Advances in Imaging and Electron Physics (eBook)

Peter W. Hawkes (Herausgeber)

eBook Download: PDF | EPUB

2001 | 1. Auflage
451 Seiten
Elsevier Science (Verlag)
978-0-08-052621-8 (ISBN)

Infos

Advances in Imaging and Electron Physics merges two long-running serials-Advances in Electronics and Electron Physics and Advances in Optical and Electron Microscopy. This series features extended articles on the physics of electron devices (especially semiconductor devices), particle optics at high and low energies, microlithography, image science and digital image processing, electromagnetic wave propagation, electron microscopy, and the computing methods used in all these domains.

Inhalt

Front Cover 1
Advances in Imaging and Electron Physics: Numerical Field Calculation for Charged Particle Optics 4
Copyright Page 5
Contents 6
Preface 12
Future Contributions 14
Acknowledgments 18
Introduction 20
Chapter I. Basic Field Equations 28
1.1 Maxwell's Equations 28
1.2 Electromagnetic Potentials 30
1.3 Variational Principles 35
1.4 Wave Equations and Hertz Vectors 39
1.5 Boundary Conditions 42
1.6 Integral Equations for Electrostatic Fields 46
1.7 Integral Equations for Magnetic Fields 52
1.8 Integral Equations for Wave Fields 55
References 56
Chapter II. Reducible Systems 58
2.1 Azimuthal Fourier-Series Expansions 58
2.2 Rotationally Symmetric Boundaries 63
2.3 Magnetic Round Lenses 66
2.4 Series Expansions 72
2.5 Planar Fields 78
References 84
Chapter III. Basic Mathematical Tools 86
3.1 Orthogonal Coordinate Systems 86
3.2 Interpolation and Numerical Differentiation 101
3.3 Modified Interpolation Kernels 113
3.4 Mathematical Representation of Curves 123
3.5 Mathematical Representation of Surfaces 129
3.6 Numerical Integration 134
References 141
Chapter IV. The Finite-Difference Method (FDM) 142
4.1 Two-Dimensional Meshes 142
4.2 Five-Point Configurations 154
4.3 Nine-Point Configurations 161
4.4 The Cylindrical Poisson Equation 172
4.5 Irregular Configurations 194
4.6 Subdivision of Meshes 212
4.7 Concluding Remarks 216
References 217
Chapter V. The Finite-Element Method (FEM) 220
5.1 Generation of Meshes 220
5.2 Discretization of the Variational Principle 227
5.3 Analysis in Triangular Elements 231
5.4 The Finite-Element Method in First Order 243
5.5 Field Interpolation 256
5.6 Solutions of Large Systems of Equations 269
References 286
Chapter VI. The Boundary Element Method 290
6.1 Discretization of Integral Equations 291
6.2 Axially Symmetric Integral Equations 311
6.3 Numerical Solution of Integral Equations 328
6.4 Special Techniques for Asymmetric Integral Equations 348
6.5 The Calculation of External Fields 362
6.6 Other Applications of Integral Equations 377
References 381
Chapter VII. Hybrid Methods 384
7.1 Combination of the FEM with the BEM 384
7.2 Combination of the FDM with the BEM 388
7.3 The Charge Simulation Method (CSM) 394
7.4 The Current Simulation Model 414
7.5 The General Alternation Method 424
7.6 Fast Field Calculation 435
7.7 Calculation of Equipotentials 445
References 453
Appendix 456
Index 460

Textauszug

Chapter I

Basic Field Equations

Erwin Kasper

Abstract

In this chapter we shall start with the presentation of Maxwell’s equations in their most general form. Although a program that could solve them for arbitrary initial boundary and material conditions would be of interest, this is hardly feasible in practice, as the amount of necessary data and computer operations would be tremendous. Therefore, we shall gradually specialize Maxwell’s equations to cases that are of importance in charged particle optics and that comprise the majority of computation problems. Throughout this volume we shall follow the standard notation in electrodynamics, presented in Table 1.1; any necessary deviations from it will be mentioned explicitly

1.1 MAXWELL’S EQUATIONS

Maxwell’s equations are partial differential equations for vectorial field functions, which all depend on the spatial position r = (x, y, z) and the time t. In the notation that is familiar in vector analysis, they are given by

Ert=−∂Brt/∂t,

  (1.1)

Hrt=∂Drt/∂t+jrt,

  (1.2)

ivDrt=ρrt,

  (1.3)

ivBrt=0

  (1.4)

Table 1.1

Standard Notations

These vector functions are interrelated by material equations, which describe the electromagnetic properties of matter on a phenomenological basis.

