*(K, K’, W ) = S(K’, K, O )
(B12)
To obtain a relationship between F and S, we exchange the variables K and K’ in (B 1 I), which yields an additional complementary equation. By adding
218
H. KOHL AND H. ROSE
these equations, using relation (B 12), and taking the imaginary part, we obtain the real part of S(K, K’, w ) : Im i{F(K, K’, w ) + F(K, K, w ) } = (-ne#l/)[l
- exp(-flhw)]
Re S(K, K’, w )
(B13)
On the other hand, by subtracting the two equations and using Eq. (B12), we obtain the imaginary part of S(K, K , w ) as 2 1 - ne0 Im - {F(K, K’, w ) - F(K’, K, a)}= [l hV 2i ~
-
exp( -flhw)] Im S(K, K , w ) (B14)
If we multiply Eq. (B14) by i and add Eq. (B13), we arrive at the following relation between the response function and the mixed dynamic form factor: S(K, K’, w ) = {ihV/2nei[1 - exp( -flhw)]}(F(K, K‘, w ) - F*(K’, K, w ) } (B 15)
In solid-state physics, we often use a dielectric function, which describes the response of the system as a function of the total potential @Iot(r, t ) = aext(r, t ) mind(r,t), where Oindis the induced potential. This dielectric functions plays the central role in the self-consistent field approximation. Here we , was intromust use the generalized inverse dielectric function ~ & , ( w ) which duced by Adler (1962) and Wiser (1963) to describe local field effects. In accordance with Adler (1962), we write
+
This relationship can be expressed as
By using the definition
THEORY OF IMAGE FORMATION
219
of the inverse dielectric function as an inverse matrix with respect to EKK,(w), we find
From the Poisson equation, we obtain {E&,(w) - dKK,}@Zt(u)
e,p$d(co) =
(B20)
K‘
Comparison of Eq. (B20) with Eq. (B8) yields F ( K , K’, W )
= E ~ K ~ { C ; ; , ’ ,( OdKK,} >)
(B2 1)
By inserting this relationship into Eq. (B15), we find
This formula enables us to determine the mixed dynamic form factor from the inverse generalized dielectric function.
C . The Mixed Dynamic Form Factor in Dipole Approximation
In the following derivation, we shall use the notation employed by Messiah (1964). As a special example we shall consider an arbitrary, rotationally symmetric system. Then an operator T can be expanded into tensor operators TF) of kth order:
We start by proving the indentity
1 (nJMITf’In’J’M’)((nJMIT$’)ln’J’M‘))* M.M‘
where InJM) denotes a state with principal quantum number n, angular momentum J , and projection M . The term ( dIT(’)IIn’J’) is the reduced matrix element. By using the Wigner-Eckart theorem (Messiah, 1964)
220
H. KOHL AND H. ROSE
and the orthogonality of the 3j symbols
for -k 5 q 5 kand ) J - J‘I 2 k 5 J
+ J‘, we find
1 ( n J M I Tf)I n’J’M’)(( nJM IT$’)In’J‘M’))* M.M ’
=
(nJI
In’J’)((nJI 1T”’)lIn’J’))* ( J M
M,M,
k J’)(.J q M’ M
k‘ q’
J’) M‘
This is precisely Eq. (C2). We now apply this identity to the sum ( n J M I Kr I n’J’M’)(n’J’M’ I K’r I n J M )
(C5)
M ,M ’
The operator r = x k r k is the sum of the position operators of the individual electrons. The z axis is assumed to point in the K direction, so that Kr = Kz. By using the rotation operator R, we can write K’r as K‘RzR-’. Since z = TL’) is the 0-th row of a dipole operator (k = l), we find 1
where R$,) denotes a row of the rotation matrix. Insertion into Eq. (C5) yields
1 ( n J M IK r 1 n’J’M’)((nJM 1 K’r In’J’M’))* M ,M ’
K K’
x
( n J M I Tb’)I n’J’M’)( ( n J M I T:) I n’J’M’))* R$‘
M ,M ’ ,q
where we have used RSb = P,(cos 9), and 9 denotes the angle between the two vectors K and K’. This relationship was employed to derive Eq. (56).
221
THEORY OF IMAGE FORMATION
D. Calculation of the Diffractogram The normalized inelastic intensity distribution per unit frequency w = E / h in the image plane of an FBEM for axial illumination can be written as
x exp{ - i[y(O, A E ) - y(@, A E ) ] } expCikp(0 - el)] d20 d2el
(D1) where
K
= k,(O,e,
+ 0),
K'
= k,(O,e,
+ el)
and
S(K, K', W ) = So
r+dm'2 S(K, K', o)dw
o-d0/2
is the mixed dynamic form factor averaged over a small frequency interval SW.
In the following, we consider w to be fixed and write I(p), thus omitting the parameter o.For w # 0, this intensity is proportional to the current density distribution in FBEM, [Eq. ( 4 3 ) ] .The diffractogram H ( 6 ) is the Fourier transform of the intensity.
H(S) =
~
~
(2.)2 k2
s
I(p) exp( -ik6p) d2p
x exp[ikp(B
S(K, K', W ) -
0' - S ) ]
x exp{ - i[y(O, A E ) - y(@, =
ki
s s
A(e)A(el)
S(K, K', w ) K
~
x exp{ -icy(& A E ) - y(@, =
k;
A E ) ] } d20 d2el d2p
A(0)A(0 - 6)
S(K, K
K
~
~
A E ) ] } S(0 - el - 6) d20 d20'
k,6, o) K2(K - k 0 6 T
x exp{ -i[y(d, A E ) - y(l0
-
-
61, A E ) ] } d20
222
H. KOHL AND H. ROSE
The first equality in Eq. (D2) shows that for 6 = 0 we obtain the integrated intensity in the image. Thus, we obtain a quantity that is proportional to the effective cross section 4k E , d28 = -- H ( 0 ) hko Eo The total number of electrons in the image (given by the right-hand side) is equal to the number that pass through the hole in the aperture screen (lefthand side). To calculate the diffractogram of the image of a single atom in FBEM, we use the dipole approximation as outlined in Subsection 1V.A and Appendix C . Setting the “strength factor” C = Sw, we find Neglecting aberrations (y excitations
(D4) H ( 4 = H 11 (6) + HL(4 obtain for the contribution of the parallel
= 0), we
and for the perpendicular excitations
The integrations over the angle /3 enclosed by the two vectors 8 and 6 yield
H 11 (4
140:
I
0
I
-
for 8, 5 6 < 20, otherwise
(D7)
223
THEORY OF IMAGE FORMATION
0
(D8)
otherwise
where 8, is the aperture angle; 8, = AE/2Eo is the characteristic scattering angle, which is defined as the ratio of the energy loss AE and twice the energy of the incident electrons E,. The remaining integrations were performed numerically and are displayed in Fig. l l . For a STEM with an extended detector, the image intensity is
x exp{ - i[yo(e) - yo(@)]} exp[ikop(8 -
e’)] d28d2W d2@ (D9)
with the scattering vectors
K
= k,[dEe,
+ (8 a)], -
K‘ = k0[8,e,
+ (0’ - a)]
(D10)
Thus, we find for the diffractogram of the STEM image HsTEM(8) = k;
s
S(K,K - ko6, W) A(e)A(e - 6 ) ~ ( @ ) K2(K - ko6)2
x exp{i[yo(8) - yo(18 - SI)]} d28d2@
This result is a generalization of Eq. (D2). Correspondingly, we obtain
and the effective cross section. between HSTEM(0)
(D11)
224
H. KOHL AND H. ROSE
Comparing Eqs. (D12) and (Dll), we find that an image yields more information than the differential cross section. The latter is related to H ( 6 = 0), whereas in an image all spatial frequencies 6 between 0 and 20, occur.
ACKNOWLEDGMENTS We would like to thank Drs. M. Achtche (Laboratoire de Physique des Solides, Orsay, France), P. E. Batson (IBM, Yorktown Heights, New York), and D. Krahl (MPG-Fritz Haber Institut, Berlin, Germany) for sending us their experimental results prior to publication and Profs. R. F. Egerton (University of Alberta, Edmonton, Canada) and J. Kiibler (Technische Hochschule Darmstadt, Germany) for fruitful discussions. In particular, we acknowledge the stimulating cooperation with Dr. C. Colliex and the members of his group from the Laboratoire de Physique des Solides in Orsay, France. Thanks are also due to Dr. A. Levensohn (Albany, New York) for his assistance in editing the manuscript. The valuable comments of Drs. R. Eusemann (Technische Hochschule Darmstadt, Germany) and M. Haider (EMBL, Heidelberg, Germany) are gratefully acknowledged.
REFERENCES Acheche, M., Colliex, C., Kohl, H., and Trebbia, P. (1985). To be published. Adler, S. (1962). Phys. Rev. 126,413. Ashley, J. C., and Ferrell, T. L. (1976). Phys. Rev. B 14, 3277. Ashley, J. C., Ferrell, T. L., and Ritchie, R. H. (1974). Phys. Rev. B 10, 554. Batson, P. E. (1982a). UItramicroscopy 9,277. Batson, P. E. (1982b). Phys. Reu. Lett. 49,936. Batson, P. E. (1985). To be published. Batson, P. E., and Treacy, M. M. J. (1980). Proc. 38th Annu. E M S A Meet. San Francisco, California, p. 126. Batson, P. E., and Treacy, M. M. J. (1982). Unpublished report. Berndt, H., and Doll, R. (1976). Optik 46, 309. Berndt, H., and Doll, R. (1978). Optik 51,93. Berndt, H., and Doll, R. (1983). Optik 64, 349. Bethge, H., and Heydenreich, J. (1982). “Elektronenmikroskopie in der Festkorperphysik.” Springer-Verlag, Berlin and New York. Born, M., and Wolf, E. (1965). “Principles of Optics,” Chap. 8.3.2. Pergamon, Oxford. Carlemalm, E., and Kellenberger, E. (1982). EMBO J . 1, 63. Castaing, R., and Henry, L. (1962). C . R. Hebd. Seances Acad. Sci. 255,76. Colliex, C. (1982). “Electron Microscopy,” 10th Int. Congr. Hamburg, Vol. I, p. 159. Colliex, C. (1984). In “Advances in Optical and Electron Microscopy,” Vol. 9, (R. Barer and V. E. Cosslett, eds.), p. 65. Colliex, C., and Mory, C. (1984). In “Quantitative Electron Microscopy,” Proc. Scott. Uniu. Summer Sch. Phys. 25fh, GIusgow (J. Chapman and A. J. Craven, eds.), p. 149. Colliex, C., Krivanek, 0. L., and Trebbia, P. (1981). Inst. Phys. ConJ Ser. 61, 183.
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ADVANCES IN ELECTRONICS AND ELECTRON PHYSICS. VOL. 65
Dimensional Terms for Energy Transport by Radiation and for Electromagnetic Quantities: Comments on the SI System BERTHOLD W . SCHUMACHER Ford Motor Company. Research Stufl Deiroit. Michigan
I. I1. I11. IV .
Why Another Discussion of Dimensional Terms? . . . . . . . . . . . . Dimensional Terms and Physical Concepts . . . . . . . . . . . . . . Some Peculiar Aspects of the SI System . . . . . . . . . . . . . . . Electric and Magnetic Quantities . . . . . . . . . . . . . . . . . Appendix 1. Magnetic Field Quantities and Their Dimensions-Energy Densities and Forces in the Fields . . . . . . . . . . . . . . . . . . . . . . A . Energy Required for Generating a Magnetic Field . . . . . . . . . . B. Field Lines and Flux . . . . . . . . . . . . . . . . . . . . . C. Flux Density . . . . . . . . . . . . . . . . . . . . . . . . D . Induced Voltages and the Transformer . . . . . . . . . . . . . . E. The Field Strength . . . . . . . . . . . . . . . . . . . . . . F . The Magnetic Moment . . . . . . . . . . . . . . . . . . . . . G . Forces on a Pole Region . . . . . . . . . . . . . . . . . . . . H . Inductance as Determined by Geometry . . . . . . . . . . . . . . I . Magnetic Tension and Magneto-Motive Force . . . . . . . . . . . J . Two Field Quantities . . . . . . . . . . . . . . . . . . . . K . Matter in the Magnetic Field. . . . . . . . . . . . . . . . . . L. Forces on Currents in the Magnetic Field-Discussion . . . . . . . . . Appendix 2. Numbers for the Characterization of Particle Beams . . . . . . A . Some General Remarks . . . . . . . . . . . . . . . . . . . B. Particle Beam Sources . . . . . . . . . . . . . . . . . . . . C. Beam Formation . . . . . . . . . . . . . . . . . . . . . . D . Disturbances to a Thermodynamic Ensemble of Particles . . . . . . . . E. Relationship between Beam and Source-Radiance Numbers . . . . . . F . Some Comparisons of Particle Beams and Light Beams . . . . . . . . Appendix 3. Numerical Values for Energy Densities . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . .
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I . W H Y ANOTHER DISCUSSION O F DIMENSIONAL TERMS? The following thoughts evolved while preparing a quantitative description of energy transport by beams of light or particles when used specifically as energy carriers. It turned out that the dimensional units currently in use. 229 Copyright C 1985 by Academic Press. Inc All rights of reproduction in any form reserved. ISBN 0-12-014665-7
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the System International or SI system, were not very helpful in this context. We may even call them an anti-mnemonic device, whereas dimensional terms should or could indeed be a mnemonic help. With very little yet well thought out change in the present system this goal could readily be achieved, together with some more general improvements. For this reason the following discussions are presented. The advantages of the suggested changes in the SI system would extend to a much broader field than the preceding starting point might suggest.
TERMS AND PHYSICAL CONCEPTS 11. DIMENSIONAL The terms that we call dimensions and we attach to the numbers expressing the size of a quantity should ideally-if we want to remain logical-be linked to our conceptual understanding of the meaning of that quantity, although in a strictly mathematical context (rather than a conceptual, physical one) this would not be necessary.* Yet, the fact remains that five kilowattseconds are something else than the five coins with which we may pay for them. The problems with dimensions have, of course, been discussed beforesome people think too often. However, some of the difficulties that people seem to have with, for instance, recognizing the proper concepts, which must be applied in preceding case of energy transport by radiation or with the subject of magnetic fields, are caused by the irrational system of units and dimensions that has been imposed thereon and not because these matters are difficult in themselves. Text books often state with respect to dimensions that they treat them with consistency. What this usually means and actually results in is a proof of the internal self-consistency of the algebra of the initially chosen equations and definitions. In fact, we get the impression that the algebra, rather than the physical meaning, has been foremost on the minds of the designers of dimensional systems. A distinction is not always made between concepts such as velocity and units of measurement, e.g., m/sec, mph, knots, or many others. Dimensional terms refer to the concepts and are not necessarily identical with
* Mathematical aspects of dimensional systems have often been discussed [see, e.g., Page (1978)l. Often the term dimension is used only with reference to the exponent of a “base” unit, so that m3 has the “dimension” + 3. This amounts to nothing but a once more abstracted algebraic accounting system without direct physical meaning. It can be useful in “dimensional analysis” of physical relationships. An article with this title by Stewart (1982), leads into these other aspects of dimensions, which are not our present concern. However, even they could benefit from the suggested revision of the system.
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units of measurement. Units based on fundamental constants, such as the Planck length or Planck mass (see Pipkin and Ritter, 1983) do not circumvent the problem of dimensionality either. The best approach appears to be to base dimensional terms on such quantities as nature has given greatest importance. Some are immediately obvious to our senses, some recognized only after a century-long process of scientific clarification, as in the case of the energy concept. We should keep in mind, however, that there is no hierarchical order in the laws of nature; they form a closed chain. Mass, energy, space, and time are dominant concepts, but none can be placed before the other. No ranking order should, therefore, be implied by the sequence in which we list dimensional terms. We cannot discuss those terms without discussing also the underlying concepts, at least to some extent. Those well known terms are mass (unit: gram or kilogram), energy (unit: Joule), and time (unit: second). From this comes energy flow (unit: Watt = Joule/sec), length, area, volume (units: m, m2, m3), velocity (no unit name given, expressed as m/sec), and change of velocity with time, called acceleration (no unit name given, expressed as m/sec2 or sometimes as multiples of the acceleration of gravity g on the earth surface). Here we see the first discrepancy between concept and dimension. The acceleration g is present and has its meaning although nothing at all needs to be moving or accelerated, i.e., needs to change its velocity. Now we get to the concept of force, which sets a mass into accelerated motion, lifts a weight, or stretches a spring (unit: Newton). In fact, a spring could very well be our primary standard of force, defined, for instance, by the compressibility of a perfect quartz crystal. At some stage of the developments we have used such crystals as time standards (if only as secondary standards). There we actually made use of its spring force and its mass. Linear compressibility, a measure of the spring force, can be measured readily and related to the standard of length, in terms of the wavelength of a spectral line. A standard spring, e.g., a quartz crystal, could be calibrated in standard units of Newtons. It is purely incidental that the present unit of force is what it is, although it is convenient to have the Newton-meter of work equated to the unit for the energy, the Joule.
The preceding difficulty generated by identifying gravimetrical attraction with acceleration would be readily avoided if we called it the gravitational attractive force per unit mass and retained the term weight for the total gravitational force on a body. There is, after all, some common sense in the term weight. Everyone knows by now that, for any particular body, it is different
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on earth, the moon, or jupiter, although the mass remains the same. We can measure weight with a spring, as we measure all other forces. The mass need not be defined by anything else but the number of atoms of a particular element in a certain volume. The prototype or standard of the kilogram actually represents nothing else, if only it were a single crystal. The old and famous query, whether inertial and gravitational mass are the same, will then merge into the more general question of whether forces of any nature-spring forces, electrical attraction or repulsion, magnetic attraction or repulsion, gravitational attraction, centrifugal force-accelerate a given mass by the same amount when we let go of the mass yet maintain the force. (It is easier to think of such an experiment than to execute it. Even for the gravitational pull on the surface of the earth it is not more than a good approximation to assume that the force remains constant.) At present, the definition of force calls it the agent for the acceleration of a body with mass. We find that indeed in some cases where forces appear there is movement as well, but in others there is none. The equivalency of the forces is taken for granted. In yet other cases, force is generated by doing work, such as friction force or muscle force. This has long delayed the logical differentiation between work and force, as is well known. Moreover, force has another seldom considered quality, it defines a straight line, e.g., if we stretch a string. The relation to the straight-line path of a light beam indicates a connection, which indeed exists. The forces between the molecules of the string are generated by the same electromagnetic fields that determine the path of the light beam. Continuing our list of concepts and dimensional terms, we then encounter the electrical effects and the problem of their quantitative description. There is the electric field and its force: the pull of the amber, the potential difference on the battery terminals, the EMF (unit: Volt), the electric charge (unit: Coulomb), and the charge flow or current (unit: Ampere). Also known since ancient times are the forces of a magnetic field. We must attribute a field strength to it; but as later studies have shown, it may be premature and not necessary to assign a separate dimension to the magnetic field strength, since magnetic fields are not independently existing entities, but always linked to electrical ones. A discussion will follow. Some of the preceding quantities are extensive, which means they can be divided. A length, a volume, and a mass can be divided (within limits). An electric current can be split into two smaller ones, a magnetic flux can be split in two branches. Energy is an extensive quantity, but it is not a substance that is carried from here to there. Let’s not revive the phlogiston theory. If we want to call it anything else to explain it better, let us be honest and say that this is just not possible. Energy is a concept in its own right.
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There are other types of quantities, called intensive, such as force, voltage, pressure, and electric and magnetic field stress or strength. They are the driving forces for any kind of flow. In a stationary system they must be balanced by counter forces. The magnetic field stress, for instance, must be taken up by the mechanical structure of the magnet. Many quantities have a definite meaning when referred to unit area, unit volume, or unit mass. This leads to various density numbers, such as Joule/m3, g/cm3, or pressure (Newton/m2 = Pascal), which got an extra name to add to the present confusion. There is a third type of quantity, not often recognized or discussed, which is neither extensive or intensive. Radiance is an important example. If we want t o describe properly the transmission of radiative energy in our threedimensional space, then we find it can be a point-to-point affair, namely where only a single quantum of light is involved. Let us remember, this process cannot be guided or controlled; it is spontaneous; it is unexplained; it is a natural phenomenon, just observed by us; or-more accurately speaking-the results are just observed by us. If a larger than the minimal quanta1 amount of energy is transmitted by radiation (whether particles or waves) then the time-energy-spatial characteristics of the beam are described by its radiance (Watt m-lsr-'). The term sr-' (per steradian) cannot be omitted because the process to be described takes place in three-dimensional space.
For black-body radiation* the radiance is: W* = aT4/.n W m-' sr-',
and for electron beams W* =j,k'$/n(kT/e) W m - 2 sr-l,
where c is the Stefan-Boltzmann constant, T the absolute temperature,& the emission current density, e the charge of the electron, and V, acceleration voltage. For details, see Appendix 2. The SI system equates radiant flux with the unit watt. This is at best misleading, because it implies a parallel flow of radiant energy (like water in a pipe), which does not exist in nature. If we refer to visible light rather than to energy when talking about radiation, then we must use the photometric units that are of a class all of their
* For other than black-body radiation the radiance is often given (and measured) per frequency or per wavelength interval. It is then called spectral radiance and the number represents the energy per unit area, unit solid angle, unit time, unit frequency, or wavelength interval.
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own, because they are tied to the response of the human eye. The International System uses the candela as base unit. The equivalent to the radiance is the luminance (unit: candela per unit area), since the term candela (=candle power or luminous intensity) is already implicitly (per definition!) measured per steradian. We can only conjecture that this has happened because light sources are often volume emitters, and it does not make sense to talk of a surface as the emitting unit. Besides, the distance from which we observe can usually be made so large compared to the size of the light source that we approach a true point source situation for which the area does not matter yet the solid angle (determined by the receiver) does. All starlight that reaches us is of this nature. The luminance (candela per square meter) is also called lumen sr-’ m-* in full analogy to the unit radiance, with the lumen proportional to the watt. The proportionality factor depends on the wavelength of light present. The eficiency of a light source may be expressed as lumen per watt. How light quanta combine to form a macroscopic beam is also unexplained, if we want to be honest. We read: “Macroscopically observable electromagnetic waves, such as those resonating in a microwave cavity, for example, are understood to be large numbers of photons all in the same state. The photon, among all the particles, is unique in having its states be macroscopically observable in this way. . .” (Good, 1974). A coherent laser beam has been called a single giant photon in a single-phase space cell; but its direction of emission can be directed at will, contrary to the photon emitted by a single atom. The laser beam has a radiance, to the atomic photon none can be attributed. The radiance concept, however, belongs to the thermodynamic concepts for the description of ensembles of many particles.* Are we truly observing particle-states? This topic has been kind of taboo for a long time. Presently it is cautiously raised again in the literature. Let us recall that light quanta don’t interact with one another. The equipartition principle cannot be applied. Therefore, Planck stipulated the “speck of black dust” on the wall of the cavity, for which he computed the blackbody radiation equilibrium of the possible number of waves reflected from the (perfectly reflecting) walls, as a function of the temperature of that speck of black dust. In the case of laser light, also generated in a cavity (a resonance cavity in this case), it is the coherent field in the mirror (acting as a kind of antenna) that assures coherence of the light across the width of the beam and causes its Gaussian intensity distribution. The radiance, which is the characteristic quantity that describes a beam carrying energy through space, is neither an extensive quantity, since it can-
* See also Lenz (1957), W y n n (1932), Brenner et al. (1984), Wolf (1982), and Bastiaans (1981).
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not be divided, nor an intensive quantity, since it is not the driving force for the energy flow. Another such quantity having an unusual dimension is action, which is the product of energy and time. It makes its most significant appearance in Planck’s constant h, the minimum quantum of action. It also determines the minimum quantum of magnetic flux. The designers of the SI system and of the MKS-Giorgi system, which are now in use, regrettably made the same mistake that the designers of the old cgs system had made-they “economized” on units. Yet, they added a few names for certain quantities, and these names took on the nature of quasidimensional terms. Electrical and magnetic quantities in particular had to be expressed by complicated combinations of dimensional factors. Since factors would often cancel, we had the ridiculous situation that capacity and inductance were once (in our lifetime) measured in cm!
111. S O M E PECULIAR ASPECTS OF T H E SI SYSTEM We note with some astonishment that, in the SI system, energy is carried as a derived unit and not as one of the base units.* Of course, energy can often be expressed by a product of factors. Treating dimensional terms like algebraic factors -canceling and rearranging them -is one of their most useful features, namely for checking relationships expressed in an equation for internal consistency. Yet, in the case of energy, it is rarely the factors, but usually the energy itself‘ that is the physically important quantity. Its present derivation from the mechanical work concept causes energy to make its first appearance as Newton-meter, as if energy had anything to do with length (or with Newton)!+ Let us show how much simpler certain relationships will become if energy were carried as the prime term in the dimensional expressions. Presently, energy carries the dimension Newton-meter, where Newton is the unit of force; but meter is here not really a distance. It is the path-traveled by that point at which the force is applied. Suppose, on the other hand, a force of 1 Newton is applied to a rigidly clamped rod (no movement, no rotation) whose length beyond the clamp is just 1 meter. Depending on the direction of the force vector, we find at the clamping point a moment (torque) of between 0 and 1 Newton-meter. In this case, the meter is a geometrical, directional quantity. No movement is involved. * See Page (1978) for a discussion of the term derived. I don’t propose, and I hope there will never be new names for the established physical quantities. The suggestion by Snedgar (1983) to call 1 Newton = 1 shove and 1 Joule = I grunt was, I hope, written and printed with “pen in cheek.”
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This well-known bewildering equivalence in the dimensions of moment of force and of energy has its source in nothing less than sloppy logic. In the first case we should never have talked of the meter in the sense of length, but rather in the sense of travel-distance. Yet, we really don't need to and should not introduce a new dimension for the latter. All we have to do is take the Joule as the dimension for energy. At present we use the Joule only as unit for the quantity of energy (see any table on the SI system). If a force of 1 Newton is required to slide a mass against the frictional forces over a distance of 1 meter then the mechanical work equivalent to 1 Joule has been expended, namely converted to heat energy of exactly the same 1 Joule. The mass need not even have moved. If it is a breaking-force on a friction-brake the mass stayed in place, only the point(s) of the application of the force has (have) moved. The counterforce, applying a torque to the brake drum, has not moved at all. The facts are simple, but their exposition must not be oversimplified. In another case, the force may not produce heat via friction but accelerate the mass, thus turning work into kinetic energy. The preceding differences, between the meter as a geometric distance and the meter-path traveled, become yet more obvious if we ask for the rate of energy (heat) production by the friction. Now it even matters in what period of time the meter has been traversed. There is the further distinction to be made that in the expression Joule = Newton-meter, the product is a scalar product; for torque, expressed as Q = N x m we have a vector product. Yet, dimensional terms are usually not marked as vectors, while the equation for a torque would usually appear as vector equation, for instance as Q = F x r. We must now discuss angles and circular paths, where even more sloppy logic has prevailed. If the kinetic energy of rotation is to be imparted to, say, a flywheel, we must apply a torque for a certain length of time, but in such a fashion that the wheel moves through a definite angle. Again, we have the situation of a force applied over a path. Wrapping a string around the wheel and hanging a weight on this string, then letting it drop a certain distance, makes the relationships quite clear. Discussing these relationships in the customary way, the angle of rotation b, which represents the path traveled, is usually replaced by the angular velocity w = db/dt rad/sec. The kinetic energy of rotation then becomes E , , ~ = 8w2/2 (measured in Joule), where 8 is the moment of inertia (0 = my2 for a thin ring). This is in full analogy to the case of linear movement, for which the = mu2/2. kinetic energy is In the case of linear movement the linear momentum or impulse is Plin= ~ E , ~ , , / U= mu. Yet, it also is mu = F . t, where F is a force applied over the time period t . Thus, Plinhas the dimension Joule/m sec-l = Joule-sec/m or N-sec. Actually, the impulse F dt and the momentum mu are different concepts,
DIMENSIONAL TERMS FOR ENERGY TRANSPORT
231
which would be immediately obvious if we had a separate name for the momentum (for instance, 1 Galileo). (see Fig. lc). The proportionality of impulse and momentum is yet another matter. In complete analogy to the preceding case, we get in the case of rotational movement the angular momentum Pro,= ~ E , , , / w= B o with the dimension Joule/rad sec- = Joule-sec/rad. In this case, too, the impulse is also equal to the product of force and time. If we refer to the moment of the force (torque) m = Y . F we get for the product of torque and time M . t = r . F . t. Therefore, the impulse is Pro,= F . t = M . t/r Newton-meter-sec/m = N-sec, just as before. We see quite clearly that in this case, N-sec corresponds to J-s/rad (the radians, the distance along the circumference of a circle, measured in meters, is the path traveled by the force). The radian is here a measure of distance, not of angle. In the term Newton-meter-sec/m, the meter/m portion represents the radian as a measure of the angle traveled; but the force point traveled a certain distance, not an angle. That this distance is expressed in radians does not permit us to cancel the travel-distance term against the unit representing the geometrical lever arm. (In the most general case, the cross product between the vectors r and F must be used, namely M = r x F . For the discussion on dimension, we may assume r and F to be perpendicular.) Let us for a moment go back to the example of wheel and weight. When the weight drops we measure the distance in meters. Yet, when the process is described in terms of rotation, these meters (along the circumference of the wheel) become a dimensionless angle, meter-circumference/meter-radius.Is this logically permitted? The relationship of angle and circumference is obvious for the circle, but not so for any other curve. (Let’s imagine, for instance, an irregular, free spinning block of matter in earth orbit with a pull string attached to a small rocket.) The length of the circumference of a circle (or the wheel in this case) is often measured in radians. Thus, the energy injected into the rotation is again force times distance, which now reads Newton-radian; but this makes sense only if we know the unit for the radians, which means the radius, and the unit in which we want to measure it. We cannot, however, introduce a second unit of length into the system. Sticking to the agreed-upon meter for distances, we must write energy
= Newton-(radians
in meter)
or
Newton-meter-radian
= Joule.
Now let us assume according to present practice that the term radian, which has no “dimension,” will be left off. Then, we obviously get Newtonmeter = Joule. While this is formally correct (a N . m is a Joule, just as a V . A . sec is a Joule) we must now, however, provide an explanation as to
238
BERTHOLD W. SCHUMACHER
how in the system of rotational movements which we are discussing, the torque (moment) is also measured in Newton-meter (namely force x leverage). The answer is that in case of rotations the dimension of the kinetic energy is indeed Joule = Newton-(radian in meter), because the geometric meter (lever arm) does not represent the unit of movement, the radian does. This is not only a problem of semantics but one of not clearly developed concepts and incompletely defined dimensional terms. It has continuously caused confusion in teaching (Edwards, 1976). Most of the preceding troubles may never have arisen if angles, namely the divergence in the direction between two points viewed from a common origin, had always been measured and referred to in degrees (the full circle equal to 360°, 400°, 64", or whatever) and not in terms of arc divided by radius (the rad). We may revert to a clearer description of the situation as follows :
To maintain logical clarity in cases where angles represent travel distances or areas whose sizes must be known, it is suggested to use new dimension symbols such as rad, and sr, for plane and solid angles, where rad, stands for meter circumference per meter radius of curvature, and sr, stands for square meter area per me m radius of curvature of a three-dimensional surface. We would then write for rotational energy (work): 1 Newton-(radian in meters) = 1 N . rad,
=
1 Joule
This expression would just be one of many energy-equivalent combination of factors. (See Fig. 1b.)
Concept
Commonly used symbols
Name(s) of concept
Unit of measure (name of unit)
Dimension
Mass Quantity of matter
m M
mass
Distance, length geometric or path traveled
s, r
distance, length radius path length
meter
m
Planar angle circular distance
u, B
angle (divergence) arc of circle
radians or deg. radius in meter
rad, degree rad,
Solid angle (Cone) Sphere = 471 Spatial period
a,71a*
solid angle
71.
-
Time, period of
t, 'I
turn time
second
turn, turn s, sec
Events/unit time
V
frequency
(Hertz)
S-'
~
kilogram mol
radius2
(+)
sr ~
'
239
DIMENSIONAL TERMS FOR ENERGY TRANSPORT Velocity, linear
U
velocity (vector) speed (scalar)
-
circular frequency rev./unit time angular velocity (vector)
2nv
ms-' m sC1
~
Circular frequency
w
Velocity, angular
W
Circular velocity
w
-
-
Acceleration
a, b, g
Energy
E, s
acceleration energy
Joule
J
Power, energy flow, power-input power-dissipation
W, N
power, wattage heat flow, etc.
(Watt)
J sC1
Force
F.
force
Newton
N
Pressure, stress
P, t , 0
pressure, stress
(Pascal)
N m-2
Electric charge field-energy/volt
4
charge (+)
A.s=(Coulomb)
JV-'=As
Electric current
I, i
current
Ampere
A
s
S-'
turns/sec -
-
rad s - ' (deg. s - I ) rad, s - '
(#I
m s-2
Electric tension
V
emf, voltage
Volt
V
Electric resistance
R
(Ohm)
V A-'
Electric conductance
u
resistance conductance
(mho) (Siemens)
A V-'
Electric capacitance charge/volt energy/volt2
C
capacitance
(Farad)
A s V-l =JVd2 = A/(V/sec)
Magnetic flux field-energy/ampere
cp
magnetic flux
(Weber) (Maxwell x 10')
J A-turn-
Magnetomotive force
Gi
mmf
A-turns (Gilbert)
A-turns
Inductance L magnetic fluxpampere energy/ampere2
inductance
(Henry)
VsA-'
Electric field stress
field strength field strength (vector)
V m-' ( # )
V m-'
Magnetic field stress
E H
A-turn m (Oersted/79.6)
A-turn m-l
Electric flux density
D
displacement (vector)
-
Magnetic flux density
B
magnetic induction magnetic flux density (Tesla) (vector) (Gauss x lo4)
(+)
= Vs/turn
=J
(+)
(#)
A-2
= V/(A/s)
(+)
J V-' m - 2 = A s m-2
(+I J A-turn- m-2 = Vs m - 2 turn-' = Vs/turn m2
FIG.la. Mechanical and electrical quantities and their dimensions. (+) means see Appendix 1 and ( # ) means that the quantity has never been given a name.
240
BERTHOLD W. SCHUMACHER
Mechanical force x linear movement Mechanical foce x rotary movement Kinetic energy, linear movement, fmu2 Kinetic energy, rotary movement, @02 Mechanical volumetric (stress) energy density, pvlM, RT, ._.. Electrical energy dissipation, power x time, Electrical energy imparted to particle
work work energy energy
N . m = Joule N . rad, = Joule kg(m s-')* = Joule
energy kinetic energy
V A sec = Joule 1 eV = 1.6 x 10- l 9 Joule
V A s m - 3 = Joule m - 3 A V S ~ =- J ~ u l e r n - ~
Electrical field energy density, E ' D / 2 = &,E2/2 = D2/2&, Magnetic field energy density, H . B / 2 = p L o H 2 / 2= B 2 / 2 p ,
FIG.1b. Energy-equivalent combinations
The values for the units of the basic dimensional terms (marked by an asterisk) in Fig. 1 have been chosen so that no fractional numbers are encountered in relationships such as the ones in Fig. lb. The units for the mass (kg), length (m), and time (sec) are artificially defined by prototype standards. (References to a length on earth, or part of the day are no longer accurate enough and meaningful.) The unit for the energy 1 J is equal to [l kg (1 m 1 sec-')2]. The unit of force 1 N is chosen so that 1 N . m which makes 1 N
=
=
1J
[l kg 1 m 1 secW2]
and only unit factors enter the equations. Since any mass is attracted by the gravitational force on the surface of the earth with a force of 1 k& per kg of mass (or a weight of 1 k&), which produces an acceleration of 9.81 m sec-* if the mass is permitted to fall freely, the relationship between kg, and the unit Newton is: 1N
=
1/9.81 kg,
=
102gf,,,,
/
In this case a number other than unity enters into the equation. While weight is a concept in its own right, it is not to be considered a dimensional quantity. Its dimension is that of a force. A dimensional equation is only complete and can only then be used in dimensional analysis for deriving new mutual relationships as in the field of similitude mechanics, if it remains correct regardless of the chosen size of the fundamental units. This is known as the n-theorem. It will not affect the arguments made in the present paper for the official use of more than three or four Force x time Mass x velocity Energyivelocity ( x 2)
impulse momentum momentum
N .sec kg.m.sec-' Joule/(m/sec)
FIG.lc. Momentum-equivalent combinations.
24 1
DIMENSIONAL TERMS FOR ENERGY TRANSPORT
dimensional terms and for the closer relationship of the used terms with the concepts that we need to formulate nature’s laws. The units Volt and Ampere have been chosen so that [ 1 V . 1 A . 1 sec] = 1 Joule, which makes the Joule the common denominator between mechanical and electrical phenomena and is in keeping with the overriding role that energy plays in all natural processes. Sometimes, in nuclear physics for instance, it is more convenient to use the electron volt (eV) in lieu of the Joule as A sec. Thus, the unit of energy. The charge of the electron is 1.60210 x Joule. weget 1 eV = 1.60210 x For rotational processes we find that 1 rad,/sec = 1/2r revolutions/sec. (The ratio of diameter to circumference of the circle, since we measure angles by radii.) The factor turn-’ (or per turn) indicates that we measure the quantity in question if we go once, and only once, around a circle in space. It is an indication of the three-dimensional character of nature. There is still the question of the orientation of the plane of the circle to be considered. The electromagnetic field energy densities and the field stresses (in vacuum) are linked by nature in a definite numerical relationship as
DIE = E~ B/H
=
=
8.85418782 x lo-’’
p o = 1.25663706 x
(J/m3)/(V/m)’
= J/V’m
= (A
sec)/(V m)
(J/m3)/(A/m)’
= J/A2m = (V
sec)/(A m)
There exists another important relationship to the speed of light co: co = 1 / J G )
=
2.997925 x 10’
m sec-’
Moreover, the square root of the ratio of pol&,, represents the impedance of the vacuum for the propagation of electromagnetic waves: Z
=
Jz 376.7 =
&/A’
m)/(J/V2 m)
= V/A =
Ohm
The special names given to certain quantities, and listed in parentheses in Fig. l a (e.g., Tesla) can be useful as short notations, but they should never be used as quasi-dimensional terms. In summary, four points can be made here: (i) Angles should not be left dimensionless. We should re-think once more what a dimension is supposed to signify. Symbols rad, and sr, are suggested when the arc-length is meant, rather than the difference in the direction of two radius vectors (which is the meaning of an angle, whether measured in degree or radians). The symbols for the angles remain rad and sr. (ii) Energy should not be measured in Newton-meter, except where work is done by linear movement against a force. Energy is a concept in its own right, for which we have already adopted the unit Joule, which should be
242
BERTHOLD W. SCHUMACHER
looked upon and elevated to a primary dimensional term. The Newton-meter should be referred to as work. In the case of electromagnetic phenomena, the Watt-second or Volt-Ampere-second can always be referred to as electrical energy, although still measured in Joule.* (iii) Force should not be defined via acceleration, but the unit Newton should be looked upon and elevated to a primary dimensional term, its unit represented by a standard spring force. (A standard yet to be defined.) (iv) The concept of acceleration should be freed of its connection with mass and force and treated as the purely geometriclkinematic concept, which the name implies, namely the change of velocity with time. We have not touched on the concept of temperature. It is clearly related to energy and quite as clearly has a dimension ofits own that cannot be derived. Using the ideal gas laws, we can write
T
= pv/R
with R a measured constant linking the energy concept with the work concept and both with temperature. A similar linkage is, of course, at the root of the second law of thermodynamics. Conversely, this shows that work (mechanical) and energy are not the same concept.? Temperature is the conceptual link. The special form of energy generally called heat, linking temperature with properties of matter, entails yet another concept. It is distinguished, for instance, by the fact that there are no oscillations of heat energy ever observed, only asymptotic processes. By contrast, kinetic and potential energy, or electric and magnetic field energy, can interact in oscillatory exchanges. The terminology we use and the dimensions we assign, should reflect, not obscure, the subtle differences in these concepts. In the present context we don’t want to expand this discussion, by going into thermodynamics with free energy, enthalpy, and entropy.
AND MAGNETIC QUANTITIES Iv. ELECTRIC
It is in the nature of any rational system of units that there are algebraic relationships that permit us to express every unit in terms of several others-the laws of nature form a closed chain. As far as the algebra is con-
* At other times the unit of electrical energy used may be the electron volt, eV. The unit measure is something quite incidental, whereas the concept of the dimensional term is fundamental. The electron volt being a composite, would not make a good dimension term. f These are not trivial matters. See, for instance, the recent discussions by Laufer (1983) or Falk et al. (1983).
243
DIMENSIONAL TERMS FOR ENERGY TRANSPORT Emphasis o n the coulomb
I
A
I
Coulomb
Weber
1
coulomb
Emphasis on the weber
1
Emphasis on the volt
ampere.sec
joule volt
I
joule ampere
volt. sec
weber
~
A.
Ampere
coulomb second
______
joule weber
ampere
Volt
joule coulomb
weber second
watt ampere
~~~~
..
~
~~
~
Some other combinations a
1 .
joule. sec
Joule. sec coulomb
..
Emphasis o n the ampere
..
watt volt
9 volt. farad
..
~
ampere. henry 0
weber henry ~
coulomb farad
volt A.
0
Farad
coulomb’ joule
joule. sec’ weber’
ampere’ sec watt
joule . volt’
coulomb volt
Henry
joule. sec’ coulomb’
weber’ joule
joule ampere’
volt’. sec watt
weber ampere
Ohm
joule. sec coulomb’
weber’ joule. sec
ampere’
volt’ watt
ampere
coulomb’ joule. sec
joule. sec weber’
ampere’ watt
watt volt’
ampere volt
~
Mho
~~
a
~
~~
~~
a
~
-
1
~
a
FIG. 2. Present use of name quantities as quasi-dimensional terms. [From Rojansky (1979).]
cerned it does not matter much what comes first and what comes second. It is the same territory, whether we cross it from east to west or west to east; yet the vistas can be drastically different at times. With this in mind, let us now consider the electric and magnetic quantities. For the SI system and the MKS-Giorgi system of units, the possible cyclical substitutions for the electric and magnetic terms are shown in the matrix of Figure 2, taken from the book of Rojansky (1979); one of the clearest expositions of the subject.* It is astonishing that none of the units was listed under a column heading “Emphasis on the Joule” (which would mean emphasis on energy). Such a column was added in Fig. 3. It is also typical that, as in most textbooks, the terms denoting the concepts, such as inductance, are not in the original * Appendix 1 brings a fuller discussion.
~
244
BERTHOLD W. SCHUMACHER
Concept Electric charge (Field energy/volt) MagneticJlux (Field energy/ampere) Electric current Electric tension, emf Driving force of current Electric field stress Magnetic tension, mmf Generating the flux Magnetic field stress Capacitance Inductance Power dissipation Energy (released or supplied or transferred) Resistance Conductance
Unit name dimension (Coulomb) A-sec (Weber) V-sec/turn A V
Relationship with emphasis on energy Joule/V Joule/V Joule/A-turn Joule/A-turn ~
VIm A-turns A-turns/m (Farad) A-sec/V (Henry) V-sec/A Watt VxA Joule W x sec (Ohm) VIA (mho)(Siemens) A/V
-
Joule/Vz ( x 2) Joule/A2 ( x 2) Joule/sec Joule W/AZ Joule/A2 sec W/V Joule/V2 sec
FIG.3. Relationship of name quantities and dimensional terms with the Joule as the dimension for energy
matrix. If we consider the concept of the magnetic flux, for example, and as its unit of measure the energy of the field per ampere of current, then we need no special unit with a special name for it; Volt and Ampere will do. The flux 4 = E,,,,~/I~ Joule/A-turn is the ratio of magnetic field energy to ampere-turn of current. Why introduce a special name for the magnetic flux? The term “magnetic flux” is not really a good one; nothing is “flowing” here, as the term might imply. Talking instead of field-lines, as Faraday did, would be better, the number of field lines being proportional to the field energy (see Eq. 12 of Appendix 1). If “Watt” were left in the system, as an additional dimensional term, simpler expressions would often result. It would, for instance, emphasize the inherent difference in the concepts of impedance (inductance) and resistance by the term in the dimension. (The difference appears in any case.) The three terms-Joule, Ampere, and Volt-can cover all other presently used electric and magnetic terms. The names in parentheses should be phased out since they are anti-mnemonic. They refer to arbitrary units, not to the concepts. If we would use the names for the concepts, e.g. magnetic flux, mag-
DIMENSIONAL TERMS FOR ENERGY TRANSPORT
245
netic flux density, magnetic field strength (stress), it would be immediately clear what is meant. Note, for instance, that no one would be tempted to look for a magnetic field strength-density. The difference between B-field and Hfield would become obvious. Electrodynamic phenomena take place in four-dimensional space. The time dimension is always present and as such an accepted dimension term, but for the space dimensions we have none. The term turn, as in ampere-turn or volt-per-turn, could be considered to represent the spatial aspects. At present, even were it matters, it is usually only implied but not explicitly stated, because turn is not considered an accepted dimensional term. What is really meant must, when the situation demands it, always be explained in the text. The problem relates to the fact that dimensional terms do not reflect the vector character of a quantity. See Appendix 1 for more details. It is purely a matter of preference and convenience (established habit) to use the Volt and the Ampere, in addition to Joule, as base units for electrical and magnetic phenomena. If we do so, it will be of considerable heuristic value that we then get Joule/V2 and Joule/A2 as the dimensions for capacitance C and inductance L, respectively. We see immediately that (within the constraints of a certain device) the field energy increases with the square of the voltage or current, the inductance and capacitance being considered constant characteristics of the device or circuit. A factor of two has to be carried 2 in numerical calculations. It is L = 24/10 and E , , ~ = LZ0/2 just as linear mo, / v , momentum p,,, = 2ckin/mand E ~ ~ = , , mu2/2 mentum Plin= ~ E ~ ~ ,angular (see Appendix 1). Let us look at the concept of inductance more closely. The energy goes into the magnetic field, therefore inductive impedance is a property of the field, not the coil. Yet there is a certain dual sense in which the word is used. This is more noticeable in the terms mutual inductance and self-inductance. Here it refers to a geometrically determined relationship between coils or parts of coils. A case could be made that, from a strictly logical point, we have here two different concepts and they should have different names. In one case, they refer to a dimension of the device, yet the energy that is involved resides entirely in the fields. Let us repeat: The ratio of the magnetic field energy to the ampere-turns that maintain the field determines what we call the total magnetic flux 6.It received a separate unit name:
14 = 1 Joule/A-turn = 1 Weber How much less mysterious would it appear if, as suggested, the magnetic flux were just introduced as the ratio of the field energy to the current that generates that field. A separate name adds mostly confusion.
246
BERTHOLD W. SCHUMACHER
In the electric case, the field energy per volt is identical in its dimension with the dimension of the charge, as is well known. This reflects the residual asymmetry of electric and magnetic phenomena. A discussion would lead deeper than possible here. The fact that the suggested dimensions for the magnetic field terms in Fig. 3 no longer contain the names of the magnetic quantities is no cause for alarm. Magnetic fields exist, as is well known today, never isolated from electric currents. For that reason even some traditional texts, such as the one of Slater and Frank (1969), did start their discussion with currents “as being in a way more fundamental.” Starting with energy is the most fundamental way of all-and the subject becomes surprisingly simple. Those illustrious men, whose names have been given to the units now in use, would not mind dropping those names, if it made the subject of their lives’ work more easily and widely understood. Adopting for the electric and magnetic units those listed in the column “Emphasis on Energy” in Fig. 3, we would add energy (Joule) and voltage (Volt) to the base units of the present SI system, rather than defining them as N . m and W/A, to say nothing of m 2 . kg . secL2and m 2 . kg . s e C 3 . A w l . Referring back to Figure 1, a summary has been provided showing the suggested set of mechanical and electromagnetic concepts with name, unit of measurement and dimension. The shift in terminology is very slight, yet the shift in thinking is significant, and the advantage is great. There is absolutely no merit in the older attempts to keep the number of base units of a dimension system rather small. But no new, not presently used terms have been introduced here, nor will any be needed. Figure l b shows combinations of terms whose dimensions are equivalent to energy or energy densities, but in themselves cannot possibly define energy. These combinations reflect, of course, the conservation of energy-one of the basic attributes of nature. The combinations in Fig. lc reflect the conservation of momentum, another basic attribute of nature. They also show the connection of force with momentum on which (quite arbitrarily) the present definition of the unit of force is based. Moreover, impulse and momentum are different concepts. The first combination is an operational definition of impulse, the second is a static definition of a quantity possessed. Only the choice of the unit of measure made them appear the same concept. Since conservation of momentum is as fundamental as conservation of energy, it is surprising that momentum did not get a name for its unit (e.g. 1 Galileo). This just shows how arbitrarily the present system has developed.
DIMENSIONAL TERMS FOR ENERGY TRANSPORT
247
As mentioned, the total flux 4 is identical with the energy per unit current generating the field. The flux density B is the energy per unit cross section and current. Both these concepts contain no reference to the volume in which the field energy is contained. Yet, it is plausible that the smaller the volume the greater the internal pressure or the internal field stresses and it is no longer just the generating current, but the current per unit length of the field lines that determines the field stress or field strength H , which is indeed measured in Ampere-turns per meter (regardless of the cross section of the field). This argument shows in a very direct manner why there are two field quantities needed, namely B and H . They refer to totally different aspects of the field. For fields in vacuum the linking factor is a dimensional constant of nature, namely pLo= 1.256637 x
Joule/A-turn m2)/(A-turn/m)
[ = J/A2 m]
Appendix 1 , which follows, presents an example of the advantages of the suggested changes, when we introduce the quantitative aspects of magnetism-a notoriously tricky subject-to a student audience, for instance. Appendix 2 presents a discussion of the terms for describing particle beams and light beams, especially as energy carriers. Appendix 3 presents a table of observed energy densities, since reference to energy as a basic quantity has been our chief concern in establishing other concepts and their dimensions and mutual relationships.
APPENDIX 1. MAGNETICFIELDQUANTITIES AND THEIR IN THE DIMENSIONS-ENERGY DENSITIESAND FORCES FIELDS A . Energy Required for Generating a Magnetic Field
As always, when we study the laws of nature, we observe some effect, then measure it in a quantitative way, thus developing in the process certain rules of how to measure it. We can then describe the effect by referring to the measured quantities by a given name. Often quite different kinds of effects are observed, all caused by the same underlying natural phenomenon. It is essential, therefore, to elucidate and formulate the mutual relationships as clearly as possible; mathematical equations alone are not enough. When we are not careful, the rules of how to measure may obscure what it was that we wanted to describe. This seems to have happened in the case of magnetics. In the following exposition of this subject we will follow a different path than standard texts.
