final - ustc

��

�� 1

1.1 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 �� . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 �� . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.2 �� . . . . . . . . . . . . . . . . . . . . . . . . . 3

�� 11

2.1 ��Æ�� . . . . . . . . . . . . . . . . . . . . . . 11

2.2 ��Æ�� . . . . . . . . . . . . . . . . . . . . . 13

2.3 �� . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.4 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.5 �� . . . . . . . . . . . . . . . . . . . . . . 17

2.6 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.7 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.8 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

�� 26

3.1 �� . . . . . . . . . . . . . . . . 26

3.1.1 �� . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.1.2 �� . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.1.3 �� . . . . . . . . . . . . . . 30

3.2 �� . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.3 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.4 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.5 �� . . . . . . . . . . . . . . . . . . . . . . . . 39

ii ��

3.5.1 �� . . . . . . . . . . . . . . . . . . . . . 39

3.5.2 �� . . . . . . . . . . . . . . . . . . . . . 42

3.6 �� . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.6.1 �� . . . . . . . . . . . . . . . . . . . . . . . 43

3.6.2 �� . . . . . . . . . . . . . . . . . . . . . . . 44

3.6.3 �� . . . . . . . . . 46

3.6.4 �� . . . . . . . . . . 50

3.6.5 �� . . . . . . . . . . . . . . 52

3.7 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.8 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.9 �� . . . . . . . . . . . . . . . . . . . . . . 58

3.9.1 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.9.2 �� . . . . . . . . . . . . . . . . . . . 63

3.9.3 ��Ǳ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.10 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.10.1 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.10.2 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.10.3 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.10.4 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.11 �� . . . . . . . . . . . . . . . . . . . . . . . . . 72

3.11.1 �� . . . . . . . . . . . . . . . . . . . 72

3.11.2 �� . . . . . . . . . . . . . . . . . . 73

3.12 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

3.12.1 �� . . . . . . . . . . . . . . . . . . . . . . . 78

3.12.2 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

3.12.3 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

�� 95

4.1 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

4.2 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

4.2.1 ��—�� . . . . . . . . . . . . . . . . . . . . . 102

4.2.2 �� . . . . . . . . . . . . . . . . . . . . . . . . . . 103

4.2.3 �� . . . . . . . . . . . . . . . . . . . . . . . 106

�� iii

4.3 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

4.3.1 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

4.3.2 �� . . . . . . . . . . . . . . . . . . . . . . . . . . 112

4.4 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

4.4.1 �� . . . . . . . . . . . . . . . . . . . . . . . . . . 113

4.4.2 �� . . . . . . . . . . . . . . . . . . . . . . . . . 115

4.5 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

4.5.1 �� . . . . . . . . . . . . . . . . . . . . . 118

4.5.2 �� . . . . . . . . . . . . . . . . . . . . . . . . . . 121

4.5.3 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

�� 129

5.1 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

5.1.1 �� . . . . . . . . . . . . . . . . . . . . . . . . . . 130

5.1.2 �� . . . . . . . . . . . . . . . . . . . . . . . . . 135

5.1.3 �� . . . . . . . . . . . . . . . . . . . . . . . . . 137

5.2 �� . . . . . . . . . . . . . . . . . . . . . . . . . 139

5.3 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

5.4 �—�� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

5.5 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

5.5.1 �� . . . . . . . . . . . . . . . . . . . . . . . . . 147

5.5.2 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

5.5.3 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

5.6 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

5.6.1 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

5.6.2 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

5.6.3 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

5.6.4 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

5.6.5 �� . . . . . . . . . . . . . . . . . . . . . . . . . 154

5.7 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

5.8 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

�� 158

6.1 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

iv ��

6.2 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

6.2.1 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

6.2.2 � �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

6.2.3 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

6.2.4 θ−Æ�� . . . . . . . . . . . . . . . . . . . . . . . . 164

6.2.5 Z−Æ�� . . . . . . . . . . . . . . . . . . . . . . . . 167

6.2.6 Grad-Shafrnov�� . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

6.3 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

6.3.1 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

6.3.2 �� . . . . . . . . . . . . . . . . . . . . . 175

6.3.3 �—�� . . . . . . . . . . . . . . . . . . . . . . . . . . 181

6.4 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

6.5 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

6.5.1 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

6.5.2 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

6.5.3 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

6.5.4 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

�� 195

7.1 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

7.1.1 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

7.1.2 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

7.1.3 �� . . . . . . . . . . . . . . . . . . . . . . . . . . 199

7.2 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

7.3 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

7.4 �� . . . . . . . . . . . . . . . . . . . . . . . . 206

7.5 Kramers-Kronig�� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

7.6 �� . . . . . . . . . . . . . . . . . . . . . . 211

7.7 �� . . . . . . . . . . . . . . . . . . . . . . 215

�� (��) 219

8.1 �� . . . . . . . . . . . . . . . . . . . . . . . . . . 219

8.1.1 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

8.1.2 �� . . . . . . . . . . . . . . . . . . . 220

�� v

8.1.3 �� . . . . . . . . . . . . . . . . . . . . . . . . . 222

8.2 �� . . . . . . . . . . . . . . . . . . . . . . . . . . 225

8.2.1 �� . . . . . . . . . . . . . . . . . . . . . . . 226

8.2.2 �� . . . . . . . . . . . . . . 227

8.2.3 �� . . . . . . . . . . . . . . . . . . . . . . . . . 229

8.2.4 �� . . . . . . . . . . . . . . . . . . . . . . . . . . 232

8.2.5 �� . . . . . . . . . . . . . . . . . . . . . . . . . . 237

8.3 �� . . . . . . . . . . . . . . . . . . . . . . . . . . 238

8.3.1 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

8.3.2 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240

8.3.3 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

8.4 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

8.5 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

�� 252

9.1 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

9.2 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258

9.3 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

9.3.1 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

9.3.2 �� K-dV�� . . . . . . . . . . . . . . . . . . . . . . . . . . . 263

9.4 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267

9.5 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272

�� 277

10.1 �� . . . . . . . . . . . . . . . . . . 277

10.1.1 �� . . . . . . . . . . . . . . . . . . . . . 277

10.1.2 �� . . . . . . . . . . . . . . . . . . . . . . . . . . 279

10.1.3 �� . . . . . . . . . . . . . . . . . . . . . . . . . . 284

10.2 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286

10.3 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290

10.3.1 �� . . . . . . . . . . . . . . . . . . . . . . . . . 290

10.3.2 �� . . . . . . . . . . . . . . . . . . . . . . 292

10.3.3 �� . . . . . . . . . . . . . . . . . . . . . . . . . . 293

10.3.4 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294

vi ��

10.3.5 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295

�� 299

11.1 �� . . . . . . . . . . . . . . . . . . . . 299

11.2 �� . . . . . . . . . . . . . . . . . . . . . . . . . . 301

11.2.1 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . 303

11.2.2 �� . . . . . . . . . . . . . . . . . . . . . . . . . . 304

11.2.3 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305

11.3 Yvon-Born-Green�� . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306

11.4 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308

11.4.1 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308

11.4.2 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311

11.5 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311

11.5.1 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312

11.5.2 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314

11.6 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316

11.7 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317

�� 319

12.1 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319

12.2 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320

12.3 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323

12.4 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324

12.5 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327

12.5.1 �� . . . . . . . . . . . . . . . . . . . . . . . 329

12.6 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329

12.7 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332

12.8 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334

12.9 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . 335

12.10 �� . . . . . . . . . . . . . . . . . . . . 338

12.10.1 �� . . . . . . . . . . . . . . . . . . . 338

12.10.2 ��Fresnel�� . . . . . . . . . . . . . . . . . . 339

12.10.3 �� . . . . . . . . . . . . . . . . . . . . 340

12.10.4 �� . . . . . . . . . . . . . . . . . . . . . . . . . . 340

�� vii

12.10.5 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34112.11 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341

viii ��

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1. �� 1980��

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4 ��

2. F. F. Chen�� 1980��Ǳ��

3. �� 1983��

4. ��(N. A. Krall)��(A. W. Trivelpieece)�� 1983��

5. ��1988��

6. ��2006��

��

1. L. Spizter, Physics of Ionized Gas (Interscience Publishers, New York, 1956). ��

��

2. C. L. Longmire, Elementary Plasma Physics (Interscience Publishers, New York,1963). ��

3. R. C. Davidson, Methods in Nonlinear Plasma Theory (Academy, New York, 1972).��

��

��Ǳ�� Physics ofPlasmas��

4. S. Ichimaru, Basic Principles of Plasma Physics (Benjamin, Massachusetts, 1973).��

5. V. L. Ginzburg, The Propagation of Electromagnetic Waves in Plasmas (Pergamon,Oxford, 1970). ��

1.2 �� 5

6. A. I. Akhiezer, et al, Plasma Electrodynamics (Pergamon Press, Oxford, 1975), Vols.I and II. ��

7. A. F. Alexandrov, L. S. Bogdankevich, and A. A. Rukhadze, Principles of Plasma

Electrodynamics (Springer, Berlin, 1984). ��

8. W. L. Kruer, The Physics of Laser Plasma Interactions (Addison-Wesley, New York,1988). ��

9. T. H. Stix, Waves in Plasmas (American Institute of Physics, New York, 1992). ��

10. S. Ichimaru, Statistical Plasma Physics (Addison-Wesley, Redwood City, 1992),Vols. 1 & 2. ��

11. J. Wesson, Tokamaks (Clarendon Press, Oxford, 1997). ��

12. T. J. M. Boyd and J. J. Sanderson, The Physics of Plasmas (Cambridge UniversityPress, Cambridge, 2003). ��

13. S. Atzeni and J. Meyer-ter-Vehn, The Physics of Inertial Fusion (Cambridge Uni-versity Press, Cambridge, 2004). ��

14. Reviews of Plasma Physics. ��/��

��Ǳ��

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�� L. D. Landau�� E. M. Lifshitz ��

6 ��

• Volume I, Mechanics

• Volume II, The Classical Theory of Fields

• Volume V, Statistical Physics I

• Volume VI, Fluid Mechanics

• Volume VIII, Electrodynamics of Continuous Media

• Volume X, Physical Kinetics

��

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� ��

a · (b× c) = b · (c× a) = c · (a× b), (1.1)

a× (b× c) = (a · c)b− (a · b)c, (1.2)

(a× b) · (c× d) = (a · c)(b · d)− (a · d)(b · c), (1.3)

∇×∇ψ = 0, (1.4)

∇ · (∇× a) = 0, (1.5)

∇× (∇× a) = ∇(∇ · a)−∇2a, (1.6)

∇ · (ψa) = a · ∇ψ + ψ∇ · a, (1.7)

∇× (ψa) = ∇ψ × a+ ψ∇× a, (1.8)

∇(a · b) = (a · ∇)b+ (b · ∇)a+ a× (∇× b) + b× (∇× a), (1.9)

∇ · (a× b) = b · (∇× a)− a · (∇× b), (1.10)

∇× (a× b) = a(∇ · b)− b(∇ · a) + (b · ∇)a− (a · ∇)b. (1.11)

1.2 �� 7

��

�� S ��Ǳ V��

Ǳ df�� ∫V

∇ ·Ad3r =∫S

A · df , (1.12)∫V

∇ψd3r =∫S

ψdf , (1.13)∫V

∇×Ad3r =

∫S

df ×A, (1.14)∫V

(φ∇2ψ +∇φ · ∇ψ)d3r =∫S

φ∇ψ · df , (1.15)∫V

(φ∇2ψ − ψ∇2φ)d3r =

∫S

(φ∇ψ − ψ∇φ) · df . (1.16)

�� C ��Ǳ S��Ǳ df��Ǳ dl��

�� ∫S

(∇×A) · df =∮C

A · dl, (1.17)∫S

df ×∇ψ =

∮C

ψdl. (1.18)

��

��

�� (ρ, φ, z)��

��

∇u =∂u

∂ρeρ +

1

ρ

∂u

∂φeφ +

∂u

∂zez, (1.19)

∇ · u =1

ρ

∂

∂ρ(ρuρ) +

1

ρ

∂

∂φ(uφ) +

∂

∂z(uz) , (1.20)

(∇× u)ρ =1

ρ

∂uz∂φ

− ∂uφ∂z

, (1.21a)

(∇× u)φ =∂uρ∂z

− ∂uz∂ρ

, (1.21b)

(∇× u)z =1

ρ

∂(ρuφ)

∂ρ− 1

ρ

∂uρ∂φ

, (1.21c)

∇2u =1

ρ

∂

∂ρ

(ρ∂u

∂ρ

)+

1

ρ2∂2u

∂φ2+∂2u

∂z2. (1.22)

8 ��

�� (r, θ, φ)��

∇u =∂u

∂rer +

1

r

∂u

∂θeφ +

1

r sin θ

∂u

∂φez, (1.23)

∇ · u =1

r2∂

∂r

(r2ur

)+

1

r sin θ

∂

∂θ(sin θuθ) +

1

r sin θ

∂uφ∂φ

, (1.24)

(∇× u)r =1

r sin θ

[∂(sin θuφ)

∂θ− ∂uθ

∂φ

], (1.25a)

(∇× u)θ =1

r sin θ

[∂ur∂φ

− ∂(r sin θuφ)

∂r

], (1.25b)

(∇× u)φ =1

r

[∂(ruθ)

∂r− ∂ur

∂θ

], (1.25c)

∇2u =1

r2∂

∂r

(r2∂u

∂r

)+

1

r2 sin θ

∂

∂θ

(sin θ

∂u

∂θ

)+

1

r2 sin2 θ

∂2u

∂φ2. (1.26)

��

�� Ǳ

��

��

∇ · E = 4πρ,

∇× E = −1

c

∂B

∂t,

∇ ·B = 0,

∇×B =1

c

∂E

∂t+

4π

cj.

�� Ǳ

∇ · E =ρ

ε0,

∇×E = −∂B∂t,

∇ ·B = 0,

∇×B =1

c2∂E

∂t+ μ0j.

1.2 �� 9

��

� c (μ0ε0)−1/2

�� E/√4πε0 E

��√ε0/4πD D

��√4πε0ρ ρ

��√μ0/4πB B

�� H/√4πμ0 H

��√4π/μ0M M

�� 4πε0σ σ

�� ε0ε ε

�� μ0μ μ

�� R/4πε0 R

�� L/4πε0 L

�� 4πε0C C

� 1.1: ��

�� (1.1)� ��

�� Ǳ

∇ · E =ρ

ε0,

��

��E�� ρ��

E → E√4πε0

, ρ→√4πε0ρ

��

∇ · E = 4πρ.

��Ǳ��

�� Ǳ

λDe =

√T

4πne2.

10 ��

��

�� m (L) 102 cm (L)

�� kg (M) 103 g (M)

�� s (T) 1 s (T)

�� Hz (T−1) 1 Hz (T−1)

� N (MLT−2) 105 dyne (MLT−2)

�� J (ML2T−2) 107 erg (ML2T−2)

�� W (ML2T−3) 107 erg/s (ML2T−3)

�� C (TI) 3×109 statcoulomb (M1/2L3/2T−1)

�� C/m3 (TIL−3) 3×103 statcoul/cm3 (M1/2L−3/2T−1)

�� A (I) 3×109 statampere (M1/2L3/2T−2)

�� A/m2 (IL−2) 3×105 statamp/cm2 (M1/2L−1/2T−2)

�� V/m (MLT−3I−1) 13 × 10−4 statvolt/cm (M1/2L−1/2T−1)

�� V (ML2T−3I−1) 1300 statvolt (MLT−3I−1)

�� C/m2 (TIL−2) 3×105 dipole moment/cm3 (M1/2I−1/2T−1)

�� C/m2 (TIL−2) 12π × 105 statvolt/cm (M1/2I−1/2T−1)

�� siemens/m (T3I2M−1L−3) 9×109 s−1 (T−1)

�� Ω (ML2T−3I−2) 19 × 10−11 s/cm (TL−1)

�� F (T4I2M−1L−2) 9×1011 cm (L)

�� Wb (ML2T−2I−1) 108 G/cm2 (M1/2L3/2T−1)

�� T (MT−2I−1) 104 G (M1/2L−1/2T−1)

�� A/m (IL−1) 4π × 10−3 Oe (M1/2L−1/2T−1)

� 1.2: ��

�� ne��

ne→ ne√4πε0

,

��

λDe =

√ε0T

ne2.

�� 1.2

��

�� 10�� 20��Ǳ��

��

��Ǳ��

��

2.1 ��Æ��

��

��

��Ǳ��

��Q��

��

∇2φ = −4πe(Zni − ne)− 4πQδ(r), (2.1)

�� δ(r)Ǳ δ−�� ni Ǳ�� ne Ǳ�� Z Ǳ��

�� φ��

��

ni = n0ie−Zeφ/T , (2.2a)

ne = n0eeeφ/T , (2.2b)

�� T Ǳ�� n0e � n0i �� Q��

12 ��

|eφ/T | � 1��

ni = n0i(1− Zeφ/T ), (2.3a)

ne = n0e(1 + eφ/T ), (2.3b)

��(2.1)�Ǳ

∇2φ−(4πZ2e2n0i

T+

4πe2n0e

T

)φ = −4πQδ(r). (2.4)

�� λD��Ǳ

1

λ2D≡ 4πZ2e2n0i

T+

4πe2n0e

T=

4πe2n0e(Z + 1)

T, (2.5)

��(2.4)�Ǳ

∇2φ− 1

λ2Dφ = −4πQδ(r). (2.6)

�� φ(r → ∞) → 0��Ǳ

φ(r) =Q

re−r/λD . (2.7)

��Q�� 0��

�� Ǳ φ = Q/εr��

� εǱ�� Q��Ǳ��

�� Ǳ��

��

��Ǳ��

��

��Ǳ��Ǳ�Ǳ�� |eφ/T | ∼ 1��

�� Ǳ��

��Ǳ

��

��Ǳ��“ �” ��

��Ǳ��

��Ǳ�

��Ǳ��Ǳ�

2.2 ��Æ�� 13

��

��

��

��

��

�� nα = nα0 exp(−Zαeφ/Tα)�

��Ǳ

λDα =

(Tα

4πe2αnα

)1/2

= 7.434× 102[Tα(eV)]1/2[nα(cm−3)]−1/2|Zα|−1 (cm), (2.8)

�� α�� ZαǱ��

�Ǳ

λ−2D =

∑α

λ−2Dα. (2.9)

2.2 ��Æ��

��

� Q ��

��Æ��

vTe = (T/me)1/2 �� λDe ��Ǳ��

��

τpe =λDe

vTe

=

√me

4πnee2. (2.10)

��

�� —�� τpe ��

�� τpe

�� τ > τpe �

L > λDe��

��Æ��(2.10)��Æ�� x = 0�� δx��

��

��Ǳ

E = 4πeneδx,

14 ��

�� ne �� x = 0��

�� x = 0� ��Ǳ

nemed2δx

dt2= −eneE = −4πn2

ee2δx,

�� Ǳ

ωpe =

√4πe2ne

me

= 5.641× 104[ne(cm−3)]1/2 (rad/sec). (2.11)

��Ǳ�Ǳ�� Ǳ��

��

��

��

2.3 ��

�� Æ�� Ǳ��

��

��Ǳ��

��

��Ǳ

n(r, t) =

N∑i=1

δ[r− ri(t)]. (2.12)

��Ǳ

nk(t) =1

V 1/2

∫V

n(r, t)e−ik·rd3r =1

V 1/2

N∑i=1

e−ik·ri(t).

� nk(t)��

nk(t) =1

V 1/2

N∑i=1

(−ik · vi)e−ik·ri,

nk(t) =1

V 1/2

N∑i=1

[−(k · vi)

2 − ik · vi

]e−ik·ri.

2.3 �� 15

��

��Ǳ

vi = − 1

m

∂

∂ri

∑j �=i

(Ze)2

|ri − rj|+ (��).

��Z��

(Ze)2

r=

1

V 1/2

∑k

4π(Ze)2

V 1/2k2eik·r,

��(�k = 0��)��

vi = − 1

m

∂

∂ri

∑j �=i

∑q

4π(Ze)2

V q2eiq·(ri−rj) = −i4π(Ze)

2

mV 1/2

∑q �=0

q

q2nq(t)e

iq·ri.

��Ǳ

nk(t) = − 1

V 1/2

N∑i=1

(k · vi)2e−ik·ri − 4πe2

mV

∑q �=0

N∑i=1

k · qq2

nq(t)ei(k−q)·ri

= − 1

V 1/2

∑i

(k · vi)2e−ik·ri − 4πe2

mV 1/2

∑q �=0

k · qq2

nq(t)nk−q(t)

� �� q = k�� nk=0 = N/V 1/2��

nk(t) +4πn(Ze)2

mnk(t) = − 1

V 1/2

∑i

−(k · vi)2e−ik·ri − 4πe2

mV 1/2

∑q �=0,k

k · qq2

nq(t)nk−q(t).

(2.13)��(2.13)��

��

��

��(2.13)��Ǳ

1

V 1/2

N∑i=1

−(k · vi)2e−ik·ri ∼ k2

T

mnk(t).

�� Ǳ

ω2pnk(t).

�� ω2p � k2T/m��

��Ǳ

k2λ2D � 1,

16 ��

��

�� k2λ2D � 1��

��

�� |e2/n−1/3T | � 1��

��

��Ǳ

��

2.4 ��

��

��

��

��—� ��Ǳ (2π�)3��

��Ǳ 1/2�� 2��

�� 0�� pF (��)��

N = 2

∫d3pd3r

(2π�)3= V

8π

(2π�)3

∫ pF

0

p2dp =V p3F3π2�3

,

��

pF = (3π2)1/3n1/3e �, (2.14)

�� ne = N/V ��Ǳ��

εF = (3π2)2/3n2/3e (�2/2me) = 0.3646

( ne

1021cm−3

)2/3eV. (2.15)

�� εF��

��Ǳ ��

��Ǳ��

Te � εF. (2.16)

��

��

n−1/3e � h

mevTe

=h√meTe

, (2.17)

2.5 �� 17

��

Te � n2/3e

h2

me∼ εF.

��(2.16)��(2.17)��Υe��Ǳ

Υe =TeεF. (2.18)

� Υe � 1��Υe � 1��—� ��Ǳ��Ǳ��

��

�� —� ��φ��Ǳ�

ne = n0eF3/2[(μ− eφ)/T ]/F3/2(μ/T ),

�� μ�� F3/2(x)��—� ��

��

λ2De =

√T 2 + ε2F4πnee2

. (2.19)

2.5 ��

��

��

�� Ǳ��

Ǳ ��Ǳ��

��Ǳ��Ǳ ne ��

��Ǳ (4πne/3)−1/3��Ǳ

Uee ∼e2

(4πne/3)−1/3, (2.20)

�� (4πne/3)−1/3Ǳ�ǱWigner-Seitz��Ǳ

Ee ∼ Te, (2.21)

��Ǳ��

Γe =Uee

Ee= 2.32

( ne

1021cm−3

)1/3(TeeV

)−1

. (2.22)

18 ��

��—�� 1��

� ��Ǳ�� 1��

��Γ > 170��

��

��Ǳ�

��

��Ǳ��

��Ǳ�� Ǳ ��

��Ǳ�� 1 � Γ � 170��

�Ǳ��

��

N =4π

3neλ

3De =

1

3√3Γ

3/2e

.

�� Γe � 1�� N � 1�� 1��

�� g�

g =1

4πneλ3De

, (2.23)

� g = 0�� g � 1

�� gǱ��

�� me��

eǱ�� Te�� neǱ��

��Ǳ��Ǳ N

�� N �� nm�� nT �� ne

��

��

limN→∞

λDe = limN→∞

√Te

4πnee2= lim

N→∞

√(nT )

4π(ne)2= λDe,

limN→∞

ωpe = limN→∞

√4πnee2

me= lim

N→∞

√4π(nee)2

(neme)= ωpe.

�� g��

limN→∞

g =1

4πλ3De

limN→∞

1

Nn= 0.

2.5 �� 19

��

��

��Ǳ��me� e� Te� ne��

��

[mxe(1/ne)

yT ze e] = 1,

��Ǳ(�� 1.1)

[e] = [��]1/2[��]3/2[��]−1,

��

x+ z + 1/2 = 0, 3y + 2z + 3/2 = 0, − 2z − 1 = 0.

��

x = 0, y = −1/6, z = −1/2.

��

(en1/6e T−1/2

e )3 = (8π3/2/neλ3De)

−1.

��(2.22)�� Te��

εF��Æ��Ǳ εF��Ǳ

Γe =Uee

εF. (2.24)

�� Ǳ��

�� Ǳ��(2.15)�� n2/3 �� n1/3 ��

��Ǳ 0��

�� e2(4πne/3)1/3 � εF��

ne �32

27π3a3B= 2.58× 1023 cm−3, (2.25)

�� aB = �2/mee

2��(2.25)��Ǳ��

20 ��

2.6 ��

��Æ��

��(��)��

��

��Ǳ��

��Ǳ

∑i

νiAi = 0, (2.26)

�� Ai�� νi��

�� 2H2+O2 − 2H2O= 0�� νi��Ǳ νH2 = 2� νO2 = 1�

νH2O = −2��

�� ∑i

νiμi = 0. (2.27)

�� μi � i��Ǳ ��

��Ǳ

μi = T lnPi + χi(T ), (2.28)

�� Pi �� i�� χi �� Pi �� P

��Ǳ Pi = ciP�� ci = Ni/N �� i��N =∑

iNi��

��Ni� i��Ǳ��χiǱ

χi(T ) = ε0i − cpiT lnT − Tζi, (2.29)

�� cpi �� ζi ��Ǳ

cp = 5/2�

�� (2.28)��(2.27)��

∑i

νiμi = T∑l

νi lnPi +∑i

νiχi = 0.

��

KP (T ) = e−∑

νiχi/T , (2.30)

2.6 �� 21

�� Ǳ

∏i

P νii = KP (T ). (2.31)

��Ǳ Pi = ciP��

∏i

cνii = P−∑νiKP (T ) ≡ Kc(P, T ). (2.32)

��(2.31)(2.32)�Ǳ��KP (T )�Kc(P, T )�Ǳ��

��

��

A0 = A1 + e−, A1 = A2 + e−, · · · ,

��(2.32)� � νn−1 = 1� νn = −1� ν = −1��

cn−1

cnc= PK

(n)P (T ), (n = 1, 2, · · · ),

�� c0 �� c1� c2� · · ·� �� c ��

K(n)P (T )�� n��

��

c = c1 + 2c2 + 3c3 + · · · .

��

�� χi�� Ǳ

χn = εn − (5/2)T lnT − T ln[gn(ma/2π�

2)3/2],

χ = −(5/2)T lnT − T ln[2(me/2π�

2)3/2],

�� ma �� me �� gn ��

�(2.30)��K(n)P (T )��Ǳ

K(n)P (T ) =

gn−1

2gn

(2π

me

)3/2�3

T 5/2eIn/T ,

�� In = εn − εn−1�� n��

��

cacic

= PK(1)P (T ) = (na + ni + ne)

ga2gi

(2π

me

)3/2�3

T 3/2eI/T , (2.33)

22 ��

� 2.1: ��—��

�� ni Ǳ�� na Ǳ�� gi Ǳ��

�� gaǱ�� I Ǳ��Ǳ

α =ni

na + ni,

��

c = ci =α

1 + α, ca =

1− α

1 + α,

��(2.33)��

α =1√

1 + PK(1)P (T )

. (2.34)

2.7 ��

��

�� Ǳ��

� �� T � εF��

�� T � εF��

��

��

�� g (�� Γ )�� Ǳ�� g → 0��

Ǳ�� (2.1)��—��Ǳ��

��(��)��

2.8 �� 23

��

��

��

��Ǳ��

��

2.8 ��

1. ��Ǳ 10 keV��Ǳ 1014 cm−3��

��

��

2. ��Ǳ��

(1) ��Ǳ ne ∼ 1021 cm−3� Te ∼ 5 keV�

(2) ��Ǳ ne ∼ 1022 cm−3� Te ∼ 50 eV�

(3) ��Ǳ ne ∼ 1026 cm−3� Te ∼ 5 eV�

��

��

3. ��

(1) � 0◦ C� 760 mm��

(2) � �(20◦ C)��Ǳ 10−3 torr��

4. ��λD�ND�

(1) �� n ∼ 1 cm−3�T ∼ 0.01 eV�

(2) �� n ∼ 106 cm−3�T ∼ 0.1 eV�

(3) �� n ∼ 109 cm−3�T ∼ 2 eV�

(4) �� n ∼ 108 cm−3�T ∼ 0.1 eV.

24 ��

5. ��

∇2φ− 1

λ2Dφ = −4πQδ(r)

�� φ(r → ∞) = 0��

6. ��ǱTe� Ti��Ǳ

Z��Ǳ ne�� λD�

7. �� Z = 1��

8. ��Ǳ ne ��Ǳ T ��

��Ǳ 0�� x = 0��Ǳ φ0��

��ǱZ = 1�

9. �� x = ±d��Ǳ φ = 0��

��Ǳ q��Ǳ n��

(1)��Ǳ

φ = 2πqn(d2 − x2);

(2) �� d > λD ��

��

10. ��Ǳ R��Ǳ

ne��Ǳ ni� ne� ni��

(1) �� r = ∞��Ǳ��

(2) �� r = 0��!��

(3) � R = 100 km�ne = 106 cm−3(�� F ��)��

11. ��Ǳ a��Ǳ ne��

��Ǳ ni�� ne = ni��

�� ξ��

2.8 �� 25

12. ��

ne = ne0 exp [−(mgz − eφ)/T ]

ni = ni0 exp [−(mgz + eφ)/T ]

�� g Ǳ�� z ��Ǳ T�

� φ��d2φ

dz2= −4πe(ni − ne).

��

13. �� 1��

�� nλ3D Ǳ�� 1.

14. ��Ǳ�(1) n = 1010 cm−3�(2) n = 1013 cm−3�(3) n = 1019 cm−3

��!��

15. �!��Ǳ 105 K��Ǳ 1012

cm−3��

16. ��Ǳ 1014 cm−3 �� 1 cm�� 1%��

��

17. ��Ǳ 3× 104 K��Ǳ 1012 cm3��Æ

��

��

��

��

��

��Ǳ��

��

��

��

��

��

��Ǳ��

��

��

��

��

3.1 ��

��

��

��—��

3.1.1 ��

�� r�� r��

�� t��

3.1 �� 27

�� v ≡ r��

��Ǳ��

��

��

S =

∫ tb

ta

L(v)dt, (3.1)

� ta� tb� �� Ǳ�L(v)��

��Ǳ��

�� S ��

��(3.1)�� L(v)dt��

��Ǳ�� xμ = (ct, r)��(�� μ = 0, 1, 2, 3)��

ds2 = dxμdxμ = c2dt2 − dr2 (3.2)

��Ǳ ds2��

dτ =√

1− v2/c2dt = ds/c,

��

L(v) = α

√1− v2

c2, (3.3)

� α��

��(3.3)�� α��

�� v/c� 1��(3.3)�� (v/c)2��

L(v) ≈ α− αv2

2c2,

��Ǳ��

��Ǳ

L(v) = −α v2

2c2. (3.4)

��

L(v) =1

2mv2 (3.5)

28 ��

�m�� (3.4)�(3.5)��α�

α = −mc2.

��

L(v) = −mc2√

1− v2/c2, (3.6)

��Ǳ

S = −mc2∫ tb

ta

√1− v2/c2dt = −mc

∫ tb

ta

ds. (3.7)

��

��

3.1.2 ��

��

��Ǳ��

p =∂L

∂v. (3.8)

��(3.6)��(3.8)��Ǳ

p =mv√

1− v2/c2. (3.9)

��Ǳ

E = p · v − L. (3.10)

�(3.6)�(3.9)��(3.10)��Ǳ

E =mc2√

1− v2/c2. (3.11)

��(3.9)�(3.11)��

p = E v

c2. (3.12)

�� Ǳ��

�(3.9)�(3.11)��Ǳ

H = c√m2c2 + p2. (3.13)

3.1 �� 29

��p/mc� 1� ��mc2��Ǳ

H =p2

2m.

��

δS = −mc∫ tb

ta

δ(ds) = 0.

�� ds = (dxμdxμ)1/2�� δ(ds) = d(δs)��

δS = −mc∫ tb

ta

dxμdδxμ

ds= −mc

∫ tb

ta

uμdδxμ.

� uμ = dxμ/ds��

δS = −mcuμδxμ|tbta +mc

∫ tb

ta

δxμduμds

ds. (3.14)

��

�� δS = 0��

� δxμ(ta) = δxμ(tb) = 0�� ta� tb�� δxμ�� δS = 0��

��duμds

= 0. (3.15)

��

��(3.1)�� a�� b��

�� S ��Ǳ��

�� δxμ(ta) = 0�� duμ/ds = 0��(3.14)�Ǳ

δS = −mcuμδxμ, (3.16)

�� δxμ(tb)�Ǳ δxμ��(3.16)��

pμ = − ∂S

∂xμ= mcuμ, (3.17)

�� pμ��

pμ = (E/c,p), pμ = (E/c,−p), (3.18)

�� pμ � pμ ��Ǳ��(3.18)Ǳ��

30 ��

3.1.3 ��

��

��Ǳ��

��Ǳ

S = −mc2∫ tb

ta

√1− v2/c2dt+

∫ tb

ta

Lintdt, (3.19)

� Lint�� S ��

��Lintdt�Ǳ��

��

��

�� xμ = (ct, r)��Aμ = (φ,A)Ǳ��

��

��

−ecAμdx

μ = e

[1

cA · v− φ

]dt,

� e��Ǳ

Lint =e

cA · v − eφ. (3.20)

��Ǳ�

L(r,v, t) = −mc2√1− v2

c2+e

cA · v − eφ. (3.21)

��Ǳ

L =1

2mv2 +

e

cA · v − eφ (3.22)

�� Ǳ

P =∂L

∂v=

mv√1− v2/c2

+e

cA = p+

e

cA. (3.23)

� p��(3.9)��

��Ǳ

E = P · v − L =mc2√

1− v2/c2+ eφ. (3.24)

3.2 �� 31

��Ǳ��

H =

√m2c4 + c2

(P− e

cA)2

+ eφ. (3.25)

��Ǳ

H =1

2m

(P− e

cA)2

+ eφ. (3.26)

3.2 ��

��

δS = δ

∫ tb

ta

Ldt = 0.

��Ǳ

L = L(r,v, t).

��Ǳ

δS =

∫ tb

ta

L(r+ δr,v + δv, t)dt−∫ tb

ta

L(r,v, t)dt

=

∫ tb

ta

[∂L

∂rδr+

∂L

∂vδv

]dt.

�� δv ≡ δ(dr/dt) = d(δr)/dt��

��

δS =∂L

∂vδr

∣∣∣∣tb

ta

+

∫ tb

ta

[∂L

∂r− d

dt

(∂L

∂v

)]δrdt. (3.27)

�Ǳ� �� δr(ta) = δr(tb) = 0�� δr��

��—��d

dt

(∂L

∂v

)=∂L

∂r. (3.28)

��(3.21)��—��(3.28)��

d

dt

(p+

e

cA)= ∇

(ecA · v − eφ

). (3.29)

��

∇(a · b) = (a · ∇)b+ (b · ∇)a+ b× (∇× a) + a× (∇× b).

32 ��

�� r� v��

e

c∇ (A · v) = e

c(v · ∇)A+

e

cv × (∇×A)

��dA

dt=∂A

∂t+ (v · ∇)A,

��(3.29)��Ǳdp

dt= −e

c

∂A

∂t− e∇φ+

e

cv × (∇×A).

��E� B�

E = −1

c

∂A

∂t−∇φ, B = ∇×A, (3.30)

��

dp

dt= eE+

e

cv ×B, (3.31)

�� p��Ǳ

p = E v

c2=

mv√1− v2/c2

.

�� p = mv��(3.31)��Ǳ

mdv

dt= eE+

e

cv ×B. (3.32)

��Ǳ�� Ǳ�� S ��

�� S ��

�� S = S(t, r)��(3.27)�� r → r+ δr��Ǳ

δS =∂L

∂v· δr.

�� S �Ǳ�� r�� r�� r → r+ δr��

δS =∂S

∂r· δr

��

∂S

∂r=∂L

∂v= P. (3.33)

3.3 �� 33

��

dS

dt= L(r,v, t).

��(3.33)��Ǳ

dS

dt=∂S

∂t+ v · ∂S

∂r=∂S

∂t+P · v.

�� dS/dt��

∂S

∂t= −P · v + L = −H(P, r, t). (3.34)

�� (3.33)� ��

∂S

∂t+H

(∂S

∂r, r, t

)= 0. (3.35)

��—��Ǳ��—��Ǳ

(∇S − e

cA)2

− 1

c2

(∂S

∂t+ eφ

)2

+m2c2 = 0. (3.36)

��Æ��Ǳ(∂S

∂xμ+e

cAμ

)(∂S

∂xμ+e

cAμ

)= m2c2. (3.37)

3.3 ��

��

��

�� Ǳ��

L = L(qi, qi, t),

� {qi}Ǳ�� {qi}Ǳ��—��Ǳ

d

dt

(∂L

∂qi

)=∂L

∂qi. (3.38)

��—��(3.38)�� qi��

�� pi�� qi�Ǳ��

34 ��

��

�� z�� ∂L/∂z = 0�� z��

�

Pz =∂L

∂vz= pz +

e

cAz = const. (3.39)

�� L��Æ ϕ��

��Æ��

Pϕ =mr2ϕ√1− v2/c2

+e

crAϕ = const. (3.40)

�� ϕ− αz�� α��

��

∂L

∂z+ α

∂L

∂ϕ= 0,

��∂L

∂z+ α

∂L

∂ϕ= const. (3.41)

�� ∂L/∂t = 0��

dH

dt= −∂L

∂t= 0, (3.42)

��Ǳ��

3.4 ��

��

��

��Ǳ��

��

��

��Ǳ

L =1

2m1r

21 +

1

2m2r

22 − U(|r1 − r2|). (3.43)

�m1,2�� U(r)��

�� R�� r��

R =m1r1 +m2r2m1 +m2

, r= r1 − r2. (3.44)

3.4 �� 35

��R�� r� ��

r1 = R+m2

m1 +m2r, r2 = R− m1

m1 +m2r. (3.45)

�(3.45)��(3.43)��

L =1

2(m1 +m2)R

2 +1

2

m1m2

m1 +m2r2 − U(r) (3.46)

��R��

∂L

∂R= (m1 +m2) R = m1r1 +m2r2

�� R = 0��(3.46)��

m =m1m2

m1 +m2

, (3.47)

��(3.46)��Ǳ

L =1

2mr2 − U(r) (3.48)

�� Ǳ��Ǳm�� U(r)��

�� (r, ϕ)��(3.48)��Ǳ

L =1

2m

(r2 + r2ϕ2

)− U(r). (3.49)

�� ϕ��

M = mr2ϕ =�, (3.50)

��Æ��Ǳ��

E =1

2m

(r2 + r2ϕ2

)+ U(r) =

1

2mr2 +

M2

2mr2+ U(r) =�. (3.51)

��

��

��

E =1

2mv20 , (3.52)

� v0��Æ��Ǳ

M = mρv0, (3.53)

36 ��

� ρ��

��(3.51)��

r =dr

dt= ±

√v20 − 2U(r)/m− ρ2v20/r

2. (3.54)

�Æ��(3.50)��Æ��Ǳ

dϕ

dt=dϕ

dr

dr

dt=ρv0r2.

��(3.54)��

dϕ

dr=

mρv0/r2√

m2v20 − 2mU(r)−m2ρ2v20/r2. (3.55)

��(3.54)�(3.55)��U(r)Ǳ

U(r) = −Ze2

r, (3.56)

� Z ��(3.56)��(3.55)�� ρ⊥ = Ze2/mv20�Æ��Ǳ

dϕ

dr=

ρ/r2√1 + 2ρ⊥/r − (ρ/r)2

,

� r��

ϕ = arccosρ/r − ρ⊥/ρ√1 + ρ2⊥/ρ2

+ constant. (3.57)

Ǳ�� f � ε�

f =ρ2

ρ⊥, ε =

√1 +

ρ2

ρ2⊥, (3.58)

��(3.57)��Ǳ ��(3.57)��Ǳ��

f

r= 1 + ε cosϕ. (3.59)

��(3.54)��Ǳ

dr

dt= ±v0

√1 + 2

ρ⊥r

− ρ2

r2.

3.4 �� 37

Ze

θs

� 3.1: ��θs��Æ�� Æ�

�� r��

t =ρ⊥v0

∫(r/ρ⊥)d(r/ρ⊥)√(r/ρ⊥ + 1)2 − ε2

,

�� r/ρ⊥ + 1 = ε cosh ξ�� τ0 = ρ⊥/v0��

��

t = τ0 (ε sinh ξ − ξ) ,

��Ǳ ��

r = ρ⊥ (ε cosh ξ − 1) , t = τ0 (ε sinh ξ − ξ) , (3.60a)

x = ρ⊥ (ε− cosh ξ) , y = ρ⊥√ε2 − 1 sinh ξ. (3.60b)

��Ǳ��

ρ⊥ =Ze2

mv20, τ0 =

ρ⊥v0, ε =

√1 +

ρ2

ρ2⊥. (3.61)

�(3.1)�� Æ θs �� ρ��

�� (3.1)� �� x−� Æ� �Ǳ

tanϕin = limξ→−∞

y

x=

√ε2 − 1,

�� x−� Æ� �Ǳ

tanϕout = − limξ→+∞

y

x= −

√ε2 − 1.

38 ��

��Æ θs�� Æ�

θs = ϕout − ϕin.

�� θs�

tanθs2

=1

tanϕin=ρ⊥ρ. (3.62)

�� ρ = ρ⊥ ��Æ �� 90◦�� ρ⊥ �� 90◦��

�� ρ⊥ ��Æ�� θs ≈ ρ/ρ⊥ � 1��

��

��Ǳ

U(r) =e2

r,

��

r = ρ⊥ (ε cosh ξ + 1) , t = τ0 (ε sinh ξ + ξ) , (3.63a)

x = ρ⊥ (cosh ξ + ε) , y = ρ⊥√ε2 − 1 sinh ξ. (3.63b)

��Ǳ

ρ⊥ =e2

mv20, τ0 =

ρ⊥v0, ε =

√1 +

ρ2

ρ2⊥. (3.64)

��Æ� ��(3.62)��

tanθs2

=ρ⊥ρ. (3.65)

�� σ �� v0��

��Æ θs Ǳ�� dN ��

��Æ�� θs → θs + dθs��n��

��Ǳ

dσ =dN

n. (3.66)

��Æ θs �� ρ��Æ�� θs → θs + dθs��

�� ρ(θs) → ρ(θs) + dρ(θs)��Æ�

� θs → θs + dθs��Ǳ dN = n2πρ(θs)dρ(θs)��Ǳ

dσ = 2πρdρ. (3.67)

3.5 �� 39

��Æ��(3.65)��

dσ =

(Ze2

2mv20

)21

sin4(θs/2)2π sin θsdθs, (3.68)

��

dσ

dΩ=

(Ze2

2mv20

)21

sin4(θs/2). (3.69)

��

3.5 ��

��

��

��

3.5.1 ��

��

��Ǳ� �� Ǳ��

�� E��Ǳ x−��Ǳ z−�� Ǳ

E = E0 cos(kz − ωt)ex, (3.70a)

B = E0 cos(kz − ωt)ey, (3.70b)

�� ω� k��Ǳ��(3.70)��A�

��

A =cE0

ωsin(kz − ωt)ex. (3.71)

�� kz − ωt��Ǳ��(3.71)�Ǳ�� η ≡ kz − ωt�

��(3.71)��Ǳ

L = −mc2(1− v2/c2)1/2 +eE0

ωvx sin η. (3.72)

40 ��

�� x� y��

P⊥ = p⊥ +e

cA = const. (3.73)

� P⊥�� x − y�� p⊥�� x − y��Ǳ

��(3.72)�� kz − ωt = k(z − ct)�� z�� t��

��

H − cPz = const. (3.74)

�

H = [m2c4 + c2(P− eA/c)2]1/2 = (m2c4 + c2p2)1/2,

�� Pz = pz �� z−��

ωt→ t, kr → r, p/mc→ p, H/mc2 → γ, eA/mc2 → A. (3.75)

��(3.73)�(3.74)��Ǳ

p⊥ +A = const, (3.76)

γ − pz = const. (3.77)

��(3.76)��Ǳ �� x− z��

��

px = −a sin η, py = 0, (3.78)

�

a =eE0

mωc(3.79)

��(3.77)��Ǳ (1+a2/2)1/2��

��

γ − pz = (1 + a2/2)1/2. (3.80)

�� γ = (1 + p2x + p2z)1/2�� pz�

pz = − a2 cos 2η

4(1 + a2/2)1/2. (3.81)

��(3.80)��Ǳ��Ǳ �

3.5 �� 41

z

x

a2

8(1+a2/2)

a√1+a2/2

k

E

� 3.2: ��8��(��Ǳ0��)�

��

p = γdr

dt= γ

dr

dη

dη

dt= (pz − γ)

dr

dη.

