Physical Principles of Electron Microscopy: An Introduction to TEM ...

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Electron Optics 37 symmetry and use cylindrical coordinates: z , r ( = radial distance away from the z-axis) and � ( = azimuthal angle, representing the direction of the radial vector r relative to the plane of the initial trajectory). Therefore, as shown in Fig. (2-7a), vz , vr and v� are the axial, radial, and tangential components of electron velocity, while Bz and Br are the axial and radial components of magnetic field. Equation (2.5) can then be rewritten to give the tangential, radial, and axial components of the magnetic force on an electron: F� = � e (vz Br ) + e (Bz vr) (2.6a) Fr = � e (v� Bz) (2.6b) Fz = e (v� Br) (2.6c) Let us trace the path of an electron that starts from an axial point O and enters the field at an angle � , defined in Fig. 2-7, relative to the symmetry (z) axis. As the electron approaches the field, the main component is Br and the predominant force comes from the term (vz Br) in Eq. (2.6a). Since Br is negative (field lines approach the z-axis), this contribution (� evzBr) to F� is positive, meaning that the tangential force F� is clockwise, as viewed along the +z direction. As the electron approaches the center of the field (z = 0), the magnitude of Br decreases but the second term e(Bz vr) in Eq. (2.6a), also positive, increases. So as a result of both terms in Eq. (2.6a), the electron starts to spiral through the field, acquiring an increasing tangential velocity v� directed out of the plane of Fig. (2-7a). Resulting from this acquired tangential component, a new force Fr starts to act on the electron. According to Eq. (2.6b), this force is negative (toward the z-axis), therefore we have a focusing action: the non-uniform magnetic field acts like a convex lens. Provided that the radial force Fr toward the axis is large enough, the radial motion of the electron will be reversed and the electron will approach the z-axis. Then vr becomes negative and the second term in Eq. (2.6a) becomes negative. And after the electron passes the z = 0 plane (the center of the lens), the field lines start to diverge so that Br becomes positive and the first term in Eq. (2.6a) also becomes negative. As a result, F� becomes negative (reversed in direction) and the tangential velocity v� falls, as shown in Fig. 2-8c; by the time the electron leaves the field, its spiraling motion is reduced to zero. However, the electron is now traveling in a plane that has been rotated relative to its original (x-z) plane; see Fig. 2-7b. This rotation effect is not depicted in Fig (2.7a) or in the other ray diagrams in this book, where for convenience we plot the radial distance r of the electron (from the axis) as a function of its axial distance z. This common convention allows the use of two-dimensional rather than three-dimensional diagrams; by effectively suppressing (or ignoring) the rotation effect, we can draw ray diagrams that resemble those of light optics. Even so, it is
38 Chapter 2 important to remember that the trajectory has a rotational component whenever an electron passes through an axially-symmetric magnetic lens. (a) (b) (c) B z F r -a 0 a z v r Figure 2-8. Qualitative behavior of the radial (r), axial (z), and azimuthal (�) components of (a) magnetic field, (b) force on an electron, and (c) the resulting electron velocity, as a function of the z-coordinate of an electron going through the electron lens shown in Fig. 2-7a. v � F z B r F � v z
Page 2 and 3: Physical Principles of Electron Mic
Page 4 and 5: Ray F. Egerton Department of Physic
Page 6 and 7: Contents Preface xi 1. An Introduct
Page 8 and 9: Contents ix 7. Recent Developments
Page 10 and 11: xii Preface textbooks, such as Will
Page 12 and 13: 2 1.1 Limitations of the Human Eye
Page 14 and 15: 4 Chapter 1 parallel beam of light
Page 16 and 17: 6 Chapter 1 Figure 1-3. One of the
Page 18 and 19: 8 Chapter 1 Figure 1-5. Light-micro
Page 20 and 21: 10 Chapter 1 Figure 1-7. Scanning t
Page 22 and 23: 12 Chapter 1 Figure 1-8. Early phot
Page 24 and 25: 14 Chapter 1 Figure 1-10. JEOL tran
Page 26 and 27: 16 Chapter 1 hydrated. But high-ene
Page 28 and 29: 18 Chapter 1 Figure 1-14. Scanning
Page 30 and 31: 20 Chapter 1 Figure 1-16. Photograp
Page 32 and 33: 22 Chapter 1 visible-light photons
Page 34 and 35: 24 Chapter 1 Typically, the STM hea
Page 36 and 37: Chapter 2 ELECTRON OPTICS Chapter 1
Page 38 and 39: Electron Optics 29 a c b object len
Page 40 and 41: Electron Optics 31 � a n 2 ~1.5
Page 42 and 43: Electron Optics 33 We can define im
Page 44 and 45: Electron Optics 35 electron source
Page 48 and 49: Electron Optics 39 As we said earli
Page 50 and 51: Electron Optics 41 2.4 Focusing Pro
Page 52 and 53: Electron Optics 43 � = [e/(8mE0)
Page 54 and 55: Electron Optics 45 image plane. Whe
Page 56 and 57: Electron Optics 47 procedure that i
Page 58 and 59: Electron Optics 49 The basic physic
Page 60 and 61: Electron Optics 51 From Eq. (2.7),
Page 62 and 63: Electron Optics 53 y +V 1 +V 2 -V 2