248
BERTHOLD W. SCHUMACHER
(i) As suggested, let us start with an observation.* In order to generate a magnetic field where there was none before, we must expend energy, yetsurprisingly, only to produce the field, not to maintain it. It appears as if the magnetic field is a kind of invisible container into which energy can be poured and retrieved again without 1oss.t (ii) Let us analyze quantitatively what happens. If a voltage V, is applied to a solenoid coil (without iron) a current starts to flow and increases with time to a maximum I , . This maximum I , is determined by the ohmic resistance R, in the coil and the internal resistance R , in the power source. We find that
I,
=
Vo/(Rc + R,)
(1)
Initially, when we first make the connection, the current starts lower. We find there exists an additional hindrance or impedance to the rise of the current. We call it the inductive impedance of the magnetic field or, for short, the inductance. As in all related situations, we find that Lenz’s law applies: The reaction to any imposed change is so as to oppose it. The effect and the magnitude of the impedance can be quantified if we look at the law according to which the coil current rises, for instance, by using an oscilloscope. We find that while the current is on the rise, a counter voltage (opposing the applied voltage V,) is induced whose magnitude is
Ynd = L ( d f / d t )
(2)
The factor L is the aforementioned inductance (different for different coils). The current rises, therefore, according to the law
I
=
Vo(l - eCRtIL)/R
(3)
where R
= Rc i- R,
The details of how we measure and verify Eqs. 2 and 3 need to be discussed, of course, but not in the present discourse on the concepts.
* This was not, historically, the first observation about magnetic fields. While the historical aspects of science must certainly be considered to gain a full understanding of a subject, we should not necessarily follow the historical path for a modern introductory exposition. Since physics is always based on observations, not mathematical axioms, I have chosen as a starting point for this new discourse on magnetism what appears to be the most significant and most consequential observation related to the magnetic field. It seems that the magnetic field even changes space-time-geometry. At least we may interpret in this way the Bohm-Aharonov effect on an electron beam path.
DIMENSIONAL TERMS FOR ENERGY TRANSPORT
249
The proportionality factor L for the induced voltage yndis entirely due to the build up of the magnetic field, and, therefore, it was introduced here as such. The value of L is different for different coils, yet the coil is actually quite incidental. It determines the shape and size of the magnetic field, but the space and the matter surrounding it will also influence the value of L. We should not get confused about the fact that there are two concepts involved and the terms impedance and inductance are sometimes used interchangeably. The conceptual difference becomes clearer when we consider, for instance, such terms as mutual inductance and self-inductance. They refer to a different concept than the impedance due to the build-up of the magnetic field, namely to a geometric relationship betwen coils or part of coils. Let us also remember that a coil wound bi-filar has no inductance and generates no magnetic field, though it may look like a solenoid. Any straight wire shows inductance, and it changes if we surround the wire by an iron ring. The inductance is not only a property of the wire, but also of its surroundings. It reminds us that an electro-magnetic wave (radio wave) is formed not only by the antenna of the transmitter but also by its interaction with the electromagnetic space (as already stressed by Heinrich Hertz). Last but not least, free space has an impedance but neither inductance nor resistance for electromagnetic waves. It will be important later to remember that the definition of L (Eq. 2 ) contains no reference to the geometry or the number of turns of the coil producing the magnetic field. (iii) Let us formulate the law which governs the build-up of a magnetic field. Based on the preceding observations and analysis we see that the instantaneous power going into the magnetic field is dE/dt
=
Vnd(t)
r(t) = LI(dZ/dt)
(4)
After the current has risen from 0 to I , we get the total energy Emagwhich has gone into the magnetic field by integrating over time, and we find P
Emag=
”
J d E = L J ZdZ = j1L Z i
for the energy in the magnetic field. It follows that the dimension for the inductance L is Joule/A2, since 2Emag 2 x Magnetic Field Energy L = ~- Joule/A2 1; Square of the Current
2 Joule/A2 carries the separate name 1 Henry. Today, inductance is usually considered a property (and quantity) that characterizes a certain coil. For a coil without enclosed matter we can indeed derive L from the coil’s geometry, as we will show later, but not when matter,
250
BERTHOLD W. SCHUMACHER
such as an iron core, is present. We may easily lose sight of the original question that led us to the definition of L, namely how much energy goes into the magnetic field. To stay consistent, we should actually say that the inductance is the measure for the energy ( x 2 ) stored as magnetic field energy for each ampere of the current squared. Here we are talking about the actual current flowing through the coil, not, as often later, about ampere-turns. The turns number is implicitly contained in the numerical value of L, just as the factor 2x was left in the definition to retain unit factors in Eq. 2. The current is fed to the coil somewhere; exactly where is so far quite incidental and unimportant. Let us remember that in the electric case, the capacitance C is the measure of the energy stored ( x 2) in an electric field for each volt-between-its-boundaries squared; namely 2E,, c=-= V:
2 x Electic Field Energy Jo ule/V2 Square of the Voltage
(7)
2 Joule/V2 carries the separate name 1 Farad. Here is a point which should be made. We would much more readily learn and connect new subject matter with already familiar concepts if we kept speaking of the inductance as Joules/A2 and the capacitance as Joules/V2, instead of coining new units. I think Faraday and Henry would agree. Nevertheless, let us now consider L and C mainly as properties of the devices, not the fields. Then the preceding dimensional equations for the inductance L can be simplified as
L=
2VAsec V Vsec = 1 Henry A/sec A A2
(8)
which would also follow directly from Eq. 2, by which it was historically defined. In either case, the term for the self-induction coefficient L is Henry (Hy) and we may also say:
A coil has an inductance of L = 1 Hy if 1 V is generated while the current changes steadily by 1 A/sec. With equal justification, we can say: If the field is generated by 1 A, or is changed by &A1 = 1 A of current, and this requires or yields $ Joule of energy (by which the field energy changes) then the field (and its generating coil) has the inductance of 1 Hy.
DIMENSIONAL TERMS FOR ENERGY TRANSPORT
251
Assigning the inductance to the coil is really only a matter of speech. As has been stressed before, the inductance is a field property and only indirectly one of the coil. The field energy is there all the time, unnoticed. It makes itself and its inertia only felt via the inductance, which appears only when something changes. Time, space, and energy are intricately woven into the electromagnetic phenomena or vice versa, if you will. Once the current I , is attained and maintained in the solenoid the field energy remains constant. In a conventional solenoid coil ohmic losses do still occur, and, therefore, energy must be supplied. If the coil is superconducting, the current, once established, keeps flowing on its own, thus maintaining the magnetic field without further energy expenditure. The latter case can serve as an analogy for the atomic or molecular currents that make permanent magnets of certain materials. Let us look at a few special cases. In any perfect conductor (e.g., a superconductor, or a high-temperature plasma) any existing magnetic field is frozen in and none can be induced from the outside. O n this basis, we can build magnetic shielding devices from superconducting metal foil, up to a certain level of flux, above which superconductivity is destroyed. (Superconductivity depends on electron pairs with opposite spin and magnetic moment, thus magnetism is involved in it from the onset.) Slugs of plasma as well as slugs of metal can be fired from a magnetic gun having a rapidly rising magnetic field, which can penetrate not as rapidly into the projectile and push it ahead of the field. Cosmic plasma clouds can hold magnetic fields that may trap or accelerate (if they are moving) charged particles of the cosmic rays. Conversely, when the matter holding the magnetic field is mechanically compressed, the field energy density increases. Mechanical flux compression is sometimes used to generate magnetic fields with higher flux. We see an extreme case in the collapse of an ordinary star to a white dwarf or a neutron star. Fields with a flux density of B = lo6 to lo9 Joule/A m2 (Tesla) seem to be generated in this process, as spectral observations show. (See below for the definition of the terms flux density, etc.) The energy placed into the magnetic field reappears when the field collapses, for instance when the solenoid current is interrupted. A voltage proportional to d I / d t is generated in the coil. Then L( can become very high since the field collapse, and therefore - d I / d t , is generally faster than the build-up. This induced voltage can produce sparks across the switch or a parallel spark gap. The prime examples in which this effect is used are ignition coils and spark plugs in a gasoline engine. When a large magnet is switched off, it is necessary to limit d I / d t and provisions must be made that the energy is dissipated in a controlled manner.
252
BERTHOLD W. SCHUMACHER
The stored field energy $LZ2 is also used in the choke coils, which smooth a rectified ac current into a more or less ripple-free dc current by feeding from this stored energy into the circuit while the ac voltage goes through zero. The time constant z with which the energy is released is given by z = L / R sec, where R is the ohmic resistance. (See also Eq. 3.) Another example: The loss-free retrieval of the magnetic field energy has recently become of importance for large-scale energy release or storage by electric utility companies. (For two reviews on this subject see Proc. ZEEE 71, 1983.) A superconducting coil generates a field with 3 0 MJ of energy ( L = 2.5 Joule/A2 at 3.5 kA of dc current.) The mean coil diameter is 3 m, the coil height 1.2 m. 10 MJoule can be withdrawn or taken up by this coil within 1 sec, practically loss-free (via a dc-to-ac converter consisting of gating devices synchronized with the main power line). This exceeds the capabilities of any other kind of energy storage method, such as compressed air or water reservoir storage. Coils for fields of 1013Joule that permit power transfer rates of 500 MW are being planned. They will have diameters of 100-1000 m and be buried in solid rock to take up the forces on the windings. B. Field Lines and Flux Let us continue with reference to another observation. As is well known, the magnetic field (as created by the solenoid and in the steady state) manifests itself in other ways too. From its effect on iron filings, which align themselves in the field, we get the image and concepts of field lines or flux. How can they be related to the field energy? It seems plausible that the total number of field lines, the total flux as we call it, (denoted by 4) is proportional to the field energy per unit current (in ampere-turns) as long as the geometry of the solenoid is not altered. Thus, by definition, a field with a total energy of 1 Joule, if created by a loop carrying 1 A-turn of current, has a total magneticfiux 4 of 1 Weber (Wb). In other words, the ratio of field energy to ampere-turns determines what we call the total flux of the field: 1 Wb
=
1 Joule/A-turn
(9)
Note that the total flux (and the associated energy of the field) is not simply a function of the field generating current. The two are linked via the inductance L of the coil. Here L depends on the size of the coil, the number of turns and the material in it, if any, as we will discuss shortly. Thus, 1 A of current can generate more or less magnetic flux, depending on other factors, such as the enclosed area and the number of times which it encircles that area. The greater the total flux, the greater the field energy. Yet,
DIMENSIONAL TERMS FOR ENERGY TRANSPORT
253
the more field energy is generated the longer it takes for I to rise to its final value (Eq. 3), because L appears correspondingly larger. Time is involved in getting the energy pushed into the magnetic field. This is akin to accelerating an inert mass to a certain kinetic energy. We can therefore also say that the magnetic field has inertia. How much flux (JoulesjAmpere-turn) a coil generates per actual flowing ampere, is determined by its inductance L, which has the dimension L = (JoulejA-t urn)/A = (2x)JoulejA
’.
How can we measure the flux? The amperes can be measured readily, and the Joules can be recorded when they are deposited. Yet there is a simpler way. We may proceed to transform Eq. (9) as follows: 1 Wb
=
1 JoulejA-turn
=
1 V A secjA-turn
=
1 V-secjturn
(10)
We find, the total flux 4 can indeed be measured directly in volt-seconds per turn. To do this, we surround the field (or solenoid) by a 1-turn wire loop that is connected to an oscillograph. (The oscillograph measures in volts, we can integrate to get volt-seconds, but the turns must be established separately by choosing the coil of 1 turn or n turns. In the latter case, the voltage readings must be divided by n.) We then move this coil out of the field completely (pull it out over the solenoid). The oscillograph will record a trace of voltage versus time that, when integrated over time, will just yield 1 volt-second for a flux of 1 Joule/A (1 Weber). A ballistic galvanometer will do the integration automatically and can, therefore, be calibrated in volt-seconds or Webers. An electronic integrating circuit can do the same. Note that the volt-second will only appear if the field through the coil is made to disappear (the coil taken out of the field)-a somewhat strange way to prove the presence of something! But then, it had also taken time to build up the field. The volt-seconds can be a useful measure in their own right. For instance, the volt-seconds of a voltage pulse to be transmitted by a pulse transformer determine its core cross section (together with the maximum permissible flux density). Note, however, that this tells us nothing yet about the Watt-seconds that can be transmitted. (See Subsection D.) Another fact is most interesting: There exists a minimum magnetic flux quantum, represented by the ratio of Planck’s action quantum h to the quantum of electrical charge e, namely: h/2e
=
2.068 x lo-’’
JoulejA
or Joule sec/A sec or Joule sec/C
We notice that the terms turn and second appear in all the preceding dimensional equations. When only the term seconds appears, as in Eq. 8, this is routinely accepted. However, in Eq. 8 we have quietly implied that the field is related to a coil with a certain number of turns. By contrast, 4 is defined per A-turn. Thus, 4 has, so far, only been introduced with respect to the static
254
BERTHOLD W. SCHUMACHER
field. The induction L was introduced with respect to a changing field. We will determine the relationships between the two in Subsection D. To keep the discussions and calculations free of implied assumptions, we must not equate Ampere-turns with Amperes, as is so often done. Volt-seconds per turn is also something else than volt-seconds. The term turn deserves to be elevated to a dimension term. It is absolutely needed to describe correctly the " three-dimensionality " of all electromagnetic phenomena and, with time included, their four-dimensional nature. C . Flux Density
So far we have not looked and did not need to know exactly how the total flux q5 is distributed. Inside the solenoid, we may assume it is uniform. We can also surmize that the flux inside the solenoid is aligned with the solenoid's axis and, therefore, has vectorial character. The alignment of the iron filings again provides an image of this concept, and it provides us with some indication of the structure of the field outside the coil. The local flux o r j u x density B in any part of the field can also be measured easily by means of a small l-turn probe coil having unit area. We find that to get maximum readings we must orient the probe coil, before we withdraw it, with its plane perpendicular to the direction of the flux lines whose direction was already revealed by the iron filings. The unit for the flux density B is then 1 Wb/m2 = 1 Joule/A-turn m2 = 1 V-sec/turn m2 = 1 Tesla
The unit of the flux density has been given the name 1 Tesla (1 T). There is a widely used older name for the flux density, namely 1 Gauss =
Tesla = lop8Joule/A-turn cm2 = lo-* V-sec/turn cm2
Another older measure for the total flux is
1 Maxwell =
Wb
=
V-sec/turn
(12a)
V-sec/turn cm2
(12b)
This makes 1 Gauss = 1 Maxwell/cm2 =
for which the name 1 line/cm2 is sometimes used. The aforementioned probe coil for measuring the flux density may be simply flipped over instead of being withdrawn, so that the direction of the field that it sees changes by 180". There are flux density meters (Gauss- or
DIMENSIONAL TERMS FOR ENERGY TRANSPORT
255
Tesla-meters) available that use a small rotating coil as the sensor. If the field is generated by an ac current and, therefore, changes its direction periodically, a stationary small coil connected to an oscilloscope will show the field and its strength. The coil can be calibrated readily by holding it close to a straight wire that carries a known current and which produces a known magnetic field at a radial distance r. It has not been necessary so far to say anything about the volume of the magnetic field or about its shape. Even the fact that #J and B are quantities with a directional character has not clearly emerged. They reveal this aspect only in connection with other vectorial quantities, e.g., the orientation of the probe coil, the required movement of the coil across the field (not along the field lines), or the interaction of two magnetic fields (the compass needle is a prime example). We have seen that the total field generated by a solenoid coil has flux lines emerging at one end and returning to the other. There are no localized, planar sources or sinks for the flux 4 or the flux density B. We say that the B-field is source-free, which is written as:
or
divB=O
V.B=O
(13)
Any vector whose divergence is zero can be represented or thought of as the rotation or curl of another vector field. We can, therefore, write B
= curl
A
(14)
Here, A is a vector function of position, a vector field. (An arbitrary constant can yet be added.) We can show that A satisfies Poisson’s equation; A can, therefore, be called a potential, the vector potential of the magnetic field. In this introductory text, however, we cannot elaborate. D . Induced Voltages and the Transformer
Measuring the flux #J or the flux density B with the probe coil, as described, yields the same integral value whether we move the coil slowly or fast. However, depending on the speed, we get greatly diflerent instantaneous voltages. (Another observational fact, found by Faraday.)
The basic luw of induction, Faraday’s law, says: The generated voltage is proportional to the rate of change of the total amount of flux that passes through a closed loop of a conductor. If the loop has n turns, then the induced voltage is n times as high. Thus, we can write
vnd
vnd= n d4fdt
(15)
256
BERTHOLD W. SCHUMACHER
It does not matter whether C#J is distributed uniformly within the loop, whether C#J changes while the loop is stationary, or whether it is just one section of the loop moving through the flux field so that the enclosed portion of it is changing. The polarity of the voltage is the opposite when the flux increases and when it decreases. If the conductor loop is closed, so that Vnd causes a current to flow and energy to be dissipated, then a reaction force will be felt which opposes the movement of the conductor. (Lenz's law) Nature opposes change, action causes reaction. The well-known eddy current brake (a copper disc moving through a magnetic field) is one example where this effect is utilized. Induced voltages and currents are, as is well known, the basis of today's alternating current technology. They are also the basis of the generation of electromagnetic waves, with subtle (yet important) differences between the two cases. We have seen (Eq. 2) that we may also write
K = L(dl/dt) Whereas Eq. 2 was derived from observing a step function response, it is nevertheless generally valid for any function of I. Of particular interest are sinusoidal functions. For instance, if
I
=I,
sin wt
then dl/dt
= Z,wt
cos(wt
+ 6)
and
6 = L o l , cos(ot + 6) If the additive phase shift factor (integration constant) 6 is zero, we see that V and I are 90" out of phase, which is typical for a purely inductive system. No energy is dissipated. The magnetic field returns, in the second half of every magnetization cycle, all the energy put into it in the first half. (There are 120 magnetic cycles for every 60 voltage cycles!) But, we don't need to know the value of L if we know the flux 4. The details of the relationship between L and 4 are discussed in Subsection H. Let us look at this in connection with a discussion of a very common, everyday device: the transformer. A sinusoidal voltage applied to the primary coil induces a magnetic field in the iron core (see Subsection K for details) and a certain primary current flows. Assuming the permissible maximum flux density is B, = 1.5 Joule/A-turn m2 (1.5 Tesla = 15 kGauss), then the flux per square centimeter of the core cross section is 1/104 of the preceding value,
DIMENSIONAL TERMS FOR ENERGY TRANSPORT
257
or B , = 1.5 x V-sec/turn cm2 (see Joule/A-turn cm2 = 1.5 x Eq. 11). For a frequency off = 60 Hz, or o = 27zf = 377 sec-', we get F,,, = n dbldt,,, = n4,o = n x 566 x Vpeakor 0.0566 V-turn (for each cm2 of core cross section). For 600 Hz it will be 10 times as much. For a core cross section of 25 cm2 (2 x 2 inch) and 600 Hz we get 14 V/turn (volt peak). When a greater voltage is applied, the maximum flux that the core can handle will be exceeded, not enough countervoltage will be induced, and the primary current will rise beyond safe limits. It will happen gradually, because B,,, is not a sharply defined value. First, losses in the core increase and its temperature increases. In a pulsed transformer this is less critical than in one for continuous duty. Thus, the practical value of B,,, will be different for the two cases. The preceding process is not dependent on an iron core. At rf-frequencies, voltages per turn can get very high without a core, into the tens of kilovolts (Tesla transformer). At rf frequencies, the normal transformer iron (grainoriented silicone iron, in laminations 2-6-mil thick) will show too much eddycurrent loss and cannot be used. Sintered powder metal ferrite cores have been developed for the high-frequency range, but their B,,, value is lower than that of transformer iron. In other words, the maximum voltage per turn of a transformer coil is entirely and solely determined by the total d 4 / d t ; i.e., by the maximum flux density B,,, and the cross section of the core, which together determine 4, and the frequency, which determines dt. While the total flux 6 determines the energy that is stored and released for each cycle per ampere-turn of current, it has nothing to do with the energy that is transmitted to the secondary coil. We must now ask how many amperes of magnetization current are needed to generate B,,, (regardless of the total flux +)? This can only be answered when we know certain properties of the core iron, as we shall see later. Let us skip ahead and say that B = p p 0 H is the ampere-turns per meter of coil lengh, and p and po are nearly constant factors. Conversely, we have H = B / p p 0 . Suppose B
=
1.5 Joule/A-turn m2
and suppose ppo = 7.5 x
J/(A-turn)2 m
then we need H
=
1.5 J/A-turn m2/7.5 x
= 200 A-turn/m
J/(A-turn)2 m
258
BERTHOLD W. SCHUMACHER
Let the primary coil be 0.5 m long and have 200 turns (400 turns/m) then the magnetization current that generates the maximum permissible flux will be 0.5 A. It flows through the 200 turns of the primary whether any secondary coil is there or not, whether it is open or connected to a load. Moreover The magnetization current (in ampere-turns) leading to saturation is independent of frequency and cross section of the core. We saw before, if the core has a cross section of 25 cm', then it supports 1.4 V/turn at 60 Hz. Therefore, 280 V peak can be applied to a 200-turn primary coil. If we ask for the energy stored per cycle, we see it is very small. At 60 Hz, with approximately 1.4 V/turn and 100 A-turns it cannot be more than 140 W, and this is for a transformer that can pass many kilowatts of power. The stored magnetic energy has nothing to do with the passage of energy from the primary to the secondary side of the transformer. The energy transmitted by a power transformer does not go through the iron core. However, in a choke coil, the stored energy will be all-important. Some energy is lost in the iron core by secondary processes such as eddy currents (see Subsection K). This also shifts the excitation current (magnetization current + iron loss current) away from the 90" phase shift. Nearly every picture in the traditional physics text books and similar pictures right on the cover of some recent electrical engineering texts, show a non-functional device and pass it off as a transformer. The primary winding is placed on one vertical leg of a square iron ring, the secondary on the other leg. The magnetic flux can only pass from one coil to the other via the horizontal sections of the iron, the maximum flux being the previously derived saturation flux. It is true that with a similar device, an iron ring with two coils side by side, Faraday discovered the laws of induction in 1831, but not the power transformer. The first power transformer on record is due to Jablochkoff in 1877 (see Drummer, 1978). In the real transformer, primary and secondary windings must lie closely coupled over the same core leg or, speaking in terms of physics, must enclose the same flux. If the secondary is open, it has no influence. A maximum voltage can be applied to the primary which generates the maximum permissible magnetization current, which is 90" out of phase with the voltage. The resulting openloop secondary voltage is also 90" out of phase with the primary voltage.
DIMENSIONAL TERMS FOR ENERGY TRANSPORT
259
If the secondary is connected to a resistive load, a primary current will flow long before the maximum permissible primary voltage (whose value does not change) is reached. As current flows in the primary, a voltage and current in the opposite sense is induced in the secondary. The magnitude of these currents depends upon the size of the resistance forming the load. When the maximum primary voltage (volts per turn) is reached, the maximum secondary voltage and current are also reached. The current can have any value that the copper wires of the coils can stand. (The internal resistance of these copper wires is additive to the resistance of the load.) The voltage and current in primary and secondary are in phase; both coils see the same dlldt (except for the small component of the magnetization current). The iron sees only the difference in primary and secondary currents. This difference must not exceed the magnetization current needed for saturation. If the primary voltage is driven higher, the higher volts per turn call for a greater flux than the core can accommodate, and the flux will spread out more and more without passing through the secondary. The transformer loses its efficiency (gradually). From the preceding discussion the core size of the transformer appears quite unimportant, and, in a way, it is. However, with a larger cross section of the core more volts per turn can be generated, less copper wire is needed, the internal resistance is lower, etc. A much smaller, lighter core and less copper wire, are sufficient if we increase the frequency. For a small, ring-shaped core, the window area limits the amount of wire that can be wound around the core. With the given conductivity of copper wire the ampere-turns are, therefore, limited by the window area. The core cross section has, as we saw, limited the volts per turn. Transformer engineers, therefore, assign a certain VA-rating to each size of core. Yet, this is nothing fundamental. It also depends on the frequency and whether or not we can fill the window with wire or need free space for insulation, as in large highvoltage transformers or high-voltage isolation transformers. Higher frequencies are used in aircraft and on board ship, where the size and weight of the transformers are most important. Another example are dcto-ac-to-dc power supplies of all sizes, known as switching power supplies, where frequencies to 3 kHz and more are used. A transformer cannot be built with superconducting coils. Residual eddy currents in the superconductors will dissipate heat and destroy the superconductivity. In this discussion we cannot go into the engineering aspects of transformer design, such as minimizing the stray flux, the weight, or the temperature rise. Let us go back to the discussion of the magnetic fields.
260
BERTHOLD W. SCHUMACHER
E. The Field Strength The second outstanding and well-known characteristic of a magnetic field -apart from carrying energy-is that it exerts forces on other magnetic fields (or magnetic bodies). If we measure these forces we find another simple relationship. For instance, theforces generating the torque on a compass needle placed inside our aforementioned solenoid coil, are (and this is an experimental, observational result) simply proportional to the current per unit length of the coil, the ampere-turns per meter. Therefore the commonly used unit for measuring the field strength is ampere-turns-per-meter generally denoted by the symbol H . Note that the diameter of the coil does not matter in this connection! Of course, the field energy and the inductance of the coil depend on it, as we saw before. A field strength of H = 79.58 A-turn/m is also called 1 Oersted, for reasons which will become clear in a moment. Conversely, l A-turn/m = 1/79.58 Oersted. Seeing that the field strength H is proportional to the current and remembering that the flux density B was also proportional to the current, namely 1 T = 1 Joule/m2A-turn, we can ask immediately for the mutual conversion factor, and we find
B/H
1.256637 x [(Joule/A-turn m2)/(A-turn/m)] = 1 x Tesla/Oersted
= p,, =
Let us now look close at the magnetic needle that we have utilized here. The magnetic characteristics of the magnet needle are defined, just as in the above case of the solenoid, by its total magnetic flux 4, and by its shape, which determines the flux distribution. The dimension of 4 is, as we remember, energy per unit current (Joules per ampere-turn), and those amperes, we may think, are flowing spontaneously as molecular currents in the iron of the magnet needle (more later). The total flux of the magnet needle can be measured by the already introduced method of the probe coil, pulled over and away from it. No new concepts are needed to deal quantitatively with this situation. Earlier we presented another argument for the existence of field stresses and for their dependence on the ampere-turns per meter, which shall be repeated here. The total flux is identical with the energy held by the field per unit current generating the field. The flux density B is the energy per unit cross section and current. Both these concepts contain no reference to the volume in which the field energy is contained. Yet it is plausible, that the
DIMENSIONAL TERMS FOR ENERGY TRANSPORT
26 1
smaller the volume the greater the internal pressure or the internal field stresses, and it is no longer just the generating current, but the current per unit length of the field lines, which determines the field stress or field strength H . Here H is indeed measured in ampere-turns per meter (regardless of the cross section of the field). This argument shows in a very direct manner why there are two field quantities needed, namely B and H . They refer to completely different aspects of the field. F. The Magnetic Moment
The torque, which we measured by turning the magnet needle when placed inside the magnetic field, is obviously the result of two forces of opposite direction acting near each end of the needle and proportional to the solenoid field H . Considering these forces, we see here an analogy to the electric case in which the electric field E exerts a force on a charge q or generates a torque (moment) on a dipole with charges of f4 at its ends. The flux 4 of the needle plays the role of the electric dipole charges q. Single magnetic charges or monopoles do not exist, as is well known. The dipole moment M depends also on the shape of the dipole, (e.g., in the ideal case on its length, namely the distance 1 by which the charges q or 4 are separated) and the size of these charges. Thus, the dipole moment is M = $1 (Joule/A-turn) m, which can be simplified to (V-sec/turn) m. In the magnetic case, however, cannot be exactly localized. Nevertheless, total flux (in Joule per ampere-turn = Weber) and in length (in meters) do indeed give a clear conceptual picture of a dipole system. (Flux = pole strength.)
+
But the factors can be and have been juggled (which is just the right word for it) in many ways. Thus for the dimension of M , we may also say that it is Joules per (Ampere-turn per meter) which means energy per unit of field strength (the amperes being those that generate the external field, never mind how the dipole flux is produced). Then M appears to be the energy that the oriented dipole holds in a field of unit strength and turned fully against the field force. This energy characterizes the dipole completely. (However, the dipole does not disappear with the external field, only the interaction energy disappears! Obviously such a definition of dipole strength is tenuous at best.) By using the older unit for the field strength, 1 Oersted = 79.58 A-turn/m, magnetic moments are often listed in terms of Joules per Oersted or erg per emu. In particular, the magnetic moment of a molecule, atom, or elementary particle may be found tabulated in these units. Since B in Tesla and H in Oersted are related by l/p0 = 79.58 x lo4, the numerical values for M remain the same, except for factors of 10, if expressed in Joules per Tesla. This must be the reason they are listed in those terms, e.g., in Document U.I.P.
262
BERTHOLD W. SCHUMACHER
20(1978) of the S.U.N. Commission of IUPAP. However, it leads to illogical dimensional factors of, for instance, A mz, which is also used for M in some tables. Illogical, because suddenly the field energy orflux field 4 (as 4/m2 = Tesla) is made responsible for the forces, whereas they are dimensionally related to the stress field H . (To say it once more, the dipole does not disappear when the B-field goes to zero.) Conversion factors are explained and listed in Subsection C, together with numerical values for the natural magnetic dipole moment of some elementary particles.
The following must be stressed. The magnetic dipole moment M is a definite, inherent, and unique property and quantity of any magnet, whereas poles are not. The poles cannot be localized and said to have a certain distance, etc. However, a U-shaped electromagnet, which is essentially a solenoid with iron core and a gap, has two pole-faces (at each side of the gap) from where the field lines emerge. The word pole is here used in a completely diferent sense, as should be obvious. The torque (moment) D, acting on the magnetic dipole moment M in a field of strength H , is given by the vector cross product: D
=
M x H
[Joule or Newton-meter]
(16)
The dipole will usually oscillate around the equilibrium position where M is parallel to H . The frequency v of these oscillations is determined by the mechanical moment of inertia of the dipole 8, which can be calculated from its shape. We find
Jz.v=J;nHle
(17)
from which we may derive M or H , if one of them is already known. Today it is of no practical importance anymore, but this was the ingenious approach of Gauss, which permitted him to measure magnetic fields and moments (taking two dipoles) in terms of mechanical units. Unfortunately, sticking to these units exclusively (the cgs system) was a very grave obstacle to a more direct understanding. For our measurements of forces we don’t need to use a compass needle (although we have seen how easy it is to measure its total flux 4 and incorporate it in our system of concepts). A small, slim, and long solenoid, generating a flux 4, emerging mainly at its ends, whose distance is 1, has the magnetic moment M = 4f.It can serve as a field probe of known characteristics. The moment of a flat rectangular coil is also easily calculated. We want to do this here only to show, or rather clarify, the confusion that may arise when using alternately B or H as the field exerting the forces on a current.
263
DIMENSIONAL TERMS FOR ENERGY TRANSPORT
A rectangular current loop (single turn) may have a width 2r, a length I, and carry a current I , . It is oriented in the field of strength H with its plane perpendicular to the field lines. The force per unit length on the sections 1 is p o H I , or BI,. (This has not yet been derived in the earlier discussions, see Subsection L.) With the lever arm of r a torque D is generated for which we can write
D
= p0r2110H
or D
=
r211,B
The factors 2rl represent the area F of the current loop. We can combine all factors beside the field strength into a term for the magnetic moment of the coil, as
M,
= p,FI, =
(V sec/Am)A m2
or
M;
=V
V sec m
= Joule/(A-turn/m) =
A-turn m2 [A m2] A sec/(V sec/turn m2) = Joule/Tesla = FI,
(Joule/A-turn)m
= Wb-m
In one case we have associated the permeability constant p o with the force law, in the other case we have associated it with the field. This leads to two different conceptual definitions for the magnetic moment M , and Mm,respectively. Only with the dimension of (Joule per ampere-turn) meter [ = Weber-meter] have we retained the actual dipole picture, namely two flux regions separated by a distance. The torque is then to be computed in two different ways, namely D
=
M',H
[Joule
=N
m]
or
D
= M,B
[Joule
=N
m]
The numerical result is the same, of course, since
M , = poM; D
= M,H
H
and
= poMA x
= B/po
B/po = M m B
Conversion factors are as follows:
]=
(Joule/A) m
l Vb secmm Joule/(A-turn/m) 1
A-turn m2 [A m2] 1.2556 x lo-.{ Joule/Tesla [J T - '1 Joule/Tesla [A m2]
1 Joule/Oersted
=
1x
1 erg/Oersted
=
1 x lo-" Joule/Tesla
1 Joule/Tesla
= 1 x lo4 Joule/Oersted = 1 x
erg/Oersted = 0.7958 x 10. (J/A) m [Wb m] [V sec m]
264
BERTHOLD W. SCHUMACHER
A free dipole, associated with a particle that also rotates (which has a spin, as we say) will not only oscillate (as does the compass needle) but will precess, like a gyroscope, when put into a magnetic field. The ultimate magnetic dipoles are found in the realm of elementary particles. The electron has a spin, a charge, and a magnetic moment. The neutron has no net charge but a spin and a magnetic moment. While we cannot elaborate here, let us nevertheless recall a few interesting facts. The dipole moment of the proton (in the water molecule, for instance) gives rise to oscillations when subjected to an external magnetic field. These oscillations can be induced and the frequency measured and the field strength can be determined. Protons are found in all organic substances and are subject to different local fields. From the frequency, damping, and relaxation time it is possible to draw conclusions about those local fields. This leads to the science of nuclear magnetic resonance and its application, especially to chemical analysis of organic substances in uioo, which is of growing importance in medical diagnostics. Even these little, elementary gyroscopes, precessing in a magnetic field, hold their absolute orientation in (the cosmic) space and can be used as reference device for detecting and measuring rotational movements, e.g., in inertial guidance systems. We cannot elaborate here. The magnetic moment of an orbital electron in an atom can be calculated like that of a current loop, making certain plausible assumptions about size and speed. This yields a value known as Bohr magneton p B = eh/47crncO.We find : pB
= 9.2733 x = 9.2733 x =
1.1653 x
= 0.5788 x
Joule/T [A m2] erg/Oersted J/(A/m) [or V s m] MeV/Gauss*
The actual magnetic moment is found to be pelectr = 9.28483 x
Joule/T [A m2]
The spin of the electron causes a magnetic dipole moment also, of the same magnitude, but its half-unit of spin (quantized, mechanical angular momentum) produces a full-unit magneton. This is accounted for by Landt's g-factor, which is g = 2.0023 for the electron.
* Nuclear physicists like to use eV or MeV as units of energy. It is: 1 Joule = 6.24 x 10l2 MeV. For best values of pg, etc., see Rev. Mod. Phys. 52, No. 2, Part 2 (1980).
DIMENSIONAL TERMS FOR ENERGY TRANSPORT
265
The magnetic moment of a particle with the mass of the proton, called a nuclear magneton, ought to be 1/1836.3 times that of the electron, namely
~ ~ 5 . 0 5 0x 8lo-’’
Joule/T = 6.3489 x = 5.0508 x erg/Oersted (or Gauss) = 3.1524 x lo-’’ MeV/Gauss
[(Joule/A)m [V sec m]
G. Forces on a Pole Region
In spite of what we have just said about the magnetic dipole being the only real physical magnet, we can talk about theforcefon one pole region of flux 4 in a field of strength H . It is simply given by
f= H
x
4 (A/m
x Joule/A = Joule/m = Newton)
Here, 4 has formerly been called the magnetic pole strength or magnetic quantity, which was never a very clear definition, and 4 is treated as a scalar and localized, whereas we know it is a flux. Let us consider another situation. Suppose 6 is the flux emerging from the flat pole piece of an electromagnet. If the pole piece area is A, then the flux density is B = 4 / A . The pole with its flux 4 faces the opposite pole, and together they produce the field H = B/po in the pole gap. Each pole with its flux density B finds itself in this field H . Thus the force per unit area (tension) . f A , is: fA = H x B = l/po x BZ = p o H 2 Newton/m2 (19) The force with which the two pole regions or pole faces are attracted to one another increases with the square of the flux or the field strength. Since, for an electromagnet, both are proportional to the current, the total force as well as the tension increases with the square of the current (as did the field energy). The attraction, the mechanical force between the two pole pieces of an electromagnet, illustrates the nature of the H field, namely its association with what one may call internal field tensions. Knowing the force per unit area, we can now once more calculate the field’s energy density, this time on the basis of mechanical measurements. Assume we increase the field volume by moving the pole pieces apart by the small distance dx. The work to be done (per unit area of the cross section) is obviously d E = fA dx. Rigorous analysis yields a factor o f f and quite generally we get for the energy density Emagof the magnetic field Emag= ) H B
A-turn/m x Joule/A-turn mz = Joule/m3
(20)
Note that H in ampere-turn per meter represents tension forces and B in (Joule per ampere-turn) per square meter represents an energy flux density.
266
BERTHOLD W. SCHUMACHER
Both quantities are proportional to the current generating the field. Thus, Emagis proportional to the square of this current, as we had found already before in the definition of the inductance L (Eq. 6).
H . Inductance as Determined by Geometry
We are now able to relate the value of the inductance of a toroidal coil with uniform flux to its size. Let L, be the inductance, I the current, n the total number of turns, N the turns per unit length, and 1 the length of the coil; therefore n = Nl. With A as the cross section of the coil, its volume is IA, and L is, as we have seen, proportional to this volume. Flux 4 and current I are related by 4 = p o H A , with H = N I = n/lI. Since
Vnd = L ( d l / d t )
[Vind= actual voltage not V/turn]
as well as [$J in V-sec/turn] Vind= n(@/dt) = n[d(BA)/dtl = p,nF(dH/dt) = p,,n2(A/l)(dl/dt) [H in A-turn/m]
it follows that L, = pon2A/1= po N2(1A)
(21)
The higher the total number of turns (n), the higher L, and the more voltage must we apply to get or change a magnetic field quickly. Yet, if n is lowered we must increase I instead to get the same field. The driving power in watts is the same. It is determined by the desired change in total field energy d 4 / d t , which is proportional to the volume of the field. This applies also to the field in the air gap of a magnetic loop with an iron core. Actually, it is most logical and useful to write L
= poN2(1A)
(22)
where N is the turns per unit length, because this shows how L is related to the independent quantities, namely proportional to the length 1 and proportional to the area A of the coil. In the other form of the Eq. (21) with the number n of the total turns, L seems to be the greater the shorter the coil, which is nonsensical. Such illogical appearing equations come about because the concept of 1 turn has no dimension just as 1 revolution (per unit time) gets (at best) the dimension of one-over-time. It is the same kind of problem that we face when
DIMENSIONAL TERMS FOR ENERGY TRANSPORT
267
talking about the spatial character of the radiance for energy transport by radiation and the angle term in its dimension. That n or N appear squared in the above equations is the consequence of L being proportional to the square of the ampere-turns. In this case, where current is quietly assumed to be unity, we simply get turns squared. It shows once more that the inductance L is primarily a quantity of the generated field and not of the geometry of the coil. The numbers n or N contribute nothing to the field geometry. The quantity (1A) is obviously the volume inside the coil. It holds most of the magnetic field. We see once more that L is proportional to the volume of the field, it is a quantity of the field. Parameter L changes when the surroundings of a coil are changed, e.g., by an iron core. The inductance per unit length of a coaxial cable with a and b as radii for the inner and outer conductors is
L,
= (Po/2.n> W
la)
(23)
If the intervening space is filled with iron of a permeability p then ppo replaces po. Similarly, ferrite beads placed over the lead wires going to a semiconductor circuit increase their inductance and help to suppress transient voltage spikes from the power supplies, etc., which could damage the circuit.
I. Magnetic Tension and Magneto-Motive Force
We have discussed the magnetic field strength H , which determines the forces that another magnet experiences in this field. Here, H is measured most conveniently in amperes per meter. Expressing it more accurately, this means the amperes in all the turns, which surround a 1-m section of a coil, are added into one value; 1000 turns/m with 0.1 A flowing are fully equivalent to 10 turns with 10 A flowing or 1 turn with 1000 A. As an interesting corollary to this fact let us mention that in transformer design it is the area of the core window that determines how many ampereturns we can send around the core, since a copper wire carries just so many amperes per unit cross section. It does not matter how we split up the copper cross section when going through the window. Only by using silver wire with its higher conductance could we get more ampere-turns in. This trick was purportedly used on the magnet of a certain isotope separator. (The silver was on loan from the US treasury.) Let us recall that the field strength of an electric field is measured in volts per meter. If we integrate over the range of the field, we get the tension or voltage (or electro-motive force = emf) between the conductors that delimit this electric field.
268
BERTHOLD W. SCHUMACHER
If we integrate over the magnetic field strength between two points in the field along any path, we get a similar quantity J H ds which has been called magnetic tension or magneto-motive force (mrnf). Its value and dimension are easily recognized if we envisage a straight, slim solenoid coil of length 1, with a current flow of I A-turn/m of its length. The field drops off to negligible values at each end of the coil, at distances small compared with 1. Therefore, we simply get between the ends of the coil a magnetic tension of
s:
H ds
= I L = I,,,
A-turns
Integrating the mmf over a closed circular path around a current carrying wire with current I , yields n
J
H ds
=I,
which looks perhaps more familiar; but a closed loop is no necessity. Equation (24) and J H ds retain their meaning in any case. Of course, I,,, is not equal to I , . The mmf is measured, as many textbooks say, in amperes. Logically and properly speaking, we should say it is measured in ampere-turns. The longer the coil the more ampere-turns and the higher the magnetic tension-although the current in amperes stays the same. The term mmf oust as emf) is mainly used for the driving forces in a closed loop path. (The voltage of a battery, for instance, as opposed to the voltage drop generated by a current in a resistor.) The driving force to which we have just alluded is, in the magnetic case, represented by the often-mentioned ampere-turns. In the case of a battery, the driving force that generates the emf, by separating electric charges which normally attract one another, is a chemical energy potential. This means that if some ions of the metal of the electrodes go into solution in the electrolyte then the total internal energy of the system is lower than before, even if the electrodes are left with an electric potential (voltage) difference between them. In a connecting wire it is the free electrons that accomplish an exchange that is not controlled by the chemical reactions. Without the wire we have a hindered equilibrium; the electric field is holding (not dissipating!) the energy released by the chemical reaction. The energy law rules everything, even charge separation and the build-up of electric fields. The preceding definition of magnetic tension already contains an instruction on how to measure it. We need now, instead of the thin planar probe coil
DIMENSIONAL TERMS FOR ENERGY TRANSPORT
269
used for measuring the flux density B, a slim solenoid of length lR and cross section F,. It should be wound around a flexible rubber strip or the like. The connecting wires must enter in the middle of a double layer of windings, so that no additional loop is formed if we bend the axis (rubber strip) of this solenoid. This kind of magnetic tension meter is known as Rogowski coil (Rogowski, 1912. Chattock, 1887). It works as follows. Part of a magnetic field may fall inside the Rogowski coil. Let this field disappear and a voltage pulse, measured in volt-seconds, is induced in each segment of the coil that is proportional to the local, previously present and enclosed flux 4. The volt-seconds of all segments add up to a value that appears at the feed wires of the coil. This value, divided by p,, and an instrument constant, namely NF,/l, (where N is the number of turns) yields immediately the H d s over the path of the coil from end to end. We soon find that only the positions ofthe end points matter, not the actual path, as long as the current loops surrounding the tension meter do not change. Placing the tension meter inside a field generating solenoid (i.e., surrounding it with another current loop) produces obviously a different value for the tension integral than placing it outside (although between the same two end points). The tension, found to be independent of the path, is typical of any scalar potential field, as we obviously have here. It follows that (in the static case) we may consider the magnetic field H , which is a vector field, as the gradient vector of a scalar potential field R. (Not to be confused with the vector potential of the B-field.) Thus H = -grad0
(26)
We cannot go into more detail here. The magnetic tension also determines an energy potential at every point of the magnetic field, just as the voltage does in an electric field. To move a pole region with flux (strength) 4 to a field point with potential I we must do work equivalent to I x 4 (A x Joule/A = Joule).
A charged particle with a certain kinetic energy moving into a magnetic field region will be deflected, as is well known. Yet, it will not gain energy, will not move up against the magnetic potential, but will be deflected so that all new velocity components follow lines of equal magnetic potential, i.e., at right angles to the H field. Magnetic electron-optical lenses are trans-gradient devices, whereas lenses for light are con-gradient devices; light rays are deflected roughly in the direction of the gradient of the refractive index. In the first case the lenses are fields whose surfaces cannot be locally shaped; in the second case they are material bodies, whose surfaces can be given any desired shape. The speed of the particle is important for its ray optics, just as in the case of light. In the magnetic lens field the total kinetic energy of the particle stays
270
BERTHOLD W. SCHUMACHER
constant. When a transverse velocity component is induced the forward velocity changes by itself, appropriately. In the electrostatic lens field it is mainly the forward velocity component that is changed, together with the kinetic energy. Since the forward velocity (mostly) determines the time of arrival, we see how in both cases the well-known time-of-flight arguments for the ray path (least time of arrival) may be developed for particle optics as for light optics. J . Two Field Quantities
The question has often been asked, why are there two field quantities needed, namely B and H , to describe one and the same field? From the preceding discussions it should be clear that they describe different aspects of the field. The magnetic field has internal energy and internal tensions. In fact, calling H field stress would have been a better choice than field strength; but this cannot be changed anymore. The best analogy is the stressed elastic body, for which we find a certain elastic deformation energy per unit volume and certain stresses (stress tensor) for every point. The two are linked by the elastic coefficients of the particular material. If those are not the same in all spatial directions (i.e., if the elastic material is not isotropic) then these elastic coefficients form a whole matrix of numbers, as is well known. In the case of the magnetic field in vacuum (or air), B and H are linearly related and have (insofar as they are vectors) the same direction. The electromagnetic space is isotropic. Thus we may write B
= p0H
(27)
For the proportionality factor p o (a constant of nature) we find po = B / H = 1.256637 x
(28)
with any one of the following dimensions: Tesla Wb/m2 Joule/A m2 - V sec/m2 - V/turn m2 PEP--A/m A/m A h A/m (A-turn/m)/sec -
flux density - induced voltage (in a loop) field strength - rate of change of field strength
-
Joule/m3 - energy density - Joule/A2 Henry inductance (A-turn/m)2 - (field stress)2 m m m
-
V Asec/A2 V/m induced electric field strength m A/sec rate of change in magnetization current
DIMENSIONAL TERMS FOR ENERGY TRANSPORT
27 1
As if that were not enough choices for the dimension of po we may also write po = B/H = 1.257 x lo-’
Gauss/(A-turn/m)
(294
p o = B/H
=
1
Gauss/Oersted
(29b)
p o = B/H
=
1x
Tesla/Oersted
(29c)
The name for po is the “permeability” of the vacuum. From the last three equations we find that
H
=W P O P
79.58 B
A-turn/m
if B is in Gauss
(294
= 795800 B
A-turn/m
if B is in Tesla
(29e)
=
This had originally led to the definition of the unit Oersted (equal to 79.58 A/m) as a measure for H . We see, then, that in vacuum the Gauss and the Oersted are numerically equal (which has caused more confusion yet). This equality, however, no longer holds if matter is introduced into the magnetic field, e.g., if we put an iron core in the solenoid. The name for p o (permeability) is a kind of crutch. We see from the dimensions (as given earlier with various degrees of cancelation of factors) that it represents an inductance per meter, or, if we refer to energy, the energy density (Joule/m3) per unit of field stress squared, (ampere-turn/meter)’. A more descriptive name for po would be “induction constant.” The different conceptual nature of B and H is also reflected in the fact that B is an energy density or flux density, a part of a total flux. It would be senseless to ask for the density of the stress quantity H . We see here the difference between an extensive and intensive entity! As an aside, we may mention here that the dielectric (or permittivity) constant c0 for the vacuum represents a capacitance per meter and also an energy density per unit of field stress squared (V/m)’ c0 = 8.854 x
Joule m-3/(V/m)2
= Joule/V’m = A
sec/V m
(30)
An electromagnetic wave in vacuum sees, therefore, the impedance
.=&J
Joule/A’m V~ Joule/V’m - A -ohm
With the measured values for p o and c0 inserted we get Z,,,
=
376.7 ohm
(32)
272
BERTHOLD W. SCHUMACHER
Television antennas and the precisely spaced double-wire cables which connect them to the receiver (known as Lecher lines) are matched to this value. Coaxial cables with 75-ohm impedance must be connected to the antenna via an impedance matching transformer. While Z,,, has the dimension of a resistance, the wave is not a current, and no energy is dissipated along the path of the wave. This leads us to another order of complexities which we will not pursue further. It was noted by Wilhelm Weber (1804-1891) that p, and their dimensions, determine a velocity, namely
I/&
E,,
in view of
(33)
= co
where c, came out equal to the velocity of light when the values for E,, and p, as they were known at the time-long before Maxwell had developed his theory-were inserted. This was the first hint at a connection between light and electromagnetic phenomena. It was, at the time, a remarkable discovery.
K, Matter in the Magnetic Field When we place an iron core inside the solenoid with which we generate the magnetic field we find that we must supply much more energy (to build up the field) until the final current I , has been attained. In other words, the coil's inductance L has increased greatly and so has the total flux 4, which represents the field energy. The current in ampere-turns per meter has not changed (it only took longer to reach the value I , ) ; therefore, H is the same as before. The increase in 4 or B is attributable to the material (the iron core) and can be represented by a number p, called the material's permeability. We can write
B = pH
Gauss (with H in Oersted)
(34)
where p gets the dimension of Gauss per Oersted, or we can write
B
= pp,H
Wb/m2 (with H in A/m)
(35)
with p a dimensionless number (of the same numerical value as before). The former linear relationship between B and H is now lost, since p depends to a great extent on H .
-
For small values of H we find for iron values of p 200, going to 4000 for larger H . For grain-oriented silicon iron p can reach 60,000. An alloy of Fe 59%Ni + 5%Mo (Supermalloy) has a maximum permeability of 800,000. The handbooks carry extensive tables for other materials.