(3.78)�(3.80)�(3.81)�� η��

x = − a√1 + a2/2

cos η, (3.82a)

z =a2

8(1 + a2/2)sin 2η. (3.82b)

��(3.82)��

��Ǳ“8” ��(3.2)�� a = 1��

��v×B/c��

��B��Ǳ ω��ǱǱω��

�� 2�� a → ∞��Ǳ√2��Ǳ 1/4��

��Ǳ 4√2�� a → 0�� Ǳ�� a/8 � 1��

��

42 ��

��

��Ǳ�

��

�� a��

�� (3.70)��Ǳ I = (c/8π)E20��

� a��

a =

[Iλ20

1.37× 1018W/cm2 · μm2

]1/2, (3.83)

�� I ��ǱW/cm2�� λ0��Ǳ μm��Ǳ 1μm�� 1018W/cm2��

3.5.2 ��

��Ǳ��Ǳ��

�� z��Ǳ

E = E0 (cos ηex + sin ηey) , (3.84a)

B = E0 (− sin ηex + cos ηey) , (3.84b)

��A�Ǳ

A =cE0

ω(sin ηex − cos ηey) . (3.85)

��Ǳ

L = −mc2(1− v2/c2)1/2 +eE0

ω(vx sin η − vy cos η). (3.86)

��(3.86)��

p⊥ +e

cA = const, (3.87a)

H − cPz = const. (3.87b)

��(3.75)��(3.87a)��Ǳ ��Ǳ

px = −a sin η, (3.88a)

py = a cos η. (3.88b)

3.6 �� 43

��Ǳ��Ǳ ��Ǳ

γ =√1 + a2. (3.89)

��

x = − a√1 + a2

cos η, y = − a√1 + a2

sin η, (3.90)

��Ǳ

r =a√

1 + a2. (3.91)

�� a → ∞�� r → λ0/2π��

��

�Ǳ

μ = eω

2π(πr2) =

ecλ04π

a2

1 + a2. (3.92)

�� μ → ecλ0/4π�

��

��

�Ǳ��

��

3.6 ��

3.6.1 ��

��E =const�� z−�� φ�� Ǳ φ = −Ez��Ǳ��A = 0�

��Ǳ

L = −mc2√

1− v2

c2+ eEz. (3.93)

�Ǳ�� x� y�� x� y��

px = p0x = const, py = py0 = const. (3.94)

��

� x−�� py0 = 0�

44 ��

�Ǳ��(3.93)��Ǳ��

H =√m2c4 + c2p2 − eEz = const. (3.95)

��Ǳ ��

z =1

eE

√m2c4 + c2p2x0 + c2p2z =

1

eE

√E20 + c2p2z,

� E20 = m2c4 + c2p2x0�� z−��—��

z−��pz =

∂L

∂z= eE,

�� pz = eEt��Ǳ pz(t = 0) = 0��

z =1

eE

√E20 + c2e2E2t2. (3.96)

��(3.12)�� x−��

x =c2p0x√

E20 + c2e2E2t2

.

�� t = 0�� x− y��

x =cp0xeE

arcsin h

(ceEt

E0

). (3.97)

��Ǳ

z =E0eE

cosh

(eE

cp0xx

)

3.6.2 ��

�� z−�� B = Bez��

��Ǳ

L = −mc2√1− v2/c2 +

e

cA · v. (3.98)

��

H =√m2c4 + c2p2 = E = const.

��

��ǱA = Bxey�Ǳ��ǱA = −Byex��ǱA = Bxey��

��Ǳ

L = −mc2√1− v2/c2 +

e

cBxvy .

3.6 �� 45

�� y� z��

Py = py +e

cBx = const, Pz = pz = pz0 = const.

��ǱA = −Byex��Ǳ

L = −mc2√1− v2/c2 − e

cByvx.

�� x� z��

Px = px −e

cBy = const, Pz = pz = pz0 = const.

��Ǳ��

��

px −e

cBy = const, py +

e

cBx = const, pz = pz0 = const. (3.99a)√

m2c4 + c2p2 = E = const. (3.99b)

��(3.99a)��

dx

dt=pxc

2

E = ωc(y − y0),dy

dt=pyc

2

E = −ωc(x− x0), (3.100)

�

ωc =ecB

E , (3.101)

�� (3.100)��Ǳ��(3.39)��Ǳ−eBy0/c� eBx0/c�� x0� y0�� (3.100)��Ǳ

d

dt[(x+ iy)− (x0 + iy0)] = −iωc [(x+ iy)− (x0 + iy0)] ,

��

x+ iy = x0 + iy0 + ae−iωct,

� a��Ǳ a = rc exp(−iα + iπ/2)�� rc� α�Ǳ

��

x+ iy = x0 + iy0 + irce−i(ωct+α),

��

x = x0 + rc sin(ωct+ α), y = y0 + rc cos(ωct+ α). (3.102)

46 ��

��Ǳ

vx = rcωc cos(ωct+ α), vy = −rcωc sin(ωct + α). (3.103)

�� v⊥�

v2⊥ = v2x + v2y = r2cω2c , (3.104)

��Ǳ

vx = v⊥ cos(ωct+ α), vy = −v⊥ sin(ωct + α).

�� (3.104)��

��

m =e

2cr× v, (3.105)

��

m = − v2⊥E2c2B

ez. (3.106)

��

��Ǳ μ��

μ =v2⊥E2c2B

. (3.107)

�� E ≈ mc2��

W⊥�

μ ≈ mv2⊥2B

=W⊥B. (3.108)

3.6.3 ��

��

��

��Ǳ��

�� z−�� y−��(3.32)��Ǳ(��)

mx =e

cyB, my = eE − e

cxB, mz = 0. (3.109)

3.6 �� 47

��

z = v�t. (3.110)

�� Ǳ

d

dt(x+ iy) + iωc(x+ iy) = i

e

mE,

� ωcǱ�� ωc = eB/mc��Ǳ

x+ iy = ae−iωct +cE

B,

� a = v⊥ exp(−iα)�� αǱ ��

x = v⊥ cosωct +cE

B, y = −rc sinωct. (3.111)

�� 2π/ωc��

〈x〉 ≡ ωc

2π

∫ 2π/ωc

0

xdt =cE

B, 〈y〉 = 0,

�

vE = 〈r〉 = cE×B

B2. (3.112)

�� vE��

�� Ǳ ��(3.111)�� Ǳ��

��

v = vE + vc, (3.113)

� vc = (v⊥ cosωct,−v⊥ sinωct)�� vc��

mdvc

dt=e

cvc ×B.

��

��Ǳ�� vE/c � 1��

��E

B� 1. (3.114)

��(3.111)�� t = 0�� Ǳ

x =v⊥ωc

sinωct+cE

Bt, y =

v⊥ωc

(cosωct− 1). (3.115)

48 ��

��(3.113)��Ǳ��Ǳ��

r(t) = ξ(t) +R(t),

� ξ(t) = (v⊥/ωc)(sinωct, cosωct)�� R(t) = (vEt,−v⊥/ωc)��

��

��

�� Ǳ

r = R(t) + ρ(t),

v = v�b+ vE + v⊥c.

��

R(t) = v�b+ vEt+ v⊥c,

ρ(t) =v⊥ωc

a,

��

��

� ��Ǳ��

��

��(3.112)��

��1��

��K �K ′� K ′��K x−�� V �

��Ǳ

Ex = E ′x, Ey =

E ′y + V B′

z/c√1− V 2/c2

, Ez =E ′

z − V B′y/c√

1− V 2/c2,

1��Ǳ

Fμν =∂Aν

∂xμ− ∂Aμ

∂xν,

�� Aμ Ǳ�� φ�� A��Ǳ Aμ = (φ,A)��

Ǳ

Fμν =

⎛⎜⎜⎜⎜⎝

0 Ex Ey Ez

−Ex 0 −Bz By

−Ey Bz 0 −Bx

−Ez −By Bx 0

⎞⎟⎟⎟⎟⎠ , Fμν =

⎛⎜⎜⎜⎜⎝

0 −Ex −Ey −Ez

Ex 0 −Bz By

Ey Bz 0 −Bx

Ez −By Bx 0

⎞⎟⎟⎟⎟⎠ .

��

3.6 �� 49

Bx = B′x, By =

B′y − V E ′

z/c√1− V 2/c2

, Bz =B′

z + V E ′y/c√

1− V 2/c2.

��K ��ǱB = Bez� E = Eey� E/B < 1��

�� V = (E/B)c��K ′��Ǳ

E′ = 0, B′ =√1− E2/B2B.

��K ′��K ′��

��K ′��

��K ��

��

��

x

z

y

x’

y’

z’

B

E

B’

V

� 3.3: ��

��K ��

��K ′��

��2��

��

2��

B2 − E2 = invariance, E ·B = invariance.

��

��Ǳ��

50 ��

3.6.4 ��

��

�� y−�� z−��

φ = −Ey, A = −Byex. (3.116)

��Ǳ

L = −mc2√

1− v2/c2 +e

cA · v − eφ

= −mc2√

1− v2/c2 − e

cByvx + eEy. (3.117)

��(3.117)�� x� z��

Px = px −e

cBy = const, (3.118a)

Pz = pz = const, (3.118b)

H =√m2c4 + c2[P− (e/c)A]2 − eEy = const. (3.118c)

Ǳ�� t = 0��

��(3.118)�Ǳ

px =eB

cy, c2p2x + c2p2y = 2mc2eEy + e2E2y2. (3.119)

Ǳ��

px,ymc

→ px,y,x, y

c/(ωc)→ x, y, ωct→ t. (3.120)

� ωc = eB/mc��(3.119)��Ǳ

px = y, p2x + p2y = 2E

By +

E2

B2y2,

�� Ǳ�� y��

px = y, py =√

2βy − (1− β2)y2. (3.121)

� β = E/B��

x =px√

1 + p2x + p2y, y =

py√1 + p2x + p2y

.

3.6 �� 51

��(3.121)� Ǳ�� y��

x =y

1 + βy, y =

√2βy − (1− β2)y2

1 + βy. (3.122)

��(3.121)�(3.122)�� x�� t�� Ǳ�� y��(3.121)�(3.122)�� β��

� β = 1��

px = y, py =√

2y, x =y

1 + y, y =

√2y

1 + y.

��

px = y, py =√2y, x =

√2y

3y, t =

√2y

3(y + 3).

�� y�� t��Ǳ��

��Ǳ

y = [3√2x/2]2/3.

� β < 1��

px = y, py =√

2βy − (1− β2)y2,

x =βπ

2(1− β2)3/2−

√2βy − (1− β2)y2

(1− β2)− β

(1− β2)3/2arcsin

β − (1− β2)y

β,

t =π

2(1− β2)3/2− β

√2βy − (1− β2)y2

(1− β2)− 1

(1− β2)3/2arcsin

β − (1− β2)y

β.

��Ǳ(x− βt)2

β2/(1− β2)+

[y − β/(1− β2)]2

β2/(1− β2)2= 1.

�� x−��Ǳ β/(1 − β2)1/2�

�� y−��Ǳ β/(1 − β2)�� Ǳ��

��Ǳ [βt, β/(1 − β2)]�� x−��Ǳ β��

��

� β > 1��

px = y, py =√

2βy + (β2 − 1)y2,

52 ��

x =− β

(β2 − 1)3/2ln

[β + (β2 − 1)y] + (β2 − 1)1/2[2βy + (β2 − 1)y2]1/2

β

+

√2βy + (β2 − 1)y2

(β2 − 1),

t =− 1

(β2 − 1)3/2ln

[β + (β2 − 1)y] + (β2 − 1)1/2[2βy + (β2 − 1)y2]1/2

β

+β[2βy + (β2 − 1)y2]1/2

β2 − 1.

��Ǳ[y + β/(β2 − 1)]2

β2/(β2 − 1)2− (x− βt)2

β2/(β2 − 1)= 1.

�� y � 1�� x ∼ y/(β2 −1)1/2�

��E/B � 1��

��E/B � 1��

3.6.5 ��

��

��

Ǳ��

�� F��

F�� Ǳ

md2r

dt2=e

cv×B+ F. (3.123)

��Ǳ�� vc�� vF�

v = vc + vF, (3.124)

�(3.124)��(3.123)��

mvc =e

cvc ×B+

e

cvF ×B+ F.

�� mvc = (e/c)vc ×B��

Ǳ �e

cvF ×B+ F = 0.

3.7 �� 53

��B�� vF�

vF =cF×B

eB2. (3.125)

��

��

��Ǳ

j =∑α

nαeαvFα =c(∑

α nαFα)×B

B2, (3.126)

� α� �� Fα = mαg��Ǳ

��Ǳ

jg = cρg ×B

B2, (3.127)

� ρ��

��

3.7 ��

��

��Ǳ��

��

��

��

dz

dt= f(z, t, θ). (3.128)

�� f �� θ��Ǳ 2π�

dθ

dt=

1

ε, (3.129)

� ε� 1��Æ��(3.128)��Ǳ��Ǳ��

�� t� θ�Ǳ��

��

z(t, θ) = Z(t) + εξ(Z, t, θ), (3.130)

54 ��

� Z(t)��ξ(Z, t, θ)� θ��

�� θ��Ǳ �

〈ξ(Z, t, θ)〉 ≡ 1

2π

∮ξ(Z, t, θ)dθ = 0.

�(3.130)��(3.128)��

dZ

dt+ ε

dZ

dt· ∇ξ + ε

∂ξ

∂t+∂ξ

∂θ= f(z, t, θ). (3.131)

��(3.131)�� ε��Ǳ�� ε

��

ξ = ξ0(Z, t, θ) + εξ1(Z, t, θ) + ε2ξ2(Z, t, θ) + · · · , (3.132a)dZ

dt= F0(Z, t) + εF1(Z, t) + ε2F2(Z, t) + · · · . (3.132b)

��(3.130)�(3.132a)��(3.131)��

dZ

dt+∂ξ0∂θ

+ ε

[dZ

dt· ∇ξ0 +

∂ξ0∂t

+∂ξ1∂θ

]+ · · · = f(z, t, θ). (3.133)

��(3.132b)��(3.133)��

F0(Z, t) +∂ξ0∂θ

+ ε

[F1(Z, t) + F0 · ∇ξ0 +

∂ξ0∂t

+∂ξ1∂θ

]= f(Z, t, θ) + ε(ξ0 · ∇)f . (3.134)

� ��(3.134)��

F0(Z, t) +∂ξ0∂θ

= f(Z, t, θ). (3.135)

�� θ��

F0(Z, t) = 〈f(Z, t, θ)〉, (3.136a)

ξ0(Z, t, θ) =

∫ θ

0

[f(Z, t, θ′)− 〈f〉]dθ′. (3.136b)

� ��(3.134)��

F1(Z, t) + F0 · ∇ξ0 +∂ξ0∂t

+∂ξ1∂θ

= (ξ0 · ∇)f (3.137)

�� θ��

F1(Z, t) = 〈(ξ0 · ∇)f〉. (3.138)

��(�� ε��)�

dZ

dt= 〈f(Z, t, θ)〉 + ε〈(ξ0 · ∇)f〉+O(ε2). (3.139)

3.8 �� 55

3.8 ��

��

��Ǳ��

��

��Ǳλ0a/2π��a� 1��

��

��

��

��Ǳ

E = E0(r) cos θ,

� E0(r)��

dθ

dt=ω

ε,

� ω ��ε��Æ��∇ × E =

(1/c)∂B/∂t��Ǳ

B = −ε cω(∇× E0) sin θ.

��Ǳ

dv

dt=

e

εm

[E0(r) cosωt− ε

1

ωv× (∇× E0(r)) sinωt

]. (3.140)

��Ǳ��Ǳ��

��Ǳ��Ǳ��

��

dr

dt= v, (3.141a)

dv

dt=

e

εm

[E0 cos θ − ε

1

ωv× (∇× E0) sin θ

]. (3.141b)

��Ǳ��

r = R+ εξ1(R,U, t, θ) + ε2ξ2(R,U, t, θ) + · · · ,

v = U+ u0(R,U, t, θ) + εu1(R,U, t, θ) + · · · ,

56 ��

��(3.141)��Ǳ

dR

dt+ ε

dR

dt· ∂ξ1∂R

+ εdU

dt· ∂ξ1∂U

+ ε∂ξ1∂t

+ εdθ

dt

∂ξ1∂θ

+O(ε2) = U+ u0 +O(ε1), (3.142)

dU

dt+dR

dt· ∂u0

∂R+dU

dt· ∂u0

∂U+∂u0

∂t+dθ

dt

∂u0

∂θ+O(ε1)

=e

εmE0(R) cos θ +

e

m

[ξ1 · ∇E0 cos θ −

1

ωu0 × (∇×E0) sin θ −

1

ωU× (∇×E0) sin θ

]+O(ε1),

(3.143)

��Ǳ��

dR

dt= U0(R,U, t) + εU1(R,U, t) +O(ε2),

dU

dt= A0(R,U, t) + εA1(R,U, t) +O(ε2),

��(3.142)�(3.143)��

U0 + ω∂ξ1∂θ

+O(ε1) = U + u0 +O(ε1), (3.144)

A0 +U0 ·∂u0

∂R+A0 ·

∂u0

∂U+∂u0

∂t+ω

ε

∂u0

∂θ+ ω

∂u1

∂θ+O(ε1)

=e

εmE0(R) cos θ +

e

m

[ξ1 · ∇E0 cos θ −

1

ωu0 × (∇×E0) sin θ −

1

ωU× (∇×E0) sin θ

]+O(ε1),

(3.145)

� ��(3.144)�ε0��

U0 + ω∂ξ1∂θ

= U + u0, (3.146)

�� θ��

U0 = U. (3.147)

��

ω∂ξ1∂θ

= u0 (3.148)

� ��(3.145)�ε−1��

ω∂u0

∂θ=

e

mE0 cos θ, (3.149)

3.8 �� 57

� ��(3.145)�ε0��

A0 +U0 ·∂u0

∂R+A0 ·

∂u0

∂U+∂u0

∂t+ ω

∂u1

∂θ

=e

m

[ξ1 · ∇E0 cos θ −

1

ωu0 × (∇× E0) sin θ −

1

ωU× (∇× E0) sin θ

](3.150)

��(3.150)�� θ��

A0 =e

m

[〈cos θξ1〉 · ∇E0 −

1

ω〈u0 sin θ〉 × ∇ ×B0

], (3.151)

��(3.149)� �� 〈u0〉 = 0��

u0 =e

mωE0 sin θ. (3.152)

��(3.148)�� 〈ξ1〉 = 0��

ξ1 = − e

mω2E0 cos θ. (3.153)

��(3.152)�(3.153)��(3.151)��

A0 = − e2

2m2ω2E0 · ∇E0 −

e2

2m2ω2E0 × (∇× E0)

= − e2

4m2ω2∇E2

0 .

��

Fp = − e2

4mω2∇E2

0 , (3.154)

��Ǳ

md2R

dt2= Fp.

�� I = cE20/8π��

Fp = − 2πe2

mcω2∇I = − e2λ2

2πmc3∇I. (3.155)

��

��

��

��Æ��

58 ��

3.9 ��

��

��

��Æ��Æ��

��Ǳ ��

��

�� —��Ǳ��

� 3.4: ��

3.9.1 ��

��Æ��

Æ��

��∣∣∣∣v⊥ωc

∇B

∣∣∣∣ � B,

∣∣∣∣ v�ωc

∇B

∣∣∣∣ � B,

� v⊥� v�� ωc��

��|(v⊥/ωc)∇B|

B,|(v‖/ωc)∇B|

B.

��Ǳ��Ǳ��

��Ǳ��Ǳ ε��

ε ∼ E

B.

�� ∣∣∣∣ ∂B

Bωc∂t

∣∣∣∣ ∼ ε2.

�� ε��

3.9 �� 59

��Ǳ

d2r

dt2=

e

mE(r, t) +

1

ε

e

mc

dr

dt×B(r, t). (3.156)

� ε��Æ��

�� ε = 1�Ǳ��

��Ǳ��(3.156)��Ǳ��

dr

dt= v, (3.157a)

dv

dt=

e

mE(r, t) +

1

εωcv × n0(r, t). (3.157b)

� ωc = eB/mc�n0 = B/B �� r��

�� Ǳ��

v = v�n0 + v⊥(n1 cos θ + n2 sin θ), (3.158)

� θ�� Æ�n1,2��

ni · nj =

{1, when i = j,

0, when i = j., n0 · (n1 × n2) = 1.

�� v��(3.158)��(3.157b)�� n0� n1 cos θ +

n2 sin θ�� −n1 sin θ + n2 cos θ�� n0� n1�� n2 ��

��

dr

dt= v�n0 + v⊥(n1 cos θ + n2 sin θ), (3.159a)

dv�dt

=e

mE · n0 + v⊥n0·(n1 cos θ + n2 sin θ), (3.159b)

dv⊥dt

= (e

mE− v�n0) · (n1 cos θ + n2 sin θ), (3.159c)

dθ

dt= −1

εωc + n1 · n2 +

1

v⊥

( emE− v�n0

)· (−n1 sin θ + n2 cos θ) (3.159d)

��Ǳ

ni =∂ni

∂t+ v�(n0 · ∇)ni + v⊥ {cos θ(n1 · ∇)ni + sin θ(n2 · ∇)ni} , (i = 0, 1, 2).

60 ��

��(3.159)��

dr

dt= v�n0 + v⊥(n1 cos θ + n2 sin θ), (3.160a)

dv�dt

= a0 + a1 cos θ + a2 sin θ + a3 cos 2θ + a4 sin 2θ, (3.160b)

dv⊥dt

= b0 + b1 cos θ + b2 sin θ + b3 cos 2θ + b4 sin 2θ, (3.160c)

dθ

dt= −1

εωc + c0 − c1 sin θ + c2 cos θ − c3 sin 2θ + c4 cos 2θ. (3.160d)

��(3.160)��Ǳ

a0 =e

mE · n0 +

v2⊥2∇ · n0,

a1 = v⊥

(∂n0

∂t+ v�T

)· n1, a2 = v⊥

(∂n0

∂t+ v�T

)· n2,

a3 =v2⊥2(T110 − T220), a4 =

v2⊥2(T120 + T210) (3.161)

b0 = −v�v⊥2

∇ · n0,

b1 =

[e

mE− v�

(∂n0

∂t+ v�T

)]· n1,

b2 =

[e

mE− v�

(∂n0

∂t+ v�T

)]· n2,

b3 = −v�v⊥2

(T110 − T220), b4 = −v�v⊥2

(T120 + T210) (3.162)

c0 = n1 ·∂n2

∂t+ v�T102 −

v�2(T210 − T120),

c1 =1

v⊥

[e

mE− v�

(∂n0

∂t+ v�T

)]· n1 − v⊥T122,

c2 =1

v⊥

[e

mE− v�

(∂n0

∂t+ v�T

)]· n2 − v⊥T211,

c3 = −v�2(T110 − T220), c4 = −v�

2(T210 + T120). (3.163)

�� ai� bi� ci � v⊥� v�� r�� t��Ǳ��

�

Tijk = ni · (nj · ∇)nk, i, j, k = 0, 1, 2, (3.164a)

T = (n0 · ∇)n0. (3.164b)

3.9 �� 61

��T��

�� ni�� Tijk ��

Tijk + Tjik = 0, T · n0 = 0,

T110 + T220 = ∇ · n0.

��(3.160)��

r = R+ ερ1(R, U�, U⊥,Θ, t) + ε2ρ2(R, U�, U⊥,Θ, t) + · · · , (3.165a)

v� = U� + εξ�1(R, U�, U⊥,Θ, t) + ε2ξ�2(R, U�, U⊥,Θ, t) + · · · , (3.165b)

v⊥ = U⊥ + εξ⊥1(R, U�, U⊥,Θ, t) + ε2ξ⊥2(R, U�, U⊥,Θ, t) + · · · , (3.165c)

θ = Θ+ εξθ1(R, U�, U⊥,Θ, t) + ε2ξθ2(R, U�, U⊥,Θ, t) + · · · , (3.165d)

dR

dt= U0(R, U�, U⊥, t) + εU1(R, U�, U⊥, t) + · · · , (3.166a)

dU�

dt= A�0(R, U�, U⊥, t) + εA�1(R, U�, U⊥, t) + · · · , (3.166b)

dU⊥dt

= A⊥0(R, U�, U⊥, t) + εA⊥1(R, U�, U⊥, t) + · · · , (3.166c)

dΘ

dt= −1

εΩ−1(R, U�, U⊥, t) + Ω0(R, U�, U⊥, t) + · · · . (3.166d)

�� ρ� ξ�� Θ��Ǳ �

��(3.165)�(3.166)��(3.160)��(��)��Ǳ

U0 = U�n0, (3.167)

A�0 = a0(R, U�, U⊥, t) =e

mE · n0 +

U2⊥2∇ · n0, (3.168)

A⊥0 = b0(R, U�, U⊥, t) = −U�U⊥2

∇ · n0, (3.169)

Ω−1 = ωc, (3.170)

62 ��

��Ǳ

ρ1 = −U⊥ωc

(n1 sin Θ− n2 cosΘ), (3.171a)

ξ�1 = − 1

ωc

(a1 sin Θ− a2 cosΘ +

a32sin 2Θ− a4

2cos 2Θ

), (3.171b)

ξ⊥1 = − 1

ωc

(b1 sinΘ− b2 cosΘ +

b32sin 2Θ− b4

2cos 2Θ

), (3.171c)

ξθ1 =

(U⊥ω2c

n1 · ∇ωc −c1ωc

)cosΘ +

(U⊥ω2c


)sinΘ (3.171d)

− 1

2ωc(c3 cos 2Θ + c4 sin 2Θ).

��Ǳ

U1 =U2⊥

2ωc

(n0 · ∇ × n0)n0 +1

ωc

(eE

m− U2

⊥2ωc

∇ωc − U2�T

)× n0, (3.172)

A�1 =U�

2ωc

n0 ·[2e

mT× E− U2

⊥ωc

T×∇ωc − U2⊥∇×T

], (3.173)

A⊥1 =U⊥2ωc

n0 ·{

1

ωc∇ωc ×

(eE

m− U2

�T

)−∇×

(eE

m− U2

�T

)+eE

m[n0 · (∇× n0)]

}

(3.174)

��Ǳ

U� +U2⊥

2ωc(n0 · ∇ × n0),

��Ǳ�� U� � ��

�Ǳ�

U⊥ − U⊥U�

2ωc(n0 · ∇ × n0),

Ǳ�� U⊥� ��

dR

dt= U�n0 +

1

ωc

(e

mE− U2

⊥2ωc

∇ωc − U2�T

)× n0, (3.175a)

dU�

dt= n0 ·

(e

mE−U2

⊥2ωc

∇ωc

)− U�

2ωcn0 ·

[(2eE

m− U2

⊥ωc

∇ωc

)×T

], (3.175b)

dU⊥dt

=U�U⊥2ωc

n0 · ∇ωc −U⊥2ωc

n0 ·[U2�

ωc

∇ωc ×T− ∇ωc

ωc

× eE

m+

e

m∇× E

]. (3.175c)

3.9 �� 63

3.9.2 ��

��

�Ǳ

d2R

dt2+d2ξ

dt2= e

[E(R) + (ξ · ∇)E+

1

2(ξξ : ∇∇)E

]+e

c

(vD + ξ

)×B. (3.176)

��

��Ǳ ��

��

0 = eE+e

2〈(ξξ : ∇∇)〉E+

e

cvD ×B. (3.177)

��(3.177)��

〈(ξξ : ∇∇)〉E =r2c2

(∂2

∂x2+

∂2

∂y2

)E =

1

2∇2

⊥E,

�∇⊥� ��

��Ǳ

R =

(1 +

r2c4∇2

⊥

)E×B

B2. (3.178)

��

�� (rc/LE)2��

��

(rc/LB)�

3.9.3 ��Ǳ

��

��

��

��Ǳdv

dt=

e

m

[E(t) +

1

cv ×B

].

�� v = v′ + vE = v′ + cE(t)×B/B2�Ǳ�� E� = 0��

dv′

dt=

e

mcv′ ×B− dvE

dt,

64 ��

��Ǳ��−mdvE/dt��

��Ǳ

vP = − mc

eB2

dvE

dt×B =

c2m

eB2

dE⊥dt

. (3.179)

��

��Ǳ��

jP =∑α

eαnαvPα =ρc2

B2

dE⊥dt

, (3.180)

� ρǱ��

�� Ǳ��

��Æ��Ǳ ω��

P = ij

ω=ρc2

B2E⊥,

��D��Ǳ

D = E+ 4πP = εE,

� ε��

��Ǳ�

ε = 1 +4πρc2

B2. (3.181)

3.10 ��

��Ǳ��

��

��

3.10.1 ��

��

��Ǳ��

3.10 �� 65

��

��

��Ǳ

mdv

dt= e

[E+

1

cv ×B(t)

].

��

�Ǳ��Ǳ

dW⊥dt

= eE · v = eE · v⊥.

�Ǳ��

dW⊥ = eE · dr⊥.

�� Ǳ

ΔW⊥ =

∫ 2π/ωc

0

eE · dr⊥ ≈ e

∮E · dr⊥

= e

∫(∇×E) · df = −e

c

∫∂B

∂t· df

=eπr2cc

dB

dt,

��

��Ǳ��Ǳ

dW⊥dt

≈ ΔW⊥2π/ωc

= μdB

dt. (3.182)

��Ǳ

W⊥ =mv2⊥2

= μB,

��

dW⊥dt

= Bdμ

dt+ μ

dB

dt. (3.183)

��(3.182)�� dμ

dt= 0.

��

66 ��

��

��(3.175b)�� Ǳ

md2s

dt2= −μ∂B

∂s. (3.184)

�� Ǳ v� = ds/dt��(3.184)�� ds/dt�

��

d

dt

(1

2mv2

�

)= m

d2s

dt2ds

dt= −μ∂B

∂s

ds

dt= −μdB

dt. (3.185)

�� Ǳ

d

dt

(1

2mv2⊥

)=

d

dt(μB) = μ

dB

dt+B

dμ

dt. (3.186)

�Ǳ��

d

dt

(1

2mv2

�+

1

2mv2⊥

)= 0 (3.187)

��(3.185)�(3.186)��(3.187)��

dμ

dt= 0.

�� Ǳ��

��

��Ǳ��

�� λ��Ǳ

�� λ�� T� λ��

��

Tdλ/dt� λ.

��Ǳ

H = H(p, q;λ). (3.188)

�� λǱ��

dEdt

=∂H

∂t=∂H

∂λ

dλ

dt. (3.189)

3.10 �� 67

��⟨dEdt

⟩≈ dλ

dt

⟨∂H

∂λ

⟩=dλ

dt

1

T

∫ T

0

∂H

∂λdt. (3.190)

�� λ�� dλ/dt��

� �� q = ∂H/∂p�� Ǳ

T =

∫ T

0

dt =

∮dq

∂H/∂p.

��(3.190)��Ǳ⟨dEdt

⟩=dλ

dt

∮dq(∂H/∂λ)/(∂H/∂p)∮

dq/(∂H/∂p). (3.191)

�� λ Ǳ��Ǳ��

��Ǳ�� p = p(q; E , λ)��E = H [q, p(q; E , λ);λ]�� λ��

∂H

∂λ+∂H

∂p

∂p

∂λ= 0,

�(∂H/∂λ)

(∂H/∂p)= −∂p

∂λ. (3.192)

�� E = H [q, p(q; E , λ);λ]�� E ��∂H

∂p

∂p

∂E = 1,

�1

(∂H/∂p)=∂p

∂E (3.193)

�(3.192)�(3.193)��(3.191)��⟨dEdt

⟩= −dλ

dt

∮dq(∂p/∂λ)∮dq(∂p/∂E) .

��∮ (∂p

∂E

⟨dEdt

⟩+∂p

∂λ

dλ

dt

)dq =

d

dt

∮p(q; E , λ)dq = 0.

��Ǳ��

dI

dt= 0,

�

I =

∮p(q; E , λ)dq/2π. (3.194)

68 ��

3.10.2 ��

��

�� Ǳ�

��

� 3.5: ��

�� E = W� + μB��

z0��

E − μB(z = 0) = 0,

�� z−�� −z0�Ǳ�� z = 0��ǱW0�

�� v�0� ��

�� Ǳ

W�0 +W⊥0 = W⊥(z = z0) (3.195)

� z0 ∈ {z|B(z) � Bmax}��

W�0 + μBmin = μB � μBmax.

��W⊥0

W�0 +W⊥0� μBmin

μBmax=Bmin

Bmax,

��v2⊥0

v2�0 + v2⊥0

� Bmin

Bmax

. (3.196)

��Æ�Æ θ0�

θ0 = arctanv⊥0

v�0, (3.197)

3.10 �� 69

v⊥

v�

��

��θc

� 3.6: �� θ < θc

�θ > π − θc ��

�� η�

η =Bmax

Bmin. (3.198)

��Æ�Æ��Æ θc��

θ0 > θc = arcsin

√1

η. (3.199)

��Æ�Æ��Æ��

�� Ǳ��

�� Ǳ

f =1

4π

∫ π−θc

θc

sin θdθ

∫ 2π

0

dϕ =

√1− 1

η. (3.200)

3.10.3 ��

��

��

J = J(E , s, t) =∮v�ds = 2

∫ s

s1

v�ds, (3.201)

70 ��

� ds�� s1,2��

��Ǳ��

τbτB

� 1,

�� τb Ǳ�� τB ��Æ��

�� J ��

�� E = mv2�/2 + μB��

v� = ±√

2

m(E − μB).

�� J ��Ǳ

J = J(E , s, t) =∫ s

s1

√2

m(E − μB)ds. (3.202)

�� t��

dJ

dt=

(∂J

∂t

)E,s

+

(∂J

∂E

)s,t

dEdt

+

(∂J

∂s

)E,t

ds

dt

= −∫ s

s1

(μ

m

∂B

∂t

)ds√

2(E − μB)/m

+

(v�v� +

μ

m

∂B

∂t+μ

mv�∂B

∂s

)∫ s

s1

ds√2(E − μB)/m

+ v�

√2

m(E − μB)− v�

∫ s

s1

μ

m

∂B

∂s

ds√2(E − μB)/m

,

�� v� = 0��

dJ

dt= −

∫ s

s1

(μ

m

∂B

∂t

)ds√

2(E − μB)/m+μ

m

∂B

∂t

∫ s

s1

ds√2(E − μB)/m

= −∫ s

s1

(μ

m

∂B

∂t

)ds

v�+μ

m

∂B

∂t

∫ s

s1

ds

v�

= −∫ τb

0

μ

m

∂B

∂tdt+

μ

m

∂B

∂t

∫ τb

0

dt

= − μ

m

[B(τb)− B(0)− τb

∂B

∂t

]

∼ O(τb/τB)2.

�� τb �� J �

��

3.10 �� 71

��

��

��

��

�� v� =√

2(W − μB)/m

��

�� L1�� L2�� B(L2) < B(L1)��W

��(��)�� μ�� v�(L2) > v�(L1)��Ǳ��

L2�� L1��

J(L2) =

∮L2

v�ds >

∮L1

v�ds = J(L1),

�� J �� J ��

��

�Ǳ J ��

�� J ��

v�L = v′�L′.

��

v′�=L

L′v� > v�.

��

3.10.4 ��

��

��

��Ǳ��

Φ =

∫B · df .

Ǳ�� Φ�� (α, β, s)�� α(r, t)�

β(r, t)�� Ǳ�� s��

�� α� β��

A = α∇β, B = ∇α×∇β.

72 ��

�� Φ� μ� J�� E �� t�� μ� J ��

Φ�� E � t��

Φ(E , t) =∮

A · dl =∮α∇β · dl =

∮αdβ.

��Ǳ

dΦ

dt=∂Φ

∂E 〈E〉� +∂Φ(E , t)∂t

,

� 〈E〉��

∂Φ

∂E =

∮∂α

∂E dβ =1

e

∮dβ

〈β〉=τpe

∂Φ

∂t=

∮∂α

∂tdβ = −1

e

∮ 〈E〉�〈β〉

dβ = −τpe〈〈E〉�〉p

� τp�� 〈〈E〉�〉p��

dΦ

dt=τpe

[〈W 〉� − 〈〈W 〉�〉p

].

�� dΦ/dt�Ǳ ��Ǳ ��

⟨dΦ

dt

⟩p

= 0.

3.11 ��

� ��

��

3.11.1 ��

��

��

2πRBt =4π

cI0,

� I0�� Ǳ

Bt(R) =2I0cR

. (3.203)

3.11 �� 73

z

R⊕�

⊗Bt

∇Bt

(a) z

R⊕ ⊕ ⊕

� � �E

⊗Bt

∇Bt

(b)

E × B

� 3.7: �� (a) �� (b) ��

��Ǳ

vD =c

eBtR(W⊥ + 2W�)n× b =

c

e(2I0/c)(W⊥ + 2W�) ez.

��(3.7)� ��

��

��Ǳ��

3.11.2 ��

��

� ��Bt��

��Æ��Bp�Æ��

��Ǳ��Æ��

Bp =2I(r)

cr=

4π

cr

∫ r

0

jϕ(r)rdr, (3.204)

� jϕ(r)�� Ǳ

B = Btet +Bpep =B0R0

Ret +Bpep.

��R = R0 + r cos θ�� θ��Æ� R0�

�� r/R0 � 1��

B = B0

(1− r

R0cos θ

)et +Bp(r)ep. (3.205)

74 ��

j

Bp Bt

� 3.8: � �� Bt��Æ��Bp��

��

��Æ��

�� !��

�� Ǳ��

�� ÆǱΔθ��

Δθ =2nπ

m,

� n�mǱ �� m��

�� !�� !��

��Δθ�� Ǳ�� Ǳ��

��ÆΔθ��

ι = limN→∞

1

N

N∑i=1

Δθi (3.206)

�� r/R0 � 1��

ι ≈∫ 2πR0

0

dθ

dzdz ≈ 2π

R0Bp

rBt

��

��Æ�� Ǳ��

��

3.11 �� 75

��

��Æ��

��

��

�� 1/R� ��

��

��

� �� Ǳ��

�� Ǳ��

��Ǳ��Ǳ��

��Ǳ��Ǳ��

��Ǳ��)�

� ��Ǳ

η =Bmax

Bmin

=R0 + r

R0 − r,

� R0�� r��

v2⊥0

v2�0 + v2⊥0

� Bmin

Bmax

��

v�0v⊥0

�√

2r

R0 − r.

��ÆǱ

θc = arcsin

√R0 − r

R0 + r.

��

Ǳ

ftrap =

√2r

R0 + r. (3.207)

��

��

��Ǳ

ω ≈ ιv�

2πR0

=Bp

Bt

v�r. (3.208)

76 ��

z

R{

d

��

��

� 3.9: ��Ǳ�� d�

�� z−��Ǳ

vD =c(2W� +W⊥)

eRBt

≈ 2cW�

eRBt

.

��Ǳ

dR

dt= ωz,

dz

dt= −ω(R− Rc) + vD,

� Rc��Ǳ

(R− Rc − vD/ω)2 + z2 = const. (3.209)

��Ǳ

d =vDω

≈ r

R

v�ωcp

, (3.210)

� ωcp = eBp/mc��(3.9)��

��

�� Ǳ

dB/ds�� Ǳ

B ≈ B0

(1− r

R0cos θ

),

3.11 �� 77

�� θ � 1��

dB

ds≈ rB0

R0

d(θ2/2)

ds. (3.211)

�� Ǳ

rdθ/ds = Bp/B (3.212)

��(3.211)�(3.212)��

θ = (Bp/rB)s. (3.213)

�(3.212)�(3.213)��(3.211)��

dB

ds=

(Bp

rB

)2rB0

R0s ≈

(Bp

rB0

)2rB0

R0s. (3.214)

�� Ǳ

d2s

dt2= − μ

m

dB

ds.

��(3.214)��

d2s

dt2= −

(v⊥Bp

rB0

)2 (r

2R0

)s.

� ωb = (v⊥Bp/rB0)(r/2R0)1/2�� ωb��

d2s

dt2= −ω2

bs.

��Ǳ

s = sb sinωbt.

��(3.213)� θ ∝ s��

θ = θb sinωbt. (3.215)

�� θb��Æ��

B(θb)

B(0)= 1 +

(v�0v⊥0

)2

(3.216)

78 ��

��

θb =v�0v⊥0

(2R0

r

)1/2

.

�� v⊥ � v��

��Ǳ

vD =cW⊥eRBt

,

��Ǳ

dr

dt= vD sin θ ≈ vDθ, (3.217)

��Æ��(3.215)��Ǳdθ

dt= ωbθb cosωbt = ωbθb

√1− (θ/θb)2. (3.218)

��(3.217)�(3.218)��dr

dθ=

vDωbθb

θ√1− (θ/θb)2

.

��

(r − r0)2 =

(θbvDωb

)2 (1− θ2/θ2b

). (3.219)

�� θbvD/ωb��

Δr =v�0ωcp

. (3.220)

� ωc = eBp/mc�

3.12 ��

3.12.1 ��

Ǳ��

��Ǳ

dr

dt= v�n0 + v⊥(n1 cos θ + n2 sin θ) (3.221a)

dv�dt

= a0 + a1 cos θ + a2 sin θ + a3 cos 2θ + a4 sin 2θ, (3.221b)

dv⊥dt

= b0 + b1 cos θ + b2 sin θ + b3 cos 2θ + b4 sin 2θ, (3.221c)

dθ

dt= −1

εωc + c0 − c1 sin θ + c2 cos θ − c3 sin 2θ + c4 cos 2θ. (3.221d)

3.12 �� 79

z

R

��

��

� 3.10: ��

�� ai� bi� ci��Ǳ

a0 =e

mE · n0 +

v2⊥2∇ · n0,

a1 = v⊥

(∂n0

∂t+ v�T

)· n1, a2 = v⊥

(∂n0

∂t+ v�T

)· n2,

a3 =v2⊥2(T110 − T220), a4 =

v2⊥2(T120 + T210)

b0 = −v�v⊥2

∇ · n0,

b1 =

[e

mE− v�

(∂n0

∂t+ v�T

)]· n1,

b2 =

[e

mE− v�

(∂n0

∂t+ v�T

)]· n2,

b3 = −v�v⊥2

(T110 − T220), b4 = −v�v⊥2

(T120 + T210)

c0 = n1 ·∂n2

∂t+ v�T102 −

v�2(T210 − T120),

c1 =1

v⊥

[e

mE− v�

(∂n0

∂t+ v�T

)]· n1 − v⊥T122,

c2 =1

v⊥

[e

mE− v�

(∂n0

∂t+ v�T

)]· n2 − v⊥T211,

c3 = −v�2(T110 − T220), c4 = −v�

2(T210 + T120).