Page 64 and 65: Electron Optics 55 point from the o
Page 66 and 67: 58 Chapter 3 3.1 The Electron Gun T
Page 68 and 69: 60 Chapter 3 The rate of electron e
Page 70 and 71: 62 Chapter 3 electron emission curr
Page 72 and 73: 64 Chapter 3 As seen from the right
Page 74 and 75: 66 Chapter 3 � r r s r � (a) (b
Page 76 and 77: 68 Chapter 3 Table 3-2. Speed v and
Page 78 and 79: 70 Chapter 3 where G is a large amp
Page 80 and 81: 72 Chapter 3 The second condenser (
Page 82 and 83: 74 Chapter 3 Figure 3-9 summarizes
Page 84 and 85: 76 Chapter 3 resolution (especially
Page 86 and 87: 78 Chapter 3 The major advantage of
Page 88 and 89: 80 Chapter 3 contrast (for a given
Page 90 and 91: 82 Chapter 3 A more correct procedu
Page 92 and 93: 84 Chapter 3 diffraction pattern (s
Page 94 and 95: 86 Chapter 3 Nowadays, photographic
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88 Chapter 3 A related concept is d
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90 Chapter 3 molecules, imparting a
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92 Chapter 3 Frequently, liquid nit
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94 Chapter 4 -e -e -e -e +Ze Figure
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96 Chapter 4 � (a) elastic z x m
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98 Chapter 4 � b a (a) (b) Figure
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100 Chapter 4 fraction of electrons
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102 Chapter 4 whose composition or
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104 Chapter 4 replica crystallites
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106 Chapter 4 4.5 Diffraction Contr
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108 Chapter 4 remain undiffracted (
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110 Chapter 4 Close examination of
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112 Chapter 4 Unfortunately, the va
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114 Chapter 4 Crystals can also con
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116 Chapter 4 A simple example of p
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118 Chapter 4 High-magnification ph
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120 Chapter 4 At this stage it is c
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122 Chapter 4 All of the above meth
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124 Chapter 4 The procedures outlin
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126 Chapter 5 specimen scan coils o
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128 Chapter 5 functions with m and
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130 Chapter 5 inelastic collisions
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132 Chapter 5 Figure 5-5. Dependenc
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134 Chapter 5 Figure 5-7. (a) Secon
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136 Chapter 5 Because each secondar
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138 Chapter 5 Backscattered electro
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140 Chapter 5 Whereas Ip remains co
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142 Chapter 5 Figure 5-14. Red, gre
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144 Chapter 5 the SE2 and SE3 compo
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146 Chapter 5 just-observable loss
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148 Chapter 5 Section 4-10. Films o
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150 Chapter 5 A backscattered-elect
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152 Chapter 5 One example of this p
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Chapter 6 ANALYTICAL ELECTRON MICRO
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Analytical Electron Microscopy 157
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Page 178 and 179:
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Page 184 and 185:
178 Chapter 7 Figure 7-1. Modified
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180 Chapter 7 7.2 Aberration Correc
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182 Chapter 7 Besides decreasing th
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184 Chapter 7 7.4 Electron Holograp
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186 Chapter 7 There are now many al
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188 Chapter 7 holography could ther
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Appendix MATHEMATICAL DERIVATIONS A
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Mathematical Derivations 193 A.2 Im
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REFERENCES Binnig, G., Rohrer, H.,
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INDEX Aberrations, 3, 28 axial, 44
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Index 199 EELS, 172 Electromagnetic
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Index 201 Principal plane, 32, 80 P
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