The electric properties are characterized by a polarization P(r, t), the spatial density of electric dipoles:

rt=ε0Ert+Prt.

  (1.5)

Similarly, the analogous magnetic property is the magnetization M (r, t), the magnetic dipole density, usually defined by

rt=μ0Hrt+μ0Mrt.

  (1.6)

With respect to applications in charged particle optics the polarization P is of little importance, as electrodes are usually conductors, and consequently all static electric fields vanish in these. In the few cases in which insulator materials are present, the assumption of proportionality is sufficient:

rt=εr−ε0Ert,

  (1.7)

whereupon Eq. (1.5) reduces to

rt=εrErt.

  (1.8)

Even in configurations with a constant dielectric coefficient ε, this becomes a spatial function because it alters discontinuously at any surface. It is now possible to eliminate the vector field D(r, t) from Maxwell’s equations, and we shall do this here.

The analogous linearizations with respect to magnetic fields would result in

rt=μr/μ0−1Hrt,

  (1.9)

rt=μrHrt.

  (1.10)

It would be a considerable simplification if these relations were valid throughout. We then speak of linear or unsaturated media.

Unfortunately, this assumption does not always hold, and there are then different steps of generalization. The simplest one is the case of isotropic nonlinear media, most favorably presented by

r=vr,|B|Br.

  (1.11)

This equation means that H and B always have the same direction, but the material factor v := μ− 1 called the reluctivity depends on the norm of B. This assumption is simplistically made in most finite-element programs for the calculation of magnetic lenses, for instance, those written by Munro [4] and Lencova [5].

This form, however, is not always sufficient. For instance, it cannot be applied to devices with permanent magnets or with magnetically anisotropic materials. In such cases, we must start from the more general material equation

r=μ0−1Br−MrB,

  (1.12)

in which M does not necessarily vanish for B = 0.

The system of basic equations is completed by a relation between the current density j and the electromagnetic field. Its simplest and most familiar form is

=kE,

  (1.13)

where k is the conductivity. We shall, however, hardly ever need this equation in charged particle optics, because here we are mainly interested in the spatial distribution j(r), producing a designed magnetic field. The determination of the voltage, to be applied to the coils, is an elementary task.

1.2 ELECTROMAGNETIC POTENTIALS

In the main course of this volume, we shall specialize to stationary, that is, time-independent fields, as these comprise most cases of technical importance; if we have to consider configurations with time-dependent fields, this will be stated explicitly.

1.2.1 Electrostatic Fields

If we disregard the electric field in the coils of magnetic devices, as is usually done, the system of Maxwell’s equations becomes uncoupled, leading to an important simplification. The electrostatic part then reduces to

E=0

  (1.14a)

ivD=ρ

  (1.14b)

=εE.

  (1.14c)

Equation (1.14a) can be integrated once by the introduction of an electrostatic potential V(r),

r=−gradVr.

  (1.15)

It is favorable to eliminate the D field completely, whereupon the system (1.14) reduces to one partial differential equation of second order for V(r),

ivεrgradVr=−ρr.

  (1.16)

Because ε > 0, this has the basic form of a self-adjoint elliptic equation. In the course of this volume, we shall encounter such a mathematical form frequently in various contexts but with different physical meanings of its variables.

Inhomogeneous dielectric properties are of little importance in charged particle optics (the discontinuity of ε at the surface between different materials must be considered by boundary conditions). With ε = const. Eq. (1.16) simplifies to Poisson’s equation. As this will appear quite frequently, we shall introduce a simplified notation for the Laplace operator,

:=∇≡divgrad.

  (1.17)

In cartesian coordinates (and only in these) this operator takes the simple form

=∂2/∂x2+∂2/∂y2+∂2/∂z2.

  (1.18)

We shall use the simpler symbol Δ where this is not misleading: otherwise we shall use the notation ∇2. Poisson’s equation now takes the familiar...

Produkt-Details

• Erscheint lt. Verlag: 5.7.2001
• Sprache: englisch
• Themenwelt: Sachbuch/Ratgeber
• Themenwelt: Mathematik / Informatik ► Informatik
• Themenwelt: Naturwissenschaften ► Physik / Astronomie ► Elektrodynamik
• Themenwelt: Technik ► Bauwesen
• Themenwelt: Technik ► Elektrotechnik / Energietechnik
• Themenwelt: Technik ► Maschinenbau
• ISBN-10: 0-08-052621-7 / 0080526217
• ISBN-13: 978-0-08-052621-8 / 9780080526218