+
DIMENSIONAL TERMS FOR ENERGY TRANSPORT
273
The maximum value of B that we can generate with water-cooled copper coils (iron core and pole pieces) is approximately 2.5 T. With superconducting coils flux densities of 8 T have been attained (in magnets for fusion research with coils of 5.5 x 3.6 m). In the large-volume (85-m3) field of the UA1 particle detector at CERN, the flux density is 0.7 T. Where is the additional energy stored, which we had to provide? The molecular magnetic dipoles in the iron, or more accurately, the spontaneous magnetization of small areas in the iron, called domains (discovered by Weiss, 1907) are normally randomly oriented, and domains are of variable size. In the externally superimposed field the domain walls shift and their magnetic orientation may change, which requires energy. The process is taking place in steps or jumps. This can be observed with a second sensing coil, connected to an audioamplifier and loudspeaker. The jumps cause a noise known as Barkhausen noise. In the electron microscope we can see the magnetic domains of a thin-film specimen, since their local magnetic fields deflect the beam and this causes changes in the contrast on the screen. The events can get more complicated in single crystals and special materials, which we cannot discuss here. In some materials the domains can be highly ordered, permitting their use as memory devices. The rearrangement of the magnetic domains is associated with some losses of energy because of friction (the noise!). Therefore, magnetization or change of magnetic orientation as it takes place in transformer iron, for instance, is associated with energy losses. The iron gets heated by it, but also, and even more so, by additional eddy current losses. In the new amorphous metals (thin foils produced from a liquid jet by rapid quenching, e.g., Fe80B20) the domain walls move easily, and a relatively low electrical conductivity keeps eddy currents low. Thus, we have a magnetically soft, low-loss material. Transformer cores can be made from it that are superior to all previous types. For instance, for a 10-kW transformer core losses are reduced from 60 to 12 W. The magnetic flux does not fully go to zero when the H field is removed. What is left is called residual flux density, remanent magnetism or remanence B, for short. It is what we see in a permanent magnet (sometimes called a hard magnetic material). A counter H field must be applied to remove the residual magnetization, and the value of this field is known as the coercive force H,. The function depicting the B values versus the H values shows what is known as a hysteresis loop. It is different for different iron alloys. Details can be complicated and are beyond the scope of the present text. By looking for the energy in the field B,, which is retained, and at the associated mmf (field H,) for generating or removing it, we realize that this stored energy is proportional to the product E, = B , H , and also represents
214
BERTHOLD W. SCHUMACHER
the energy loss going through a cycle of the hysteresis loop. There are distinct values B, and H , associated with a maximum value of the energy product per unit volume E , , that ~ ~ can ~ be achieved for a particular type of magnet. It is also desirable that H , be high, so that the magnet is not easily demagnetized. Typical values for E , , are ~ ~ shown ~ in the following tabulation. Material Steel, 1 % C Cobalt steel (36 % Co) Alnico V Ba-Ferrite (oriented) Magnequench [GM, 19831
B* H, (Gauss) (Oersted) 9000 9750 12,500 3800 -
51 240 6000 2000 ~
Er,,,
(Ga x Oe x .20 .93 5.0 3.25 30.0
Bd
Hd
(Gauss)
(Oersted)
5900 6300 10,200 2000
34 148 490 1625
~
-
1 Gauss x Oersted = 79.58 x Joule/m3 1 MegaGauss x Oersted = 7.9 kJoule/m3
In crystals of rare-earth/cobalt [RCo,] magnets (as today used in light weight earphones) the reorientation of the magnetic axis may require as much as 200,000 Oersted. The higher the energy product for a permanent magnet, the greater the power per unit weight of an electric motor built with it. The recently developed magnequench material, with its extremely high energy product, gets its properties partly from thermal treatment. (One never knows in advance where progress comes from.) An increase in temperature lowers the flux; a decrease in temperature raises it ( 2% per "C for barium ferrite). Once more we have an effect due to an energy equilibrium, magnetic energy versus thermal energy. There is also a maximum magnetic moment ps that can be attained at saturation. Approximate values are shown in the following tabulation N
Iron 1700 Cobalt 1400 Nickel 480
[cgs/cm3]
The magnetic moment per unit volume pm/Vis called the magnetization M. Since p, can be measured in A m2 and V in m3, we get M in A/m. We find, the magnetization of the earth is -80 A/m (with a moment of pm = 11 x 1OI6 V sec m = 1.4 x 10" Joule/T). A steel magnet may reach 7.5 x 10' A/m. (Sometimes M is found expressed in units of Tesla.) The iron core of a solenoid, in a macroscopic sense, appears to act as a kind of conductor for the B field by letting it come out at its ends. It appears as if there were magnetic poles present. In a magnetic loop of iron with a small air gap it then seems as if the whole ampere-turns were wound over the
DIMENSIONAL TERMS FOR ENERGY TRANSPORT
275
air gap, producing a very strong H field there. (The principle of the electromagnet.) High-permeability materials act as a kind of short circuit for a magnetic field and are used for magnetic shielding. Ferro-magnetism is a collective phenomenon and disappears when thermal mouement (the kinetic energy!) of the atoms exceeds a certain limit. The temperature T, where this happens is known as the Curie temperature. Approximate values are 1100 K for iron, 1390 K for cobalt, 630 K for nickel, 510 K for barium-ferrite, and 290 K for gadolinium. (Another process controlled by energy !) Other materials, called paramagnetic, also pull the B field in but not as pronounced as the ferromagnetic materials. Another class of materials, called diamagnetic, tend to expel1 the field (superconductors, for instance). Diamagnetism is universal-an expression of Lenz’s law on the atomic scale. Paramagnetic effects are simply overcompensating for it. Mechanical stresses can affect the value of p ; conversely, a magnetic field can change the crystalline dimensions (magnetostriction). Referring the magnetic data to the individual atoms and molecules leads to interesting information about the atomistic structure of the materials, which is even today not yet fully elucidated by theory. We cannot and need not go into details in the present context. Let us finally add the following. While the B field is free of planar sources and remains so in the presence of materials (normal component at any interface is constant), the H field, in the presence of matter, has sources at the interfaces; the divergence of H is equal to -div M , where M is called the magnetization vector. We cannot go into further details. The tangential component of H at a pole face is continuous, as long as we don’t cross a current sheet on its surface. The H field inside a solenoid, with or without matter, is always the same and is determined solely by the ampere-turns. Inside an iron core, however, the value of H may be diminished by back-flowing H lines from the pole-faces of the magnet, depending on it shape. L. Forces on Currents in the Magnetic Field-Discussion
We saw already earlier that the field strength H determines the forces on the magnetic dipole represented by a compass needle or by a current loop. Let us look at the situation more closely in a qualitative discussion. The quantitative side, the equations for special cases, are found in any physics text book and shall not be reiterated in the present context. When we consider the fact that a solenoid coil, generating and holding a magnetic field, encloses a region of higher energy density in space (energy which has been pushed in there when the current was switched on), then it
276
BERTHOLD W. SCHUMACHER
will not appear surprizing that the coil is subject to a kind of internal pressure. In fact, a coil which is subjected to a sudden, very high current (which it could not sustain in a dc mode because of excessive ohmic heating) may simply explode. A memorable example was a high-voltage transformer coil (built by the lowest bidder) subjected to the surge current from an arc-over in an electron gun. By using another expression to describe what goes on, we may say the bundle of magnetic field lines wants to expand, just as it indeed does where it leaves the solenoid. This is only one special case of a quite general rule, namely that parallel field lines repulse one another, and they push the wire loop (coil) that generated them outward. The forces that act between straight wires that carry electric currents were studied and quantified by Ampere. The field lines generated by a straight current conductor are circles around the conductor. By looking along the conductor from the positive pole of the battery towards the negative pole, Ampere stipulated that the direction of the field lines shall be that of a clockwise arrow, as if generated by the movement of a right-handed screw (but the field lines form circles, not a helix). This convention is still adhered to today. If a compass needle is brought close to the wire and we look again from the positive pole of the battery along the wire, we will see the north pole of the needle point in the clockwise direction. The northern magnetic pole of the earth is the south pole of a magnet, that end of a magnet needle which points to geographic north is called a magnetic north pole; very simple. Suppose there are two parallel currents. The two circles of field lines cancel one another in the space between the currents and add up in the space outside the currents. As a consequence, we may say, they pull the currents together. Quite generally speaking, parallel currents attract one another, anti-parallel currents repulse one another. Note, opposite elements of a wire loop (solenoid) always carry anti-parallel currents and are, therefore, pushed outwards! The superconducting energy storage coils mentioned before must be placed in an underground tunnel in solid rock, which can take up the radial forces. Pulsed magnetic fields are routinely used in industry to form and join sheet metal parts. The induced field pushes the part against a die, just as hydraulic fluid pressure would do. Two parallel streams of electrons repulse one another because of their electrostatic fields. Their current is, however, a function of velocity as well as charge. For the same charge per unit length, the current (and, therefore, the magnetic attraction) increases with the velocity of the electrons. If the speed of light is reached, both forces cancel one another exactly (Weber, 1856). If some positive ions are left in the path of these streams, then a contraction
DIMENSIONAL TERMS FOR ENERGY TRANSPORT
277
may occur. This is known as the relativistic pinch eflect; it is independent of current. Suppose the current is carried by a liquid conductor, such as a mercury column in a glass tube. If this current becomes strong enough (a few hundred amperes) its magnetic field pinches off the liquid column interrupting the current (Northrup, 1907). The same can happen in a gaseous plasma when the current is increased in an attempt to get higher plasma temperatures (Bennnett pinch; Bennett, 1934). But the pinched current path, to the sorrow of the plasma physicists, is normally extremely unstable and short-lived. This magnetic pinch effect depends on the current and is quite different from the aforementioned relativistic pinch effect, which is independent of the value of the current. The magnetic pressure p , exerted by the self-field of a current I , flowing in a conductor of radius r is J/m3 = N/m2 The magnetic forces on a liquid electrical conductor can be utilized to generate a pumping action. The liquid sodium coolant of some nuclear reactors is pumped by such magneto-hydrodynamic pumps, while in a closed loop of pipes. The liquid sodium in a beta”-alumina/sodium heat engine/ battery is pumped in a similar manner. Here are a few other examples of the effects of magnetic fields. Every day in electric arc welding, the skilled welder positions his electrode so that the magnetic field, generated by the current which flows through it, does not blow the arc or the molten metal (which also carries some current) into an undesirable direction. Tubular parts can be welded by first heating them with an arc traveling in a gap left between the parts. The arc is driven by a radial magnetic field that is generated by properly located coils. The molten ends are then pushed together and fused. In switches for interrupting high currents, the contacts and electrodes are so arranged that, when an arc is drawn as the contacts separate, the arc is blown out of the gap by the self-magnetic field of the current. The circular magnetic field lines surrounding, for instance, a current in a gaseous conductor or plasma can be said to exert a pressure, not unlike any other gas pressure. The ratio of thermal (gas) pressure to magnetic pressure, usually denoted by /3, is an important characteristic number for a plasma generated by an electric current flowing in it. From the preceding measurements on the magnetic field using a probe coil, we have seen that voltages are generated when a wire is moved through a magnetic field, which was Faraday’s great discovery. This is the basis of all power generating equipment in practical use today. When the arrangements are just right, the generated voltage (and current caused by it) can generate, pm = fpo((l,/r)2
278
BERTHOLD W. SCHUMACHER
or at least enhance, the magnetic field that is needed. This is the dynamo principle invented by Siemens in 1879. An essential feature of these mechanical generators is the predetermined path that the conductor must follow-in spite of the reaction forces acting on it. 'Overcoming them with the mechanical driving forces is the process that turns mechanical energy into electrical energy. We can also generate a voltage by shooting a gaseous conductor through a magnetic field. For instance, a jet flame seeded with cesium, which has the lowest ionization potential, has been used. A voltage appears at those of the flame's surfaces that are perpendicular to the field direction and the flow velocity vector. Such devices are known as magneto-hydrodynamic or M H D generators. The basic idea sounds deceptively simple, but now the conductor (the gas or flame jet) tends to move out of the field, as soon as a noticeable current is drawn. This is akin to electron optical effects on particle currents. It has so far prevented the development of a practical commercial MHD power station, apart from some other engineering difficulties, mainly with high-temperature materials. (Experimental systems, producing tens of megawatts of electrical power, have been built. The energy efficiency and the life of the equipment are the problem.) The reason that electrostatic machines cannot compete in everyday use with magnetic machines for the generation of electrical power, or as motors, lies very deep. The electric field energy densities attainable in air are so much lower than magnetic field energy densities that electrostatic machines get very big compared with magnetic ones of the same power rating. Only for the generation of very stable medium-high dc voltages (megavolt range) and low currents, do certain electrostatic machines, notably the Van-de-Graff generator and its derivatives, still have a role to play. The highest electric field strengths and energy densities are found in molecular structures. It is the energy for the chemical bond, and again it is far higher than the energy densities due to magnetism. Therefore the gasoline burning motor can be smaller and lighter than an electro-motor of the same continuous duty rating. Typical values for energy densities are listed in Appendix 3,
APPENDIX 2. NUMBERSFOR THE CHARACTERIZATION OF PARTICLE BEAMS A . Some General Remarks Particle beams, such as electron or ion beams, have been called multiparameter radiation because particle energy and particle momentum change differently when the velocity changes. The energy is proportional to the square of the velocity, the momentum is directly proportional to it, as we would
DIMENSIONAL TERMS FOR ENERGY TRANSPORT
279
expect for any particle with finite rest mass. Both are proportional to the mass. The velocity is infinitely variable. Momentum and charge determine the deflection in a magnetic field, whereby the particle neither loses nor gains energy. Energy and charge determine the deflection in an electric field, and the particle may loose or gain kinetic energy. Only for charged particles do we have focusing and deflection devices (in the form of electric and magnetic fields). No optical elements are available for neutral particle beams. By contrast, in the case of light, the frequency determines the energy and characterizes a light quantum fully. In vacuum, the velocity is independent of the energy, only in matter (glass, etc.) does it become a function of energy. (Dispersion of the colors.) The mass is zero. The multiparameter particle beam has alternately a rectangular and a Gaussian power density distribution in its two conjugated cross sections (source and pupil of the system), whereas a laser beam (single parameter light), in the fundamental mode, is Gaussian everywhere. Charged particle beam optics, in its analogy with light optics, is “momentum optics,” but the particle energy makes itself felt in many ways: The thermal energy that the particles possess as they come from the source remains superimposed on all momentum-optical beam manipulations, e.g., the acceleration of the particles in an electric field. The thermal energy causes an elementary beam cone to be emitted from each single point of the source. At larger distances from a finite size source any beam exhibits a similar thermal beam spread. A parallel beam without a finite angular aperture cannot carry any energy-a basic thermodynamic fact. The number characterizing a thermionic beam, in its most efficient, Gaussian characteristic mode, is the radiance W z with the dimension W m-’ sr-l (also used for beams of light). A Gaussian beam goes through an optical system exactly as a “ray” (line) of geometrical optics does, when just object and image planes are considered. Note that all electron optical trajectory calculations apply equally to the thermal rays and to the principal rays of geometrical optics, but the distinction is of great help towards the understanding of different systems and their thermodynamic limitations. B. Particle Beam Sources
Charged particles can be drawn from the plasma (ionized gas) of an electrical gas discharge, contained in a space with a small aperture at which an electric field is applied from the outside. A positive field electrode extracts
280
BERTHOLD W. SCHUMACHER
electrons, a negative one extracts gaseous ions. The thermal energy or temperature of the extracted particles is practically the same, the particles being in equilibrium in the plasma and the extraction field pulling the electrons out or pushing them back. The temperature is usually higher than for thermionic sources, but the current density can also be higher, especially with magnetic constriction of the source plasma in the (single) extraction hole. More widely used, particularly as sources for electron beams, are the thermionic sources, e.g. a tungsten surface maintained at 3000 K or other substances such as oxides or borides that already emit at lower temperatures. Ions can also be obtained from some surfaces. A thermionic cathode emits electrons spontaneously. The velocity distribution of the emitted electrons is a Maxwell-Boltzmann equipartition distribution (in velocity space!) This leads to a radial distribution of the emission angles, e.g., the current per unit angle, which is a Gaussian probability distribution (in normal three-dimensional or configuration space). The emission current density is described by the Richardson-Dushman equation jk= ATZexp( -eO/kT) A m-2
where Tis the absolute temperature, A = 120 A cm-’deg-’ (theoretical value), 8 is the work function of the cathode material and the emitting crystal plane, but effected by surface layers, and e is the charge of the electron. (Sometimes eO in electron-volts is called the work function.) The most probable thermal kinetic energy of the emitted electron is $mV:h = kT. The related most probable thermal voltage is given by
Kh = kT/e = TI11604
V
E
=
(2)
The current density distribution across a flat cathode of constant temperature is uniform, j, = constant. (By contrast, the output intensity across a single mode laser source is Gaussian.) Ions may be drawn from a plasma boundary which is uniform and also yields a constant value of j,. Only jk and yh together characterize a particle beam source. Both are functions of temperature that are unrelated to one another, but we may combine them both to form the number:
R,* = j,/(nkT/e) A m-’ sr-’ V - ’
(3)
It may be called ray-value-per-volt for the source, as will be shown below. The voltage referred to here is, in general, the acceleration voltage that is employed to form a beam. In Eq. 3 it is the most probable thermal voltage of the electrons emitted into the half-space (with 2n sr). Yet R:is a characteristic
DIMENSIONAL TERMS FOR ENERGY TRANSPORT
28 I
term describing just the source. Inserting the value for j, as function of temperature we get (Schumacher, 1982) R,*= A/(nkT/e)T2 exp(-ee/kT)
= A*Texp(-ee/kT)
(4)
with A*
= A/arke =
A/3694
=
3.25 x lo-'
Acm-' sr-lv-ldeg-'
(5)
or R,*= 3694j,/T
(6)
We see from Eq. (4) that the ray-value-per-volt for a thermionic cathode, perhaps surprizingly, increases with temperature. Equation (6) is misleading in this respect, since j, is itself implicitly a function of temperature. For a plasma-type cathode or ion source we must expect more complex relationships.
C. Beam Formation In an acceleration field with parallel field lines the particles move as illustrated in Fig. 4. A beam at a large distance from the (transparent) field electrode then shows an angular current density distribution which is Gaussian, as shown in Fig. 5. If now the beam is focused to form a cross-over (a pupil of the optical system) the radial current density distribution j(r) in the cross-over plane is Gaussian as well, while the current per unit angle in the cross-over plane is constant between f a , (but obviously different for each r-value). The angle a, = D,/2L is under the operator's control. Figure 6 illustrates these relationships. Focusing action was added to the system in Fig. 5, by means of a properly shaped acceleration field. The angular Gaussian current density distribution is now transformed into a radial Gaussian current density distribution (per unit area, not angle) in the cross-over or focal plane (actually a pupil of the system, a constant velocity focus). The image of the source lies beyond the cross-over plane and its current density is constant, the current per unit angle again being Gaussian, as it was at the source. A radial energy component has now been given to the principal rays as well, namely by the optical action of the accelerating field. Yet the thermal radial energy component remains superimposed, the thermal particles forming the elementary ray cone around the principal rays, even where those are optically bent. We can say: No beam can be characterized without reference to angular aperture (current per unit angle) and to current density (current per unit area). This is expressed in the dimension of the radiance number.
282
BERTHOLD W. SCHUMACHER SOURCE a t t e m p e r a t u r e T w i t h e m i s s i o n current density
jk
A m-2
A t source p o t e n t i a l
PRINCIPAL RAYS, namely t h o s e which a r e e m i t t e d w i t h o u t r a d i a l e n e r g y compon e n t and a r e t h e r e f o r e l e a v i n g normal t o the s u r f a c e , i.e. p a r a l l e l t o the
THERMAL RAYS, h a v i n g a r a d i a l e n e r g y component when e m i t t e d and f o l l o w i n g a p a r a b o l i c p a t h when a c c e l e r a t e d .
FIELD ELECTRODE Beyond t h e ( t r a n s p a r e n t ) f i e l d e l e c t r o d e t h e p a r t i c l e s w i t h t h e most probable thermal energy: m h
n
/I
t r a v e l a t an a n g l e t o t h e z - a x i s :
Go= Vth is t h e ' t h e r m a l v o l t a g e '
Vo i s t h e a c c e l e r a t i n g v o l t a g e
FIG.4. Simple diode with parallel acceleration field for the particles. If the field electrode is transparent (e.g., a fine wire mesh) the particles will retain thereafter the velocity and direction which they had. Scattering and optical effects by the wire mesh (or an acceleration electrode with a central hole) are obviously different and unrelated matters, falling under the category of ray optics.
The relevant terms are summarized in Fig. 7. All values whose symbols are shown with a bar are averages over a distribution function. For instance, & is the diameter of the beam at the l/e value of the maximum intensity. Numerical values for W,*and R,*are listed in the tables of Appendix 3. For light beams it is not necessary to distinguish between averages and geometrically defined quantities. Yet, while black-body radiation fills an aperture evenly (e.g., sun light falling on a burning glass), a laser beam is Gaussian and the l/e diameter of the distribution must be used to define the radiance. More complex beams (light or particles) do not have a uniquely definable radiance and require more elaborate descriptions. Only in two conjugated planes or their images, namely source plane and cross-over plane, can a thermionic beam be described simply by three
&
DIMENSIONAL TERMS FOR ENERGY TRANSPORT
(screen)
.
cathode
I
7
,
6% Of 'Current
~~~~~
, / z
.-~
. / / .
?< + :--
I
---
-
-\=
.I '
I
'th
GEJ I
1
1
Principal
--.--+=+-+--------'= . .\. ;.' ....
Cathode Potential
283
-
Rays
-7
.... . .1
Non-Pr incipl Rays
-
w
"0
Undefined "Wing"
+iConstant
Plane (Voltage Vo)
I
z =I o
Angular Curre Distribution at "Inf rnity"
zA
FIG.5. Current density distribution as function of angle along the beam in the drift region beyond the field electrode and at infinity after an initial parallel acceleration of the particles (electrons, in this case). At z + D,,the current density as a function of angle is Gaussian. In the plane of the field electrode (anode) which is not a conjugated plane of the system, the current distribution per unit area and unit angle cannot be described by one, two, or even three simple numbers.
numbers, namely voltage, total power, and radiance. In any other cross section the beam can only be represented by six coordinate numbers and an intensity number (six-dimensional phase space representation). A graphical display is possible, but it requires two three-dimensional plots with xx' and yy' as coordinate axes, the intensity at each point plotted as z values. These values can, in principle, be measured for any cross-sectional plane, assuming V, is closely the same for each particle. A simpler form of plotting at least some of this information is the display of the area of a phase space cross section within which the beam passes. For a circular symmetrical beam the coordinates are r and dr/dz (angular aperture). In the general case the area is an ellipse, but this assumes sharp boundaries of the beam in r and drldz. As we have seen, a Gaussian beam does not have such boundaries. The phase
284
BERTHOLD W. SCHUMACHER
AUXILIARY
FIELD ELECTRODE
PRINCIPAL RAYS
PARTICLE
' TRAJECTORIES
2
-
LINES
VOLTAGE 1, 150 kV
GROUND
Lenee region with focal Length L
SOURCE
J
GROUND
CURRENT PER UNIT AREA
so
i.
AP Integrated over
1o = ?4 D 2k j k
CURRENT PER UNIT ANGLE [and area] for any I r , r ) IRE L a t i ve va tuee]
-
r
PHASE-SPACE CROSS SECTIONS [with intensity Contour tinas]
NO SHARP BOUNDARY
FIG.6. Focusing action added to the system of Fig. 5.
space cross section areas are, therefore, at least unbounded in one coordinate. Three such phase space cross sections are included in Fig. 6. Only in the case of a thermionic beam do we know the intensity distribution for the cross sections in the conjugated planes; in any other case, intensities as function of coordinates must be shown separately or the cross sectional phase space area does not tell us much.
-
DIMENSIONAL TERMS FOR ENERGY TRANSPORT
285
Key Terms for the Characterization of a Particle Beam and its Source
Current
I, -+
Cross section Angular aperture
Power density N* (W m-’)
2, or D, an
or 8,
+ Radiance W* m-’ sr-l)
(w
W * / V i = Radiance/Vi = Source characteristic R:(W m-’sr-’ or R*/Vb = Ray-value/Vo = R:(A m-’sr-l V - ’ )
V-’)
The Radiance/Volt W*/V, = R* (A m-’ sr-I) is known in the literature as Richtstrahlwert, best translated ray-value of a beam. It changes with V, just as the radiance does, yet V, does not appear in its definition. Thus, its roots and meaning are not immediately apparent. It does not characterize either the beam or the source completely. R:, however, is independent of V, and a true characteristic number of the source alone. A name for R:, the source characteristic, has not been established. It would seem appropriate to call it the ray-value-per-volt for the source.
FIG.7. Terms and numbers required to describe a particle beam and its source in its most efficient Gaussian mode.
The radiance (and the inverse of it, the normalized emittance, which is sometimes used) are also only fully meaningful for a Gaussian beam. Any other beam cannot be so simply described. Many arguments could be avoided if this fact would always be kept in mind. The equations for the radiance will be discussed below. D. Disturbances to a Thermodynamic Ensemble of Particles
The particles in a thermionic beam represent an ensemble in the sense of a thermodynamically uniform group, a thermodynamic ensemble. The reason is, obviously, that they have a common origin. A focal spot with Gaussian intensity distribution, as is typical for this ensemble, carries the highest possible information content per unit beam energy (at a given beam voltage) and is fully described by a single radiance number. This thermodynamic ensemble has the highest possible radiance. (We may also say, it has the lowest possible entropy.) Any change of the internal beam intensity distribution will diminish the radiance (and the information content if the beam is used as a probe). Any disturbance introduced by the aberrations of optical elements in the beam path is, in principle, reversible in a subsequent stage of optical elements (example: image restoration). Scattering of the beam particles causes an irreversible change, an increase in entropy. (If the scattering is accompanied by
286
BERTHOLD W. SCHUMACHER
energy loss, then the scattered particles may be filtered out by an energyselective element. This is a different situation.) It is generally not permitted to treat scattering as a diffusion process, although this has often been done. E. Relationship between Beam and Source-Radiance Numbers
Particle beams get their energy not from the source, as light beams do, but from the subsequent acceleration of the particles in an electric field. The particle energy is proportional to the acceleration voltage V,; the particle velocity is proportional to the square root of the voltage. There exists an analogy between the refractive index n, in the light optical case and even to the extent that the energy density in a light beam is proportional to n:. While the total energy of a beam is proportional to V, the radiance is proportional to V g . In Fig. 7 the definitions of the terms radiance, radiance per volt, and rayvalue-per-volt have been explained. In Fig. 8 the quantitative relationships are shown, with reference to the geometric terms as defined in Fig. 6. The radiance W,*is not only determined by the source but also by the square of the acceleration voltage Vo. To get a characteristic number which describes the source itself we can divide W,*by V g . In other words, we find the radiance for a 1-V beam from that source, R:. Its dimension can be either W m-2 sr-' V 2or A m - 2 sr-l V - ' with emphasis either on the energy or on the current carried in the beam. The numerical value remains the same. Figure 8 provides a quantitative description of a thermionic particle beam at its conjugated cross sections, namely cross-over (focus) and source, or their Beam power (invariant)
Watt
W" = I , vo
Radiance (invariant)
Watt m-'sr-'
1st Form 2nd Form 3rd Form (at the virtual source)
4th Form 5th Form 6th Form 7th Form (at image of the source) (at the cross-over) Watt m-'
Power density (locally different) N*
= j i Vo
N&,*
= j,,,
Vo = nu; W,*
FIG.8. Quantitative description of a thermionic beam.
DIMENSIONAL TERMS FOR ENERGY TRANSPORT
287
images. (The term virtual source means the rays are drawn as if the source were at the same potential as the accelerated particles.) Quantities and symbols used here are defined in Figures 3 and 7. The term (d-,~,),which appears in the denominator of some equations is called the emittance of the beam (not of the source!). It is very confusing when used in isolation and should be avoided. It changes with W ; . In light optics the angle-area product (D,B,)’is known as optical invariant but it too changes, namely with the refractive index. For electron beams the maximum current density is often described by what is known as the Langmuir limit, namely j,,, = j k ( l
+ eVo/kT)cc;
A m-’
For higher values of V, the 1 can be neglected and we get the equations of above. We may write, for instance,
and since 8,
< 1, we get
which follows also from the second and fifth form of the equations. Once more the interdependence of current densities and aperture angles is shown. The meaning of the term or concept of radiance can be further illustrated by saying that the higher the radiance the smaller the angular aperture (e.g. U, in Fig. 6) and therefore the greater the depth of focus for a beam of a certain energy. The operator can always change the focus and thereby a, and decrease or increase the power density N 2 a x .But never can the radiance be increased unless the particle energy (beam voltage) is also increased. In general, all kinds of effects are constantly reducing the radiance. It has been said that the radiance of a light beam from an incandescent source and that from a beam of spontaneously emitted thermionic electrons (0.1-0.5-V beam) is of the same order of magnitude (see Appendix 3), and that, therefore, we have an almost complete identity of electron optics and light optics. This ignores the fact that higher-voltage beams have higher radiance, whereas that of light cannot be increased. The associated greater information content of any electron microscope picture is today well known. Of course, the shorter wavelength of the electrons permits this, but in itself it poses only a limit at the extremes of magnification where diffraction effects influence (diminish) the beam radiance.* *See also Lenz (1957). Wigner (1932), Brenner et al. (1984), Wolf (1982), and Bastiaans ( 198 1).
288
BERTHOLD W. SCHUMACHER
The energy of a particle beam is imparted to it by the acceleration field. The energy, which we must supply to generate the free particles in the first place, is small, often quite negligible. For instance, a cathode heated by 200 W can deliver a beam with 60-kW beam power or more, if the acceleration voltage is 165 kV or greater. Here the conversion of electrical power to beam power is better than 99% efficient. F. Some Comparisons of Particle Beams and Light Beams
It was recognized by Clausius in 1864 that the radiance is a ray invariant along any narrow aperture light ray, even if the refractive index ni changes. This can be expressed by
Here W is the energy traveling in the ray cone, assumed to be constant (no absorption, no reflection losses), dA represents the area through which W passes (e.g., dA = nr2 for a round spot of radius r), dR is the angular aperture in sr (divergence or convergence, dR = nu2 for a circular cone) of the rays passing through dA, n, is the refractive index of the medium through which the energy travels, and W: is the radiance in vacuum with n, = 1. When the light cone enters a medium of different refractive index the area dA is the same for both sides. How, then, can W*change? As is well known, cone angle, wavelength, and the speed of light change. From inside the other medium we measure a different radiance, namely W*(ni) = n:W,*
(8)
This represents, for instance, the radiance of an imbedded black body. (The possible mode of oscillations in a cavity is, just as the spatial energy density, proportional to n:.) It is another explanation of the fact that an oil imersion type of microscope objective yields higher resolution. The liquid in the eye may be mentioned here too. It is also a factor to be considered in heat exchange calculations, e.g. for a glass melting furnace. The spatial energy density E , , i.e., the volume density of energy traveling with speed ci in the light cone, is given by the ratio of energy flow [W m-' sr-l] divided by the flow speed [m sec-'1 E,
= W*(ni)/ci = n:W,*/c,
J m-3 sr-l
(9)
It is proportional to n; just as the spatial mode density in a black body cavity. (Use has been made here of the facts that ci = c,/ni, and Izi = Izo/ni.)
289
DIMENSIONAL TERMS FOR ENERGY TRANSPORT
The term sr - * in the dimension may seem odd. We are not accustomed to giving dimensions to angles. The angular term disappears if we integrate the energy flow over the half-space or assume a plane wave, but nature does not provide those, and we would lose information about the anatomy of the energy flow. The sr-l is a reminder that we live, and that all radiation propagates in a three-dimensional space. It is unfortunate that the Stefan-Boltzmann constant 0 is defined for the half-space instead for the steradian. No physical, flat (surface) light source emits into the half-space with equal radiance for all angles to the normal. Per / . The radiance of black body radiation is steradian we get CJ*= a W; = 0 ~ 4 / n
with CJ
=
5.67 x lo-’’ Joule sec-’ cm-’ KP4.
Max Planck in his book “The Theory of Heat Radiation” used the letter K for W,*,without ever using the term radiance, but alternately the terms “specific intensity,” “brightness,” and “Helligkeit,” and “Flachenhelle”. The present term used in German literature is Strahldichte. (The literal meaning of this latest term is equally obscure in itself and requires definition, as do all the others.) The term Radiance has the advantage of a good technical term, namely no connotations in everyday language. The term (nzdA dSZ) is variously called optical invariant, throughput, ktendue, or lichtleitwert. where The speed of the particles in a particle beam is proportional to r/I is the potential difference between the source and the point or space of observation. From this fact alone we can recognize that there exists an ana(Glaser, 1933). On this basis, we logy between the refractive index ni and can derive an equivalent to Eq. (7), namely
fi,
fi
which we had called ray-value and which is obviously an invariant of the beam wherever the voltage (energy, momentum) is the same. But Eq. (10) is not the energy, it is the momentum. We are in the momentum-optical regime. The energy of a light beam cannot change while it is on its way. Only energy density and radiance can change with the refractive index, as we have seen. In case of a particle beam, its energy changes, as well as its radiance, when the refractive index (square root of F) changes. Only R,* = R,YF which we called ray value per volt is a true systems invariant, insofar as it is describing the source.
290
BERTHOLD W. SCHUMACHER
We may, nevertheless, ask for the equivalent to Eq. (9) which becomes:
It represents the spatial momentum density per steradian, since V1l2 is proportional to the velocity u. Whatever the mass of the particle, the ratio of energy over velocity yields the momentum, and pi is the spatial momentum density. Most interesting is the appearance of the factor V3/’, but in connection with a term per-steradian (as part pf R,Y).This factor is known as the perveance factor from space charge limited current flow, yet here it comes from the geometrical optics of realworld, three-dimensional beams. Asking for the spatial density of the kinetic energy and the particle velocity (the two being linked, in this case), we must fall back on the energy equivalency-the kinetic energy is equal to the “electrical” energy. It is eV, = mv2/2 or v =.-/, The volumetric (spatial) density (of any quantity) is always equal to the ratio of the rate of flow to the velocity. For particle beams, the energy is codetermined by the velocity or vice versa. We can write [W m-’ sr-’]:[m sec-’1 = [Joule m-3 sr-’1. To obtain the numerical value we must express e/m (= 1.759 x lo-’’ A sec/kg for electrons) in electrical units. It is m = 2eV0/u2 [Joule (m/sec)-’]. Thus we get
with R,*= lo5 A m-’sr-’ V-’ and V, E
~
=
= 1.7 ~ x~ lo-’ ,
1 V we get: ~
J~m-~sr-’
The flux (beam power) is proportional to the accelerating voltage V,; the velocity is proportional to V;”. It follows that the spatial energy density is increasing with V;”. Thus, for V, = 150 kV and the same &*value as before, we get a value E , , ~ = 387 qol, A 10-fold increase in R,* results in another 10-fold increase in E , , ~ . These values may be compared with those for light (Appendix 3). We have stayed strictly within the limits of geometrical optics. Nothing was assumed about source properties, such as yield or yield as function of area and angle; nothing was assumed about thermal energy, etc. All these factors are implicitly contained in W* and R,*. It should not be implied, however, that W* and R,* are the same in all directions or for every point of a beam cross section; in fact, they are usually
DIMENSIONAL TERMS FOR ENERGY TRANSPORT
29 1
functions of direction and location.* They can only be applied to a large beam as a whole if the source emits uniformly, and the “optics” handles all parts of the beam uniformly. Fortunately, this is often the case and beam description becomes simple. Unfortunately, it is not often recognized that not every beam can be described by this simple terminology. No forced simplification can give an adequate description of more complex beams. Ultimately we must resort to a set of two three-dimensional phase space plots in xx’ and yy’ to describe a complex beam fully. All descriptions break down for beams of just a few electrons or quanta. We cannot apply thermodynamics or statistics to ensembles of one or two. A very weak beam of light cannot be focused. If the scattered electrons in an electron microscope do not fill a Gaussian envelope (which the scattering laws do not guarantee), then the resolution is impaired.t For light we can also derive a momentum density. While for particles we have referred to a constancy of charge (current) flowing in the beam, for light we must refer to a constancy of energy (quanta) flow. We get, as usual, the momentum density by dividing the energy density by the velocity, namely G = &?/cO [J m P 3sr-’ rn-’ sec = Newton-sec/m3 sr] (12) We cannot ignore the angular aperture by which we focus or into which the energy is emitted; the term sr-l appears again. A word about laser beams may be added here. The laser beam gets its energy from the accumulation of many events of stimulated emission from numerous previously exited atoms, e.g., in a gas discharge. Since these excitation energy levels are in the order of the chemical binding energy per atom, the laser, especially the gas laser, is not an efficient energy store or converter,
* The simple concept of a uniform ray value R (see Fig. 7) has been extended to a crosssection dependent ray-value function by Lenz (1957). It can even be applied to describe the scattering of electron beams and to derive an approximate solution to the scattering (diffusion) equation in closed form. Of course, we have then left the field of basic thermodynamic concepts and entered into the realm of descriptive/predictive mathematics. t The description also breaks down for source areas and dimensions in the order of the wavelength of the radiation, when interference effects become noticeable. This led to a whole new science of mathematical optics for ray optics as well as for wave optics. It is inherently connected to the fact that when quantum phenomena become noticeable, the Boltmann distribution factor of E/kTmust be replaced by H / k T , where H is the Hamiltonian function of the system, related to the energy E, but not as readily calculated (Wigner, 1932). The equations for the optical cases are discussed by Brenner et al. (1984), Wolf (1982), and Bastiaans (1981). References to earlier work can be found there. These matters do not affect our thermodynamic concepts and discussions of the same, as long as the source areas studied in any experiment are uniform, extending over dimensions of many 2 s . To repeat, statistical and thermodynamic concepts cannot be applied to only a small number of entities; however, this does not diminish the fundamental importance of these concepts.
-
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nor can the energy storage density be high. Therefore high-energy lasers become very large physically. (For particle beams, on the other hand, no intermediate storage of the energy is required at all.) The stimulated radiation has the direction of the stimulating radiation, because of the required conservation of momentum. In the fundamental mode, the emission density has a radial Gaussian distribution. At one cross section the beam has its smallest diameter, a parallel envelope, and the wavefronts are plane but not uniform in intensity. This narrowest cross section may lie inside or outside the laser resonator cavity. From there, the beam spreads according to the laws of light diffraction with the smallest diameter acting as the aperture. The l/e diameter of the beam can be taken as the size of this aperture. Because of the Gaussian intensity distribution across this aperture the diffraction pattern has no wings or fringes but has again a Gaussian intensity distribution for every cross section, yet now the wave fronts have become curved. The originally Gaussian radial intensity distribution stays Gaussian. This is distinctly different from what we find in a Gaussian particle beam (see, for instance, Fig. 6). If the source intensity distribution is not a Gaussian, the beam is not either. The beam from a semiconductor injection laser, for instance, shows the diffraction pattern most clearly because of the smallness of the source. Consequently, the angular aperture of this kind of laser beam is large, more than 15". The laser light is spatially and temporally coherent, as if, as has been said, the whole light wave were one giant photon (but one whose energy can now be divided without changing its wavelength?). By contrast, in a particle beam the corpuscles travel at large distances from one another, even at rather high currents. Electrons are Fermions and cannot occupy the same phase space cell, as can Bosons (light quanta). Therefore no laserlike equivalent is possible for particle beams.
APPENDIX3. NUMERICAL VALUES FOR ENERGYDENSITIES Maximum Field Energy Densities The maximum values of practically achievable energy densities in electric and magnetic fields are determined by materials properties. Some typical values are Electric field in air with E = 5 kV/cm E , ~= 1.1 x Joule/cm3 Condensor with E = 100 kV/cm E , ~= 4.4 x Joule/cm' Internal molecular fields (with E = lo9 V/cm, E , ~= 4.4 x lo4 Joule/cm3 (= 10.6 kcal/cm3) representing chemical energy densities)
DIMENSIONAL TERMS FOR ENERGY TRANSPORT
293
Energy (density) for freezing/melting water 335 Joule/g = 335 Joule/cm3 SiO, 2100 Joule/g = 5460 Joule/cm3 iron 1300 Joule/g = 10,220Joule/cm3 Energy (density) for raw materials production steel 20,000 Joule/g polyethylene 90,000 Joule/g aluminum 300,000 Joule/g Energy (density) in storage systems (practically realized values) HV capacitor 0.2 Joule/g 79 Joule/g Pb/H,SO, battery NijCd battery 120 Joule/g HCOOH/O, battery 600 Joule/g Ag/Zn battery ( 1700 Joule/g active ingredient) 400 Joule/g TNT (decomposition energy) 5000 Joule/g Aviation gasoline (combined energy, less weight of combustion air) 84,000 Joule/g Flywheel, steel 170 Joule/g Stored energy, CO, laser gas 6 Joule/g = 0.08 J o ~ l e / c m - ~ Escape energy from earth
63,000 Joule/g
Magnetic field in air, H = 1.6 x lo4 A/cm E , = 1.6 x Joule/cm3 Energy stored in permanent magnet (Alnico V) 4.8 x lo-' Joule/cm3 Joule/cm3 Magnetic energy density in field of collapsed star
-
Pulsed magnetic fields have been generated in the range of 100 Joule/A m2 [Tesla] Fields in the order of 10 Joule/A m z can be held for times of 1 sec. The lowest field detectable with a Josephson junction is -4 x lo-', Joule/A m2. The magnetic field of the earth is -5 x Joule/A m2 [ = 0.5 Gauss]
1 man-hour of heavy labor
- 550,000 Joule
Density of radiant energy (at a focus) Power density at the focus: (radiance x aperture) Spatial energy density, light Spatial energy particle beams
N*
W m-2
E* =
W,+nag n: W,+/c,
E* =
(R:/J2e/m)V212
=
IR lamp Sun light Blueish star Light from atomic explosion HeNe laser (1 mW, 0.5-mm diameter beam) CO, laser (100 W, perfect single mode) Microwave beam (large phased array antenna)
Joule m - sr Joule m - 3 sr-'
Radiance W t W cm-* s r - l
Energy density Joule sr-'
2 2000 lo6 1010-lo'2 2 x 105 3 x lo8 5 x 107
6x 6 x lo-* 3 x 10-5 0.3 -30 6x 1 x lo-, 0.3 x 10-3
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Electron beam Tungsten cathode (1 V) Tungsten cathode (150 kV) LaB, cathode (150 kV) W-Th cathode (150 kV) Field emitter (10 PA, 17 kV) Proton beam (Duoplasmatron source) (200 kV)
10 10 100 2
5 x 10-5
10 2.2 x 1 0 ’ O 2.2 x 10” 4.4 x 109 10’2-1016
1.7 x 10-7 6.6 x 6.6 x 1.3 x 10-5 -
2.0 x lo6
Electromagnetic radiation density needed in a focus for the onset of voltage breakdown in air Safe energy densities for reflection by a mirror or for transmission through a window CO, laser light NaCl window Copper mirror Storable inversion energy, per unit beam cross section avoiding internal breakdown, limit for CO, laser
-3.0 x lo5 Joule
-2.0 Joule cm-, -4.0 Joule cm-’
-
5.0 Joule cm-
REFERENCES Bastiaans, M. J. (1981). Opt. Acta 28, 1215. Brenner, K.-H. (1984). Opt. Acta 31,231. Dummer, G. W. A. (1978). “Electronic Inventions and Discoveries,” 2nd ed. Pergamon, Oxford. Edwards, R. A. (1976). Phys. Bull. May 1976, 192. Falk, G., Hermann, F., and Schmid, G . B. (1983). Am. J . Phys. 51, 1074. Good, H. R., Jr. (1974). In “The Encyclopedia of Physics” (R. M. Besanpon, ed.) 2nd ed., p. 700. Van Nostrand-Reinhold, Princeton, New Jersey. Laufer, G. (1983). Am. J. Phys. 51,42. Lenz, F. (1957). Habilitationsschrft T. H. Aachen. Page, C. H. (1978). Am. J. Phys. 46,79 and references therein. Pipkin, F. M., and Ritter, R. C. (1983). Science, 219,913. Rojansky, V. (1979). “Electromagnetic Fields and Waves.” Dover, New York. Schumacher, B. W. (1982). Optik 62,249. Slater, J. C., and Frank, N. H. (1969). “Electromagnetism.” Dover, New York. Snedegar, W. H. (1983). Am. J. Phys. 51, 684. Stewart, J. W. (1982). I n “The Encyclopedia of Physics.” p. 247. McGraw-Hill, New York. Wigner, E. (1932). Phys. Rev. 40,749. Wolf, E. (1982). JOSA 72, 343.
ADVANCES IN ELECTRONICS AND ELECTRON PHYSICS. VOL . 65
Image Calculations in High-Resolution Electron Microscopy: Problems. Progress. and Prospects D. VAN DYCK University of Antwerp Rijksuniversitair Centrum Antwerp. Belgium
I. Introduction . . . . . . . . . . . . . . . . . . . . . . I1. Principles of Image Formation . . . . . . . . . . . . . . . 111. Present Status of Many-Beam Electron-Diffraction Calculations . . . A . History . . . . . . . . . . . . . . . . . . . . . . . B. The Multislice Method . . . . . . . . . . . . . . . . . C. Computations . . . . . . . . . . . . . . . . . . . . D . Use of the Fast Fourier Transform . . . . . . . . . . . . E. Special-Purpose Hardware . . . . . . . . . . . . . . . F . Discussion . . . . . . . . . . . . . . . . . . . . . . IV. Systematic Approach to the Electron-Diffraction Problem . . . . . A . General Theory . . . . . . . . . . . . . . . . . . . . B. High-Energy Approximation . . . . . . . . . . . . . . C. Validity of the High-Energy Approximation . . . . . . . . . D. Discussion . . . . . . . . . . . . . . . . . . . . . . V. Study of Existing Many-Beam Diffraction Formulations . . . . . . A . Differential Equations . . . . . . . . . . . . . . . . . . B. MatrixMethods . . . . . . . . . . . . . . . . . . . . C. The Iterative Method . . . . . . . . . . . . . . . . . . D. Direct Integrating Methods . . . . . . . . . . . . . . . E. SliceMethods . . . . . . . . . . . . . . . . . . . . F . Discussion . . . . . . . . . . . . . . . . . . . . . . VI. A New Formulation: The Real-Space Method . . . . . . . . . . A . Principles . . . . . . . . . . . . . . . . . . . . . . B . Comparison with Other Slice Methods . . . . . . . . . . . C. Numerical Procedure . . . . . . . . . . . . . . . . . . D . Analysis of Input Parameters . . . . . . . . . . . . . . E. Discussion . . . . . . . . . . . . . . . . . . . . . . VII . Recent Developments, Unsolved Problems, and Prospects for the Future A . The Real-Space Patching Technique . . . . . . . . . . . . B. Considerations Concerning the Periodic-Continuation Method . . C. Atomic Column Approximation . . . . . . . . . . . . . D. Absorption. Inelastic Scattering, and Thermal Diffuse Scattering . . E . The Reverse Problem: Direct Structure Retrieval . . . . . . . F . Upper Layer Lines and Beam Tilt . . . . . . . . . . . . . G . Application to General Quantum-Mechanical Problems . . . . . H. Conclusion and Prospects for the Future . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . .
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I. INTRODUCTION
High-resolution electron microscopy is definitely breaking through in both materials science and biology. It is now quite well established that the interpretation of the high-resolution electron micrographs in terms of the microstructure of the specimen is more reliable if the experimental images are compared with computer simulations in which the dynamic electron diffraction in the specimen as well as the image formation in the electron microscope are simulated for all plausible trial structures. Today, most of the specialized electron microscopic laboratories in the world have their own image calculation programs. There is now a growing tendency to perform image simulations interactively and closer to the electron microscope, e.g., on a small laboratory computer with display facilities. Interactive programs have been written and complete, dedicated, simulation hard- and software is available. However, the early expectations regarding the simulation technique, which ran rather high after its first successes in the early 1970s, have yet to be fulfilled. Indeed, since its introduction, the simulation technique has hardly undergone any further development. Simulation is sometimes used more as a gadget than as a tool. For instance, the comparison between experimental and simulated images is often done visually, i.e., qualitatively, without even taking account of the characteristics of the photographic material. Furthermore, a number of experimental parameters, such as specimen thickness, absolute value of defocus, and angle of beam convergence, which are needed as input parameters in the calculations, are often not determined, determined with insufficient accuracy, or even treated as adjustable parameters (Howie 1983). On the other hand, the number of experimental and simulated images of the same specimen, such as through-focus or thickness series, is often insufficient. Furthermore, the normalization of the total diffraction intensity is still often used as a criterion for the selection of input parameters, such as slice thickness and number of beams. However, while this criterion is questionable, other simple criteria are still lacking. It is apparent that some of the image simulations reported in the literature are performed using input parameters that obviously lead to erroneous simulations. As a result, the reliability of the high-resolution image simulation technique for the determination of the microstructure is still relatively poor and can only be used with some confidence in cases where the structure is already sufficiently known and slight modifications of the structure are tested by trial and error. Hence the use of the technique is seldom convincing. On the other hand, the simulation technique would be much more powerful if the contrast of the experimental images were to be measured quantitatively, e.g., using microdensitometry or on-line image capturing and digitalization. In that
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case, the experimental and simulated images could be compared by using least-squares techniques, and iterative image calculations could lead to structural refinements. However, at this stage the computational procedures are still too time-consuming and expensive. For example, a typical image calculation performed on a large mainframe computer can cost somewhere between $100 and $1000 of computer time. Hence, least-squares matching or applications to more complicated systems, such as extended defects in crystals, or to convergent beam electron diffraction are often restricted. Although most of the current many-beam diffraction computations are based on the multislice theory, which dates from 1957, this does not mean that the theory itself is completely satisfactory, but rather that it has not been tested to its extremes. Rigorous approaches however are scarce. Furthermore, there are still a number of problems that are not quite solved or understood, or even not recognized, such as the influence of higherorder Laue zones, the effect of absorption, the influence of inelastically scattered electrons, the presence of anomalous fringes in some high-resolution images [e.g., (001) rutile], and the validity of the approximations, such as the high-energy approximation. There is also some hope that in the future, the reverse problem can be solved, i.e., how to retrieve the specimen structure directly from the electron micrographs without trial-and-error procedures. For this purpose the dynamic diffraction must be “reversed,” which seems to be a tedious problem in which questions of uniqueness are expected to occur. Nowadays, the situation of high-resolution electron microscopy with image simulation is comparable to the situation in the early days of x-ray diffraction, when the power of the new technique was recognized but the applications were limited by theoretical and computational problems. In this work, we sketch the principles of image simulation, compare different computational methods, and discuss recent developments and possible future prospects. 11. PRINCIPLES OF IMAGE FORMATION
Since the famous paper of Scherzer (1949), the principles of image formation in the electron microscope have.been well established and will be briefly sketched below (for a more general treatment see, e.g., Van Dyck, 1978a or Spence, 1981). Calling q(x, y ) the wave function at the exit plane of the object, the diffraction pattern, i.e., the wave function in the focal plane of the objective lens is, then, according to Abbe’s theory, given by its Fourier transform
F u, v
CdX,
Y)l
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where u, v are the reciprocal coordinates that characterize the diffracted beams. In the second part of the image formation, before undergoing a second Fourier transform, the diffracted beams are selected for imaging by an aperture function A(u, v) and suffer a phase shift ~ ( uu,) due to the spherical aberration of the objective lens and the defocusing of the microscope. The aperture function A(u, v) can eventually be modulated by a damping factor in order to account, in a first approximation, for the effects of beam divergence and chromatic aberration (see, e.g., Spence, 1981). Finally, the wave function in the image plane is given by
In the hypothetical case of an ideal electron microscope without aberrationphase shift, and aperture, the image intensity I+(x, y)I2 of a pure phase object q(x, y ) = exp[i+(x, y)] would show no contrast at all. For the study of phase objects, such as thin biological or material foils, the inevitable occurrence of a phase shift due to spherical aberration, which is inherent in the use of magnetic lenses, is in fact a fortunate coincidence. Indeed, as shown by Scherzer (1949), by proper defocusing of the objective lens (called the “Scherzer” focus), the phase shift of the spherical aberration can be compensated in such a way that the central beam is approximately retarded over n/2 in phase with respect to the major part of the other diffracted beams. Hence, the electron microscope can be considered as a kind of phase contrast microscope with the image intensity roughly proportional to the projected electrostatic potential of the specimen. However, for thicker specimens and other focus values, the interpretation of the images is often less straightforward and must rely on computer simulations. By using fast Fourier transform routines, the computer simulation of the imaging process using Eq. (1) does not require stringent computer demands. Typical calculation times are on the order of seconds. The image intensity I $(x, y)I2 can be displayed as a halftone picture by using several techniques such as overprinting, photowriting, electrostatic printing, storage CRT, and television screen, depending whether the simulations are performed in batch or in real time. The most laborious part of the simulation process is the computation of the electron wave function q(x, y) at the exit plane of the specimen. Due to the complexity of the electron-object interaction, which is a fully dynamic diffraction process in which many beams can be involved, the computer demands and costs can be so high that they put serious restrictions on the use of the simulation technique. This is, for instance, the case when complicated objects such as extended defects in crystals are involved, in which sometimes 10,000 or more beams take part in the diffraction process, or when the convergence of the incident beam is treated rigorously by computing
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separately the images corresponding with all different directions of incidence and adding the image intensities afterwards. In this work we focus our attention primarily on the problems emerging from the computation of q(x, y) and discuss the most recent developments in the theory and the computer algorithms for the simulation of the many-beam electron diffraction process. 111. PRESENT STATUS OF MANY-BEAM ELECTRON-DIFFRACTION CALCULATIONS A . History
The theory of dynamic elastic electron scattering in a perfect crystal has a long, involved, and even devious, history (Moodie, 1981). The first, and still one of the most complete treatments, was already provided by Bethe as early as 1928. Inspired by Ewald’s work on the dynamic scattering of electromagnetic waves and Born’s theory of quantum scattering, he treated the electronscattering problem starting from the Schrodinger equation by using a Fourier expansion for the periodic crystal potential and a plane wave expansion for the electron wave function equivalent to the concept of Bloch waves. In this way he obtained a set of coupled linear dispersion equations for the plane wave expansion coefficients that can be put into matrix form and, in the approximation of forward scattering, can be reformulated as an eigenvalue problem. However, since the dimensions of the dynamic matrix are proportional to the number of plane waves under consideration, this formulation is only appropriate when the number of beams involved is relatively small. Since Bethe’s original approach, (Bethe, 1928) several many-beam electron-diffraction theories have been developed, some of which were adaptations of existing methods of band theory. Before the famous Kyoto Conference (1961), many of the interconnections between the various formulations were not known or obscure (Moodie, 1981). However, they all turned out to be reformulations of the original Bethe theory. In the early 1950s, experiments showed that many-beam effects could be significant and, with increasing accelerating voltage, hundreds and even thousands of simultaneous reflections could occur. However until 1955, the most reliable many-beam diffraction theory available was still the kinematic theory (Sanders and Goodmann, 1981). Hence, the need for alternative formulations became urgent. In 1950, Sturkey presented his scattering matrix theory (Sturkey, 1962).In 1955, Fujimoto, using the work of Niehrs and Wagner (1959), showed that the scattering matrix theory could be derived directly from the Bethe dispersion equations. Howie and Whelan (1961) used a totally different approach.
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D. VAN DYCK
They started from Darwin’s treatment of x-ray scattering and obtained a system of coupled first-order differential equations known as the system of Howie and Whelan, which is also equivalent to the eigenvalue method derived from Bethe’s dispersion equations. This approach turned out to be extremely valuable for the calculation of the contrast of electron microscopic images in two-beam or three-beam situations. In combination with the column approximation, it has rendered immense service to electron microscopic studies of crystal defects. At the same time, Tournarie (1960, 1961, 1962) started by Fourier transforming the Schrodinger equation using a semireciprocal notation, and, in a short and clear treatment, he derived a similar set of coupled first-order differential equations. However, none of these treatments was particularly suitable for the description and calculation of many-beam diffraction situations. The first (and for a long time only) method capable of treating many-beam diffraction was the well-known multislice method of Cowley and Moodie (1957), the starting point of which differed completely from the other existing theories. Originally, the concept of the multislice theory was inspired by previous work of Cowley and Moodie (1951) on Fourier images in optics. Here a crystal is considered as a succession of two-dimensional slices in the direction of the incident beam. Each slice acts as a pure phase grating. According to the laws of physical optics, the propagation of electrons from one slice to the next is governed by the Fresnel formula. The whole diffraction process can thus be described as a succession of phase-grating transmissions and propagations (see Section 111). The multislice theory gained considerable interest in the early 1970s when the Australian group (see Alpress et al., 1972) succeeded in implementing the multislice theory for computer simulation (Lynch and O’Keefe, 1972; OKeefe, 1973) of high-resolution images. The agreement between the famous experimental images of Iijima (complex oxides) and the first computer simulations was so striking that it could convince the most sceptical theoretician. However, because the multislice theory was in principle derived from a physico-optical formalism, its relationship to the other existing diffraction theories has been rather obscure for some time. Goodman and Moodie (1974) in an excellent paper elucidated the interrelationship between the various existing theories and the multislice theory, starting from Schrodinger’s equation. Independently and using the totally different but equivalent approach of the Feynman path integral formalism of quantum mechanics, Van Dyck (1975) showed the equivalence between the multislice formulas and the system of Howie and Whelan. Furthermore, the path integral concept provides much complementary insight into the diffraction process. The relationship between the multislice and the Darwin methods was established by Spence and Whelan (1975).
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It seems surprising that, since that time, only relatively little progress has been made in the field of many-beam diffraction calculations. The major improvement in multislice calculations was introduced by Ishizuka and Uyeda (1977) by making use of the fast Fourier transforms for the calculation of the convolution products that describe the electron propagation between the slices. Most of the current image simulation programs are still based on this method; other attempts at speeding up the calculations were proposed by Van Dyck (1979). A possible promising alternative, the real-space (RS) method, was first introduced by Van Dyck in 1980. In the beginning, this method was subject to criticism (Self, 1982; Self et al., 1983; Kilaas and Gronsky, 1983) and some computational difficulties needed to be resolved. Recently (Van Dyck and Coene, 1984; Coene and Van Dyck, 1984a,b), the last difficulties were resolved and the method seems to be competitive, especially when very complicated systems are involved. In the following we discuss the different many-beam diffraction theories with respect to their numerical performance.
B. The Multislice Method Although the multislice formula can be derived from quantum-mechanical principles (Goodman and Moodie, 1974; Van Dyck, 1975), we follow a simplified version of the more intuitive original optical approach (Cowley and Moodie, 1957). Part of the discussion has already been presented elsewhere (Van Dyck, 1980); a more rigorous treatment is given in Section V. Consider a plane wave, incident on a thin specimen foil and nearly perpendicular to the incident beam direction z. If the specimen is sufficiently thin, we can assume the electron to move approximately parallel to z so that the specimen acts as a pure phase object with transmission function
with
where E is accelerating voltage, A = l/k is the electron wavelength, and V,(x, y ) = f V(x, y , z) dz is the specimen potential projected along z. A thick specimen can now be subdivided into thin slices, perpendicular to the incident beam direction. The potential of each slice is projected into a plane which acts as a two-dimensional phase object. Each point (x, y ) of the exit plane of the first slice can be considered as a Huyghens source for a secondary spherical wave with amplitude q(x, y ) (Fig. 1).
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9
---E
E
E
E
FIG. 1. Schematic representation of the propagation effect of electrons between successive slices of thickness E. [From Van Dyck (1983).]
Now the amplitude 4(x’,y’) at the point x’, y’ of the next slice can be found by the superposition of all spherical waves of the first slice, i.e., by integration over x and y, yielding exp(2nikr) dx dy When Ix - x’ I 6 E and ly - y’( e E, with approximation can be used, i.e.,
E
the slice thickness, the Fresnel
so that
[(x
-
x‘)’
+ (y
-
y’)’]
CALCULATIONS IN HIGH-RESOLUTION ELECTRON MICROSCOPY
303
which, apart from constant factors, can be written as a convolution product: 4(x, y ) = exp[ioI/,(x, y ) ] * exp[ink(x2
+ y2)/&]
(3) The propagation through the vacuum gap from one slice to the next is thus described by a convolution product in which each point source of the previous slice contributes to the wave function in each point of the next slice. The motion of an electron through the whole specimen can now be described by an alteration of phase object transmissions (multiplications) and vacuum propagations (convolutions). In the limit of the slice thickness E tending to zero, this multislice expression converges to the exact solution of the nonrelativistic Schrodinger equation in the forward-scattering approximation. C. Computations For numerical computations, the wave function and the slices are sampled by a two-dimensional network of N closely spaced points. In order to represent the local fluctuations in the wave function and the phase object, the distances between adjacent points are typically on the order of 10 pm for heavy atoms to 50 pm for light atoms. The required computer memory is proportional to N . Typical slice thicknesses are on the order of 0.1-1 nm. For small N the computations can be performed at 32-bit precision. For large N (e.g., 4096), 48-bit precision is recommended so as to avoid the accumulation of rounding-off errors (Lynch, 1982).Hence, with a computer memory on the order of 64 kBytes, common in most mini- or microcomputers, a multislice calculation can be performed for up to about 1000 points, which corresponds approximately to a real-space area (or projected unit cell in the case of crystals) of about 0.4 x 0.4 nm for heavy materials to 2 x 2 nm for light materials. If a fast disk memory is used to store intermediate data, the number of points can eventually be increased by a factor of two or more. In order to handle larger areas within the same computer memory such as for the simulation of extended defects in crystals, the specimen can be subdivided into partly overlapping patches in which the calculations are done separately by using the method of periodic continuation after which the images are joined together (Olsen and Spence, 1981). Using this technique, the memory requirements are not stringent even for image simulation in larger areas. For most purposes, a 64-kByte computer proves satisfactory. However, the overlap areas between adjacent patches should be large enough to avoid scattering from adjacent patches into the image of the nonoverlap region, although this can depend on whether we want to know all the coordinates of all the atoms in the defect. Since the width of these overlap areas is proportional to the crystal thickness, this method cannot be extended to thicker specimens without increasing the computer memory. A much more stringent requirement is
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D. VAN DYCK
the computing time necessary for the calculation of Eq. (3). The major part is required for the calculation of the convolution products, with time proportional to N 2 , whereas the time for the direct products with the phase objects is proportional to N and, thus, is not dominant. In practice, most of the computations are carried out in reciprocal space by using the Fourier transform of Eq. (3). Here the sampling points are transformed into diffracted beams and the direct and convolution products are interchanged with the total calculation time remaining on the same order of magnitude. When the specimen is not periodic in the x, y-plane, such as for a crystal containing one or more defects, the diffraction pattern is continuous. By sampling the reciprocal space with discrete beams, an artificial periodicity is created in real space, which is termed the periodic continuation. This approximation is valid when the periodicity is large enough, i.e., when the defect is surrounded by enough perfect crystal area to avoid scattering from adjacent periods (unit cells). The area plays the same role as the overlap area used in the method of Olson and Spence, so that in this case also the validity of the approximation decreases with increasing crystal thickness. For most of the image simulations thus far, the reciprocal version of the multislice method has been used. Various programs, written by different authors (e.g., P. Fejes, K. Ishizuka, M. A. O’Keefe, and A. J. Skarnulis) are available. Typical calculation times on mainframe computers are of the order of 10 sec/slice for 1000 beams (17 sec on IBM 370,7 sec on Amdahl, 90 sec on CDC 3600, 20 sec on CDC 6600, and 3 sec on CDC 7600). These values should be multiplied by a factor between 10 and 100 for minicomputers (e.g., 460 sec on Prime 750). D. Use of the Fast Fourier Transform A very elegant way to increase the speed of the computations was introduced by Ishizuka and Uyeda (1977). Here each propagation in Eq. (3) is computed in reciprocal space and each phase-object transmission is computed in direct space. The links are made by a fast Fourier transform (FFT) with a calculation time proportional to N log, N . The number of beams needs to be a power of two in each direction and must be increased by a factor of two, because of aliasing effects (Ishizuka and Uyeda, 1977). However, the total calculation time can be reduced considerably especially for large N . Calculations have been performed for up to 256 x 256 beams. Typical calculation times for the FTT multislice are on the order of 1 sec/slice for 1000 beams on a large mainframe (2.8 sec on IBM 370,0.6 sec on Amdahl, 0.3 sec on CDC 7600) in comparison to 45 sec on H P 21 MX minicomputer.
CALCULATIONS IN HIGH-RESOLUTION ELECTRON MICROSCOPY
305
E . Special-Purpose Hardware The computation of the FFT multislice can be carried out almost entirely using vector arithmetic. Hence, it is very suitable for a minicomputer equipped with a dedicated hardware F F T processor or an array processor with sufficient dynamic range, by which the computation can be speeded up by another order of magnitude (even faster than a large mainframe). Typical calculation times are on the order of 0.1 sec/slice for 1000 beams C0.3 sec on Prime 750 with Floating point systems A P 120 (Skarnulis et al., 1981); 0.025 sec on P D P 1134 with Analogic A P 400; 0.1 sec on P D P 1134 with SKY MNK (estimated)]. As follows from Fig. 2 (Rez, 1980), the computing time for the A P 400 (as compared with the CDC 7600) is no longer dominated by the Fourier transforms but rather by the setup of the phase object and propagator, which are still calculated by the host computer. The calculation time for the phase object is proportional to the number of atoms times the number of beams and is on the order of 1-100 sec on a mainframe.
SLICES FIG. 2. Calculation time for a 1000-point multislice: solid line, PDP 1103 with AP 400; dotted line CDC 6600 mainframe; dashed line, CDC 7600 mainframe. [From Rez (1980).]
306
D. VAN DYCK
F . Discussion
At this point, a number of problems are not yet elucidated. (i) The image calculation technique is still laborious and expensive when complicated systems are involved. For example, the simulations computed for the single-split interstitial in gold, using periodic continuation, have taken 8 h on a UNIVAC 1100 (Fields & Cowley, 1978). Often the simulation of convergent beam electron diffraction (CBED) is unfeasible because of unreasonably large computer times. Although the image simulation technique can be compared to curve fitting in other scientific work, a least-squares refinement between computed images of trial structures and the corresponding experimental micrographs still belongs to science fiction. The future certainly lies in minicomputers (e.g., 32-bit) equipped with array processing facilities, which are becoming continually cheaper. Probably, dedicated image processing systems can also be used efficiently for image simulation. However, any speeding up of the computational algorithms, including the setting-up time for the phase object, would still be very welcome. (ii) In the propagator convolution, an electron is allowed to travel from any point x on one slice to any point x’ on the next slice (Fig. 1). The number of paths N 2 dominates the computing time. However, for most of these paths the Fresnel approximation ( x - x’( 4 E, ( y - y’( 4 E is violated, so that the form of the propagator in Eq. (3) is, in principle, not valid. According to Berry (1971), the effect of these paths is canceled by their rapidly oscillating phases and should be, in principle, unimportant. However, this statement is not necessarily true for numerical computations. For instance, suppose the object is sampled with a square mesh of size 6 x 6 nm. If the slice thickness E is chosen so that, by accident, it obeys the relation k6’/& = 2n with n integer, then exp[ink(x2 y ’ ) / ~ ]= 1 for all paths in the propagator equation [Eq. (3)], i.e., every point (x, y) contributes to every point (x’,y’) on the next slice with the same weight (unity). Hence, the wave function in the next slice becomes constant and all information of the scattering in previous slices is lost. This is an artificial resonance effect, comparable to what is known in multislice theory as pseudo-upper-layer line reinforcement. The choice of the mesh size and slice thickness thus requires special attention. In practical multislice calculations, we use the requirement that ks2/E < 1, which can be derived in different ways. It is related to Heisenberg’s inequality since l/k6’ is related to the excitation error of the outermost Bragg beam, which in fact is the uncertainty on the wave vector, (see Fig. 5), and E is the uncertainty in the position of the scattering. However, the effect of the Fresnel approximation on the propagator as a function of mesh size and slice thickness should be further elucidated. Furthermore, if the propagator paths are restricted to a small
+
CALCULATIONS IN HIGH-RESOLUTION ELECTRON MICROSCOPY
307
cone (only in adjacent points), which is a reasonable assumption for fast electrons, then the artificial resonance effects are avoided and at the same time the computing time might be reduced to a value proportional to N . (iii) By projecting the slice potential onto a two-dimensional plane, the information about the atom coordinates along z is lost. When, in the case of a crystal, the slice thickness is a multiple of the unit cell parameter c along z, the potential is in fact considered as independent of z throughout the whole crystal. In reciprocal space, here only the zeroth-order Laue zone is excited. In order to obtain information about the upper-layer lines (ULL) or higher-order Laue zones (HOLZ), the slices must be made much smaller and the computation times consequently become larger, e.g., if the nth layer lines are important, the potential should be sampled at intervals Az = c/2n. However, in practice it is sometimes necessary to take the slice thickness E much less than Az in order to obtain a convergent multislice calculation. A recent comparison between multislice and Bloch wave calculations by Shannon (1979) shows that in the case of heavy atoms, such as gold, slice thicknesses on the order of 0.03 nm, which is less than the atomic diameter, must be used. If atom sectioning proves necessary, the multislice method is not very practical. In that case, it should be worth searching for a slice method that accounts for ULL interactions to such an extent that atom slicing can be avoided. (iv) Another problem that has not been fully elucidated is how to know that the slice thickness E and the interval 6 are adequately chosen. In principle we could reduce E and 6 until satisfactory convergence is obtained. Frequently, the normalization of the total diffracted intensity is used as a test for normalization; however, c and 6 may not be considered independently. Indeed, normalization can be obtained for any 6 (i.e., any number of beams) in the limit for E tending to zero (Anstis, 1977) and for any E in the limit for 6 tending to zero (see Section VI). Shannon (1 979) has proposed a rather complicated convergence criterion in which E and 6 are coupled. Another possibility is that E and 6 be coupled by the relation derived in point (ii). However, further investigation of the problem is required. (v) In the derivation of the multislice formulas, forward scattering of the high-energy electrons is assumed. However, thus far no convincing proof has been given for the validity of this approximation, especially when thicker foils are involved. (vi) With the current many-beam diffraction theories, it is still impossible to include inelastically scattered electrons in the calculations. However, since the mean-free-path for inelastic scattering is on the order of tens of nanometers, this omission might put serious restrictions on the maximum foil thickness in the calculations.
308
D. VAN DYCK
(vii) The effect of the thermal motion of the atoms is usually included by multiplying the atomic scattering factors with appropriate Debye-Waller factors. However, this technique stems from x-ray diffraction in which the kinematically scattered x-ray “sees” a spatially averaged atom. However, at present it is not clear whether this approximation still holds for dynamic electron scattering. In the following section we treat the diffraction problem in a more systematic way in order to search for an optimal computational algorithm and to elucidate some of these problems. Iv. SYSTEMATIC APPROACH TO THE ELECTRON-DIFFRACTION PROBLEM A . General 7heory’
For a general treatment of electron diffraction, we start from the timeindependent Schrodinger equation. Contrary to the practice of describing diffraction in reciprocal space, we treat the problem in real space. This approach will prove to have certain advantages (Van Dyck, 1980). Consider an electron with wave vector k and energy on the order of 100 keV-1 MeV, which is incident on a thin specimen foil (Fig. 3) with a local potential V(r). The wave function $(r) of the electron in the specimen is then governed by At,h(r) 4n2k2$(r) V(r)$(r) = 0 (4) with V(r) = (2rne/ti2)V(r) (5)
+
+
and in which rn and k are the relativistic values for the electron mass and wave vector, respectively (Fujiwara, 1961). The incident electron can be represented by a plane wave $(r) = exp(2nik * r) The problem now consists of computing the wave function of the electron from Eq. (4) at the exit plane of the foil (Fig. 3). For high-energy electrons ( 2 100 keV), the influence of the foil can be considered as a perturbation, i.e., V(r) 4 4n2k2,so that it is convenient to rewrite the wave function as a modulated plane wave $(r) = exp(2nik * r)4(r) (6) Van Dyck (1979,1980,1983); Van Dyck and Coene (1984).
CALCULATIONS IN HIGH-RESOLUTION ELECTRON MICROSCOPY
0
309
z
FIG.3. Geometry of the problem.
The modulation function &r), which is in fact the real-space representation of a depth-dependent Bloch wave, will vary relatively smoothly as a function of the crystal depth. Substitution of Eq. ( 6 ) into Eq. ( 4 ) yields A+(r)
+ 471ik - V+(r) + V(r)$(r)
=0
(7)
When the incident beam is nearly parallel to the surface normal e , , as is the case in most of the experimental situations, the normal component k, of the wave vector is large compared with the parallel component kx,. Hence, it is appropriate to separate Eq. (1) into
where V,, and Ax), are, respectively, the gradient and Laplacian operators in the coordinates, x, y , which are chosen in the foil plane. This equivalent form of Schrodinger's equation requires two boundary conditions for the wave function both at the entrance ( z = 0) and exit ( z = z ) planes of the foil and appears difficult to solve when the specimen is not a perfect and simple crystal. However, by using the high-energy approximation outlined in Subsection IV.B, this equation can be transformed into a firstorder differential equation that requires only one boundary condition at the entrance face.
B. High-Energy Approximation In high-energy electron diffraction where the specimen potential is small compared with the energy of the incident electrons, it is generally assumed that the z component of wave vector k, is large and that the wave function
310
D. VAN DYCK
4(r) varies smoothly with z so that a 2 4 / d z 2 can be neglected with respect to the term 4nik, d 4 / a z resulting in a first-order differential equation
or in shorthand notation W r ) / d z = ACA
+ V(r)l4(r)
(10)
with il= i/4nkZ A = Axy 4nikXy* V,,
+
This approximation is sometimes called the forward-scattering approximation. To the knowledge of the author, Schiske (1950) in his quantum-mechanical treatment of electron motion in magnetic lenses was the first to approximate the time-dependent Schrodinger equation by a parabolic equation including only the first derivative with regard to z, which he termed “paraxial Schrodinger equation.” Equation (10) can now be integrated readily from the entrance face to the exit face. It is the basic equation for the dynamic diffraction calculations. As will be shown later, nearly all existing diffraction formulations are implicitly or explicitly based on this equation. In case of normal incidence Eq. (10) is equivalent with the time-dependent Schrodinger equation in two dimensions (Van Dyck, 1980; Gratias and Portier, 1983). Hence, methods for solving Eq. (10) can probably be used for the solution of genera1 quantum-mechanical problems (see Section V1I.G). C. Validity ofthe High-Energy Approximation
Although the high-energy approximation is intuitively sound, we give, for completeness, a more rigorous proof of its validity, which is not essential for the remainder of this work and may be skipped by the reader. In the case of normal incidence, we represent the wave function as the sum of two particular solutions of the general differential equation [Eq. (411, one corresponding to forward-scattered electrons and one corresponding to backscattered electrons = e2nikz 4(,.) + ,-Znikz W) (1 1) 4(r) obeys the differential equation [Eq. (8)] in shorthand notation
4’ = A(A + V ) 4 + $6”
(12)
CALCULATIONS IN HIGH-RESOLUTION ELECTRON MICROSCOPY
31 1
and similarly for 8,
e = +A + v)e
-
AW
with
2 = il4nk Since the high-energy approximation is based on the idea that the wave vector k is large (small wavelength), it is appropriate to expand Eqs. (12) and (13) in powers of the wavelength. If we successively substitute 4‘ [left-hand side of Eq. (12)] into [right-hand side of Eq. (12)], we obtain
4’ = [A(A + v) + 22v
+ 23(v” + (A + v)*)+ o(n4)14
(15)
and similarly for Eq. (1 3),
If the potential varies slowly enough over one wavelength distance, i.e., 23V”
22V’ < 2v
then it may be assumed that the series in powers of A can be truncated after a few terms, and the second-order differential equations [Eqs. (1 5) and (16)] can be approximated by first-order differential equations. If Eq. (15) is truncated after the first-order term, the usual high-energy equation is obtained. In the vacuum space at the entrance side of the crystal (Fig. 2), the wave function can be written as tj
= exp(2xikz)
+ exp( -2nikz)
R(r)
(17)
where exp( -2nikz)R(r) represents the reflected waves. Here R(r) obeys Eq. (1 6) in which V(r) = 0, i.e., R‘ = -AAR
+ o(23)
(18)
This approximation is equivalent to the parabolic approximation for the Ewald sphere. In the vacuum space at the exit side of the crystal, the wave function takes the form $(r)
= exp(2nikz)
T(r)
corresponding to the transmitted electrons. By using the same approximation, we similarly obtain for T(r):
T’ = AAT
+ 0(A3)
(19)
312
D. VAN DYCK
From the continuity requirements of the wave function and its derivative at the boundary planes z = 0 and z = d, we obtain at z = 0
+ R(O)= +(o) + e(o)
(20)
+ R'(0) = 2nik[4(O) - O(O)] + @(O) + W(0)
(21)
1 2nik[1
-
R(O)]
atz=d
+
exp(2nikd) 4(d) exp( -2nikd) O(d) = exp(2nikd) T(d) x 2nik[exp(2nikd) 4(d) - exp( - 2nikd) O(d)] exp(2nikd) @(d) + exp( - 27cikd)W(d) = 2nik exp(2nikd) T(d) exp(2nikd) T'(d)
+
+
(22)
(23)
By eliminating R(0) from Eqs. (20) and (21) and using Eqs. (12), (13), (14), and (18), we obtain
+
1 - 4(0) = - 2A2A4(0) - 'A V 4 ( 0 ) A2 VO(0)
+ O(A3)
(24)
and similarly from Eqs. (22) and (23), using Eqs. (12), (13), (14), and (19), we obtain exp(-2nikd) O(d) = -A2Vexp(2nikd) 4(d) + A2(2A + V ) x exp( - 2nikd) O(d) O(A3)
+
(25)
so that finally,
4(0) = 1 + A ~ V ( O+) o(n3) and exp( - 2nikd) e(d) = - A2V(d) exp(2nikd) 4(d) + 0(i3)
(26)
i.e., the amplitude of the backscattered electron wave function is small and on the order of A2V. The wave function at the exit plane of the crystal is then [Eqs. (22) and (26)], $(d) = [I - A2V(d)
+ o(n3)] exp(2nikd) 4(d)
(27)
Similar to the results obtained in reciprocal space (Van Dyck, 1976), the realspace expression [Eq. (27)] shows two differences when compared with the high-energy wave function as obtained from Eq. (10). a. Amplitude Correction - A2 V ( d )
This correction stems from the backscattered electrons. It is dependent on the value of the potential at the boundary plane and its effect is negligible.
CALCULATIONS I N HIGH-RESOLUTION ELECTRON MICROSCOPY
313
b. Corrected Differential Equation The first-order differential equation [Eq. (15 ) ] , which determines $(d), is slightly different from the high-energy differential equation [Eq. (lo)] in which the potential V is replaced by V
+ AV’ + A2V” + A2(A + V)’ + ...
(29)
As follows from Eq. (29), the first-order correction on Vis of the form AV‘ and is, thus, of the same magnitude as if the whole foil were shifted in the z direction over the very small distance I A 1, which can be assumed to be negligible. The same argument holds for the term 1’V”. O n the other hand, the secondorder term A2(A + V)’ accumulates on integration and might become appreciable for thicker foils. In reciprocal space this correction term was estimated to become important for foils on the order of hundreds of nanometers thick. Although a careful analysis of this correction term in real space requires a separate investigation, it can be concluded that the high-energy equation [Eq. (lo)] will be sufficiently accurate for the usual dynamic calculations. This equation will be used in the remainder of this chapter.
D. Discussion Equation (10) is the basic equation from which most of the current dynamic diffraction formulations can be deduced. The operator A
a2
=
ax
+
a2 .-
ay2
+ h i k , a + 47ciky a ~
ax
-
aY
(in Cartesian coordinates) describes the propagation of the electron in free space and V(r) describes the scattering of the electron at the potential. Notice that AV(r> # V(r)A, i.e., the operators do not commute. The physical meaning of Eq. (10) becomes clear if we study both parts of the right-hand side of the equation separately, which correspond to two simple but physically different processes. If the propagation term is neglected, the basic equation is reduced to
which can be solved readily
3 14
D. VAN DYCK
and from Eq. (6), we have for the wave function $(r)
= exp(2nik * r)
exp( A j I V ( r ) d r )
This is the well-known phase-grating expression in which the projected crystal potential acts as a phase grating for the incident plane wave and which has also been introduced in the physico-optical approach [Eq. (2)]. If the scattering term is neglected, the basic equation becomes a4(r)/az = AA4(r) (33) This is a diffusion type of differential equation in which time t is replaced by foil depth z , thus expressing the diffusion of the electron on propagating through the foil. This physical process can be illustrated clearly in the one-dimensional case, assuming normal incidence (k, = 0, k, = k), Eq. (33) becomes
This equation describes a kind of one-dimensional Brownian motion (apart from the imaginary factor i). Imagine the foil subdivided into thin slices of equal thickness E parallel to the entrance plane (Fig. 4). The wave function in a point x of the slice at depth z E can be calculated from
+
4(x, z
+
E) =
4(x, z )
+ 84 aZ (x, + O2(E) + ... E -
2)
The second-order derivative term in x can be calculated from the wave function in the adjacent points x + 6, x - 6 by using 2) ax2
- +(x + 6, Z )
+ 4(x - 8, Z ) - ~ h2
( xZ ),
+ OZ(6) +
* * '
FIG.4. Schematic representation of the diffusion of the electrons between successive slices. [From Van Dyck (1980).]
CALCULATIONS IN HIGH-RESOLUTION ELECTRON MICROSCOPY
315
so that finally,
ie _______
-t 47ck6’
[qqx
+ 6, z) + 4(x - 6, z)] + OZ(E,6) + ... +
The wave function at a point x in slice z E can thus be calculated by using a linear combination of the wave function at three adjacent points x 6, x, and x - 6 of the previous slice. The propagation of the electron from slice to slice can thus be described as a random walk process. Hence, if an electron starts from the origin, in the entrance plane, and diffuses through a large number of thin slices, its wave function at the exit plane is given by the Gaussian distribution function
+
as indicated schematically in Fig. 4. The total wave function at the exit plane is then obtained by adding up the probability amplitudes @(x‘,0) for starting at points x’,each multiplied by its properly shifted Gaussian distribution, yielding
or in convolution notation
$(x, z) =
exp(i7ckx2/z)* $(x, 0)
It is clear that the solution of the general diffusion equation Eq. (33), which can be written formally as
can be constructed in the same way. Hence, the phase grating as well as the Fresnel expression for the propagation used in the physico-optical approach of Subsection 1II.B is established on a more rigorous quantum-mechanical basis. Moreover, it is now clear that phase grating and propagator are intimately coupled in one differential equation [Eq. (lo)] for which the multislice expression [Eq. (3)] is only an approximate solution.
316
D. VAN DYCK
v. STUDY OF
EXISTINGMANY-BEAM FORMULATIONS DIFFRACTION A. Diferential Equations
1. Real-Space Representation The starting point from which most of the current dynamic diffraction formulations can be derived is the high-energy equation (10)
and
A
a2
=
~
+
ax2
+ 4nik, a + 47cik,
a2 -
ay2
d
-
-
ax
aY
(37)
We now search systematically for the most efficient formulations and computational algorithms for the solution of Eq. (35). It is interesting to note that Eq. (35) is also equivalent to the time-dependent Schrodinger equation in which the time t is replaced by the depth z in the specimen (Van Dyck, 1980; Gratias and Portier, 1983). Effective computer algorithms might probably also find applications for the solution of more general time-dependent quantum-mechanical problems. This aspect will be discussed in Subsection V1I.G.
2. Reciprocal-Space Representation If the potential V(r) is periodic in three dimensions, such as is the case in perfect crystals, it is appropriate to expand V(r) in its Fourier series V(r)
=
1 5 exp[hig - r]
(38)
8
where g represents the vectors of the reciprocal lattice. By writing
m)= 1 4,w exp(2nig
*
r>
8
and substituting in Eq. (35), we obtain, after identification of the terms,
(39)
CALCULATIONS IN HIGH-RESOLUTION ELECTRON MICROSCOPY
3 17
3
I
crystal foil
I
I
sphere
FIG.5. Ewald sphere and extinction distance.
where 4,(z) is the amplitude of the diffracted beam g at depth z and s, is the excitation error, approximately equal to the distance between the reciprocal point g and the Ewald sphere, measured along e, (Fig. 5). Equation (40) is the well-known system of coupled first-order differential equations of Howie and Whelan (1961), originally derived for a distorted crystal. Denoting by @(z) the column vector of diffraction amplitudes $&z), T, the diagonal matrix with elements
and V, the matrix with elements Vg,g,= inVg-g’
Eq. (40) can be written in matrix notation as
dW)ldz
=
CT +
~ldm
(43)
or
d+(z)ldz
= Sd4Z)
(44)
with S=T+V
(45)
where S is called the dynamic matrix or sometimes the scattering matrix; T can be considered as the Fourier transform of A, which represents the propagation of the electron in free space, i.e., the electron remains in the beam g but receives a phase shift. V can be considered as the Fourier transform of V(r),
318
D. VAN DYCK
and Vg,grrepresents the matrix element (probability amplitude) for scattering from beam g’ into g. It is interesting to note that, henceforth, each step in the description in real space has its analog in reciprocal space and vice versa, which can be obtained simply by Fourier transforming; direct products and convolution products are then interchanged. This duality can provide much complementary insight into the diffraction process. 3. Mixed Representation In some cases, e.g., when the potential of the foil is not periodic along the z axis, it can be more convenient to expand V(r) and #(g) in a two-dimensional instead of three-dimensional Fourier series V(r)
=
1 VG(z)exp(2niG
*
R)
G
and #(r)
=
c &(z) exp(2niG - R )
(47)
G
+
where r = R ze, and g = G + gzez, i.e., the reciprocal vector G belongs to the [OOl] zone. Note that the Fourier coefficients VG(z)of the potential are now dependent on z so as to include the influence of the higher-order Laue zones (upper layer lines). Hence, no generality is lost as compared to the other representation. Substitution of Eqs. (46) and (47) in the high-energy equation now results in
where sG is the excitation error as indicated in Fig. 5. This is the system of coupled differential equations in mixed representation as originally derived by Tournarie (1960, 1961, 1962). It can also be put into matrix form, similar to the foregoing,
d#(z)ldz = CT + V(Z)l#(Z) (49) in which the matrix V ( z )is now dependent on z. In case only the [OOl] Laue zone is excited, i.e., only the projected potential is considered, the matrix Vis independent of z and Eq. (49) becomes d+(z)ldz = (7-+ ~ > + ( z )
(50)
which is similar to Eq. (43). Another kind of mixed representation has been introduced by Howie and Basinski (1968) for the treatment of electron dif-
CALCULATIONS IN HIGH-RESOLUTION ELECTRON MICROSCOPY
3 19
fraction in a simple crystal containing a lattice defect. If the crystal is not drastically deformed by the defect, it is appropriate to expand the crystal potential and the wave function in a Fourier series as for the perfect crystal, hereby allowing the Fourier coefficient to vary with position (x, y ) so as to account for the influence of the defect. Hence, a system similar to Eq. (40) is obtained, in which partial derivatives with respect to x and y occur. Spence (1978) treated this problem by using a kinematic approximation for the scattering at the defect, thereby obtaining a drastic gain in computing time. Neither of these methods will be discussed here.
B. Matrix Methods In principle, the solution of Eq. (43) or (44) can formally be written in matrix form as
with '1
1
OI 4(0) = 0
I] 0 and where the exponential of a matrix is defined in the usual sense by its power series. This solution was first introduced by Sturkey in the early 1950s (Sturkey, 1962). Sturkey formulated the diffraction process as a unitary transformation that rotates the initial vector +(O) in the n-dimensional diffraction space to the final vector @(z).If absorption is neglected, the scattering matrix S is imaginary and, from simple matrix theory, it can be shown that
l4(Z>l2
= &)4*(4
=
6(0)4*(0) = l4(0)l2 = 1
i.e., the total diffracted intensity remains constant. Since no intensity is lost, this formulation can be termed self consistent. It is very important to note that, since the total intensity is maintained, irrespective of the number of beams g considered (even in a two-beam case), the normalization test cannot be used as a criterion for determining the number of diffracted beams that contribute to the diffraction process. Another matrix method consists of diagonalizing the scattering matrix S by means of a unitary transformation matrix U , i.e.,
U-'SU=D
or
S = UDU-'
(52)
320
D. VAN DYCK
where D is a diagonal matrix with elements yn. The columns of the matrix U are the eigenvectors On of S , where y, are the corresponding eigenvalues, i.e.,
and by using Eq. (52), we obtain from Eq. (44) +(z)
= exp(Sz)+(O) =
U exp(Dz)U-'+(O)
(54)
with eDz a diagonal matrix with diagonal elements eYnz.Equation (53) is the eigenvalue problem that can be derived directly from Bethe's dispersion equation in the forward-scattering limit (Howie, 1978). It has also been used by Sturkey (1957) and Niehrs and Wagner (1955), whereas Eq. (54) has been introduced by Fujimoto (1959). The eigenvalue method is also self consistent in the sense that the total diffracted intensity remains unity (no absorption) irrespective of the number of beams. The eigenvalue method has the advantage that once the eigenvalues y, have been calculated, b(z) can be obtained directly for all thicknesses z and for all +(O). Furthermore, in situations where only a few beams are excited, the method with concepts such as Bloch waves and dispersion surface offers much physical insight into phenomena such as the critical voltage effect (e.g., Gevers et al., 1974 and many others). With modern diagonalization programs, the eigenvalue problem is still tractable when the number N of diffracted beams does not exceed a few hundred. Eigenvalue methods can be used for image simulation in simple perfect crystals, possibly matched with slice calculations for the crystal parts containing defects. Since the memory requirements for matrix methods is on the order of N 2 and the calculation time at least on the order of N 3 , the matrix methods become unmanageable when the number of beams is very large, as is usually the case for image simulations in more complex systems.
C . The Iterative Method When the number of diffracted beams becomes so large that full matrix expressions cannot be handled, we can expand Eq. (51) by using only one vector at the time, i.e., starting with +(O) and ending with +(z). An example of such a method is the iterative method (Van Dyck, 1978a,b, 1979). It is constructed from the definition of the exponential matrix exp[(T
+ l/)z] = 1 ( T +n!v)nzn OU
n=O
(55)
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321
By a suitable iterative procedure, it can be computed in vector form b y expanding
where Onis defined from Eq. (55) and (56) as
with
From Eq. (57), we have the recurrence relation
which can be computed by means of three memory vectors [remember that multiplication with V, which is a convolution, can always be speeded up using fast Fourier transforms (FFTs)]. Since the order in powers of z is, in principle, unlimited, this method is by far the best when very accurate results are required, since it takes only two FFT per order. Furthermore, it is normalized, so that the total intensity of the diffracted beams equals unity for n -+ co.This can serve as a criterion for the accuracy but is worthless as an estimation of the number of beams required. This method has been used successfully for some years in o u r laboratory. Disadvantages of the method are It is only applicable to perfect crystal parts. (2) The method can be applied for one crystal thickness z at a time. (3) Especially when thick crystals and large excitation errors are involved, the values of the elements of @ can rise seriously before converging to their limiting values, thus requiring a larger number of significant digits in the calculation. (1)
D. Direct Integrating Methods
In principle, the differential equations [Eq. (35), (43), (49), or (SO)] can be integrated directly from the entrance plane to the exit plane of the specimen. For this, standard routines such as Runge-Kutta or predictor-corrector methods can be used. Specifically for this purpose, a special Runge-Kutta method has been constructed with minimal computer memory requirements (Van Dyck, 1978a; Van Dyck et al., 1980).
322
D. VAN DYCK
If the excitation errors in the diagonal matrix Tare large, so as to include diffracted beams with a large Bragg angle, it is appropriate to use a kind of perturbation technique by replacing 4(z)
= exp(W
(59)
and substituting in Eq. (43), yielding dO(z)/dz
=
Q(z) O(z)
with Q(z) = exp( - Tz) V exp( Tz) The new differential equation allows larger integration steps to be used and the integration proceeds faster. However, it can be shown (Van Dyck, 1979) that Runge-Kutta methods are somewhat slower and more memory-consuming than the slice methods of the same order. Hence, slice methods are still preferred. E . Slice methods
The best candidates for image simulation are the slice methods that, starting from the boundary condition at the entrance plane, integrate Eq. (35), (43), or (49) or expand Eq. (51) step by step throughout the crystal, thereby keeping the memory requirement proportional to N (the number of beams or sampling points). The accuracy and/or speed of slice methods can best be compared to the expansion of the exact solution in one slice in powers of A and V(or Tand V ) , which are proportional to the wavelength (A = l/k) and converge rapidly for fast electrons (Van Dyck, 1980). Several slice methods are discussed in the literature. a. Multislice Method Here the solution within one slice is separated into scattering and propagation
exp(Vz) exp(Tz) = exp[(T
+ V)z] + &VT - TV)’ + ...
(62)
if the projection plane is the front plane of the slice or
exp(Tz) exp(Vz) = exp[(T
+ V)z] + &TI/
-
+
~ ) z ’
if the projection plane is the back plane of the slice. Both slice methods are, thus, expansions of first order in z. Since they do not contain ULL effects within one slice, it is not clear whether the slice must be made sufficiently thin to take into account these effects.
CALCULATIONS IN HIGH-RESOLUTION ELECTRON MICROSCOPY
6. Improved Multislice Method metrical center of the slice, i.e.,
323
By placing the projection plane in the geo-
exp($Tz) exp( Vz) exp(+Tz) = exp[(T
+ V)z] + 03(Tz, Vz)
exp()Vz) exp(Tz) exp()Vz)
+ V)z] + 03(Tz, Vz)
or = exp[(T
(63)
the multislice formulas are transformed into expansions of second order, just as the first-order rectangular integration rule can be transformed into the second-order trapezium integration rule by starting and ending with one-half of an integration step (Van Dyck, 1978b). This improvement can be easily introduced into the existing multislice programs. However, ULL effects within the slice are not included. c. Third-Order Slice Method Van Dyck (1978b, 1979) introduced a third-order slice method that consists of an alternation of functions of T and V, suitable for the use of fast Fourier transforms, i.e.,
exp($aTz) exp(aVz) exp($Tz) exp(bVz) exp(3bTz) = exp(T + V)z 04(Tz, Vz)
+
(64)
with
a
= (3
f i,,/3)/6
and
b
=
1
-
a
The iterative method [Eq. (57)] and the direct integrating method [Eq. (60)] can also be considered as higher-order slice methods. Higher-order methods provide a higher precision for the same slice thickness; they increase the accuracy for the same slice thickness, but they do not increase the speed when the accuracy of the second-order method is sufficient. However, since the differences between the first-order and the second-order multislice is simply equivalent to the shift of the whole crystal by half a slice, which can hardly be observed experimentally, it can be expected that the accuracy of the second-order method will be sufficient for most experimental situations. Higher-order methods also allow a larger slice thickness to be used for obtaining the same precision. However, this benefit is only hypothetical when the ULLs within the slice are important. It will be better to search for a second-order slice method that accounts for ULL interactions within the slices.
324
D. VAN DYCK
F. Discussion The diffraction process can be described adequately by one of the four equivalent equations
In the projection approximation
a&)/az
= (T
+ V)&)
(50)
In principle all the methods discussed earlier are mathematically equivalent in the sense that in the proper limits (e.g., infinite number of beams, zero slice thickness, etc.), they all yield the same “exact” solution. The main difference between these methods lies in the case of treating many beams with respect to computing time and memory. At present the optimal speed/accuracy compromise can be obtained by the slice methods. With slice methods, the wave function can be obtained for a number of foil thicknesses in the course of one calculation, which is advantageous for image simulations. When ULL effects are negligible, the multislice method is sufficiently accurate. However, for a number of possible applications, the present computational algorithms are still too slow and any speeding up would be very welcome. Furthermore, a number of problems and questions raised in Section 111 are still open. In the next section we follow a rigorous but more systematic approach to finding a slice method and a computational algorithm that is optimized with respect to speed/accuracy. We also try to answer some of the open questions. VI. A NEW FORMULATION: THE REAL-SPACE METHOD A . Principles’
The power of a particular method for solving a quantum-mechanical problem is strongly dependent on the choice of a suitable basis. For instance, when electron diffraction is described in a simple perfect crystal, the crystal
* Van Dyck (1980, 19083); Van Dyck and Coene (1984); Coene and Van Dyck (1984a).
CALCULATIONS IN HIGH-RESOLUTION ELECTRON MICROSCOPY
325
potential, as well as the wave function of the electron, can be described by using a small number of Fourier terms (plane waves), in which case it is advantageous to describe the diffraction process in reciprocal space. However, this benefit disappears when large and complicated (artificial) unit cells are involved or when the periodicity is lost, as is the case for diffraction at extended crystal defects. By using the fast Fourier transform (see Section 111) in the multislice programs, the phase-grating transmission is described in real space, while the propagation is still described in reciprocal space. However, the use of the fast Fourier technique has some disadvantages, e.g., the aliasing effect, the requirement that the number of sampling points in each direction needs to be a power of 2, the calculation time ( N log, N ) and the memory requirements. Hence, if complicated structures are involved, it can be more attractive to describe the diffraction process entirely in real space, where the forward-scattering character of the electrons during propagation can be exploited to speed up the calculations and reduce the memory requirements. For this purpose we try to construct a slice method as a solution of Eq. (35) in real space. Since, for high-energy electrons, A is a small quantity, it seems logical to expand the solution of Eq. (35) in one slice of thickness E in powers which converges faster than the Taylor expansion in powers of E. of i, For this purpose, we start from the integral equation form of Eq. (35):
4(x, Y, E ) = 4 ( x , Y , 0) + 2
s:
[A
+ V(x, Y , z > l $ ( x , Y , z ) d z
(65)
It is interesting to note that Eq. (65) is, in principle, equivalent to the Green’s function approach in the forward-scattering limit. This approach has been used by Fujiwara (1959) and Ishizuka and Uyeda (1977). By using the stationary-phase approximation, the latter authors have rederived the multislice expression. Expanding Eq. (65) in powers of A by iteration yields
[A
+ V ( x , Y , 211 dz
+ ,I2 j i [ A + V ( X ,y , z ) ] dz
By calling
+ V ( x , y , z’)] dz‘
326
D. VAN DYCK
the projected potential of the slice at the point (x,y) and
‘zV(x, y , z ) dz
=qx,y) =
z
the center of potential of the slice at the point ( x , y), and using the identities
1:
j I V ( x , y , z ) dz
V ( x ,y , z’) dz‘
1). j;qx,
y, z’) dz’
and writing in shorthand 4(x, y, E ) +(E)
=
{ + LAE + 1
+ A’A(E
-
=
1 Vi 2
=-
= ( E - T)V,
4(~),we have explicitly from Eq. (66)
(AAE)Z l2V; +nv,+- 2 2
~
F)V,
+ A2FVpA + 03(1) + .-.}4(0)
Since A and V are noncommuting, it is advantageous to write the expansion as an alternative of a minimal number of functions of Vand A
These functions can be determined by expanding them as powers of A and V and identifying Eq. (70) with Eq. (69). The higher-order powers that are not uniquely determined will be chosen so as to present the functions in closed form. The simplest expression that expands Eq. (69) up to second order is the following:
4 ( ~= ) exp[fAVp(l + d)] exp(lAe) exp[$AV,(l
- d)]4(0)
(71)
where d
= d(x, Y ) =
c.0,Y)
-
(E/2)1/(&/2)
is the “potential eccentricity.” It is the relative deviation between the center of potential Z at the point (x,y) and the center of the slice 42. The wave function at the exit plane of the foil can then be calculated by repeated application of Eq. (71) in successive slices. This will be the basis for the real-space method. As explained in Section 111, the propagation part exp(lAe) can be calculated in reciprocal space and the phase-grating product in real space, both linked by fast Fourier transforms. However, it is possible to set up a new computational algorithm in which all the calculations are performed in real space.