�� r�� v⊥� v��Æ θ�

80 ��

��(3.221)��

r = R+ ερ1(R, U�, U⊥,Θ, t) + ε2ρ2(R, U�, U⊥,Θ, t) + · · · , (3.222a)

v� = U� + εξ�1(R, U�, U⊥,Θ, t) + ε2ξ�2(R, U�, U⊥,Θ, t) + · · · , (3.222b)

v⊥ = U⊥ + εξ⊥1(R, U�, U⊥,Θ, t) + ε2ξ⊥2(R, U�, U⊥,Θ, t) + · · · , (3.222c)

θ = Θ+ εξθ1(R, U�, U⊥,Θ, t) + ε2ξθ2(R, U�, U⊥,Θ, t) + · · · , (3.222d)

dR

dt= U0(R, U�, U⊥, t) + εU1(R, U�, U⊥, t) + · · · , (3.223a)

dU�

dt= A�0(R, U�, U⊥, t) + εA�1(R, U�, U⊥, t) + · · · , (3.223b)

dU⊥dt

= A⊥0(R, U�, U⊥, t) + εA⊥1(R, U�, U⊥, t) + · · · , (3.223c)

dΘ

dt= −1

εΩ−1(R, U�, U⊥, t) + Ω0(R, U�, U⊥, t) + · · · . (3.223d)

�� ρ� ξ��

〈ρ〉 = 〈ξ〉 = 0. (3.224)

��(3.222)�(3.223)��(3.221a)�(3.221b)�(3.221c)��ε1��

U0 + εU1 − Ω−1∂ρ1

∂Θ+ ε

(U0 ·

∂ρ1

∂R+ A�0

∂ρ1

∂U�

+ A⊥0∂ρ1

∂U⊥+ Ω0

∂ρ1

∂Θ− Ω−1

∂ρ2

∂Θ

)

= U�n0 + U⊥(n1 cosΘ + n2 sinΘ) + εU�(ρ1 · n0)

+ εU⊥(ρ1 · ∇)n1 cosΘ + εU⊥(ρ1 · ∇)n2 sinΘ

+ εξ�1n0 + εξ⊥1(n1 cosΘ + n2 sinΘ) + εU⊥ξθ1(−n1 sinΘ + n2 cosΘ),

3.12 �� 81

A�0 + εA�1 − Ω−1∂ξ�1∂Θ

+ ε

(U0 ·

∂ξ�1∂R

+ A�0∂ξ�1∂U�

+ A⊥0∂ξ�1∂U⊥

+ Ω0∂ξ�1∂Θ

− Ω−1∂ξ�2∂Θ

)

= a0 + a1 cosΘ + a2 sin Θ + a3 cos 2Θ + a4 sin 2Θ

+ ε

(ρ1 ·

∂a0∂R

+ ξ�1∂a0∂U�

+ ξ⊥1∂a0∂U⊥

)

+ ε

(ρ1 ·

∂a1∂R

+ εξ�1∂a1∂U�

+ εξ⊥1∂a1∂U⊥

)cosΘ− εa1ξθ1 sin Θ

+ ε

(ρ1 ·

∂a2∂R

+ εξ�1∂a2∂U�

+ εξ⊥1∂a2∂U⊥

)sinΘ + εa2ξθ1 cosΘ

+ ε

(ρ1 ·

∂a3∂R

+ εξ�1∂a3∂U�

+ εξ⊥1∂a3∂U⊥

)cos 2Θ− 2εa3ξθ1 sin 2Θ

+ ε

(ρ1 ·

∂a4∂R

+ εξ�1∂a4∂U�

+ εξ⊥1∂a4∂U⊥

)sin 2Θ + 2εa4ξθ1 cos 2Θ,

A⊥0 + εA⊥1 − Ω−1∂ξ⊥1

∂Θ+ ε

(U0 ·

∂ξ⊥1

∂R+ A�0

∂ξ⊥1

∂U�

+ A⊥0∂ξ⊥1

∂U⊥+ Ω0

∂ξ⊥1

∂Θ− Ω−1

∂ξ⊥2

∂Θ

)

= b0 + b1 cosΘ + b2 sinΘ + b3 cos 2Θ + b4 sin 2Θ

+ ε

(ρ1 ·

∂b0∂R

+ ξ�1∂b0∂U�

+ ξ⊥1∂b0∂U⊥

)

+ ε

(ρ1 ·

∂b1∂R

+ εξ�1∂b1∂U�

+ εξ⊥1∂b1∂U⊥

)cosΘ− εb1ξθ1 sin Θ

+ ε

(ρ1 ·

∂b2∂R

+ εξ�1∂b2∂U�

+ εξ⊥1∂b2∂U⊥

)sin Θ + εb2ξθ1 cosΘ

+ ε

(ρ1 ·

∂b3∂R

+ εξ�1∂b3∂U�

+ εξ⊥1∂b3∂U⊥

)cos 2Θ− 2εb3ξθ1 sin 2Θ

+ ε

(ρ1 ·

∂b4∂R

+ εξ�1∂b4∂U�

+ εξ⊥1∂b4∂U⊥

)sin 2Θ + 2εb4ξθ1 cos 2Θ,

��(3.222)�(3.223)��(3.221d)��ε0��

− 1

εΩ−1 + Ω0 − Ω−1

∂ξ⊥1

∂Θ

= −1

εωc − ρ · ∇ωc + c0 − c1 sinΘ + c2 cosΘ− c3 sin 2Θ + c4 cos 2Θ,

��

U0 = U�n0, A�0 = a0, A⊥0 = b0, Ω−1 = ωc, (3.225)

82 ��

−Ω−1∂ρ1

∂Θ= U⊥(n1 cosΘ + n2 sin Θ), (3.226a)

−Ω−1∂ξ�1∂Θ

= a1 cosΘ + a2 sin Θ + a3 cos 2Θ + a4 sin 2Θ, (3.226b)

−Ω−1∂ξ⊥1

∂Θ= b1 cosΘ + b2 sinΘ + b3 cos 2Θ + b4 sin 2Θ, (3.226c)

−Ω−1∂ξ⊥1

∂Θ= −rho · ∇ωc − c1 sinΘ + c2 cosΘ− c3 sin 2Θ + c4 cos 2Θ, (3.226d)

U1 = U⊥〈(ρ1 · ∇)(n1 cosΘ + n2 sinΘ)〉+ 〈ξ⊥1(n1 cosΘ + n2 sin Θ)〉

+ U⊥〈ξθ1(−n1 sinΘ + n2 cosΘ)〉, (3.227)

A�1 = ∇a1 · 〈ρ cosΘ〉+ ∂a1∂U�

〈ξ�1 cosΘ〉+ ∂a1∂U⊥

〈ξ⊥1 cosΘ〉 − a1〈ξθ1 sin Θ〉

+∇a2 · 〈ρ sinΘ〉+ ∂a2∂U�

〈ξ�1 sinΘ〉+ ∂a2∂U⊥

〈ξ⊥1 sinΘ〉+ a2〈ξθ1 cosΘ〉

+∇a3 · 〈ρ cos 2Θ〉+ ∂a3∂U�

〈ξ�1 cos 2Θ〉+ ∂a3∂U⊥

〈ξ⊥1 cos 2Θ〉 − 2a3〈ξθ1 sin 2Θ〉

+∇a4 · 〈ρ sin 2Θ〉+ ∂a4∂U�

〈ξ�1 sin 2Θ〉+ ∂a4∂U⊥

〈ξ⊥1 sin 2Θ〉+ 2a4〈ξθ1 cos 2Θ〉, (3.228)

A⊥1 = ∇b1 · 〈ρ cosΘ〉+ ∂b1∂U�

〈ξ�1 cosΘ〉+ ∂b1∂U⊥

〈ξ⊥1 cosΘ〉 − b1〈ξθ1 sinΘ〉

+∇b2 · 〈ρ sinΘ〉+ ∂b2∂U�

〈ξ�1 sinΘ〉+ ∂b2∂U⊥

〈ξ⊥1 sinΘ〉+ b2〈ξθ1 cosΘ〉

+∇b3 · 〈ρ cos 2Θ〉+ ∂b3∂U�

〈ξ�1 cos 2Θ〉+ ∂b3∂U⊥

〈ξ⊥1 cos 2Θ〉 − 2b3〈ξθ1 sin 2Θ〉

+∇b4 · 〈ρ sin 2Θ〉+ ∂b4∂U�

〈ξ�1 sin 2Θ〉+ ∂b4∂U⊥

〈ξ⊥1 sin 2Θ〉+ 2b4〈ξθ1 cos 2Θ〉. (3.229)

��(3.226)��(3.224)��

ρ1 = −U⊥ωc

(n1 sin Θ− n2 cosΘ), (3.230a)

ξ�1 = − 1

ωc

(a1 sin Θ− a2 cosΘ +

a32sin 2Θ− a4

2cos 2Θ

), (3.230b)

ξ⊥1 = − 1

ωc

(b1 sinΘ− b2 cosΘ +

b32sin 2Θ− b4

2cos 2Θ

), (3.230c)

ξθ1 =

(U⊥ω2c


)cosΘ +

(U⊥ω2c


)sinΘ (3.230d)

− 1

2ωc(c3 cos 2Θ + c4 sin 2Θ).

3.12 �� 83

��(3.230)��

〈ρ cosΘ〉 = U⊥2ωc

n2,

〈ρ sinΘ〉 = −U⊥2ωc

n1,

〈ρ cos 2Θ〉 = 〈ρ sin 2Θ〉 = 0,

〈ξ�1 cosΘ〉 = 1

2ωca2 =

U⊥U�

2ωcT · n2,

〈ξ�1 sinΘ〉 = − 1

2ωca1 = −U⊥U�

2ωc �

T · n1,

〈ξ�1 cos 2Θ〉 = 1

4ωc

a4 =U2⊥

8ωc

(T120 + T210),

〈ξ�1 sin 2Θ〉 = − 1

4ωca3 = −U2

⊥8ωc

(T110 − T220),

〈ξ⊥1 cosΘ〉 = 1

2ωc

b2 =1

2ωc

( emE− U2

�T)· n2,

〈ξ⊥1 sin Θ〉 = − 1

2ωcb1 = − 1

2ωc

( emE− U2

�T)· n1,

〈ξ⊥1 cos 2Θ〉 = 1

4ωcb4 = −U�U⊥

8ωc(T120 + T210),

〈ξ⊥1 sin 2Θ〉 = − 1

4ωc

b3 =U�U⊥8ωc

(T110 − T220),

〈ξθ1 sinΘ〉 = 1

2ωc

(U⊥ωc

n2 · ∇ωc − c2

)

=1

2ωc

[(U⊥ωc

∇ωc −eE

mU⊥+U2�

U⊥T

)· n2 + U⊥T211

]

〈ξθ1 cosΘ〉 = 1

2ωc

(U⊥ωc

n1 · ∇ωc − c1

)

=1

2ωc

[(U⊥ωc

∇ωc −eE

mU⊥+U2�

U⊥T

)· n1 + U⊥T122

],

〈ξθ1 sin 2Θ〉 = − c44ωc

=U�

8ωc(T210 + T120),

〈ξθ1 cos 2Θ〉 = − c34ωc

=U�

8ωc(T110 − T220),

�� ∂n0/∂t��

∇a1 · 〈ρ cosΘ〉+∇a2 · 〈ρ sinΘ〉 = U2⊥U�

2ωc

[n2 · ∇(T · n1)− n1 · ∇(T · n2)] .

84 ��

∇b1 · 〈ρ cosΘ〉+∇b2 · 〈ρ sinΘ〉

=U⊥2ωc

{n2 · ∇

[( emE− U2

�T)· n1

]− n1 · ∇

[( emE− U2

�T)· n2

]}

∂a1∂U�

〈ξ�1 cosΘ〉+ ∂a2∂U�

〈ξ�1 sinΘ〉 = U2⊥U�

2ωc[(T · n1)(T · n2)− (T · n2)(T · n1)]

= 0,

∂b1∂U�

〈ξ�1 cosΘ〉+ ∂b2∂U�

〈ξ�1 sinΘ〉 = U⊥U2�

ωc

[(T · n1)(T · n2)− (T · n2)(T · n1)]

= 0,

∂a1∂U⊥

〈ξ⊥1 cosΘ〉+ ∂a2∂U⊥

〈ξ⊥1 sinΘ〉

=U�

2ωc

{(T · n1)

[( emE− U2

�T)· n2

]− (T · n2)

[( emE− U2

�T)· n1

]}

=U�

2ωc

(T× e

mE)· n0,

∂b1∂U⊥

〈ξ⊥1 cosΘ〉+ ∂b2∂U⊥

〈ξ⊥1 sinΘ〉 = 0 + 0 = 0,

− a1〈ξθ1 sin Θ〉+ a2〈ξθ1 cosΘ〉

= −U⊥U�

2ωc

(T · n1)

[(U⊥ωc

∇ωc −eE

mU⊥+U2�

U⊥T

)· n2 + U⊥T211

]

+U⊥U�

2ωc(T · n2)

[(U⊥ωc

∇ωc −eE

mU⊥+U2�

U⊥T

)· n1 + U⊥T122

]

= −U⊥U�

2ωc

[T×

(U⊥ωc

∇ωc −eE

mU⊥+U2�

U⊥T

)]· n0 −

U2⊥U�

2ωc[(T · n1)T211 − (T · n2)T122]

= −U⊥U�

2ωc

[T×

(U⊥ωc

∇ωc −eE

mU⊥

)]· n0 −

U2⊥U�

2ωc[(T · n1)T211 − (T · n2)T122] ,

− b1〈ξθ1 sin Θ〉+ b2〈ξθ1 cosΘ〉

= − 1

2ωc

[( emE− U2

�T)· n1

] [(U⊥ωc

∇ωc −eE

mU⊥+U2�

U⊥T

)· n2 + U⊥T211

]

+1

2ωc

[( emE− U2

�T)· n2

] [(U⊥ωc

∇ωc −eE

mU⊥+U2�

U⊥T

)· n1 + U⊥T122

]

= − 1

2ωc

n0 ·{( e

mE− U2

�T)× U⊥ωc

∇ωc

}

− U⊥2ωc

{[( emE− U2

�T)· n1

]T211 −

[( emE− U2

�T)· n2

]T122

}

3.12 �� 85

∇a3 · 〈ρ cos 2Θ〉+∇a4 · 〈ρ sin 2Θ〉 = 0 + 0 = 0,

∇b3 · 〈ρ cos 2Θ〉+∇b4 · 〈ρ sin 2Θ〉 = 0 + 0 = 0,

∂a3∂U�

〈ξ�1 cos 2Θ〉+ ∂a4∂U�

〈ξ�1 sin 2Θ〉 = 0 + 0 = 0,

∂b3∂U�

〈ξ�1 cos 2Θ〉+ ∂b4∂U�

〈ξ�1 sin 2Θ〉

= −U⊥U2⊥

16ωc

(T110 − T220)(T120 + T210) +U⊥U2

⊥16ωc

(T120 + T210)(T110 − T220)

= 0,

∂a3∂U⊥

〈ξ⊥1 cos 2Θ〉+ ∂a4∂U⊥

〈ξ⊥1 sin 2Θ〉

= −U�U2⊥

8ωc

(T120 + T210)(T110 − T220) +U�U

2⊥

8ωc

(T120 + T210)(T110 − T220)

= 0,

∂b3∂U⊥

〈ξ⊥1 cos 2Θ〉+ ∂b4∂U⊥

〈ξ⊥1 sin 2Θ〉

=U2�U⊥

16ωc(T110 − T220)(T120 + T210)−

U2�U⊥

16ωc(T120 + T210)(T110 − T220)

= 0,

− 2a3〈ξθ1 sin 2Θ〉+ 2a4〈ξθ1 cos 2Θ〉

= −U2⊥U�

8ωc(T110 − T220)(T210 + T120) +

U2⊥U�

8ωc(T210 + T120)(T110 − T220)

= 0,

− 2b3〈ξθ1 sin 2Θ〉+ 2b4〈ξθ1 cos 2Θ〉

=U2�U⊥

8ωc(T110 − T220)(T210 + T120)−

U2�U⊥

8ωc(T120 + T210)(T110 − T220)

= 0

86 ��

��

U1 =U2⊥

2ωc

[(n2 · ∇)n1 − (n1 · ∇)n2 − T211n1 + T122n2]

+1

2ωc

{[( emE− U2

�T)· n2

]n1 −

[( emE− U2

�T)· n1

]n2

}

− 1

2ωc

[(U2⊥ωc

∇ωc −e

mE+ U2

�T

)· n2

]n1

+1

2ωc

[(U2⊥ωc

∇ωc −e

mE+ U2

�T

)· n1

]n2

=U2⊥

2ωc(n0 · ∇ × n0)n0 +

1

ωc

(eE

m− U2

⊥2ωc

∇ωc − U2�T

)× n0,

A�1 =U2⊥U�

2ωc

[n2 · ∇(T · n1)− n1 · ∇(T · n2)] +U�

2ωc

(T× e

mE)· n0

− U⊥U�

2ωc

[T×

(U⊥ωc

∇ωc −eE

mU⊥

)]· n0 −

U2⊥U�

2ωc[(T · n1)T211 − (T · n2)T122]

=U�

2ωc

n0 ·[T×

(2e

mE− U2

⊥ωc

∇ωc

)]

+U2⊥U�

2ωc[n2 · ∇(T · n1)− n1 · ∇(T · n2) + (T · n2)T122 − (T · n1)T211]

=U�

2ωc

n0 ·[T×

(2e

mE− U2

⊥ωc

∇ωc

)]− U2

⊥U�

2ωc

n0 · ∇ ×T

=U�

2ωcn0 ·

[2e

mT×E− U2

⊥ωc

T×∇ωc − U2⊥∇×T

],

A⊥1 =U⊥2ωc

{n2 · ∇

[( emE− U2

�T)· n1

]− n1 · ∇

[( emE− U2

�T)· n2

]}

− 1

2ωc

n0 ·{( e

mE− U2

�T)× U⊥ωc

∇ωc

}

− U⊥2ωc

{[( emE− U2

�T)· n1

]T211 −

[( emE− U2

�T)· n2

]T122

}

= −U⊥2ωc

n0 ·{( e

mE− U2

�T)× ∇ωc

ωc

}

− U⊥2ωc

n0 ·{∇×

( emE− U2

�T)− E(n0 · ∇ × n0)

}

= −U⊥2ωc

n0 ·[( emE− U2

�T)× ∇ωc

ωc+∇×

( emE− U2

�T)− eE

m(n0 · ∇ × n0)

].

��

U =

[U� +

U2⊥

2ωc(n0 · ∇ × n0)

]n0 +

1

ωc

(eE

m− U2

⊥2ωc

∇ωc − U2�T

)× n0, (3.231)

3.12 �� 87

dU�

dt=

(eE

m− U2

⊥2ωc

∇ωc

)· n0 +

U�

2ωc

[2e

mT× E− U2

⊥ωc

T×∇ωc − U2⊥∇×T

]· n0, (3.232)

dU⊥dt

=U⊥U�

2ωc

∇ωc · n0

+U⊥2ωc

[∇ωc

ωc

×( emE− U2

�T)−∇×

( emE− U2

�T)+ (n0 · ∇ × n0)

eE

m

]· n0

(3.233)

��

U� +U2⊥

2ωcn0 · ∇ × n0 → U�, (3.234)

U⊥ − U�U⊥2ωc

n0 · ∇ × n0 → U⊥, (3.235)

��

dR

dt= U�n0 +

1

ωc

(e

mE− U2

⊥2ωc

∇ωc − U2�T

)× n0, (3.236a)

dU�

dt= n0 ·

(e

mE−U2

⊥2ωc

∇ωc

)− U�

2ωcn0 ·

[(2eE

m− U2

⊥ωc

∇ωc

)×T

], (3.236b)

dU⊥dt

=U�U⊥2ωc

n0 · ∇ωc −U⊥2ωc

n0 ·[U2�

ωc∇ωc ×T− ∇ωc

ωc× eE

m+

e

m∇× E

]. (3.236c)

�� ∂ωc/∂t�

��(3.236)��

d

dt

(mU2

⊥ωc

)= −mU

2⊥

ω2c

(n0 · ∇ × eE

m+∂ωc

∂t

)= 0.

3.12.2 ��

��

[(n1 · ∇)n2 − (n2 · ∇)n1] = T112n1 − T221n2 − n0(n0 · ∇ × n0) (3.237)

�� n0 = n1 × n2��

∇× n0 = ∇× (n1 × n2) = n1∇ · n2 − n2∇ · n1 + (n2 · ∇)n1 − (n1 · ∇)n2,

�� n0��

n0 · ∇ × n0 = T021 − T012.

88 ��

��

[(n1 · ∇)n2 − (n2 · ∇)n1] = T112n1 + T212n2 + T012n0 − T121n1 − T221n2 − T021n0,

�� T212 = T121 = 0��

[(n1 · ∇)n2 − (n2 · ∇)n1] = T112n1 − T221n2−n0(n0 · ∇ × n0).

��

[(n1 · ∇)a2 − (n2 · ∇)a1] = a1T112−a2T221 − a0(n0 · ∇ × n0) + n0 · ∇ × a. (3.238)

�� ai = a · ni��

[(n1 · ∇)(a · n2)− (n2 · ∇)(a · n1)]

= a · (n1 · ∇)n2 + n2 · (n1 · ∇)a− a · (n2 · ∇)n1 − n1 · (n2 · ∇)a

= a · [(n1 · ∇)n2 − (n2 · ∇)n1] + [n2 · (n1 · ∇)a− n1 · (n2 · ∇)a] ,

��(3.237)��

n2 · (n1 · ∇)a− n1 · (n2 · ∇)a = (n1 × n2) · (∇× a),

��

[(n1 · ∇)a2 − (n2 · ∇)a1] = a· [T112n1 − T221n2−n0(n0 · ∇ × n0)] + n0 · ∇ × a

= a1T112 − a2T221 − a0(n0 · ∇ × n0) + n0 · ∇ × a.

��

n0 · (n0 · ∇)(∇× n0) = n0 · ∇ ×T− (n0 · ∇ × n0)∇ · n0. (3.239)

��∇(n0 · n0) = 2(n0 · ∇)n0 + 2n0 × (∇× n0) = 0��

∇×T = −∇× [n0 × (∇× n0)]

= −{n0∇ · (∇× n0)− (∇× n0)(∇ · n0) + [(∇× n0) · ∇]n0 − (n0 · ∇)(∇× n0)} ,

��∇ · ∇ × n0 = 0� n0 · [(∇× n0) · ∇]n0 = 0��

n0 · ∇ ×T = (n0 · ∇ × n0)(∇ · n0) + n0 · (n0 · ∇)(∇× n0).

3.12 �� 89

ds

b

db

B

�c

dθ

� 3.11: ��

3.12.3 ��

�� (n0 · ∇)�� n0 · n0 = 1��

0 = (n0 · ∇)(n0 · n0) = 2n0 · (n0 · ∇)n0 = 2n0 ·T,

��T = (n0 ·∇)n0� ��(3.11)��(n0 ·∇)n0 = ∂n0/∂s

�� Ǳ

ds = �cdθ.

�� c�� n0��Ǳ

|dn0| = dθ =ds

�c.

� dn0��Ǳ�� Ǳn��

T =∂n0

∂s= − n

�c. (3.240)

90 ��

��

1. ��

2. ��dH

dt= −∂L

∂t.

� L�H ��

3. !��

A = Breϕ/2, φ = −E0r2/2a.

��Æ��Æ

ΔϕT = π[1− (1− 4eE0/maω2c )

−1/2].

4. �� z−��H − cPz ��

��H �� Pz �� z−��

5. ��(3.63)��

6. �� z−�� t = 0��

��Ǳ �� Ǳ p0�

7. ��Ǳ

L(x, x) =x2

2− 1

2ω2x2.

!��—��

8. �� E = Eey ��B = Bez� E = B��

��

9. ��

B = Bez, E =Ea

arer.

�� z−��Æ�Ǳ

ϕ = −cEa

aB.

�� 3��

3.12 �� 91

10. !��By�∂By/∂xǱ��B = (0, By, B0)��

��

11. �� E = E0 cos(ωt + θ0)��E0 � θ0 Ǳ��

��E0Ǳ�� E0(0) = 0�!��

��

12. ��

(1) 0.5G��Ǳ 104 eV��

(2) B = 5× 10−5 G��Ǳ 3× 105 m/s��

(3) ��Ǳ 104 eV� He+��B = 500G.

13. �� n(r)��Ǳ

λ�� ∂n/∂r = −n/λ�

(1) ��E = −∇φ�� λ��

(2) �� vE = vth�� rc = 2λ�

14. ��Q�� Te = Ti = 0.2 eV��Ǳ 2× 103

G�� Ǳ

n = n0 exp[exp

(−r2/a2

)− 1

],

�� a = 10−2 m�n0 = 1017 m−3��n = n0 exp(eφ/T )�

(1) �� (vE)max�

(2) � (vE)max�� vg � �

(3) � BǱ"��K+(A = 39)� �� a�

15. ��Ǳ 0.3 G� �� 1/r3��

�� 1 eV�� 3 × 104 eV�� r = 5R (RǱ��)��Ǳ n = 105 m−3

�

(1) ��

92 ��

(2) ��

(3) ��

(4) ��

16. �� Ǳ I�� t = 0�Ǳ�

�� φ��Ǳ�� I��

(1) �� I� B�� vE�� v∇B�

�� vR��

(2) � I = 500 A� φ = 560 V�� Ǳ 1 mm�!�� 1 cm�� 10 cm�� φǱ �

17. ��Rm = 5��

ǱW = 103eV� ��v⊥ = v��Vm = 104m/s��L = 1010m�

(1) ��μ��

(2) �� #��

18. �J��

19. �� μ�� T⊥ �� B ��

��Ǳ�� ! ��

��

20. ��B��

n(r) = n0 exp(−r2/r20), ni = ne = n0 exp(eφ/T ).

(1) �� vE �� v∇p��

(2) ��

21. �� B = 4 × 103 G� n0 = 1016 m−3� T = 0.25 eV�� B�#�#�

22. ��ωc�ωp��

3.12 �� 93

23. !��∇ · E = 4πρ�∇ ·B = 0�� Ǳ��

��ǱǱ��

24. !��

25. ��B = Bz(x)ez �� E = Eyey ��

��

26. �Ǳ 10 keV��Ǳ 5× 1020 m−3��B = 5 T��

27. �� 5V/m��

(1) ��

(2) ��

28. �� T = 10 keV�� n = 1020 m−3�� B = 4 T�� Rc = 5 m�

(1)��

(2)"��

29. �� 1 �� 8 T�Ǳ 1 T�! ��

30. ��Ǳ 5��Ǳ n = 1020 m−3�� TD = TT = 50

keV�Te = 10 keV��

31. ��

jφ = j0φ(1− r/a), (0 � r � a),

��Aφ�Æ��Bθ�

32. �� ǱR0 = 5 m� a = 1.5 m��

94 ��

33. ��E��B�� W⊥��

��

34. ��

35. ��

�Ǳ��(inverse Faraday effect)��Ǳn��(� ��ǱM = nμ��ǱB= 4πM�)

36. �� E = E0[cos(kz − ωt)ex + sin(kz − ωt)ey]��

�� B = B0ez ��Ǳm�� e��

37. !� ��

38. �� Ǳ 0.351 μm��1015W/cm2 �� 10.6μm��

39. �� Ǳ

μ =mU2

⊥2B

.

��(3.175)��dμ

dt= 0.

��

��Ǳ��

��

��Ǳ��

��Ǳ��

��Ǳ��

��

��

��

��

��

��−Z��

��

4.1 ��

��

∇ ·B = 0, ∇×E = −1

c

∂B

∂t, (4.1a)

∇ · E = 4πρ, ∇×B =1

c

∂E

∂t+

4π

cj, (4.1b)

�� E� B�� j� ρ��

�� (φ,A)��

E = −∇φ− 1

c

∂A

∂t, (4.2a)

B = ∇×A, (4.2b)

96 ��

��(4.2)��(4.1)��(4.1a)�Ǳ��(4.1b)��Ǳ

∇2φ+1

c

∂

∂t(∇ ·A) = −4πρ, (4.3a)

∇2A− 1

c2∂2A

∂t2= −4π

cj+∇(∇·A) +

1

c

∂

∂t(∇φ). (4.3b)

��1

c

∂φ

∂t+∇ ·A = 0, (4.4)

��(4.3)��

∇2φ− 1

c2∂2

∂t2φ = −4πρ(r, t), (4.5a)

∇2A− 1

c2∂2A

∂t2= −4π

cj(r, t), (4.5b)

��(4.5)��Ǳ

φ(r, t) =

∫ρ(r′, t− |r− r′|/c)

|r− r′| d3r′ + φ0, (4.6a)

A(r, t) =1

c

∫j(r′, t− |r− r′|/c)

|r− r′| d3r′ +A0, (4.6b)

�� φ0 � A0 ��

�� j� ρ�� φ0 = 0� A0 = 0��

��

��

��Æ� r′ ��

��Æ�� r��

r � r′.

��

|r− r′| ≈ r − r′ · n,

�� n = r/r��(4.6)��Ǳ

φ(r, t) ≈ 1

r

∫ρ(r′, t− r/c+ r′ · n/c)d3r′, (4.7a)

A(r, t) ≈ 1

cr

∫j(r′, t− r/c+ r′ · n/c)d3r′. (4.7b)

4.1 �� 97

z

y

x

r

r′

r-r′

o

� 4.1: �� r�� r′��

��

��(4.7b) ��(�� r′)��A�� t− r/c

��ǱA = A(r, t− r/c)�Ǳ��

�(4.7b)��

A(r, t) ≈ 1

cr

∫j(r′, t′)δ[t′ − (t− r/c+ r′ · n/c)]d3r′dt′. (4.8)

��Ǳ

j(r, t) = ev(t)δ[r− r0(t)], (4.9)

�� r0(t) �� v(t) ��(4.9)��(4.8)� ��A(r, t)��Ǳ��

A(r, t) =e

cr

∫v(t′)δ[r′ − r0(t

′)]δ[t′ − (t− r/c+ r′ · n/c)]d3r′dt′

=e

cr

∫v(t′)δ[t′ − (t− r/c+ r0(t

′) · n/c)]dt′. (4.10)

� δ−��δ[f(x)] =

δ(x− x0)

|f ′(x0)|, (4.11)

�� f(x)�� x0 �� f(x)��

��Ǳ

A(r, t) =e

cr

v(t′)1− n · v(t′)/c. (4.12)

98 ��

�� t′��

t′ − 1

cn · r0(t′) = t− r/c. (4.13)

��Ǳ��Ǳ n��

��

Ǳ

E = B× n.

��Ǳ� ��

A��(4.7b)��(4.7b)��(4.2b)��

B(r, t) = ∇×A(r, t− r/c) ≈ 1

c

∂A

∂t×n (4.14)

��(4.14)�� 1/r2��Ǳ

B =1

c

∂A

∂t×n, E = B× n. (4.15)

��Ǳ

S =c

4πB2n. (4.16)

�� I�� r2dΩ��

��

dI =c

4πB2(t)r2dΩ. (4.17)

�� dΩ�� Æ��(4.17)��r2dΩ��

��

��

��Æ dΩ��Ǳ

dE =

∫ ∞

−∞dIdt =

c

4π

∫ ∞

−∞B2(t)r2dtdΩ. (4.18)

Ǳ��B��

B =

∫ ∞

−∞Bωe

−iωtdω

2π, (4.19)

�� Bω ��

Bω =

∫ ∞

−∞B(t)eiωtdt. (4.20)

4.1 �� 99

�� B−ω = B∗ω��

��

dEndΩ

=c

4π

∫ ∞

−∞|Bω|2r2

dω

2π=

c

(2π)2

∫ ∞

0

|Bω|2r2dω, (4.21)

�d2EnωdωdΩ

=c

(2π)2|Bω|2r2, (4.22)

��(4.22)��

��

(4.14)��Bω �� (∂A/∂t)ω ��

Bω =1

c

(∂A

∂t

)ω

× n. (4.23)

�� (∂A/∂t)ωǱ

(∂A

∂t

)ω

=

∫ ∞

−∞

∂A(r, t− r/c)

∂teiωtdt,

�� τ = t− r/c�� Ǳ

(∂A

∂t

)ω

= eiωr/c∫ ∞

−∞

∂A(r, τ)

∂τeiωτdτ. (4.24)

�(4.12)��(4.24)� �� τ ��Ǳ�� t′��

(∂A

∂t

)ω

=e

creiωr/c

∫ ∞

−∞

d

dt′

[v(t′)

1− n · v(t′)/c

]eiω[t

′−n·r0(t′)/c]dt′. (4.25)

�� Ǳ

Bω =e

c2reiωr/c

∫ ∞

−∞

d

dt

[v(t)× n

1− n · v(t)/c

]eiωt−ik·r0(t)dt. (4.26)

�� k = ωn/c ��(4.26)��dv/dt = 0��(4.26) ��Ǳ 0��

��(4.26)� ��

Bω = ieω

c2reiωr/c

∫ ∞

−∞[n× v(t)]eiωt−ik·r0(t)dt. (4.27)

100 ��

�(4.26)�(4.27)��(4.22)��

d2EnωdωdΩ

=e2

4π2c3

∣∣∣∣∫ ∞

−∞

d

dt

[v(t)× n

1− n · v(t)/c

]eiωt−ik·r0(t)dt

∣∣∣∣2

(4.28)

=e2ω2

4π2c3

∣∣∣∣∫ ∞

−∞[n× v(t)]eiωt−ik·r0(t)dt

∣∣∣∣2

. (4.29)

��(4.28)��

d

dt

[v(t)× n

1− n · v(t)/c

]≈ v(t)× n,

��

k · r0(t) ∼ω

cvt� ωt,

��(4.28)��Ǳ

d2EnωdωdΩ

=e2

4π2c3

∣∣∣∣∫ ∞

−∞v(t)× neiωtdt

∣∣∣∣2

(4.30)

�� Ǳ

d = er, (4.31)

��(4.30)��Ǳ

d2EnωdωdΩ

=1

4π2c3

∣∣∣∣∫ ∞

−∞d× neiωtdt

∣∣∣∣2

. (4.32)

�Ǳ��

��

d2EnωdωdΩ

=1

4π2c3

∣∣∣dω × n∣∣∣2 . (4.33)

��Æ��dEωdω

=2

3πc3

∣∣∣dω

∣∣∣2 (4.34)

��Ǳ 2π/ω0��Ǳ��

��Ǳ��

B(t) =∞∑

n=−∞Bne

−inω0t, (4.35)

4.2 �� 101

��Bn�� Ǳ��B��

� Bn = B∗−n�� Ǳ

Bω =∞∑

n=−∞

∫ ∞

−∞Bne

i(ω−nω0)tdt = 2π∞∑

n=−∞δ(ω − nω0)Bn. (4.36)

�(4.35)��(4.18)��

dEndΩ

=c

4πr2∫ ∞

−∞

∞∑n,m=−∞

Bn ·Bme−i(n+m)ω0tdt

=c

4πr2∫ ∞

−∞dt

∞∑n=−∞

|Bn|2.

��,dPn

dΩ=

c

2πr2

∞∑n=1

|Bn|2. (4.37)

�� n� ��Ǳ ω = nω0��Ǳ

d2Pnω

dωdΩ=

c

2πr2

∞∑n=1

δ(ω − nω0) |Bn|2 , (4.38)

�� ω� ��

��

��Ǳ

Bn = ie

c2r

nω20

2πeinω0r/c

∫ 2π/ω0

0

n× v(t)einω0[t−n·r0(t)/c]dt

= ie

c2r

nω20

2πeinω0r/c

∮einω0[t−n·r0/c]n× dr0, (4.39)

��ǱdP

dΩ=

∞∑n=1

e2n2ω40

(2π)3c3

∣∣∣∣∮einω0[t−n·r0/c]n× dr0

∣∣∣∣2

. (4.40)

4.2 ��

��

��(4.38)��

��

102 ��

Ze

−e γγ

r

� 4.2: ��

4.2.1 ��—��

�� Ǳ

d = −ere + Zeri. (4.41)

�� re � ri ��

�� R��

r��

d = (Z − 1)R−(

1

me

+Z

mi

)emr.

��m = memi/(me +mi)��

��

d = −(

1

me

+Z

mi

)emr. (4.42)

��mi � me��m ≈ me��

��Ǳ

d ≈ −er (4.43)

��Ǳ��

(d)ω =

∫de

iωtdt. (4.44)

��Æ��

(x)ω =

∫ ∞

−∞xeiωtdt, (y)ω =

∫ ∞

−∞yeiωtdt.

�� (3.60)��Ǳ

x = − 1

m

∂U

∂x=Ze2

m

x

r3=ρ⊥τ 20

ε− cosh ξ

(ε cosh ξ − 1)3, (4.45a)

y = − 1

m

∂U

∂y=Ze2

m

y

r3=

√ε2 − 1ρ⊥τ 20

sinh ξ

(ε cosh ξ − 1)3. (4.45b)

4.2 �� 103

�(4.45)��(4.44)��

(x)ω =ρ⊥τ0

∫ ∞

−∞

ε− cosh ξ

(ε cosh ξ − 1)2eiν(ε sinh ξ−ξ)dξ, (4.46)

(y)ω =

√ε2 − 1ρ⊥τ0

∫ ∞

−∞

sinh ξ

(ε cosh ξ − 1)2eiν(ε sinh ξ−ξ)dξ. (4.47)

�� ν ≡ ωτ0��

d

dξ

1

ε cosh ξ − 1= − ε sinh ξ

(ε cosh ξ − 1)2,

(y)ω ��(4.47)��

(y)ω = i

√ε2 − 1

ερ⊥ω

∫ ∞

−∞eiν(ε sinh ξ−ξ)dξ.

��

Kiν(z) =1

2e−νπ/2

∫ ∞

−∞eiz sinh ξ−iνξdξ, (4.48)

K ′iν(z) =

1

2e−νπ/2

∫ ∞

−∞

z − ν cosh ξ

(z cosh ξ − ν)2eiz sinh ξ−iνξdξ, (4.49)

��Kν(z)�� (x)ω � (y)ω ��ǱKiν ��

(x)ω = 2ρ⊥ωeνπ/2K ′iν(νε), (4.50)

(y)ω = i2

√ε2 − 1

ερ⊥ωeνπ/2Kiν(νε). (4.51)

��(4.50)�(4.51)��∣∣∣dω

∣∣∣2 = e2m2[|(x)ω|2 + |(y)ω|2

]= 4e2ρ2⊥ω

2eνπ[K ′2

iν(νε) +ε2 − 1

ε2K2

iν(νε)

].

��(4.34)�� ρ�� v0��

��

dEωdω

=16

3(2π)c3e2v2eνπν2

[K ′

iν(νε)2 +

ε2 − 1

ε2Kiν(νε)

2

]. (4.52)

4.2.2 ��

��Ǳ��

��(4.52)��Ǳ��

104 ��

��

��

��(4.52)��Ǳ��Ǳ��

��Ǳ v�� ρ → ρ+ dρ��Ǳ 2πnivρdρ�� ni ��

�� ρ�� 2πnivρdρ��

�� v → v + d3v��Ǳ

fe(v) =ne

(2π)3/2v3eexp

(−v2/2v2e

), (4.53)

�� ve = (Te/me)1/2��ne��

�� ω → ω + dω��

��Ǳ

dPω

dω=

∫d3v

∫ ∞

0

2πρdρdEωdω

fe(v)v

=16ninee

2

3c31

(2π)3/2v3e

∫v3e−v2/2v2e eνπd3v

×∫ ∞

0

ν2[K ′

iν(νε)2 +

ε2 − 1

ε2Kiν(νε)

2

]ρdρ. (4.54)

��(4.54) �� ρ �� ε � ρ �� ε =√1 + ρ2/ρ2⊥��

∫ ∞

0

ν2[K ′

iν(νε)2 +

ε2 − 1

ε2Kiν(νε)

2

]ρdρ

= ρ2⊥

∫ ∞

1

[K ′

iν(νε)2 +

ε2 − 1

ε2Kiν(νε)

2

](νε)d(νε)

= ρ2⊥

∫ ∞

ν

[K ′

iν(z)2 +

(1− ν2

z2

)Kiν(z)

2

]zdz.

Ǳ��Kiν ��

d2Kiν

dz2+

1

z

dKiν

dz−(1− ν2

z2

)Kiν = 0,

��

z

[K ′2

iν +

(1− ν2

z2

)K2

iν

]=

d

dz(zKiνK

′iν) .

4.2 �� 105

�� ∫ ∞

ν

[K ′

iν(z)2 +

(1− ν2

z2

)Kiν(z)

2

]zdz = −νKiν(ν)K

′iν(ν),

��Ǳ

dPω

dω= −16nine

3c32

(2π)1/2v3e

∫v5e−v2/2v2eeνπνρ2⊥Kiν(ν)K

′iν(ν)dv. (4.55)

��(4.55) �� v ��

�� ν � 1��

Kiν(ν)��

eνπ/2Kiν(ν) =

∫ ∞

0

cos [ν (sinh ξ − ξ)] dξ,

�� sinh ξ − ξ � 1�� sinh ξ � ξ��

��

eνπ/2Kiν(ν) ≈∫ ∞

0

cos(ν sinh ξ)dξ = K0(ν) ≈ ln2

γν,

�� ln γ = 0.5772 . . .��

eνπ/2K ′iν(ν) ≈ −1

ν.

��(4.55)��

dPω

dω=

32neni

3c3Z2e6

(2π)1/2v3em2e

∫ ∞

0

ve−v2/2v2e ln2mv3

γωZe2dv, for ω � τ−1

0 . (4.56)

��

dPω

dω=

32e6Z2neni

3(2π)1/2c3m2eve

ln

[mev

3e

ωZe2

(2

γ

)5/2]. (4.57)

�� ν � 1�� sinh ξ − ξ � 1��

�� ξ � 1��

eνπ/2Kiν(ν) ≈∫ ∞

0

cos

[1

6νξ3]dξ =

21/3π

32/3ν1/3Γ(2/3),

eνπ/2K ′iν(ν) ≈ −

∫ ∞

0

ξ sin

[1

6νξ3]dξ = −22/331/6Γ(2/3)

2ν2/3,

��(4.55)��

dPω

dω=

32π

3c3Z2e6

(6π)1/2v3em2e

∫ve−v2/2v2edv, for ω � τ−1

0 . (4.58)

106 ��

2Δτ

a(t)

� 4.3: ��Δτ��

Ǳ��

��

��Ǳmv2min/2 = �ω � Te��

dPω

dω=

32πe6Z2neni

3(6π)1/2c3m2eve

exp (−�ω/Te) . (4.59)

��(1) � Z2 ��

��Z−��Z �

��(2) � T1/2e ��Ǳ

��(3) �� (4) ��

4.2.3 ��

�� |t| < Δτ �

��

{v(t) = 0, when |t| < Δτ,

v(t) ≈ 0, when |t| > Δτ.

��4.3�� Ǳ��

��

4.2 �� 107

��(4.30)��

d2EnωdωdΩ

=e2

4π2c3

∣∣∣∣∫ ∞

−∞v(t)× neiωtdt

∣∣∣∣2

(4.60)

�� ωΔτ � 1��(4.60)��Ǳ

d2EnωdωdΩ

≈ e2

4π2c3

∣∣∣∣∫ Δτ

−Δτ

v(t)× neiωtdt

∣∣∣∣2

≈ e2

4π2c3

∣∣∣∣∫ Δτ

−Δτ

v(t)× ndt

∣∣∣∣2

=e2

4π2c3|Δv × n|2 , (4.61)

�� Δv = v(Δτ) − v(−Δτ) ≈ Δv(+∞) − Δv(−∞) ��

�(4.61)��Æ��dEωdω

=2e2

3πc3|Δv|2 .

��Ǳ

|Δv| = 2v sinθs2. (4.62)

�� θs��Æ��Æ��Ǳ

tanθs2

=ρ⊥ρ,

��

sinθs2

=ρ⊥√ρ2 + ρ2⊥

.

��Ǳ

dEωdω

=8Z2e6

3πc3m2v21

ρ2⊥ + ρ2.

��Ǳ

dPω

dω=

∫dEωdω

fe(v)v2πniρdρd3v

=32Z2e6nine

3(2π)1/2c3m3/2e T

1/2e

∫ ∞

0

e−v2/2vdv

∫ ρmax

0

ρdρ

ρ2 + ρ2⊥

=32Z2e6nine

3(2π)1/2c3m3/2e T

1/2e

∫ ∞

0

e−v2/2v ln

√1 +

ρ2max

ρ2⊥dv. (4.63)

108 ��

�� ωΔτ � 1�

��Ǳ

|v| = ρ⊥τ 20 (ε cosh ξ − 1)2

,

�� ξ �� t�Ǳ t = τ0(ε sinh ξ − ξ)�� ξ = 0��

��Ǳ ρ⊥/τ 20 (ε − 1)2�� cosh ξ = 3��Ǳ��

�� | sinh ξ − ξ| ∼ 1��

Δτ ∼ τ0ε = τ0

√1 +

ρ2

ρ2⊥. (4.64)

� ωΔτ � 1��(4.63)��Ǳ√1 +

ρ2max

ρ2⊥∼ 1

ωτ0. (4.65)

�(4.65)��(4.63)� ��

dPω

dω=

32Z2e6nine

3(2π)1/2c3m2eve

∫ ∞

0

e−v2/2v lnmv3ev

3

ωZe2dv

=32Z2e6nine

3(2π)1/2c3m2eve

ln

[mev

3e

ωZe2

(2

γ

)3/2]. (4.66)

��Ǳ��(4.56)��(4.66)��(4.66)��(4.56)��

Ǳ��Ǳ √1 +

ρ2max

ρ2⊥=

2

γ

1

ωτ0, (4.67)

��(4.56)��

4.3 ��

��

��Ǳ��

��

4.3 �� 109

z

y

x

B

n

θ

o

� 4.4: ��

4.3.1 ��

��4.4�� z−�� n� x − z �� n� z−��ÆǱ θ��

n = (sin θ, 0, cos θ).

�� Ǳ

v(t) = c (β⊥ cosωct,−β⊥ sinωct, β�) ,

r0(t) = c

(β⊥ωc

sinωct,β⊥ωc

cosωct, β�t

).

�� β⊥ = v⊥/c��β⊥ = v�/c��

��

n× v(t) = c (β⊥ cos θ sinωct, β⊥ cos θ cosωct− β� sin θ,−β⊥ sin θ sinωct) ,

k · r0(t) =ω

ωc

β⊥ sin θ sinωct + ωtβ� cos θ.

��(4.27)��

(Bω)x = ieω

creiωr/cβ⊥ cos θ

∫ ∞

−∞ei(1−β� cos θ)ωt−i(ω/ωc)β⊥ sin θ sinωct sinωctdt, (4.68a)

(Bω)y = ieω

creiωr/c

∫ ∞

−∞ei(1−β� cos θ)ωt−i(ω/ωc)β⊥ sin θ sinωct (β⊥ cos θ cosωct− β� sin θ) dt,

(4.68b)

(Bω)z = −ieωcreiωr/cβ⊥ sin θ

∫ ∞

−∞ei(1−β� cos θ)ωt−i(ω/ωc)β⊥ sin θ sinωct sinωctdt. (4.68c)

110 ��

��

e−iz sinφ =∞∑

n=−∞Jn(z)e

−inφ, (4.69)

�� δ−��δ(ω) =

1

2π

∫ ∞

−∞eiωtdt, (4.70)

�� Jn(z)��(4.68) ��Ǳ

Bω = 2π

∞∑n=−∞

δ [(1− β� cos θ)ω − nωc]Bn

=2π

1− β� cos θ

∞∑n=−∞

δ

[ω − nωc

1− β� cos θ

]Bn. (4.71)

��(4.71) � δ−��

ω =nωc

1− β� cos θ. (4.72)

��(4.71) �� BnǱ

(Bn)x = ieω

creiωr/c

β⊥ cos θ

2i{Jn+1[(ω/ωc)β⊥ sin θ]− Jn−1[(ω/ωc)β⊥ sin θ]} , (4.73a)

(Bn)y = ieω

creiωr/c

{β⊥ cos θ

2[Jn+1[(ω/ωc)β⊥ sin θ] + Jn−1[(ω/ωc)β⊥ sin θ]]

−β� sin θJn[(ω/ωc)β⊥ sin θ]} , (4.73b)

(Bn)z = −ieωcreiωr/c

β⊥ sin θ

2i{Jn+1[(ω/ωc)β⊥ sin θ]− Jn−1[(ω/ωc)β⊥ sin θ]} . (4.73c)

��

Jn−1(z) + Jn+1(z) =2n

zJn(z),

Jn−1(z)− Jn+1(z) = 2J ′n(z),

��(4.73)��

(Bn)x = −eωcreiωr/cβ⊥ cos θJ ′

n[(ω/ωc)β⊥ sin θ], (4.74a)

(Bn)y = ieω

creiωr/c

[ nωc

ω tan θ− β� sin θ

]Jn[(ω/ωc)β⊥ sin θ], (4.74b)

(Bn)z =eω

creiωr/cβ⊥ sin θJ ′

n[(ω/ωc)β⊥ sin θ]. (4.74c)

4.3 �� 111

�� Ǳ

dPobs

dΩ= lim

T→∞1

2T

∫ T

−T

cr2

4πdt

∫ ∞

−∞Bωe

−iωtdω

2π·∫ ∞

−∞Bω′e−iω′tdω

′

2π

= limT→∞

1

2T

∫ T

−T

cr2

4πdt

∞∑n,m=−∞

Bn ·Bm

(1− β� cos θ)2e−i(n+m)ωct/(1−β� cos θ)

=cr2

2π

∞∑n=1

|Bn|2(1− β� cos θ)2

. (4.75)

��(4.75)��

��Ǳ

dP

dΩ=cr2

2π

∞∑n=1

|Bn|2(1− β� cos θ)

. (4.76)

��(4.75)�(4.76)�� 1− β� cos θ��

��

t′(1− β� cos θ)−c

ωc

sinωct′ = t− r/c,

��Ǳ Δt′ = 2π/ωc��Ǳ

Δtobs = Δt′(1 − β� cos θ)��

�� 1 − β� cos θ��

��

�� (4.74)��(4.75)��

dP

dΩ=

∞∑n=1

e2n2ω2c

2πc(1− β� cos θ)3

{β2⊥J

′2n

[nβ⊥ sin θ

1− β� cos θ

]+

(cos θ − β�

sin θ

)2

J2n

[nβ⊥ sin θ

1− β� cos θ

]}.