CALCULATIONS IN HIGH-RESOLUTION ELECTRON MICROSCOPY
327
B. Compurison with Other Slice Methods In comparison with the usual multislice (MS) expression [Eq. (62)], the new expression [Eq. (7 l)] shows two corrections (a) Upper layer line effects, which account for the z dependence of the potential, are partly introduced by a kind of dipole expansion of the slice potential, using the concept of potential eccentricity 6. (A brief discussion will be given in Section V1I.F.) In the MS method, the slice potential is projected along z (the so-called projection approximation), which is a kind of monopole expansion that does not take into account the ULL effects within the slice. In order to account for ULL effects, the slices must be made sufficiently thin. However, in two cases the correction term 6 disappears so that the original MS expression becomes a second-order method, apart from the first and last slice (i) when the center of the slice can be chosen in the center of the potential; (ii) in the case of a perfect crystal, where the c parameter is small enough to be taken as slice thickness. Hence, the correction term - 6 of one slice is canceled by the term + 6 of the next slice. In this case the ULL effect is of the same magnitude as the scattering of one slice of unit cell thickness. (b) The first-order MS expression is transformed into second order by simply starting and ending with one-half of a phase grating, which is analogous to the transformation of the first-order rectangular integration rule into the second-order trapezium rule by shifting over one-half an integration step. In the case of a perfect crystal for which the periodicity along the beam direction is small enough to be taken as slice thickness, the second-order expression yields the same results as the multislice expression, apart from the first and last slice. This difference is expected to be large when thicker slices are used, as in the case of light scattering materials (e.g., oxides). Figure 6 shows a comparison between the accuracy obtained with the MS method and the new second-order method in the case of a light-scattering material (SiF,). From this it is clear that for medium foil thickness (below 10 nm) and when very accurate results are required, as is the case for the simulation of dark-field or weak-beam images, the accuracy of the secondorder method is one order of magnitude better for the same slice thickness. However, when normal accuracy is sufficient or when high-scattering materials, such as Au alloys, are involved for which the slice thickness needs to be much smaller, the accuracy of the second-order method is not much different from that of the first-order multislice. In cases where the unit cell periodicity along the beam direction is larger (e.g., > 1 nm) or in case of disorder, the
328
D. VAN DYCK
a
10
13
18
24 g M (nm-’)
270
I
540
I
1080
FIG.6. Accuracy Sz versus foil thickness for different values of the slice thickness E and number of beams. Computations are performed for SiF,[OOl] using 200-keV electrons. Full horizontal scale is 30 nm. Dashed line represents second-order method; solid line represents firstorder multislice method. For a definition of S2see Section VI.D.2. [From Van Dyck and Coene (1984).]
second-order expression may be much more suitable, but this effect is still under investigation. When ULL effects are neglected, i.e., the potential is assumed to be independent of z, the solution of the basic equation Eq. (35) can be written in closed form as
dJ(d= expCWA + V(X, Y)ldJ(O)
CALCULATIONS IN HIGH-RESOLUTION ELECTRON MICROSCOPY
329
or, in shorthand notation,
4(4 = expCWA + V14(0)
(72) All slice methods are based on expansions of Eq. (72) within one slice. An interesting way of comparing different expansions makes use of the (A, V ) matrix as shown in Table I. Here the terms in the expansion of Eq. (72) are schematically arranged in increasing powers of V and increasing powers of A and the symbols have the meaning (A, V ) = (A2, V ) = (A, V z )=
A’&’ ~
2!
(AV
+ VA)
A2s3 7 (A’V + AVA + VA’)
7(AV’ + VAV + V’A)
A2E3
For the multislice expression, we have (see also Section V.E) exp(2sA) exp(AsV) = exp[Ae(A
+ V ) ] + $I’E~(AV
-
VA)
+ 03(A)
so that the first horizontal and vertical rows are properly included, but the second-order mixing term (A, V ) is only partially included. For the new expression [Eq. (71)] of second order, neglecting ULL effects (6 = 0), we have
exp(4hV) exp(AsA) exp(&V)
= exp[Ae(A
+ V ) ]+ 0 3 ( A )
Hence, the first horizontal and vertical rows of Table I are included as well as the first mixing term (A, V ) .The quality of an expansion is mainly determined by the importance of the omitted terms in Table I. However, the relative magnitude of these terms depends largely on the local potential in the foil.
TABLE I CLASSIFICATION OF THE TERMS OF THE TAYLOR FOR EXP[AE(A + V ) ] I N POWERS OF EXPANSION B A N DV I
V
V2
v 3
...
I A A2 A3
V (A, V ) (A2, V )
V2 (A, V 2 )
v3
...
A A2 A3
...
...
...
1
...
...
330
D. VAN DYCK
For instance, between the atoms, the potential is rather small and smooth and here the electron will not be scattered over large angles. Hence, in reciprocal space, the major diffracted beams will be close to the transmitted beam and their excitation errors will be small (Fig. 7a, region I). In a similar fashion, the term A, which is the real-space transformation of the diagonal matrix of the excitation errors, will be small so that AE g 1. In that case, the
'1
Incident High Energy Electrons
ev
[AJI
eA
I
It
m
I
i
z
z+ f
L Real-Space Electron propagatlon
FIG.7. (a) Regions of reciprocal lattice points, corresponding to increasing excitation errors (distances to the Ewald sphere). (b) Schematic representation of the local calculation of the propagation effect using a linear combination of the values of the wave function at adjacent points.
CALCULATIONS IN HIGH-RESOLUTION ELECTRON MICROSCOPY
331
leading terms in Table I are mainly in the first horizontal row, which is simply the phase-grating expression exp(AEV) = 1 LEV + ( 1 2 / 2 ! ) ~ 2+~ z. . .
+
i.e., for smooth potentials the phase-grating expression might be accurate. Closer to the atoms, the potential change becomes larger and so do the maximum beam angles and excitation errors (Fig. 7a, region 11). In this case the mixing terms in Table I become more important. Close to the atom cores, the potential change is larger and scatters the electrons over a wider angle, corresponding to a larger excitation error, so that the powers of A in Table I become increasingly important (Fig. 7a, region 111). It is clear that the atom cores provide the toughest challenge for a computational method. However, the closer to the atom core, the smaller the scattering area and the smaller the number of scattered electrons. Thus, paradoxically, most of the computational difficulties are imposed by the outermost diffracted beams that carry only a minor part of the total intensity. On the other hand, a correct calculation of the electron wave function at the atom core where the Coulomb potential is infinite requires an infinite number of beams, thus making calculation by any of the current computational procedures impossible. In the original proposal for the real-space method (Van Dyck, 1980) it was argued that, since Eq. (71) is a second-order expansion, it suffices to retain only the terms of Table I up to second order, so that the propagator, which is given by the first column of Table I, can be truncated as exp(1eA) z 1 + h A + $(AeA)’ (73) in order to cut the computational speed. However, as pointed out by Self (1 982), this approximation can lead to computational divergencies that stem from the fact that near the atomic core the electrons are scattered into beams with a large Bragg angle and a large excitation error. In principle, the intensity of these beams is damped by destructive interference caused by the rapid oscillation of eLEAfor large values of A. However, by truncating Eq. (73) after the second-order term, the beams are not necessarily damped but can even be artificially amplified. This effect can be elucidated using a simple analogy: The second-order expansion for exp(-z) z 1 - z + $2 diverges for large z, whereas exp( - z) vanishes. In principle, these computational divergencies can be removed by using a very small slice thickness E, so that the second-order expansion converges even for the outermost beams. However, in this way the computing time becomes unnecessarily large, and the method is no longer competitive with conventional multislice calculations.
332
D. VAN DYCK
A more effective way of avoiding the divergencies without increasing the computing time is to use a better approximation for the propagation and/or by artificially eliminating the outermost beams from the diffraction process. In numerical calculations in reciprocal space, this problem is usually overcome by restricting the beams in the calculations to within a so-called dynamic aperture, centered around the transmitted beam (low-pass filter). The size of this dynamic aperture is determined by the criterion that the total neglected intensity must not exceed a certain tolerance and depends on the scattering power of the atoms and the thickness of the foil. In real space, this approximation corresponds to a smoothing of the wave function and the phase grating at the atom core by a kind of convolution with the Fourier transform of the dynamic aperture function. A similar, but smaller, smoothing effect occurs when using analytical approximations for the atom form factors (e.g., Doyle-Turner and Smith-Burge) or Debye-Waller temperature factors. A proper smoothing in real space enables us to increase the sampling distance and to decrease the calculation time. We will not go into the details of such a smoothing procedure. C . Numerical Procedure
For the computation of the wave function, the foil is divided into thin slices with thickness E, typically on the order of 0.1 to 1 nm, parallel to the entrance and exit faces of the foil. In each slice the wave function is calculated from the wave function in the previous slice by using Eq. (71):
4(x, Y, z + E ) = exp()lV,(x, Y ) C ~+ x e x P p p ( x , Y)C1
+, y)I} exp(lA~)
-
d(x, Y)l}4(X3 Y, z )
The wave function and the phase grating are sampled by a two-dimensional network of closely spaced points in the (x, y) plane of the slice. The phase grating can be directly calculated in real space by using the analytical Fourier transform of the Doyle-Turner expression for the atomic form factors (Doyle and Turner, 1968; Coene and Van Dyck, 1984a). Afterwards, the phasegrating function is smoothed in order to avoid computational divergencies near the atom core. The spacing between the sampling points is related to the fluctuations in the wave function and the potential and is on the order of 10 to 100 pm. Multiplication with the phase-grating factors expC(WVp(1 - 41
and
exPC(wyo(1
+ 611
is done by multiplying the value of the wave function in each mesh point with the corresponding value of the phase-grating function. The computation of the propagator elAa,which is a convolution operator, requires some special
CALCULATIONS IN HIGH-RESOLUTION ELECTRON MICROSCOPY
333
attention. As shown in Section IV.D, the propagator describes the motion of the free electron between successive slices as a kind of complex random walk in which each point of the previous slice contributes to each point of the next slice. However, it is known that most of the high-energy electrons scatter forward within a cone with an apex on the order of the maximum Bragg angle (--1 1 ,oo l o rad) similar to the Takagi triangle (see, e.g., Subsection VII.B, Matsuhata et a!., 1984; Humphries and Spence, 1979; Spence et al., 1978). Hence, it can be proposed that the wave function after propagation through a slice may be evaluated from a linear combination of the values of the wave function at a small number of adjacent points in the previous slice, as shown in Fig. 7b. In practical calculations, the maximum diffraction angle between two slices is much larger than the maximum Bragg angle in order to account for the destructive interference of the outermost beams. By using this procedure for the calculation of the propagator, the total calculation time becomes proportional to N , the number of sampling points, which is of the same magnitude as the number of diffracting beams and smaller than the N log, N of the multislice calculations, especially for large N (large unit cells). In the original approach (Van Dyck, 1980), the linear combination was chosen so as to represent the propagator up to the second order in 1.However, as explained earlier, this algorithm can lead to computational difficulties at the atom cores and can only yield reliable results if the number of dynamical reflections (or sampling points) is increased and the slice thickness reduced. (Kilaas and Gronsky, 1983). Recently (Coene and Van Dyck, 1984b), a more systematic approach was introduced. For the sake of simplicity, we restrict ourselves to the onedimensional situation. In case of an orthogonal sampling of real space, the two-dimensional propagation operator can always be uncoupled as the product of two one-dimensional propagation operators. If the wave function after propagation from slice z to slice z + E is calculated from a linear combination of the values of the wave function before propagation in a limited number of sampling points (Fig. 7), 4 ( x n >z
+ E) = C a p 4 ( x n + ~ P
6Z>, = C a p + ( x n + p ,
Z)
(74)
P
with x, the sampling points and 6 the sampling distance, then the propagation in reciprocal space is described by the Fourier transform of Eq. (74), thus yielding
334
D. VAN DYCK
The coefficients up can be obtained by using a least-squares technique, i.e., by minimizing
where the integration is carried out over the reciprocal area S = ( - 1/6, and aM/aa, = o yields for the coefficients 1
ap =
+ 1/6),
lS
2 f ( dexp( - 2ZiPW dg
i.e., the coefficients can be determined as the Fourier coefficients of the function f(g). In principle, the most natural choice for up is obtained by taking f ( g ) as the exact analytical propagation function, i.e., in case of normal electron incidence, f ( g ) = exp[ - i m g 2 / k ] (76) As pointed out by Self (1982), the coefficients ap obtained in this way are related to the Fresnel integrals C(x) and S(x) and decrease slowly with p . Hence, a large number of terms are required in Eq. (74) in order to obtain sufficient accuracy, so this algorithm is far from being competitive. The difficulty stems from the fact that, by discrete sampling of real space with a sampling interval 6, the reciprocal space is artificially made periodic in 216. Indeed, the Fourier approximation of the propagator [Eq. (75)] has this periodicity, whereas the exact function Eq. (76) does not, so that their behavior at the boundaries g = f 1/6 is entirely different, which causes the difficulties. A much better accuracy/speed compromise can be obtained by replacing the exact function, by a functionf(g) that differs from Eq. (16) only near the boundaries, where it has the desired continuity properties but where the intensity of the diffracted beams is negligible.
D. Analysis of Input Parameters 1 . Error Tests Used in Current Multislice Calculations
In multislice calculations, the choice of slice thickness and number of beams is generally connected with the normalization criterion, i.e., when absorption is neglected, the total intensity in the included diffracted beams at a certain thickness is compared to unity as a measure for the amount of electrons being lost by the numerical calculation. It is generally believed that 90%
CALCULATIONS I N HIGH-RESOLUTION ELECTRON MICROSCOPY
335
is a reliable threshold value, and if normalization is below this value, more beams and thinner slices are to be used. However, some authors (Lynch, 1971; O'Keefe, 1973) allow much lower normalization tests of about 70%, which is then justified by the assumption that even for such a diminishing accuracy the relative values of the diffracted beams are not drastically altered. Bursill et al. (1977, 1978) proposed a set of four tests on the calculations. First, a normalization test on the phase-grating function is carried out, and the preceding normalization test on the electron wave function is taken into account. The authors then considered an independent convergence test for an increasing number of beams and, separately, a decreasing slice thickness. The general validity of the normalization criterion has been critically questioned by Anstis (1977). In his paper, he proved that the multislice procedure converges to a normalized result for the slice thickness tending to zero, regardless of the number of included dynamic reflections. On the other hand, the analytical form of the slice method is normalized independent of the slice thickness. These statements show that the normalized criterion cannot be a satisfactory test for both the number of beams and the slice thickness, if considered independently. Shannon (1978) has presented a more sensitive but complicated test combining the slice thickness and the number of beams by checking some particular multislice results with eigenvalue calculations. 2. S2 Test Criterion
Since most of the dynamic calculations are used for high-resolution image simulations, a suitable error criterion should reflect the visibility of the calculation error in the final image. The final imaging procedure uses only a limited number of beams, selected by the objective aperture of the microscope (radius rA);these beams are situated close to the optical axis. The outermost reflections do not contribute directly to the HREM images, although they are more sensitive to change in the numerical parameters. The S2 test criterion (Coene and Van Dyck, 1984a) is defined as s2(E, g M ,
z, r A )
=
1
Id)R(g, z ,
-
4 k z>12
lgI5rA
where 4(g, z ) is the amplitude of the reflection g at a depth z calculated with a particular choice of input parameters (c, gM), where gM is the absolute value of the largest g in the calculations; and 4R(grz ) is the corresponding amplitude obtained with a reference calculation ( E very small, gM very large). As follows from Parseval's theorem, S2 also represents a kind of average quadratic deviation from the exact wave function in a discrete sampled real space, so that it can be considered as a measure of the contrast difference in the final image.
336
D. VAN DYCK
3. Study of the Input Parameters Coene and Van Dyck (1 984a,b) have calculated a large number of images for systems with high (Au,Mn) and with low (SiF,) scattering power and a range of choices for the input parameters. By visually comparing these images with the corresponding reference calculation it can be concluded that, in contrast to the usual normalization test, S 2 is a more faithful measure of the visible contrast error, whereby S2 = 1 x can be used as a safe criterion. Using this criterion, a rule of thumb for accurate calculations can be established
where E is the slice thickness, k is the wave function, and 6 = 1/29, the sampling distance in real space. Equation (6) also ensures that the influence of pseudo-upper-layer reinforcements (Shannon, 1978; Lynch, 1971) is fully suppressed. In the literature, however, some authors have performed computations with input parameters that violate the compulsory rule [Eq. (77)] so that the reliability of such calculations is doubtful. In practice, the value 6 = 1/29, is mainly determined by the scattering power of the material. The slice thickness and g, are then obtained from Eq. (77). For instance, in the case of Au,Mn, gM is typically on the order of 50 nm- (6 = 20 pm), thus yielding a very small slice thickness E of about 20 pm (200-keV electrons). For SiF, on the other hand, g, zz 20 nm- ‘(6 % 50 pm) and E z 0.25 nm. It may seem surprising that the number of reflections that needs to be included in the dynamic calculations according to the S2 criterion clearly exceeds the number of observable reflections in the experimental diffraction pattern. However, before these outermost beams are extinguished by the destructive interference beyond a certain thickness, they may have been scattered into lower-order reflections so that their inclusion in the dynamical calculations is necessary. E . Discussion
The real-space method is mathematically sound and, in the limit, yields the “exact” solution of the high-energy diffraction equation. The forwardscattering character of the electrons can be exploited to decrease the computing time. Although the performance and accuracy are more critically dependent on the choice of the input parameters, such as number of beams and slice thickness, the computational algorithm can be tuned by using rules of thumb like Eq. (77). Since the difference of accuracy between 32- and 64-bit calculations is found to be smaller than the critical Sz value (Coene and Van Dyck (1984b), the optimized real-space calculations (the setup of the phase
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grating not included) can be performed with 32-bit precision even in the case of the high-scattering material Au and for foil thicknesses up to 30 nm. However, 32-bit calculations are roughly twice as fast and require only half of the computer memory. In this way, calculation times on the order of 1 sec/slice (for 1000 sampling points) can be achieved on a 16-bit minicomputer, which is competitive when compared with other methods (see Section 111). Since the calculation time is proportional to the number of sampling points, i.e., to the sampled area, the method is very suitable for handling complicated systems. Indeed, the operator describing the propagation of the electron operates locally and is independent of the crystal structure. Hence, as opposed to fast Fourier techniques, all restrictions concerning the size of the unit have disappeared (see Subsection VILA). VII. RECENT DEVELOPMENTS, UNSOLVED PROBLEMS, AND PROSPECTS FOR THE FUTURE3
A . T h e Real-Space Patching Technique4
In the real-space method, very extensive structures, as required for calculating defect images, can be divided into a set of adjoining, nonoverlapping regions, called patches. The dynamic calculation for a thickness increment of one slice can be performed in each patch separately in the central computer memory. The other patches are stored on fast disk memory. The high-energy electron scattering, represented in a multiplication with the phase-grating expression, is a point-to-point operation and is implemented in a simple way. Since the electron propagation is calculated locally, it requires only some extra consideration at the edges of each patch where the information of the adjacent sampling points in the neighboring patches is needed. It should be noted that a small but not insignificant fraction of the computing time is required by the input/output of the patches from disk. This patching procedure is in principle also possible in FFT-multislice calculations. However, the convolutions calculated by using fast Fourier transforms do not operate locally, so that the input/output time becomes proportional to the product of the number of patches with the number of sampling points. Hence, the applications of a patching method in the FFTmultislice calculations is not so efficient. Another kind of patching technique was introduced by Olsen and Spence (1981). Here the dynamic diffraction is treated for the whole crystal thickness 3Van Dyck ( 1 984). 4Coene and Van Dyck (1984).
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D. VAN DYCK
in each patch separately by using the multislice method of periodic continuation (see Subsection V1I.B). Then the obtained images are joined together. However, the periodic continuation artificially influences the image near the boundaries of each patch. Although this effect can be eliminated by introducing an overlap area between adjacent patches, the effective area of each patch is reduced, so that the total computing time increases, especially for thicker crystals. For the simulation of HREM images of a crystal defect, the FFTmultislice algorithm inevitably needs the periodic continuation of the defect, sometimes by artificially introducing returning defects (Wilson and Spargo, 1982). In real-space calculations, on the other hand, the defect area need not be embedded in an artificially repeated supercell, but can be surrounded by a perfect matrix, for which the wave function is calculated separately. Figure 8a shows an example of a very extensive structure, originating
1 1 1
z: 0
1
z=t
FIG.8. (a) High-resolution image of the overlap area of two coaxial Au,Mn variants. The images of the respective variants are visible at the extremes of the image, where the white dots correspond to the Mn columns. (b) Schematic drawing of the overlap area. [From Coene et al. (1985).]
339
340
D. VAN DYCK
from the overlap between two coaxial Au,Mn variants with the common axis along the electron beam. The compound Au,Mn is a column structure consisting of columns of Au and Mn. When the crystal is oriented with the columns parallel to the electron beam, the Mn columns often appear as white dots. Hence, it has been proposed [Van Tendeloo and Amelinckx (1978a,b)] that the white dots in the overlap area stem from overlapping Mn columns. The model used for image simulation consists of two overlapping wedges of both variants (Fig. 8b). A projection of the model structure along the beam axis is shown in Fig. 9. Simulations are performed for different specimen thicknesses, wedge angles, and focus values. Some of the results are shown in Fig. 10, from which it is clear that the white dots indeed correspond with
- 50 - 60
- 70 - 80 - 90 defocus(nm) FIG.10. Simulated images of the overlap area for different focus values. The wedge angle is 38" and the foil thickness 8.1 nm. The width of the overlap area is also indicated. [From Coene et al. (1985).]
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overlapping Mn columns. More work must still be done in order to elucidate the structure completely.
B. Considerations Concerning the Periodic-Continuation Method Apart from the real-space programs, the current computer programs for the simulation of high-resolution electron micrographs are devised for perfect crystals. Hence, the images of isolated defects are simulated by using the method of periodic continuation (Grinton and Cowley, 1971) in which the defects are repeated periodically, yielding a perfect crystal with a very large unit cell. Intuitively, this method is justified when the distance between adjacent defects is large enough to avoid overlap between their images. When the width of the defect image is large, as is the case for large defocus values, under conditions in which Fresnel-like fringes appear, the image simulations often require prohibitively large computer demands. Most of the computing time is required for the calculation of the amplitudes of the dynamically diffracted electron beams within the crystal. However, for thin crystals, it can be shown that the broadening of the defect images due to diffraction in the crystal is much smaller than that due to the electron transfer through the microscope, especially when large defocus values are used for imaging. Figure 1 l a shows the model system used for the calculations. It consists of a Au,Mn matrix in which periodic antiphase boundaries (APB) are introduced parallel to the incident beam. The distance between successive APBs is chosen as 1.277 nm, which leads to a nominal composition of Au,,Mn,; it corresponds to an actual one-dimensional periodic antiphase boundary structure found in the Au-Mn system. This particular model structure was selected in the first place because the scattering power is expected to be large and to provide a challenge for image simulation programs. The electrons crossing the APB plane can be visualized by calculating the electron density at the exit surface of the defective crystal by using the method of periodic continuation (Fig. 11b) and by subtracting from this the electron density calculated for the perfect crystal shifted along the APB. This image can then be displayed as a halftone picture (Fig. 1lc). It represents in fact an image of the deviation from the column approximation (Howie and Whelan, 1961). Figure 12 shows how the width of the APB image increases approximately linearly with increasing thickness. Similarly, it can be shown that in the electron microscope, the image widens further with defocus (Matsuhata et al., 1984). In both cases, the proportionality factor is about Since in most of the experimental situations the crystal thickness is smaller than the defocus value, it is profitable to perform the image simulation for isolated defects in
A.
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D. VAN DYCK
(b)
(C)
FIG. 11. (a) Structure model, consisting of a perfect Au,Mn matrix with periodically introduced APB. (b) Electron density at the exit face of the foil (thickness, 8 nm). (c) Difference image using the method outlined in the text. [From Matsuhata et al. (1984).]
two stages so as to reduce the computing time and the required memory. For the simulation of the dynamic diffraction in the crystal, the artificial periodof the icity can be taken as small as possible, e.g., somewhat larger than crystal thickness. For the simulation of the transfer through the electron microscope, the artificial periodicity can be increased to a distance somewhat larger than of the largest defocus value. A similar reasoning holds mutatis mutandis also for the real-space method. It should be noted, however, that the technique cannot be applied for the calculation of the diffraction pattern. Indeed, as pointed out by Wilson and Spargo (1982), the simulation of the diffraction amplitudes requires special attention. In order to simulate the diffuse intensity scattered from an isolated defect by discrete sampling in reciprocal space, the scattering between successively repeated defects should be separated, which puts some restrictions on the nature of these defects and on their separation distance.
&
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FIG. 12. Width of the APB image as a function of thickness. Thicknesses are (from left to right) 8, 16,24, 32,40, and 48 nm. [From Matsuhata et al. (1984).]
C . Atomic Column Approximation Recently, a large amount of high-resolution electron microscopic work has been done on substitutional alloy systems with a column structure (e.g., Amelinckx, 1978; Van Sande et al., 1979; Schryvers et al., 1984; for a more complete reference list see Van Dyck et al., 1982). It has been shown by Van Dyck et al. (1982) that, under certain suitable experimental conditions, the high-resolution electron images of substitutional alloy systems can be interpreted directly in terms of the atomic columns, viewed along the beam direction. This feature is related to the fact that electron scattering is forward (see Subsection VII.B), and that for thin specimens, the wave function at the exit face reveals a local nature (Humphries and Spence, 1979; Spence et al., 1978; Bourret et al, 1983). This leads to the idea of the atomic column approximation in which the electron scattering is treated dynamically within the atomic columns but scattering between columns is neglected. Hence, for the image calculations in complicated alloy systems with a column structure, it suffices to perform the dynamic calculations for each type of atomic column only and to assemble the wave functions at the exit plane afterwards. This procedure can yield a drastic gain in computing time, especially for very complicated column structures (Matsuhata et al., 1983b). Figure 13 shows the intensity at the exit plane of the Au,Mn column structure for various crystal thicknesses, calculated with a full dynamic calculation and with the atomic column approximation (200 keV). Figure 14 shows the image after the electron microscopic transfer. From this it is clear that even for such heavily scattering material, the atomic column approximation holds to a thickness of somewhat less than 20 nm (Matsuhata et al., 1983b). It would be very interesting if the method could still be improved somewhat, so as to hold for thicknesses that cover most of the experimental situations. This work is still in progress.
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D. VAN DYCK
FIG. 13. Electron density at the exit plane of a Au,Mn crystal, viewed along the atomic columns, for different foil thicknesses (a) using a complete dynamic calculation and (b) using the atomic column approximation.
D . Absorption, Inelastic Scattering, and Thermal DifSuse Scattering
The importance of absorption and inelastic scattering is generally underestimated and often ignored in simulation programs. However, in biological materials for instance, the mean-free-path for inelastic scattering is on the order of tens of nanometers so that the image simulation for thick crystals is probably only faithful if the effect of absorption is properly taken into account. In electron-diffraction calculations in reciprocal space, anomalous absorption is often introduced in a first approximation by using imaginary Fourier coefficients for the crystal potential (Yoshioka, 1957). In this way, the inelastically scattered electrons are eliminated from the diffraction process. The real-space method is particularly suitable for the investigation of absorption effects. By Fourier transforming the absorption coefficients into real space, an absorption potential is obtained that consists of bell-shaped imaginary absorption functions localized at the atom positions. The absorption
FIG.14. High-resolution image of a Au,Mn crystal as a function of foil thickness, calculated as for Fig. 13: (a) complete dynamic calculation and (b) atomic column approximation. (Cs = 1.1 nm, aperture = 7 nm-’, defocus = -45 nm).
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functions are composed of two major parts stemming from inner shell electron excitation and phonon excitation, respectively, whereas the plasmon contribution can be considered as constant over the crystal. The first part is dependent only on the atom type and can be tabulated, e.g., in the form of a sum of Gaussian functions. These values can be used for all real-space highresolution image calculations. The phonon contribution, however, which is sharply peaked, is also dependent on the structure of the crystal. Hence complete models for the absorption potential are only available for simple crystals, such as FCC metals (Humphries and Hirsch, 1968; Radi, 1970). In order to study the relative importance of the different inelastic processes, we performed dynamic calculations for Au[001] using different assumptions for the absorption potential (Matsuhata et al., 1983a). Figure 15 shows the calculated electron density at the Au atom core, where the sensitivity to absorption is maximal, as a function of crystal thickness. Calculations were performed using the model of Humphries-Hirsch with electron-phonon interaction (solid line), without electron-phonon interaction (dotted line), and without absorption (dashed line). It is interesting to note that, without absorption, the electron density at the atom core oscillates with crystal thickness with a periodicity of about 4 nm. This effect will be discussed in Subsection V1I.G. From Fig. 15 it is clear that calculations without absorption are only valid for small thicknesses and that the electron-phonon interaction constitutes the major part of the absorption. Unfortunately, an accurate calculation of the
20
10
0
0
2 0 nm
FIG. 15. Electron density at the atom core of Au [Ool] as a function of crystal thickness. Solid line uses the model of Humphries-Hirsch with electron-phonon interaction; dotted line uses the same model without electron-phonon interaction; and dashed line represents model without absorption.
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D. VAN DYCK
electron-phonon absorption potential requires a knowledge of the lattice dynamics (phonons) in the crystal, which in turn requires a knowledge of the crystal structure. A single atom absorption function is no longer valid (Whelan, 1984). Hence, it might be intrinsically impossible to account properly for absorption in crystals with an unknown structure, which poses a serious problem. Furthermore, in electron image simulation, the inelastically scattered electrons are considered as lost for imaging. For thick crystals, it might also be worthwhile to account for the inelastically scattered electrons. However, since these electrons cannot be considered as coherent with the incident electrons, the computation can be a tremendous task. Although some theoretical progress has been made in the past (Rez, 1978; Serneels et al., 1980), there still remains a lot of work to be done (Howie, 1983). More experimental data, such as energy-filtered high-resolution images, would also be very welcome. Another problem that has not yet been treated adequately is the influence of the thermal motion of the atoms on the diffraction process. This effect is usually introduced by using a technique adapted from x-ray diffraction, i.e., by multiplying the atomic scattering factors with a Debye-Waller factor. In a sense, the atom is replaced by a kind of spatially averaged atom. In principle, the Debye-Waller factor can only be used to calculate the intensity of Bragg reflections when the diffraction is kinematic. However, at present, it is not clear whether the approximation also holds when the diffraction is highly dynamic. Furthermore, the diffuse intensity scattered in between the Bragg reflections cannot be treated. In principle, the thermal diffuse scattering (TDS) can be calculated by using a very large unit cell in which all the atoms may have different displacements. In order to account for the temporal averaging during exposure on the photographic plate, different images must be calculated and added. At present, such procedure is still limited by the computer demands.
E . The Reverse Problem: Direct Structure Retrieval Instead of using trial-and-error simulation techniques, which are only capable of testing small modifications of approximately known structures and are very time consuming, it would be much more interesting to search for methods that could extract the structural information directly from the electron micrographs. Such a direct method should proceed in two stages. First, the effects of the electron microscope transfer must be eliminated, i.e., we must return from the image to the exit plane of the specimen. Although much progress has been achieved in this field (e.g., Saxton, 1978; Kirkland, 1982; Saxton and Stobbs, 1984), especially for thin specimens, there still remains work to be done.
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In the second stage, the problem of deriving the crystal potential from the knowledge of the wave function at the exit plane of the crystal foil should be solved. Although it is sometimes argued that this problem cannot be solved mathematically due to questions of uniqueness, we show that in principle direct procedures are possible. Using the notation of Section IV and neglecting ULLs, the wave function at the exit plane of the crystal is given by 4(z)
= exPrMA
+ Vl 4 0
(78)
with $(O) = 1 for a normal incident plane wave. The problem now is how to derive the projected potential V, which is a two-dimensional real function, from the knowledge of 4(r), which is a two-dimensional complex function. If the crystal foil is sufficiently thin, so that the propagation effect can be neglected, the wave function can be expressed by the phase grating 4(z)
= exp(AzV)
(79)
from which V can be derived in a direct manner. If the crystal is thicker so that the phase grating does not hold, we start from the identity exp(AzV) = exp(AzV)
+ 4(z)
-
exp[Az(A
+ V)]
(80)
which can be used to construct a recursive algorithm yielding successive i.e., values V " , which converge toward V for n -+a, exp(AzV"+')
= exp(AzV")
+ 4(z)
-
exp[Az(A
+ V")]
(81)
with V o = 0. Note that V' is the result obtained by using the phase-grating expression. It is clear that if V" converges toward a limit, then Eq. (80) is satisfied. The limit thus presents the projected potential V if the solution of Eq. (78) is unique. In order to test this statement, we proceed as follows. First, the wave function 4 ( r ) is calculated with the RS method, starting from a known crystal potential V. In practice, we use the crystal SiF,, the projected model of which is depicted in Fig. 16. The recursive procedure [Eq. (Sl)] is started, and the successive results for V" are compared with V. Figure 17 shows the results for V' (dotted line), V 2 (dashed line), and V (solid line) taken along a trace that intersects the Si atoms (the origin is taken at the Si atoms and the two peaks are located at the F atoms). Crystal thicknesses are 1.0 and 1.6 nm, respectively. From this, it is clear that Eq. (78) converges to the exact solution for crystal thicknesses that exceed the phase-grating limit. This work is still in its initial stage. In the near future we must study other recursive algorithms instead of Eq. (78), which are expected to converge for larger crystal thicknesses, such as exp(lzV"+')
= exp( - AzA/2)$(z)
x exp[dz(A
-
exp( - AzA/2)
+ V " - ' ) ] + exp(AzV")
(82)
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D. VAN DYCK
0
0
0
0
FIG. 16. Model of SiF,. Black circles represent Si atoms, and white circles represent F atoms; a=0.383 nm; c=0.540 nm.
It should be noted that in practical situations the crystal thickness z is not known and the wave function +(z) is only known to a certain resolution. These problems are still being studied. Another interesting approach that allows direct structure information is the so-called improved phase-grating approximation (Van Dyck, 1983). Substituting +(r) = expCWl (83) into the high-energy diffraction equation [Eq. (35)] yields, for normal incidence (in shorthand notation), ae/az =
n[(ve)z + A e + vl
(84)
FIG.17. Retrieval of the potential for SiF,. The abscissa represents the line intersecting the Si atoms (see Fig. 16.). Dotted line represents first iteration; dashed line represents second iteration; and solid line represents exact potential; t is the crystal thickness; (a) t = 1.0 nm and (b) t=1.6nm.
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where V and A are the gradient and Laplacian operators in the (x,y) plane, respectively, and 1 is defined as in Eq. (36). Equation (84) can also be written as an integral equation
O(d) = O(0)+ 1
Jod + Jod V dz
[(Ve)’
1
+ Ae] dz
where d is the foil thickness. When ULL effects are neglected, V is independent of z, so that B(d)
=
O(0)
+ 1Vd + 1
[(VO)’
+ AO] dz
Jod
(85)
Expansion of 0 in powers of 1 now yields with B(0) = 0 B(d) = 1Vd
+ (1’d2/2)AV+
(A3d3/3)(VV)’ + (d3d3/6)A2V+ O(A4)) (86)
which can be approximated up to second order as
e(d) =
+(i~/2)p
(87)
where A V = p is proportional to the projected charge density in the (x,y ) plane and hence, from Eq. (83),
4 = exp(Al/d + (1’d2/2)p)
(88)
An approximation of Eq. (88) can also be derived by using the Bloch wave formalism (Pirouz, 1981), so that in this way a direct link can be made between Bloch-wave and real-space theory. Since from Eq. (36) 1 is an imaginary constant, the electron density at the exit face of the crystal is given by
14(d)12 = exp(L2d2p)= 1 + L2d2p
(89)
Whereas for a pure phase grating the electron density remains unity, there is now a term, proportional to the projected charge density and the square of the crystal thickness, which is positive at the atom core and negative in the electron cloud. That is, if 141’ is visualized as a halftone picture, such as in Fig. 14, the atom core is imaged as white peak surrounded by a dark ring. This effect was observed by Ishizuka and Uyeda (1977) by using multislice calculations on Cu phthalocyanine, where it was attributed to a lens effect of the atoms. From these calculations, it can be estimated that in such material Eq. (88) holds to about 15 nm for 500 keV electrons, which is far better than the pure phase grating. Equation (88) or (89) can also be used to find an approximate solution to the reverse problem. Indeed from the phase of the wave function at the exit plane we know dV, and from the amplitude we know d’p, from which, by
350
D. VAN DYCK
solving Poisson’s equation, we obtain V. Hence, the projected potential is obtained in two ways and the ratio yields the crystal thickness. As a test for the validity of Eq. (88), this ratio must be constant over the whole exit face. Deviations are expected to start from the atom cores on increasing crystal thickness.
F . Upper Layer Lines and Beam Tilt Most of the algorithms currently in use for image simulation are based on the projection approximation in which ULLs or HOLZs are neglected (Spence, 1981). Although Lynch (1971) argued that ULLs are important for Aurlll], it is generally believed that the projection approximation is valid when the incident beam is nearly parallel to a zone axis and when the periodicity c along the beam direction is not too large. This is also clear from Subsection VLB, where it is shown that the ULL effect is of the same order of magnitude as the scattering of a slice of unit cell thickness. However, there are a number of experimental situations in which ULL effects can be important. First, when the periodicity c is large (Nihoul and Cesari, 1984) or when disorder is present along the c axis. If the crystal planes are perpendicular to the zone axis, we can use either the usual multislice simulation programs, provided c is divided in several slices, or the real-space method corrected for potential eccentricity [Eq. (71)], which allows larger slice thicknesses to be used. However, a problem arises when the crystal planes are not perpendicular to the zone axis as is the case for monoclinic crystals viewed along the monoclinic axis. Here the two-dimensional unit cell of the projected potential can be small if the projection extends over the whole c, but may be infinitely large for each of the subdividing slices (Wood, 1982). Hence, although the three-dimensional unit cell may have relatively small dimensions, the use of the multislice programs will require the use of the periodic continuation technique. In this case the use of the real-space method can be much more appropriate. Another experimental situation where ULL effects can be important is the imaging of surface structures by using dynamically forbidden reflections (e.g., Yagi, 1984). When the c axis is not too large, the real-space formula can be used, since the surface structures are reflected in the potential eccentricity 6 of the first and last slice. Another very important experimental situation in which ULL effects can also be important arises when the incident beam is inclined from the zone axis so that the Ewald sphere cuts the higher-order Laue zones, as is the case of convergent electron beam diffraction, STEM, and high-resolution electron microscopy with a convergent incident beam (open condensor). Although the multislice method (Ishizuka, 1982) and the real-space method can in principle be adapted for beam tilt, the current programs are
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not adequate for larger tilt angles. If the various incident beams of the illumination core are incoherent, different images must be calculated for each of these directions of incidence, and the images must be superimposed afterwards. This procedure, however, is very time consuming and prohibitive. Perturbation methods (e.g., Von Hugo, et al., 1984) are promising, but more theoretical work must be done.
G. Application to General Quantum-Mechanical Problems As already stated by Van Dyck (1980), and Gratias and Portier (1983), the high-energy diffraction equation for normal incidence is of the same type as the time-dependent Schrodinger equation in two dimensions, provided the depth z is replaced by the time t. This can easily be demonstrated by the following example. Imagine a column structure such as Au,Mn. If ULLs are neglected, the potential is only dependent on (x,y), and, on moving through the crystal, the electron sees atom “tubes.” If we calculate the wave function at successive crystal depths, we obtain a time sequence of the motion of an electron in a twodimensional assembly of atom potentials. The electron density at various depths in Au, Mn is shown in Fig. 18 (200 keV). This figure can be compared with Fig. 13. From this, it is clear that at the atom core of Au the intensity
FIG.18. Electron density at the exit face of Au,Mn (cf. Fig. 13) for different thicknesses. The Mn positions are located at the dots in the corners. The Au positions are located at the four dots in the center of the squares.
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D. VAN DYCK
increases and decreases periodically with the depth with a periodicity of approximately 4 nm, which is in agreement with the oscillations observed in Fig. 15. This phenomenon can be explained intuitively as due to an oscillator-like behavior of the electron in the electrostatic potential of the atom. The periodicity of 4 nm corresponds with a time period of about 2 x lo-’’ and a frequency of about 5 x 1OI6 Hz. A similar behavior is observed at the Mn atom cores but, due to the smaller electrostatic potential, the periodicity is much larger. It would be interesting to study the relation between this phenomenon and the interaction between Bloch waves as well as the correspondence with atomic and molecular orbitals [as investigated by Buxton et al., (1978)l or Bloch wave channeling (Humphries, 1980). From the foregoing, it is clear that the slice methods, and in particular the real-space method, which is optimized for solving the basic equation [Eq. (35)] by depth slicing, can also be used to solve the time-dependent Schrodinger equation by time slicing, i.e., the slice methods are in fact numerical approximations to the Feynman path integrals (Feynman and Hibbs, 1965; Van Dyck, 1975). Hence, the real-space method might be put into good use for the solution of more general quantum-mechanical problems, but this requires further investigation. H . Conclusion and Prospects for the Future For the present type of image simulations in crystals, the current multislice programs are still adequate if the unit cell is not too large. When large unit cells, defects, disorder, and upper layer lines are involved, the real-space method, combined with the patching technique (Subsection VILA), seems to be more appropriate. Probably the simulation technique can be extended to other areas, such as the diffuse scattering from thermal atom motion or substitutionally disordered systems. However, in order to make image simulation more reliable, a more quantitative comparison between experimental and simulated images, combined with a least-squares refinement, is highly desirable. For this purpose, a speeding up of the computational algorithms is very welcome. In the real-space method some extra speeding up can be expected from better sampling schemes that take into account the local nature of the potential and probably also from the improved phase grating approach [Eq. (85)]. In this respect, a dedicated image computer can be put into very good use either for the treatment of the digitized experimental images or for the calculation of the simulated images. Here the disposal of an array processor is also advantageous. More studies in the near future must be concentrated toward the effect of beam tilt, for the treatment of convergent beam electron diffraction, and toward inelastic scattering. Also, a real-space description of the image forma-
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tion in the electron microscope can yield some benefits (Marks, 1984). It is our belief however, that, on a long term, the direct structure retrieval methods, combined with on-line image capturing, filtering, and processing devices, are the most promising.
ACKNOWLEDGMENT The author is indebted to Dr. Coene, Dr. Matsuhata, Prof. R. Serneels, Prof. J. Van Landuyt, Dr. G. Van Tendeloo, and Prof. S. Amelinckx for many fruitful discussions and for the use of photographic materials.
REFERENCES Alpress, J. G., Hewat, E. A., Moodie, A. F., and Sanders, J. V. (1972). Acta Crystallogr. Sec. A 28, 528. Amelinckx, S. (1978). Chem. Scr. 14, 197. Anstis, G. R. (1977). Acta Crystallogr. See. A 33, 844. Berry, M. V. (1971). J. Phys. C4,697. Bethe, H. A. (1928). Ann. Phys. 87,55. Bourret, A., Thibault-Desseaux, J., DAnterroches, C., Penisson, J. M., and De Crecy, A. (1983). J. Microsc. 129. 337. Bursill, L. A., and Wilson, A. R. (1977). Acta Crystallogr. See. A 33,672. Bursill, L. A,, and Wood, G. J. (1978). Philos. Mag. A 38,673. Buxton, B. F., Loveluck, J. E., and Steeds, J. W. (1978). Philos. Mag. A 3, 259. Coene, W., and Van Dyck, D. (1984a). Ultramicroscopy. 15,41. Coene, W., and Van Dyck, D. (1984b). Ultramicroscopy. 15,287. Coene, W., Van Dyck, D., Van Tendeloo, G. and Van Landuyt, J. (1985) Phil. Mag. A. In press. Cowley, J. M., and Moodie, A. F. (1951). Proc. Phys. Soc. B 70,486. Cowley, J. M., and Moodie, A. F. (1957). Acta Crystallogr. 10,609. Doyle, P. A,, and Turner, P. S. (1968). Acta Crystallogr. Sec. A 24,390. Feynman, R. P., and Hibbs, A. R. (1965). “Quantum Mechanics and Path Integrals,” McGrawHill, New York. Fields, P. M., and Cowley, J. M. (1978). Acta Crystallogr. See. A 34, 103. Fujimoto, F. (1959). J. Phys. Soc. Jpn. 14,1558. Fujiwara, K. (1959). J. Phys. Soc. Jpn. 14, 1513. Fujiwara, K. (1961). J. Phys. Soc. Jpn. 16,2226. Gevers, R., Serneels, R., and David, M. (1974). Phys. Status Solidi B 66,471. Goodman, P., and Moodie,A. F. (1974). Acta Crystallogr. See. A 30,280. Gratias, D., and Portier, R. (1983). Acta Crystallogr. Sec. A 39, 576. Grinton, G. R., and Cowley, J. M. (1971). Optik 34,221. Howie, A., (1978). in “Diffraction and Imaging Techniques in Material Science,” Vol 2 (S. Amelinckx. R. Gevers, and J. Van Landuyt, eds.), p. 457. North-Holland Publ., Amsterdam. Howie, A. (1983). J. Microsc. 129,239. Howie, A., and Basinsky, Z. S. (1968). Philos. Mag. 17 1039.
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Howie, A., and Whelan, M. J. (1961). Proc. R. SOC.(London) Ser. A 263,217. Humphries, C. J. (1980). Proc. 6th Int. ConJ HVEM Antwerp, (P. Brederoo and J. Van Linduyt, eds.) p. 68. Humphries, C. J., and Hirsch, P. B. (1968). Philos. Mag. 18, 115. Humphries, C. J., and Spence, J. C. H. (1979), Proc. 37th E M S A Meet. Baton Rouge, Louisiana, p. 554. Claitor’s Publishing Division. Ishizuka, K. (1982). Acra Crystallogr. 138,773. Ishizuka, K., and Uyeda, N. (1977). Acta Crystallogr. See. A 33,740. Kilaas, R.,and Gronsky, R. (19x3). Ultramicroscopy 11,289. Kirkland, E. J. (1982). Ul/ramicroscopy 9,45,65. Lynch, D. F. (1971). Acta Crystallogr. See. A 27,399. Lynch, D. F. (1982). Proc. Austr. Conf: Electron Microsc., 7th. Lynch, D. F., and OKeefe, M. A. (1972). Acta Crystalogr. See. A 28,536. Marks, L. D. (1984). Ultramicroscopy 12,237. Matsuhata, H., Van Dyck, D. and Coene, W. (1983a). Proc. Joinr Meet. Belg. Ger. SOC.Electron Microsc. Antwerp, p. 135. Matsuhata, H., Van Dyck, D., and Coene, W. (1983b), Proc. Joint Meet. Belg. Ger. Soc. Electron Microsc., Antwerp, p. 136. Matsuhata, H., Van Dyck, D., Van Landuyt, J., and Amelinckx, S. (1984). Ultramicroscopy 13, 343.
Moodie, A. F. (1981). In “Fifty Years of Electron Diffraction,” (P. Goodman, ed.), p. 327. D. Reidel, Dordrecht. Niehrs, H., and Wagner, F. N. (1955). 2. Phys. 143,285. Nihoul, G., and Cesari, C. (1984). Phys. Status Solid; A 81.87. OKeefe, M. A. (1973). Acta Crystallogr. See. A 29,389. Olsen, A., and Spence, J. C. H. (1981). Philos. Mag. A 43, 945. Pirouz, P. (1981). Acta Crystallogr. See. A 37,465. Radi, G. (1970). Acta Crystallogr. Sec. A 26.41. Rez, P., (1978). Inst. Phys. ConJ Ser. 41,61. Rez, P. (1980). Proc. Annu. Meet., Electron Microsc, SOC.Am. p. 180. Sanders, J. V., and Goodman, P. (1981). In “Fifty Years of Electron Diffraction,” (P. Goodman, ed.), p. 281. D. Reidel, Dordrecht. Saxton, W. 0. (1978). In “Advances in Electronics and Electron Physics,” Supplement 10 Academic Press, New York. Saxton, W. 0.and Stobbs, W. M. (1984). Proc. EUREM 8, Budapest 1,287. Scherzer, 0. (1949). J. Appl. Phys. 20,20. Schiske, P. (1950)). Ph.D. Thesis, University of Vienna. Schryvers, D., Van Landuyt, J., and Amelinckx, S. (1984). Proc. EUREM 8, Budapest I , 891. Self, P. G. (1982). J. Microsc. 127,293. Self, P G., OKeefe, M. A., Buseck, P. R., Spargo, A. E. C. (1983). Ultramicroscopy 11,35. Serneels, R., Haantjens, D., and Gevers, R. (1980). Pilos. Mag. A 42, 1. Shannon, M. D. (1978). Inst. Phys. Con$ Ser. 41,41. Skarnulis, A. J., Wild, D. L., Anstis, G. R.,Humphries, C. J., and Spence, J. C. H. (1981). Inst. Phys. ConJ Ser. 61,347. Spence, J. C. H. (1978). Acra Crystallogr. Sec. A 34, 112. Spence, J. C. H. (1981). “Experimental High-Resolution Electron Microscopy.” Clarendon Press, Oxford. Spence, J. C. H., and Whelan, M. J. (1975). Acta Crystallogr. See. A 31, 242. Spence, J. C. H., OKeefe, M., and Iilima, S. (1978). Phillos. Mag. A 38,463. Sturkey, L. (1957). Acta Crystallogr. 10, 858.
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Sturkey, L. (1962). Proc. Phys. Soc. 80, 321. Tournarie, M. (1960). Bull. Soc. Franc. Miner. Cryst. 83, 179. Tournarie, M. (1961). C. R. Acad. Sci. 252, 1961. Tournarie, M. (1962). J. Phys. Soc. Japan 17, Suppl. B. 11. 98. Van Dyck, D. ( 1975) Phys. Status Solidi B 72,32 1. Van Dyck, D. (1976) Phys. S/atus Solidi B 77,301. Van Dyck, D. (1978a). In “Diffraction and Imaging Techniques in Material Science,” Vol. I , (S. Amelinckx, R. Gevers, and J. Van Landuyt, eds.), p. 355. North-Holland Publ., Amsterdam. Van Dyck, D. (1978b). Proc. Inter. Congr. Electron Microsc., 9th, Toronto, Vol. 1, (J. M. Sturgess, ed.) p. 196. Van Dyck, D. (1979). Phys. Status. Solidi A 52,283. Van Dyck, D. (1980).J. Microsc. 119, 141. Van Dyck, D. (1983). J. Micrsoc. 132, 31. Van Dyck, D. (1984). Proc. EUREM 8, Budapest 1,261. Van Dyck, D., and Coene, W. (1984). Ultramicroscopy. 15.29. Van Dyck, D., DeRidder, R., and DeSitter, J. (1980). J. Comp. Appl. Math. 6(1), 83. Van Dyck, D., Van Tendeloo, G., and Amelinckx, S. (1982). Ulrrumicroscopy 10,263. Van Sande, M., Van Tendeloo, G., Amelinckx, S., and Airo, P. (1979). Phys. Sfatus Solidi A 54, 499. Van Hugo, D., Kohl, H., and Rose, H. (1984). Proc. EUREM8, Budapest 1,135. Van Tendeloo, G., and Amelinckx, S. (1978a). Phys. Slat. Solidi A 47, 555. Van Tendeloo, G., and Amelinckx, S. (1978b). Phys. Stat. Solidi A 49, 337. Whelan, M. J. (1984). Proc. EUREM 8, Budapest 1,429. Wilson, A. R., and Spargo, A. E. (1982). Philos. Mag. A 46, 435. Wood, G. J. (1982). Private communication. Yagi, K. (1984). Proc. I n / . Congr. Crysrullogr. 13th. p. C2. Yoshioka, H. (1957). J. Phys. Soc. Jpn. 12,618.
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ADVANCES IN ELECTRONICS AND ELECTRON PHYSICS, VOL. 65
Theory of Surface Electronic Structure* E. WIMMERt Department of Physics and Astronomy Northwestern University Evanston, Illinois and Institutfir Physikalische Chemie Universitiit Wien Vienna, Austria
H. KRAKAUER Department of Physics College of William and Mary Williamsburg, Virginia
A. J. FREEMAN Department of Physics and Astronomy Northwestern University Evanston, Illinois
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Thin-Slab Approximation . . . . . . . . . . . . B. Local-Spin-Density-Functional (LSDF) Theory . . . 111. Approach and Methodology . . . . . . . . . . . . IV. Examples of Applications . . . . . . . . . . . . .
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A. Aluminum surfaces . . . . . . . . . . . . . . . B. Transition Metals . . . . . . . . . . . . . . . C. Magnetic Transition-Metal Surfaces and Interfaces . . . D. Energetics of Surfaces: All-Electron Total Energy Approach References . . . . . . . . . . . . . . . . . . .
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I. Introduction
11. Theoretical Framework.
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* Parts of this work were supported by the National Science Foundation (DMR grants no. DMR 81-20550 and DMR 82-16543) and the Office of Naval Research (N00014-81-K-0438). t Present address: CRAY Research, Inc. Mendota Heights, Minnesota. 351 Copyright 0 1985 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-014665-7
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I. INTRODUCTION The past decade has witnessed a dramatic increase of detailed information, much of it not understood, about surfaces, overlayers, and interfaces due to the development and refinement of surface-sensitive experimental techniques. These advances in research have been spurred by the enormous technological relevance of surface and interface phenomena in areas such as microstructure electronics, catalysis, and corrosion. In turn, these developments have challenged present theoretical understanding and have encouraged the development of realistic theoretical methods with which to analyze and interpret critically this rapidly growing body of information. Since many of these phenomena are directly related to the electronic structure at the surface, attention has focused on extending well-known theoretical methods for calculating bulk properties to treat surfaces. This timely and successful pursuit resulted from the fact that considerable activity has been focused in recent years on quantum mechanical theories, which describe the dynamics of many-interacting electrons in an external potential, for example, due to the atomic nuclei. An important breakthrough was the development of densityfunctional theory (Hohenberg and Kohn, 1964; Kohn and Sham, 1965), which states that under certain conditions all ground-state properties of an interacting electron gas are completely determined by the charge density of the electrons. In principle, density-functional theory permits the accurate evaluation of the electronic part of the total energy and, consequently, after including the Coulombic interaction between the nuclei, it enables the evaluation of all structural properties. At present, the most precise way to evaluate the electronic structure is to solve the Kohn-Sham equations in the local-density approximation. However, in the period of time just after the development of density-functional theory, the numerical methods for solving the Kohn-Sham equations were so crude that resulting errors completely obscured the effects of the physical approximations. Only recently has it become possible, due to enormously increased computer power and sophisticated combined theoretical and computational approaches, to solve these equations in a much more exact fashion. Thus, first-principles energy-band studies for surfaces have demonstrated a fair degree of sophistication in tackling a number of complex problems involving the electronic structure of free surfaces (including magnetic properties, surface reconstruction, and relaxation), chemisorption bonding of atomic absorbates, and interface phenomena. A principal aim of such calculations is to achieve self-consistency between the calculated electronic charge density and the input potential. Such effects are especially important for systems with sizeable charge transfer between the atomic constituents.
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While the electronic properties of semiconductor and simple metal surfaces have been the subject of extensive study, it is only recently that attempts have been made to deal theoretically with the additional complexity arising from the localized d-electrons in noble and transition metals. For example, a review article by Appelbaum and Hamann (1976) needed only about one page to describe the few realistic calculations for transition metal surfaces that had been reported at that time, and none of these was self-consistent. [A review by Inglesfield (1982) has served to update this situation.] It is precisely the d-band electrons, however, which play such a crucial role in such interesting and important surface phenomena as surface reconstructions, differences between surface and bulk magnetic properties, and catalytic activity (especially the group VIII transition metals and their alloys). While the importance of the d-band electrons in these phenomena is generally accepted, the details of their influence are less clear. The aim of this chapter is to describe a newly developed, highly accurate, and unified method for calculating surfaces, the full-potential linearized augmented plane wave (FLAPW) method (Wimmer et al., 1981a; Posternak et al., 1980; Krakauer et al., 1979a,b; Jepsen et al., 1978; Hamann et al., 1981). This is a unified method in that it can easily treat not only simple metals and semiconductors but also transition-metal surfaces. It is also unified in the sense that it is capable of treating molecular absorption on surfaces and also the extreme limit of the isolated molecule and the clean surface. In this approach, which represents a major advance in that the local density equations are solved without any shape approximations to the electronic potential or electronic charge density, a new technique for solving Poisson’s equation (Weinert, 1981) for a general charge density and potential is implemented. Results obtained for thin metal films and for nearly free molecules (Wimmer et al., 1981a) demonstrate the high degree of accuracy possible with this method. The development of accurate methods to solve the local-density equations has brought about increasing interest in using these methods to determine the total energy and related properties (such as equilibrium phases, lattice constants, and force constants), since the total energy is a fundamental quantity in density-functional theory. For all-electron methods, there has been the major problem in evaluating the total-energy expression, which arises from numerical problems introduced by the necessity of canceling the very large (positive) kinetic and large (negative) potential energy contributions. As is well known, this problem becomes very severe for heavy atoms since the core electrons are responsible for the largest part of the total energy. A successful solution to this difficulty has been presented in the form of a new formalism for determining highly accurate total energies of solids within density-functional theory. In this approach (Weinert et al., 1982), all necessary
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terms are easily obtained from the energy-band calculations. A major feature of this all-electron method is the explicit algebraic cancellation of the numerical Coulomb singularities in the kinetic and potential energy terms that leads to good numerical stability. As implemented in the FLAPW film approach (Weinert et al., 1982), the method allows us to treat the total energy of an all-electron system to high accuracy without resorting to frozen-core, pseudopotential, or other approximations. The organization of this chapter is as follows. Section I1 discusses the general theoretical framework for the FLAPW calculations: the thin-slab structural model for surfaces and the local-(spin)-density-functional [L(S)DF] approximation and the limitations of each of these. Section I11 presents the FLAPW methodology. Finally, Section IV presents a selection of applications and examples that illustrate its applicability to a variety of surface and interface phenomena. 11. THEORETICAL FRAMEWORK
Any theoretical approach must address two basic problems. Within the Born-Oppenheimer approximation, the first (and more difficult problem) is the treatment of many-body electron-electron interactions. The second problem is to devise a structural model of the surface for which realistic calculations can be performed. By realistic, we mean calculations comparable to state-of-the-art bulk calculations. The structural problem is discussed first. A . Thin-Slab Approximation
Our focus in this chapter is the treatment of extended surfaces with or without ordered overlayers of absorbed atoms. Probably the most successful structural model uses a thin slab to simulate both surface and bulk effects on an equal footing. Calculations in this model are performed for a finite-thickness slab that is infinitely periodic in the plane parallel to the surface. Typically, slabs 5-13 atomic-layers thick are used. The features of this model have been discussed at length by Appelbaum and Hamann (1976). To ensure accurate results, the slab should be thick enough so that the electronic structure in the interior of the slab resembles closely the expected bulk structure. Thicker slabs a h o reduce size-effect energy splittings between surface states that are localized on the upper and lower surfaces. For metals with welllocalized surface states (such as transition metals), these requirements are satisfied for slabs as thin as five layers. If it is necessary to identify more extended surface states, thicker slabs must be used. The desirability of thicker
THEORY OF SURFACE ELECTRONIC STRUCTURE
36 1
slabs for the preceding reasons must be balanced, however, by the practical consideration that the magnitude of the calculation (measured by the amount of computer memory and computer time required) increases rapidly with increasing slab thickness (roughly between N 2 and N 3 , where N is the number of layers in the slab). B. Local-Spin-Density-Functional (LSDF) Theory
Major progress in the last 15 years has been made in the theoretical understanding of ground-state properties of many-electron systems. This achievement is based largely on the success of LSDF theory (Hohenberg and Kohn, 1964; Kohn and Sham, 1965; von Barth and Hedin, 1972; Gunnarsson et al., 1972). The LSDF theory introduces an effectioe one-electron potential, which is a function of the charge density p(r) and the spin density n(r) = pf(r) - d r ) :
[For paramagnetic systems, n(r) = 0 and V,fff = Vf,,.) These lead to oneparticle Schrodinger-like equations for spin-up and spin-down one-particle states (in Rydberg atomic units): [- V 2
+ VI#)(r)
-
c:(J)]$!(J)(r) = 0
(2)
In LSDF theory, the one-particle energies & t ( l and ) states $if(J), essentially have no physical meaning other than specifying the charge and spin densities p f ( J ) ( r= )
1~ ! ( l ) ( r ) * $ ~ ( J ) ( r ) O ( ~El~f ( 1 ) ) -
(3)
i
where E , is the Fermi energy and the step function O(E, - E ! ( ~ ) )ensures that the summation is over occupied states only. The essential feature of Eqs. (2) and (3) is that the potential VL#) must be self-consistent with the charge and spin densities. Thus, these equations must be solved by iteration until selfconsistency is achieved. The effective potential can be expressed as the sum of two terms: Vdff(r)= Kou,[p(r)1 + V::')cp(r), 4 r ) l
(4)
where V,,,, is the classical Hartree potential due to the charge distribution of all the electrons and nuclei in the system, and V::') is the exchange-correlation potential and includes all many-body effects on ground-state properties. The functional form for V,"" is usually determined by calculations for the homogeneous electron gas at different densities and spin densities (von Barth
362
E. WIMMER, H. KRAKAUER, A N D A. J. FREEMAN
and Hedin, 1972; Gunnarsson et al., 1972). In the course of iterating Eqs. (2)-(4), large changes occur from one iteration to the next in Vcou, due to sizeable charge rearrangements when self-consistency is approached. For low-symmetry systems like surfaces, these must be strongly damped in order for the iterative process to converge. Typically, this is done by using some sort of attenuated feedback method. The LSDF theory has been used successfully to understand many bulk ground-state properties. For example, Moruzzi et al. (1978)have calculated ground-state properties for elements in the periodic table through the 4d series, and the agreement with experiment is typically about 10% for such properties as binding energies, lattice parameters, and bulk modulus. Similarly, the theory of itinerant electron magnetism has been considerably advanced in recent years by the success of ab initio self-consistent band-theory calculations in providing a quantitative understanding of the bulk groundstate properties to the ferromagnetic transition metals iron, cobalt, and nickel. [Indeed, the calculations (Moruzzi et al., 1978) predict that these are the only elemental ferromagnets.] These calculations have been remarkably successful in obtaining good agreement with such experimental quantities as magnetization, neutron-form factors, hyperfine field, lattice parameter, bulk modulus, cohesive energies, and Fermi surface properties. This is particularly impressive considering that all many-electron effects are included only through an effective one-electron local potential. This achievement is a major confirmation of the utility LSDF theory, which provides the formal justification for using the single-particle picture [Eqs. (2)-(4)] to determine groundstate properties. While the ground-state properties are now quantitatively understood on this basis, this is not true, unfortunately, for the elementary excitations of many-electron systems as measured, for example, by photoemission. Spindensity-functional theory provides the formal justification for using singleparticle band structures and charge densities to obtain ground-state properties only. There is, however, no formal sanction to interpret the band structure in terms of effective independent quasi-particle states (Kleinman and Mednick, 1981;Treglia et al., 1980;Liebsch, 1981), although the agreement (especially in metals) is sometimes surprisingly good. Of the three ferromagnetic elements, for example, the limitations of spin-density-functional theory are most apparent in Ni where photoemission experiments (Eastman et al., 1980;Plummer and Eberhardt, 1980) map out energy bands that differ considerably from those predicted by theory (Wang and Callaway, 1977). Thus, for example, the average ferromagnetic spin splitting of the Ni bands is predicted to be about twice as large as what is actually observed. In order to improve the theoretically obtained excitation energies, further many-body corrections must be applied. Strictly speaking, however, the com-
THEORY OF SURFACE ELECTRONIC STRUCTURE
363
monly derived perturbation-type many-body corrections (Kleinman and Mednick, 1981) apply to the Hartree-Fock approximation and not energy bands determined by using LSDF theory. Given the good agreement with experiment often obtained (especially in metals) using LSDF, however, it is desirable to perform the most accurate LSDF calculation that is possible within the spirit of using the best possible results for the homogeneous interacting electron gas. [It has been shown, however, (Perdew and Norman, do not 1982) for tiqhrly bound electrons that the exact orbital energies )l(!& accurately predict these excitation energies. By exact, we mean the ci derived from the exact functional form of ViL1).] Furthermore, there are indications that some discrepancies (e.g., band gaps in semiconductors and insulators) between LSDF calculations and experiment are due (to a significant extent) to so-called self-interaction (Perdew and Zunger, 1981; Gunnarsson and Jones, 1981) and other effects (Alonso and Girifalco, 1978; Langreth and Mehl, 1981) that are related in part to neglected nonlocal contributions to the correlation potential. These authors have investigated possible ways to incorporate these effects into a form that is practical for extended systems, i.e., a form as tractable as the LSDF approach. Thus, before considering possible many-body corrections, it is important to optimize the LSDF calculations. Vosko et a/. (1980) have recently presented an improved expression for the spin-dependent exchange-correlation potential, which is based on more accurate numerical results (Ceperley, 1978; Ceperley and Alder, 1980) over a wide range of densities for the interacting electron gas. Accuracy over a wide range of densities is especially desirable in surface calculations, where the charge density varies rapidly at the vacuum interface. Definitive results using some of these new approaches have not yet been obtained, however. 111. APPROACH AND METHODOLOGY
Currently, one of the most successful structural models for ab initio surface calculations is the single-slab (or thin-film) geometry. A film thickness of five to ten atomic layers is usually sufficient to obtain bulk-like properties in the center of the film and, consequently, true surface phenomena on the two film- vacuum interfaces. As discussed in the previous section, density-functional theory (Hohenberg and Kohn, 1964; Kohn and Sham, 1965) provides an elegant and powerful framework to describe the electronic structure of condensed systems (e.g., bulk crystals, surfaces, interfaces). In its local approximation, density-functional theory leads to Schrodinger-like one-particle equations (Kohn-Sham equations) containing an effective potential energy operator that is determined by the self-consistent charge distribution. Thus, the local density-functional one-particle equations must be solved iteratively.
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E. WIMMER, H. KRAKAUER, AND A. J. FREEMAN
One of the most accurate and powerful schemes to solve the local (spin) density (LSD) one-particle equations for the film geometry is the all-electron full-potential linearized-augmented-plane-wave (FLAPW) method (Wimmer et al., 1981a). The basic idea in this variational method is the partition of real space into three different regions, namely, spheres around the nuclei, vacuum regions on both sides of the film, and the remaining interstitial region. In each of these regions the “natural” form of the variational basis functions is adopted, i.e., plane waves in the interstitial region, a product of radial functions and spherical harmonics inside the spheres, and in the vacuum a product of functions that depend only on the coordinate normal to the film and two-dimensional plane waves. Each of these basis functions is continuous in value and derivative across the various boundaries. This is possible because inside the spheres (and analogously in the vacuum) two radial functions for each l-value are used, namely, the solution of the radial Schrodinger equation for the current potential and its energy derivative. In the FLAPW method, no shape approximations are made to the charge density and the potential. Both the charge density and the effective one-electron potential are represented by the same analytical expansions, i.e., a Fourier representation in the interstitial region, an expansion in spherical harmonics inside the spheres, and in the vacuum two-dimensional Fourier series in a set of planes parallel to the surface. The generality of the potential requires a method to solve Poisson’s equation for a density and potential without shape approximations. This is achieved by a new scheme that goes beyond the Ewald method (Weinert, 1981).The key idea in this new scheme is the observation that the potential outside a sphere does not depend on the actual charge density inside the sphere but only on its multipole moments. Now, Poisson’s equation is solved straightforwardly when the charge density is given in a Fourier representation. Because of the sharp structure of the charge density in the core region (including the nuclear charge), a Fourier expansion of the total density would be extremely slowly convergent. However, since the potential outside the sphere depends on the charge inside only through the multipole moments, the true charge density can be replaced by a smooth density that has a rapidly converging Fourier series and the same multipole moments as the true density. With this replacement of the density inside the spheres, we have a Fourier expansion of a charge density that gives the correct potential outside and also on the sphere boundaries. To find the potential inside the sphere, we are faced in a final step with a standard boundary-value problem of classical electrostatics, which can be solved from the original charge densities inside the spheres and the potential on the sphere boundaries by a Green’s function method. Thus, the FLAPW method allows a fully self-consistent solution of the LSD one-particle equations for the film geometry and yields charge densities
THEORY OF SURFACE ELECTRONIC STRUCTURE
365
and spin densities close to the LSD limit. Aside from the total charge density, the key quantity in density-functional theory is the total energy corresponding to the ground-state charge density. We have presented a new scheme (Weinert et al., 1982) to calculate accurate and stable all-electron densityfunctional total energies and have applied it within the FLAPW method. The capability of total energy calculations for various geometrical arrangements provides us with a powerful theoretical tool to study the energetics and, at least in principle, the dynamics of surfaces and overlayers. We now highlight the key formulas of the FLAPW method without making use of special symmetries, such as inversion symmetry or mirror reflection on the central plane of the film. In practice, these symmetries are of great advantage, since the inversion symmetry makes the Hamiltonian and overlap matrix real and the mirror reflection breaks these matrices into two blocks according to even and odd states. For a thin-film geometry and within the LDF approach, the wave functions for each state are solutions of the one-particle equations where k is a vector of the two-dimensional first Brillouin zone and i is a band index. The effective potential V,,, is given as the sum of the electrostatic Coulombic potential, related to the charge density by Poisson’s equation, and the local exchange-correlation potential as obtained from many-body theory. In Eq. (9,Rydberg atomic units are used (.ti2/2m= 1, e2 = 2). In the FLAPW method, the wave function of each state is expanded variationally in the reciprocal lattice
where each of the basis functions is an augmented-plane-wave given by for r E interstitial
(7a)
1 CAL(Kj)uI(EI,4 + 4 , ( K j ) W , , r)lYL(4
for r E sphere (7b)
Q-’’2
exp(iKj r)
4 W j ) =‘
C CAq(Kj)Uq(Ev, 2 ) + Bq(Kj)Gq(Ev, 211 4
with Kj = k + G j . Here, G j is a vector of a three-dimensional reciprocal lattice defined in terms of the auxiliary periodicity domain 0” (Fig. 1). The reason for choosing d larger than D is simply to gain a greater variational freedom in the basis functions. (For d = D, the charge density would have, at the vacuum boundary, an artificial zero slope in the direction perpendicular to the surface.) Here, i2 is the volume of the unit cell between the vacuum
366
E. WIMMER, H. KRAKAUER, A N D A. J. FREEMAN
f t t D 7 2
1 - D72 FIG. 1. Thin-film geometry as used in the FLAPW method. Here, I, 11, and 111 are the spheres, the interstitial, and the vacuum regions, respectively. Note that the vacuum regions start at D/2; d provides an auxilliary periodicity domain as discussed in the text. [After Krakauer et al. (1979b).]
boundaries at fD/2; u,(E,, r ) are solutions of the radial Schrodinger equation obtained with the actual spherical part of the effective potential inside a sphere for a fixed energy E,, and zi,(E,, r ) is the energy derivative of this radial function. The coefficients AL(Kj) and B,(Kj) are determined by the requirements that the plane wave Eq. (7a) be continued smoothly in value and derivative across the sphere boundaries. Similarly, in the vacuum, u,(E,, z ) are solutions of the equation [ -(dz/dz2)
+ V(Z)
E,
+ (k + K,)2]~,(E,,
=0
(8) where V ( z ) is the component of the effective potential in the vacuum, which depends only on the distance perpendicular to the surface, E , an energy parameter for the vacuum analogous to the parameters E , inside the muffin-tin spheres, zi,(E,, z ) the energy derivative of the function uq(Ev,z), and K, denotes a two-dimensional (i.e., parallel to the surface) reciprocal lattice vector. The matching coefficients A,(Kj) and B,(Kj) are determined by the continuity conditions of 4(Kj) across the vacuum boundaries at & D/2. In the FLAPW method for thin films, the electronic charge density is represented in each of the three spacial regions by the natural representation, namely, -
Z)
for r €sphere
I?
-
p,(z) exp(iK, r)
for r E vacuum
THEORY OF SURFACE ELECTRONIC STRUCTURE
367
The electrostatic potential is obtained from the electronic charge density and the nuclear charges by solving Poisson's equation using the technique described by Weinert (1981) as implemented into the FLAPW method (Wimmer et ul., 1981a). The exchange-correlation potential is calculated from the local electronic charge density by a least-squares-fitting technique where the root-mean-square deviation is usually about 1 mRy in the interstitial region and better than 0.1 mRy inside the spheres and in the vacuum. Finally, the effective one-electron potential (as the sum of the electrostatic Coulombic potential and the exchange-correlation potential) is represented in the form completely analogous to the charge density as given by Eqs. (9a)-(9c). The LD one-particle equations [Eq. (5)] are now solved iteratively. A starting density in the form of Eqs. (9a)-(9c) is constructed from a superposition of self-consistent atomic densities. From this density the corresponding potential is calculated, which defines the effective one-particle operator in Eq. (5). Using the expansion [Eq. (6)] and the explicit form [Eqs. (7a,b)] of the basis functions, the coefficients c i j of Eq. (6) are obtained via a Rayleigh-Ritz variational procedure. These coefficients now define the film wave functions $i(k), which in turn yield, according to Fermi-Dirac statistics, a new charge density p"r>
= e2
1 JBZ$f(k. r)$i(k, r) d 2 k
(10)
occ
where the summation runs over all occupied states. The density of the core electrons is obtained by solving fully relativistically a free atom-like problem using the current effective potential. This completes one iteration cycle. The new density is fed back and self-consistency is achieved when p' = p , i.e., when the output density is equal to the input density. In practice, self-consistency is assumed when the potentials corresponding to the input and output densities differ on the average by less than about 1 mRy. The self-consistency procedure is accelerated by using an attenuated feedback, i.e., the new input density is a mixture of, say, 95 % of the input density and 5 % output density of the previous iteration. Faster convergence is achieved by employing a more sophisticated scheme (Andersen, 1965) involving the input and output densities of two previous iterations.
Iv. EXAMPLES OF APPLICATIONS In the modern theory of the electronic structure and the electronics of condensed systems, simple metals such as Na and A1 became the show horses of new approaches. Thus, it is almost unavoidable to start a discussion of the applications and examples with a simple metal such as aluminum.
368
E. WIMMER, H. KRAKAUER, A N D A. J. FREEMAN
A . Aluminum Surfaces 1. Surface States and Surface Resonance States
a. Al(001) The first directly observed occupied surface state on any simple nearly-free-electron (NFE) metal was reported by Gartland and Slagsvold (1 978) using angle-resolved photoemission measurements on the (001) face of aluminum. A dominant surface-sensitive peak was interpreted as emission from a two-dimensional band of surface states. Spectra recorded along the f - R line in the two-dimensional Brillouin zone yielded an experimental dispersion relation for this peak that is parabolic with an effective mass m* = (1.03 & 0.l)m. The nature and the origin of these surface states and surface resonance states are readily seen when we start our considerations with the bands of a three-dimensional empty lattice, project these bands onto the two-dimensional surface Brillouin zone, proceed from there on to the projected bands of a realistic bulk A1 calculation, and then compare with the results of a surface (single-slab) calculation. Figure 2 shows projected bulk free-electron bands,
(Caruthers et al., 1973), for a few fixed values of k,, the z component of the three-dimensional Bloch momentum. Here, k l l and k, refer to the slabadapted (three-dimensional) Brillouin zone (BZ), and the three-dimensional reciprocal lattice vector G refers to the primitive BZ. In Fig. 2b we have reproduced from Caruthers et al. (1973) those projected bulk bands with the same values of k , displayed in Fig. 2a. Absolute bulk band gaps are represented by the shaded regions. The band structure for the nine-layer Al(001) film, shown in Fig. 2c, was calculated non-self-consistently (Krakauer et al., 1978) for 41 k,,points along f - R to about 10 mRy accuracy using the film-muffin-tin (FMT) potential. The plus and minus signs in Fig. 2c label states that are, respectively, symmetric and antisymmetric with respect to z reflection (i.e., in the central plane of the film). We want to emphasize that the film bands, although seemingly complicated at first glance, are essentially free-electron-like as can be seen from the following. Turning on the bulk crystal potential causes bands with the same value of k, in Fig. 2a to repel one another (all the bands shown in Fig. 2 have the same two-dimensional A 1 symmetry, i.e., there are no A2 bands in the energy range shown), and this causes bulk energy gaps to appear. The resulting band structure, pictured in Fig. 2b, remains predominantly parabolic as expected for a NFE metal-like aluminum. Introducing the perturbation due to the presence of the surface destroys periodicity in the z direction, with the result that k, is no longer a
THEORY OF SURFACE ELECTRONIC STRUCTURE
369
0 1.0
0 1.0
0 10
2 8
.2 .8
5 .8
'.2 1.0
85
M
:
i
0
-0.8-
I
2
.2
0 - o'6 -.5 I=
0
R
I
I=
I
R
FIG.2. (a) Free-electron bulk bands. (b) Projected bulk bands [After Caruthers et al. (1973)l. Absolute bulk band gaps are represented by the shaded regions. A partial Bragg-reflection gap is represented by vertical cross hatching. (c) Band structure for a nine-layer Al(001) film. The relevant pair of surface states and surface resonances is identified by the heavy lines. The numbers that label the bands in (a) and (b) represent values of k , in units of 27c/A, where A is the bulk lattice parameter. [After Krakauer et al. (1978).]
good quantum number. Crossings of bulk bands with different k, in Fig. 2b must now become anticrossings in the film calculation, Fig. 2c. The only allowed crossings in Fig. 2c are between states of different z-reflection symmetry. As the film becomes thicker, however, k, is more nearly a good quantum number, and these additional sharp anticrossings begin to look more and more like true crossings. In view of these remarks, we can see the close similarity of the film bands (Fig. 2c) with the projected bulk bands (Figs. 2a and 2b). The other new feature in the film calculation is the occurrence of surface states and surface resonances. For comparison with experiment, the relevant pairs of surface states and surface resonances are identified in Fig. 2c by heavy lines, which clearly have a free-electron-like dispersion. The surface states start at and run up to about k = (0.5,O) within a region corresponding to the bulk band gap in Fig. 2b. These surface states then persist as a pair of surface resonances into a region corresponding to the continuum of bulk states and within the partial
r
370
E. WIMMER, H. KRAKAUER, A N D A. J. FREEMAN
Bragg reflection gap in Fig. 2b (represented by the vertical cross hatching), again becoming true surface states as they enter the smaller absolute band gap near R. In agreement with the results of Caruthers et al. (1973), we see two pairs of surface states in this gap; one of these pairs is the continuation of resonance states. While these surface states have been found previously (Boudreaux, 1971; Caruthers et al., 1973), Krakauer et al. (1978) focused for the first time on the existence of the surface resonance. The dispersion of the heavy lines in Fig. 2c agrees extremely well with the experimental dispersion relation. The difference between the Fermi energy and the energy of the surface state (average of symmetric and antisymmetric state) at is 2.97 eV, which compares excellently with the experimental value (Gartland and Slagsvold, 1978) of 2.8 rt 0.2 eV. The ratio m*/m is found to be 1.04 f 0.03 and Gartland and Slagsvold (1978) report 1.03 rt 0.1. The agreement with experiment is remarkably good here. The experiment, which is sensitive to states localized near the surface, thus detects a true surface state for k,,less than about (0.5,O) and a surface resonance for larger k ,confirming the interpretation of Gartland and Slagsvold (1978). Contour plots of the true surface state at fi (Fig. 3a) and the surface resonance state (Fig. 3b) midway between and X show the similarity of these
r
,
r
r
FIG. 3. Contour plots of the charge density for (a) the clean A1 surface state at and (b) the Successive contours differ by 0.2 in units of clean A1 surface resonance state midway between electrons per bulk A1 unit cell, [After Krakauer et al. (1981).]
rx.
THEORY OF SURFACE ELECTRONIC STRUCTURE
37 1
states, the localization on the surface atoms, and the rather slow decay into the bulk. The formation of surface resonances in partial Bragg reflection gaps in NFE metals is a particularly striking and illustrative example of a general mechanism that can also be present in more complex systems. Thus, we could generally expect to find surface resonances at points in the two-dimensional BZ, where (in corresponding projected bulk band structure for a single twodimensional symmetry type, e.g., A , , or A 2 along + X)a bulk band edge of one character is embedded in a continuum of bulk states of another character. In their general review, Appelbaum and Hamann (1976) refer to this type of situation. While in NFE-like metals it is particularly easy to map such partial gaps (since the character of the bulk states is simply related to their plane-wave composition), a similar mapping for general systems would require inspection of projections of individual bulk bands. This could prove useful not only as a relatively simple model for predicting the occurrence of this class of surface resonances in complicated calculations of surface electronic structure, but also as a guide for experiments searching for states that are localized near the surface.
r
h. Al( 1 1 I ) Angle-resolved photoemission studies by Hansson and Flodstrom (1978) have revealed the band structure of surface states for several low-index Al surfaces including the ( 1 11) face. Theoretically, after early calculations by Boudreaux (197 1)and Carutherset al. ( 1974),Chelikowsky et al. ( 1 975) found from self-consistent, semiempirical pseudopotential studies states at fi that decay very slowly toward the bulk interior and some surface states at K in the two-dimensional BZ. Wang et al. (1981a) performed the first ah initio self-consistent all-electron calculation using the linearizedaugmented-plane-wave (LAPW) method. In addition to the surface states at R, Wang et u1. (1981a) found another set near that are in very good agreement with the position and k dispersion observed by photoemission (Hansson and Flodstrom, 1978). Figure 4 gives the band structure of the surface states of the Al( 1 1 1 ) face. For reference, the bottom of the conduction band is indicated by full lines and the insert shows the two-dimensional Brillouin zone of the Al(111) surface. The charge density of the surface state at K = 0.58 eV below the Fermi energy is shown in Fig. 5. This state is, to a large extent, localized in the plane of the surface atoms and shows an interesting amplitude in the second layer below the surface. In contrast to this situation, the surface state at T M is essentially localized between the surface and subsurface atoms (Fig. 6). The existence of surface states near PM can be understood by considering certain features of the three-dimensional band structure of the bulk Al. First we plot, as Fig. 7, the cross section of the three-dimensional Brillouin
a
3 72
E. WIMMER, H. KRAKAUER, AND A. J. FREEMAN
4 .O 2 .o
-> -
0
a
-2.0
>
(3
lx
-4.0
W
W
.
-6.0
'
/I
-8.0
-I 0.0 -12.c -14.0
T
R
Tr
i=
FIG.4. Band structure of nine-layer Al(111) films. Larger crosses inside circles denote the position of the surface states with more than 60% of the electronic charge on the first two layers. The dashed line near R shows the states with an electronic charge slightly less than 60% on the first two layers. Insert is the two-dimensional BZ of the Al(111) surface. [After Wang et a/. (1981a).]
zone in the plane with k , as the vertical axis and k, the two-dimensional wave vector under consideration, along the horizontal axis. The other (dashed) lines in this figure arise from the intersection of the second Brillouin zone with the plane containing z and k (in this case, k = k , , is parallel to [2TT]). For k at A?, the two band gaps corresponding to the boundary of the first three-dimensional Brillouin zone have different magnitudes of k,. The gap at the X point with k , = 2n/$a will be crossed by the continuum of states around k, % - 2 n / a a , which would have the same energy in the nearly-free-electron approximation. Similarly, the gap at the L point with kL = -n/$a is crossed by the states around k , FZ n / f i a . This analysis,
373
THEORY OF SURFACE ELECTRONIC STRUCTURE
i FIG. 5. Contour plot of the charge density for the surface state at I? with E = -0.58eV. The area shown is near the surface and on the vertical plane through the two inequivalent threefold sites. The steps between successive contours are 0.045e/at. [After Wang et al. (1981a).]
based on the shape of the boundary of the Brillouin zone and the nearly-freeelectron approximation, agrees with the numerical results for the bulk A1 band structure as shown in Fig. 8. This figure, drawn by using the data listed by Snow (1967), shows the projection of the three-dimensional energy band onto one point M of the two-dimensional Brillouin zone. When k decreases from A to 1 T M , the two band gaps occurring at the U and K points on the boundary of the first Brillouin zone have the same k, magnitude, and the crossing will become smaller. This is verified by projecting the three-dimensional energy band onto the 3 rR point, as shown in Fig. 8b. The smaller crossing gives rise to the possibility of the existence of a more localized surface state (or a surface-resonance state). In addition to this group of surface states, the preceding most-localized surface states near K also exist inside the band gap, which arises from the V’,, component of the potential. This is quite natural because, as is well
374
E. WIMMER, H. KRAKAUER, A N D A. J. FREEMAN
FIG. 6. Contour plot of the charge density for the surface state with K at a r M and E = - 1.05eV. The area shown is near the surface and on the vertical plane through the two inequivalent threefold sites. The steps between successive contours are 0.045 e/at. [After Wang et al. (1981a).]
known, V,,, is the largest potential component in A1 and so the band gap is widest at these points. The same situation also occurs for Al(100) and (1 10) surfaces (Hansson and Flodstrom, 1978; Krakauer et al., 1978,1979b).
2. Charge Densities: Screening at the Surface The fundamental quantity in density-functional theory is the charge density. Thus, we present in Figs. 9 and 10 these charge densities for the Al(001) and the Al(111) surfaces, respectively, in a plane perpendicular to the surfaces. It is amazing to observe how rapidly the influence of the metal-vacuum interface is screened out: even the subsurface atoms have an almost bulk-like electronic environment. The bonding characteristics of the inner layers of the slab are typically metallic with a fairly constant charge density between the
THEORY OF SURFACE ELECTRONIC STRUCTURE
/
375
/
/’
I
I I
I
I
\
I
\
I
\ \ \
/
\
/
/
FIG. 7. The intersection of the three-dimensional Brillouin zone with the plane containing the film normal [ I 1 I ] and the interesting two-dimensional vector K = [fll]. Solid and dashed lines are the intersection with the boundary of first and second three-dimensional Brillouin zones, respectively. [ A h Wang et ul. (I98la).]
atoms (the average charge density is three electrons per bulk unit cell) with slight directional bonding along the body diagonals. There are substantial differences, however, in the surface layer. Unlike the bulk, there is a rapid variation in p(r) in the surface interstitial region, with p(r) decreasing in magnitude toward the vacuum. Proceeding outward into the vacuum from the surface, p(r) falls off sharply and soon “heals” the discrete atomic nature of the surface, i.e., p(r) becomes nearly constant with respect to translation parallel to the surface. This sizeable redistribution of charge near the surface is, of course, associated with the formation of the surface dipole layer, which sensitively determines the work function. The role of surface dipoles for the work function will be discussed later in connection with Cs on W(OO1). It is interesting to note that the charge density near the Al( 1 11) surface, averaged in planes parallel to the surface, exhibits no Friedel oscillations in agreement with a simple jellium model but in disagreement with the semiempirical pseudopotential calculation by Chelikowsky et al. (1 975).
376
E. WIMMER, H. KRAKAUER, AND A. J. FREEMAN
E
0
-112 kZ
-
l '2 LIIII
J3 x 27r/a)
E
U
1/2 k z (J3 x 2s/ol FIG. 8. Projection of the three-dimensional bulk bands onto two-dimensional G points. (a) Gat M and (b) kat 3 PM. Data points are taken from Snow (1967).
3. 2p Core-Level Shifts and Crystal Field Splitting at the Al(001) Surface Eberhardt et al. (1 979) found from x-ray photoemission spectroscopy (XPS) measurements on an Al(001) surface that, quite surprisingly, the 2p line from the surface layers was not shifted within an experimental uncertainty of 40 meV compared to the bulk signal, but the line was markedly broadened by 100-200 meV. These authors attributed the broadening to a crystal-field splitting of the initial states in the surface layer and supported this idea by a simple atomic model calculation.
+
THEORY OF SURFACE ELECTRONIC STRUCTURE
377
FIG. 9. Contour plot of the self-consistent total valence charge density for the upper half of the clean nine-layer AI(001) slab. Successive contours differ by 0.4in units of electrons per bulk A1 unit cell. [After Krakauer ef a!. (1981).]
Although a comprehensive interpretation of photoemission spectra must take into account the complicated scattering mechanisms of the photoelectron, electron-hole interactions, secondary-processes-like Auger transitions, and many-body relaxation effects, the starting point of a theoretical investigation must be an accurate calculation of the initial ground state. As it turns out in many cases, the electronic structure of the initial ground state alone determines the main features of the photoemission spectrum. Wimmer et al. (1 98 1b) have carried out a self-consistent calculation for the nine-layer Al(001) film using the full-potential linearized augmented-plane-wave (FLAPW) method. They find for the A1 2p states in the surface layer a shift to smaller binding energies of 120 meV accompanied by a crystal-field
378
E. WIMMER, H. KRAKAUER, AND A. J. FREEMAN
FIG. 10. Contour plot (in steps of 0.27e/at.) of the self-consistent total charge density for the outer three layers of a nine-layer Al( 1 1 1) film. The area shown is on a vertical plane through the two inequivalent threefold sites. The area labeled Al is the core region with a very high charge density. [After Wang et al. (1981a).]
splitting of 39 meV for the 2p states. The core-level shift in the subsurface layer is decreased to 50 mRy and vanishes in the third layer. A pronounced crystal-field splitting is found only for the surface layer. Simultaneously and independently, Chiang and Eastman (1 98 1) derived from surface-sensitive - 57 meV and a photoelectron partial yield spectra a 2p core-level shift of much smaller surface-sensitive broadening than did Eberhardt et al. (1979). The theoretical results indicate that both effects (shift and splitting) are important in analyzing the experimental data.
-
THEORY OF SURFACE ELECTRONIC STRUCTURE
319
I
I
surface
-69.4
-69.2
-69.0
I
-68.8
E (eV) FIG 1 I . FLAPW one-electron energy eigenvalues for the 2p core stales in a nine-layer Al(001) film. The values are given with respect to the vacuum. The dashed lines are the values for the 2p,,, states without crystal-field splitting. [After Wimmer et al., (1981b).]
The FLAPW one-electron energy eigenvalues for the 2p core states including the crystal-field splitting are plotted in Fig. 11. We observe a shift of 120 meV to smaller binding energies for the 2p,,, and 2p,,, levels in the surface layer compared to the central (bulk-like) layer. A similar shift, although reduced to 50 meV, is also found for the subsurface layer (S-l). In layer S-2, the surface-induced shift is almost completely screened, but we notice a slight oscillation in layers S-2, S-3, and the central layer. The 1s and 2s core states show essentially the same shifts as the 2p levels. As expected, the crystal-field splitting is localized to the surface layer, and almost no splitting is found for the subsurface layers. It is interesting to note that in the case of A1 not only the core levels of the surface atoms but also those of the subsurface atoms are shifted. This indicates that for a nearly-free-electron metal the influence of the surface is screened off after two atomic layers.
-
4. Initial Oxidution of’ A1 Surfuces A fundamental understanding of the process of metallic oxidation remains as one of the basic problems in surface chemistry and physics, one which has wide-reaching consequences of great technological importance. Over the last few years, evidence has accumulated which indicates, not surprisingly, that the complex process of oxidation may proceed through stages that are conceptually quite distinct. In particular, the initial oxidation state
380
E. WIMMER, H. KRAKAUER, A N D A. J. FREEMAN
(perhaps involving chemisorbed overlayers of oxygen) has been extensively studied experimentally, usually on well-characterized single-crystal surfaces. One of the most notable examples, which has been the subject of intense experimental investigation (Grepstad et al., 1976; Gartland, 1977; Hofmann et al., 1979; Flodstrom et al., 1976,1978; Yu et al., 1976; Bachrach et al., 1978; Eberhardt and Kunz, 1978; Martinson and Flodstrom 1979a,b; Bianconi and Bachrach, 1979; Bianconi et al., 1979a,b; Eberhardt and Himpsel, 1979; den Boer et al., 1984; Johansson and Stor, 1979; Barrie, 1973), is the study of the initial oxidation of A1 surfaces. Recent work on well-characterized single-crystal surfaces has shown that there are basic differences in oxygen chemisorption on the low-index face of Al. While the (1 11) surface seems to be the best characterized face, less is known about oxide formation on the (100) and (110) faces. Based on Auger electron spectroscopy (AES) and work-function measurements, Gartland (1977) proposed that thin ( - 5 A) islands of Al,O,-like oxide form on the (100) face. While the work function changes only slightly (-0.1 eV) with oxygen exposure on the (111) and (110) surfaces, the work function decreases almost linearly (Grepstad et al., 1976; Gartland, 1977; Hofmann et al., 1979) with coverage on the Al(100) surface to a saturation difference of 0.5-0.8 eV. This behavior was suggested by Gartland (1977) to correlate with an island growth mechanism for this surface. Oxygen absorbed outside the surface, however, would be expected to cause an increase in work function as a result of its high electronegativity. Thus, the large decrease of the work function on the (100) surface has been taken as evidence for incorporation of oxygen atoms on the (100) surface-the most likely site being the fourfold hollow position. The oxidation of the (100) surface may then proceed by filling the fourfold sites together with the formation of A1,0, islands (Martinson and Flodstrom, 1979a). A recent surface extended appearance potential fine structure (den Boer et al., 1980) (EXAPFS) study also suggests the filling of the fourfold hollow site, since the A1-0 spacing for 1.5-monolayer coverage is found to be 1.98 k 0.05 A, which is consistent with a bond length of 2.02 A for the fourfold site. Photoemission studies (Flodstrom et al., 1976, 1978; Yu et al., 1976; Bachrach et al., 1978; Eberhardt and Kunz, 1978; Martinson and Flodstrom, 1979a,b; Bianconi et al., 1979a,b; Eberhardt and Himpsel, 1979) of the valence and core regions have also revealed qualitative differences between the three low-index faces of A1 on exposure to oxygen. Substrate core chemical shifts upon chemisorption of oxygen and A1 have been observed (Flodstrom et al., 1976, 1978; Eberhardt et al., 1979) on all three faces, and there are significant differences between the different faces. The (1 1 1) surface is characterized by the appearance of two chemically shifted oxygen-derived A1 2p core-level peaks at greater binding energies. Below 100 L on the (1 11) face, a
THEORY OF SURFACE ELECTRONIC STRUCTURE
38 1
peak first appears at 1.4 eV toward greater binding energy, which is interpreted as an ordered chemisorption phase acting as a precursor stage for oxidation (Flodstrom et al., 1978). At higher exposures, a peak at 2.7eV greater binding energy begins to grow, indicating the formation of the bulklike Al,O,-phase (Barrie, 1973). Low-energy electron diffraction (LEED) measurements (Flodstrom et al., 1978) also support the formation of a (1 x 1) chemisorbed phase at low coverages on the (1 11) face as does a surface EXAFS study of this surface (den Boer et al., 1980). These two chemically shifted peaks have also been observed on the (100) and (1 10) faces (Eberhardt and Kunz, 1978) with the difference that both peaks seem to be present with nearly the same relative intensities at low coverages. Eberhardt and Kunz (1978) concluded that, taken together with the work-function decrease, the existence of the 1.4-eV peak on the Al(001) surface is consistent with 0 atoms having penetrated into the bulk. As noted earlier, EXAPFS studies support this conclusion (den Boer et al., 1980). More recently, a surface soft-x-ray absorption (SSXA) spectroscopy study has been reported for the three low-order faces (Bianconi et al., 1979a,b). By using an interatomic Auger transition A1(2p)-0(2s), evidence for an oxygen chemisorption phase below 50 L and the existence of unoccupied interface states on these surfaces was presented. a. AI(001) A few theoretical calculations have been carried out for A1 surfaces exposed to oxygen. Lang and Williams (1975, 1978) performed selfconsistent jellium calculations for an isolated 0 atom absorbed onto a jellium substrate. Messmer and Salahub (1977) and Salahub et al. (1978) reported self-consistent cluster calculations designed to model possible surface site configurations for the absorbed oxygen atoms and concluded that the coplanar fourfold hollow site was the most likely. Harris and Painter (1976) performed a non-self-consistent calculation for a similar but smaller cluster. Non-self-consistent slab or thin-film calculations have been performed for the (001) surface by Painter (1 978) and Batra and Ciraci (1977). Krakauer et al. (1981) presented the first self-consistent calculation for an ordered (1 x 1) oxygen monolayer on the Al(001) surface aimed at investigating the initial oxidation of this surface. In order to assess changes, they first studied a clean nine-layer Al(001) slab and then the same slab with a (1 x 1) monolayer of oxygen atoms on each surface located in the fourfold hollow sites, i.e, coplanar with the surface A1 atoms. Contour plots of the self-consistent charge density of the oxidized AI(001) surface (Fig. 12a) and the difference between this charge density and the self-consistent result for the clean Al(001) surface (Fig. 12b) and superposed atomic charge densities (Fig. 12c) give insight into O-induced charge rearrangements. Upon going in toward the center of the film in Fig. 12a, the
382
E. WIMMER, H. KRAKAUER, AND A. J. FREEMAN
FIG. 12. Contour plots of (a) the self-consistent total valence charge density for the upper half of the A1(001)/0 slab, (b) the difference between the A1(001)/0 self-consistent total charge density and that for clean A1(001), and (c) the difference between the A1(001)/0 self-consistent total charge density and the non-self-consistent starting charge density (constructed as a superposition of overlapping spherical atomic charge densities) for the A1(001)/0 slab. Successive contours differ by 0.4 in units of electrons per bulk Al unit cell. [After Krdkauer et al. (1981).]
charge density becomes essentially identical to its bulk value by the third layer in from the surface. Near the center of the film, the bonding is essentially identical to that exhibited by the clean A1 surface. This is emphasized by the difference contour plot in Fig. 12b, which reveals only very small changes in the interior of the nine-layer A1 slab. To appreciate fully the significance of this surface-screening effect, compare Figs. 12b and 12c. The superposition charge density (Fig. 12c) exhibits substantial differences throughout the slab, but after achieving self-consistency for both the clean and oxygen-covered surfaces, only minimal differences are evident in Fig. 12b. Thus, the nine-layer slab of A1 is seen to be completely adequate for treating both the bulk and surface electronic structure of the oxidized surface. Between the second and third layers in from the surface, however, departures from bulk behavior become apparent (Fig. 12). The second-layer A1 atom and especially the surface A1 atom show marked changes in the charge density, reflecting the strong influence of the neighboring 0 atom in the surface layer. It is apparent from Figs. 12b and 12c that the bonding is largely
THEORY OF SURFACE ELECTRONIC STRUCTURE
383
ionic in nature, i t . , the radius of the nearly spherical charge density centered on the 0 atom ( - 1.3 A) in Fig. 12b is quite close to the ionic radius of 0’(1.4 A) (Slater, 1972). Another interesting feature in Figs. 12b and 12c is the pileup of bonding charge just below the A1 surface layer. This feature is also apparent in Fig. 12a, reaching a maximum of about 4.2 e/cell. The pile-up of electronic charge just above the surface A1 atom is accompanied by a depletion of electronic charge just above the surface A1 atom. This results in the formation of a dipole moment (opposing the usual surface dipole layer), which tends to reduce the work function as compared to that for the clean A1 surface, as discussed later. Another effect of the charge transfer onto the 0 atom is that the O(2p) states experience an increased screening of the 0 nucleus. This results in the expansion of the O(2p) density, which is strikingly pictured in Fig. 12c for the O(p,, p,) states (i.e., the contours near the 0 nucleus have large negative values, indicating the movement of electrons away from the nucleus compared to the free atom). In addition, the downward bulging of the contours below the 0 atom in Fig. 12c reveals the expansion of the 0(2p,) states as well as the effects of interaction with the substrate. The substantial change in the surface charge density between the clean surface and the 0-covered surface leads to a reduction in the work function of 0-Al(001) as compared to AI(001) by 0.6 eV. This value is in good agreement with the 0-saturated value of 0.5-0.8 eV for the Al(001) surface (Grepstad ef ul., 1976). This is a significant result, since the saturation change in work function on Al( 110) and Al( 11 1) surfaces is only on the order of about 0.1 eV (Grepstad et al., 1976; Gartland, 1977). Thus, the anomalous behavior of the work function for the (001) surface is seen to be associated with the following: (1) The open structure of the (001) surface, which permits the 0 atoms to be incorporated into the fourfold hollow site in a coplanar position with respect to the A1 surface atoms. Coplanar absorption seems unlikely in the hollows of the more open (1 10) surface, as this would require a 2.43-A A1-0 bond length between the 0 atom and the subsurface A1 atom, a value much smaller than the shortest bond length, (1.86A) (Slater, 1972) in bulk A1’0,. (2) The large charge transfer onto the 0 atom and the accompanying formation of a dipole moment localized on the surface A1 atom (Fig. 12b), which opposes the usual surface dipole barrier formed on the clean A1 surface. All of these results are consistent, therefore, with earlier suggestions that the 0 atoms are absorbed into the fourfold hollow sites of the surface layer. An overview of the electronic structure for the 0-covered surface is given by the layer-by-layer density of the states (DOS) present in Fig. 13. Since the
384
E. WIMMER, H. KRAKAUER, A N D A. J. FREEMAN
0 atoms are located precisely on the A1 surface layer, there are five layers going from the center of the film out toward the surface (note the change of the scale in the top panel of Fig. 13). Each of the layer-projected DOS curves shown in this figure has been smoothed with a Gaussian of full width at half maximum equal to 0.3 eV. The overall shape of the DOS for the innermost layers is parabolic, as is expected for a nearly-free-electron substrate metal like Al. There are prominent peaks in the surface layer DOS at about - 8.0 and - 10.0 eV below the Fermi energy E,, which are due to the O(2p) bands. The overall width of the O(2p) bands is seen to be about 3.5 eV, and the split peak structure indicates
FIG. 13. Layer-projected density of states for the A1(001)/0 slab. Note the change of scale for the surface-layer density of states. [After Krakauer et al. (1981).]