(4.77)�� (4.77)��Ǳ

dP

dΩ=

∞∑n=1

dPn

dΩ. (4.78)

�� n� ��(4.77)�� β� = 0��

Ǳ��(4.77)�� β� = 0� β⊥ = β��

dP

dΩ=

∞∑n=1

e2n2ω2c

2πc

[β2J ′2

n (nβ sin θ) + tan−2 θJ2n(nβ sin θ)

](4.79)

112 ��

�� Ǳ��

ω = nωc.

��(4.77)��Æ��

Pn =∞∑n=1

2e2ω2c

(1− β2� )3/2β⊥c

[β2⊥nJ

′2n

(2nβ⊥√1− β2

�

)− (1− β2)n2

∫ β⊥/√

1−β2�

0

J2n(2nξ)dξ

].

(4.80)�� β� = 0��

Pn =

∞∑n=1

2e2ω2c

βc

[β2nJ ′

2n (2nβ)− (1− β2)n2

∫ β

0

J2n(2nξ)dξ

]. (4.81)

��

∞∑n=1

∫ x

0

n2J2n(2nz)dz =x3

6(1− x2)3,

∞∑n=1

nJ ′2n(2nz) =

z

2(1− z2)2,

��

P =2

3

e2ω2cβ

2⊥

(1− β2)2c. (4.82)

4.3.2 ��

��(4.77)��(4.53)��Ǳ

d2P

dωdΩ=

2πnec3

(2π)3/2v2e

∫ ∞

0

β⊥dβ⊥

∫ ∞

−∞dβ� exp

[− (β2

⊥ + β2�)

2(Te/mec2)

]e2ω2

2πc(1− β� cos θ)

×∞∑n=1

δ

[ω − nωc

1− β� cos θ

]{β2⊥J

′2n [(ωβ⊥/ωc) sin θ] +

(cos θ − β�

sin θ

)2

J2n [(ωβ⊥/ωc) sin θ]

}.

�� β⊥ � 1�β� � 1�� Ǳ� nβ⊥ � 1��

��

Jn(z) =zn

n!2n+ o(zn),

4.4 �� 113

-e(ω0,k0)

incident�wave

scattering�wave

(ωs,ks)

� 4.5: ��

�� cos θ � β��

d2P

dωdΩ≈ 2πnec

3

(2π)3/2v3e

e2ω2

2πc

∫ ∞

0

β⊥dβ⊥

∫ ∞

−∞dβ� exp

[− (β2

⊥ + β2�)

2(Te/mec2)

]

×∞∑n=1

δ [ω − nωc − ωβ� cos θ]

{n2n

(n!)222nβ2n⊥(cos2 θ + 1

)sin2(n−1) θ

}

=nee

2ω

(2π)3/2ve

1 + cos2 θ

cos θ

∑n

n2n

2nn!

(Temec2

)n

sin2(n−1) θ exp

[− (ω − nωc)

2

2(Te/mec2)(ω cos θ)2

].

�� Te/mec2 � 1�� n��

�� Ǳ

d2P2/dωdΩ

d2P1/dωdΩ∼ 8Temec2

.

��Ǳ T ∼ 5 keV��d2P2/dωdΩ

d2P1/dωdΩ∼ 0.08.

4.4 ��

4.4.1 ��

�� Ǳ��Ǳ��

��

��

��

��—��Ǳ��

114 ��

��

��Ǳ

mev = −eE0 cos[k0 · r0(t)− ω0t], (4.83)

�� r0�� E0 cos (ω0t− k0 · r)��ω0� k0��

��Ǳ��

�� Ǳ��

�Ǳ��(4.26)�� Ǳ

Bωs =−ec2r

eiωsr/c

∫ ∞

−∞v(t)× neiωst−iks·r0(t)dt, (4.84)

�� ks = (ωs/c)ns��(4.83)��(4.84)��

Bωs =(e2/mec

2)

r(E0 × ns)e

iωsr/c

∫ ∞

−∞cos[k0 · r0(t)− ω0t]e

iωst−iks·r0(t)dt. (4.85)

�Ǳ�� Ǳ��

��

r0(t) = vt, (4.86)

�� v�� (4.86) ��(4.85)� ��

Bωs = πrer(E0 × ns)e

iωsr/c {δ[ωs(1− ns · v/c)− (ω0 − k0·v)]

+δ [ωs(1− ns · v/c) + (ω0 − k0·v)]} . (4.87)

�� re = e2/mec2 ��Ǳ��

�(4.87)��(4.36)��(4.38)��

d2Pnsωs

dΩdωs

=c

8πr2e(E0 × ns)

2δ[ωs(1− ns · v/c)− (ω0 − k0·v)]. (4.88)

��ωs > 0��(4.88)��

ωs =ω0 − k0 · v1− ns · v/c

. (4.89)

��

�Ǳ ω0 − k0 · v��ǱǱω0 − k0 · v��

4.4 �� 115

�Ǳ��Ǳ (ω0 − k0 · v)/(1 − ns·v/c)��(4.89)��Ǳ��Ǳ��(4.89)��Ǳ

ωs ≈ ω0 + [(ω0/c)ns − k0] · v = ω0 + k · v, (4.90)

��

k = (ω0/c)ns − k0 (4.91)

�Ǳ��

��Ǳ��(4.88)��Æ��

dPns

dΩ=

c

8πr2e(E0 × ns)

2. (4.92)

��Ǳ��Æ��

dσT

dΩ=dPn/dΩ

cE20/8π

= r2e sin2 θ, (4.93)

�� θ�� n��Æ��Æ��

��

σT =8π

3r2e . (4.94)

4.4.2 ��

��

��

��Ǳ��

��Æ��

�� N ��

��(4.85)��

Bωs =rer(E0 × ns)e

iωsr/c

N∑i=1

∫ ∞

−∞cos[k0 · ri(t)− ω0t]e

iωst−iks·ri(t)dt. (4.95)

�� ri�� i��Ǳ��

��

n(r, t) =N∑i=1

δ(r− ri(t)). (4.96)

116 ��

�� n(r, t)�� Ǳ

nk(t) =N∑i=1

e−ik·ri(t). (4.97)

��(4.97)��(4.95)��Ǳ

Bωs =1

2

rer(E0 × ns)e

iωsr/cN∑i=1

∫ ∞

−∞

[eik0·rie−iω0t + e−ik0·rieiω0t

]eiωst−iks·ridt

=1

2

rer(E0 × ns)e

iωsr/c

∫ ∞

−∞

[nks−k0(t)e

i(ωs−ω0)t + nks+k0(t)ei(ωs+ω0)t

]dt. (4.98)

��Æ dΩ��Ǳ

dE =c

4πr2dΩ

∫Bωs ·B−ωs

dωs

2π. (4.99)

�(4.98)��(4.99)��

dEns

dΩ=cr2e16π

(E0 × ns)2

∫ ∞

−∞

dωs

2π

∫ ∞

−∞

[nks−k0(t)e

i(ωs−ω0)t + nks+k0(t)ei(ωs+ω0)t

]dt

×∫ ∞

−∞

[n−ks−k0(t

′)e−i(ωs+ω0)t′ + n−ks+k0(t′)e−i(ωs−ω0)t′

]dt′. (4.100)

��

dEns

dΩ=cr2e16π

(E0 × ns)2

∫ ∞

−∞

dωs

2π

×∫dtdt′

[〈nks−k0(t)n

∗ks−k0

(t′)〉ei(ωs−ω0)(t−t′) + 〈nks+k0(t)n∗ks+k0

(t′)〉ei(ωs+ω0)(t−t′)].

(4.101)

��

〈nk(t)n∗k′(t′)〉 = 0, when k = k′.

�Ǳ ∫ ∞

−∞

dωs

2π〈nks+k0(t)n

∗ks+k0

(t′)〉ei(ωs+ω0)(t−t′)

=

∫ −∞

∞(−)

dωs

2π〈n−ks+k0(t)n

∗−ks+k0

(t′)〉e−i(ωs−ω0)(t−t′)

=

∫ ∞

−∞

dωs

2π〈n∗

ks−k0(t)nks−k0(t

′)〉ei(ωs−ω0)(t′−t).

4.4 �� 117

�� k�� ω�

k = ks − k0, ω = ωs − ω0,

��(4.101)��Ǳ

dEns

dΩ=cr2e8π

(E0 × ns)2

∫ ∞

−∞

dω

2π

∫dtdt′〈nk(t)n

∗k(t

′)〉eiω(t−t′)

=cr2e8π

(E0 × ns)2

∫ ∞

−∞

dω

2π

∫ ∞

−∞dt

∫ ∞

−∞dτ〈n∗

k(t)nk(t + τ)〉eiωτ . (4.102)

�� 〈n∗k(t)nk(t+ τ)〉�� t��

�� τ��

dPns

dΩ=cr2e8π

(E0 × ns)2

∫ ∞

−∞

dω

2π

∫ ∞

−∞dτ〈n∗

k(t)nk(t + τ)〉eiωτ .

��Ǳ

d2Pnsω

dωdΩ=cr2e8π

(E0 × ns)2N

1

2πN

∫ ∞

−∞dτ〈n∗

k(t)nk(t+ τ)〉eiωτ .

�� S(k, ω)�

S(k, ω) =1

2πN

∫ ∞

−∞dτ〈n∗

k(t)nk(t+ τ)〉eiωτ , (4.103)

��Ǳ

d2Pnsω

dωdΩ= I0Nr

2e sin

2 θS(k, ω). (4.104)

�� I0 = cE20/8π�� θ��Æ�

��

�� Ǳ

〈n∗k(t)nk(t+ τ)〉 =

⟨N∑

i,j=1

eik·[ri(t)−rj(t+τ)]

⟩=

⟨N∑i=1

e−ik·vjτ

⟩+

⟨N∑

i,j=1i �=j

eik·[ri(t)−rj(t+τ)]

⟩

�� Ǳ��

〈n∗k(t)nk(t+ τ)〉 =

⟨N∑i=1

e−ik·vjτ

⟩=

N

(2π)3v3e

∫e−v2/2v2ee−ik·vτd3v

=N

(2π)1/2ve

∫ ∞

−∞e−v2

�/2v2ee−ikv�τdv�.

118 ��

�� Ǳ

S(k, ω) =1

(2π)3/2ve

∫ ∞

−∞dτeiωτ

∫ ∞

−∞e−v2

�/2v2ee−ikv�τdv�

=1

(2π)1/2ve

∫ ∞

−∞δ(ω − kv�)e

−v2�/2v2edv�

=1

(2π)1/2kveexp

[− ω2

2k2v2e

]. (4.105)

��Ǳ

d2Pnωs

dωsdΩ=

1√2πkve

I0r2eN sin2 θ exp

[− ω2

2k2v2e

]. (4.106)

��

��

��

4.5 ��

��

��

��

4.5.1 ��

��

d2F

dz2+

1

z

dF

dz−(1 +

ν2

z2

)F = 0. (4.107)

�� Iν(z)� Kν(z)��Ǳ��

��Kv(z)��Ǳ

Kν(z) =1

2

∫ ∞

−∞e−z cosh ξ−νξdξ, (−π/2 < arg z < π/2). (4.108)

��(4.108)�Kiν(z)��Ǳ

Kiν(z) =1

2

∫ ∞

−∞e−z cosh ξ−iνξdξ. (4.109)

4.5 �� 119

��(4.109)��Kiν(z)�K ′iν(z)��

Kiν(z) =1

2e−νπ/2

∫ ∞

−∞eiz sinh ξ−iνξdξ, (4.110)

K ′iν(z) =

1

2e−νπ/2

∫ ∞

−∞

z − ν cosh ξ

(z cosh ξ − ν)2eiz sinh ξ−iνξdξ. (4.111)

��(4.109) �� t = ξ + iπ/2�� cosh(t −iπ/2) = −i sinh t��

Kiν(z) =1

2e−νπ/2

∫ ∞+iπ/2

−∞+iπ/2

eiz sinh t−iνtdt. (4.112)

��

∫ ∞+iπ/2

−∞+iπ/2

eiz sinh t−iνtdt =

∫ ∞

−∞eiz sinh ξ−iνξdξ.

� ξ−�� exp (iz sinh ξ − iνξ)�� ξ−�� ∫

C

eiz sinh t−iνtdt = 0,

��??�� Ǳ

C : −∞ → ∞ → ∞+ iπ/2 → −∞+ iπ/2 → −∞. (4.113)

��

∫ ∞

−∞eiz sinh ξ−iνξdξ =

∫ ∞+iπ/2

−∞+iπ/2

eiz sinh ξ−iνξdξ +

∫ −∞+iπ/2

−∞eiz sinh ξ−iνξdξ

+

∫ ∞

∞+iπ/2

eiz sinh ξ−iνξdξ.

��

∫ −∞+iπ/2

−∞eiz sinh ξ−iνξdξ =

∫ ∞

∞+iπ/2

eiz sinh ξ−iνξdξ = 0.

Ǳ��

ξ = ρeiφ,

� ρ→ ∞�� 0 < Im ξ < π/2��

0 < ρ sin φ < π/2, ρ→ ∞.

120 ��

Re ξ

Im ξ

0

C

iπ/2

φ

ρ

� 4.6: � ξ�� C�

�� ρ� 1�� φ� 1��

ξ = ρ+ iρφ.

��

∫ ∞

∞+iπ/2

eiz sinh ξ−iνξdξ = limρ→∞

∫ 0

π/2ρ

eiz sinh(ρ+iρφ)−iν(ρ+iρφ)ρeiφidφ

= limρ→∞

∫ 0

π/2ρ

eiz sinh ρ cos(ρφ)−z cosh ρ sin(ρφ)−iνρ+νρφeiφid(ρφ)

= limρ→∞

i

∫ 0

π/2

eiz sinhρ cos θ−iνρ−iνρe−z cosh ρ sin θ+νθdθ

�� (0, π/2)�� sin θ�� exp(−z cosh ρ sin θ)�� ρ��

�� ∫ ∞

∞+iπ/2

eiz sinh ξ−iνξdξ = 0.

�� ∫ −∞+iπ/2

−∞eiz sinh ξ−iνξdξ = 0.

��(4.110)��(4.109)�� z > 0��K ′

iν(z)��Ǳ�

��

K ′iν(z) = −1

2

∫ ∞

−∞cosh ξe−z cosh ξ−iνξdξ. (4.114)

�(4.114)��

K ′iν(z) =

1

2

∫ ∞

−∞

iν sinh ξ − z

(z sinh ξ + iν)2e−z cosh ξ−iνξdξ.

4.5 �� 121

�� t = ξ + iπ/2��

K ′iν(z) =

1

2e−νπ/2

∫ ∞+iπ/2

−∞+iπ/2

z − ν cosh t

(z cosh t− ν)2eiz sinh t−iνtdt.

��

∫ ∞

−∞

z − ν cosh t

(z cosh t− ν)2eiz sinh t−iνtdt =

∫ ∞+iπ/2

−∞+iπ/2

z − ν cosh t

(z cosh t− ν)2eiz sinh t−iνtdt.

�� (ν/z) < 1�Ǳ��

∫C

z − ν cosh t

(z cosh t− ν)2eiz sinh t−iνt = i2πRes

[z − ν cosh t

(z cosh t− ν)2eiz sinh t−iνt, t = i arccos

ν

z

].

�� C ��(4.113)��

Res[z − ν cosh t

(z cosh t− ν)2eiz sinh t−iνt, t = i arccos

ν

z

]

= limt→i arccos(z/ν)

d

dt

[(t− i arccos

ν

z)2

z − ν cosh t

(z cosh t− ν)2eiz sinh t−iνt

]

= 0.

��

∫ −∞+iπ/2

−∞

z − ν cosh t

(z cosh t− ν)2eiz sinh t−iνtdt =

∫ ∞

∞+iπ/2

z − ν cosh t

(z cosh t− ν)2eiz sinh t−iνtdt = 0

��(4.111)��

4.5.2 ��

Ǳ�� (4.76)��Æ��

sin θ′ =

√1− β2

� sin θ

1− β� cos θ, cos θ′ =

cos θ − β�1− β� cos θ

,

sin θ =

√1− β2

� sin θ′

1 + β� cos θ′, cos θ =

cos θ′ + β�1 + β� cos θ′

.

122 ��

��

sin θdθ =1− β2

�

(1 + β� cos θ′)2sin θ′dθ′,

1− β� cos θ =1− β2

�

1 + β� cos θ′,

cos θ − β�sin θ

=(1− β�)

1/2

tan θ′,

nβ⊥1− β� cos θ

sin θ =nβ⊥√1− β2

�

sin θ′,

�� Ǳ

Pn =e2n2ω2

c

(1− β2� )

2

∫ π

0

{β2⊥J

′2n

[nβ⊥ sin θ′√

1− β2�

]+

1− β2�

tan2 θ′J2n


1− β2�

]}(1 + β� cos θ

′) sin θ′dθ′

=e2n2ω2

c

(1− β2� )2

∫ π

0

{β2⊥J

′2n


1− β2�

]+

1− β2�

tan2 θ′J2n


1− β2�

]}sin θ′dθ′.

��

Jn−1(z)− Jn+1(z) = 2J ′n,

Jn−1(z) + Jn+1(z) =2n

zJn,

��

J2n−1(z) + J2

n+1(z) = 2J ′2(z) + 2n2

z2J2n(z),

��

β2⊥J

′2n


1− β2�

]+

1− β2�

tan2 θ′J2n


1− β2�

]

=β2⊥2

[J2n−1 + J2

n+1 − 21− β2

�

β2⊥ sin2 θ′

J2n

]+

1− β2�

tan2 θ′J2n

=β2⊥2

(J2n−1 + J2

n+1

)− (1− β2

�)J2

n

��

Pn =e2n2ω2

c

(1− β2� )2

∫ π

0

{β2⊥2

[J2n−1

(nβ⊥ sin θ′√

1− β2�

)+ J2

n+1


1− β2�

)]

−(1− β2�)J2

n


1− β2�

)}sin θ′dθ′.

4.5 �� 123

��

J2n(z) =

1

π

∫ π

0

J0 (2z sin ξ) cos(2nξ)dξ,

��

Pn =e2n2ω2

c

(1− β2� )2

1

π

∫ π

0

sin θ′dθ′∫ π

0

dξJ0

(2nβ⊥ sin θ′√

1− β2�

sin ξ

)

×[β2⊥2

cos[2(n− 1)ξ] +β2⊥2

cos[2(n + 1)ξ]− (1− β2�) cos(2nξ)

]

=e2n2ω2

c

(1− β2� )2

I,

��I1��Ǳ

I1 =1

π

∫ π

0

sin θ′dθ′∫ π

0

dξJ0

(2nβ⊥ sin θ′√

1− β2�

sin ξ

)

×[β2⊥2

cos[2(n− 1)ξ] +β2⊥2

cos[2(n+ 1)ξ]− (1− β2�) cos(2nξ)

],

��

∫ π/2

0

J0(z sin θ′) sin θ′dθ′ =

sin z

z.

��

I =

√1− β2

�

2nβ⊥

1

π

∫ π

0

dξ sin

(2nβ⊥ sin ξ√

1− β2�

)β2⊥

sin ξ

{cos[2(n− 1)ξ] + cos[2(n+ 1)ξ]− 2(1− β2

�) cos 2nξ

}.

��

J2n(z) =1

π

∫ π

0

cos(z sin θ) cos(2nθ)dθ,

��∫ z

0

J2n(ξ)dξ =1

π

∫ z

0

dξ

∫ π

0

cos(ξ sin θ) cos(2nθ)dθ =1

π

∫ π

0

sin(z sin θ)

sin θcos(2nθ)dθ.

��

I =

√1− β2

�

2nβ⊥

∫ 2nβ⊥/√

1−β2�

0

{β2⊥[J2(n−1)(ξ) + J2(n+1)(ξ)

]− 2(1− β2

�)J2n(ξ)

}dξ.

124 ��

��

J2(n−1)(ξ) + J2(n+1)(ξ) = 4J ′′2n(ξ) + 2J2n(ξ),

��

I =

√1− β2

�

2nβ⊥

∫ 2nβ⊥/√

1−β2�

0

dξ{4β2

⊥J′′2n(ξ) + 2β2

⊥J2n(ξ)− 2(1− β2�)J2n(ξ)

}

=

√1− β2

�

2nβ⊥

[4β2

⊥J′2n

(2nβ⊥√1− β2

�

)− 4n(1− β2)

∫ β⊥/√

1−β2�

0

J2n(2nξ)dξ

].

�� Ǳ

Pn =2e2ω2

c

(1− β2� )3/2β⊥

[β2⊥nJ

′2n

(2nβ⊥√1− β2

�

)− (1− β2)n2

∫ β⊥/√

1−β2�

0

J2n(2nξ)dξ

].

4.5.3 ��

�� γ � 1�� γ = (1 −β2)−1/2 ��

�� n � 1��

��Ǳ�� β� = 0�

��

J2n(2nβ) =1

π

∫ π

0

cos [2n(ξ − β sin ξ)] dξ, (4.115)

� n � 1��(4.115) �� ξ � 1��

�

J2n(2nβ) ≈1

π

∫ ∞

0

cos

[2n

(ξ − βξ + β

ξ3

6

)]dξ. (4.116)

�� β → 1� 1 − β ≈ 1/2γ2��

��

J2n(2nβ) ≈1

π

∫ ∞

0

cos

[2n

(1

2γ2ξ +

ξ3

6

)]dξ =

1

n1/3Ai(n2/3γ−2),

�� Ai(x)��Ǳ

Ai(x) =1

π

∫ ∞

0

cos

(xξ +

ξ3

3

)dξ. (4.117)

4.5 �� 125

u = n2/3γ−2��

J ′2n(2nβ) ≈ − 1

n2/3Ai′(u),

∫ β

0

J2n(2nξ)dξ ≈1

2n

∫ ∞

u

Ai(u′)du′.

�� n� ��Ǳ

Pn ≈ 2e4B2

m2c3γ2

[−n1/3Ai′(u)− n1/3

2u

∫ ∞

u

Ai(u′)du′]

= −2e4B2u1/2

m2c3γ

[Ai′(u) +

1

2u

∫ ∞

u

Ai(u′)du′].

� u� 1��

Pn ≈ −2e4B2n1/3

m2c3γ2Ai′(0) = 0.52

e4B2n1/3

m2c3γ2, when 1 � n� γ3.

� u� 1��

Pn ≈ e4B2n1/2

2√πm2c3γ5/2

exp

[−2

3nγ−3

], when n� γ3.

�� Ǳ

n ∼ γ3.

��

ω ∼ eB

mcγγ3 =

eB

mcγ2.

�� ωc = eB/mcγ��

��

��

Ai(u) =1

π

√u

3K1/3

(2

3u3/2

),

Ai′(u) = − u√3πK2/3

(2

3u3/2

),

126 ��

� 4.7:

�

Ai′(u) +1

2u

∫ ∞

u

Ai(u′)du′ =u√3π

[−K2/3

(2

3u3/2

)+

1

2

∫ ∞

u

u′1/2K1/3

(2

3u′3/2

)du′]

=u√3π

[∫ ∞

ξ

K ′2/3(x)dx+

1

2

∫ ∞

ξ

K1/3(x)dx

]

=u√3π

[∫ ∞

ξ

(−1

2K1/3 −

1

2K5/3

)dx+

1

2

∫ ∞

ξ

K1/3(x)dx

]

= − u

2√3π

∫ ∞

ξ

K5/3(x)dx,

�� ξ = 2u3/2/3 = (2/3)nγ−3 = (2/3)(ω/ωc)γ−3� � Pn��

dP = Pndn = Pndω

ωc,

��

dP = dω

√3

2π

e3B

mc2F

(2ω

3ωcγ3

),

��

F (x) = x

∫ ∞

x

K5/3(ξ)dξ.

��(4.7)�� F (x)�� x = 0.29��

Ǳ 0.92�� F (x)��

�� 0.44ωcγ3�� Ǳ n ≈ 0.44γ3�

��

1. ��(4.6)��Lienard-Wiechert�

φ =e

R− v ·R/c, A =ev

c(R− v ·R/c) ,

4.5 �� 127

��R = r− r0(t′)� t′Ǳ��

2. �� a(t)�� Ǳ aω��∫ ∞

−∞a2(t)dt =

∫ ∞

−∞|aω|2

dω

2π,

�� aω ��Ǳ

aω =

∫ ∞

−∞a(t)eiωtdt.

[�� δ−�� δ(ω) =∫∞−∞ exp(−iωt)dt/(2π)�]

3. ��(4.18)��(4.21)�

4. �� z < 0��Ǳ 0��Ǳ t0��

�� v��

�� t < t0��

��

5. � β ��Ǳ v��

��

6. ��Æ�� Ǳ��Æ��

�� dEω/dω��

2ωK1(ω) =

∫ ∞

−∞

1

(t2 + 1)3/2e−iωtdt,

2ωK0(ω) =

∫ ∞

−∞

t

(t2 + 1)3/2e−iωtdt.

�

7. ��Ǳ��

A =∞∑l=1

An exp(−ilω0t),

��2π/ω0��

Al =ω0

2π

e

creilω0r/c

∮exp [ilω0(t− r0 · n/c)] dr0,

�� r0(t)��

128 ��

8. �� (4.74)�

9. ��Ǳ

ωs − ω0 = (ks − k0) · v,

�� ω0,s� k0,s�� v��

��

��

��Ǳ��Ǳ��

��Ǳ��

��Ǳ��Ǳ��

��

��Ǳ�Ǳ��

��Ǳ��

��

f =n(r, t)

(2π)3/2v3T (r, t)exp

{− [v − u(r, t)]2

2v2T (r, t)

}, (5.1)

�� vT (r, t) = [T (r, t)/m]1/2 ��T (r, t)��

��

��

��

��

��Æ�� L�Æ�� T �� λ

�� τ��

L� λ, T � τ, (5.2)

��

��Ǳ��

��

��Ǳ��MHD�� Ǳ��

��

��

130 ��

z

y

x

dfu

V(ρ,u, ε)o

� 5.1: �� V ��

��

5.1 ��

��Ǳ��

��

��

��

��

5.1.1 ��

��

��Ǳ��

��

��

��

��(5.1)�� V��

�� S��Ǳ df�� V ��

�� V ��Ǳ ∮S

ρu · df =∫V

∇ · (ρu)d3r, (5.3)

5.1 �� 131

�� ρ�� u�� V ��

��Ǳ

− d

dt

∫V

ρd3r = −∫V

∂ρ

∂td3r. (5.4)

�Ǳ�� V �� V��

�(5.3)��(5.4)��

− d

dt

∫V

ρd3r =

∮S

ρu · df ,

∫V

[∂ρ

∂t+∇ · (ρu)

]d3r = 0.

�Ǳ V �� Ǳ��

�∂ρ

∂t+∇ · (ρu) = 0. (5.5)

�� ρu��

��

�� ρ =��(5.5)��Ǳ

∇ · u = 0. (5.6)

��Ǳ�

��

�� V �� V ��

��Ǳd

dt

∫V

ρud3r =

∫V

∂

∂t(ρu)d3r.

��Ǳ�� V ��

��Ǳ ∮S

ρuu · df =∫V

∇ · (ρuu)d3r.

�� V ��

�� V ��

��Ǳ ∮S

Pdf =

∫V

(∇P )d3r,

132 ��

�� P ��

��ǱF�� V ��Ǳ∫V

ρFd3r,

��

−∫V

∂

∂t(ρu)d3r =

∮S

ρuu · df+∮S

Pdf−∫V

ρFd3r.

�Ǳ V � ��

∂

∂t(ρu) +∇ · (ρuu) +∇P = ρF. (5.7)

��(5.5)��(5.7)��

ρ

[∂u

∂t+ (u · ∇)u

]= −∇P + ρF. (5.8)

��(5.8)�Ǳ��Ǳ��

��

Πik = Pδik + ρuiuk (5.9)

�Ǳ

Π = P I+ ρuu,

�� I��(5.9)Ǳ�Ǳ��(5.7)��Ǳ��

∂

∂t(ρu) +∇ ·Π = 0. (5.10)

��

�� V ��

�� ∫V

(1

2ρu2 + ρε

)d3r,

�� ε��Ǳ�� V ��

��Ǳ� ∮S

(1

2ρu2 + ρε

)u · df =

∫V

∇ ·[(

1

2ρu2 + ρε

)u

]d3r,

5.1 �� 133

�� V ��

�� V ��Ǳ

∮S

Pu · df =∫V

∇ · (Pu) d3r.

�Ǳ��

− d

dt

∫V

(1

2ρu2 + ρε

)d3r =

∮S

(1

2ρu2 + ρε

)u · df+

∮S

Pu · df .

�� V ��

∂

∂t

(1

2ρu2 + ρε

)+∇ ·

[(1

2ρu2 + ρε+ P

)u

]= 0. (5.11)

�� h��Ǳ

h = ε+ P/ρ,

��Ǳ∂

∂t

(1

2ρu2 + ρε

)+∇ ·

[(1

2ρu2 + ρh

)u

]= 0. (5.12)

��Ǳ��

��(5.12)��Ǳ

Q = ρ(u2/2 + h

)u. (5.13)

��(5.12)��Ǳ��(5.12)�� Ǳ

ρ

[∂ε

∂t+ (u · ∇)ε

]− P

ρ

[∂ρ

∂t+ (u · ∇)ρ

]= 0. (5.14)

�� V = 1/ρ�� Ǳ

dε

dt+ P

dV

dt= 0, (5.15)

�� d/dt��

d

dt=

∂

∂t+ u · ∇.

134 ��

��

Tds = dε+ PdV, (5.16)

�� s��(5.15)��Ǳ

ds

dt= 0. (5.17)

��

��

ds =δQ

T,

��Ǳ�

��Ǳ��Ǳ��

�� δQ = 0��(5.17)��(5.17)Ǳ��

��

��(5.5)��(5.8)��(5.12)�� ρ��

� u�� ε��Ǳ��(�� )�� (�� ρ�� ε�� P�� u)�� P�� ρ�� ε��

�� P�� V �� T ��

P = P (V, T ) (5.18)

�Ǳ��

��Ǳ��

�Æ��Ǳ��

��

Ǳ�� Ǳ�

P = nT =ρT

m, (5.19)

��m�� n��

��Ǳ

s = − 1

mlnP +

cPm

lnT +1

m(cP + ξ), (5.20)

5.1 �� 135

�� cP �� ξ��

cp =5

2, ξ =

3

2ln

m

2π�2. (5.21)

�� s =const��(5.20)�

T cP /P = const.

��(5.19)��

Pρ−γ = const, (5.22)

�� γ = cP/(cP − 1) = cP/cV �� cV ��

5.1.2 ��

�� (5.7)��

�� Ǳ

��Ǳ

Πik = Pδik + ρuiuk + πik = −σik + ρuiuk. (5.23)

�� πik �Ǳ��

�Ǳ�� πik ��

��

�� ∂ui/∂xk��Æ��Ω��

��Ǳ u = Ω× r��Ǳ��

∂ui∂xk

+∂uk∂xi

�� u = Ω× r�Ǳ�� πik ��

�� Ǳ

πik = −η(∂ui∂xk

+∂uk∂xi

− 2

3δik∂ul∂xl

)− ζδik

∂ul∂xl

, (5.24)

�� η� ζ ��Ǳ�� ζ ��Ǳ��

��

��

136 ��

��(5.24)�(5.23)��

ρ

[∂u

∂t+ (u · ∇)u

]= −∇P + η∇2u+ (ζ + η/3)∇ (∇ · u) + ρF. (5.25)

��—��

��Ǳ��

��

��Ǳ

Ekin =1

2ρ

∫V

u2d3r.

��Ǳ

Ekin =1

2

∫∂

∂t(ρu2)d3r =

∫ρu · ∂u

∂td3r.

��—��(5.25)��

∂ui∂t

= −uk∂ui∂xk

− 1

ρ

∂P

∂xi− 1

ρ

∂πik∂xk

,

��Ǳ

∂

∂t

(ρu2/2

)= ρui

[−uk

∂ui∂xk

− 1

ρ

∂P

∂xi− 1

ρ

∂πik∂xk

]

= −ρuk∂

∂xk(uiui/2)− ui

∂P

∂xi− ui

∂πik∂xk

= −ρuk∂

∂xk

[u2/2 + P/ρ

]− ∂

∂xk(uiπik) + πik

∂ui∂xk

.

��∇ · u = 0�� Ǳ

∂

∂t

(ρu2/2

)= − ∂

∂xk

[ρuk

(u2/2 + P/ρ

)+ uiπik

]+ πik

∂ui∂xk

. (5.26)

�� ρuk(u2/2 + P/ρ)��

�� uiπik ��Ǳ��(��)� ��Ǳ��

��(5.26)��

Ekin = −∮ [

ρuk(u2/2 + P/ρ

)− uiσ

′ik

]dfk +

∫πik

∂ui∂xk

d3r, (5.27)

5.1 �� 137

��Ǳ��

Ekin =

∫πik

∂ui∂xk

d3r =1

2

∫πik

(∂ui∂xk

+∂uk∂xi

)d3r.

��∇ · u = 0��Ǳ πik = −η(∂uk/∂xi + ∂ui/∂xk)�

��

Ekin = −η2

∫ (∂ui∂xk

+∂uk∂xi

)2

d3r. (5.28)

�� Ekin < 0��(5.28)�� η��

5.1.3 ��

�� (5.12)�� Ǳ��

��

��

q = −κ∇T, (5.29)

�� κ�Ǳ� ��

��Ǳ��Ǳu · π [��(5.26)]��Ǳ

∂

∂t

(1

2ρu2 + ρε

)+∇ ·

[(1

2ρu2 + ρh

)u+ u · π − κ∇T

]= 0, (5.30)

��Ǳ

Q =(u2/2 + h

)ρu+ u · π − κ∇T. (5.31)

��(5.30)Ǳ��Ǳ��

ρT

(∂s

∂t+ u · ∇s

)= −πik

∂ui∂xk

+∇ · (κ∇T ). (5.32)

��(5.32)�� Ǳ��

��Ǳ

ρε =3

2nT, (5.33)

138 ��

�� n��T��

�Ǳ��(5.33)��Ǳ�� g = 1/4πnλ3D �� (5.33) ��(5.30)��

3

2n

(∂T

∂t+ u · ∇T

)= −P∇ · u− πik

∂ui∂xk

+∇ · (κ∇T ). (5.34)

��

�� κ� η�� ζ�

��Ǳ��Ǳ

S =

∫ρsd3r,

��Ǳ

S =

∫∂(ρs)

∂td3r.

��∂(ρs)

∂t= −s∇ · (ρu) + ρ

∂s

∂t.

��(5.32)�Ǳ

ρT

(∂s

∂t+ u · ∇s

)=η

2

(∂ui∂xk

+∂uk∂xi

− 2

3δik∇ · u

)2

+ ζ(∇ · u)2 +∇ · (κ∇T ), (5.35)

��

∂(ρs)

∂t= −s∇ · (ρu)− ρu · ∇s + 1

T∇ · (κ∇T )

+η

2T

(∂ui∂xk

+∂uk∂xi

− 2

3δik∇ · u

)2

+ζ

T(∇ · u)2.

��Ǳ

S = −∫

∇ · (ρsu) d3r +∫

1

T∇ · (κ∇T )d3r +

∫ζ

T(∇ · u)2d3r

+

∫η

2T

(∂ui∂xk

+∂uk∂vi

− 2

3δik∇ · u

)2

d3r

= −∫

∇ · (ρsu) d3r +∫

∇ ·(κ∇TT

)d3r +

∫κ(∇T )2T 2

d3r

+

∫ζ

T(∇ · u)2d3r +

∫η

2T

(∂ui∂xk

+∂uk∂vi

− 2

3δik∇ · u

)2

d3r.

5.2 �� 139

u(a, t)

a

r(a, t)

z

y

x

o

� 5.2: � ��Ǳ�� a��

��Ǳ��

�� Ǳ��

S =

∫κ(∇T )2T 2

d3r +

∫ζ

T(∇ · v)2d3r +

∫η

2T

(∂ui∂xk

+∂uk∂vi

− 2

3δik∂ul∂xl

)2

d3r. (5.36)

��Ǳ�� S > 0�� κ� η�� ζ ��

5.2 ��

��

��Ǳ�Ǳ�� Ǳ�Ǳ��

�Ǳ��

�� Ǳ� �� Ǳ� ��

��

��

�� x ��Ǳ�

�� Ǳ a��

�� x�� (τ, a)�� (t, x)��Ǳ

t = τ, x = a+ ξ(a, τ), (5.37a)

ξ(a, τ) =

∫ τ

0

u(a, τ ′)dτ ′. (5.37b)

140 ��

�� ξ(a, τ)��Ǳ�� Ǳ a��Ǳ τ �� u(a, τ)��Ǳ�

� Ǳ a�� (5.37)��

∂

∂x=

[1 +

∫ τ

0

∂u(a, τ ′)∂a

dτ ′]−1

∂

∂a, (5.38a)

∂

∂t=

∂

∂τ− u(a, τ)

[1 +

∫ τ

0

∂u(a, τ ′)∂a

dτ ′]−1

∂

∂a. (5.38b)

�� (5.38)��

∂

∂t+ u

∂

∂x=

∂

∂τ. (5.39)

�� (5.38a)�(5.39)��Ǳ

∂ρ

∂τ+ ρ

[1 +

∫ τ

0

∂u(a, τ ′)∂a

dτ ′]−1

∂u

∂a= 0.

��∂

∂τ

∫ τ

0

∂u(a, τ ′)∂a

dτ ′ =∂u(a, τ)

∂a,

��Ǳ∂

∂τ

{ρ

[1 +

∫ τ

0

∂u(a, τ ′)∂a

dτ ′]}

= 0.

��

ρ(a, τ)

[1 +

∫ τ

0

∂u(a, τ ′)∂a

dτ ′]= ρ(a, 0). (5.40)

��Ǳ

ρ

[1 +

∫ τ

0

∂u(a, τ ′)∂a

dτ ′]∂u

∂τ= −∂P

∂a,

��(5.40)��

∂u

∂τ= − 1

ρ(a, 0)

∂P

∂a. (5.41)

��(5.17)��

∂s(a, τ)

∂t= 0. (5.42)

��

P (a, τ)[ρ(a, τ)]−γ = P (a, 0)[ρ(a, 0)]−γ. (5.43)

5.3 �� 141

��(5.40)�(5.41)�(5.43)� �� ρ(a, τ) � u(a, τ)��

u(a, τ)�� (5.37)��

��

��

�� Ǳ��

��

5.3 ��

Ǳ��

��

�Ǳ�� Ǳ ��

��Ǳ��

��

ρ = ρ0 + δρ, P = P0 + δP,

�� ρ0� P0�� δρ� δP��

��

∂δρ

∂t+ ρ0∇ · u = 0.

��

ρ0∂u

∂t= −∇δP.

�� s(ρ, P ) =const��Ǳ��ǱP = P (s, ρ)�

�Ǳ��Ǳ

δP =

(∂P

∂ρ

)s

δρ.

��Ǳ

ρ0∂u

∂t= −

(∂P

∂ρ

)s

∇δρ.

�� u�� cs =√

(∂P/∂ρ)s��

∂2

∂t2δρ− c2s∇2δρ = 0. (5.44)

142 ��

�� cs��Ǳ�

��

�� Pρ−γ =const��(∂P

∂ρ

)s

=γP

ρ,

�� P = nT = ρT/m��m��

��

cs =

√γT

m. (5.45)

�� γ = cP/cV = 5/3��Ǳ

cs =

√5T

3m. (5.46)

�� exp[i(k ·r−ωt)]�� (5.44)��

ω2 = k2c2s. (5.47)

��

Ǳ��

�� Ǳ��

Ǳ(��)�

δE = δ

(ρε+

1

2ρu2)

≈ ∂(ρε)

∂ρδρ+

1

2

∂2(ρε)

∂ρ2(δρ)2 +

1

2ρ0u

2.

��

∂(ρε)

∂ρ= ε+ ρ

(∂ε

∂ρ

)s

,

�� dε = Tds− PdV = Tds+ (P/ρ2)dρ�� (∂ε/∂ρ)s = P/ρ2��

��

[∂(ρε)/∂ρ]s = ε+ P/ρ = h.

[∂2(ρε)/∂ρ2]��Ǳ

[∂2(ρε)/∂ρ2]s = [∂h/∂ρ]s =

(∂h

∂P

)s

(∂P

∂ρ

)s

= c2s(∂h/∂p)s.

5.4 ��—�� 143

�� dh = Tds + V dP = Tds + (1/ρ)dP�� (∂h/∂P )s = 1/ρ�

��

[∂2(ρε)/∂ρ2]s = c2s/ρ.

��

δE = h0δρ+c2s2

(δρ)2

ρ0+

1

2ρ0u

2.

��∫δρdV = 0��Ǳ

∫V

[1

2ρ0u

2 +c2s2ρ0

(δρ)2]d3r.

� ��Ǳ��ǱE�

E =1

2ρ0u

2 +c2s2ρ0

(δρ)2 (5.48)

��Ǳ�� (x − cst)��

� δρ = ρ0u/cs��Ǳ

E = ρ0u2. (5.49)

��Ǳ

Q = ρ[u2/2 + h

]u ≈ ρhu = (h0ρ+ δP )u,

�� h0ρu��Ǳ��Ǳ��

�� h0(δρ)��Ǳ

Q = δPu. (5.50)

�� δP = csρ0u��Ǳ

Q = csρ0u2n = csEn. (5.51)

5.4 ��—��

��

��(��)�

144 ��

ρ2

ρ1

g∇ρ

� 5.3: ��—��

��

��—��—��

��

��(��)��5.3��Æ��

��Ǳ��

∂ρ

∂t+∇ · (ρu) = 0, (5.52a)

ρ∂u

∂t+ ρ(u · ∇)u = −∇P + ρg, (5.52b)

�� gǱ��Ǳ��(5.52a)�(5.52b)��

∂δρ

∂t+∇ · (ρ0u) = 0, (5.53a)

ρ0∂u

∂t= −∇δP + δρg, (5.53b)

�� δρ� δP ��

Ǳ��Ǳ�

dρ

dt=∂ρ

∂t+ u · ∇ρ = 0. (5.54)

��(5.54)Ǳ��

∂δρ

∂t+ u · ∇ρ0 = 0. (5.55)

5.4 ��—�� 145

��(5.55) ��(5.53a)��

∇ · u = 0. (5.56)

��

�� z−�� g = −gez�� ∂ρ0/∂x = ∂ρ0/∂y = 0�� ∂ρ0/∂z �= 0��

��(5.53b)�(5.55)�(5.56)��

ρ0∂ux∂t

= −∂δP∂x

, (5.57a)

ρ0∂uy∂t

= −∂δP∂y

, (5.57b)

ρ0∂uz∂t

= −∂δP∂z

− δρg, (5.57c)

∂δρ

∂t+ uz

∂ρ0∂z

= 0 (5.57d)

∂ux∂x

+∂uy∂y

+∂uz∂z

= 0. (5.57e)

��Ǳ��

(ux, uy, uz, δP, δρ) = [Ux(z), Uy(z), Uz(z), Q(z), R(z)]eikxx+ikyy−iωt, (5.58)

��(5.57)�Ǳ

−iωρ0Ux = −ikxQ, (5.59a)

−iωρ0Uy = −ikyQ, (5.59b)

−iωρ0Uz = −∂Q∂z

− Rg, (5.59c)

−iωR + Uz∂ρ0∂z

= 0, (5.59d)

ikxUx + ikyUy +∂Uz

∂z= 0. (5.59e)

��(5.59a)�(5.59b)�(5.59e)��Ux� Uy�

ωρ0∂Uz

∂z= −ik2Q,

�� k2 = k2x + k2y��(5.59c)�(5.59d)��R�

∂Q

∂z= iωρ0Uz + i

g

ωUz∂ρ0∂z

.

146 ��

��Q�� Uz ��

∂

∂z

(ρ0∂Uz

∂z

)= k2ρ0Uz

(1 +

g

ω2ρ0

∂ρ0∂z

). (5.60)

��∂ρ0/∂z = 0�

��(5.60)��Ǳ∂2Uz

∂z2= k2Uz. (5.61)

��

Uz = Ae+kz +Be−kz.

��Ǳ 0��

�� Uz ��

Uz =

{We+kz, z < 0,

We−kz, z > 0, (5.62)

��W = Uz(z = 0)��

�� z��Ǳ��

��(5.60)��∫ 0+

0− dz��

−kW (ρz>0 + ρz<0) =k2Wg

ω2(ρz>0 − ρz<0).

��

ω2 = −kg (ρz>0 − ρz<0)

(ρz>0 + ρz<0). (5.63)

��A��Ǳ��

A =(ρz>0 − ρz<0)

(ρz>0 + ρz<0). (5.64)

� ρz>0 > ρz<0�� ω2 < 0��

�� Ǳ

γ = |ω| =√kgA. (5.65)

5.5 ��

��

��

��

��

5.5 �� 147

shock front

D

shocked

ρ1,u1, T1

unshocked

ρ0,u0, T0

� 5.4: ��

5.5.1 ��

��

��Ǳ��

∂ρ

∂t+∂(ρu)

∂x= 0.

�� D ��

� ξ = x−Dt��Ǳ

d

dξ[ρ(u−D)] = 0.

��∫ D+0

D−0

d

dξ[ρ(u−D)] dξ = 0.

��

ρ1(u1 −D) = ρ0(u0 −D), (5.66)

�� 1�� 0��

��(5.66)��Ǳ

ρ1u1 = ρ0u0. (5.67)

��

P1 + ρ1u21 = P0 + ρ0u

20, (5.68)

��

(ρ1u21/2 + ρ1ε1 + P1)u1 = (ρ0u

20/2 + ρ0ε0 + P0)u0. (5.69)

148 ��

��(5.69)��(5.67)��

u21/2 + h1 = u20/2 + h0. (5.70)

�� h = ε+ P/ρ��

5.5.2 ��

��(5.67)�(5.68)�(5.70)�� V =

1/ρ��(5.67)�V0V1

=u0u1,

��(5.67)�(5.68)��

u20 = V 20

P1 − P0

V0 − V1,

u21 = V 21

P1 − P0

V0 − V1.

��Ǳ

u0 − u1 =√(P1 − P0)(V0 − V1).

��Ǳ

ε1(P1, V1)− ε0(P0, V0) =1

2(P0 + P1)(V0 − V1). (5.71)

��Ǳ

h1 − h0 =1

2(P1 − P0)(V0 + V1) (5.72)

��Ǳ��

P1 = H(V1, P0, V0). (5.73)

5.5.3 ��

��Ǳ��

ε = cV T =1

γ − 1PV, h = cpT =

γ

γ − 1PV.

5.5 �� 149

V

P� 5.5: ��Ǳ��

��

P1

P0

=(γ + 1)V0 − (γ − 1)V1(γ + 1)V1 − (γ − 1)V0

,

V1V0

=(γ − 1)P1 + (γ + 1)P0

(γ + 1)P1 + (γ − 1)P0,

T1T0

=P1V1P0V0

,

u20 =V02[(γ − 1)P0 + (γ + 1)P1],

u21 =V02

[(γ + 1)P0 + (γ − 1)P1]2

[(γ − 1)P0 + (γ + 1)P1].

�� P1/P0 → ∞��

ρ1ρ0

=V0V1

=γ + 1

γ − 1,

�� γ = 5/3��Ǳ4��

��

�� Pρ−γ =const��Ǳ��

��Ǳ(u0c0

)2

=(γ − 1) + (γ + 1)P1/P0

2γ> 1,

(u1c1

)2

=(γ − 1) + (γ + 1)P0/P1

2γ< 1.

150 ��

�� u0��

��

P1/P0 ≈ 1�� u0 ≈ u1 ≈ c0 ≈ c1��

��

��

S1 − S0 = cV lnP1V

γ1

P0Vγ0

= cV ln

{P1

P0

[(γ − 1)(P1/P0) + (γ + 1)

(γ + 1)(P1/P0) + (γ − 1)

]γ}.

�� S1 ≈ S0��

S1/S0 = cV ln

[P1

P0

(γ − 1

γ + 1

)γ]→ ∞.

��

��Ǳ��

��Ǳ��

��

5.6 ��

��

��

��

fα =nα(r, t)

(2π)3/2v3Tα(r, t)

exp

{−m[v − uα(r, t)]

2

2Tα

}, (α = e, i).

��

��

��Æ�

��Ǳ

τe =3m

1/2e T

3/2e

4√2πZ2e4ni ln Λ

, (5.74a)

τi =3m

1/2i T

3/2i

4√πZ4e4ni ln Λ

, (5.74b)

5.6 �� 151

�� ln Λ��Ǳ

λe = veτe =3T 2

e

4√2πZ2e4ni ln Λ

,

λi = viτi =3T 2

i

4√πZ4e4ni ln Λ

.

��

L� max(λe, λi), Δτ � max(τe, τi). (5.75)

5.6.1 ��

��

��∂nα

∂t+∇ · (nαuα) = 0, (α = e, i). (5.76)

�� nα��uα��

��Ǳ

∂nα

∂t+∇ · (nαuα) = Sα,

�� Sα�� /��/��Ǳ�

5.6.2 ��

��Ǳ��

��Ǳ��

��Ǳ

mαnα

(∂uα

∂t+ uα · ∇uα

)= Zαenα

(E+

1

cuα ×B

)−∇Pα −∇ · πα +Rα, (5.77)

�� πα��(��—��—��)�� Rα��

��

��

|ωcατα| � 1,

152 ��

�� ωcα��

�� z−��

πzz = −η0Wzz,

πxx = −1

2η0(Wxx +Wyy)−

1

2η1(Wxx −Wyy)− η3Wxy,

πyy = −1

2η0(Wxx +Wyy)−

1

2η1(Wxx −Wyy) + η3Wxy,

πxy = πyx = −η1Wxy +1

2η3(Wxx −Wyy),

πxz = πzx = −η2Wxz − η4Wyz,

πyz = πzy = −η2Wyz + η4Wxz.

��

Wik =∂ui∂xk

+∂uk∂xi

− 2

3δik∇ · u.

��Ǳ

ηi0 = 0.96niTiτi,

ηi1 =3

10

niTiω2ciτi

, ηi2 = 4ηi1,

ηi1 =1

2

niTiωci

, ηi4 = 2ηi3.

��(Z = 1)Ǳ

ηe0 = 0.73neTeτe,

ηe1 = 0.51neTeω2ceτe

, ηe2 = 4ηe1,

ηe1 = −1

2

neTe|ωce|

, ηe4 = 2ηe3.

��

Re +Ri = 0.

��Re �� u = ue − ui ��

��

Re= Ru +RT ,

5.6 �� 153

�� Z = 1��

Ru = −mene

τe(0.51u� + u⊥) = ene

(j�σ�

+j⊥σ⊥

),

RT = −0.71ne∇�Te −3

2

ne

|ωce|τeb×∇Te.

�� σ�� σ⊥�� j��

��j⊥�� b��

5.6.3 ��

��

�� Ǳ

3

2nα

(∂Tα∂t

+ uα · ∇Tα)

= −Pα∇ · uα −∇ · qα − παik∂uαi∂xk

+Qα, (5.78)

��Ǳ

qe = qeu + qe

T ,

��

qeu = neTe

(0.71u� +

3

2

b× u

|ωce|τe

),

qeT =

neTeτeme

(−3.16∇�Te −

4.66

ω2ceτ

2e

∇⊥Te −5

2|ωce|τeb×∇Te

).

��Ǳ

qi =nTiτimi

(−3.9∇�Ti −

2

ω2ciτ

2i

∇⊥Ti −5

2ωciτib×∇Ti

).

��

Qi =3me

mi

n

τe(Te − Ti) ,

��

Qe = −Re · u−Qi =j2�

σ�+j2⊥σ⊥

+1

enej ·RT − 3me

mi

n

τe(Te − Ti) .

�� Ǳ�Ǳ Braginskii�� S. I.Braginskii��

154 ��

5.6.4 ��

�� Ǳ��

��

��

��

∇×B =4π

cj,

��

∇× E = −1

c

∂B

∂t.

��

��

Zni ≈ ne.

��

��

E+1

cui ×B =

1

c

1

enej×B− ∇Pe

ene− ∇ · πe

ene+

Re

ene. (5.79)

��Ǳ��

5.6.5 ��

��

��

��

∂nα

∂t+∇ · (nαuα) = 0,

nαmα

(∂uα

∂t+ uα · ∇uα

)= eαnα

(E+

1

cuα ×B

)−∇Pα.

�� Ǳ��

��

Pαn−γα =�.

5.7 �� 155

�� γ = 3�� γ = 1��

��

∇×B =1

c

∂E

∂t+

4π

c

∑α

eαnαuα,

∇×E = −1

c

∂B

∂t,

∇ ·B = 0,

∇ · E = 4π∑α

eαnα.

5.7 ��

��

��

��

��

��

��

ρ = mini +mene ≈ mini,

ρu = miniui +meneue ≈ miniui.

��u = ui��

��

∂

∂t(mini) +∇ · (miniui) = 0.

�∂ρ

∂t+∇ · (ρu) = 0.

��

ρ

(∂u

∂t+ u · ∇u

)= (Zni − ne)E+

1

c(Zeniui − eneue)×B−∇(Pe + Pi).

�� Zni ≈ ne��

�� Ǳ��

��

j = Zeniui − eneue

156 ��

��

ρ

(∂u

∂t+ u · ∇u

)= −∇P +

1

cj×B,

�� P = Pe + Pi��

�� Ǳ

∇×B =4π

cj,

∇× E = −1

c

∂B

∂t.

��

j = σ

(E+

1

cu×B

).

��

E = −1

cu×B.

��

∂ρ

∂t+∇ · (ρu) = 0,

ρ

(∂u

∂t+ u · ∇u

)= −∇P +

1

cj×B,

∇×B =4π

cj,

∇× E = −1

c

∂B

∂t,

E = −1

cu×B.

5.8 ��

1. ��(5.5)��(5.7) ��(5.8)�

2. ��(5.5)��(5.7)��(5.11)��(5.14)��

5.8 �� 157

3. ��Ǳ� ��

4. �� Ǳ��ǱP = nT + aT 4/4��

Ǳs = (1/m) ln(T 3/2/n) + aT 3/nm��TǱ��nǱ��

a = 4π2/45�3c3��

5. � ��ǱP0��Ǳρ0��

ǱUs� ��ǱUp��

��

��

��

��

��

6.1 ��

��Ǳ

∂B

∂t= ∇× (u×B) . (6.1)

�� C��

��6.1��C ��Ǳ

Φ =

∫S

B · df ,

�� S �� C ��Φ��

��Ǳ��

��

��ǱdΦ

dt=

∫S

∂B

∂t· df +

∮B · (u× dl).

�� C ��Ǳ�� S ��(6.1)��

dΦ

dt=

∫S

[∂B

∂t−∇× (u×B)

]· df = 0.

��

��

��

6.1 �� 159

B

u(r, t)

dl

S

C

� 6.1: ��C ��

��

∇× (a× b) = (b · ∇) a+ (∇ · b) a− (a · ∇)b− (∇ · a)b,

��(6.1)��Ǳ∂B

∂t= (B · ∇)u− (u · ∇)B− (∇ · u)B.

��d

dt=

∂

∂t+ u · ∇,

��Ǳ

dB

dt= (B · ∇)u+ (∇ · u)B. (6.2)

��Ǳ

∇ · u = −1

ρ

dρ

dt. (6.3)

�(6.3)��(6.2)��

d

dt

(B

ρ

)=

(B

ρ· ∇)u (6.4)

�� δl��

��Ǳ t��Ǳ u��

��Ǳu+ δl · ∇u��6.2�� dt��Ǳ dt(δl · ∇)u��

δl��Ǳdδl

dt= (δl · ∇)u. (6.5)

160 ��

u + δl · ∇u

u

Bδl

� 6.2: ��δl��Ǳ��Ǳ��

��(6.4)�� δl�� B/ρ��Ǳ δl�

B/ρ��ǱǱ��

Ǳ�Ǳ��Ǳ��

��Ǳ��Ǳ��B/ρ��

��

��

��

�� δl� B/ρ��

��6.2��Ǳ dS0��Ǳl0��

��Ǳ��

� B0dS0 = BdS��Ǳ��Ǳ��

�� ρ0l0dS0 = ρldS��

B0

ρ0l0=B

ρl. (6.6)

�� ρ =const��(6.6)�Ǳ

B = (B0/l0)l, (6.7)

��

6.2 �� 161

� 6.3: ��

6.2 ��

��

��

��

��Ǳ��

��

��

��Ǳ��

6.2.1 ��

��Ǳ

1

cj×B−∇P = 0, (6.8)

��

B · ∇P = 0, j · ∇P = 0, (6.9)

��Ǳ�� j��

� B��

��

��

P (x, y, z) = const.

��

162 ��

��(6.8)��B��

j⊥ =cB×∇P

B2. (6.10)

��

��

��

��Ǳ−∇P/(ne +

ni)�� ne,iǱ��

�� Ǳ

ve,iD = − c∇P ×B

(ne + ni)ee,iB2,

�� ee,i��Ǳ��

jD =∑e,i

ne,iee,ive,iD = −c∇P ×B

B2. (6.11)

��

��

��

� ��

j =c

4π∇×B = αB,

�� α�� Ǳ��

��

��

6.2.2 ��

�� u = 0��(??)��

∂

∂xi

[P +

(B2

8πδik −

1

4πBiBk

)]= 0. (6.12)

(6.12)��

��Ǳ��Ǳ��

6.2 �� 163

��Ǳ��

�(��)��

β =P

P +B2/8π. (6.13)

�� β ��Ǳ��

��

��

� �� Ǳ� ��

��Ǳ V ��Ǳ

F =1

4π

∫V

∇ · (BB) dV =1

4π

∮S

B (B · df) ,

�� S ��Ǳ�� V Ǳ��

��6.4��

1

4π

∮S

B (B · df) = 1

4π[B2 (B2S2)−B1 (B1S1)] ,

�� B1,2� S1,2�� V ��Ǳ��

��B2S2 = B1S1 = Φ��

�� Ǳ

Fc =Φ

4π(B2 −B1) .

�� Ǳ��6.4��

��Ǳ��

��

6.2.3 ��

��(6.12)�� (6.12)��Ǳ

∂Πik/∂xi = 0, (6.14)

�� Πik ��Ǳ��

Πik = Pδik +1

8πB2δik −

1

4πBiBk. (6.15)

164 ��

B1Φ

B2Φ

B2

B1

dS1 dS2

� 6.4: ��

��(6.14)��

∂

∂xi(Πikxk) = Πii.

�� S �� V ��∫V

ΠiidV =

∫V

∂

∂xi(Πikxk) dV =

∮S

Πikxkdfi. (6.16)

�(6.15)��(6.16)��∫V

(3P +

1

8πB2

)dV =

∮S

[(P +

1

8πB2

)r− 1

4π(B · r)B

]· df . (6.17)

��

�� P = 0��

�� P = 0�� 1/r3��(6.17)��Ǳ��

��Ǳ��

��

��(6.17)��(6.17)��

6.2.4 θ−Æ��

�� Ǳ��

�� θ−Æ�� z−Æ��Æ��

�� r�� (er, eθ, ez)�

��Ǳ4π

cj = −dBz

dreθ +

1

r

d(rBθ)

drez. (6.18)

6.2 �� 165

B(r)

j

� 6.5: θ−Æ��

�(6.18)��(6.8)��Ǳ

d

dr

[P +

B2θ +B2

z

8π

]= − B2

θ

4πr. (6.19)

�� θ−Æ��Æ��Ǳ�Bθ = 0��Bz = B��

��θ−Æ��P (r) +

B2(r)

8π=B2

0

8π. (6.20)

�� B0��

��(6.20)��

��

��Ǳ

μα = −W⊥α

Bb,

��W⊥α� α�� b��MǱ

M =∑

naμα = −b

B

∑α

naW⊥α.

��

M = −b

B

∑α

nαT = −b

BP,

166 ��

�� T ��P =∑

α nαT��

��Ǳ

Bs = 4πM = −4πP

Bb. (6.21)

��

��(6.21)�� β ��ǱB0��ǱBd��

�

P +(B0 +Bd)

2

8π=B2

0

8π,

��Bd��

Bd

(1− 1

2

Bd

B0

)=

4πP

B0

.

�� β��Bd/B0 � 1�� Bd ≈ 4πP/B0 = Bs��

��

��

js = c∇×M = −c∇×(P

Bb

)= cb×∇

(P

B

), (6.22)

�Ǳ�� z ��∇× b = 0��

�� θ−Æ��

Vα∇B =

cW⊥α

eαB2b×∇B,

�� Ǳ

jd =∑α

eαnaVα∇B =

c

B2b×∇B

∑α

nαW⊥α.

��

jd =cP

B2b×∇B. (6.23)

�(6.22)�(6.23)��

j = cb× ∇PB

=cB×∇P

B2.

��Ǳ��

6.2 �� 167

j(r)

B

� 6.6: z−Æ��

6.2.5 Z−Æ��

��Z−Æ��Ǳ�Bz = 0��Æ��Bθ = B��6.6�� (6.19)��Ǳ

dP (r)

dr= − B

4πr

d(rB)

dr. (6.24)

�� r2�� r� r = 0�� r = a��

(r2P )r=a − 2

∫ a

0

rPdr = − 1

8π(rB)2r=a.

�� Ǳ��

2

∫ a

0

rPdr =1

8π(rB)2r=a. (6.25)

��(6.18)��

4π

cj =

1

r

d

dr(rB),

�� r�� r� r = 0��r = a��

(rB)r=a =4π

c

∫ a

0

rj(r)dr. (6.26)

��(6.26)��

I0 = 2π

∫ a

0

rj(r)dr. (6.27)

168 ��

��(6.25)�(6.26)�(6.27)��

2

∫ a

0

rPdr =I20

2πc2. (6.28)

��

〈P 〉 = 1

πa2

∫ a

0

2πrPdr,

��(6.28)��ǱI20 = 2πa2c2〈P 〉.

��Ǳ� Z−Æ��

��

P = ne(r)T + ni(r)T = ne(r)T (1 + 1/Z),

�� Z �� ne,i��

I20 = 4πc2T (1 + 1/Z)

∫ a

0

ne(r)rdr = 2c2T (1 + 1/Z)Ne, (6.29)

�� Ne = 2π∫ a

0nerdr��

6.2.6 Grad-Shafrnov��

��

�Æ�� (R, φ, z)��Æ�� Ǳ

Ψ(R, z) =

∫ R

0

2πR′Bz(R′, z)dR′, (6.30)

�� Bz �� z��Bz ��Ǳ

Bz =1

2πR

∂Ψ

∂R. (6.31)

BRǱ�� Ψ�� ∇ ·B = 0��

1

R

∂

∂R(RBR) +

∂Bz

∂z= 0.

��(6.31)��

BR = − 1

2πR

∂Ψ

∂z. (6.32)

6.2 �� 169

��(6.31)�(6.32)��

B · ∇Ψ = 0. (6.33)

��Ψ =��Ǳ��Ψ��Æ��

�� Ψ�� Ǳ

I(R, z) =

∫ R

0

2πR′jz(R′, z)dR′. (6.34)

��

jz =1

2πR

∂I

∂R. (6.35)

��∇ · j = 0��

��(6.35)��Ǳ�� jR��Ǳ I � ��

jR = − 1

2πR

∂I

∂z. (6.36)

�� I =��Ǳ

j · ∇I = 0. (6.37)

��Ǳ��Ǳ

jR = − c

4π

∂Bφ

∂z, (6.38a)

jφ =c

4π

(∂BR

∂z− ∂Bz

∂R

)(6.38b)

jz =c

4πR

∂

∂R(RBφ) . (6.38c)

�(6.31)�(6.32)��(6.38b)��ǱΨ� ��

jφ = − c

8π2R

[∂2Ψ

∂R2− 1

R

∂Ψ

∂R+∂2Ψ

∂z2

]. (6.39)

��(6.38c)�� 2πR�� R��Bφ ��Ǳ I �

��

Bφ =2I

cR. (6.40)

��ǱΨ� I � ��

jR = − 1

2πR

∂I

∂z, jφ = − c

8π2RΔΨ, jz =

1

2πR

∂I

∂R, (6.41)

170 ��

BR = − 1

2πR

∂Ψ

∂z, Bφ =

2I

cR, Bz =

1

2πR

∂Ψ

∂R, (6.42)

��

Δ∗Ψ ≡ ∂2Ψ

∂R2− 1

R

∂Ψ

∂R+∂2Ψ

∂z2.

�� B�� j��

B · ∇P = 0, j · ∇P = 0.

�(6.35)�(6.36)�� j · ∇P = 0��

jR∂P

∂R+ jz

∂P

∂z=

1

2πR

(−∂I∂z

∂P

∂R+∂I

∂R

∂P

∂z

)= 0.

��

∇I ×∇P = eφ

(∂I

∂z

∂P

∂R− ∂I

∂R

∂P

∂z

)= 0.

��∇Ψ×∇P = 0��

∇Ψ ‖ ∇P, ∇I ‖ ∇P. (6.43)

��Ǳ�� P� Ψ� I

��Æ�� P ��

� � I ��ǱÆ�� Ψ� ��

P = P (Ψ), I = I(Ψ). (6.44)

��(6.8)��Ǳ

jφBz − jzBφ = c∂P

∂R, (6.45a)

jRBφ − jφBR = c∂P

∂z, (6.45b)

jzBR = jRBz. (6.45c)

�Ǳ�� B�� j�� P ��Ǳ Ψ� ��(6.45)��Ǳ��Æ�� Ψ��(6.39)�(6.40)��(6.45a)�� I

� P �� Ψ� ��

Δ∗Ψ+8π2

c2dI2

dΨ+ 16π3R2dP

dΨ= 0. (6.46)

6.3 �� 171

�� Grad-Shafranov�� jφBz�� Æ��

�� jzBφ��Æ��

��

Grad-Shafranov �� Ψ �� Grad-Shafranov �� Grad-Shafranov ��I � P �� Ψ� ��6.7��

� 6.7: ��

6.3 ��

��

��

��

��Ǳ

��

��

��

6.3.1 ��

��

172 ��

��

��

��

��

∂ρ1∂t

= −∇ · (ρ0u), (6.47a)

ρ0∂u

∂t= −∇P1 +

1

4π[(∇×B0)×B1 + (∇×B1)×B0] , (6.47b)

∂B1

∂t= ∇× (u×B0), (6.47c)

∂P1

∂t= −u · ∇P0 − γP0∇ · u, (6.47d)

�� γ = cp/cv �� r0��

�� Ǳ ξ(r0, t)��

��Ǳ

u(r, t) =∂

∂tξ(r0, t), (6.48)

�� r = r0 + ξ(r0, t)��(6.48)��(6.47a)��(6.47c)��(6.47d)��Ǳ � ξ� ��

ρ1 = −∇ · (ρ0ξ), (6.49a)

B1 = ∇× (ξ ×B0), (6.49b)

P1 = −ξ · ∇P0 − γP0∇ · ξ. (6.49c)

��(6.49)��(6.47b)�� ξ��

ρ0∂2ξ

∂t2= ∇ (ξ · ∇P0 + γP0∇ · ξ) + 1

4π(∇×B0)× [∇× (ξ ×B0)]

+1

4π{∇ × [∇× (ξ ×B0)]} ×B0. (6.50)

��

��(6.50) ��

�� S0Ǳ��

� ��

6.3 �� 173

� S0Ǳ ��B0 · ∇P0 = 0��

�Ǳ�� S0��Ǳ n0��(6.12)��n0��

n0 · ∇(P0 +

B20

8π

)=

1

4πn0· (B0 · ∇)B0. (6.51)

�� (B0 · ∇)B0��(6.51)��(6.51)��

P0 +B2

i0

8π=B2

e0

8π,

�� Bi0� Be0��

��

��

S1�Ǳ��

P0 + P1 +1

8π(Bi0 +Bi1)

2 =1

8π(Be0 +Be1)

2 . (6.52)

��

r = r0 + ξnn0,

�� r0�� S0�� n0�� ξn = ξ · n0��

��

P0(r) = P0(r0) + (ξ · ∇)P0(r0),

P1(r) = −(ξ · ∇)P0(r0)− γP0(r0)∇ · ξ,

[Bi0(r) +Bi1(r)]2 = B2

i0(r0) + (ξ · ∇)B2i0(r0) + 2Bi0(r0) ·Bi1(r0)

[Be0(r) +Be1(r)]2 = B2

e0(r0) + (ξ · ∇)B2e0(r0) + 2Be0(r0) ·Be1(r0)

�� (6.52)��

−γP0∇ · ξ +1

4πBi0 ·Bi1 +

1

8πξn∂B2

i0

∂n=

1

4πBe0 ·Be1 +

1

8πξn∂B2

e0

∂n, (6.53)

�� S0��

��

��

n ·Be = n ·Bi,

174 ��

�� S ��

d

dt

∫S

Be · df =d

dt

∫S

Bi · df .

��Ǳ��

∫S

[∂Be

∂t−∇× (u×Be)

]· df = 0.

� S ��

n · ∂Be

∂t= n · ∇ × (u×Be).

�� Ǳ

n0 ·Be1 = n0 · ∇ × (ξ ×Be0) (6.54)

��

��

∇×Be1 = 0. (6.55)

�� ψ��

Be1 = ∇ψ. (6.56)

��(6.55)��Ǳ∇ ·B1e = 0��

∇2ψ = 0. (6.57)

�� (6.54)��

n0 · ∇ψ = n0 · ∇ × (ξ ×Be0). (6.58)

��

n0 · ∇ψ = 0. (6.59)

6.3 �� 175

��

��Ǳ��

��Ǳ

exp(−iωt)�� ξ��

ξ(r, t) =∑n

ξn(r)e−iωnt.

��

−ω2nρ0ξn(r) = F[ξn(r)].

��Ǳ�� Æ�� ωn ��Æ�� ξn ��

ωn��Æ��Æ� ω2n > 0��

��Æ� ω2n < 0��

6.3.2 ��

��

��Ǳ a��Ǳ

B0i = (0, 0, B0), B0e = (0, Bφ(r), Bz),

�� B0 � Bz �� Bφ(r)��Æ��

��Ǳ I��Æ��Ǳ

Bφ(r) =2I

cr.

��

�Ǳ P0�Ǳ��Ǳ��

��(6.47b) ��Ǳ��∇×B0 = 0��

ρ0∂2ξ

∂t2= −∇P1 +

1

4π(∇×B1i)×B0i. (6.60)

Ǳ��∇ · ξ = 0��Ǳ

P1 = −(ξ · ∇)P0 − γP0∇ · ξ = 0,

176 ��

��Ǳ

B1i = ∇× (ξ ×B0i) = (Bi0 · ∇)ξ = B0∂ξ

∂z.

�� ξ��

ξ(r, t) = ξ(r) exp (imφ+ ikz − iωt) , (6.61)

��Ǳ

Bi1 = ikB0ξ.

��(6.60)�Ǳ

−ω2ρ0ξ =ikB0

4π(∇× ξ)×B0i. (6.62)

��(6.62)��

B0i · ∇2ξ = 0, (6.63)

��

∇2ξz = 0. (6.64)

�� (6.61)��(6.64)��[d2

dr2+

1

r

d

dr−(k2 +

m2

r2

)]ξz(r) = 0, (6.65)

��(6.65)�� 1��

� Im(x) � Km(x) �� Im(x) � x = 0 �� x 1��

Im(x) ∼ exp(x)/√2πx�Km(x)� x = 0��

K0(x) ∼ ln2

γx, Km(x) ∼

(n− 1)!

2

(x2

)−m

,

x 1��Km(x) ∼√π/2x exp(−x)��

��(6.65)��Ǳ

ξz(r) = ξz(a)Im(kr)

Im(ka). (6.66)

1��Ǳ [d2

dr2+

1r

d

dr+(

1 − m2

r2

)]f(r) = 0.

6.3 �� 177

��(6.62)��Ǳ

−ω2ρ0ξr = ikB2

0

4π(∇× ξ)φ. (6.67)

��

(∇× ξ)φ =∂ξr∂z

− ∂ξz∂r

= ikξr − kξz(a)I ′m(kr)Im(ka)

,

��(6.67)��

ξr(r) = − ik2(B20/4π)ξz(a)

k2(B20/4π)− ρ0ω2

I ′m(kr)Im(ka)

. (6.68)

�� ψǱ��

ψ(r, t) = ψ(r) exp (imφ + ikz − iωt) .

�� ψ(r)�� ξz(r)��

[d2

dr2+

1

r

d

dr−(k2 +

m2

r2

)]ψ(r) = 0.

��

ψ(r) = ψ(a)Km(kr)

Km(ka). (6.69)

�� Ǳ��

�� (6.53)��

1

4πBi0(a) ·Bi1(a) =

1

4πBe0(a) ·Be1(a) +

1

8πξr(a)

∂B20e

∂r

∣∣∣∣r=a

.

��∝ 1/r��

∂

∂rB2

e0(r)

∣∣∣∣r=a

= −2B2

φ(a)

a.

�

Be0(a) ·Be1(a) = [Bez +Bφ] · ∇ψ = i[kBz +

m

aBφ

]ψ(a),

Bi0(a) ·Bi1(a) = ikB20ξz(a),

��ikB2

0

4πξz(a) =

i

4π

[kBz +

m

aBφ(a)

]ψ(a)− 1

4πaξr(a)B

2φ(a). (6.70)

178 ��

�� (6.58)� ��

n0 ·Be1 = er · ∇ψ = kψ(a)K ′

m(ka)

Km(ka),

n0 · [∇× (ξ ×Be0)] = er · ∇ × [(ξφBz − ξzBφ)er − ξrBzeφ + ξrBφez]

= i(kBz +

m

aBφ

)ξr(a).

��

i(kBz +

m

aBφ

)ξr(a) = kψ(a)

K ′m(ka)

Km(ka)(6.71)

�� (6.70)�(6.71)�� ξz(a)� ψ(a)��

��Ǳ��

4πρ0ω2

k2= B2

0 −[Bz +

m

kaBφ(a)

]2 I ′m(ka)Km(ka)

Im(ka)K ′m(ka)

−B2φ(a)

I ′m(ka)(ka)Im(ka)

. (6.72)

�� ω2 > 0��

��(6.72)��Ǳ��(�Ǳ I ′m/Im > 0�K ′

m/Km < 0)�Ǳ�� k · B0e = kBz + (m/a)Bφ = 0��Ǳ��Ǳ

��Ǳ��

��

��

��(6.72)��Bz = 0�m = 0��(6.72)�Ǳ

ω2 =B2

0k2

4πρ0

[1−

B2φ(a)

B20

I ′0(ka)(ka)I0(ka)

]. (6.73)

��

I ′0(x)xI0(x)

� 1/2,

��(6.54)��

ω2 � B20k

2

4πρ0

[1−

B2φ(a)

2B20

].

��

B20 >

1

2B2

φ(a) =2I2

c2a2, (6.74)

6.3 �� 179

� 6.8: ��

��6.8� ��

��Bz = 0�m = 1��

Kn−1(x)−Kn+1(x) = −2n

xKn(x),

Kn−1(x) +Kn+1(x) = −2K ′n(x),

��(6.72)�Ǳ

ω2 =B2

0k2

4πρ0

[1 +

B2φ

B20

I ′1(ka)K0(ka)

(ka)I1(ka)K ′1(ka)

]. (6.75)

��K0(ka)/K1(ka) < 0�� ka→ 0��

�� I1(x) ≈ (x/2)�K0(x) ≈ ln(2/γx)�K1(x) ≈ 1/x��

ω2 =B2

0k2

4πρ0

(1−

B2φ

B20

ln2

γka

). (6.76)

��Ǳ��Ǳ��

� m = 1��

��Ǳ��

��

180 ��

� 6.9: ��(kink instability).

�� Bz �= 0�� ka � 1��

�� I ′m(x)/Im(x) ≈ m/x�K ′m(x)/Km(x) ≈ −m/x��(6.72)��

4πρ0ω2 = k2B2

0 +(kBz +

m

aBφ

)2− m

a2B2

φ. (6.77)

��

ω2 � ω2min =

B2φ

4πρ0a2

(m2B2

0

B2z +B2

0

−m

). (6.78)

��Ǳ ka = −mBzBφ/(B2z + B2

0)�� ω2�� ω2min��

��

P0 +B2

0

8π=B2

z +B2φ(a)

8π,

��Æ��Bz Bφ(a)��

β = P0/[(B2z +B2

φ)/8π] ≈ P0/(B2z/8π)�� ω2

min��Ǳ��

ω2min =

B2φ

4πρ0a2m

(1− β

2− βm− 1

), (6.79)

�� β �� m = 1�m = 2��

�� Bz/Bφ��m = 1�m = 2��

��

�� L��Ǳ kmin = 2π/L��m = 1��

6.3 �� 181

��(6.77)Ǳ

4πρ0ω2 = k2min(B

20 +B2

z ) + 2kmin

aBzBφ.

��Æ�� Bz Bφ�� β ��

�� B0 ≈ Bz��Ǳ�� ω2 > 0� ��

Bφ

Bz

< kmina =2πa

L. (6.80)

�� Kruskal-Shafranov��Æ��

6.3.3 ��—��

��—��Ǳ

B0(z) = (Bx(z), By(z), 0),

��Ǳ

g = (0, 0,−g).

�� x− y��

u(r, t) = u(z) exp(ikxx+ ikyy − iωt).

��

∇ · u = ik · u+duzdz

= 0,

�� k = (kx, ky, 0)�

��Ǳ

ρ0∂u

∂t= −∇

(P1 +

1

4πB0 ·B1

)+

1

4π(B0 · ∇B1 +B1 · ∇B0) + ρ1g, (6.81)

��Ǳ

∂P1

∂t= −u · ∇P0 = −uzP ′

0,

∂B1

∂t= B0 · ∇u− u · ∇B0 = i(k ·B0)u− uzB

′0,

∂ρ1∂t

= −u · ∇ρ0 = −uzρ′0,

182 ��

��(6.81)��

−ρ0ω2u = k

[iuzP

′0 +

1

4π(k ·B0)(B0 · u) +

i

4πuz(B0 ·B′

0)

]− 1

4π(k ·B0)

2u

+ ez

[(uzP

′0)

′ − i

4π[(k ·B0)(B0 · u)]′ +

1

4π[uz(B0 ·B′

0)]′ + uzρ

′0g

](6.82)

�� k��

−[ρ0ω

2 − (k ·B0)2/4π

](k · u) = k2

[iuzP

′0 +

1

4π(k ·B0)(B0 · u) +

i

4πuz(B0 ·B′

0)

].

�� k · u��

−iuz[ρ0ω

2 − (k ·B0)2/4π

]= k2

[iuzP

′0 +

1

4π(k ·B0)(B0 · u) +

i

4πuz(B0 ·B′

0)

],

��

1

4π(k ·B0)(B0 · u) = −iuz

k2[ρ0ω

2 − (k ·B0)2/4π

]− iuzP

′0 −

i

4πuz(B0 ·B′

0).

��(6.82)� z−�� uz ��

d

dz

[(ρ0ω

2 − (k ·B0)2

4π

)duzdz

]− k2

(ρ0ω

2 − (k ·B0)2

4π

)uz − k2g

dρ0dz

uz = 0. (6.83)

�� (6.83)�Ǳ

u′′z − k2uz = 0,

�� Ǳ

uz =

{uz(0)e

−kz, z > 0

uz(0)ekz, z < 0

. (6.84)

�(6.84)��(6.83)�� z� z = 0−� z = 0+��

−k[(ρ1 + ρ2)ω

2 − (k ·B0)2

2π

]uz(0)− k2g(ρ1 − ρ2)uz(0) = 0.

��

ω2 = −kg(ρ1 − ρ2)

ρ1 + ρ2+

(k ·B0)2

2π(ρ1 + ρ2). (6.85)

� k ·B0 �= 0��

6.4 �� 183

6.4 ��

��

��Ǳ

Q = Q0 +Q1ei(k·r−ωt),

�� Q1 �� Q0�� k� ω��

��Ǳ��Ǳ��Ǳ�

��

Q1/Q0 � 1.

��Ǳ��

��

∂ρ1∂t

= −ρ0∇ · u, (6.86a)

ρ0∂u1

∂t= −∇P1 +

1

cj1 ×B0, (6.86b)

P1ρ0 = γP0ρ1, (6.86c)∂B1

∂t= ∇× (u×B0), (6.86d)

��∂

∂t= −iω, ∇ = ik,

��Ǳ��

ωρ1 = ρ0k · u, (6.87a)

ωρ0u1 = kP1 +i

cj1 ×B0, (6.87b)

P1ρ0 = γP0ρ1, (6.87c)

ωB1 = −k× (u×B0) (6.87d)

��Ǳ

j1 = ic

4πk×B1, (6.88)

��(6.87c)��(6.87b)��

ωu1 = kc2s(ρ1/ρ0) +1

4πρ0B0 × (k×B1),

184 ��

z

y

x k

B0θ

� 6.10: ��

�� c2s = (∂P/∂ρ)s = γP0/ρ0��

��

ωB1 = −k× (u×B0), (6.89a)

ωρ1 = ρ0k · u, (6.89b)

ωu = kc2s(ρ1/ρ0) +B0 × (k×B1)/4πρ0. (6.89c)

�(6.89a)��(6.10)��Ǳ x−�� k�� B0��Ǳ x− y��(6.89a)��Ǳ

ωB1y = −kB0xuy + kB0yux,

ωB1z = −(k ·B0)uz.

�(6.89b)��(6.89c)��

ux = k2c2sux/ω2 − kB0yB1y/4πωρ0,

uy = −kB0xB1y/4πωρ0,

uz = −(k ·B0)B1z

4πωρ0.

��Ǳ��

uz = −(k ·B0)B1z

4πωρ0, B1z = − 1

ω(k ·B0)uz; (6.90)

6.4 �� 185

ωB1y = −kB0xuy + kB0yux, (6.91a)

ux = k2c2sux/ω2 + kB0yB1y/4πωρ0, (6.91b)

uy = −kB0xB1y/4πωρ0. (6.91c)

��(6.90)��

ω2 =(k ·B0)

2

4πρ0. (6.92)

�� k��ÆǱ θ��

ω2 = k2v2A cos2 θ, (6.93)

�� vA =√B2

0/4πρ0 �� E = u× B/c� �

��Ǳ�

E =1

cuzB0 (sin θex − cos θey) .

�� Ǳ��

��

��

�� ρ1 = 0� ��

Ǳ��

−∇B2

8π≈ − 1

8π∇(B2

0 + 2B0 ·B1) = 0.

��

1

4πB · ∇B ≈ 1

4π

(B0x

∂

∂x+B0y

∂

∂y

)Bz1ez =

i

4πkB0xB1z.

�� θ = 0��

ω2 = k2v2A.

�� vA =√B0/4πρ0 �� Ǳ B2

0/4π�

��Ǳ ρ0�� θ = 0�� y−��

� z−�� y−��

186 ��

B = B0ex + B1ez

uz = −cEy/B0

� 6.11: �� Ǳ��

�� y−�� z−��(6.91)��

ω2

k2=

B20x

4πρ0+

B20y

4πρ0(1− k2c2s/ω2). (6.94)

��

(ω/k)2f,s =1

2

{v2A + c2s ±

[(v2A + c2s

)2 − 4c2sv2A cos2 θ

]1/2}. (6.95)

��Ǳ��

��Ǳ��Ǳ��

��Ǳ

− 1

8π∇B2 = − 1

4π∇(B0 ·B1) = − ik

4πB0yB1yex,

� �Ǳ

1

4πB · ∇B =

1

4π

(B0x

∂

∂x+B0y

∂

∂y

)(B1yey)

=ik

4πB0xB1yey.

��(θ = 0)��

(ω/k)2f = max(v2A, c2s), (ω/k)

2f = min(v2A, c

2s),

��Ǳ ��Ǳ ��Ǳ��

��

ux = k2c2sux/ω2,

uy = −kB0B1y/4πωρ0, ωB1y = −kB0uy,

6.5 �� 187

��Ǳ��

��

��Ǳ ��Ǳ ��

��(θ = π/2)��Ǳ

(ω/k)2f = v2A + c2s, (ω/k)2s = 0.

��Ǳ

ωB1y = kB0ux,

ux = k2c2sux/ω2 + kB0B1y/4πωρ0,

uy = 0.

��

��

6.5 ��

� ��Ǳ ��

��(��)�� Ǳ��

��

��

6.5.1 ��

��Ǳ

∂ρ

∂t= −∂(ρuk)

∂xk,

∂(ρui)

∂t= − ∂

∂xk(Pδik + ρuiuk + δikB

2/8π − BiBk/4π),

∂

∂t(ρu2/2 + ρε+ B2/8π) = − ∂

∂xk[(ρu2/2 + ρε+ P )uk + ukB

2/4π − Bk(B · u)/4π].

��

��

[ρun]1 = [ρun]0 = j, (6.96)

188 ��

[(u2/2 + ε+ P/ρ)ρun + un(B2 − B2

n)/4π −Bn(Bt · ut)/4π]1

= [(u2/2 + ε+ P/ρ)ρun + un(B2 −B2

n)/4π − Bn(Bt · ut)/4π]0, (6.97)

[P + ρu2n + (B2t − B2

n)/8π]1 = [P + ρu2n + (B2t − B2

n)/8π]0, (6.98)

[ρunut − BnBt/4π]1 = [ρunut − BnBt/4π]0 (6.99)

�� n �� t��

∇ ·B = 0��

[Bn]1 = [Bn]0 = Bn. (6.100)

��Ǳ��E = −u × B/c�

��

[Bnut −Btun]1 = [Bnut −Btun]0. (6.101)

��(6.96)�(6.100)��(6.99)��Ǳ

(Bt1 −Bt0) =4π

Bn

j(ut1 − ut0). (6.102)

��(6.101)��Ǳ(V1Bt1 − V0Bt0) = (Bn/j)(ut1 − ut0), (6.103)

�� V = 1/ρ��(6.102)�(6.103)��Bt1 −Bt0��

� V1Bt1 − V0Bt0�� Bt1�Bt0��B1�B0

� ��

��(6.102)�(6.103)��

j2(V1Bt1 − V0Bt0) =B2

n

4π(Bt1 − Bt0). (6.104)

��(6.97)��Ǳ

j(h1 − h0) +1

2j3(V 2

1 − V 20 ) +

1

2j(u2t1 − u2t0) +

j

4π(V1B

2t1 − V0B

2t0)

=Bn

4π(Bt1 · ut1 −Bt0 · ut0),

�� (j/2)(u2t1 − u2t0)− (B2n/32π

2j)(B2t1 − B2

t0)��

j(h1 − h0) +1

2j3(V 2

1 − V 20 ) +

j

4π(V1B

2t1 − V0B

2t0)−

B2n

32π2j(B2

t1 − B2t0)

= −1

2j

[(ut1 −

Bn

4πjBt1

)2

−(ut0 −

Bn

4πjBt0

)2].

6.5 �� 189

��(6.102) ��Ǳ��

(h1 − h0) +1

2j2(V 2

1 − V 20 ) +

1

4π(V1B

2t1 − V0B

2t0)−

B2n

32π2j2(B2

t1 − B2t0) = 0. (6.105)

��(6.98)��

j2 =P1 − P0 + (B2

t1 − B2t0)/8π

V0 − V1(6.106)

��(6.105)��

ε1 − ε0 =1

2(P1 + P0)(V0 − V1) +

1

16π(V0 − V1)(Bt1 −Bt0)

2. (6.107)

��

6.5.2 ��

��Ǳ��Bt = 0� ut1 = 0�

��Ǳ�

[ρu]1 = [ρu]0 = j,

[P + ρu2]1 = [P + ρu2]0,

[(u2/2 + ε+ p/ρ)ρu]1 = [(u2/2 + ε+ p/ρ)ρu]0,

��

6.5.3 ��

��Ǳ��Bn = 0� Bt1� Bt0

��Ǳ

[ρu]1 = [ρu]0 = j,

[(u2/2 + ε+ P/ρ)ρu+ uB2/4π]1 = [(u2/2 + ε+ P/ρ)ρu+ uB2/4π]0,

[P + ρu2 +B2/8π]1 = [P + ρu2 +B2/8π]0,

��Ǳ

[(u2/2 + ε+ P/ρ) +B2/4πρ]1 = [(u2/2 + ε+ P/ρ)ρu+B2/4πρ]0.

190 ��

�

r =ρ1ρ0,

��u1u0

=ρ0ρ1

=1

r,

��(6.101)��

[Bu]1 = [Bu]0,

��B1

B0=u0u1

= r.

��Ǳ��Ǳ

ε =PV

γ − 1=

P

(γ − 1)ρ,

��M�

M =u0cs,

�� c2s = (∂P/∂ρ)s = γP0/ρ0��

�M2

2(1− 1/r2) =

1

γ − 1(R/r − 1) + (r − 1)

v2Ac2s, (6.108)

�� R = P1/P0��

M2(1− 1/r) =1

γ(R− 1) +

1

2(r2 − 1)

v2Ac2s

(6.109)

��R��

r2 − rM2(γ − 1) + 2 + γv2A/c

2s

(γ − 2)(v2A/c2s)

+γ + 1

γ − 2

M2

(v2A/c2s)

= 0. (6.110)

��Ǳ

r1r2 =γ + 1

γ − 2

M2

(v2A/c2s).

� γ < 2�� Ǳ��

�r > 1��

γ + 1

2− γ

M2

(v2A/c2s)

= r2 + rM(γ − 1) + 2 + γv2A/c

2s

(2− γ)(v2A/c2s)

> 1 +M2(γ − 1) + 2 + γv2A/c

2s

(2− γ)(v2A/c2s)

6.5 �� 191

shock front

D

B1 B0

� 6.12: ��(c2s +

v2A)1/2��

��

M2

(v2A/c2s)> 1 +

1

(v2A/c2s),

�

u20 > v2A + c2s. (6.111)

�� (v2A + c2s)1/2��(6.111)��

��

6.5.4 ��

��Æ�� 0� π/2��Ǳ��

�� u�� B��Ǳ�

��

u0 ×B0 = 0.

192 ��

��Ǳ x−��Ǳ y−��

u0y = u0xB0y/Bx.

�� B0x = B1x = Bx��Ǳ�

u1y = u1xB1y/Bx.