THEORY OF SURFACE ELECTRONIC STRUCTURE
385
a crystal-field splitting of the O(2p) states. These peaks are in good agreement with photoemission measurements (Bachrach et al., 1978; Eberhardt and Kunz, 1978). In addition to this structure, there are narrow peaks in the surface DOS above E,.. These structures are related to unoccupied oxygeninduced surface resonance states. We believe that they correspond to the unoccupied interface DOS at 1.0 eV (S1 and S2 in the works of Bianconi et al., 1979a,b) observed by surface soft-x-ray absorption (SSXA) spectroscopy. All of these oxygen-related structures are superimposed on the parabolic A1 background DOS. In the first layer in from the surface (S-1 in Fig. 13), the oxygen-related structures are greatly reduced in magnitude, and they have nearly vanished by layer S-2, thus reflecting the surface-localized nature of the chemical bond between the 0 atoms and the A1 substrate. This behavior is also reflected in the rapid healing of the charge density to bulk A1 character shown in Fig. 12. b. Al( 111) Experiments have established that a well-defined chemisorption phase exists on the Al( 111) surface at low oxygen exposure: The oxygen atoms form a p(1 x 1) overlayer (Martinson and Flodstrom, 1979a) and are located in the threefold hollow sites. This unique phase has a characteristic 1.4-eV shift to higher binding energy (Flodstrom et al., 1978) of the A1 2p core level. After high exposure ( > 50 L), another A1 2p resonance shifted 2.6 eV to higher binding energy appears and increases in intensity with exposure. Since the same 2.6-eV chemical shift in bulk A1,0, oxide (Barrie, 1973), its appearance is considered to be an indication of the formation of a layer with Al,O,-like oxide structure. The existence of a unique chemisorption phase on the Al(111) surface is also consistent with previous measurements of the variation of striking coefficient and work function with exposure on different surfaces (Gartland, 1977). Experimentally, the vertical position of the oxygen atom is, however, less certain. Gartland (1977) concluded from the observed increase of the work function that the oxygen is on top of the metal surface, because the inward dipole layer formed by the outward electron transfer from metal substrate to electronegative adatoms is consistent with experimental results. This conclusion is also quite reasonable from the atomic arrangement of the Al(111) surface. For fcc aluminum, (1 11) is the most compact surface and the in-plane distance between the threefold center and its adjacent A1 atoms is only 1.65 A, which is in bulk A1,0,. It is quite improbable for the oxygen atom to incorporate with or penetrate through the surface imperfections. Recently, extended x-ray absorption fine structure (EXAFS) has been used to determine the chemisorption bond length of oxygen on Al(111) surfaces (Johansson and Stohr, 1979). The 0-A1 length is found to be 1.79 _+ 0.05 A. This 0-A1 bond length corresponds to d, the distance between oxygen overlayer and metal surface layer, which is equal to 0.70 A. The 0-0 distance
386
E. WIMMER, H. KRAKAUER, A N D A. J. FREEMAN
is consistent with a p(1 x 1) structure. However, an analysis (Martinson et al., 1979; Yu et al., 1980) of the LEED intensity of oxygen on the Al(111) surface gives d = 1.33-1.468, which is quite different from the EXAFS value and corresponds to an 0 - A 1 bond length equal to 2.12-2.2& which is even larger than the bulk A1,0, bond length. Ab initio electronic structure calculations by Wang et al. (1981d) performed for both Al-0 distances clearly show better agreement in the work function and the position and dispersion of the O(2p) bands using a layer distance d of 0.70A. Thus, the theoretical study gives strong support for the EXAFS results. For the geometry with d = 0.70A, Wang et al. (1981~)obtained a charge density as shown in Fig. 14a. The ionic character of the adatom is easily seen from the fact that the radius of the concentrated region of nearly spherical charge density centered on the 0 atom (- 1.38) is quite close to the ionic radius (Slater, 1972) of 0’- (1.48). The total charge within the spherical region around the 0 atom (Y = 1.3 A) is 9.24e. This behavior contrasts with that of the ion core of A1 (as evidenced from the spherical charge density centered on the A1 atom), which does not change appreciably even for the surface A1 atoms in close contact with the adatom. The transfer of electronic
FIG. 14a. Contour plot of self-consistent total charge density shown for a vertical plane through the two adjacent inequivalent threefold sites (in steps of 0.3 e/Al at.) for (a) the five-layer Al(111) film with one oxygen overlayer on each side. [After Wang et al. (1981c).]
387
THEORY OF SURFACE ELECTRONIC STRUCTURE
charge obtained by the electronegative 0 atoms comes from the Al conduction bands. The charge transfer accompanying chemisorption can be seen clearly from the contour plot of the difference of charge density between the oxygen chemisorbed and clean films shown in Fig. 14c. As found for the clean Al( 1 1 1 ) surface (Wang et al., 198la), active surface states exist in k near K and the point at three-quarters of the way from f' to M . The charge density of these states is concentrated in real space at the threefold hollow sites near the metal surface, as denoted by A in Fig. 14c. Since these states mix easily with the 0 states, it becomes energetically favorable for the oxygen to locate at these threefold hollow sites during chemisorption. As a result of this mixing, the region above the oxygen (A1 in Fig. 14c) acquires additional electronic charge, while the region below oxygen (A2 in Fig. 14c) gets less charge (or
VACUUM
(b)
0.3
FIG. 14b. Contour plot of self-consistent total charge density shown for a vertical plane through the two adjacent inequivalent threefold sites (in steps of 0.3 e/Al at.) for (b) the clean five-layer Al(1II) film. [After Wang et ul. (1981c).]
388
E. WIMMER, H. KRAKAUER, A N D A. J. FREEMAN
FIG. 14c. Contour plot of self-consistent total charge density shown for a vertical plane through the two adjacent inequivalent threefold sites (in steps of 0.3 e/Al at.) for (c) the contour plot (in steps of 0.2 e/Al at.) of the difference of charge density of oxygen chemisorbed and clean five-layer AI(11 I ) films. [After Wang et al. (1981c).]
indeed may even lose some electrons). This outward electron transfer corresponds to an additional inward dipole layer and leads to an increase in the work function. The theoretical value of the work function found by Wang et al. (1981~)is 5.4 eV for their model system, or a 0.7 eV increase over that for the clean Al(111) work function (4.7 eV) (Wang et al., 1981a). Note also that electrons have been attracted to the 0 2 -ion from the region outside it, as shown by the decrease of charge density in the regions B1 and B2 in Fig. 14c. Wang et al. (1981~)report that the surface states that exist on the clean Al( 111) surface play an important role in the chemisorption process: The hybridization of the A1 2p states with these A1 surface states leads to the chemisorption states with the largest binding energy. These authors also find that the local density functional one-particle eigenvalues for the 0 2p states
THEORY OF SURFACE ELECTRONIC STRUCTURE
389
can be compared quantitatively with photoemission experiments, if effects of electron-hole relaxation and correlation effects of 2.2 eV are taken into account. In addition, along the symmetry line T (from to A),where polarized angle-resolved photoemission data are available (Eberhardt and Himpsel, 1979), the theoretical dispersion of the odd 0 2p, states and of the antibonding 0 2p,-like states are in coincidence with the experimentally observed odd-symmetry states and the even-symmetry band with lower binding energy. These theoretical results suggest that another even band observed in photoemission at higher binding energy consists, in fact, of two bands, namely, the 0 2p,-like and the 0 3p,-like bonding states. The calculated work function increases by 0.7 eV upon addition of an oxygen overlayer to Al(111). This result is compatible with the experimental increase 0.1 eV at higher oxygen exposure, if another accompanying process (namely, the formation of the Al,O,-like oxide structure) is taken into consideration. This process has been shown to lead to a decrease of work function by 0.6 eV (Krakauer et a/., 1981). 5. Sodium Chrmisorption on the Al(001) Surface The discussion of the chemisorption of an electronegative element, such as oxygen, has revealed (cf. the previous sections) that the adsorbate-induced changes in the electronic structure are essentially localized to the first two surface layers of the substrate and that the strong oxygen-aluminum interaction gives rise to pronounced charge rearrangements in the surface region mainly due to the filling of 0 2p states. This situation is now contrasted with the chemisorption of an alkali atom such as sodium on the Al(001) surface. Benesh et ul. (1981) performed an LAPW calculation of a c(2 x 2) Na overlayer on the Al(001) surface. In this coverage, one Na atom corresponds to two surface atoms. These authors find an occupied pocket of Na-3s derived states near the center of the Brillouin zone. Their layer-projected density of states reveals a Na-induced bonding and an antibonding peak at 1 and 3 eV above the Fermi energy. The result for a single Na atom adsorbed on a jellium surface (Lang and Williams, 1977) also shows a maximum in the density of states but without a splitting into two peaks. We may conclude that the self-consistent jellium model correctly yields the position of the adsorbedatom resonance peak, but the absence of atomic structure in the model substrate suppresses the appearance of bonding-antibonding splitting that is characteristic of the chemical bond. Contour plots of the charge densities of the clean and Na-covered Al(001) surface (Fig. 15a,b) reveal that upon deposition of Na the charge density in the interface region is increased. Together with the large unoccupied parts in the Na-projected density of states, this may be taken as an indication of important ionic components in the Na-A1 interaction. However, it is difficult in
390
E. WIMMER, H. KRAKAUER, AND A. J. FREEMAN
FIG. 15. (a) Total valence charge density contours in (100) plane of the three-layer Al film (vacuum is at the top). Successive contours are separated by units of 0.2 electrons per bulk unit cell. (b) Total valence charge density contours in (100) plane of the five-layer Al(001):Na film. The Na atom is chemisorbed in the fourfold hollow site above the lower Al atom. [After Benesh et a/.(198 l).]
this case to describe the charge transfer in terms of an LCAO picture because of the diffuse character of the alkali valence states. Essentially, the Na 3s electrons lose their identity upon chemisorption, and only quantities such as the total electronic charge density (cf. Fig. 15a,b) remain rigorously meaningful. Unfortunately, Benesh et al. (1981) used a film of only three layers of Al to represent the substrate. Hence, their work does not allow us to assess the screening of the Na-induced changes in the surface electronic structure. B. Transition Metals
The last few years have witnessed a dramatic increase in experimental and theoretical studies of transition-metal surfaces. Electronic states that are localized near the surface are among the most prominent of the observed spectroscopic features. Experimentally, however, there is often little difference between surface states (SS), which decay exponentially into the bulk, and surface resonances (SR), which can be described as a hybrid of a true surface state and a bulk state. If SS and SR are relatively well understood on semiconductor and simple metal surfaces (Appelbaum and Hamann, 1976), it is only relatively recently that attempts have been made to deal with the addi-
THEORY O F SURFACE ELECTRONIC STRUCTURE
39 1
tional complexity arising from d electrons in noble and transition metals. So far, self-consistent calculations have been performed for only a few surfaces of the 3d metals [Sc (Feibelman and Hamann, 1979), Ti (Feibelman et al., 1979), Fe (Wang and Freeman, 1981; Ohnishi et al., 1983), Ni (Wang and Freeman, 1980, Jepsen et a!., 1980, 1982; Freeman et a/., 1982a; Krakauer et al., 1983; Wimmer et al., 1983), Cu (Gay et al., 1977, 1979; Appelbaum and Hamann, 1978)], and the 4d metals [Nb (Louie et al., 1976, 1977), Mo (Kerker et al., 1978, 1979), Pd (Louie, 1978, 1979), Ag (Appelbaum and Hamann, 1978)l. However, the most extensively studied metallic surface resonances occur on the W(OO1) surface (Feuerbacher and Willis, 1976; Swanson and Crouser, 1966, 1967; Plummer and Gadzuk, 1970; Waclawski and Plummer, 1972; Feuerbacher and Fitton, 1972; Weng et a!., 1978; Holmes and Gustafsson, 1981; Campuzano et a/., 1981; Kirschner et a/., 1981; Hussain et al., 1982; Tung and Graham, 1982), and the origin and nature of these states have been long-standing problems. The wealth of detailed experimental information on W(OOl), reviewed later, thus presents a well-charted area in which to explore the theory of surface electronic structure. 1. The Clean W(OO1) Surjuce An anomalous peak in the W(OO1) field-emission energy distribution (FEED) was first discovered by Swanson and Crouser (1966), and a similar feature was subsequently observed on the (001) face of Mo (Swanson and Crouser, 1967). The W(OO1) peak was originally interpreted in terms of the relativistic bulk-band structure (Swanson and Crouser, 1966), but a later FEED study by Plummer and Gadzuk (1970) identified it as a surface state. The state on W(OO1) was also reported as the first observation of a metallic surface state using photoemission (Waclawski and Plummer, 1972; Feuerbacher and Fitton, 1972). Weng et al. (1 978) have recently presented a very complete review of the SS and SR on the (001) surfaces of Mo and W, utilizing both angle-resolved photoemission (ARP) and FEED. They concluded that many-body effects (e.g., final-state relaxation, plasma interaction) and d-band edge effects are not required to explain any aspect of the photoemission results from these resonances. Experimentally, then, three bands of surface resonances are observed in the A R P experiments (Weng et al., 1978) on W(00 1) : (i) A high-lying SR is located 0.3 eV below the Fermi energy. The intensity of this resonance peaks at normal exit (kil = 0) and decreases rapidly as I k I increases (off-normal emission). This state has even parity with respect to mirror plane symmetry (e.g., d: or s-orbital character). (ii) For emission angles greater than 2", an additional second high-lying resonance appears as a shoulder about 0.8 eV below the first high-lying resonance. Its photoemission intensity is strictly zero at normal exit ( k l l = 0),
392
E. WIMMER, H. KRAKAUER, A N D A. J. FREEMAN
but increases gradually with increasing polar angle reaching a maximum at Weng et al. (1978) suggest that this resonaround 10" (Ik,,I z 0.2-0.3 k'). ance has odd parity with respect to a (1 10) mirror plane although they note evidence that it may contain a small component of even parity. (iii) The third band is similar to the first but is located about 4.2eV below E,. Experimentally, the dispersion is less than 0.3 eV for each band. The theoretical interpretation of these spectroscopic features [especially the first high-lying band (i)] in terms of initial-state properties of surface states and surface-resonance states has been quite controversial. Feder and Sturm (1975) found an SR in a relative or filled spin-orbit gap at k = (0,O) near E , as well as a true SS in a spin-orbit symmetry gap just below the relative gap. Both of these states have A, symmetry; however, Hermanson (1977) showed that this was not consistent with the ARP observations of Feuerbacher and Fitton (1974) for the high-lying resonance band. Kasowski (1975) found an SS of the correct symmetry in the vicinity of E , at kll = 0, but only for a contracted surface. [It has since been suggested (Weng et al., 1978), however, that this state be identified with the low-lying (4.2-eV) resonance at kll = 01. Noguera et al. (1977) also required a surface contraction (for Mo) to obtain an SS at kll = 0 near E,. Modinos and Nicolaou (1976) by contrast, interpreted the high-lying resonance (1) as being due to SR states near EF, which they found for finite k # 0 (i.e., near k = 0). Similar results have been obtained by other non-self-consistent calculations (Weng et al., 1978; Kar and Soven, 1976; Smith and Mattheiss, 1976; Desjonqueres and CyrotLackman, 1976; Bisi et al., 1977; Laks and Goncalves da Silva, 1978; Grise et al., 1979), but comparison with the most recent experimental results leads to the conclusion (Weng et al., 1978) that while these calculations can be interpreted as correctly predicting both the existence and symmetry of the second high-lying resonance (1 1) and the low-lying resonance (iii), they fail to explain the first high-lying resonance. [Inglesfield's calculation (1978a,b) yields an unoccupied SS state near EF at kll = 0; this result, however, is in contradiction with the experimental observation that this state exists below E,.] This somewhat unusual situation is explained by the fact that the existence of the high-lying surface resonance is very sensitive to the potential near the surface, as discussed in Section 111. Indeed, Kerker et al. (1978, 1979) demonstrated that self-consistency was required in order to get an occupied highlying resonance at k l l= 0 for Mo(001), as well as the other resonances (ii) and (iii). However, Kerker et al. (1978,1979) show only the second high-lying resonance along the TR direction, which is in contradiction to the experimental observation of three bands. Posternak et al. (1980) presented the first self-consistent band structure calculation for the W(OO1) surface using the LAPW method (Krakauer et al.,
,
THEORY OF SURFACE ELECTRONIC STRUCTURE
393
1979b) for thin films. The calculations were performed for a seven-layer film and included all relativistic corrections except the spin-orbit interaction. These calculations provided, for the first time, a complete description of all three experimentally observed resonance bands. Self-consistency was found to be essential in obtaining accurate work function results: The theoretical value of 4.60 0.05 eV (Ohnishi et al., 1984) is in excellent agreement with the experimental value of 4.63 eV (Billington and Rhodin, 1978). Ohnishi et al. (1984) investigated the electronic structure of the W(OO1) surface by precise FLAPW calculations for films consisting of one, three, five, seven, and nine layers. This work demonstrates that a film of only five layers is sufficiently thick to describe the energies and dispersions of the characteristic surface states and surface-resonance tables. a. Charge Density and Work Function Since the charge density is of fundamental importance in local density-functional theory, we examine first the total self-consistent charge density and the resulting work function as this is a sensitive test of the quality of the results obtained. Figure 16 shows the total valence charge density p ( r ) (i.e., the core charge density has been subtracted) in a (1 10) plane for positive z values. The charge density is seen to fall off smoothly as we progress normal to the surface into the vacuum and to heal very rapidly to bulk-like character on going away from the surface into the solid. Each layer of the film is approximately neutral in charge (including the vacuum charge as part of the surface layer). The bonding characteristics are also evident in Fig. 16. In the interior atoms, the bonding xy, xz, yz (t,,)d orbitals form fairly localized lobes pointing along the body diagonals to the nearest-neighbor atoms. In addition, there is a rather uniform metallic bonding charge density in the interstitial regions. These features persist up to the second layer from the surface. In the surface layer, there are marked changes in the bonding character. Compared to an interior atom, substantial weight has been removed from the lobes pointing toward the missing nearest neighbor above the film, although the maximum value of the d-bond charge density is still about the same in these directions. By contrast, the maximum value in the downward pointing lobes is somewhat reduced as a consequence of an increased charge transfer into this bond from the surface atom. Finally, there is a large and rapid variation in p(r) in the interstitial region of the surface layer, thus demonstrating the importance of the warping contributions to the potential. There is a sizeable redistribution of charge near the surface associated with the formation of the dipole layer, which sensitively determines the work function @. As expected, self-consistency is found to be crucial for obtaining an accurate value of CD. Ohnishi et al. (1984) found CD = 4.6 0.05 eV in excellent agreement with experiment (Billington and Rhodin, 1978).
394
E. WIMMER, H. KRAKAUER, AND A. J. FREEMAN
FIG. 16. Contour plot of the total valence charge density. Successive contours are separated by units of 0.8 electrons per bulk unit cell. [After Posternak et al. (1980).]
b. Surface States and Surface Resonance States Since most of the eigenvalue information plotted as band structure is complex and not easily assimilated (in such plots, the three-dimensional bulk-band structure is essentially projected onto the two-dimensional Brillouin zone, which tends to obscure details about surface states as well as the underlying bulk bands), we focus first on the information contained in the theoretical layer-by-layer density of states (DOS). The local density of states (LDOS) for each layer, shown in Fig. 17, was obtained using the two-dimensional analogy (Wang and Freeman
THEORY OF SURFACE ELECTRONIC STRUCTURE
395
I .c
0.5
0.0 W v1
+
5
0.5
lL
0 h I-
5 0.0
z
w n
0.5
0.0
I
CENTER
I
0.5
0.0 ENERGY ( e V )
FIG. 17. Layer-projected density of states [or the seven-layer film. Here S-1 is the first layer in from the surface and S-2 is the second layer in from the surface. [After Posternak et al. (1980).]
1979) of the bulk linear analytic tetrahedron method (Jepsen and Andersen, 1971; Lehmann and Taut, 1972). The LDOS shown were obtained using 15 k points in the irreducible of the two-dimensional BZ and then smoothed with a Gaussian (FWHM = 0.3 eV). The general trend seen is that of an approach to bulk behavior in going from the surface to the central layer, since the latter indeed displays the characteristic structure of the tungsten bulk DOS. The bandwidth is already achieved by the second layer. There are two prominent peaks in the surface LDOS (Fig. 17) at - 0.5 and - 4.4 eV. As will be shown shortly, the one located at about 0.5 eV below the Fermi energy is
396
E. WIMMER, H. KRAKAUER, AND A. J. FREEMAN 2
0 7 % -
EF
\ k 1
"
- 2
-2 > 22
I
-4
/ ....... 1
W
5
-6
-8
- 10
T
Z
R
Y
F
I
L
T FIG. 18. Surface states and surface-resonance states for the W(OO1) slab. Solid and dashed curves refer to the uncontracted surface, as discussed in the text. Dotted curves show the effect of a 6% contraction. The bottom of the conduction band is outlined along the lower portion of the figure. [After Posternak et al. (1982).]
related partly to a short SS line of states around (k = 0) and partly to a double SR line of states for k > 0; the second, located at -4.4 eV, corresponds to a low-lying SR band. These resonances are found in both angleintegrated and angle-resolved photoemission (Weng et d., 1978). In order to obtain information about SS and SR, we display in Fig. 18, along the highsymmetry directions, those SS and SR that have a localization greater than 55% in the two outermost layers. Analysis of these results shows that they provide good agreement with detailed ARP measurements (Weng et al., 1978) of all three surface-resonance bands described earlier. (i) There is an extremely localized state just below E , near T, which has a small upward dispersion along TR and TR with A l and C, symmetry in agreement with experiment. It disappears about one-third of the way between r 8 and f A . This is the state that is seen as a sharp peak in photoemission spectra at normal exit. It is a true SS only at f where it exists in a bulk A, symmetry gap. Figure 19 shows the origin of the surface state at r. We have displayed all of the film energy eigenvalues at r as though they were derived from the projection of the three-dimensional crystal-band structure along T H in the bulk bcc Brillouin zone (open circles are even and solid circles are odd with
THEORY O F SURFACE ELECTRONIC STRUCTURE
r
a
397
A
FIG. 19. Film-derived bulk energy bands along TH showing the origin of the very localized surface state 0.3 eV below E , at i=(k,, = 0). [After Posternak et al. (1980).]
respect to z reflection). The general shape of the bulk bands along T H is recognized, and we note the near degeneracy of the film A;, and A5 bands at r, which reflect the exact degeneracy of the bulk bands at the AL5 point. (The lower symmetry due to the presence of the surface does not require this degeneracy, similarly, to the films A , and A2 band at the HI, point). This figure shows that two states from the upper A , branch are shifted downward into a A1 symmetry gap, which give rise to the pair of SS odd and even with respect to z reflection. As mentioned, this SS is very sensitive to the actual potential. Our starting potential, for example, does not yield this SS. These results confirm early predictions by Caruthers and Kleinman (1975) for SS at on transition-metal bcc (001) surfaces. In calculations for Fe(001), they found that both the existence and symmetry of SS at depend crucially on details of the potential and are very dependent on small changes in the potential. On the W(OO1) surface studied here, the SS just below E , is extremely localized in the surface layer ( - 93 %) and projects quite far out into the vacuum region (see Fig. 20). Thus, it is perhaps not surprising that the conditions for the formation of this state are sensitive to the behavior of the potential near the
r
398
E. WIMMER, H. KRAKAUER, A N D A. J. FREEMAN
VACUUM
FIG. 20. Charge-density contour plot for the very localized surface state 0.3 eV below E , at by units of 0.4 electrons per bulk unit cell. [After Posternak et al. (1980).]
F(k,,= 0). Successive contours are separated
surface. The surface potential in turn depends on the delicate rearrangement of electronic charge at the surface, which leads to the formation of the surfacedipole barrier and the correct work function. The treatment of this electronic screening is, of course, beyond the scope of non-self-consistent calculations, and it explains the general failure of such calculations to predict correctly the high-lying SS. (ii) We also find a pair of SR (for k , , # 0) about 0.5 eV below the SS described earlier. Along the symmetry lines r 8 and FA, one of the pair of SR is symmetric (A, and C,,respectively) and the other is antisymmetric (A, and Z,, respectively) with respect to the corresponding mirror planes [a (100) plane along fX and a (1 10) plane along FA]. Along the rh? direction, these states have a small upward dispersion (in agreement with experiment) that cuts E , about half-way between and M . Along the FX direction, however,
THEORY O F SURFACE ELECTRONIC STRUCTURE
399
these states have a small downward dispersion [similar dispersion was found along r R in a self-consistent calculation for M o (Kerker et al., 1978, 1979)], whereas the experimentally determined dispersion (Weng et al., 1978) is slightly upward toward E,. We do not believe that this is a size effect due to the use of a thin film (Kerker et al., 1978, 1979), since Grise et al. (1979) found a slight upward dispersion along of their A SR state for both five- and seven-layer films as well as for a 39-layer film. It is important to note that the SR is about 20-25 % more localized in the two outermost layers than is the C1 SR. (Below E,, the localization in the topmost two layers is typically greater than 90% for the C, SR.) Taken together with the fact that the C, and Z2 SR have nearly the same energy and dispersion, this could explain the weak shoulder found in Fig. 9 of Weng et al. (1978) when the vector potential A is parallel to k along FA?. The presence of this shoulder led Weng et a!. (1978) to suggest that this SR might possibly contain a small component of even (XI) parity, although the predominant component seemed to be of odd ( C J parity. The results of Posternak et al. (1980) show that the C, SR is less localized than the Zz SR and would thus be expected to give a smaller contribution to the photoemission intensity. Grise et a!. (1979) invoked spin-orbit coupling to explain this observation. Whereas, they find only a single SR with odd parity (E2) when spin-orbit coupling shows that this state acquires a small component of even parity. By contrast, we find two SR, one of each parity, without resorting to the spin-orbit interaction. (iii) Finally, there is a low-lying SR with very flat dispersion along and part of the way along X M at about -4.2 eV. This state is also found part of the way along fiM, but is much less localized and shows a greater dispersion. The symmetries are A,, y2,and C,, in agreement with the experimentally determined (AICl) low-lying resonance. This SR band corresponds to the low-lying peak in the surface LDOS seen in Fig. 17.
fix z2
fix
In addition to these three bands of SR that have been experimentally observed, we find additional SR, which are shown in Fig. 18. The low-lying A , SR about -4.5eV, continues as a F2 SR fading away about half-way between ZM. To our knowledge, this y2 SR has not yet been observed. Just above this state, there is a y, SR. Near E , there is another pair of SR, one of which is symmetric (Fl)and the other antisymmetric (TJ with respect to the (010) mirror plane. Along fiM there is an unoccupied SR with El symmetry that comes down to within 1 eV of E , . c. Relaxation and Reconstruction It seems well established that the surface layer of clean tungsten (001) undergoes a continuous reversible displacive phase transition when the temperature is lowered below about 400 K (Felter er ul., 1977; Debe and King, 1977). In spite of several recent
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E. WIMMER, H. KRAKAUER, A N D A. J. FREEMAN
experimental investigations (Stevens and Russell, 1980; Campuzano et al., 1980, 1981), the nature of the driving force behind the transition remains unclear. This is partly due to the fact that most experiments permitting the determination of the atomic arrangements for the surface do not lead to consistent interpretations of the reconstruction. The situation is still further complicated by the noticeable differences between the available angleresolved photoemission (ARP) spectra (Campuzano et d., 1980, 1981; Holmes and Gustafsson, 1981), even for the unreconstructed phase. A precise determination of electronic surface states close to the Fermi energy E , is also required in order to support a well-known theoretical model for the reconstruction, thus suggesting that these SS could drive the instability in chargedensity-wave (CDW) type mechanism (Felter et al., 1977; Debe and King, 1977; Tosatti, 1978; Krakauer et al., 1979a). At and above room temperature, the clean surface is assumed to retain the (1 x 1) bulk symmetry, with, however, a contraction of the topmost surfacelayer spacing (Reed and Russell, 1979). While Lee et al. (1977) have reported by analysis of low-energy electron diffraction (LEED) data a contraction of 11 %, Feldman et al. (1977) have deduced on the basis of back-scatteringchanneling experiments an upper limit of 6 % for this relaxation. Recent all-electron FLAPW total energy calculations (Fu et al., 1984a) predict a multilayer relaxation with the first interlayer spacing reduced by 5.7 %. We shall return to this point in Subsection 1V.D. Complementing these structural determinations, Weng et al. (1978) have presented an extensive study of electronic states that are localized near the (001) surface of W above room temperature, utilizing both ARP and field emission energy distribution (FEED). Until quite recently, no comparison has been done between ARP spectra for the high-temperature phase and the low-temperature reconstructed phase. In fact, most available information was of a structural nature. On the basis of LEED intensity studies, Debe and King (1977) and Felter et al. (1977) have independently shown that the clean surface reconstructs by lowering the temperature below 300 K to a c(2 x 2) structure. Two possible models for the surface reconstruction involve: (i) A parallel shift of the surface atoms involving alternating displacements by about 0.15-0.30A along the [llO] directions, thus forming zigzag chains with a c(2 x 2) R 45" unit cell and p2mg two-dimensional space-group symmetry (Debe and King, 1977; Barker et al., 1978). (ii) The perpendicular-shift model in which alternate W atoms are vertically displaced. Felter et al. (1977) originally proposed the perpendicular-shift model, but Debe and King (1977) and Barker et al. (1978), in further LEED study, concluded that the parallel shift model was probably more correct. Unfortunately, other experimental techniques have not led systematically to
THEORY OF SURFACE ELECTRONIC STRUCTURE
40 1
the same conclusion. Tsong and Sweeney (1979) using field-ion microscopy (FTM) found an image structure that is consistent with a (1 x 1) structure at 21 K. They suggest either that no atomic rearrangement takes place a t this temperature, or that shifts in the (1 1) direction, if any, are too small to be observed in FIM. Furthermore, using FIM in the range 15-460 K, Melmed et al. (1979) found results that support the perpendicular-shift model. This same model is consistent with the surface-weighted effective potentials, obtained above and below the phase-transition temperature by Stevens and Russell (1980) from the surface-resonance-band structure. Finally, using high-energy ion scattering in combination with channeling, Stensgaard et al. (1979) concluded that neither of these two models proposed for the c(2 x 2) surface structure is valid. The basis of their argument is that models implying a displacement of all surface atoms are inconsistent with their ion-scattering data. Recently, Tung and Graham (1982) concluded from FIM measurements that the W(OO1) surface is reconstructed over the temperature range 15-580 K and that the reconstructed surface contains an alternating vertical component to the displacements of the W surface atoms. From a theoretical point of view, the parallel-shift model was initially favored by a mechanism for the reconstruction proposed by Tosatti (1 978). The electronic surface-resonance states near E , might drive the phase transition through a surface-phonon softening and a gapping of the two-dimensional Fermi surface. In order to check the validity of this theory, Krakauer et al. (1979a) computed, on the basis of their LAPW band structure, the surface-generalized susceptibility for the seven-layer film. They found, by adding local-field effects and taking matrix elements into account, that surface-resonance-state coupling does lead to a significant peak in the susceptibility at W, which is due to two-dimensional Fermi surface nesting of the SR states cutting E , midway between and nT. This could result in a surface-phonon softening that is compatible with the parallel-shift model reconstruction. Using a Green’s functions technique in a non-self-consistent surface calculation, Inglesfield (1978a,b) also computed the surface-response function, but his results, in contradiction with those of Krakauer et al. (1979a), show only an extremely small Kohn anomaly due to SR states. Very recently, Campuzano et al. (1980, 1981) presented a detailed study of ARP data throughout the surface Brillouin zones (SBZ) obtained from both high- and low-temperature W(OO1) structures. Simultaneously, Holmes and Gustafsson ( 198l ) , using high-resolution ARP with polarized radiation, also gave the dispersion and the symmetry of SR states along in the hightemperature phase. Although these two studies do not agree in several important respects, perhaps due to resolution characteristics and band assignments, there is evidence that the k,,-dependentbehavior of SS and SR state near E , along rR is much more complicated than was expected from both
402
E. WIMMER, H. KRAKAUER, A N D A. J. FREEMAN
self-consistent pseudopotential calculation for molybdenum (Kerker et al., 1978, 1979). The main point is that the two-dimensional Fermi-surface nesting in the FM direction, which was a striking feature of the surface-band structure of Posternak et al. (1980), is called into question by these two ARP experiments. The high-temperature surface structure may thus be more complicated than the simple ( 1 x 1) structure previously assumed. In order to obtain further theoretical insight into this problem, the simplest (and most natural) step is to retain the (1 x 1) bulk symmetry with a contraction of the topmost surface-layer spacing. As mentioned earlier, this situation was considered until now as the most plausible one for the hightemperature phase. Furthermore, a relaxation is often a partial ingredient of a more complicated structure. Posternak et al. (1982) presented an LAPW band-structure calculation for a seven-layer film similar in all respects to the calculation reported earlier (Posternak et al., 1980) but included a contraction of the topmost layer by 6%. In addition to the previously mentioned problems, this study also provides information regarding the sensitivity of the surface-band structure to general atomic displacements of such an amplitude. The surface states and surface-resonance states for the relaxed surface are shown in Fig. 18 as dotted lines, where the difference to the unrelaxed surface is significant. We note that the contraction does not change appreciably the localization of the SS and SR. The low-lying SR are slightly lowered in energy (-0.15 eV), while the SR in the vicinity of E , show both upward and downward shifts. There is no appreciable change in the dispersion of all the states. In particular, the pair of SR along Z, which is relevant to the proposed mechanism (Krakauer et al., 1979a) for the phase transition, does not change significantly. It intersects E , slightly closer to M than in the unrelaxed case, so that we may expect a lowering of the intraband part of the surface-generalized susceptibility at A. Posternak et al. (1982) also found that a relaxation of 6 % of the surface layer did not significantly affect the work function nor surface core-level shifts. Let us briefly summarize the results at this point. Posternak et a!. (1982) find that a contraction by 6 % of the topmost interlayer spacing does not significantly change the location and dispersion of SS and SR bands. In particular, a simple contraction cannot explain the differences found between the theoretical results (Posternak et al., 1980, 1982) and two recent ARP measurements (Campuzano et al., 1981; Holmes and Gustafsson, 1981) at room temperature. Restricting the discussion to the direction, there are two major discrepancies.
c
(i) Posternak et al. (1980, 1982) find a very localized even SS located 0.3 eV below E , at and crossing the Fermi level at k,,= 0.1 k l ,a result that is compatible with the observations of Campuzano et al. (1981).
r
THEORY OF SURFACE ELECTRONIC STRUCTURE
403
r,
(ii) Posternak et al. (1980, 1982) also find a pair of SR, absent at about 0.5 eV below the SS described earlier and cutting E , at k , , = 0.8 k‘. The highest of these states has even symmetry, and the lowest has odd symmetry.
Holmes and Gustafsson (1981) find this doublet with much less dispersion crossing the Fermi level at 1.2A-I. In contrast, Campuzano et al. (1981) obtained a dispersion for their even state that is in good agreement with those of Posternak, while they find a much flatter odd state (possibly containing some extra even component) extending nearly the full length of the X symmetry line. Clearly, these experimental data do not both support the electronic mechanism for the reconstruction; but before attempting to find out which factor drives the W(OO1) phase transition, it seems essential, in a first step, to get a proper understanding of the high-temperature structure on the basis of the conflicting results described earlier. Concerning the LAPW method of calculation, we shall mention two approximations which, in principle, might affect the location and dispersion of SR. (i) Finite-thickness slab approximation. Ohnishi et al. (1984) demonstrated that five-layer films gave essentially the same results as films with seven and nine layers. Furthermore, Grise et al. (1979), using a non-selfconsistent energy-band scheme including spin-orbit interaction, do not find major changes in the dispersion of their SS and SR by going from a five-layer to a 39-layer film. (ii) Neglect of spin-orbit interaction. Recently, Mattheiss and Hamann (1 984) performed a self-consistent scalar-relativistic calculation of the electronic structure of a seven-layer W(OO1) film. Using a nonorthogonal tight-binding fit, these authors extended the calculation to a 19-layer W(OO1) film including spin-orbit coupling. From this approach Mattheiss and Hamann ( 1 984) found “significant changes” due to the inclusion of spinorbit coupling. The discrepancies with photoemission data for surfaceresonance states along I:are diminished but not removed. As a further possible cause of the discrepancies between photoemission results and the theoretical electronic structure, many-body final-state effects and multiple scattering of the photoelectrons detected at the surface from underlying bulk layers may be significant. Finally, let us stress the fact that the actual structure of the W(OO1) surface in the high-temperature phase is unclear and may be quite different from the relaxed-surface model studied here. In this respect, two different suggestions have been proposed. (i) Holmes and Gustafsson (1981) and Walker et al. (1981) argue that the surface layer on W(OOl), though keeping a (1 x 1) structure, undergoes a uniform lateral shift in the (lo) direction.
404
E. WIMMER, H. KRAKAUER, AND A. J. FREEMAN
(ii) Another model for the high-temperature structure, which is consistent with the phase-transition mechanism, LEED, and ion-scattering data, was suggested by Stensgaard et al. (1979). Here, the transition observed upon cooling is characterized as a disorder-order process of the type also discussed by Debe and King (1977). At high temperature, the surface atoms are already displaced in a c(2 x 2) structure but only inside small domains. There is no long-range order, and the short-range order is not compatible with the coherence length consistent with LEED experiments. The diffraction spots are then produced by atoms in underlying layers. Cooling increases the long-range order giving rise to the sharper spots of the c(2 x 2) structure. To account for further increases in half-order spot intensities, a change in atom displacements may be involved. This model is compatible with a microscopic theory of CDW developed by McMillan (1 977) for transition-metal dichalcogenides, which assumes that the coherence length at finite temperatures is short and that the dominant entropy is the lattice entropy rather than the electronic one. Let us note that the temperature-induced effects observed by Campuzano et al. (1980, 1981) on SS do not seem to support this disorderorder mechanism for the phase transition. However, further experimental investigations, together with first-principle band-structure calculations for the other models of the unreconstructed phase, are required in order to settle these questions. 2. The Cesiated W(OO1) Surface a. Work-Function Lowering Cesiated metal surfaces are important both for their prototypical properties and for their technological applications. Thus, the lowering of the work function by the deposition of an overlayer of Cs on a metal surface like W, first discovered by Kingdon and Langmuir (1923) and still widely used in a host of experiments, has widespread applicability in such areas as thermionic conversion (Hatsopoulos and Gyftopoulos, 1979), ion propulsion (Forbes, 1968), and recently as negative hydrogen (deuterium) ion sources for magnetic fusion reactors (Hiskes et al., 1976; Hiskes, 1979, Hiskes and Schneider, 1981). Despite its long history and obvious importance, the mechanism for the work-function lowering still remains a challenge to electronic structure calculations. Because of the complexity of rigorous calculations for this type of system, only calculations on a jellium model have been carried out to describe the work-function changes induced by alkali absorption (Lang, 1971). Although these calculations give a qualitatively correct description of the work-function changes, important structures due to the atomistic nature of the surface and interface atoms are not included in these calculations. In particular, an understanding of the role played by the rich localized W d surface states (Posternak et al., 1980, 1982) requires a more detailed description of the surface.
THEORY OF SURFACE ELECTRONIC STRUCTURE
405
Wimmer et al. (1982, 1983) presented results of the first rigorous, fully selfconsistent, all-electron calculation of Cs chemisorbed on a W(OO1) surface using the highly accurate FLAPW method for thin films (Wimmer et al., 1981a). The chemisorption process is characterized by comparing the results of independent self-consistent calculations for a five-layer W(OO1) slab, an unsupported Cs monolayer, and Cs in a c(2 x 2) coverage on W(OO1) at distances ranging from 2.6 to 2.9 A. The hypothetical c(2 x 2) coverage has been chosen since it is computationally the best accessible and combines the features of the completed Cs monolayer (which is hexagonal) with geometrical aspects of Cs in fourfold hollow sites as found for the lower-coverage p(2 x 2) structure. Wimmer et al. (1982, 1983) find that the reduction in the work function @, upon cesiation, arises from a multiple dipole formation at the surface and interface layers. A characteristic feature of the clean W(OO1) surface is the pronounced spill-out of electrons from the surface layer into the vacuum, which leads to the formation of a strong surface dipole. This surface dipole is reduced in the cesiated system: Cs forms a metallic overlayer with its valence electrons polarized toward the W surface, thus reducing the spill-out into the vacuum. Correlated with the polarization of the Cs valence electrons, the Cs 5p semicore electrons are markedly counterpolarized. Because of the strong metallic screening, changes in the electronic environment are localized to the Cs overlayer and the Cs-W interface layer. The net result of these polarizations is a reduction of the effective electrostatic surface barrier. As a result, all W states, and also the Fermi level, are shifted to smaller binding energies with respect to the vacuum (or, equivalently, the vacuum level is lowered with respect to the Fermi energy of the substrate) and consequently the work function is reduced by 2.0-2.5 eV, depending on the Cs-W distance used. Although crucial for the determination of the lowering of the work function upon cesiation, there are no accurate experimental determinations of the distance between the Cs and W atoms. It is therefore necessary to carry out the electronic structure studies for various assumed Cs-W distances. The spill-out of the electrons on clean surface and the associated dipole layer, which is modified and reduced upon cesiation, is important to an in-depth understanding of the mechanism of the lowering of the work function. One way to monitor this dipole layer is a comparison of the calculated charge density (Fig. 21a) and that of an ideal surface formed by cleaving a bulk crystal along the boundaries of Wigner-Seitz cell of the bulk-like atoms in the center of the clean W(OO1) film (Fig, 21b). The difference between the charge densities of the calculated and the ideal surfaces (Fig. 21c) reveals that the electronic environment of essentially only the surface atoms is affected by the spill-out of electrons into the vacuum. The electrostatic dipole barrier associated with that spill-out, shown in Fig. 2112, is found to be 5.5 eV. The resul-
406
E. WIMMER, H. KRAKAUER, AND A. J. FREEMAN
FIG. 21. Total charge density on (a) a clean W(OO1) surface, (b) an “ideal” surface, and (c) the difference between (a) and (b), showing the spill-out of electrons into the vacuum. The units are elat;. Dotted contour lines indicate loss of electrons. The vertical scale at the left gives the distance from the surface W atoms in atomic units. [After Wimmer et al. (1983).]
tant work function of 4.77 eV of the five-layer W(OO1) slab agrees quite well with experiment (4.63 k 0.02 eV); the CD of an independent FLAPW calculation (Ohnishi et al., 1984) for a seven-layer W(OO1) slab (4.63 eV) gives excellent agreement with experiment. Let us now focus on how the charge density at the W surface is modified upon cesiation since this will give insight into the dipoles that counteract and reduce the original spill-out dipole. To this end, we subtract from the total charge density of the cesiated W surface (Fig. 22a) the charge density of the clean W surface and the charge density of a Cs monolayer [with the same Cs-Cs spacings as in Cs c(2 x 2) on W]. The resulting charge-density difference (Fig. 22b) reveals the charge redistribution upon cesiation. Two main effects become obvious: a loss of electrons in the region between and outside the Cs atoms combined with a pronounced increase in electronic charge near the surface W atoms and a dramatic polarization of the Cs 5p semicore electrons. Both effects are associated with the formation of dipole layers that influence the work function of the system. The region between and outside the Cs atoms is the domain of Cs valence electrons (originating from the atomic 6s functions). Upon deposition of the Cs overlayer on the W surface, the Cs valence electrons are polarized toward the W surface leading to an increase of electronic charge in the interface region between the Cs and surface W atoms. An analysis of the projected density of states by atom type and orbital angular momentum for the clean and cesiated
THEORY OF SURFACE ELECTRONIC STRUCTURE
407
FIG. 22. (a) Total electronic charge density for Cs c(2 x 2) on W(OO1) and (b) the difference charge density between that of Cs-W and the superposed density of clean W(OO1) and a Cs monolayer (in units of 10- elat;). The vertical scale at the left gives the distance from the surface W atoms in atomic units. [After Wimmer et al. (1982).]
W surface reveals that the contribution from the surface W atoms remains the dominant component near the Fermi energy and that the Cs d character of the states near E , is greatly enhanced compared with that of an unsupported Cs monolayer. Together, these observations indicate a polarized metallic bond with a tendency toward the formation of a covalent bond between Cs and the surface W atoms. We are now in a position to understand the changes in the potential induced by the cesiation of the W surface. In Fig. 23, we show the effective oneelectron potential for the clean and the cesiated (Fig. 24) W surface together with their corresponding Fermi energies and work functions for a Cs-W distance of 2.6 A. The analysis of these results shows that the potential near the W atoms is shifted by almost the same constant amount (2.00 eV) by which the work function is reduced [a = 2.77 eV for cesiated W(OO1) surface]. This shift is caused by the change in the effective surface dipole layer: The polarization of the Cs valence electrons toward the surface W atoms gives rise to a dipole barrier that counteracts the original spill-out dipole and, thus, reduces the work function. By contrast, the dipole layer associated with the
408
E. WIMMER, H. KRAKAUER, AND A. J. FREEMAN
FIG. 23. Effective one-electron potential in the (110) plane normal to the surface for a clean W(OO1). Here Q, denotes the work function. [After Wimmer et al. (1983).]
FIG. 24. Etrective one-electron potential for c(2 x 2) Cs on W(OO1) (d = 2.60A) in the (1 10) plane perpendicular to the surface. Here @ denotes the work function of this system. [After Wimmer et a!. (1983).]
THEORY OF SURFACE ELECTRONIC STRUCTURE
409
polarization of the Cs 5p semicore electrons acts in the same direction as the spill-out dipole and tends to increase the work function. The resulting effect of these multiple dipole formations is a net reduction of the effective surface dipole and hence a lowering of the work function. The Cs-induced changes in the electrostatic potential, i.e., the formation of a dipole layer that counteracts the spill-out dipole, is shown in Fig. 25 in the form of the difference between the Coulombic potential of the cesiated system and the superposed Coulombic potentials of the clean W and the unsupported Cs monolayer. We observe clearly a dipole barrier due to the polarization of the Cs valence electrons and a superposed structure due to the Cs 5p semicore polarization. The net result is an almost constant shift by 2eV inside the Cs- W interface region. Further, as found from independent self-consistent calculations for increased W-Cs distances from 2.6 to 2.75 and to 2.90& there is an increase of the spatial extension of the polarized Cs overlayer and hence an enhancement of the counteracting dipole. As a result, the work function is further reduced to 2.55 and 2.28 eV, respectively.
x
0 0 0 ° 00
FIG. 25. Electrostatic Coulombic potential barrier originating from the polarizations in a c(2 x 2) Cs overlayer ( d = 2.60 A) on a W(OO1) surface in the (1 10) plane perpendicular to the surface as indicated by the insert in the upper right corner. [After Wimmer et ai. (1983).]
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E. WIMMER, H. KRAKAUER, AND A. J. FREEMAN
b. Alkali Transition-Metal Surface Bonding As discussed in Subsection IV.B.1, the W(OO1) surface exhibits a rich structure of surface states and surface resonance states. In particular, the wave function of the surface state at f 0.3 eV below the Fermi energy projects far out into the vacuum (cf. Fig. 20) and is, therefore, expected to interact strongly with overlayer atoms. Thus, we now focus on the eigenvalues at f near the Fermi energy and compare the states of the clean five-layer W(OO1) slab, the Cs monolayer, and of the cesiated W surface for three different heights of the Cs atoms, all shifted to the same Fermi energy. Furthermore, we classify the states according to their atomic (i,e., layer) and /-projected characters. The most striking observation (cf. Fig. 26) is the interaction between the W(d) SS and the Cs-s conduction band, which brings about the dramatic lowering in energy of f , SS. In the unoccupied part of the energy spectrum, we find interactions of Cs p- and d-like states with the W(d) SS. As expected, the energies of these hybridized states show a pronounced dependence on the Cs-W separation. Further insight into the mechanism of the stabilization of W-d states at the surface, which we have discussed so far in terms of characteristic parts of the energy band structure, is provided by a study of the one-particle charge
-
w
0
-
EF
-
_.
-
-
--
W(d) - CS(S)
-0.1
-
- CS(4 -
I
Cs-monolayer
Clean W(OO1)
2.90
I 2.75
-
I 2.60 d(Cs- W) in
Cs c(2 X 2) on W(OO1) 26. Atomic (layer) and I-projected character of the states at for W(OO1), a Cs monolayer, and c(2 x 2) Cs on W(OO1) for three different heights of the Cs atoms. All states are referred to the same Fermi energy. For clean W(OOl), states from have been backfolded to (cf. Fig. I). [After Wimmer et al. (1983).] FIG.
41 1
THEORY OF SURFACE ELECTRONIC STRUCTURE
density. When Cs is deposited on the W(OO1) surface, the wave function of the SS at f 0.3 eV below E , (Fig. 27, left panel) overlap with those of the conduction electrons of Cs (Fig. 28 shows the bottom of the conduction band in a Cs monolayer) to form a new bonding state, which is depicted in Fig. 27. The shape of this W(d)-Cs(s) hybridized state suggests important polarizedcovalent contributions to the bonding mechanism between Cs and the W(OO1) surface. It is apparent from the one-particle charge density plots of Figs. 26 and 27 that the Cs 6s electrons, because of their extended character, lose their identity when Cs is absorbed on the transition metal surface and form a new hybridized state. When the wave function of this new state is described by an angular momentum representation near the Cs nucleus, we find that its s-component has - due to core orthogonalization - a nodal structure, which resembles that of an atomic 6s function. Despite this resemblance, we should consider this state as a new entity, unique to the chemisorbed Cs-W(OO1) system and describe it as a hybrid between a W(d) SS and a Cs valence state. Thus, the energetic stabilization of the SS by 1 eV is explained by the formation of a polarized chemical bond between this surface state and Cs valence electrons. The energetic stabilization of the other W(d) SS and SR states, which was discussed earlier and found to be
rl
r,.
Fic;. 27. Single-particle density of the adsorption-sensitive SS The lowest contour and the contour spacings are 0.001 e/Bohr3. In the three-dimensional plots shown below, the cut-off is at a density of0.030e/Bohr3. [After Wimmer et al. (1983).]
412
E. WIMMER, H. KRAKAUER, AND A. J. FREEMAN
FIG. 28. Single particle density of the f,state at the bottom of the conduction band for a Cs monolayer. The density is given in units of electrons per Bohr3. [After Wimmer et a/. (1983).]
less pronounced when compared with the TI SS, indicates similar although weaker bonding mechanisms for these other SS and SR states that have predominantly d,, and d,, character and project out into the vacuum to a lesser extent. This theoretical result has been confirmed experimentally by Soukiassian et al. (1983, 1984) by electron-loss spectroscopy and angle-resolved photoemission. They observe that upon cesiation the TI SS of the clean W(OO1) surface is shifted by 1 eV (but not broadened). A similar effect has been observed also for the system Cs on Mo (001) (Soukiassian et al., 1982, 1985) and for Cs on Ta(001) (Soukiassian et al., 1985). Finally, we return in our discussion of the cesiated W(OO1) surface to the fundamental quantity of density-functional theory, namely, the charge density. A comparison of the high, localized valence charge density of the clean W(OO1) surface (Fig. 29a) and the low, extended and weakly bound valence charge density of a Cs monolayer (Fig. 29e) makes it immediately obvious that the valence charge density in the surface and interface region is by far dominated by the W-d states. Figures 29b-d show the valence-band charge density for the cesiated W(OO1) surface. The first surprising observation is the fact that the region near the Cs nuclei is gaining electrons. Thus, the quantum mechanical result clearly contradicts the simple classical picture of a chemisorbed Cs' ion. When we compare the charge density in the region between the Cs adatoms for the unsupported Cs monolayer and the chemisorbed Cs
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413
FIG. 29. Valence change density in units of 10-’e/Bohr3 of (a) the clean W(OO1) surface, (b)-(d) the W(OO1) surface with a c(2 x 2) Cs overlayer for three different heights of the Cs atoms, and (e) an unsupported Cs monolayer. [After Wimmer et al. (1983).]