��Ǳu1yu0y

=u1xu0x

B1y

B0y=

1

r

B1y

B0y. (6.112)

��(6.99)��

u1yu0y

− 1 =v2Au20

(B1y

B0y

− 1

), (6.113)

�� v2A = B20/4πρ0 �� (6.112)��(6.113)��

B1y/B0y��u1yu0y

=u20 − v2Au20 − rv2a

=1

r

B1y

B0y

, (6.114)

��(6.97)��

ρ1u1x

(u212

+γ

γ − 1

P1

ρ1

)= ρ0u0x

(u202

+γ

γ − 1

P0

ρ0

),

��

P2

P1= r +

(γ − 1)ru202c2s

(1− u21

u20

)

= r + r(γ − 1)

2

u20c2s

[1− cos2 θ

r2− sin2 θ

(u20 − v2Au20 − rv2a

)2]. (6.115)

��(6.114)�B1y

B0y= 1 +

(r − 1)u20(u20 − rv2A)

,

�� u20 > rv2A > v2A�� B1y > B0y�� B1x = B0x��

B1 > B0��(6.114)Ǳ��Ǳ

B1y

B0y

=1

1 + (r − 1)u20/r(v2A − u20)

,

�� u20 < v2A < rv2A�� B1y < B0y��

�� u20 = rv2A� � B0y = 0� u20 = v2A� � B1y = 0��

��Ǳswitch on�switch off��

6.5 �� 193

� 6.13: ��switch on�switch off��

��

1. ��

Tik =

(P +

B2

8π

)δik −

BiBk

4π

��Ǳ��Æ�

⎛⎜⎜⎝

P +B2/8π 0 0

0 P +B2/8π 0

0 0 P − B2/8π

⎞⎟⎟⎠ .

2. ��Ǳ104G��β = 1%��

Ǳn = 1018m−3��Ǳ105eV��

3. ��n = 7 × 1021m−3��Ǳ��Ǳ1keV�10keV��

4. ��Ǳ0.1m�Æ��1016��

�Ǳ107K�

194 ��

5. ��

(B · ∇)j = (j · ∇)B, ∇ · (B×∇p) = 0.

6. 2 × 106A��Ǳ1m��Ǳn =

1020m−3��ǱT = 10keV�� (1) ��(2)j×B��

7. ��

∇×B = αB,

��α�� αǱ��B��Helmholtz��

(∇2 + α2)B = 0.

8. �� Ψ/Ψ0 =

1

2

(bR2

0 +R2)z2 +

1

8(a− 1)(R2 − R2

0)2,

dP

dΨ= − a

16π3Ψ0,

dI2

dΨ= − c2

8π2bR2

0Ψ0

��Grad–Shafranov��a�b�R0��Ψ =const��Ǳ��R = R0�z = 0��

��

9. ��Ǳb��(6.72)�Ǳ

4πρ0ω2

k2= B2

0 −[Bz +

m

kaBθ(a)

]2 I ′m(ka)Im(ka)

[Km(ka)I

′m(kb)− Im(ka)K

′m(kb)

K ′m(ka)I

′m(kb)− I ′m(ka)K ′

m(kb)

]

− B2θ(a)

ka

I ′m(ka)Im(ka)

.

[�� Ǳ0��∂ψ/∂r|r=b = 0�]

10. ��

11. � �Ǳ��Ǳ

ε =1

γ − 1

P

ρ.

��u20/c2s > 1��P1/P0 → ∞��

��Ǳρ1/ρ0 = (γ + 1)/(γ − 1)��c2s��

��

��

��Ǳ��

��Ǳ��

��Ǳ��

��Ǳ��

��

7.1 ��

7.1.1 ��

��

∇×E = −1

c

∂B

∂t, ∇ ·B = 0, (7.1a)

∇×B =1

c

∂E

∂t+

4π

cj, ∇ · E = 4πρ. (7.1b)

�� ρ�� j�� E� B��

��Ǳ��

��

��

�� j(t, r)��Ǳ

j(t, r) =

⟨∑α

eαvαδ(r− rα)

⟩, (7.2)

�� vα�� eα �� δ(r)��δ−�� 〈· · · 〉��

196 ��

��dvα

dt=

eαmα

E(t, r), (7.3)

��

��

vα(t) =eαmα

∫ t

E[t′, r(t′)]dt′. (7.4)

��(7.4)��Ǳ t��

��

��(7.4)��(7.2)��Ǳ��

��Ǳ��Ǳ

��Ǳ��

j(r, t) = σE(r, t).

��

��Ǳ��

ji(r, t) =

∫ ∫ t

−∞Sik(r, r

′, t, t′)Ek(r′, t′)dt′d3r′, (7.5)

�� ji �� i�� Ek �� k�� Sik ��

��Ǳ��

�� Sik(r, r′, t, t′)�� r − r′��

� t− t′�� r− r′ = ρ� t− t′ = τ��(7.5)��Ǳ

ji(r, t) =

∫ ∞

0

∫Sik(ρ,τ)Ej(r− ρ, t− τ)d3ρdτ. (7.6)

��Ǳ�� Sik(ρ,τ)��

��Ǳ��

�Ǳ�� Sik ��Ǳ�

�� j� E��Ǳ��

j(r, t) =

∫j(k, ω)ei(k·r−ωt)dωd

3k

(2π)4, j(k, ω) =

∫j(r, t)ei(ωt−k·r)dtd3r,

E(r, t) =

∫E(k, ω)ei(k·r−ωt)dωd

3k

(2π)4, E(k, ω) =

∫E(r, t)ei(ωt−k·r)dtd3r,

7.1 �� 197

�� j(k, ω)� E(k, ω)��

��Ǳ

ji(k, ω) = σik(k, ω)Ek(k, ω), (7.7)

�� σik(k, ω)�� Sik(ρ,τ)��

σik(k, ω) =

∫ ∞

0

∫Sik(ρ, τ)e

i(ωτ−k·ρ)d3ρdτ. (7.8)

��Ǳ��Ǳ��

��Ǳ��

�Ǳ�� j�� E�� Sik(ρ, τ)��Ǳ��

��(7.8)��

σik(−k,−ω) = σ∗ik(k, ω) (7.9)

7.1.2 ��

�� —�� P��

∂P

∂t= j, (7.10a)

∇ ·P = −ρ, (7.10b)

�� ∂ρ/∂t + ∇ · j = 0��

(7.10a)�(7.10b)�� —��D��

�� P��Ǳ

D = E+ 4πP. (7.11)

��D��(7.1)��

∇× E = −1

c

∂B

∂t, ∇ ·B = 0 (7.12a)

∇×B =1

c

∂D

∂t, ∇ ·D = 0. (7.12b)

��(7.5)��(7.10)��Ǳ��D�� E��

��

198 ��

��Ǳ� ��

��Ǳ

Di(r, t) = Ei(r, t) +

∫ t

−∞

∫Kik(r, r

′, t, t′)Ek(r′, t′)d3r′dt′, (7.13)

��Kij ��

��(7.13)��D�� E��Ǳ t

��Ǳ��

��Ǳ��D(r,t)�� r

��

��D�� E��Ǳ��

��Ǳ��

��

��(7.13)��Kik ��

r− r′�� t− t′�� r− r′ = ρ� t− t′ = τ��(7.5)��Ǳ

Di(r, t) = Ei(r, t) +

∫ ∞

0

∫Kij(ρ, τ)Ej(r− ρ, t− τ)d3ρdτ. (7.14)

��D�� E��Ǳ��

D(r, t) =

∫D(k, ω)ei(k·r−ωt)dωd

3k

(2π)4, D(k, ω) =

∫D(r, t)ei(ωt−k·r)dtd3r,

E(r, t) =

∫E(k, ω)ei(k·r−ωt)dωd

3k

(2π)4, E(k, ω) =

∫E(r, t)ei(ωt−k·r)dtd3r,

��D� E��Ǳ

Di(k, ω) = εik(k, ω)Ek(k, ω), (7.15)

�� εik(k, ω)��Ǳ

εik(k, ω) = δik +

∫ ∞

0

∫Kik(ρ, τ)e


�� k�

�Ǳ��D�� E��Kik(ρ, τ)��Ǳ��

��(7.16)��

εik(−k,−ω) = ε∗ik(ω,k). (7.17)

7.1 �� 199

��Ǳ��

��

�� (7.7)�(7.10)��(7.11)��

Di = εikEk = Ei + 4πPi = Ei + i4π

ωji = δikEk + i

4π

ωσikEk,

��

εik(ω,k) = δik + i4π

ωσik(ω,k). (7.18)

7.1.3 ��

�� εik(k, ω)�� k��

��Ǳ��Ǳ��Ǳ�� k��

�� k��

��Ǳ

εik(k, ω) = εt(k, ω)(δik − kikk/k2) + εl(k, ω)kikk/k

2, (7.19)

� εt(k, ω)� εl(k, ω)�� ω�� k��

��Ǳ��(7.19)��E�� k��

��Ǳ

D = εlE,

�� E�� k��Ǳ

D = εtE.

�� εl � εt ��Ǳ�� k = 0��

��Æ��Ǳ��

εik(k = 0, ω)��Ǳ�� εik(k = 0, ω)��Ǳ ε(ω)δik ��

� ε(ω)�� k = 0�� εl� εt��

εt(k = 0, ω) = εl(k = 0, ω) = ε(ω). (7.20)

��(7.17)�� εl��

εt��

εl(k,−ω) = ε∗l (k, ω), εt(k,−ω) = ε∗t (k, ω). (7.21)

200 ��

�� εl� εt��Ǳ��

εl(k, ω) = ε′l(k, ω) + iε′′l (k, ω), εt(k, ω) = ε′t(k, ω) + iε′′t (k, ω).

��(7.21)�� εl,t �� ω ��

��

ε′l,t(k,−ω) = ε′l,t(k, ω), ε′′l,t(k,−ω) = −ε′′l (k, ω). (7.22)

��

��

��

��

�� Ǳ vT /νc�� vT ��

�� νc��Ǳ vT/ω�� ω��

��Æ��Ǳ

L ∼ min (vT/νc, vT/ω) .

�� kL � 1�� kL � 1��

�� νc � ω�� L ∼ vT/ω��

��

�� vT = 0��

��

�� 10−10 m�� x—��Ǳ��

�� Ǳ��

��D��P��P��

�� EǱ��B� ��

��(7.12a)�� B�� E��

�� E��Ǳ��Ǳ��

�� H��M = 0� B = H��

��

��E��B��D�

��H��Ǳ��

��Ǳ��Ǳ��Ǳ��Ǳ��Ǳ��Ǳ

7.2 �� 201

��M�Ǳ��Ǳ�� P�Ǳ��

��

j = c∇×M+∂P

∂t,

��H�Ǳ

B = H+ 4πM,

��D�Ǳ

D = E+ 4πP,

��(7.1b)��Ǳ

∇×H =1

c

∂D

∂t, ∇ ·D = 0.

Ǳ��D�H� E� B��Ǳ��

�� ε�� μ��Ǳ��

�� ε� μ��

D(k, ω) = ε(k, ω)E(k, ω), H(k, ω) =1

μ(k, ω)B(k, ω). (7.23)

��Ǳ��Ǳ��

�� εl �� εt��

�� (εl(k, ω), εt(k, ω))��

(ε(k, ω), μ(k, ω))��

7.2 ��

��(7.12)��

−∇ ·( c

4πE×B

)=

1

4π

(E · ∂D

∂t+B · ∂B

∂t

). (7.24)

�� (7.24)��

��(7.24)��Ǳ��Ǳ��Ǳ�� Ǳ�

��

��(7.24)��Ǳ��

202 ��

��

��

��

�(7.24)��Q� ��

��

��Ǳ

E(r, t) =1

2

[E0e

i(k·r−ωt) + E∗0e

−i(k·r−ωt)],

�� E0 ��(7.15)��Ǳ

Di(r, t) =1

2

[εij(ω,k)E0je

i(k·r−ωt) + εij(−ω,−k)E∗0je

−i(k·r−ωt)]

=1

2

[εij(ω,k)E0je

i(k·r−ωt) + ε∗ij(ω,k)E∗0je

−i(k·r−ωt)],

��(7.17)��Ǳ

∂Di

∂t=

1

2

[−iωεij(ω,k)E0je

i(k·r−ωt) + iωε∗ij(ω,k)E∗0je

−i(k·r−ωt)].

��Ǳ�� (7.24)��Ǳ��

Q =iω

16π

[ε∗ij(ω,k)− εji(ω,k)

]E0iE

∗0j . (7.25)

�� εij ��Ǳ��

εij = εHij + εAij ,

��

εHij =1

2

(εij + ε∗ji

), εAij =

1

2

(εij − ε∗ji

).

��

ε∗ij = εHji − εAji,

��(7.25)��Ǳ

Q = − iω

8πεAjiEiE

∗j , (7.26)

��

��

εij(ω,k) = εt(ω, k)(δij − kikj/k2) + εl(ω, k)kikj/k

2,

7.3 �� 203

��Ǳ

εAij(ω,k) = iε′′t (ω, k)(δij − kikj/k2) + iε′′l (ω, k)kikj/k

2.

��Ǳ

Q =ω

8πε′′l (ω, k)|El|2. (7.27)

��Ǳ

Q =ω

8πε′′t (ω, k)|Et|2. (7.28)

�� Q > 0��

� ��Ǳ��

ε′′l,t(ω, k) > 0, for ω > 0. (7.29)

�� ±∞�� ω ��

��Ǳ��

7.3 ��

��(7.24)��Ǳ��(7.24)��

�� Ǳ��

��

�� Ǳ

E(t, r) = E0(t, r)ei(k0·r−ω0t), B(t, r) = B0(t, r)e

i(k0·r−ω0t), (7.30)

�� E0(t, r) � B0(t, r) ��(7.30)��(7.24)��Ǳ

1

16π

[(E+ E∗) ·

(∂D

∂t+∂D∗

∂t

)+ (B+B∗) ·

(∂B

∂t+∂B∗

∂t

)].

��Æ � 2π/ω0��

1

16π

[E · ∂D

∗

∂t+ E∗ · ∂D

∂t+B · ∂B

∗

∂t+B∗ · ∂B

∂t

].

204 ��

��

∂Di(r, t)

∂t=

∂

∂t

∫Dω,k;ie

i(k·r−ωt)d3kdω

(2π)4

=

∫−iωεik(ω,k)Eω,k;ke

i(k·r−ωt)d3kdω

(2π)4, (7.31)

�� Eω,k;k��

Eω,k;k =

∫E0k(t, r)e

i(k0·r−ω0t)e−i(k·r−ωt)d3rdt

= E0k(ω − ω0,k− k0) (7.32)

�� E0k(ω − ω0,k − k0)��

�� k0 � ω0 ��(7.31)��ωεik(ω,k)� (ω0,k0) ��

∂Di(r, t)

∂t≈ −i

∫ [ω0εik(ω0,k0) + (ω − ω0)

∂(ωεik)

∂ω

+(k− k0) ·∂(ωεik)

∂k

]Eω,k;ke

i(k·r−ωt)d3kdω

(2π)4, (7.33)

�(7.32)��(7.33)��∫E0k(ω − ω0,k− k0)e

i[(k−k0)·r−(ω−ω0)t]d3kdω

(2π)4= E0k(t, r),

��

∂Di(r, t)

∂t=

[−iω0εik(ω0,k0)E0k +

∂(ωεik)

∂ω

∂E0k

∂t− ∂(ωεik)

∂k· ∂E0k

∂r

]ei(k0·r−ω0t). (7.34)

��(7.34)��

E · ∂D∗

∂t= Ei

∂D∗i

∂t

= iω0ε∗ikE0iE

∗0k +

[∂(ωε∗ik)∂ω

]ω=ω0k=k0

E0i∂E∗

0k

∂t−[∂(ωε∗ik)∂k

]ω=ω0k=k0

· E0i∇E∗0k,

E∗ · ∂D∂t

= E∗i

∂Di

∂t

= −iω0εikE∗0iE0k +

[∂(ωεik)

∂ω

]ω=ω0k=k0

E∗0i

∂E0k

∂t−[∂(ωεik)

∂k

]ω=ω0k=k0

· E∗0i∇E0k.

7.3 �� 205

��

ε∗ik = εki,

��

E · ∂D∗

∂t+ E∗ · ∂D

∂t=∂(ωεik)

∂ω

∂ (E∗0iE0k)

∂t− ω

∂εik∂k

· ∇ (E∗0iE0k) .

Ǳ��

B · ∂B∗

∂t+B∗ · ∂B

∂t=

∂

∂t(B0iB

∗0i) .

��

1

16π

∂

∂t

[∂(ωεik)

∂ωE∗

0iE0k +B0iB∗0i

]+∇·

[c

8πRe (E∗

0 ×B0)−ω

16π

∂εik∂k

E∗0iE0k

]= 0. (7.35)

��Ǳ

U =1

16π

[∂(ωεik)

∂ωE∗

i Ek +BiB∗i

], (7.36)

��Ǳ�� 0��Ǳ

S =c

8πRe (E∗ ×B)− ω

16π

∂εik∂k

E∗i Ek. (7.37)

��Ǳ

∂U

∂t+∇ · S = 0. (7.38)

��Ǳ��

∂U

∂t+∇ · (Uvg) = 0. (7.39)

��

�� vg �� vg ��

�� (7.38)�(7.39)��Ǳ

vg =S

U. (7.40)

��Ǳ

U =1

16π

∂(ωεl)

∂ω|El|2, (7.41)

206 ��

U =1

16π

∂

∂ω

[ω

(εt −

k2c2

ω2

)]|Et|2. (7.42)

��Ǳ��

∂(ωεl)

∂ω> 0,

∂

∂ω

[ω

(εt −

k2c2

ω2

)]> 0. (7.43)

��Ǳ

S = − ω

16π

∂εl(ω, k)

∂k|El|2n, (7.44)

�� n = k/k�� Ǳ

S = − ω

16π

∂

∂k

[εt(ω, k)−

c2k2

ω2

]|Et|2n (7.45)

7.4 ��

�� εij(ω,k)��(7.16)�� Kij �� τ ��(7.16)��

εij(ω,k) = δij +

∫ ∞

0

∫Kij(τ,ρ)e


�� Kij(τ,ρ) �� τ ��

�� τ → ∞ �� Kij(τ,ρ) ��

�� εij(ω,k) ��

ω = ω′ + iω′′ ��Ǳ��

(7.46)�� exp(−ω′′τ)�� Kij(τ,ρ)��

�� ω′′ > 0��(7.46)��( ω′′ = 0 )Ǳ�� ω = 0��

��(7.46)��Ǳ��

��

��(7.13)�� t

�Ǳ�� t�Ǳ��Ǳ��

��Ǳ��

(7.46)�� τ �� 0�∞��−∞�∞�

7.4 �� 207

��(7.46)��

εij(−ω∗,−k) = ε∗ij(ω,k). (7.47)

��(7.17)��

εl,t(−ω∗, k) = ε∗l,t(ω, k). (7.48)

��

εl,t(iω′′, k) = ε∗l,t(iω

′′, k). (7.49)

��(7.49)�� εl,t��

�� εij ��

�� δij��

εij(|ω| → ∞,k) = δij, (Imω > 0). (7.50)

�� ω′′ → +∞��e−ω′′τ ��(7.46)�� ω′�

�� eiω′τ�(7.46)��Ǳ��

�� εl,t��(1)��(2)��Ǳ��(3) εl,t(|ω| → ∞, k) = 1�� εl,t��

��(1)�� k� ��

�� εl,t��Ǳ��(2) �� εl,t(ω = 0, k)��

�� 1�(3) ��Ǳ��

1

2πi

∫C

dα(ω)

dω

dω

α(ω)− a. (7.51)

�� α(ω) = εl,t(ω, k)− 1�� C Ǳ��

�� εl,t(ω, k)− 1 − a��

��1�� aǱ�� εl.t(ω,k)� ω��

�� εl,t(ω, k)− 1 − a� ω ��Ǳ��

1��f(z)��C��

f(z)�C��C��

1

2πi

∫C

f ′(z)

f(z)dz = N − P,

��N�P��f(z)�C��(��k��k��)�

208 ��

�(7.51)�� εl,t(ω, k)− 1− a��Ǳ

�� a��

Re ω

Im ω

0

C

a a0

Im α

Re α

C ′

0

� 7.1: �� C (��)�� α�� C ′�

Ǳ��(7.51)��Ǳ��

1

2πi

∫C

dα(ω)

dω

dω

α(ω)− a=

1

2πi

∫C′

dα

α− a.

�� C ′�� α��C �� α�� α : C → C ′�

��(7.50)�� ω �� α = 0��

ω = 0��α0 = εl,t(ω = 0, k)−1�ω��

� α�� α�� 7.1��Ǳ�� ε′l,t(−ω, k) = ε′(ω, k)� ε′′l,t(−ω, k) = −ε′′l,t(ω, k)�� ε′′l,t(ω > 0, k) > 0�� α = 0�� α = α0 �� C ′ �� α

��(7.51)��Ǳ 1 ( 0 < a < α0 )��Ǳ 0 (a < 0� a > α0 )�� 0� α0 �� a�� εl,t(ω, k) = 1 + a

��

��Ǳ��

��Ǳ��

� a�� εl,t(ω, k) = 1 + a��Ǳ

εl,t(ω = 0, k) > 1�� εl,t(ω = 0, k)��

� 1�� εl,t(ω, k) = 0��Ǳ��

��Ǳ��Ǳ 1�

7.5 Kramers-Kronig�� 209

Im z

Re z

C

ω

� 7.2: ��(7.52)��

7.5 Kramers-Kronig��

�� ∫C

εij(z,k)− δijz − ω

dz, (7.52)

�� ω�� C �� ω′�� ωǱ��

�� C ��

�� ∫C


dz = 0. (7.53)

�� εij(|ω| → ∞,k) → δij��(7.53)��Ǳ 0�

��

P

∫ ∞

−∞


dz − iπ [εij(ω,k)− δij ] = 0,

�� P �� ε′ij(ω,k)�� ε′′ij(ω,k)�

��

ε′ij(ω,k)− δij =1

πP

∫ ∞

−∞

ε′′ij(z,k)

z − ωdz, (7.54a)

ε′′ij(ω,k) = −1

πP

∫ ∞

−∞

ε′ij(z,k)− δij

z − ωdz, (7.54b)

�� Kramers-Kronig��Kramers-Kronig��

��

210 ��

�� εij ��

ε′l,t(ω, k)− 1 =1

πP

∫ ∞

−∞

ε′′l,t(z, k)

z − ωdz, (7.55a)

ε′′l,t(ω, k) = −1

πP

∫ ∞

−∞

ε′l,t(z, k)− 1

z − ωdz. (7.55b)

�� ε′′l,t(ω, k)� ω��(7.55a)��Ǳ�

ε′l,t(ω, k)− 1 =2

πP

∫ ∞

0

zε′′l,t(z, k)

z2 − ω2dz. (7.56)

��Ǳ��

��

ω → ∞��

εt,l ∼ 1− 4πnee2

meω2= 1−

ω2pe

ω2,

��(7.56)��Ǳ

2

πP

∫ ∞

0

zε′′l,t(z, k)

z2 − ω2dz = − 2

πω2

∫ ∞

0

zε′′l,t(z, k)dz.

��

∫ ∞

0

ωε′′l,t(ω, k)dω =π

2ω2pe. (7.57)

�� ∫ ∞

0

ω Im1

εl,t(ω, k)dω = −π

2ω2pe (7.58)

��

• ��εik��Ǳ

εik(ω,k) = εl(ω, k)kikkk2

+ εt(ω, k)

(δik −

kikkk2

).

• ��εl,t(|ω| → ∞, k) = 1�

•εl,t(−ω, k) = ε∗l,t(ω, k).

7.6 �� 211

ε′l,t(ω, k)− 1 =1

πP

∫ ∞

−∞

ε′′l,t(z, k)

z − ωdz,

ε′′l,t(ω, k) = −1

πP

∫ ∞

−∞

ε′l,t(z, k)− 1

z − ωdz.

∫ ∞

0

ωε′′l,t(ω, k)dω =π

2ω2pe,∫ ∞

0

ω Im1

εl,t(ω, k)dω = −π

2ω2pe

• ��ε′′l,t(ω > 0, k) > 0

• ��

∂(ωε′l)∂ω

> 0,∂

∂ω

[ω

(ε′t −

k2c2

ω2

)]> 0.

��

• ��

7.6 ��

��

��Æ��

��

��

��

��Ǳ��Ǳ��

��

��

��

212 ��

��Ǳ v��Ǳ E��

��ǱdEdt

= eE · v.

�� E��

��EǱ 0��

��Ǳ 0�

��

��Ǳ

F ≡ −dEdl

= −dEdt

dt

dl= −e

vE · v. (7.60)

Ǳ�� E��Ǳ

��

��

Ǳ��

∇× E = −1

c

∂B

∂t, ∇ ·B = 0 (7.61a)

∇×B =1

c

∂D

∂t+

4π

cjext, ∇ ·D = 4πρext, (7.61b)

�� jext� ρext��Ǳ

jext = qvδ(r− vt), ρext = qδ(r− vt). (7.62)

��Ǳ��

Eω,k = Elω,k + Et

ω,k. (7.63)

��

k× Etω,k =

ω

cBω,k, k ·B = 0, (7.64a)

k×Bω,k = −ωcDω,k − i

4π

cjω,k, k ·Dω,k = −i4πρω,k, (7.64b)

�� jω,k� ρω,k��

jω,k =

∫jext(r, t)ei(ωt−k·r)dtd3r = 2πqvδ(ω − k · v),

ρω,k =

∫ρext(r, t)ei(ωt−k·r)dtd3r = 2πqδ(ω − k · v).

7.6 �� 213

��Dω,k;i = εij(ω,k)Eω,k;j��(7.63)��

εij =kikjk2

εl(ω, k) +

(δij −

kikjk2

)εt(ω, k),

��Ǳ

Dω,k = εlElω,k + εtE

tω,k. (7.65)

��(7.65)��(7.64)��

Elω,k = −i 4π

kεlρω,k,

Etω,k = −i4π

c

ω

c

jω,k − (k · jω,k)k/k2εtω2/c2 − k2

��Ǳ

Eω,k = Elω,k + Et

ω,k

= −i2(2π)2qδ(ω − k · v)[ω

c2v

εtω2/c2 − k2− ω2

c2k/k2

εtω2/c2 − k2+

k/k2

εl

].

��

E(r, t) =

∫Eω,ke

i(k·r−ωt)dωd3k

(2π)4.

��Ǳ

F = −qv

∫v · Eω,ke

i(k·r−ωt)dωd3k

(2π)4

= i2(2π)2q2

v

∫ [ω

c2v2

εtω2/c2 − k2− ω2

c2(k · v)/k2εtω2/c2 − k2

+(k · v)/k2

εl

]

× δ(ω − k · v)ei(k·r−ωt)dωd3k

(2π)4.

��r = vt�� ω ��

��

F = i2q2

(2π)2v

∫v2 − (k · v)2/k2

εt(k · v, k)(k · v)2 − k2c2(k · v)d3k

+ i2q2

(2π)2v

∫(k · v)

k2εl(k · v, k)d3k. (7.66)

��

214 ��

��Ǳ��Ǳ��

�(7.66)��Ǳ�� Ǳ��

��

��

� ω = kvμ�� μ�� k�� v��Æ��

F = iq2

πv2

∫dk

k

∫ kv

−kv

ω

εl(ω, k)dω.

��Ǳ

F = − q2

πv2

∫dk

k

∫ kv

−kv

Im

[1

εl(ω, k)

]ωdω.

�� v � (T/m)1/2��

�� ±∞ ��(7.58)��

F =q2ω2

pe

v2

∫ kmax

kmin

dk

k. (7.67)

��(7.67)��

��Ǳ��

��

��Ǳ��Ǳ��

��

ωp − k · v = 0,

�� ωp�� kmin = ωp/v��Ǳ

Fcollective =q2ω2

pe

v2

∫ kD

ωp/v

dk

k=q2ω2

pe

v2lnkDv

ωp,

�� kD = 2π/λD� λD ��

k > kD ��Ǳ��Ǳ��

�� kmax��

��

kmax = αmv2/|qe|2,

7.7 �� 215

��m�� α��Ǳ1��

Findividual =q2ω2

pe

v2lnkmax

kD,

��Ǳ

F = Fcollective + Findividual =q2ω2

pe

v2lnkmaxv

ωp. (7.68)

7.7 ��

��

��(7.12)��

ik×E = iω

cB, ik ·B = 0,

ik×B = −iωcD, ik ·D = 0,

��

Di = εij(ω,k)Ej.

��

k(k · E)− k2E+ω2

c2ε : E = 0,

��ε��(kikj − k2δij +

ω2

c2εij

)Ej = 0. (7.69)

�� Ǳ��

��(7.69)��Ǳ��

det

(kikj − k2δij +

ω2

c2εij

)= 0. (7.70)

��

��(7.69)��

��

�� εl(ω, k)�� εt(ω, k)��

εij(ω,k) = εt(ω, k)(δij − kikj/k2) + εl(ω, k)kikj/k

2

216 ��

��(7.69)��{[

ω2

c2εt(ω, k)− k2

] (δij − kikj/k

2)+ω2

c2εl(ω, k)kikj/k

2

}Ej = 0. (7.71)

�� k��Ǳ��E��Ǳ��

��

E = El + Et, El · k = Elk, Et · k = 0.

��(7.71)��Ǳ��

ω2

c2εl(ω, k)El = 0,[

ω2

c2εt(ω, k)− k2

]Et = 0.

��Ǳ

εl(ω, k) = 0, (7.72a)

ω2

c2εt(ω, k)− k2 = 0. (7.72b)

��(7.72a)��(7.72b)��

7.7 �� 217

��

1. ��Ǳ εij(ω,k)�� q ��

��Ǳ

φ(r) =q

2π

∫exp(ik · r)kikjεij(0,k)

d3k.

2. ��

εij = εt(δij − kikj/k2) + εl(kikj/k

2),

�� kikjεij = k2εl�

3. ��ω = 0�Ǳ

εl(0,k) = 1 +1

(kλD)2,

�� q��

4. ��(7.23)�� ε(ω, k)�� μ(ω, k)�Ǳ��

Ǳ�� εl(ω, k)�� εt(ω, k)��

��

ε(ω, k) = εl(ω, k),

1

μ(ω, k)= 1 +

ω2

c2k2[εl(ω, k)− εt(ω, k)] .

5. ��(7.33)��(7.34)�

6. �� ω → ∞��

εt,l = 1−ω2pe

ω2,

�� ∫ ∞

−∞ω Im

[1

εl,t(ω, k)

]dω = −πω2

pe.

[�� ∫C

1/εl,t − 1

z − ωdz,

�� C ��ω��]

218 ��

7. ��∫ ∞

0

[εl,t(iω, k)− 1] dω =

∫ ∞

0

ε′′l,t(ω, k)dω.

[�� z−�� ∫C

z [εl,t(z, k)− 1]

z2 + ω2dz,

�� C Ǳ��]

8. ��Ǳ

U =1

16π

∂(ωεl)

∂ω|El|2,

S = − ω

16π

∂εl(ω, k)

∂k|El|2n,

�� n�� Ǳ

vg = ndω(k)

dk

∣∣∣∣εl[ω(k),k]=0

.

9. ��Ǳ

U =1

16π

∂

∂ω

[ω

(εt −

k2c2

ω2

)]|Et|2,

S = − ω

16π

∂

∂k

[εt(ω, k)−

k2c2

ω2

]|Et|2n,

�� n�� Ǳ

vg = ndω(k)

dk

∣∣∣∣εl[ω(k),k]−c2k2/ω2=0

.

�� (��)

��Ǳ��

��

��Ǳ��

��

8.1 ��

��

��

��

8.1.1 ��

Ǳ��Ǳ��

��Ǳ��

��Ǳ��Ǳ��

��

��Ǳ��

��Ǳ

��

��

∂nα

∂t+∇ · (nαuα) = 0, (8.1a)

mαnα

(∂uα

∂t+ uα · ∇uα

)= −∇Pα + eα

(E+

1

cuα ×B

), (8.1b)

d

dt

(Pαn

−γαα

)= 0, (8.1c)

220 ��(��)

� α�� nα�� uα�� Pα�

�� γα��

��(8.1)��Ǳ��

��

��

��

��

� ��

∇× E = −1

c

∂B

∂t, ∇ ·B = 0 (8.2a)

∇×B =1

c

∂D

∂t, ∇ ·D = 0. (8.2b)

8.1.2 ��

�� Ǳ��

�� εij��Ǳ��Ǳ�

��

��(8.1)��

∂nα1

∂t+ nα0∇ · uα = 0,

nα0mα∂uα

∂t= nα0eαE−∇Pα1,

Pα1/Pα0 = γαnα1/nα0.

��∂/∂t = −iω�∇ = ik��

nα1

nα0=

k · uα

ω,

uα = ieαmαω

E+ kγαPα0

nα0mαωnα1,

��

uα − γαv2Tαk(k · uα)/ω

2 = ieαmαω

E,

� vTα = (Tα/mα)1/2��Æ�� Ǳ��(

δij − γαv2Tα

ω2/k2kikjk2

)uαj = i

eαmαω

Ei,

8.1 �� 221

��

uαi = ieαmαω

(δij +

γαv2Tα

ω2/k2 − γαv2Tα

kikjk2

)Ej.

��Ǳ

ji =∑α

eαnα0uαi = i∑α

nα0e2α

mαω

[(δij −

kikjk2

)+

ω2/k2


kikjk2

]Ej

�� ∂P/∂t = j�� P = ij/ω��

Pi = −∑α

nα0e2α

mαω2

[(δij −

kikjk2

)+

ω2/k2


kikjk2

]Ej.

��D = E+ 4πP��

Di = δijEj −∑α

4πnα0e2α

mαω2

[(δij −

kikjk2

)+

ω2/k2


kikjk2

]Ej .

��

εij = δij −∑α

ω2pα

ω2

(δij −

kikjk2

)−∑α

ω2pα

ω2

ω2/k2


kikjk2

=

[1−

∑α

ω2pα

ω2

](δij −

kikjk2

)+

[1−

∑α

ω2pα

ω2 − γαk2v2Tα

]kikjk2

��

εl(ω, k) = 1−∑α

ω2pα

ω2 − γαk2v2Tα

, (8.3a)

εt(ω, k) = 1−∑α

ω2pα

ω2. (8.3b)

�� ω → ∞��

εl,t ∼ 1−∑

α ω2pα

ω2≈ 1−

ω2pe

ω2,

�� k → 0��

εl = εt = 1−∑

α ω2pα

ω2≈ 1−

ω2pe

ω2.

�� ω2��

��

��

��

��

222 ��(��)

8.1.3 ��

��(8.3)��(7.72)��

��

��

ω2pi =

4πni0Z2e2

mi=Zme

miω2pe,

��

c2i = γiv2i , c

2e = γev

2e ,

��Ǳ

εl(ω, k) = 1−ω2pi

ω2 − k2c2i−

ω2pe

ω2 − k2c2e= 0. (8.4)

��Ǳ ��

�∂E/c∂t��

��(8.4)��

ω2 = ±

√(ω2pi + k2c2i + ω2

pe + kc2e2

)2

− k4c2i c2e − k2ω2

pic2e − k2ω2

pec2i

+ω2pi + k2c2i + ω2

pe + kc2e2

. (8.5)

��(8.5)��Ǳ�� (8.4)�� ωpi � ωpe��(8.4)��(8.4)�Ǳ

1−ω2pe

ω2 − k2c2e= 0.

��

ω2 = ω2pe + k2c2e = ω2

pe

(1 + γek

2λ2De

). (8.6)

��(8.5)��Ǳ��(8.6)��

8.1 �� 223

��Ǳ��

��ueui

= − mi

Zme

1− k2c2i /ω2

1− k2c2e/ω2≈ − mi

Zme

.

��Ǳ��

��Ǳ��(8.6)��γe��Ǳ3.��Ǳ��(8.4)�Ǳ

−ω2pi

ω2 − k2c2i−

ω2pe

ω2 − k2c2e= 0.

��

ω2 = k2ω2pec

2i + ω2

pic2e

ω2pe + ω2

pi

≈ k2(ZγeTe + γiTi

mi

). (8.7)

��Ǳ��

�

cs =

√ZγeTe + γiTi

mi. (8.8)

��(8.7)��∣∣∣∣ ω2

pi

ω2 − k2c2i

∣∣∣∣� 1.

��(8.7)��

k2λ2De � 1,

��

��

��Ǳ�� Ǳ��

��Ǳ

ueui

= − mi

Zme

1− k2c2i /ω2

1− k2c2e/ω2= 1.

��

��

��Ǳ��

γe = 1, γi = 3.

224 ��(��)

��

��(8.4)�Ǳ

1−ω2pi

ω2 − k2c2i= 0.

��

ω2 = ω2pi + k2c2i = ω2

pi

(1 + γik

2λ2Di

). (8.9)

��Ǳ��

�(8.9)�� ∣∣∣∣ ω2pe

ω2 − k2c2e

∣∣∣∣� 1.

��(8.9)��

k2λ2De � 1,

��

��

�Ǳ

ueui

= − mi

Zme

1− k2c2i /ω2

1− k2c2e/ω2≈ 1

γek2λ2De

� 1,

��

��7.3��Ǳ

U =1

16π

∂(ωεl)

∂ω|El|2

=ω2

8π

∑α

ω2pα

(ω2 − k2c2α)2|El|2 .

��(8.6)��

U =1

16π

[1 +

(1 + 2γek

2λ2De

)]|El|2 ,

��Ǳ |El|2/16π� ��Ǳ (1 + 2γek

2λ2De) |El|2 /16π�� k → 0��

��

8.2 �� 225

��

��(7.72)�(8.3b)��

ω2

c2

[1−

∑α ω

2pα

ω2

]− k2 = 0,

��

ω2 = k2c2 +∑α

ω2pα ≈ k2c2 + ω2

pe. (8.10)

��Ǳ��

��

��7.3��Ǳ

U =1

16π

∂

∂ω

[ω

(εt −

k2c2

ω2

)]|Et|2

=1

16π

(1 +

ω2pe

ω2+k2c2

ω2

)|Et|2 .

��

��Ǳ |Et|2 /16π��Ǳ (ωpe/ω)2 |Et|2 /16π�

��Ǳ (kc/ω)2 |Et|2��

U =1

8π|Et|2 .

�� k → 0��

��

8.2 ��

��

��Ǳ��Ǳ�

��

��Ǳ��Ǳ��

�� k�Ǳ��B0�

226 ��(��)

8.2.1 ��

��

��Ǳ

∂nα

∂t+ nα0∇ · uα = 0, (8.11a)

mα∂uα

∂t= eα

(E+

1

cuα ×B0

)− 1

nα0

∇Pα1, (8.11b)

Pα1

Pα0= γα

nα1

nα0. (8.11c)

��

uα = ieαE

mαω+ i

ωcα

ωuα × b+

c2eω2/k2

n(n · uα), (8.12)

� b�� n�� ωca��

ωcα =eαB0

mαc,

��

Ǳ�� z−�� k�x− z��

��

Δα;ijuα;j = ieαmαω

Ei, (8.13)

��Δα��Ǳ

Δα;ij =

⎛⎜⎜⎝

1− k2xc2α/ω

2 −iωcα/ω −kxkzc2α/ω2

iωcα/ω 1 0

−kxkzc2α/ω2 0 1− k2zc2α/ω

2

⎞⎟⎟⎠ . (8.14)

��(8.13)��Ǳ

uα;i = ieαmαω

Δ−1α;ijEj , (8.15)

�Δ−1α �Δα�� Ǳ

Δ−1α;ij =

1

|Δα|

⎛⎜⎜⎜⎜⎜⎝

1− k2zc2α

ω2iωcα

ω

(1− k2zc

2α

ω2

)kxkzc

2α

ω2

−iωcα

ω

(1− k2zc

2α

ω2

)1− k2c2α

ω2−iωcα

ω

kxkzc2α

ω2

kxkzc2α

ω2iωcα

ω

kxkzc2α

ω21− ω2

cα

ω2− k2xc

2α

ω2

⎞⎟⎟⎟⎟⎟⎠, (8.16)

8.2 �� 227

|Δα|� Δα��

|Δα| =(1− ω2

cα

ω2

)(1− k2zc

2α

ω2

)− k2xc

2α

ω2. (8.17)

��Ǳ

ji =∑α

eαnα0uα;i =∑α

ie2αnα0

mαωΔ−1

α;ijEj ,

��

εij = δij −∑α

ω2pα

ω2Δ−1

α;ij , (8.18)

8.2.2 ��

��

��

��Æ�cαǱ��Δ−1α ��Ǳ��

Δ−1α;ij =

1

1− ω2cα/ω

2

⎛⎜⎜⎜⎝

1 iωcα

ω0

−iωcα

ω1 0

0 0 1− ω2cα

ω2

⎞⎟⎟⎟⎠ . (8.19)

�� Ǳ��

εij =

⎛⎜⎜⎝

S −iD 0

iD S 0

0 0 P

⎞⎟⎟⎠ , (8.20)

��

S = 1−∑a

ω2pα

ω2 − ω2cα

, D =∑α

ω2pαωcα

ω(ω2 − ω2cα)

, P = 1−∑α

ω2pα

ω2. (8.21)

��B0 → −B0�� ωcα → −ωcα��

εij(ω,k;B0) = εji(ω,k;−B0).

��

�� εij = ε∗ji�

228 ��(��)

� ω → ∞��

S ∼ 1−∑

α ω2pα

ω2≈ 1−

ω2pe

ω2, (8.22a)

D ∼∑

α ω2pαωcα

ω3∼ 0, (8.22b)

P = 1−∑α

ω2pα

ω2≈ 1−

ω2pe

ω2, (8.22c)

�� εij ∼ (1− ω2pe/ω

2)δij�� ω → 0��

limω→0

S = 1 +∑α

ω2pα

ω2cα

= 1 +4πρc2

B20

= 1 +c2

v2A, (8.23a)

limω→0

D = − limω→0

∑α

ω2pα

ωωcα

= −4πc

B0

limω→0

∑α nαeαω

= 0, (8.23b)

limω→0

P = − limω→0

ω2pα

ω2= −∞. (8.23c)

��

��ÆǱ θ��(??)��∣∣∣∣∣∣∣∣

S − n2 cos2 θ −iD n2 sin θ cos θ

iD S − n2 0

n2 sin θ cos θ 0 P − n2 sin2 θ

∣∣∣∣∣∣∣∣= 0, (8.24)

� n = ck/ω��(8.24) ��

An4 − Bn2 + C = 0, (8.25)

��

A = P cos2 θ+ S sin2 θ, B = SP (1+ cos2 θ) + (S2 −D2) sin2 θ, C = P (S2 −D2). (8.26)

Ǳ��

R = S +D = 1−∑α

ω2pα

(ω + ωcα)ω, (8.27a)

L = S −D = 1−∑α

ω2pα

(ω − ωcα)ω. (8.27b)

8.2 �� 229

��(8.25)��(8.26)��

A = P cos2 θ + S sin2 θ, B = SP (1 + cos2 θ) +RL sin2 θ, C = PRL. (8.28)

��(8.25)�� Ǳ��

tan2 θ = − P (n2 −R)(n2 − L)

(n2 − P )(Sn2 − RL).(8.29)

��(8.25)��Æ��

n2 =1

2A

(B ±

√B2 − 4AC

). (8.30)

� n2 > 0�� n2 < 0��

��Ǳ��n2 = 0��Ǳ��

�� n2 = ∞��Ǳ��

8.2.3 ��

��

��(8.25)�� C = 0��C = PRL��

��Ǳ

P = 0, R = 0, or L = 0.

��

� P = 0��Ǳ��

ωp =√ω2pe + ω2

pi ≈ ωpe. (8.31)

� R = 0��Ǳ��

ωR ≈√ω2pe + ω2

ce/4 + |ωce|/2. (8.32)

� L = 0��Ǳ��

ωL ≈√ω2pe + ω2

ce/4− |ωce|/2. (8.33)

��Ǳ��

��(8.32)�(8.33)�� ωR ��

|ωce|� ωL�� |ωce|�

230 ��(��)

�� n → ∞��(8.30)�� A → 0 ��

�(8.28)�� A��A = 0��

tan2 θ = −PS. (8.34)

��Ǳ��(8.34)�� P � S � ��

� �� θ = 0��

Ǳ P = 0� S = ∞��Ǳ

ωres = ωpe, |ωci|, |ωce|, (8.35)

��Ǳ�� θ = π/2��

��Ǳ P = ∞� S = 0��Ǳ

ωres = 0, ωLH, ωHH , (8.36)

��

ωLH =ω2pe + ω2

pi + ω2ce + ω2

ci

2−

√(ω2pe + ω2

pi + ω2ce + ω2

ci

2

)2

− ω2peω

2ci − ω2

piω2ce − ω2

ceω2ci

≈ [ω2ci + ω2

piω2ce/(ω

2pe + ω2

ce)]1/2, (8.37)

ωHH =ω2pe + ω2

pi + ω2ce + ω2

ci

2+

√(ω2pe + ω2

pi + ω2ce + ω2

ci

2

)2

− ω2peω

2ci − ω2

piω2ce − ω2

ceω2ci

≈ (ω2pe + ω2

ce)1/2. (8.38)

��Ǳ ��

��

ωLH =

{(ω2

ci + ω2pi)

1/2, when ω2ce � ω2

pe,

(|ωceωci|)1/2, when ω2ce � ω2

pe.

�� x−��

�� ωpe ��B0��

�� y−��

232 ��(��)

��(8.25)�� n = n(ω, θ)��⎛⎜⎜⎝

S − n2 cos2 θ −iD n2 sin θ cos θ

iD S − n2 0

n2 sin θ cos θ 0 P − n2 sin2 θ

⎞⎟⎟⎠

⎛⎜⎜⎝

Ex

Ey

Ez

⎞⎟⎟⎠ = 0. (8.39)

��

��

uα;i = ieαmαω

Δ−1α;ijEj ,

��

�� (8.39)��

limn→∞

Ex

Ez

= − limn→∞

P − n2 sin2 θ

n2 sin θ cos θ=

sin θ

cos θ=kxkz,

limn→∞

Ey

Ex

= − limn→∞

iD

S − n2= 0,

��

��

�� θ = 0��

��

��Ǳ��

��(��)�� Ǳ��

8.2.4 ��

�� θ = 0��(8.39)Ǳ⎛⎜⎜⎝

S − n2 −iD 0

iD S − n2 0

0 0 P

⎞⎟⎟⎠

⎛⎜⎜⎝

Ex

Ey

Ez

⎞⎟⎟⎠ = 0. (8.40)

�� (8.40)�� (

S − n2 −iDiD S − n2

)(Ex

Ey

)= 0, (8.41a)

PEz = 0. (8.41b)

8.2 �� 233

z

y

xR L

B0

kR kL

� 8.2: ��

��

�

P = 0, or ω2 =∑α

ω2pα.

��Ǳ��

��Ǳ��

��

��

��Ǳ

n2 = S ±D. (8.42)

��(8.41a)��Æ��8.2��Ǳ��

��

Ey

Ex= −i, for n2 = S +D = R,

Ey

Ex= i, for n2 = S −D = L.

��(8.42)��

234 ��(��)

��

�� ω → 0��(8.22)��

ω2 = k2v2A

1 + v2A/c2, (8.43)

�� (8.43)�� 1Ǳ 1 + v2A/c

2��

� �� v2A/c2��

��

�� uαi = i(eα/mαω)Δ−1α;ijEj �� ω → 0�

��

limω→0

(eα/mαω)Δ−1α;ij =

⎛⎜⎜⎝

0 c/B0 0

−c/B0 0 0

0 0 ieα/maω

⎞⎟⎟⎠ ,

��Ǳ(uαx

uαy

)=

(0 c/B0

−c/B0 0

)(Eαx

Eαy

),

�� uα = cE×B0/B20�

� �� Ǳ��

�� D → 0��

�� Ǳ��

��

��

R = 1−ω2pe + ω2

pi

(ω − |ωce|)(ω + ωci).

�� |ωci|�� |ωce|�� ω2

pe/|ωce|��

ω =k2c2|ωce|ω2pe

, for |ωci| � ω � |ωce|, ω2pe/|ωce|. (8.44)

��Ǳ��

vg =dω

dk=

2c

ωpe

√|ωce|ω, (8.45)

8.2 �� 235

��

��

��

��

��

��

��

��

��Ǳω�� s��Ǳ

t =

∫s

ds

vg=

∫s

ωpe(s)ds

2c√ω|ωce(s)|

.

��

��

ω2 = k2c2 + ω2pe

(1± ωce√

k2c2 + ω2pe

), for ω � |ωce|, (8.46)

� +�� −��

��

�� x−��

E(z = 0, t) = E0exe−iωt =

1

2E0 [(ex + iey) + (ex − iey)] e

−iωt.

��Ǳ d��Ǳ

E(z = d, t) =E0

2(ex + iey)e

ikRd−iωt +E0

2(ex − iey)e

ikLd−iωt.

��E� x� y��Ǳ

Ex(z = d)

Ey(z = d)= −iexp(ikRd) + exp(ikLd)

exp(ikRd)− exp(ikLd)

=1

tan[(kL − kR)d/2],

�� x−��ÆǱ

ϕ = arctanEy

Ex

=kL − kR

2d. (8.47)

236 ��(��)

+ =

⊗ B0

EL ER Eout

� 8.3: ��

� 8.4: ��

��Æ��

� ��

��

kL − kR =ω

c(nL − nR) ≈

ω2pe|ωce|

cω2√1− ω2

pe/ω2.

��ÆǱ

ϕ =(ne/nc)

2√

1− ne/nc

eB0d

mec2. (8.48)

��

8.2 �� 237

8.2.5 ��

� θ = π/2��(8.39)Ǳ⎛⎜⎜⎝

S −iD 0

iD S − n2 0

0 0 P − n2

⎞⎟⎟⎠

⎛⎜⎜⎝

Ex

Ey

Ez

⎞⎟⎟⎠ = 0. (8.49)

��Ǳ

n2 = P

n2 = RL/S.

�� n2 = P ��

��

��

��Ǳ��Ǳ

ω2 = ω2pe + c2k2.

�� n2 = RL/S ��

��Ǳ��Ex

Ey

= iS − n2

S= i

D

S.

�� Ǳ�� Ǳ��

��

�� ω → 0��

R → S, L→ S, S → 1 +c2

v2A, D → 0

��

ω2 = k2v2A

1 + v2A/c2. (8.50)

�� Ǳ��

��

��

�(8.49)�� x−��Ǳ��y−�� x−�� Ǳ��

238 ��(��)

� 8.5: ��

��

��

D → 0, S → 1, R→ 1, L→ 1,

��

�� tan2 θ = −P/S�� S = 0��Ǳ Ey/Ex = −iS/D = 0��

��Ǳ�� S = 0�

1−ω2pe

ω2 − ω2ce

−ω2pi

ω2 − ω2ci

= 0.

�� Ǳ�( )�� ǱωHH � ωLH�

��

ω < ωLH ��Ǳ� �� Ǳ �� ωL < ω < ωHH

��Ǳ��Ǳ��Ǳ�� ω > ωR��Ǳ��

��Ǳ��

8.3 ��

��

��

��Ǳ�� Ǳ�

8.3 �� 239

8.3.1 ��

�� (8.16)�� kz = 0��

Δ−1α;ij =

1

1− ω2cα/ω

2 − k2c2α/ω2

⎛⎜⎜⎜⎜⎝

1 iωcα

ω0

−iωcα

ω1− k2c2α

ω20

0 0 1− ω2cα

ω2− k2c2α

ω2

⎞⎟⎟⎟⎟⎠ (8.51)

�(8.51)��(8.18)��

εij =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

1−∑α

ω2pα

ω2 − ω2cα − k2c2α

−i∑α

ωcαω2pα

ω(ω2 − ω2cα − k2c2α)

0

i∑α

ωcαω2pα


1−∑α

ω2pα(ω

2 − k2c2α/ω2)

ω2(ω2 − ω2cα − k2c2α)

0

0 0 1−∑α

ω2pα

ω2

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

≡

⎛⎜⎜⎝

S ′ −iD′ 0

iD′ S ′′ 0

0 0 P

⎞⎟⎟⎠ .

��Ǳ ⎛⎜⎜⎝

S ′ −iD′ 0

iD′ S ′′ − n2 0

0 0 P − n2

⎞⎟⎟⎠

⎛⎜⎜⎝

Ex

Ey

Ez

⎞⎟⎟⎠ = 0 (8.52)

��Ǳ

(P − n2)[S ′(S ′′ − n2)−D′2] = 0.

��Ǳ��

n2 = (S ′S ′′ −D′2)/S ′,

� ��

S ′ = 1−∑α

ω2pα


≈ 1 +∑α

ω2pα

ω2cα

= 1 +c2

v2A,

S ′′ = 1−∑α

ω2pα(ω

2 − k2c2α)

ω2(ω2 − ω2cα − k2c2α)

≈ 1 +∑α

ω2pα

ω2cα

− k2

ω2

∑α

ω2pαc

2α

ω2cα

= 1 +c2

v2A− c2sv2An2

240 ��(��)

D′ =∑α

ωcαω2pα


≈ −∑α

ω2pα

ωω2cα

= 0.

��Ǳ

ω2 = k2(v2A + c2s)/(1 + v2A/c2),

� c2s = [(Z +1)γT/mi]1/2��

��

8.3.2 ��

��

��(8.52)��

S ′ = 1−∑α

ω2pα


= 0 (8.53)

�� D′ � S ′ Ǳ��Ǳ�� Ey = 0�� Ex ��Ǳ 0�

�� x−��(8.53)��Ǳ�� Ǳ��Ǳ��

�(8.53)��Ǳ��

ω2 = ω2ce + ω2

pe + k2c2e = ω2HH + k2c2e. (8.54)

�� ω2 � ω2ce��

S ′ ≈ 1 +ω2pe

ω2ce + k2c2e

−ω2pi

ω2 − ω2ci − k2c2i

= 0

��

ω2 = ω2ci + k2c2i +

ω2ce + k2c2e

ω2HH + k2c2e

ω2pi. (8.55)

� ω2pe � ω2

ce�

ω2 ≈ ω2ci + k2c2i +

ω2ce + k2c2e

ω2pe + k2c2e

ω2pi = ω2

LH + k2c2s.

� ω2pe � ω2

ce�

ω2 ≈ ω2ci + ω2

pi + k2c2i = ω2LH + k2c2i .

8.3 �� 241

z,B0

x

k

wave front

� 8.6: ��Ǳ��Ǳ��

8.3.3 ��

�� Ǳ��

��Ǳ ωci� ��

�� ω2 ≈ ω2ci ��

��

tan2 θ = −PS

= −1−

∑a ω

2pα/ω

2

1 −∑

a ω2pα/(ω

2 − ω2cα)

≈ −ω2pe/ω

2ci

ω2pi/(ω

2 − ω2ci)

=ω2pe

ω2pi

ω2ci − ω2

ω2≈ 2mi

Zme

ωci − ω

ω.

�� θ �= π/2�� (Zme/2mi) tan2 θ � 1��

��Ǳ ω ≈ ωci�� θ �= π/2��

��Ǳ��

��

��Ǳ��Æ��

��

∇ ·D = 0.

� ��

kiDi(ω,k) = 0.

��Di = εijEj��

εijkiEj = 0.

242 ��(��)

��Ej = (kj/k)E��

εijkikjE = 0.

�� E �= 0��

εijkikj = 0.

�Ǳ�� x− z��Ǳ

εxxk2x + εzzk

2z + 2εxzkxkz = 0.

Ǳ�� k2zc2e/ω

2ci � 1��

εzz = 1−ω2pi

ω2−ω2pe

ω2

1− ω2ce/ω

2 − k2xc2e/ω

2

(1− ω2ce/ω

2)(1− k2zc2e/ω

2)− k2xc2e/ω

2

≈ 1−ω2pi

ω2+

ω2pe

k2zc2e

,

εxx = 1−ω2pi

ω2 − ω2ci

−ω2pe

ω2

1− k2zc2e/ω

2

(1− ω2ce/ω

2)(1− k2zc2e/ω

2)− k2xc2e/ω

2

≈ 1−ω2pi

ω2 − ω2ci

+ω2pe

ω2ce

≈ 1−ω2pi

ω2 − ω2ci

,

εxz = −ω2pe

ω2

kxkzc2e/ω

2

(1− ω2ce/ω

2)(1− k2zc2e/ω

2)− k2xc2e/ω

2

≈ −ω2pe

ω2ce

kxkz,

��Ǳ (1−

ω2pi

ω2 − ω2ci

)k2x +

ω2pe

c2e= 0.

�� ω2pe/k

2xc

2e � 1��

ω2 = ω2ci + k2xc

2s.

8.4 �� 243

8.4 ��

��

��

��—��—��

�� z = 0��Ǳ

ε =

{ε2 = 1− ω2

pe/ω2, when z < 0,

ε1 = 1, when z > 0.

��

��

��

∇×E = −1

c

∂B

∂t, ∇ ·B = 0, (8.57a)

∇×B =1

c

∂D

∂t,∇ ·D = 0. (8.57b)

��

E = (Ex, 0, Ez)e−κ|z|eiqx−iωt, (8.58a)

B = (0, By, 0)e−κ|z|eiqx−iωt, (8.58b)

��Ǳ ω�� x−��Ǳ q��

��Ǳ κ��(8.58)��(8.57)��

iqB1y = −iωcε1E1z, (8.59a)

iqB2y = −iωcε2E2z, (8.59b)

κ1B1y = −iωcε1E1x, (8.59c)

−κ2B2y = −iωcε2E2x, (8.59d)

�(8.57b)��

−κ1E1x − iqE1z = iω

cB1y, (8.60a)

κ2E2x − iqE2z = iω

cB2y, (8.60b)

244 ��(��)

��

iqE1x − κ1E1z = 0, (8.61a)

iqE2x + κ2E2z = 0. (8.61b)

��

��

B1y = B2y, (8.62a)

E1x = E2x, (8.62b)

ε1E1z = ε2E2z. (8.62c)

��(8.59a)�(8.59c)�(8.60a)��

κ1 =

√q2 − ε1

ω2

c2=√q2 − ω2/c2. (8.63)

��(8.59b)�(8.59d)�(8.60b)��

κ2 =

√q2 − ε2

ω2

c2=√q2 − εω2/c2. (8.64)

��(8.61)��(8.60)��(−κ1 +

q2

κ1

)E1x = i

ω

cB1y,(

κ2 −q2

κ2

)E2x = i

ω

cB2y.

��(8.62)��

κ1(1− q2/κ21) + κ2(1− q2/κ22) = 0. (8.65)

�(8.63)�(8.64)��(8.65)��ε1κ1

+ε2κ2

= 0. (8.66)

��(8.67)��

ω2 =ω2pe + 2q2c2 −

√ω4pe + 4q4c4

2. (8.67)

��(8.7)��8.8��

��

8.4 �� 245

0 2 4 6 80

0.2

0.4

0.6

0.8

q/(ωpe

/c)

ω/ω

pe

� 8.7: ��

� 8.8: ��

246 ��(��)

8.5 ��

��

��Ǳ�

��

��B0� z−�� x−�� k� y − z�

�� k = kyey + kzez�� ky � kz��

��

ω/kz � vA,

��Ǳ��Ǳϕ��

E = −∇φ.

��Ǳ

me∂ue

∂t= e∇φ− e

cue ×B0 −

∇Pe

n0(x).

� ��z��

me∂uez∂t

= e∂φ

∂z− Te∂n1e/∂z

n0(x).

��

n1e = n0eφ

Te.

��∂n1i

∂t+ ui · ∇n0 + n0∇ · ui = 0,

min0∂ui

∂t= −en0∇φ+ n0

e

cui ×B0.

��Ǳ�� exp[i(kyy + kzz − ωt)]��Ǳ

−iωn1i + uixdn0(x)

dx+ in0k · ui = 0,

−imin0ωui = −ien0φk+ en0ui ×B0/c.

�� x−�� uix = cEy/B0�� y−�� kz �� z−�� z−��Ǳ

0 = −iωn1i + uixdn0(x)

dx+ in0k · ui ≈ −iωn1i +

−ikycϕB0

dn0(x)

dx,

8.5 �� 247

��Ǳ

n1i = −kycφωB0

dn0

dx

�� n1e = n1i��

n0eφ

Te= −kycφ

ωB0

dn0

dx,

�� φ�Ǳ��

ωi = − kyTec

en0B0

dn0

dx=

kyv2Te

|ωce|Ln

,

�� Ln��Æ�

Ln = −(

1

n0

dn0

dx

)−1

> 0.

�� Ǳ

VD =c(−∇Pe/n0)×B0

(−e)B20

= −cTedn0/dx

eB0n0ey =

v2Te

|ωce|Lney,

��

ω = k ·VD = ωi.

�� ωi = kyvD �Ǳ��

248 ��(��)

��

1. ��

(1) ω2 = k2c2/(1 + c2/v2A) (� ��)�

(2) ω = k2c2ωcl/ω2p (��)�

(3) ω2 − ωω2p/(ω − ωce) = k2c2, ω → ωce (��)�

2. ��

3. ��Ti = 0�� Ǳ

ω = kcs(1 + k2λ2De)−1/2.

4. �� Ǳ

E21

4π=

1

2w,

B21

4π=

1

2w(1− ω2

pe/ω2),

1

2meneu21 =

1

2wω2pe

ω2,

��

w =E2

1

4π+B2

1

4π+

1

2mnu21.

�E1�B1�u1��

5. ��

6. ��u0��

7. ��(5)��

∇ · (εE) = 0,

��ε� ��

8. ��−mnνev�� Ǳ��Im(ω)��

8.5 �� 249

9. ��Ǳn0�

(1) ��

(2) ��Ti = 0�Te �= 0��Boltzmann��Poisson��

10. ��Ǳκ��m3�

��n0��K+�κn0��Cl−�1 − κn0��κ =

0.6��0.03m��n0��

11. �8cm��8mm��

(1) ��1/10��

��(��2π��)�

(2) ��

12. ��ξ��ω2 = ω2

pe + ω2ce�

13. ��

14. ��B��Æθ��

(1) ��

(2) ��ω2��θ → 0�θ → π/2��

�ω��

(3) ��x = cos θ�y = 2ω2/ω2HH�a = ω2

HH/2ωcωp��Ǳ��

�(y − 1)2

12+x2

a2= 1.

(4) ��ωc > ωp��ω1(��)��ωp(��θ > 0)��ω2(��)��ωc�ωHH��ωp > ωc�

�ω1 �ωc�ω2�ωp�ωHH��

250 ��(��)

15. ��x = 0��x��

��x = 0��

16. ��

17. ��

18. ��

19. ��Ǳ ��(ωceωci)1/2��

��

20. ��Ǳ

k2zc2

ω2=

ω2pi

ω2ci − ω2

− k2rc2

2ω2+

[ω4pi

(ω2ci − ω2)2

+

(k2rc

2

2ω2

)2]1/2

,

��kz�kr��Ǳ��

21. ��

22. ��Ex�Ey��

F (ω)(Ex − iEy) = 0, G(ω)(Ex + iEy) = 0,

��R�F (ω) = 0�G(ω) �= 0��L�G(ω) = 0�F (ω) �= 0�

23. ��ω = ωce/2��

24. ��ω � ωce��ω1/2�

25. �� (��)��

26. �103G��8mm��100cm��π/4��

27. ��Ǳn0��M1��Ǳ(1− ε)n0��M2��

�Ǳεn0��Ti1 = Ti2 = 0�Te �= 0�

(1) ��

8.5 �� 251

(2) �εǱ��ω2��

28. ��k��B0��ω0 = (ωceωci)

1/2��

�νe � ω0�νe � ω0��

29. ��

(1) ��n ∼ 107m−3�B ∼ 10−3G�

(2) ��n ∼ 1012m−3�B ∼ 10G�

(3) ��n ∼ 1011m−3�B ∼ 1G�

(4) ��n ∼ 1020m−3�B ∼ 104G�

(5) ��n ∼ 1028m−3�B ∼ 103G.

30. ��

31. �� M1�M2(M1 > M2)��ǱA1�A2(A1 +

A2 = 1)�

(1) �� ω2

pe�ωce(A1ωci+Aωci2)�ωciωci2(A1ωci2+A2ωci1)/(A1ωci1+A2ωci2)�

(2) �(1)��

32. �� Ǳ100G.

33. ��

34. ��B0��

35. ��Ǳ��Ǳ

q =ω

c

√ω2pe − ω2

ω2pe − 2ω2

.

36. ��ω/q < c�

��

��

��

��

��Ǳ��

��

��—��Ǳ��

��∂u

∂t+ u

∂u

∂x= F. (9.1)

�� F Ǳ�� u∂u/∂xǱ��

��Ǳ��Ǳ

u = cos kx,

��

u∂u

∂x= −k cos kx sin kx = −k

2sin 2kx.

��

��

Ǳ��

��(9.1)�� u0�� u0 ��

�(9.1)��∂u1∂t

+ u0∂u1∂x

= F.

F Ǳ 0��Ǳ��

u(x, t) = u0 + u1 cos k(x− u0t),

9.1 �� 253

�� u0�� u0��

��

��

9.1 ��

��Ǳ��

��Ǳ��

�� x−��Ǳ∂u

∂t+ u

∂u

∂x= − e

me

E, (9.2)

�� u�� E��

∂E

∂x= 4πe(n0 − ne), (9.3)

�� n0�� ne��

∂ne

∂t+∂(neu)

∂x= 0. (9.4)

��

∇×B =1

c

∂E

∂t+

4π

cj,

� x−��∂E

∂t= 4πeneu. (9.5)

��(9.3)�� u��(9.5)��

∂E

∂t+ u

∂E

∂x= 4πen0u. (9.6)

��d

dt=

∂

∂t+ u

∂

∂x,

��(9.2)�(9.6)��Ǳ

du

dt= − e

meE,

dE

dt= 4πeneu. (9.7)

��

d2u

dt2+ ω2

peu = 0, (9.8)

254 ��

�� ω2pe = 4πn0e

2/me ��

��Ǳ��

�(9.7)�(9.8) �Ǳ��Ǳ��Æ��Ǳ��

��(9.2)�(9.3)�(9.4)�(9.6)��Ǳ�� (a, τ)�� (x, t)��Ǳ

t = τ, x = a + ξ(a, τ), (9.9)

�� ξ(a, t)��Ǳ��Ǳ a��Ǳ τ ��

ξ(a, τ)� Ǳ

ξ(a, τ) =

∫ τ

0

u(a, τ ′)dτ ′, (9.10)

�� u(a, τ)��Ǳ a�� 5.2��(9.5)��(9.6)��Ǳ

∂

∂τu(a, τ) = − e

meE(a, τ), (9.11)

∂E

∂τ= 4πen0u(a, τ). (9.12)

∂

∂τ

{n(a, τ)

[1 +

∫ τ

0

∂u(a, τ ′)∂a

dτ ′]}

= 0. (9.13)

��(9.11)�(9.12) ��E��

∂2

∂τ 2u(a, τ) + ω2

peu(a, τ) = 0. (9.14)

��Ǳ��

�(9.14)��(9.14)��Ǳ

u(a, τ) = U(a) cosωpeτ + V (a) sinωpeτ. (9.15)

�(9.15)��(9.11)��

E(a, τ) =ωpeme

e[U(a) sinωpeτ − V (a) cosωpeτ ] . (9.16)

��(9.13)�� τ ��

n(a, τ) =n(a, 0)[

1 +1

ωpe

∂U(a)

∂asinωpeτ +

1

ωpe

∂V (a)

∂a(1− cosωpeτ)

] . (9.17)

9.1 �� 255

��(9.15)�(9.16)�(9.17)��U(a)� V (a)��U(a)��Ǳ��(9.15)��

U(a) = u(a, 0). (9.18)

V (a)��Ǳ�� t = 0��Ǳ

∂E(a, 0)

∂a= 4πe [n0 − n(a, 0)] .

��(9.16)� ��

∂V (a)

∂a= ωpe

[n(a, 0)

n0

− 1

]. (9.19)

��Ǳ��

t = τ, x = a+ ξ(a, τ), (9.20a)

ξ(a, τ) =U(a)

ωpesinωpeτ +

V (a)

ωpe(1− cosωpeτ). (9.20b)

��(9.20)�� (a, τ) Ǳ�� (x, t)��

��(9.15)�(9.16)�(9.17)��(9.20)�� a��

�� x��

�� Ǳ

��Ǳ��Ǳ

n(a, 0) = n0(1 + Δcos ka). (9.21)

u(a, 0) = U(a) = 0. (9.22)

�(9.21)�(9.22)�(9.15)�(9.16)�(9.17)��

u(a, τ) =ωpe

kΔsin ka sinωpeτ, (9.23a)

E(a, τ) = −me

eω2pe

Δ

ksin ka cosωpeτ, (9.23b)

n(a, τ) = n01 + Δcos ka

1 + Δcos ka(1− cosωpeτ). (9.23c)

256 ��

��Ǳ

t = τ, kx = ka+ α(τ) sin ka, (9.24)

��

α(τ) = 2Δ sin2(ωpeτ/2).

�� |Δ| � 1�� a ≈ x��(9.23)� Δ��

��

u(x, t) =ωpe

kΔsin kx sinωpet,

E(x, t) = −me

eω2pe

Δ

ksin kx cosωpet,

n(x, t) = n0(1 + Δcos kx cosωpet).

��ǱǱ��

��

�� ωpet = π/2� 3π/2�� n(x, t) = n0� ωpet = π �� α(τ) = 2Δ��

kx = ka + 2Δ sin ka�� Δ > 0�� kx = (2n + 1)π��

� ��

nmax(ωpet = π) =1−Δ

1− 2Δn0.

Δ = 0.45�� 5.5n0��

�� 9.1��

��

��(9.24)�� sin ka� cos ka� kx��

sin ka�Ǳ kx��

sin ka(x, t) =

∞∑n=1

an(t) sinnkx. (9.25)

��Ǳ

an(t) = (−1)n+1 2

nα(t)Jn[nα(t)], (9.26)

�� Jn(x)��

9.1 �� 257

� 9.1: Δ = 0.45��

�(9.25)�(9.26)�(9.23)��

u(x, t) =ωpe

kΔ

∞∑n=1

(−)n+1 2

nα(t)Jn[nα(t)] sinnkx sinωpet, (9.27a)

E(x, t) = −me

e

ω2pe

kΔ

∞∑n=1

(−)n+1 2

nα(t)Jn[nα(t)] sinnkx cosωpet, (9.27b)

n(x, t) = n0 +2n0Δ

α(t)

∞∑n=1

(−)n+1Jn[nα(t)] cosnkx cosωpet (9.27c)

��

Ǳ k��

� �� Ǳ�� t > 0��

�� 50%��Ǳ��

�� 1 > |Δ| > 1/2��Ǳ� |Δ| > 1/2��

��Ǳ��

�� |Δ| > 1/2��

� �� Ǳ��

��—��Æ��

�9.2��Δ = 0.45� Δ = 0.6��

��

��Ǳ��

258 ��

� 9.2: Δ = 0.45�Δ = 0.6��

��Ǳ

��a < b��

��Ǳ��

a + ξ(a, τ) < b+ ξ(b, τ).

��

∂ξ(a, τ)

∂a> −1. (9.28)

��(9.28)��(9.28)��

9.2 ��

��Ǳ��

��

��

��

��

��(9.9)�(9.10)��Ǳ

∂p

∂τ= −eE. (9.29)

��(9.29)�� τ �� (9.12)��

∂2p

∂τ 2= −4πe2n0u = −ω2

pe

p√1 + (p/mc)2

. (9.30)

9.2 �� 259

�� u = p/me

√1 + (p/mec)2�Ǳ��

��

ωpeτ → τ, p/mec→ p,

��(9.30)�Ǳ∂2p

∂τ 2+

p√1 + p2

= 0. (9.31)

�� p � 1��(9.31)��

∂2p

∂τ 2+ p = 0.

��(9.31)��

1

2

(∂p

∂τ

)2

+√

1 + p2 = C(a), (9.32)

�� C(a)�� a��ǱC =√1 + p2m(a)�

�� pm(a)��Ǳ�� a��(9.32)��

∫dp√

(1 + p2m)1/2 − (1 + p2)1/2

=√2τ.

��Ǳ

T =√2

∫ pm

−pm

dp√(1 + p2m)

1/2 − (1 + p2)1/2, (9.33)

��Ǳ

ω =2π

T=

π√2∫ pm0

dp/√(1 + p2m)

1/2 − (1 + p2)1/2. (9.34)

��

I[pm(a)] =

√2

π

∫ pm

0

dp√(1 + p2m)

1/2 − (1 + p2)1/2, (9.35)

�� pm(a)�� ω = 1/I(pm)��

��

260 ��

��(9.35)�� 1� pm � 1��

I[pm(a)] ≈2

π

∫ pm(a)

0

dp

(p2m − p2)1/2[1− (p2m + p2)/4]1/2

≈ 2

π

∫ pm

0

1 + (p2m + p2)/8

(p2m − p2)1/2dp

= 1 +3

16p2m(a).

��Ǳ

ω = 1− 3

16p2m(a). (9.36)

��

��

�� pm � 1��

I(pm) ≈√2

π

∫ pm

0

dp√pm − p

=2√2

πp1/2m

��Ǳ

ω =π

2√2p

1/2m

. (9.37)

��Ǳ��

��

��Δ��

n =n0(1 + Δcos ka)

1 + Δcos ka[1− cosωτ − (3/8)Δ2τ sin2 ka sinωτ ], (9.38)

��

ω = 1− 3

16Δ2 sin2 ka,

kx = ka+ 2Δ sin2 ωτ sin ka.

��(9.38)��Δ��

��

��

9.3 �� 261

9.3 ��

9.3.1 ��

��(�� )��

��

��Ǳ�� x−��Ǳ

0 = e∂φ

∂x− 1

ne

∂Pe

∂x.

�� ∂Pe/∂x =

Te∂ne/∂x� ��

ne = n0 exp

(eφ

Te

). (9.39)

�� φ��Ǳ

∂ni

∂t+

∂

∂x(niui) = 0, (9.40a)

∂ui∂t

+ ui∂ui∂x

= − e

mi

∂φ

∂x. (9.40b)

��Ǳ��

��∂2φ

∂x2= −4πe(ni − ne). (9.41)

��

x

λDe→ x, ωpit→ t,

ni

n0→ n,

eφ

Te→ φ,

uics

→ u,

��(9.40)��(9.41)�Ǳ

∂2φ

∂x2= eφ − n, (9.42a)

∂n

∂t+∂(nu)

∂x= 0, (9.42b)

∂u

∂t+ u

∂u

∂x= −∂φ

∂x. (9.42c)

262 ��

�� (φ, n, u)�� η = x−Mt��M ��

�� ∂/∂z = d/dη� ∂/∂t = −Md/dη��(9.42)�Ǳ

φ′′ = exp(φ)− n, (9.43a)

−Mn′ + (nu)′ = 0, (9.43b)

−Mu′ + uu′ = −φ′. (9.43c)

��(9.43b)��n(u−M) = −M. (9.44)

�� |η| → ∞� n = 1� u = 0��(9.43c)��

1

2(u−M)2 = −φ+

1

2M2. (9.45)

�� φ(|η| → ∞) = 0��(9.44)�(9.45)�� u�

n =M

(M2 − 2φ)1/2. (9.46)

��(9.46)�(9.43a)�� φ��

d2φ

dη2= exp(φ)− M

(M2 − 2φ)1/2. (9.47)

�� φ′(|η| → ∞) = 0��

1

2φ′2 + V (φ) = 0, (9.48)

�� V (φ)Ǳ

V (Φ) = 1− eφ +M2[1−

√1− 2φ/M2

], (9.49)

��(9.48)��Ǳ 1��Ǳ 0�� V (x)��

��Ǳ1

2x2 + V (x) = 0.

�� |t| → ∞�� x → 0�� V (x)��

��Ǳ�� V (x)��

�� x � M2/2��Ǳ x(|t| → ∞) = 0�� (0,M2/2)��

�� (0,M2/2)�� V (x)��

��

V (x = 0) =dV (x)

dx

∣∣∣∣x=0

= 0,d2V (x)

dx2

∣∣∣∣x=0

= 1/M2 − 1

9.3 �� 263

−0.4 −0.2 0 0.2 0.4 0.6 0.8 1 1.2−0.4

−0.35

−0.3

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

x

V(x

)

� 9.3: M = 1.58�� V (x)�

��M > 1�� x = 0�� V (x) � ��M < 1�� x = 0��

� V (x)� ��M < 1�� V (x)� (0,M2/2)��

�� 0��M �� 1�M > 1�� (0,M2/2)��

��9.3�M = 1.58��Ǳ��

Ǳ 0�� V (M2/2) � 0��M � ��

M � 1.5852�

��

(1, 1.5852)�� 1.2564�

�� 1� δM = M − 1��

δM � 1�� V (φ)��

V (φ) ≈ −φ2δM +1

3φ3,

��(9.48)�Ǳ1

2φ′2 =

1

3φ2(3δM − φ).

��

φ = 3δMsech2(√δM/2η). (9.50)

��Ǳ 3δM��Ǳ (δM)−1/2��

��9.4��

9.3.2 �� K-dV��

��

��K-dV�� ε ≡ δM��

264 ��

−20 −10 0 10 200

0.05

0.1

0.15

0.2

0.25

0.3

0.35

η

Φ(η

)

� 9.4: δM = 0.1��

�� (ε� 1)��(9.50)�� Ǳ√2

2

[ε1/2(x− t)− ε3/2t

].

�� (9.42)��

ξ = ε1/2(x− t), τ = ε3/2t, (9.51)

��Ǳ�

∂

∂x= ε1/2

∂

∂ξ,∂

∂t= ε3/2

∂

∂τ− ε1/2

∂

∂ξ. (9.52)

�� n�φ� u�� εǱ��

n = 1 + εn1 + ε2n2 + · · · , (9.53a)

φ = εφ1 + ε2φ2 + · · · , (9.53b)

v = εu1 + ε2u2 + · · · . (9.53c)

��(9.52)��(9.53)��(9.42a)�Ǳ

ε∂2

∂ξ2(εφ1 + ε2φ2 + · · ·

)= 1+(εφ1+ε

2φ2+· · · )+ 1

2!(εφ1+· · · )2+· · ·−(1+εn1+ε

2n2+· · · ).

��

ε(n1 − φ1) + ε2(∂2φ1

∂ξ2+ n2 − φ2 −

1

2φ21

)+ · · · = 0.

ε� ε2��Ǳ��

n1 = φ1,

9.3 �� 265

∂2φ1

∂ξ2+ n2 − φ2 −

1

2φ21 = 0.

��(9.42b)��Ǳ

ε5/2∂n1

∂τ− ε3/2

∂n1

∂ξ− ε5/2

∂n2

∂ξ+ ε3/2

∂u1∂ξ

+ ε5/2∂(n1u1)

∂ξ+ ε5/2

∂u2∂ξ

+ · · · = 0.

��

ε3/2(∂u1∂ξ

− ∂n1

∂ξ

)+ ε5/2

[∂n1

∂τ− ∂n2

∂ξ+∂(n1u1)

∂ξ+∂u2∂ξ

]+ · · · = 0.

ε3/2� ε5/2��Ǳ��

∂u1∂ξ

− ∂n1

∂ξ= 0,

∂n1

∂τ− ∂n2

∂ξ+∂(n1u1)

∂ξ+∂u2∂ξ

= 0.

��(9.42c)��Ǳ

ε5/2∂u1∂τ

− ε3/2∂u1∂ξ

− ε5/2∂u2∂ξ

+ ε5/2u1∂u1∂ξ

+ · · · = −ε3/2∂φ1

∂ξ− ε5/2

∂φ2

∂ξ+ · · · ,

��

ε3/2(∂φ1

∂ξ− ∂u1

∂ξ

)+ ε5/2

(∂u1∂τ

+ u1∂u1∂ξ

− ∂u2∂ξ

+∂φ2

∂ξ

)+ · · · = 0.

ε3/2� ε5/2��Ǳ��

∂φ1

∂ξ− ∂u1

∂ξ= 0,

∂u1∂τ

+ u1∂u1∂ξ

− ∂u2∂ξ

+∂φ2

∂ξ= 0.

��

φ1 = n1,∂n1

∂ξ=∂u1∂ξ

,∂u1∂ξ

=∂φ1

∂ξ, (9.54)

∂2φ1

∂ξ2− φ2 −

1

2φ21 + n2 = 0, (9.55a)

∂n1

∂τ− ∂n2

∂ξ+∂(n1u1)

∂ξ+∂u2∂ξ

= 0, (9.55b)

∂u1∂τ

− ∂u2∂ξ

+ u1∂u1∂ξ

+∂φ2

∂ξ= 0. (9.55c)

266 ��

��(9.54)��

φ1 = n1 = u1. (9.56)

��

�(9.56)��(9.55)��

∂2n1

∂ξ2− φ2 −

1

2n21 + n2 = 0, (9.57a)

∂n1

∂τ− ∂n2

∂ξ+ 2n1

∂n1

∂ξ+∂u2∂ξ

= 0, (9.57b)

∂n1

∂τ− ∂u2

∂ξ+ n1

∂n1

∂ξ+∂φ2

∂ξ= 0. (9.57c)

��(9.57a)� ξ��

∂3n1

∂ξ3=∂(φ2 − n2)

∂ξ+ n1

∂n1

∂ξ. (9.58)

��(9.57b)��(9.57c)��

2∂n1

∂τ+∂(φ2 − n2)

∂ξ+ 3n1

∂n1

∂ξ= 0. (9.59)

��(9.58)�(9.59)�� n1��

∂n1

∂τ+ n1

∂n1

∂ξ+

1

2

∂3n1

∂ξ3= 0. (9.60)

�� K-dV�� D. J. Korteweg� G. de Vries� 1895��(9.56)��Ǳ�� (9.60)��.

K-dV��(9.60)��Ǳ��

η = ξ −Mτ,

��K-dV��Ǳ

−Mn′ + nn′ +1

2n′′′ = 0.

��

−Mn +1

2n2 +

1

2n′′ = A.

9.4 �� 267

�� η = ±∞�� n1 = n′1 = n′′

1 = 0��A = 0� ��

��

1

4

(dn

dη

)2

+1

6n3 − 1

2Mn2 = 0.

��

∫dn

n√M − n/3

=√2η.

��Ǳ−2arc tanh(√

1− n/3M)/√M��

n = 3Msech2

(√M

2η

).

��

9.4 ��

�� 1015 W�� 1021 W/cm2� ��

��

�� Ǳ��Ǳ��

��

� ��

∇× E = −1

c

∂B

∂t, ∇ ·B = 0 (9.61a)

∇×B =1

c

∂E

∂t− 4π

cenu, ∇ · E = 4πe(n0 − n). (9.61b)

��Ǳ∂p

∂t+ (u · ∇)p = −eE− e

cu×B. (9.62)

��(9.61)�(9.62)��Ǳ�� x− V t�� x−�� V Ǳ��Ǳ

��

ωpe

c(x− V t) = η, V/c = β, (9.63)

268 ��

�� ωpe = 4πn0e2/meǱ��

∂

∂x=ωpe

c

d

dη,∂

∂t= −ωpeβ

d

dη.

��Ǳ

ωpe

cex × E′ =

ωpe

cβB′,

ωpe

cex ·B′ = 0. (9.64a)

ωpe

cex ×B′ = −ωpe

cβE′ − 4π

cenu,

ωpe

cex ·E′ = 4πe(n0 − n). (9.64b)

�� η��Ǳ

−ωpeβp′ +

ωpe

c(ex · u)p′ = −eE− e

cu×B. (9.65)

�� E� � B Ǳ��

��(9.62)�� (9.64a)��

B =1

βex × E, (9.66)

��(9.64b)�� ex��

ωpe

cβ(ex · E′) = −4π

cen(ex · u). (9.67)

��(9.64b)��(9.67)��

n =n0β

β − (ex · u)/c. (9.68)

��(9.64b)�� ex��

ωpe

cex × (ex ×B′) = −ωpe

cβ(ex × E′)− 4π

cen(ex × u)

��(9.64a)��

B′ = −4πen(ex × u)

ωpe(β2 − 1). (9.69)

��(9.65)�� ex��

−ωpe(β − ex · u/c)(ex × p′) = −e(ex × E)− e

cex × (u×B)

= −eβB− e

c[(ex ·B)u− (ex · u)B]

= −e(β − ex · u/c)B,

9.4 �� 269

B =ωpe

eex × p′. (9.70)

��(9.70)� η��(9.69)��

(ex × p)′′ =e

ωpe

B′ = − 4πne2

ω2pe(β

2 − 1)(ex × u). (9.71)

��(9.68) �� n��

(ex × p)′′ = −mc β

β2 − 1

ex × u/c

β − ex · u/c. (9.72)

��(9.65)�� ex�� η��

−ωpe[(β − ex · u/c)(ex · p′)]′ = −e(ex · E′)− e

cex · (u×B)′. (9.73)

��(9.70)��(9.73)��Ǳ

−ecex · (u×B)′ = −ωpe

c[u · p′ − (u · ex)(ex · p′)]′ ,

��(9.73)��

[β(ex · p′)− (u · p′)/c]′ =e

ωpe(ex · E′). (9.74)

��(9.67)��(9.74)��(9.68) �� n��

[β(ex · p′)− (u · p′)/c]′ = −mc ex · u/cβ − ex · u/c

. (9.75)

��(9.72)�(9.75)�� p/mec→ p��Ǳ�

d2

dη2

(βpx −

√1 + p2

)+

px

β√1 + p2 − px

= 0, (9.76a)

d2pydη2

+β

β2 − 1

py

β√1 + p2 − px

= 0, (9.76b)

d2pzdη2

+β

β2 − 1

pz

β√1 + p2 − px

= 0. (9.76c)

��Ex = 0��(9.67)�� ǱǱ�� ǱǱ��

��(9.76a)��d2

dη2

√1 + p2 = 0.

270 ��

��Ǳ√1 + p2 = C1η + C2�� C1 � C2 ��

��

p2 = p20 = const., (9.77)

��(9.76b)�(9.76c)��

d2pydη2

+1

(β2 − 1)√

1 + p20py = 0, (9.78a)

d2pzdη2

+1

(β2 − 1)√1 + p20

pz = 0. (9.78b)

��(9.77)��(9.78)�� Ǳ

px = p0 cos(ωη + α), py = ±p0 sin(ωη + α), (9.79)

�� α�� ω�Ǳ

ω =1

(β2 − 1)1/2(1 + p20)1/4. (9.80)

��(9.63)��

ω =β

(β2 − 1)1/2(1 + p20/m2c2)1/4

ωpe. (9.81)

�� p0��

��

ω

k=

c

ε1/2, (9.82)

�� Ǳω

k= V = βc,

��(9.81)� β Ǳ�� ω� ωpe�� p0��

β2 =1

1− (ω2pe/ω

2)(1 + p20/m2c2)1/2

,

��Ǳ�

ε = 1−ω2pe

ω2(1 + p20/m2c2)1/2

. (9.83)

9.4 �� 271

��Ǳ��(9.64a)�� (9.70)��

E = βωpe

ep′,

�� (9.79)� ��

Ex = −ωep0 sin[ω(x/V − t)], (9.84a)

Ey = ±ωep0 cos[ω(x/V − t)], (9.84b)

�� p0 ��Ǳ p20 = e2E20/m

2�� E0 Ǳ��

�� ε��Ǳ

ε = 1−ω2pe

ω2[1 + (eE0/mωc)2]1/2. (9.85)

��(9.84)�� Ǳ��

��

�� ε > 0� ε = 0��

�� ne = nc��

�� (9.85)��Ǳ

εt = 1− ne

nc

√1 + Iλ2/1.38× 1018

,

�� I Ǳ ��ǱW/cm2� λǱ ��Ǳ μm��Ǳ 1μm�� 1.38 × 1020W/cm2�� 10��Ǳ��(relativistic transparency)��Ǳ�� ε��

��

�� Ǳ

��

��

��Ǳ��Ǳ�� ux = 0��

��(9.67)� Ex = 0��

�� x− y��

�Ǳ�� x− y�� p� 1��

272 ��

� py �� px��Ǳ

d2pxdη2

+pxβ2

=1

β

[pyd2pydη2

+

(dpydη

)2], (9.86a)

d2pydη2

+1

β2 − 1py = 0. (9.86b)

��

px = − p20β

(3β2 + 1)cos[2η/(β2 − 1)1/2

], (9.87a)

py = p0 sin[η/(β2 − 1)1/2

], (9.87b)

��(9.86)��

9.5 ��

��φ�ϕ��Ǳ

φ = ϕ+ z sinϕ, (9.88)

��ϕ�0��2π��φǱ�0��2π��sinϕ�φ��

Ǳ2π��sinϕ�Ǳφ��

sinϕ =

∞∑n=1

fn(z) sin nφ. (9.89)

��fn��

�Ǳ��

eiz sinϕ =

∞∑n=−∞

Jn(z)einϕ, (9.90)

Jn−1(x) + Jn+1(x) =2n

xJn(x), (9.91)

��

Jn(x+ y) =∞∑

k=−∞Jk(x)Jn−k(y). (9.92)

9.5 �� 273

�� (9.92)��

Jn

(m∑i=1

xi

)=

∞∑k1=−∞

· · ·∞∑

km−1=−∞Jn−k1−...−km−1(x1)Jk1(x2) . . . Jkm−1(xm) (9.93)

eiϕ =

∞∑n=−∞

aneinφ, (9.94)

��Ǳ

an =1

2π

∫ 2π

0

eiϕe−inφdφ =1

2π

∫ 2π

0

eiϕe−inφ dφ

dϕdϕ, (9.95)

��φ�ϕ��(9.88)��

dφ

dϕ= 1 + z cosϕ = 1 +

z

2eiϕ +

z

2e−iϕ,

��(9.95)��

an =1

2π

∫ 2π

0

eiϕ(1 +

z

2eiϕ +

z

2e−iϕ

)e−inφdϕ

=1

2π

∫ 2π

0

[z2+ eiϕ +

z

2ei2ϕ]e−inφdϕ.

��

eiφ = ei(ϕ+z sinϕ) = eiϕeiz sinϕ = eiϕ∞∑

n=−∞Jk(z)e

ikϕ =∞∑

k=−∞Jk−1(z)e

ikϕ,

n > 0��

e−inφ =

[ ∞∑k=−∞

Jk−1(z)e−ikϕ

]n

=∞∑

m=−∞e−imϕ

∞∑k1,...kn−1=−∞

Jm−n−(k1−1)−···−(kn−1−1)(z)Jk1−1(z) . . . Jkn−1(z)

=∞∑

m=−∞e−imϕJm−n(nz).

274 ��

n < 0��

e−inφ = ei|n|φ =

[ ∞∑k=−∞

Jk−1(z)eikϕ

]|n|

=∞∑

m=−∞eimϕ

∞∑k1,...kn−1=−∞

Jm−|n|−(k1−1)−···−(kn−1−1)(z)Jk1−1(z) . . . Jk|n|−1(z)

=∞∑

m=−∞eimϕJm−|n|(|n|z)

��n > 0��Ǳ

an =[z2J−n(nz) + J1−n(nz) +

z

2J2−n(nz)

]

=(1− n)

nJ1−n(nz) + J1−n(nz)

= (−)n−1 1

nJn−1(nz)

n < 0��Ǳ

an =[z2J−|n|(|n|z) + J−1−|n|(|n|z) +

z

2J−2−|n|(|n|z)

]

=(−1− |n|)

|n| J−1−|n| + J−1−|n|

= (−)|n|1

|n|J|n|+1(|n|z)

��(9.94)��

sinϕ =

∞∑n=1

an sinnφ+

−∞∑n=−1

an sin nφ

=∞∑n=1

(−)n−1 1

nJn−1(nz) sin nφ−

∞∑n=1

(−)n1

nJn+1(nz) sin nφ

=∞∑n=1

(−)n−1 1

n[Jn−1(nz) + Jn+1(nz)] sin nφ

=

∞∑n=1

(−)n−1 2

nzJn(nz) sin nφ.