[cf. the contour lines of 1 x l o p 3e / ( a . ~ .in ) ~Figs. 29b-e], we observe a polarization of the Cs valence electrons toward the W surface. Clearly, the Cs valence electrons lose their identity as they penetrate into the high-density region of the W(d) electrons. From Fig. 29 it also becomes apparent why the region near the Cs nuclei is gaining electrons: The electrons that spill out into the vacuum on a clean W surface are attracted by the Cs nuclei. The wave functions of these electrons obtain their nodal structure near the Cs nuclei due to the orthogonalization to the Cs-core electrons. Clearly, this charge near the Cs nuclei is increased as the Cs atoms are brought close to the surface (cf. Fig. 29). C . Magnetic Transition-Metal Surfaces and Interfaces 1. Ferromagnetic Fe(001) Surface
It is natural to begin our discussion of magnetic surfaces with the classic ferromagnet bcc iron, which has a bulk magnetic moment of 2.12 ,ue (Danan et al., 1968). In an earlier theoretical study of surface states, surface magnetization, and electron spin polarization of the Fe(001) surface, Wang and Freeman (1 98 1 ) used an LCAO thin-film method and found an enhancement of the surface magnetism and strong Friedel-type oscillations in the spin density. In this LCAO calculation, a small variational basis set for the wave functions and a superposition model of spherical charges were used. Both computational restrictions can cloud the significance of the results, particularly for delicate quantities such as spin densities. Therefore, Ohnishi et al. (1983) undertook a reexamination of the electronic and magnetic surface properties of Fe(001) using the highly accurate FLAPW method.
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E. WIMMER, H. KRAKAUER, A N D A. J. FREEMAN
As in the earlier LCAO calculation, the surface is represented by a sevenlayer slab and the spin-polarized exchange-correlation potential given by von Barth and Hedin (1972) is used. The FLAPW result of a surface-induced enhancement of the magnetic moment agrees with that of the earlier LCAO calculation. However, in this new calculation no significant Friedel oscillation is found. The FLAPW results for the magnetic moments going from the bulk-like center of the seven-layer film to the surface are 2.27,2.39, 2.35, and 2.96 ,uB,i.e., the moment at the surface is increased by 30 % compared to the center. A study of the density of states decomposed into atomic (i.e., layer) and 1-like components gives insight into the mechanism of the surfaceinduced enhancement of the magnetism. The DOS in the central layer of the seven-layer Fe(001) film (lower panels of Fig. 30) is very close to the DOS of bulk bcc Fe [compare, e.g., with Moruzzi et al. (1978), p. 1701: The DOS shows a three-peak structure, so typical for the bcc structure, with a pronounced minimum below the highest peak. The d-band for the majority spin MINORlTY
SPIN
PIN
s '
3t C .F
_____I
!L 'F
c=b
4 1
-8
E (eV) FIG. 30. Atomic and I-projected densities of states for the center (lower panels) and surface (upper panels) atoms in a seven-layer Fe(001) slab. [After Ohnishi et al. (1983).]
THEORY O F SURFACE ELECTRONIC STRUCTURE
415
has a small unoccupied part, whereas for the minority DOS the Fermi energy falls into the characteristic minimum leaving about 30% of the minority d-band unoccupied. The resulting spin imbalance is reflected in the large magnetic moment of bcc iron. The DOS for the surface layer (top panels of Fig. 30) is dramatically changed compared with the bulk-like DOS of the central layer. Due to the reduced symmetry and fewer number of nearest neighbors in the surface layers, the characteristic three-peak bcc structure is lost and the d-band is narrowed. As a consequence, the majority d-band is now almost completely filled. For the minority DOS, we observe (upper right panel in Fig. 30) a peak just at the Fermi energy with its center of gravity slightly above E,. The states leading to this peak in the surface DOS fall in the minimum of the bulk-like DOS (lower right panel in Fig. 30) and can be identified as surface states. These surface states are also present in the majority DOS and are shifted to lower energies by an exchange splitting of about 2 eV. The net result of the surface-induced d-band narrowing and the occurrence of surface states is a larger spin imbalance corresponding to an enhancement of the magnetic moment in the surface. The theoretical result of a large enhancement of the surface magnetic moment for the Fe(001) from 2.27 (bulk) to 2.96 pB (surface) appears to agree with the enhanced magnetism inferred from Mossbauer experiments (Tyson et al., 1981). We now consider the spin density of the Fe(001) surface, which is shown for the (1 10) plane perpendicular to the surface in Fig. 31. The greatest part is dominated by positive spin densities with only small pockets of negative spin densities between the atoms. In the bulk-like center the shape of the spin density shows important nonspherical components originating from orbitals of t,, symmetry. Surprisingly, this bulk-like shape is found even for the iron atoms just one layer below the surface, and only the surface atoms exhibit a different shape of the spin density that is markedly more spherical. The polarization on the vacuum side of the surface atoms is found to be positive. In conclusion, the theoretical all-electron self-consistent results for the Fe(001) surface demonstrate, contrary to the earlier work (Wang and Freeman, 1981), that the seven-layer film is a very good model of the Fe(001) surface, with both the bulk and surface properties well represented. 2. Ferromagnetic Ni(001) Surface Since the report of magnetically dead layers on a Ni(001) surface (Liebermann et al., 1970), this system has attracted a host of experimental and theoretical investigations, partly with contradicting conclusions. It is now established, however, that the clean Ni(OO1) surface is not magnetically dead. This result is based on experiments using spin-polarized field emission (Landolt and Campagna, 1977), spin-polarized photoemission (Moore and
416
E. WIMMER, H. KRAKAUER, AND A. J. FREEMAN
S
s-l
s-2
C FIG. 31. Spin density of a seven-layer Fe(001) slab in the (1 10) plane perpendicular to the surface in units of e / ( a . ~ . Dotted )~ contours indicate negative spin densities. [After Ohnishi et nl. (1983).]
Pendry, 1978), electron-capture spectroscopy (Rau, 1982), and polarizedelectron diffraction (Feder et al., 1983). It seems that the experimental results showing magnetically dead layers have been clouded by difficulties in the sample preparation and characterization partly due to interface problems. This wealth of experimental information provided a challenging subject for theoretical studies in terms of accurate a6 initio calculations. The first steps in this direction were made by Dempsey and Kleinman (1977) who reported on a parametrized, non-self-consistent Ni surface calculation. Wang and Freeman (1980) presented the first self-consistent study of a nine-layer slab of Ni(001) using the linear-combination of atomic orbitals discretevariational-method (LCAO-DVM). They found that the surface was not magnetically dead, but that the magnetic spin moment for the surface atoms was reduced compared with the bulk-like center of the film. However, as stated above, these LCAO-DVM calculations have possible limitations in the
THEORY O F SURFACE ELECTRONIC STRUCTURE
417
variational freedom of the basis and the representation of the charge density. Jepsen et al. (1 980, 1982) concluded from their five-layer LAPW film calculations that the surface magnetic moment is slightly increased, which is in agreement with theoretical-computational studies by Freeman et al. (1982a) and Krakauer et al. (1983). Recent highly accurate FLAPW studies on a seven-layer Ni(OO1) slab (Freeman er a!., 1982a,b; Wimmer et al., 1984) shed new light onto the magnetism of the Ni(OO1) surface: (i) The magnetic moments in atomic-nearestneighbor volumes are found to be (from the central layer to the surface) 0.56, 0.59, 0.60, 0.68pB,i.e., there is an enhancement of the magnetic moment of -20% of the surface compared to bulk; (ii) no Friedel oscillation of the moment going from the surface to the center of the seven-layer film; and (iii) the majority surface state A?, lies 0.14 eV below the Fermi energy, i.e., there is no majority-spin d-hole at h?. In all three points, the FLAPW results disagree with the early pioneering calculations on a nine-layer Ni(OO1) film of Wang and Freeman (1980). Differences between the two approaches can be seen in the spin densities [compare Fig. 32 with Fig. 7 of Wang and Freeman], which
S
s-l
s-2
C FIG. 32. Spin density of a seven-layer Ni(001) slab in the (100) plane perpendicular to the surface in units of 10-4e/(a.u.)3.[After Wimmer et al. (1984).]
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E. WIMMER, H. KRAKAUER, A N D A. J. FREEMAN
in the FLAPW case shows much more negative values between the atoms. The theoretical result for the work function (5.37 eV) agrees well with the experimental value of 5.22 eV. The decomposition of the majority and minority charges into l-projected partial charges inside atomic spheres gives insight into the mechanism of the surface-induced enhancement of the magnetic moments. The dominant partial charge inside the Ni spheres has d-character. For the majority spin, this d-like charge is increased for the surface atom compared with the interior of the film and the s,p-charge is decreased; in other words, for the atoms with a reduced number of nearest neighbors the charges become more d-like. For the minority spin, we find a similar trend, but here also the d-charge decreases in the surface leading to an increased d-moment. The majority d-band is completely filled giving a rather small density of states at the Fermi energy of 16.9 states/Ry, whereas the only partially filled minority d-band gives a high density of states at the Fermi energy of 175.8 states/Ry. The exchange splitting varies from 0.19eV for a state near the bottom of the d-band with s-mixture to 0.69eV for a pure d-state about 1 eV below E,. For the M , surface states with the majority state just below E,, the unoccupied minority state is found to be 0.78 eV higher in energy. The spin densities for the seven-layer Ni(001) slab, shown in Fig. 32 exhibit bulk-like features for all atoms except those at the surface. We observe a localized positive spin density inside atomic spheres of a radius of about 2 a.u. and a pronounced negative spin density in the interstitial region. The spin density inside the atomic spheres includes important nonspherical components originating from wave function of t Z gsymmetry. The dominant component of this spin density maps out the top part of the completely filled d-band, which has no occupied counterpart in the minority spin system. The shape of the spin density of the surface atoms is quite different from the bulk-like atoms in the interior of the film particularly on the vacuum side where the spin density shows an egg-like shape. The spin density in the vacuum region is slightly negative. We consider now the spin densities at the nuclei for the seven-layer Ni(001) slab, which give the contact polarization hyperfine field. For all layers the total spin density at the nucleus is negative. The value for the surface layer is reduced by 15 % compared to the center of the film. A decomposition of the contact spin density into core and valence contributions shows that the core part scales with the moment in the spheres, i.e., we find an increase for the surface, whereas the valence part becomes positive for the surface atoms. The ratio of the core polarization at the nucleus and the magnetic moment due to the d-valence electrons is nearly constant. The FLAPW results indicate a surface-induced core-level shift to smaller binding energies for the 3p,,, states of 0.30 and 0.40eV for majority and
THEORY OF SURFACE ELECTRONIC STRUCTURE
419
minority spin, respectively. For the ls1,* states, the core-level shifts are 0.35 eV to smaller binding energies for both spin directions. The surface-induced core-level shifts are caused by a global electrostatic shift of the potential to smaller limiting energies. This effect also leads to the formation of the R,surface states, which is split off from the top of the d-band and shifted to smaller binding energies by about 0.3 eV. Thus, there is a close correspondence between surface states and core-level shifts on the Ni(001) surface (Wimmer, 1984). It is remarkable that for both Fe and Ni the reduced number of nearest neighbors on the surface leads to an enhancement of the magnetic moment. This trend becomes even more obvious and a simple picture emerges when we include linear chains and free atoms in the consideration: Going from bulk to a (001)-surface, then to a linear chain (Freeman and Weinert, 1982; Weinert and Freeman, 1983a), and finally to the free atom, the moments are 0.56,0.68, 1.1, and 2.0 for Ni and 2.27, 2.96, 3.3, and 4.0 for Fe. Thus, as the dimensionality is decreased, the magnetic moments approach the values of the free atoms. 3. Induced Magnetism on the Pt(001) Surface So far, we have discussed two ferromagnetic surfaces, Fe(001) and Ni(001), and have found in both cases a surface-induced enhancement of the magnetic moment. We now consider a case of a paramagnetic transition metal surface and study the response to an external magnetic field. Platinum is an interesting candidate, since recent lg5Pt NMR results (Yu et al., 1980; Yu and Halperin, 1981; Stokes et al., 1981) on very small particles (d 20-3OOw) indicate that due to the large surface their electronic and magnetic properties are quite different from bulk Pt metal. The experiments suggest that there is a Knight shift distribution related to the surface and a shift toward positive values. Using the FLAPW method, Wang et al. (1983) and Weinert and Freeman (1983b) performed an all-electron local spin density calculation for a five-layer Pt(001) slab. An external magnetic field of 1 mRy was applied during the self-consistency procedure. Since we are interested here in energies in the milliRydberg range, a high degree of accuracy and self-consistency is required from the calculations. The results show that the induced magnetic moment decreases as we go from the center to the surface of the Pt film, thus indicating a reduced magnetic susceptibility at the surface. This in turn causes a decrease in the core-polarization contribution to the Knight shift. The positive valence contribution, on the other hand, increases at the surface. The net result is a positive Knight shift at the surface. The surface-induced decrease in the magnetic moment can be anticipated and interpreted by a study of the atomic and l-projected densities of states, which exhibit a decrease in
-
420
E. WIMMER, H. KRAKAUER, A N D A. J. FREEMAN
the density of states in the surface atoms at the Fermi energy due to a narrowing of the d-band. In addition, we notice for the surface atom an increase in s,p-character relative to the center of the film of the DOS at the Fermi energy. The decreased d-density of states accounts for the reduced susceptibility and consequently for the reduced core-polarization part of the Knight shift. The increased s-character is responsible for the enhanced valence contribution to the Knight shift. The resulting positive Knight shift for the surface atoms explains the experimental results (Stokes et al., 1981) for small Pt particles. 4. Magnetism on the Ni-Cu Interface One of the best studied interfaces between magnetic and nonmagnetic metals is that of the Ni-Cu system. Because of the good match of their lattice constants [a(Ni) = 3.526 A and a(Cu) = 3.615 A], overlayers and interfaces are readily accessible experimentally. However, questions about the magnetism of Ni overlayers on a Cu surface are still open. Interface phenomena also play a key role in layered coherent modulated structures (CMS) (Hilliard, 1979), a promising new kind of synthetic materials for which the Ni-Cu system is the prototype. Wang et al. (1981b,e, 1982) and Freeman et al. (1982a) carried out the first ab initio determination of the electronic structure and magnetism of Ni overlaps on Cu(OO1). They used the linearized augmented plane wave (LAPW) thin-film method to obtain accurate self-consistent spin-polarized semirelativistic energy band solutions for Ni overlayers on a Cu(OO1) substrate, consisting of a five-layer Cu(OO1) slab plus one or two p( 1 x 1) layers on Ni on either side, referred to as Ni-Cu and 2Ni-Cu, respectively. The spatial distribution of the spin density in both cases shows that the magnetization is localized in the Ni layers. As seen in Fig. 33, the magnetization is essentially zero on the Cu layers. The vacuum and interstitial regions are slightly polarized in the opposite direction, similar to that reported for the clean Ni(OO1) film show in Fig. 32. An examination of the layer-by-layer magnetic moments (contributed by the electrons inside the touching muffin-tin spheres) indicates that the magnetic moment of the surface Ni layer of the 2Ni-Cu film increases by about 10 % to 0.69 pB compared to the bulk value (Freeman et a/., 1982a). The moment of the interface layer Ni(1) of the 2Ni-Cu film decreases by 24%, and the Ni layer of the Ni-Cu system decreases by 37 % (to 0.39 pB) compared to the calculated bulk value. From an orbital angular-momentum decomposition, we find that the contribution to the moments arises almost completely from the d-like component, similar to that in the clean Ni film. An analysis of the theoretical results leads to two observations that relate to the problem of surface magnetism. First, the surface and interface affect the
THEORY OF SURFACE ELECTRONIC STRUCTURE
42 1
VACUUM
FIG. 33. Spin densities for two layers of Ni on both sides of a Cu(OO1) five-layer film. The units are 10-4e/(a.u.)3.[After Wang et a/. (1982).]
total number of electrons so that, in general, the surface atoms have fewer electrons and the Ni atoms in contact with the Cu substrate have more electrons due to charge transfer from Cu to Ni. Second, the change in the total number of electrons arises almost completely from the minority spin electrons, and this leads to the decrease of the moment with an increase of the total number of electrons. The influence of charge transfer and the change of bonding on the position of the 3d majority and minority spin bands are not independent; their splitting will change through the exchange-correlation. However, the exchange splitting ( A E ) is not the same for different states. For the Ni-Cu film, and f,, 0.28 and 0.37 eV for the exchange splitting is 0.39 and 0.35 eV for MI and M 3 , and 0.38 eV for X2.The moment of the various layers is found to be proportional to the exchange splitting. However, the ratio A E / p , the socalled Stoner-H ubbard parameter, remains almost unchanged and is also close to the value for the isolated monolayer and bulk Ni.
r4
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E. WIMMER, H. KRAKAUER, AND A. J. FREEMAN
One of the interesting results of our analysis is the correlation of the magnetic moment with the number of p-like electrons. At the surface this dehybridization of the s-, p-, and d-electrons acts to increase the magnetic moment and is related to the d-band narrowing seen there. This dehybridization is also related to the simultaneous removal of p-electrons from the muffin-tin (MT) spheres (where the d-electrons are mainly localized) as they spill out into the vacuum region. It is not surprising that the number of p-electrons in the MT region is correlated with the degree of dehybridization; in the free Ni atom, the p-orbitals are completely unoccupied. There is a remarkable correlation between the total s and p charge of both spins, q, + qp (p, and pp are essentially zero), and the magnetic moment. Since q, is relatively unchanging ( - k 0.03 electrons), this is, indeed, a correlation with qp. In the unsupported Ni monolayers, where the electrostatic shift mechanism is absent, this dehybridization accounts for the large increase of pLacompared to the bulk value. In all cases, the total charge in each vacuum region (there are two per slab) is equal(to within 0.01 electrons) to the loss of p-electrons from the MT spheres. In the 2Ni-Cu slabs, other effects are seen; charge transfer (about 0.1 electrons) into the d-bands of the interfacial Ni layer reduces the magnetic moment to 0 . 4 8 in ~ ~this layer. Since this atom still has a coordination number of 12, the dehybridization should not take place, as is also seen. By contrast, the surface Ni atoms of the 2Ni-Cu slab show some dehybridization and can also have an upward electrostatic shift. Indeed, the Ni(S) atom does show a 3p3,, core-level shift of 0.26 eV to reduced binding energy relative to the Ni(1) atom. Both these effects are consistent with the increased moment of 0.69 pB. Finally, it is interesting to compare these results with those for Ni-Cu modulated structures obtained by Jarlborg and Freeman (1980). Their model uses a unit cell for the modulated Cu-Ni structure along [loo], and contains, in all, eight layers with no local distortion considered. The result for 1 or 2 Ni layers (Ni,Cu, and Ni,Cu,) shows a strong quenching of the moment, but no dead layers. More Ni layers (Ni3Cu, and Ni,Cu,) also give some reduction, but the variation of moments between different layers is negligible. Thus, they find reduced magnetic moments on Ni when it is layered with Cu but no magnetically dead Ni layers when Ni is deposited onto a Cu(100) substrate. D. Energetics of Surfaces: All-Electron Total Energy Approach
The intense experimental interest in surface problems, such as chemisorption, surface reconstruction and relaxation, and dynamics, has produced a wealth of data, much of it still not understood. Complicating the theoretical understanding is the fact that often the important structural parameters needed are not available experimentally. In order to solve this problem theo-
THEORY OF SURFACE ELECTRONIC STRUCTURE
423
retically, we can use the general principle of minimization of the total energy to determine the stability of a system. Density-functional theory (Hohenberg and Kohn, 1964; Kohn and Sham, 1965) provides an elegant framework in which the total energy of solid-state systems can be obtained for any geometrical configuration of the nuclei. With the advent of accurate methods to solve the local-density (LD) one-particle equations, there has been increasing interest (Janak, 1974; Moruzzi et al., 1978; Zunger and Freeman, 1977; Ihm et al., 1979; Ferrdnte and Smith, 1978; Wendel and Martin, 1979) to use these methods to determine the total energy and related properties, such as equilibrium phases, lattice constants, and force constants of both bulk solids and surfaces. The major problem in any straightforward application of the total-energy expressions involves numerical problems arising from the cancellation between the very large kinetic- and potential-energy contributions (Janak, 1974). The problem obviously becomes more severe for heavier atoms since the (chemically inactive) core electrons are responsible for the largest part of the total energy. To avoid this problem, one successful approach has been to remove the core electrons from the problem as is done in the pseudopotential method (Ihm et al., 1979). Within an all-electron approach, by using the muffin-tin approximation (Moruzzi et al., 1978), Janak (1974) has obtained an algebraic cancellation of part of the core contributions in the expressions for the total energy and pressure. Weinert et al. (1982) go beyond these treatments and consider the total energy using an all-electron, general-potential approach. A key feature of this new approach is the high accuracy that results from an explicit cancellation of the Coulomb singularities in the kinetic and potential-energy terms arising from the nuclear charge. As an example of the applicability of this method to solid-state systems, Weinert et al. (1982) have implemented it in the FLAPW method (Wimmer et al., 1981a) for thin films. Results are presented for characteristic problems: (i) the equilibrium distance in a monolayer of covalently bonded graphite for which comparisons of the calculated equilibrium structural properties and cohesive energy can be made with experiment; (ii) the relaxation of the W(OO1) surface; and (iii) surface energies of the W(OO1) and V(OO1) surfaces. 1. Equilibrium Geometry and Cohesive Energy of a Graphite Monolayer As stated, one important motivation for determining the total energy is as
a tool to treat problems primarily in surface physics. Because of the complexity of the problem, it is important to demonstrate the accuracy of the method
424
E. WIMMER, H. KRAKAUER, A N D A. J. FREEMAN
for systems that have experimentally known parameters. For this reason, Weinert et al. (1982) have chosen as an illustrative application a graphite monolayer with which to test the applicability of their method for determining the ground-state properties of surfaces and thin films. Graphite is a layered compound that has very strong interactions in the plane, but only very weak interplane interactions, i.e., the structural parameters in the plane are nearly independent of crystal thickness (Donohue, 1974). A central question in the theory of cohesion is the relative stability of different crystal structures. For this reason Weinert et al. (1982) have determined the total energy per atom versus nearest-neighbor distance for both the square and hexagonal lattices of carbon atoms. The results are presented in Fig. 34. As expected, the hexagonal lattice is energetically favored (by 3.3 eV/at.) compared to the square lattice. The calculated equilibrium nearest-neighbor distances for the two lattice types are substantially different. However, the density of atoms is nearly the same: The equilibrium planar area per atom is 2.60 and 2.65A2 for the hexagonal and square lattices, respectively. The calculated bond length for the hexagonal graphite monolayer
-
-e
\
-75.3-
\
Y
x
0
tC
YI
-
-75.5
1.3
1.5
d
(Q
1.7
FIG. 34. Total energy per atom versus nearest-neighbordistance for square and hexagonal monolayers of carbon atoms. [After Weinert et al. (1982).]
THEORY OF SURFACE ELECTRONIC STRUCTURE
425
of 2.450 A is contracted by -0.4 % compared to the experimental value of 2.461 A for bulk graphite Donohue, 1974). The fact that the monolayer is contracted with respect to the bulk is consistent with the experimental observation that the in-plane thermal expansion coefficient is negative below about 400°C (Ludsteck, 1972). The amount of contraction for a monolayer cannot be estimated easily, but should be on the order of a few tenths of a percent (or -0.006 A, in this case). These arguments suggest that the agreement with experiment for the lattice constant is better than the 0.4 % disagreement compared to bulk. Perhaps, more importantly, the results show that local density functional theory is able to predict correctly the small contraction of the in-plane lattice parameter. The absolute values of the experimental and calculated LD total energies differ by 10 eV/at., which by comparing ionization potentials is seen to arise mainly from the self-energy of the 1s electrons. This problem of local density does not affect cohesive energies as can be seen from frozen-core results where errors in the core are canceled explicitly. We now compare the calculated cohesive energies (defined as the difference between the total energy per atom, corrected for the zero-point energy of the monolayer, and the spin-polarized free atom) with the experimental value. Since we are neglecting the interlayer binding, we should expect to obtain a slightly smaller cohesive energy (on the order of a few tenths of an electron volt) than is found for bulk graphite. What is found, however, is that the result of calculations using a limited variational basis is in fortuitously excellent agreement with experiment, while the converged LD result is overbound by 1.3 eV/at.). The tendency for LD to overestimate the cohesive energy is in agreement with the results of Moruzzi et al. (1978) on a large number of bulk metals. These authors suggest that the problem lies in the atomic calculations, a suggestion that is confirmed also for the carbon atom (Weinert et al., 1982). The results for the graphite monolayer show that the use of the allelectron total-energy formalism (Weinert et al., 1982) in conjunction with an accurate all-electron method, such as FLAPW, leads to the accurate determination of structural parameters. The high degree of internal numerical stability allows us to treat even quite large systems composed of transition metals, thereby opening the possibility of accurately describing the energetics and dynamics of surface processes.
-
2. Multilayer Relaxation of the W(OO1) Surface Relaxation and reconstruction play a fundamental role in the physics and chemistry of surfaces and thus have been the object of intense experimental efforts. Their observation in such diverse systems as semiconductors and clean and adsorbate-covered metal surfaces indicates that these structural
426
E. WIMMER, H. KRAKAUER, A N D A. J. FREEMAN
changes can be considered to be the rule rather than the exception. In some cases, such as the Al(110) (Nielsen et al., 1982), the Cu(ll0) (Adams et al., 1982), or the V(100) (Jensen et al., 1982) and Re(010i) (Davis and Zehner, 1980) surfaces, it has been possible recently to present experimental evidence for multilayer relaxation effects. Earlier theoretical attempts to calculate surface-relaxation effects have been limited to semiempirical tight-binding calculations (Allan and Lannoo, 1973; Treglia et ul., 1983; Stephenson and Bullett, 1984) or to simplified model Hamiltonians (Barnett et al., 1983).The accurate experimental determination of multilayer relaxation effects have called attention to the need for precise theoretical determinations of the energetics and detailed information on the driving mechanism behind the observations. First-principles self-consistent calculations have confirmed a damped oscillatory multilayer relaxation of the Al(110) surface (Ho and Bohnen, 1984). O n the other hand, for the important class of transition-metal surfaces, no ab initio study of any surface multilayer relaxation has been reported and little is known about the energetics of such a process. For well-studied surfaces such as W(OOl), no multilayer relaxation has been observed and even the extent of the relaxation of just the first layer has been a matter of controversy. As discussed in Subsection 1V.B.l.c, the values of the low-energy electrondiffraction (LEED) analyses for this contraction vary between 4.4 f 3 % and 11 f 2 %. Backscattering-channeling experiments (Feldman et al., 1977) using mega-electron-volt ions lead to the conclusion that the value for the contraction does not exceed 6 %. A recent spin-polarized LEED study (Feder and Kirschner, 1981) suggests a value of 7.0 f 1.5%. Fu et al. (1984) presented the first all-electron local density functional study of the energetics of the multilayer relaxation process on the W(OO1) surface employing the FLAPW total-energy approach. They predict a contraction of the topmost layer by 5.7 % accompanied by an outward relaxation of the second and third layers leading to an increase of the second and third interlayer spacings by 2.4 % and 1.2%, respectively. Surprisingly, they find that the relaxation of the second and third interlayer spacings does not influence the equilibrium spacing between the two topmost layers, i.e., keeping the inner layers unrelaxed leads to practically the same equilibrium distance between the first and second layers as is found for the fully relaxed system. Thus, the equilibrium between the adjacent layers appears to be governed by highly screened local interactions. This decoupling of the relaxation for the topmost and inner layers is the more remarkable, since 25 % of the total relaxation energy of 0.06 eV originates from the relaxation of the inner layers. In order to explore the energy hypersurface that determines the multilayer relaxation process in the case of the five-layer film, the interlayer spacings between first and second ( d 1 2 )and second and third layers ( d 2 3 ) are varied independently. For the seven-layer film, three interlayer spacings, d , , , d 2 3 ,
THEORY O F SURFACE ELECTRONIC STRUCTURE
427
and d,,, are treated as independent quantities. The energy hypersurface is scanned by varying d with a step width of 1.5 % of the bulk lattice constant. Near the equilibrium geometry, these discrete points of the energy hypersurface are parabolically fitted as shown in Fig. 35. The root-mean-square (rms) value of this parabolic fit is less than 0.2 mRy, showing that, as expected, the system behaves harmonically around its equilibrium positions. Furthermore, this small rms value demonstrates the high-numerical precision and stability of the all-electron FLAPW approach. The electronic origin of the relaxation on a transition-metal surface such as the W(OO1) surface may be understood by considering the simultaneous effects of bonding of localized d-electrons and delocalized sp-electrons. In the bulk of a transition metal, the bond formation (Pettifor, 1977) driven by d-electrons tends to decrease the interatomic distances while the freeelectron-like sp-electrons minimize their contribution to the total energy by expanding the system, which decreases their kinetic energy. The balance between these two mechanisms leads to the bulk equilibrium geometry. At the surface, the d-d bonding energy is increased by enhanced d-d bonding, while the sp-electrons can maintain their low kinetic energy by extending further out into the vacuum. As a consequence of this increased spill-out of electrons into the vacuum, the topmost interlayer spacing is contracted relative to the bulk layer spacing. This balance between these two mechanisms also results in a work function (a= 4.6eV) that shows very little variation (less than 0.1 eV) for all the relaxation processes studied here, i.e., the enhancement of the d-d bonding enhances 0,while the rearrangement of the sp electrons decreases the surface-dipole layer as the surface layer spacing is contracted. Furthermore, the surface-relaxation energy (0.06 eV) amounts to only 2 % of
FIG.35. Energy hypersurface for a seven-layer W(OO1) film as a function of the relative changes (in percentage) of the first two interlayer spacings. For this plot, the third interlayer spacing is kept at its bulk value. The minimum in the total energy (indicated by the star) is set equal to zero, i.e., subtracting 226144.0175 Ry. The open circles indicate the geometries where self-consistent calculations have been performed. [After Fu et al. (1984).]
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E. WIMMER, H. KRAKAUER, A N D A. J. FREEMAN
the surface energy (Fu et al., 1985). Thus, the surface-relaxation mechanism does not lead to a significant change in the surface energy. 3. Surface Energies of Transition Metals: W(OO1) and V(OO1) The energy to form a surface plays an important role in many physical and chemical processes on solid surfaces such as fracture, catalysis, and epitaxial growth. Unfortunately, experimental measurements of the surface energy are difficult to perform. They are mostly constrained to the determination of surface energies at high temperatures and are subject to numerous errors due to surface-active contaminants. For example, although the tungsten surface has been the most studied metal surface in the last decade, the experimentally measured surface energies at high temperature scatter widely from 1.8 J/mZ to 5 J/m2 (Barbour et al., 1960; Bettler and Barnes, 1968; Dranova et al., 1970; Hodkin et al., 1970; Allen, 1972). For other important metals such as vanadium, no measurements of surface energies have been reported. In principle, total-energy calculations for thin films and bulk materials can give surface energies straightforwardly by comparing the total energy per atom in a film and in the bulk or by comparing two films of different thicknesses. In practice, however, both procedures requires high precision and numerical stability, and thus represent challenging problems. Recently, Fu et al. ( 1 985) reported theoretical determinations of the surface energies of W(100) and V(100) based on all-electron self-consistent firstprinciples calculations within local-density-functional theory by using the FLAPW total energy method. The W(OO1) and V(OO1) surfaces are described in a single-slab geometry with five and seven atomic layers. The convergence of the surface energy with respect to film thickness depends on how well the inner layers of the film approach the bulk. This is examined by comparing the total energy of the bulk atoms and the corresponding value obtained from the total energy difference between the five- and seven-layer films. Thus, Fu et al. (1985) have incorporated results obtained with both the FLAPW thin-film method (Wimmer et al., 1981a) and an independent FLAPW bulk method (Jansen and Freeman, 1984) into the surface energy determinations. As it turns out, five- and seven-layer W(OO1) films are sufficiently thick to derive meaningful values for the surface energy: The total energy per bulk W atoms as obtained from the film-FLAPW results is - 32306.350 Ry, which agrees to within 0.003 Ry with the bulk-FLAPW results. The value for the surface energy of the W(OO1) surface is found to be 230 mRy (5.1 J/m2) (Fu et al., 1985), where the contribution from the surface relaxation is only about 4.5 mRy. For the surface energy of V(OOl), Fu et al. (1985) obtained a value of 3.4 J/m2. A comparison of the theoretical value at T = 0 K with the measured surface energies at high temperatures for W(OO1) suggests a large sur-
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face entropy (AS 1.5 x Jm-’K-’ ). This implies large lattice anharmonicity and emphasizes the possible disordered nature of the W(OO1) surface at high temperatures. In the near future, the all-electron local-density-functional total-energy approach, combined with calculations of forces, applied to problems of surface structures, chemisorption, and catalysis can be expected to enlarge substantially our microscopic understanding of surfaces that display such a great variety of interesting and technologically important phenomena.
ACKNOWLEDGMENTS We are grateful to our colleagues and collaborators (S. R. Chubb. C. L. Fu. H. J. F. Jansen. D. D. Koelling, S. Ohnishi, M. Posternak, D. S. Wang, and M. Weinert) who participated in the work reported here and in discussions.
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A
Acceleration, 23 1-232, 240, 242 Acceleration voltage, particle beam, 286,288 Action, 235 Alkali transition-metal surface bonding, 410-413 Aluminum surfaces, 368- 390 charge densities, 374-375 chemisorption phase, 385-388 layer-by-layer density of states, 383 - 385 oxidation, 379-389 sodium chemisorption, 389- 390 surface states and surface resonance states, 368-374 2p core-level shifts and crystal field splitting, 376-379 Ampere, 24 I , 243 - 245 Ampere-turn, 244 - 245, 253- 254 magnetic field strength measurement, 260-261 Analytical electron microscopy, I74 Angle, dimensional terms, 238, 241 Area, 23 I , 233
B Barium ferrite, in a magnetic field, 274-275 Barkhausen noise, 273 Black-body radiation, radiance, 233, 288289 C
Candela, 234 Capacitance, 245, 250 CBED, see Convergent beam electron diffraction Cesiated tungsten surfaces, 404 -4 I3 alkali bonding, 410-413 work-function lowering, 404-409 Charged particle beam optics, 279 Chemisorption aluminum surfaces, 385-390 cesiated tungsten surface, 405 435
Cobalt Curie temperature, 275 in a magnetic field, 274-275 spin-density-functional theory, 362 Computer output devices, ink jets used in, 103- I07 Convergent beam electron diffraction (CBED), 306 Copper-nickel interface, magnetism on, 420-422 Coulomb, 243 Curie temperature, 275 D
Density-functional theory, 358 - 359, 423; see also Local-spin-density-functional theory charge densities, 374-375 Diamagnetic material, 275 Dielectric constant, 27 I -272 Dimensional terms for energy transport, 229-294 electric and magnetic quantities, 242 -247 energy densities, numerical values for, 292-294 magnetic field quantities and their dimensions, 247-278 particle beams, numbers for the characterization of, 278-292 physical concepts and, 230-235 SI system, peculiar aspects of, 235-242 Dipole moment, 26 1 -264 Domain, 273 Dyadic function, definition and properties of, 7-9 Dyadic Green’s function, 1-90 calculations asymptotic expression for media with four plane and parallel layers, 3 1 asymptotic expression for media with three cylindrical concentric layers, 31-32 asymptotic expression for media with three plane and parallel layers, 28-3 1
436
INDEX
Dyadic Green's function (Continued) media with four plane and parallel layers, 18-23 media with three cylindrical concentric layers, 23-28 media with three plane layers, 14 - I8 expansion, 1 1 - 12 cylindrical concentric layered media, 13-14 media with parallel plane layers, 12- 13 free-space dyadic Green's functions, 9 - 1 I introduction, 3 -4 microstrip antennas, analysis of conclusions, 88 - 89 microstrip disk antenna, 32-42 microstrip ring antenna, 42 - 55 microstrip wraparound antenna, 5 5 -66 Dynamo principle, 278
E EELS, see Electron energy loss spectroscopy Electrical effects quantitative description, 232, 235, 240247 table, 238-239 Electric arc welding, 277 E1ectron magnetic moment, 264 surface electrons, see Surface electronic structure Electron beam radiance, 233 sources, 280 Electron diffraction general theory, 308- 309 high-energy approximation, 309 - 3 13 many-beam diffraction, 299 -308 computations, 303 - 304 differential equations, 3 16- 3 19 direct integrated methods of calculation, 321-322 fast Fourier transform, 304-305, 325, 337-338 history, 299 - 30 1 iterative method of calculation, 32032 I matrix methods of calculation, 3 19-320 mixed representation, 3 18- 3 19, 324 multislice method, 300- 303, 322- 323, 325,327-331, 337-338
real-space representation, 3 16, 324- 340 reciprocal space representation, 3 16318,324 slice methods of calculation, 322- 323, 327-332, 352 special-purpose hardware for computations, 305 study of existing formulations, 3 16-324 periodic-continuation method of calculation, 34 1 -342 real-space method, 324-340 comparison with slice methods, 327332 error tests in multislice calculations, 334-335 input parameters, analysis of, 334- 336 numerical procedure, 332- 334 patching technique, 337-340 principles, 324-326 S2test criterion, 335 Schrodinger equation, 308 - 309 Electron energy loss spectroscopy (EELS), 173 Electron microscopy image calculations in high resolution microscopy, 295 - 355 absorption inelastic scattering, 344 - 346 atomic column separation, 343 beam tilt, 350-351 direct structural retrieval, 346 -350 future prospects, 352-353 general quantum-mechanical problems, 351-352 introduction, 296-297 man y-beam electron-diffractioncalculations, 299 - 308 periodic-continuation method, 34 1 - 342 phase-grating approximation, 348 - 349 real-space patching techniques, 337 - 340 systemic approach to electron-diffraction problem, 308 - 3 15 thermal diffuse scattering, 346 upper layer lines, 350 image formation, principles of, 297 -299 image formation by inelastically scattered electrons, 173-227 amorphous sphere, 208 assemblies of atoms, 205 -21 1 conclusion, 2 13- 2 14 diffractograms, 200 - 20 1, 22 1 - 224 in dipole approximation, 219-220
437
INDEX generalized dielectric function, 183185,216-219 intensity distribution, 195-208, 2 13 introduction, 173- 174 mixed dynamic form factor, 174- 185, 195, 213, 216-219 numerical results, 195- 2 13 phase problem in electron scattering, 174-180 single atom, image ofa, 195-205 surface plasmon, image of a, 2 I I -2 I4 theory of image formation, 185- 195 thin crystal, 208 -2 I 1 thin foil, 206 - 207 transition rate, calculation of, 2 14- 2 15 Electron scattering, see also Electron microscopy, image formation by inelastically scattered electrons absorption inelastic scattering, 344- 346 phase problem, 174- 180 thermal diffuse scattering, 346 Electron volt. 241 Electrostatic lens, 270 Electrostatic machine, 278 ELNES, see Energy-loss near edge structure Emittance, particle beam, 287 Energy, 231 -232 electric and magnetic quantities, 244 - 247 magnetic field, generation of, 247-252 particle beams, 279 radiance, 233 SI system, 235 - 242 Energy bands, in theory of surface electronic structure, 359, 362 aluminum, 368 - 369,37 1-373 Energy densities, numerical values for, 292294 Energy flow, 23 I Energy-loss near edge structure (ELNES),206 Energy transport, dimensional terms for, see Dimensional terms for energy transport Entropy, particle beam, 285 EXELFS, see Extended energy-loss fine structure Extended energy-loss fine structure (EXELFS), 206 Extensive quantity, 232, 27 1 F Farad, 243 FBEM, see Fixed-beam electron microscope
Ferromagnetic materials Curie temperature, 275 spin-density-functional theory, 362 surface electronic structures, 4 13-4 19 Fixed-beam electron microscope (FBEM) amorphous sphere, image of an, 208 diffractograms, 200-20 I , 22 1 - 222 image formation, 189- 19 1 lattice image, 21 1 normalized intensity distribution in image of thin foil, 207 phase contrast, 194 reciprocity theorem, 192- 193 single atom, image of a, 195-200 surface plasmon, image of a, 2 12 FLAPW, see Full-potential linearized augmented plane wave method Force, 231 -232, 240,242 momentum, connection with, 246 Full-potential linearized augmented plane wave (FLAPW) method, surface calculations, 359-360, 364-367 aluminum surfaces, 377-379 cesiated tungsten surfaces, 405 ferromagnetic iron surface, 4 13-4 15 ferromagnetic nickel surfaces, 4 17 -4 19 platinum surfaces, 41 9 surface energies oftungsten and vanadium, 428-429 tungsten surfaces, 426-428
G Gauss, 27 I Graphite monolayer, equilibrium geometry and cohesive energy of, 423-425
H Heat, 242 Henry, 243, 250 Hysteresis loop, 273-274 1
Impedance, 244-245,248-249,271-272 Impulse, 236-237, 240, 246 Inductance, 245, 248-251 geometry as determinate of, 266-267 matter in the magnetic field, 272 Inelastic electron scattering, image formation by, see Electron microscopy, image for-
438
INDEX
Inelastic electron scattering (Continued) mation by inelastically scattered electrons Ink-jet printing, 9 1 - 171 applications, 103- 1 1 1, 164, I66 carpet printing, 11 1, 163 ceramics industry, 166 computer terminals, 103- 107 document printers, 107- 108 facsimile printers, 108- 1 1 1 printing codes, 108 routing information, 108 special purpose printers, I 1 I conclusions, I63 - 166 continuous-jet systems, 93, 97- 103, 163165 break-up of a continuous fluid jet, 132140 compound jet, I38 - I40 drop formation, 132- 136 electrostatic ink-jet formation, 102- 103 Hertzmethod, 93, 100-102, 152-161 ink properties, 137- 138 mechanical valves, 103 oscillograph recorders, I4 1 - 142 satellite drops, 136- 137 Sweet method, 93,98-100, 142- 152 conventional printing methods versus ink jet, 91 -92, 166 drop-on-demand systems, 93 -97, I63 165 computer terminals, 103- 107 drop shaping, refill, and reverberations, 127-131 facsimile transceiver, 1 10 ink properties, 131 piezo-ceramic transducer, 1 13- 1 I5 principle of, I 1 I - 1 13 wave propagation in ink channel, 112115 transformation of pressure waves into velocity, 122- 127 introduction, 9 1 - 1 1 1 mechanical valves, 103, I I I , 161 - 163 methods, review of, 92- 103 Intensive quantity, 233, 27 1 Ion beam sources, 280 Iron Curie temperature, 275 in a magnetic field, 272-275
spin-density-functional theory, 362 surface electronic structures, 4 13-4 15
J Joule, 236-238,240-246
K Kinetic energy, 236
L Laser, 29 1 -292 radiance, 234 Length, 231-232,240 Light comparison of light beam with particle beam, 288-292 radiance, 233 - 234, 287 Linear compressibility, as unit of measurement, 23 1 Local-spin-density-functional (LSDF) theory, surface electronic structure, 36 1 363 Local-spin-density (LSD) one-particle equations, 364-365, 367 LSDF theory, see Local-spin-density-functional theory Luminance. 234
M Magnequench material, 274 Magnetic effects, quantitative description of, 232,235,240-247 Magnetic field, 244-246 energy required for generation of, 247-252 forces on currents in, 275-278 matter in the, 272-275 two-field quantities, 270-272 Magnetic field energy, 244-247 Magnetic field strength, 244, 247, 260-26 1, 267-268,270-275 matter in the magnetic field, 272-273 Magnetic flux, 244-245 field lines and, 252-254 forces on a pole region, 265 Magnetic flux density, 244, 246, 251, 254255,269-275 matter in the magnetic field, 272-273 Magnetic lens, 269
439
INDEX Magnetic moment, 26 1 -265 maximum, 274 Magnetic pinch effect, 277 Magnetic tension. 267 - 270 Magnetic transition-metal surfaces and interfaces, 4 1 3 - 422 Magnetization, 274 Magneto-hydrodynamic (MHD) generator, 27 8 Magneto-motive force, 267 Magnetostriction, 275 Many-beam electron diffraction, see Electron diffraction Mass, 231 -233, 240 Measurement, units of, 230-23 1 Mechanical effects, dimensional terms, table, 238-239 Meter, in energy relationships, 235 -237 MHD generator, see Magneto-hydrodynamic generator Mho, 243 Microstrip antenna, 3 application, 5 -6 cavity model with conducting magnetic side walls, 6 - 7 description, 4 - 5 Microstrip disk antenna radiation pattern, effect of dielectric substrate on, 32-42 equivalent magnetic current density, 33-34 radiated fields, 34 -4 1 surface-wave excitation, influence on efficiency and directivity of space waves, 66-88 directivity of antenna, 87-88 integral representation of H , and H,, 8 1 82 integral representation of the fields, 67 integration of H4, 68 -81 poles in p plane, 82-85 radiation efficiency of space waves, 85 88 Microstrip ring antenna radiation characteristics, effects of dielectric cover on, 42- 55 equivalent magnetic current density, 42-44 radiated fields, 44- 55 Microstrip wraparound antenna
radiation pattern, influence of dielectric substrate on, 55-66 equivalent magnetic current density, 56-57 far fields radiated from magnetic current ring, 58-59 far fields radiated from wraparound antenna, 59-66 MKS-Giorgi system, 235, 243 Momentum, 236-237,240 force, connection with, 246 Mutual inductance, 245, 249 N Nearly-free-electron (NFE) metal surface states, 368- 37 I Needle printer, see Wire-matrix printer Newton, in energy relationships, 235, 240, 242 Newton-meter, 235 -242 NFE metal, see Nearly-free-electron metal Nickel Curie temperature, 275 spin-density-functional theory, 362 surface electronic structures, 4 15-4 19 Nickel-copper interface, magnetism on, 420-422 Nuclear magnetic resonance (NMR), 264 Nuclear magneton, 265
0 Oersted, 26 1, 27 1 Ohm, 243 Oxidation, aluminum surfaces, 379-389
P Paramagnetic material, 275 Particle beam, 278-279 comparison with light beam, 288-292 disturbances to thermodynamic ensemble Of, 285 - 286 formation, 281 -285 radiance numbers, 286-288 sources, 279-28 I Permanent magnet, 273-274 Permeability, 27 1-272, 273 Permittivity constant, 27 1-272 Perveance factor, 290
440
INDEX
Photon, 234 Plasma, particle beam source, 279 -28 1 Platinum surfaces, induced magnetism on, 4 19-420 Pole region, forces on, 265-266 Proton, dipole moment, 264
R Radian, 237 Radiance, 233-234,288-290 particle beams, 279,281 -283,285-288 Radiance per volt, 285-286 Radiation, dimensional terms, 233 Ray-value-per-volt, 280-281, 285-286,290 Reciprocity theorem, in light optics, 191- 193 Refractive index, 288-289 Relativistic pinch effect, 276-277 Remanent magnetism, 273 Resistance, 244 Rogowski coil, 269 Rotation, dimensional terms for energy transport, 237-238,241
applications, 367 - 429 approach and methodology, 363-367 energetics of surfaces, 422-429 graphite monolayer, 423 -425 introduction, 358-360 iron surfaces, 4 13-4 15 local-spin-density-functional theory, 36 1 363 magnetic transition-metal surfaces and interfaces, 4 13- 422 nickel-copper interface, 420-422 nickel surfaces, 4 15- 4 19 platinum surfaces, 4 19- 420 theoretical framework, 360- 363 thin-slab approximation, 360- 36 1 transition metals, 390-422 tungsten surfaces, 39 1-4 13 multilayer relaxation, 425 -428 surface energies, 428-429 vanadium surfaces, 428 System International, see SI system T
S Scanning transmission electron microscope (STEM) amorphous sphere, image of an, 208 diffractogram, 223 image formation, 185- 189 lattice image, 208 - 2 10 reciprocity theorem, 192- 193 single atom, image of a, 200-205 spectrometer acceptance angle, dependence on, 213-214 surface plasmon, image of a, 2 12 Scattering, electron, see Electron scattering Scattering, particle beam, 285 -286 Self-inductance, 245, 249 SI system, 230 electric and magnetic quantities, 242 - 247 energy transport, dimensional terms for, 233-235 peculiar aspects of, 235-242 Spring force, 23 1,242 STEM, see Scanning transmission electron microscope Superconductivity, 25 1 Surface electronic structure, 357-434 aluminum surfaces, 368 - 390
Temperature, 242 Tesla, 254, 26 1 - 262 Thermionic source of electron beams, 28028 I Thin film, surface electronic structure, theory of, 363-367 Time, 23 1,240 Torque, 236-238,260-263 Transformer, 256-259 ampere-turns, 267 Transition metal surfaces, 359, 390-422 ferromagnetic iron surface, 4 13-4 15 ferromagnetic nickel surface, 415-419 magnetic transition metals, 41 3 -422 nickel -copper interface, 420 - 422 platinum surfaces, 4 19-420 thin slabs, 360 tungsten surfaces, 39 1 - 4 13 multilayer relaxation, 425 -428 surface energies, 428 -429 vanadium surfaces, 428 Tungsten surfaces, 39 1-4 I3 alkali bonding, 410-413 cesiated W(OO1) surface, 404-413 charge density and work function, 393 clean W(O0 1) surface, 39 1 -404
441
INDEX rnultilayer relaxation of W(O0 1) surface, 425-428 relaxation and reconstruction, 399 -404 surface energies of W(OOl), 428-429 surface states and surface resonance states, 394-399 work-function lowering, 404-409 Turn, concept of, 245,254
V Vanadium, surface energies of, 428 Velocity, 23 I , 240 Volt, 24 1, 243 - 246 Voltage generation, 277-278 induced, 255-259 particle beam, 280,283, 286
Volt-ampere, 242 Volt-second, 253-254 Volume, 231 -233
W Watt, 244 Watt-second, 242 Weber, 243,245 Weight, 231 -232, 240 Wire-matrix printer, 1 1 1 - 1 12 Work. 240-242
X X-ray absorption near-edge structure (XANES), 206
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