��(9.89)��Ǳ

fn(z) = (−)n−1(2/nz)Jn(nz). (9.96)

9.5 �� 275

��

1. �� B = B0ez��

��x��t� ��

��x−��∂2

∂τ 2ux(a, τ) + ω2

HHux(a, τ) = 0,

��ω2HH = ω2

pe + ω2ce��(a, τ)��(9.9)�(9.10)��

��

2. �� Ǳ

ω = ωpe

(1− 3

16u2m

),

��um��

3. ��x − y��

�(9.76)��Ǳ

d2pxdη2

+pxβ2

=1

β

[pyd2pydη2

+

(dpydη

)2],

d2pydη2

+1

β2 − 1py = 0.

��

px(0) = − p20β

(3β2 + 1), px(0) = 0,

py(0) = 0, py(0) = p0(β2 − 1)−1/2,

��

4. K-dV��V (x, τ)Ǳ��

∂2

∂x2ψ(x, τ) + [E − V (x, τ)]ψ(x, τ) = 0,

��EǱ��V (x/τ)��Æ��Æ�E��

�τ� ��V (x, t)��K-dV��

∂V

∂τ− 6V

∂V

∂x+∂3V

∂x3= 0,

276 ��

�ψ(x, τ)��x��|x| → ∞��Æ�E��τ�[ ��V (x, τ) Ǳψ(x, τ)��K-dV��

dE

dτψ2 =

∂Q

∂x,

��Q�ψ��]

��

��

��

��

��

��

��

10.1 ��

��

��

��Ǳ��

��Wentzel-Kramers-Brillouin (WKB)��Ǳ��

10.1.1 ��

��

��Ǳ��

��Ǳ

E = E(r) exp (−iωt) , B = B(r) exp(−iωt). (10.1)

��

��Ǳ��

∂ue

∂t= − e

mE(r)e−iωt. (10.2)

278 ��

��

j = −ene(r)ue =iω2

pe(r)

4πωE, (10.3)

�� ωpe(r)�� r��

��

∇×E = iω

cB, (10.4a)

∇×B = −iωcε(r)E, (10.4b)

��

ε(r) = 1−ω2pe(r)

ω2= 1− ne(r)

nc

, (10.5)

��Ǳ��

nc = ω2me/4πe2 (10.6)

��Ǳ ω�� (10.4)��

∇2E−∇(∇ · E) + ω2

c2εE = 0, (10.7a)

∇2B+ω2

c2εB+

1

ε∇ε× (∇×B) = 0. (10.7b)

��

�(10.7)��(1)��(2)��Æ�

��

��

��(10.3)��

∇ · j = i

4πω∇ ·[ω2pe(r)E

].

��∂

∂t(−eδne) +∇ · j = 0,

�� δne��

eδne = −∇ ·[ω2pe(r)

4πω2E

]= −∇ ·

[ne

4πncE

].

10.1 �� 279

�� ∇ · E = −4πeδne��

∇ · (εE) = 0.

��Ǳ

∇ · E = −E · ∇εε

=E · ∇ne

nc(1− ne/nc). (10.8)

��

�� δne��

�� ω�� ω = ωpe(r)��

��

�� Ǳ��

10.1.2 ��

��Ǳ��

�� z�� z−��x−��

ne(r) = ne(z), ε = ε(ω, z),

E = exE(z) exp (−iωt) , B = eyB(z) exp(−iωt).

�(10.7a)��Ǳd2

dz2Ex +

ω2

c2εEx = 0. (10.9a)

�(10.7b)��Ǳ

d2

dz2By +

ω2

c2εBy −

1

ε

dε

dz

dBy

dz= 0, (10.10a)

�� (10.9a)��

��WKB��

��WKB�� x−�� Ex = E�

�(10.9a)��d2E

dz2+ω2

c2ε(ω, z)E = 0. (10.11)

280 ��

��(10.11)��

E(z) = E0(z) exp

[iω

c

∫ z

ψ(z′)dz′]. (10.12)

�(10.12)��(10.11)��

c2

ω2

d2E0

dz2+ i

2c

ωψdE0

dz− ψ2E0 + i

c

ω

dψ

dzE0 + εE0 = 0 (10.13)

�� c/ω = λ0/2π�� λ0 �� d/dz ∼ 1/L�� L��

��Æ��(10.13)��

c2

ω2

d2E0

dz2∼(λ0L

)2

E0,c

ωψdE0

dz∼(λ0L

)ψE0,

c

ω

dψ

dzE0 ∼

(λ0L

)ψE0.

��Æ� L�� λ0 �� λ0/L

�� ε = λ0/L��(10.13)�� ε0�Ǳ ��

�� ψ��

−ψ2E0 + εE0 = 0. (10.14)

��

ψ(z) =√ε(ω, z). (10.15)

�� ε1�Ǳ ��

2ψdE0

dz+dψ

dzE0 = 0. (10.16)

��Ǳ

E0(z) =const√ψ. (10.17)

� ��(10.15)�(10.17)��(10.12)��(10.36)�WKB��

E(z) =EFS

ε1/4exp

(iω

c

∫ z√ε(ω, z′)dz′

), (10.18)

�� EFS ��Ǳ

B = −i cω∇× E ≈ eyEFSε

1/4 exp

(iω

c

∫ z√ε(ω, z′)dz′

). (10.19)

��Ǳ��

��

10.1 �� 281

��(10.18)� ε−1/4 ��

��(�7.3��)��Ǳ

S =1

8π

kc2

ω|E|2. (10.20)

�� ε− c2k2/ω2 = 0�� kc2/ω = ε1/2c��

��Ǳ

S =c

8πε1/2|E|2. (10.21)

�Ǳ��

��c

8πε1/2|E|2 = c

8πE2

FS,

�� |E| = EFS/ε1/4�

��WKB��(10.11)�� (λ0/L)2

� (c2/ω2)(d2E0/dz2)��WKB�� Ǳ

∣∣∣∣d2E0

dz2

∣∣∣∣�∣∣∣∣ωc

dψ

dzE0

∣∣∣∣ =∣∣∣∣2ωc ψ

dE0

dz

∣∣∣∣ (10.22)

� E0(z)��(10.18)��(10.22)��

∣∣∣∣∣5

16

1

ε2

(dε

dz

)2

− 1

4

1

ε

d2ε

dz2

∣∣∣∣∣�∣∣∣∣ ω2c

1

ε1/2dε

dz

∣∣∣∣ . (10.23)

��

∣∣∣∣∣5

16

1

ε2

(∂ε

∂z

)2∣∣∣∣∣�

∣∣∣∣ ω2c1

ε1/2∂ε

∂z

∣∣∣∣ , (10.24a)

∣∣∣∣141

ε

∂2ε

∂z2

∣∣∣∣�∣∣∣∣ ω2c

1

ε1/2∂ε

∂z

∣∣∣∣ , (10.24b)

��(10.23)�� λ(z) = 2π/(ωε1/2/c)��

�(10.24)�� ∣∣∣∣λ∂ε∂z∣∣∣∣� 2π|ε|. (10.25)

��(10.25)�WKB��

282 ��

��

�� z0�� ε(ω, z0) = 0��ǱǱ��

��WKB�� (10.25)��WKB�� z0��(10.11)��Ǳ�� ε(ω, z)��

�� ε(ω, z)��

ε(ω, z) = ε(ω, z0) + (z − z0)∂ε/∂z + · · · ,

� ∂ε(z0)/∂z �= 0��Ǳ��

��Ǳ��

ne =

{0, z < 0,

nczL, z > 0,

��Ǳ

ε = 1− z

L. (10.26)

��(10.11)Ǳd2E

dz2+ω2

c2

(1− z

L

)E = 0. (10.27)

�� ξ�

ξ = (ω2/c2L)1/3(z − L),

��(10.27)�Ǳd2E

dξ2− ξE = 0. (10.28)

��Ǳ��

��

��(10.28)�� Ǳ�� Ai(x)��

�� Bi(x)��

E(ξ) = αAi(ξ) + βBi(ξ).

�� α� β �� ξ → ∞�� Bi(ξ) → ∞��Ǳ�� β = 0��Ǳ

E(ξ) = αAi(ξ).

10.1 �� 283

�� (10.26)�� z = 0�� ξ = −(ωL/c)2/3�� (ωL/c) 1��

��Ai(ξ)��

Ai(−ξ) ∼ |ξ|−1/4

√π

cos

(2|ξ|3/2

3− π

4

), when ξ 1,

��

Ez=0 =α√

π(ωL/c)1/6

{exp

[i

(2

3

ωL

c− π

4

)]+ exp

[−i(2

3

ωL

c− π

4

)]}

=α

2√π(ωL/c)1/6

ei(2ωL/3c−π/4)

[1 + exp

{−i(4

3

ωL

c− π

2

)}].

�� z = 0��Ǳ��Ǳ��

�� Φ��

��Ǳ

Ez=0 = EFS {1 + exp (−iΦ)} ,

�� EFS ��

α = 2√π(ωL/c)1/6EFSe

−i(2ωL/3c−π/4),

Φ =4

3

ωL

c− π

2.

�� α��Ǳ

E(ξ) = 2√π(ωL/c)1/6EFSe

−i(2ωL/3c−π/4)Ai(ξ). (10.29)

�� ξ ≈ −1� (z − L) = −(c2L/ω2)1/3 ��

��Ǳ

∣∣∣∣Emax

EFS

∣∣∣∣2

≈ 3.6(ωL/c)1/3.

��ǱB = −i(c/ω)(∂E/∂z)ey��

B(ξ) = −i2√π(c/ωL)1/6EFSAi

′(ξ). (10.30)

�� ξ = 0��Ǳ

|B (ξ = 0) | = 2√π(c/ωL)1/6EFSAi

′(0) = 0.9175(c/ωL)1/6. (10.31)

284 ��

10.1.3 ��

��Æ��

��Ǳ��

��Æ��Ǳ�� s��

�� p��

��Ǳ s��

�� z��Ǳ� y − z �

�� x−��Ǳ

∂2Ex

∂y2+∂2Ex

∂z2+ω2

c2ε(ω, z)Ex = 0 (10.32)

�Ǳ�� z−�� y−��ky = (ω/c) sin θ�� θ��Æ��

Ex = E(z) exp

(iω sin θ

cy

),

��

d2E(z)

dz2+ω2

c2[ε(z)− sin2 θ

]E(z) = 0. (10.33)

��(10.11)�� ε(z)− sin2 θ = 0��

�� ε = 1 − ne(z)/nc��

��Ǳ

ne(z)/nc = cos2 θ,

��Ǳ�� ne = ncz/L��

� z = L cos2 θ�

��Ǳ p��

� p��E · ∇ne �= 0��

��Ǳ ��

��

��Ǳ�� p��

��

10.1 �� 285

�� y − z��Ǳ�� p��Ǳ

E = Eyey + Ezez.

��Ǳ

∂2Ey

∂y2+∂2Ey

∂z2+ω2

c2ε(ω, z)Ey +

∂ ln ε(z)

∂z

∂Ey

∂y= 0, (10.34)

∂2Ez

∂y2+∂2Ez

∂z2+ω2

c2ε(ω, z)Ez +

∂

∂z

[Ez∂ ln ε(z)

∂z

]= 0. (10.35)

�� x−��Ǳ∂2Bx

∂y2+∂2Bx

∂z2− ∂ ln ε

∂z

∂Bx

∂z+ω2

c2ε(ω, z)Bx = 0. (10.36)

��(10.36)��

Bx = B(z) exp [i(ωt− sin θωy/c)] ,

��

d2B

dz2− ∂ ln ε(z)

∂z

dB

dz+ω2

c2[ε(ω, z)− sin2 θ

]B = 0. (10.37)

��∇×B = iωεE��

Ez =sin θB(z)

ε(z).

�� ε(z) = 0�� ε→ 0��

��

��B(z = L)��

��B(z = L cos2 θ)�� Ǳ s

��

B(z = L) ≈ e−βB(z = L cos2 θ) = 0.9EFS(c/ωL)1/6e−β,

�� β��Ǳ

β =

∫ L

L cos2 θ

ω

c

√z

L− cos2 θdz = (2ωL/3c) sin3 θ.

��Ǳ

B(z = L) = 0.9EFS(c/ωL)1/6 exp

(−2ωL sin3 θ

3c

).

286 ��

��Ǳ

Ez(z = L) =Ed

ε(z),

��

τ = (ωL/c)1/3 sin θ,

��

Ed =EFS√2πωL/c

φ(τ).

�� φ(τ) = 2.3τ exp(−2τ 3/3)�

�� Ǳ

ε = 1−ω2pe

ω(ω + iν),

�� ν�� ν � ω��Ǳ

Iabs = limν→0

∫ ∞

0

νE2

z

8πdz =

1

8πlimν→0

∫ ∞

0

νsin2 θB2(z)

|ε|2 dz

=1

8πlimν→0

∫ ∞

0

ν

(1− z/L)2 + (z/L)2(ν/ω)2sin2 θB2(z)dz

=ωL

8πlimν→

∫(ν/ω)2

(1− z/L)2 + ν2sin2 θB2(z)d(z/L)

=ωL

8π

∫πδ(z/L− 1) sin2 θB2(z)d(z/L)

=ωL

8E2

d .

��Ǳ I = cE2FS/8π��Ǳ

fabs =IabsI

=φ2(τ)

2.

10.1�� (ωL/c)1/3 sin θ��Æ

Ǳ ��Ǳ ��Ǳ��

��Æ��

��Ǳ�� Æ��

10.2 ��

��

��

10.2 �� 287

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

(ωL/c)1/3 sinθf ab

s

10.1: ��(ωL/c)1/3 sin θ��

�� z−��

��

��Ǳ��Ǳ��

�� vg = c(1− ne/nc)1/2��

��

vph = c(1− ne/nc)1/2.

�� ne/nc � 1��

��

��Ǳ

γ =1√

1− v2ph/c2=

ωl

ωpe,

�� ωl �� ωl/ωpe 1��

��

��Ǳ��

��

��Ǳ

∂δn

∂t+ ne∇ · u = 0, (10.38a)

me∂u

∂t= −eE+ Fp, (10.38b)

∇ · E = −4πeδn, (10.38c)

288 ��

�� δn�� ne �� Fp ��(�3.10�)�

Fp = − e2

4meω2∇E2

l , (10.39)

�� El��Ǳ

I =c

8πE2

l ,

�� Ǳ

Fp = −∇ I

2cnc

.

�� φp��

φp = − I

2cenc, (10.40)

��Ǳ

Fp = e∇φp. (10.41)

�� φ�� E = −∇φ� ��(10.38b)��

me∂

∂t∇ · u = −e∇ · E+∇ · Fp

= e∇2φ+ e∇2φp.

��(10.38a)� ��(10.38c)� �� u�� δn��

�� (∂2

∂t2+ ω2

pe

)∇2φ = −ω2

pe∇2φp.

�� φ��

(∂2

∂t2+ ω2

pe

)φ = −ω2

peφp. (10.42)

��(10.42)��

�� z−�� φp ��

�� z − vgt�� z �� t�Ǳ�� φp��

��

φp(r, z, t) = φ0(r)f0(z − vgt) (10.43)

10.2 �� 289

�� f0�� φ0��(10.43)��(10.42)��

φ = φ0(r)f(z − vgt), (10.44)

�� f �� (∂2

∂t2+ ω2

pe

)f = −ω2

pef0. (10.45)

��

ξ = z − vgt, τ = t,

��(10.45)�Ǳ (d2

dξ2+ k2p

)f(ξ) = −k2pf0(ξ). (10.46)

�� kp = ωpe/vg�� ξ → +∞�� f0(ξ) = 0��

��Ǳ�� f(ξ) → 0��

� �Ǳ

f(ξ) = kp

∫ ∞

ξ

f0(ξ′) sin[kp(ξ − ξ′)]dξ′. (10.47)

��

f0(ξ) = Imax exp

(− ξ2

2L2

).

��(10.47)��

f(ξ) = i

√2π

4(kpL)Imaxe

−ikpξ−(kpL)2/2

×[ei2kpξ erf

(ξ/L+ ikpL√

2

)+ erf

(−ξ/L+ ikpL√

2

)− ei2kpξ + 1

].

�� erf(x)�� ξ → −∞��

f(ξ) =√2πImaxkpLe

−(kpL)2/2 sin kpξ. (10.48)

�(10.48)�� kpL�� kpL = 1��Ǳ

fmax/Imax =√2πe−1/2 = 1.52.

10.2�� kpL = 1��

290 ��

-3 -2 -1 0 1

-1

0

1

kpξ/2π

f(ξ)

/I max

激激激激

10.2: kpL = 1��

��

10.3 ��

��kpL 1��

��

��Ǳ��

��

��

��(��)�

10.3.1 ��

��

��(��)��

��

��

��

B = ∇×A, (10.49a)

E = −∇φ− 1

c

∂A

∂t. (10.49b)

10.3 �� 291

��

∇×B =1

c

∂E

∂t+

4π

cj,

��

(1

c2∂2

∂t2−∇2

)A = −1

c

∂

∂t∇φ+

4π

cj. (10.50)

��Ǳ��

��

j = jt + jl.

��

∇ · jt = 0. (10.51)

��Ǳ

0 =∂ρ

∂t+∇ · j = ∂ρ

∂t+∇ · jl.

��

∇2φ = −4πρ.

� ��

∇ ·(∂

∂t∇φ− 4πjl

)= 0.

��∂

∂t∇φ = 4πjl. (10.52)

��(10.50)�Ǳ (1

c2∂2

∂t2−∇2

)A =

4π

cjt. (10.53)

Ǳ��A · ∇ne = 0��

��

∂ut

∂t≈ − e

mE =

e

mc

∂A

∂t,

�� ut = (e/mc)A��Ǳ

jt = −eneut = −e2ne

mcA, (10.54)

292 ��

��A��Ǳ

(1

c2∂2

∂t2−∇2

)A = −4πe2

mc2neA. (10.55)

��Ǳ��

A = A0 +As,

��

ne = ne0 + δne,

��Ǳ

(∂2

∂t2− c2∇2 + ω2

pe

)As = −4πe2

mδneA0. (10.56)

��

��

��

��

��

10.3.2 ��

��Ǳ��

��Ǳ��Ǳ

∂ne

∂t+∇ · (neue) = 0,

∂ue

∂t+ ue · ∇ue = − e

me

(E+

1

cue ×B

)− ∇Pe

neme

.

��ǱǱ��Ǳ��ue = uet + uel��

B =∇×A =mec

e∇× uet =

mec

e∇× ue,

��∇× vel = 0��

−ue · ∇ue − ue × (∇× ue) = −1

2∇ (ue · ue) ,

10.3 �� 293

��Ǳ

∂uel

∂t=

e

me∇φ− 1

2∇(uel +

eA

mec

)2

− ∇Pe

neme.

��

∂ne1

∂t+ ne0∇ · uel = 0,

∂uel

∂t=

e

me∇φ− e2

m2ec

2∇ (A0 ·As)

2 − 3v2Te

ne0∇δne.

��Ǳ γ = 3� v2Te = Te/meǱ��

�� φ��

�� (∂2

∂t2− 3v2Te∇2 + ω2

pe

)δne =

n0e2

m2ec

2∇2(A0 ·As).

��

��

10.3.3 ��

��

�e

m∇φ− e2

m2c2∇ (A0 ·As)

2 − v2Te

n0∇δne = 0,

��

��Ǳ γ = 1��

��Ǳ∂δni

∂t+ ni0∇ · uil = 0,

∂uil

∂t= −Ze

mi

∇φ,

�� Ǳ��Ǳ��

��

∂2δni

∂t2− ni0Ze

mi∇2φ = 0

��

δne = Zδni

294 ��

��Ǳ(∂2

∂t2− c2s∇2

)δne =

Zn0e2

memic2∇2(A0 ·As).

��Ǳ��

��Ǳ��

10.3.4 ��

��

��

��

Ǳ��

��Ǳ(∂2

∂t2− c2∇2 + ω2

pe

)As = −4πe2

meδneA0,(

∂2

∂t2− 3v2Te∇2 + ω2

pe

)δne =

n0e2

m2ec

2∇2(A0 ·As).

��Ǳ

A0 = A0 cos(k0 · r− ω0t),

��

(ω2 − k2c2 − ω2pe)As(k, ω) =

4πe2

2me

A0[δne(k− k0, ω − ω0) + δne(k+ k0, ω + ω0)],

(ω2 − 3k2v2Te − ω2pe)δne(k,ω) =

k2e2n0

2m2ec

2A0 · [As(k− k0, ω − ω0) +As(k+ k0, ω + ω0)].

��Ǳ (k, ω)��Ǳ (k− k0, ω − ω0)

� (k + k0, ω + ω0)��Ǳ

��Ǳ��

�� δne(k− 2k0, ω − 2ω0)�

δne(k+ 2k0, ω + 2ω0)��

ω2 − ω2epw =

ω2pek

2v2osc4

[1

D(ω − ω0,k− k0)+

1

D(ω + ω0,k+ k0)

].

��

D(ω,k) = ω2 − k2c2 − ω2pe,

10.3 �� 295

vosc =eE0

meω0c=

eA0

mec2.

��

�� 1/D(ω + ω0,k+ k0)��

(ω2 − ω2epw)[(ω − ω0)

2 − (k− k0)2 − ω2

pe] =ω2pek

2v2osc4

.

��

ω = ωepw + iγ, γ � ωepw

��

(ωepw − ω0)2 − (k− k0)

2 − ω2pe = 0.

��

γ =kvosc4

[ω2pe

ωepw(ω0 − ωepw)

]1/2.

��

��

Ǳ νe��Ǳ νs��

(γ + νe)(γ + νs) = γ20 .

��Ǳ

γ0 ≥√νeνs.

10.3.5 ��

��

��Ǳ(∂2

∂t2− c2∇2 + ω2

pe

)As = −4πe2

meδneA0,(

∂2

∂t2− c2s∇2

)δne =

Zn0e2

memic2∇2(A0 ·As).

��

ω2 − k2c2s =k2v2oscω

2pi

4

[1

D(ω − ω0,k− k0)+

1

D(ω + ω0,k+ k0)

].

296 ��

��Ǳ

γ =1

2√2

k0voscωpi√ω0k0cs

.

��Ǳ

γ >√νiνs.

��

��(two plasmon decay instability)��(Langmuir decay instability)��(ion-acousticdecay instability)��Ǳ��Ǳ��

��Ǳ��

�� Ǳ

EM → EM + IAW (stimulated Brillouin scattering);

EM → EM + EPW (stimulated Raman scattering);

EM → EPW + EPW (two plasmon decay instability);

EM → EPW + IAW (ion-acoustic decay instability);

EPW → EPW + IAW (Langmuir decay instability).

��

��

��

��

��

��

��

��(10.28)��

E(ξ) =

∫C

F (s)esξds. (10.57)

��(10.28)��∫C

(s2 − ξ)F (s)esξds = 0. (10.58)

10.3 �� 297

��

− F (s)esξ∣∣C+

∫C

[s2F (s) +

dF (s)

ds

]esξds = 0. (10.59)

��F (s)��

dF

ds+ s2F = 0.

��Ǳ

F (s) = exp(−s3/3). (10.60)

��

F (s)esξ∣∣C= exp(−s3/3 + sξ)

∣∣C= 0. (10.61)

�� s−��

−π6< arg(s) <

π

6,π

2< arg(s) <

5π

6, − 5π

6< arg(s) < −π

2, (10.62)

� |s| → ∞��exp(−s3/3 + sξ)��Ǳ ��

��

��Ǳ

fn(ξ) =1

2πi

∫Cn

e−s3/3+sξds, n = 1, 2, 3. (10.63)

n = 1��ǱAi(ξ)�Ǳ�� ξ��

��

Ai(ξ) =2

π

∫ ∞

−∞cos

(x3

3+ ξx

)dx. (10.64)

��(10.63)��Ǳ��

f2(ξ) = e−i2π/3Ai(ξe−i2π/3),

f2(ξ) = ei2π/3Ai(ξei2π/3).

��

Bi(ξ) = i[f2(ξ)− f3(ξ)],

Ǳ��Ai(ξ)��10.3��

298 ��

−15 −10 −5 0 5

−0.4

−0.2

0

0.2

0.4

0.6

x

Ai(x

)

−15 −10 −5 0 5−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

x

Bi(x

)

10.3: ��Ai(x)��Bi(x)��

��Ǳ��

Ai(ξ) =

⎧⎪⎪⎨⎪⎪⎩

√ξ

3

[I−1/3

(2

3ξ3/2)− I1/3

(2

3ξ3/2)]

, for ξ > 0,√|ξ|3

[J1/3

(2

3|ξ|3/2

)+ J−1/3

(2

3|ξ|3/2

)], for ξ < 0,

Bi(ξ) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

√ξ

3

[I−1/3

(2

3ξ3/2

)+ I1/3

(2

3ξ3/2)]

, for ξ > 0,√|ξ|3

[J−1/3

(2

3|ξ|3/2

)− J−1/3

(2

3|ξ|3/2

)], for ξ < 0.

� ξ → +∞��

Ai(ξ) ∼ ξ−1/4

2√πexp

[−2ξ3/2

3

],

Bi(ξ) ∼ ξ−1/4

2√πexp

[2ξ3/2

3

].

� ξ → −∞��

Ai(ξ) ∼ |ξ|−1/4

√π

cos

(2|ξ|3/2

3− π

4

),

Bi(ξ) ∼ |ξ|−1/4

√π

sin

(2|ξ|3/2

3− π

4

).

�� ξ = −1.018793��Ǳ

Aimax = 0.5356.

��

��

��Ǳ��Ǳ��

��Ǳ��

��

��

�� g��Ǳ��

��

11.1 ��

��N ��

��

wN = AN exp(−EN/T ), (11.1)

�� EN � N �� T �� AN ��

��Ǳ��

EN =

N∑i=1

p2i

2m+ ΦN , (11.2)

�� pi �� ΦN ��

��N ��Ǳ

wN =1

N !

1

(2π�)3N1

QN(V, T )exp

[−(

N∑i=1

p2i2m

+ ΦN

)/T

], (11.3)

�� Ǳ��QN(V, T )�� Ǳ

QN(V, T ) =1

N !

∫exp

[− 1

T

(N∑i=1

p2i /2m+ ΦN

)]d3Npd3Nr

(2π�)3N, (11.4)

300 ��

�� d3Npd3Nr � d3p1d3r1 · · · d3pNd3rN ��(11.4)��

��Ǳ��

∫ ∞

−∞exp

(− p2

2mT

)dp =

√2πmT,

��Ǳ

QN(V, T ) =(2πmT )3N/2

N !(2π�)3N

∫exp [−ΦN/T ] d

3Nr =1

N !λ3NTZN(V, T ), (11.5)

��

λT =2π�√2πmT

(11.6)

��ZN(V, T )�Ǳ�� Ǳ

ZN(V, T ) =

∫e−ΦN/Td3Nr. (11.7)

��QN(V, T )��Ǳ

F (V, T ) = −T lnQN (V, T ). (11.8)

��Ǳ

dF (V, T ) = −SdT − PdV, (11.9)

�� S �� P ��(11.8)(11.9)��

S = −(∂F

∂T

)V

, P = −(∂F

∂V

)T

. (11.10)

��

��

��

��Ǳ

ΦN = 0,

��(11.7)��ǱZ idN = V N

��Ǳ

QidN(V, T ) =

V N

N !λ3NT.

11.2 �� 301

�� lnN ! ≈ N lnN −N ��

F id = −NT[ln

V

λ3TN+ 1

]. (11.11)

��(11.10)��

S = N

(ln

V

λ3TN+

5

2

), P =

NT

V= nT. (11.12)

�� n = N/V ��

��Ǳ

QN (V, T ) = QidN(V, T )

ZN(V, T )

V N. (11.13)

��Ǳ

F = F id + F ex, (11.14)

�� F ex�� Ǳ

F ex = −T lnZN(V, T )

V N. (11.15)

11.2 ��

��N ��

��

� N ��(11.3)��N ��

DN(r1, · · · , rN) =∫wN

N∏i=1

d3pi=exp(−ΦN/T )

ZN(V, T ). (11.16)

��N ��DN ��

��Ǳ��

��

�� s��

Gs(r1, · · · rs), s = 1, 2, · · · (11.17)

302 ��

��

1

V sGs(r1, · · · rs)d3r1 · · ·d3rs

�� s�� r1 → r1 + d3r1, · · · , rN → rs + d3rs��

�� Gs��DN ��Ǳ

Gs(r1, · · · rs) = V s

∫V

· · ·∫V

DN (r1, · · · , rN)N∏

i=s+1

d3ri. (11.18)

��Gs��

�

A =∑

1≤i1<···<is≤N

a(ri1 , · · · ris), (11.19)

�� a � N �� s �� (11.19)��N !/s!(N − s)! �� s �� DN ��

�(11.19)��Ǳ

〈A〉 =∫V

· · ·∫V

ADN(r1, · · · , rN)N∏i=1

d3ri

=N !

V ss!(N − s)!

∫V

· · ·∫V

a(r1, · · · rs)Gs(r1, · · · rs)s∏

i=1

d3ri. (11.20)

�Ǳ��

��Ǳ

Φ =∑

1�i<j�N

φ(rij), (11.21)

�� φ(rij)� i�� j �� rij ��

��(11.20)��Ǳ

U = 〈Φ〉 = N(N − 1)

2V 2

∫V

∫V

φ(r12)G2(r1, r2)d3r1d

3r2

= Vn2

2

∫φ(r)G2(r)d

3r.

��

�� G2(r1, r2) = G(r12)��

��

N, V → ∞, limN,V→∞

N

V= n. (11.22)

11.2 �� 303

��

��G2��

11.2.1 ��

��(11.14)(11.15)��Ǳ��

�� λ�� 0� 1��

��Ǳ

λΦ = λ∑

1�i<j�N

φ(rij). (11.23)

�� λ = 0�� λ = 1��

��(11.8)��(11.23)��Ǳ

F (V, T ;λ) = −T ln

[1

N !λ3NT

∫V

· · ·∫V

exp (−λΦ/T )N∏i=1

d3ri

], (11.24)

��(11.24)�� λ��

∂F (λ)

∂λ=

∫V· · ·∫VΦexp (−λΦ/T )

∏i

d3ri

∫V· · ·∫Vexp (−λΦ/T )

∏i

d3ri=V

2n2

∫φ(r)G2(r;λ)d

3r, (11.25)

�� G2(λ; r)��Ǳ(11.23)��

G2(r;λ) = V 2

∫V· · ·∫Vexp (−λΦ/T )

N∏i=3

d3ri

∫V· · ·∫Vexp (−λΦ/T )

N∏i=1

d3ri

.

��(11.25)�� λ�� F (V, T ;λ = 0)��

�� F id��(11.23)��Ǳ

F = F id +V

2n2

∫ 1

0

dλ

∫V

φ(r)G2(r;λ)d3r. (11.26)

��(11.15)��

F ex =V

2n2

∫ 1

0

dλ

∫V

φ(r)G2(r;λ)d3r. (11.27)

304 ��

11.2.2 ��

��G2(r;λ)�� P = −(∂F/∂V )T �

��(11.26)�� Ǳ��Ǳ�

��

��QN ��(11.8)��(11.10)��Ǳ

P = T∂ lnQN

∂V= T

1

QN

∂QN

∂V. (11.28)

Ǳ�� ∂QN/∂V��

QN (V, T ;λ) = λ3N∫

· · ·∫

V

exp

[−

∑1�i<j�N

φ(λrij)/T

]N∏i=1

d3ri, (11.29)

�� λ�� λ = 1�� QN(V, T ;λ = 1) = QN(V, T )�

�λ �= 1��(11.29)��r→ λr��

QN(V, T ;λ) =

∫· · ·∫

V ′

exp

[−

∑1�i<j�N

φ(rij)/T

]N∏i=1

d3ri = QN (V′, T ).

��V ′ ��V ��V ′

V= λ3.

��

∂QN (V, T )

∂V= lim

V ′→V

QN (V′, T )−QN(V, T )

V ′ − V= lim

λ→1

QN (V, T ;λ)−QN(V, T ; 1)

V λ3 − V

= limε→0

QN (V, T ; 1 + ε)−QN (V, T ; 1)

V (1 + 3ε)− V=

1

3V

[∂QN (V, T ;λ)

∂λ

]λ=1

. (11.30)

�QN(V, T ;λ) ��λ ��

∂QN (V, T ;λ)

∂λ=

3N

λQN − λ3N

T

∫· · ·∫

V

∑1�i<j�N

rijφ′(λrij)e−

∑1�i<j�N φ(λrij)/T

N∏i=1

d3ri.

�� λ = 1��

∂QN (V, T ;λ)

∂λ

∣∣∣∣λ=1

= 3NQN − N(N − 1)

2T

∫· · ·∫

V

rijφ′(rij)e−

∑1�i<j�N φ(rij)/T

N∏i=1

d3ri

= 3V QN

[n− n2

6T

∫rdφ(r)

drG2(r)d

3r

],

11.2 �� 305

��Ǳ

P =T

3V QN

[∂QN (V, T ;λ)

∂λ

]λ=1

= nT − n2

6

∫rdφ(r)

drG2(r)d

3r. (11.31)

11.2.3 ��

��Ǳ��W �� V ��

��W ��

fW (r) =

{1, if r ∈ W,

0, if r /∈ W.(11.32)

��W ��Ǳ

NW =

N∑i=1

fW (ri). (11.33)

��Ǳ

〈NW 〉 = N

V

∫V

fW (r)G1(r)d3r = n

∫W

G1(r)d3r.

�� G1 ��

�� V ��G1 = 1��

〈NW 〉 = nW�

�W ��Ǳ

⟨(NW − 〈NW 〉)2

⟩=

∫V

· · ·∫V

{N∑i=1

[fW (ri)− α]

}2

DN

N∏i=1

d3ri, (11.34)

�� α =∫WG1d

3r/V��(11.34)��

⟨(NW − 〈NW 〉)2

⟩= 2

∫V

· · ·∫V

DN

∑1�i<j�N

[fW (ri)− α] [fW (rj)− α]

N∏i=1

d3ri

+

∫V

· · ·∫V

DN

N∑i=1

[fW (ri)− α]2N∏i=1

d3ri

=N(N − 1)

V 2

∫V

∫V

G2(r1, r2) [fW (r1)− α] [fW (r2)− α] d3r1d3r2

+N

V

∫V

G1(r) [fW (r)− α] d3r,

306 ��

� ��

N(N − 1)

V 2

∫V

∫V

G2(r1, r2) [fW (r1)− α] [fW (r2)− α] d3r1d3r2

=N(N − 1)

V 2

∫W

∫W

[G2(r1, r2)−G1(r1)G1(r2)]d3r1d

3r2,

N

V

∫V

G1(r) [fW (r)− α] d3r = 〈NW 〉(1− 〈NW 〉

N

),

��W ��Ǳ

⟨(NW − 〈NW 〉)2

⟩=N(N − 1)

V 2

∫W

∫W

[G2(r1, r2)−G1(r1)G1(r2)]d3r1d

3r2+〈NW 〉(1− 〈NW 〉

N

).

��W Ǳ��

��

〈δn2〉 = n + n2

∫[G2(r)− 1]d3r. (11.35)

11.3 Yvon-Born-Green��

��

��

��

��

��(11.16)�� ri (1 � i � s)��

∂DN

∂ri+

1

T

∂ΦN

∂riDN = 0. (11.36)

�� V s�� rs+1, · · · , rN ��

� �(11.18)��

∂Gs

∂ri+

1

TV s

∫V

· · ·∫V

∂ΦN

∂riDN

N∏i=s+1

d3ri = 0, (1 � i � s). (11.37)

�� ∂ΦN/∂ri��

∂ΦN

∂ri=

∂

∂ri

∑1�k<j�s

φ(|rk − rj|) +∂

∂ri

∑s+1�j�N

φ(|ri − rj |)

≡ ∂Φs

∂ri+

∂

∂ri

∑s+1�j�N

φ(|ri − rj|),

11.3 Yvon-Born-Green�� 307

�� Φs =∑

1�k<j�s φ(|rk − rj|)� s��(11.37)��Ǳ

V s

∫V

· · ·∫V

∂ΦN

∂riDN

N∏i=s+1

d3ri = V s

∫V

· · ·∫V

[∂Φs

∂ri+

∂

∂ri

∑s+1�j�N

φ(|ri − rj|)]DN

N∏i=s+1

d3ri

=∂Φs

∂riGs +

N − s

V

∫V

∂φ(|ri − rs+1|)∂ri

Gs+1d3rs+1.

��(11.37)��Ǳ

∂Gs

∂ri+

1

T

∂Φs

∂riGs +

1

T

N − s

V

∫V

∂φ(|ri − rs+1|)∂ri

Gs+1d3rs+1 = 0, (1 � i � s). (11.38)

�� (11.36)�� s��

��(11.38)�� s�� s + 1��

�� s = 1, · · · , N�� N ��

��DN ��

��(11.38)��(11.38)��

��(11.38)� Ǳ��(11.38)� ��Ǳ�� (11.38)��

��N ��

��(11.38)��

∂Gs

∂ri+

1

T

∂Φs

∂riGs +

n

T

∫V

∂φ(|ri − rs+1|)∂ri

Gs+1d3rs+1 = 0, (1 � i � s). (11.39)

��Ǳ Yvon-Born-Green�� Ǳ YBG�� (11.39)��

��

��

Gs(r1, · · · rs)−s∏

i=1

G1(ri) → 0, for all the |ri − rj | → ∞. (11.40)

��

limV→∞

1

V

∫V

G1(r)d3r = 1, (11.41)

308 ��

limV→∞

1

V

∫V

Gs+1(r1, · · · rs+1)d3rs+1 = Gs(r1, · · · rs). (11.42)

11.4 ��

�Ǳ YBG��—��

Æ�� r0�� Æ��

�� Lennard-Jones��

φ(r) = 4u(R−12 − R−6), R = r/r0.

�� u��

�� 1/r6��

��

��

�� r0�� n1/3 ��Ǳ��

nr30 � 1�� 1��

��(11.39)� ��nr30�� nr30Ǳ��

��(11.39)��(11.31)��

11.4.1 ��

��Ǳ��(�� nr30 )��

Gs = G(0)s + nG(1)

s + n2G(2)s + · · · (11.43)

��(11.43)��YGB�� (11.39)�� n��

∂G(0)s

∂ri+

1

T

∂Φs

∂riG(0)

s = 0, (1 � i � s) (11.44)

∂G(1)s

∂ri+

1

T

∂Φs

∂riG(1)

s +1

T

∫∂φ(|ri − rs+1|)

∂riG

(0)s+1d

3rs+1 = 0, (1 � i � s) (11.45)

��Ǳ

G(0)s (r1, · · · rs)−

s∏i=1

G(0)1 (ri) → 0, for all the |ri − rj| → ∞, (11.46)

11.4 �� 309

G(1)s (r1, · · · rs)−

s∑i=1

G(1)1 (ri)

∏j �=i

G(0)1 (rj) → 0, for all the |ri − rj| → ∞, · · · (11.47)

��

limV→∞

1

V

∫V

G(0)1 (r)d3r = 1, lim

V→∞1

V

∫V

G(1)1 (r)d3r = 0. (11.48)

��

G(0)s = C(0)

s (r1, . . . , rs) exp (−Φs/T ) ,

��(11.44)��

∂C(0)s

∂ri= 0, (1 � i � s).

�� C(0)s ��(11.46)�� C

(0)s = [C

(0)1 ]s��

�(11.48)�� C(0)1 = 1��

G(0)s = exp (−Φs/T ) , G

(0)1 = 1. (11.49)

��(11.49)��(11.45)��

∂G(1)s

∂ri+

1

T

∂Φs

∂riG(1)

s +1

T

∫∂φ(|ri − rs+1|)

∂riexp (−Φs+1/T ) d

3rs+1 = 0. (11.50)

��Ǳ��

1

T

∂φ(|ri − rs+1|)∂ri

e−Φs+1/T = −e−Φs/T∂

∂riexp

{− 1

T

s∑j=1

φ(|rj − rs+1|)}

= −e−Φs/T∂

∂ri

s∏j=1

[1 + f(|rj − rs+1|)] ,

��

Φs+1 = Φs +

s∑j=1

φ(|rj − rs+1|), (11.51)

��

f(r) = exp [−φ(r)/T ]− 1. (11.52)

��(11.50)��Ǳ

∂G(1)s

∂ri+

1

T

∂Φs

∂riG(1)

s = e−Φs/T∂

∂ri

∫ s∏j=1

[1 + f(|rj − rs+1|)] d3rs+1. (11.53)

310 ��

��(11.53)��

1 +

s∑j=1

f(|rj − rs+1|)

�� f(r)� � r��∫f(r)d3r��

��

∂

∂ri

∫ [1 +

s∑i=j

f(|rj − rs+1|)]d3rs+1 = 0.

��(11.53)��

∂G(1)s

∂ri+1

T

∂Φs

∂riG(1)

s = e−Φs/T∂

∂ri

∫ { s∏j=1

[1 + f(|rj − rs+1|)]− 1−s∑

j=1

f(|rj − rs+1|)}d3rs+1.

(11.54)��

G(1)s = C(1)

s (r1, · · · , rs) exp (−Φs/T ) , (11.55)

��(11.54)��C(1)s ��

∂C(1)s

∂ri=

∂

∂ri

∫ { s∏j=1

[1 + f(|rj − rs+1|)]− 1−s∑

j=1

f(|rj − rs+1|)}d3rs+1.

��

C(1)s =

∫ { s∏i=1

[1 + f(|ri − rs+1|)]− 1−s∑

i=1

f(|ri − rs+1|)}d3rs+1 + ks,

�� ks Ǳ��

��Ǳ��Ǳ

G(1)s = e−Φs/T

∫ { s∏i=1

[1 + f(|ri − rs+1|)]− 1−s∑

i=1

f(|rs − rs+1|)}d3rs+1. (11.56)

��(11.49)(11.56)��

Gs = e−Φs/T

{1 + n

∫ [ s∏i=1

[1 + f(|ri − rs+1|)]− 1−s∑

i=1

f(|ri − rs+1|)]d3rs+1

}

(11.57)��G2��(11.57)��s = 2��

G2(r) = e−φ(r)/T

[1 + n

∫f(|r− r′|)f(r′)d3r′

]. (11.58)

11.5 �� 311

11.4.2 ��

��(11.58)��(11.31)��

P

nT= 1− n

6T

∫rdφ

dre−φ/T

[1 + n

∫f(|r− r′|)f(r′)d3r′

]d3r. (11.59)

��

− 1

T

dφ

dre−φ/T =

d

dre−φ(r)/T =

d

drf(r),

��(11.59)��Ǳ

− n

6T

∫rdφ

dre−φ(r)/T d3r = −n

2

∫f(r)d3r.

��(11.59)��Ǳ

− n2

6T

∫∫rdφ

dre−φ(r)/T f(|r− r′|)f(r′)d3rd3r′ = n2

6

∫∫rdf(r)

drf(|r− r′|)f(r′)d3rd3r′.

� r��∫∫rdf(r)

drf(|r− r′|)f(r′)d3rd3r′ = −2

∫∫f(r)f(|r− r′|)f(r′)d3rd3r′,

��

p

nT= 1− β1

2n− 2β2

3n2 − · · · , (11.60)

�� β1 β2Ǳ

β1 =

∫f(r)d3r, β2 =

1

2

∫∫f(|r− r′|)f(r)f(r′)d3rd3r′. (11.61)

�� β1 β2�Ǳ��

11.5 ��

��

��Ǳ��

φ(r) ∝ 1

r.

��

��∫f(r)d3r��

��

312 ��

11.5.1 ��

�� Ǳ��

� ��Ǳ��Ǳ�

��Ǳ��

��(11.39)��Ǳ��(11.39)�� s = 2 i = 1�� s = 2�� Φ2 = φ(r12) = e2/r12��

∂G2

∂r1+

1

T

∂φ(r12)

∂r1G2 +

n

T

∫V

∂

∂r1φ(r13)G3d

3r3 = 0. (11.62)

Ǳ��

G2(r1, r2) = G1(1)G1(2) + C(1, 2)

G3(r1, r2, r3) = G1(1)G1(2)G1(3) + C(1, 2) + C(2, 3) + C(1, 3) +R(1, 2, 3),

�� C �� R� ��

�� C → 0� ��

�� R → 0�

��G1 = 1��

∂C(1, 2)

∂r1+

1

T

∂φ(r12)

∂r1[1 + C(1, 2)] +

n

T

∫V

∂φ(r13)

∂r1[C(2, 3) +R(1, 2, 3)]d3r3 = 0.

Ǳ��

��

∂C(1, 2)

∂r1+

1

T

∂φ(r12)

∂r1[1 + C(1, 2)] +

n

T

∫V

∂φ(r13)

∂r1C(2, 3)d3r3 = 0. (11.63)

��

∂2C

∂2r1+

1

T

∂2φ(r12)

∂r21[1 + C(1, 2)] +

1

T

∂φ

∂r1· ∂C∂r1

+n

T

∫V

∂2φ(r13)

∂r21C(2, 3)d3r3 = 0.

��∂2φ(r12)

∂r21= −4πe2δ(r1 − r2),

∂2φ(r13)

∂r21= −4πe2δ(r1 − r3),

��

∂2C(1, 2)

∂2r1+

1

T

∂φ

∂r1· ∂C(1, 2)

∂r1− 4πne2

TC(1, 2) =

4πe2

T[1 + C(1, 2)]δ(r1 − r2).

11.5 �� 313

�� C(1, 2)�� C(1, 2) = C(r12)��Ǳ

∂2C(r)

∂2r+

1

T

∂φ(r)

∂r· ∂C(r)

∂r− 4πne2

TC(r) =

4πe2

T[1 + C]δ(r).

��

1

r2d

dr

(r2dC

dr

)− e2

T

1

r2dC

dr− C

λ2D=

4πe2

T[1 + C]δ(r). (11.64)

�� λD ��

�� Æ�� e2/T��

n−1/3�� λD��

� r > λD�� r�� r/λD → r��

d2C

dr2+

2

r

dC

dr− g

1

r2dC

dr− C = 0.

�� g = 1/4πnλ3D ��

�� g � 1�� g��

d2C

dr2+

2

r

dC

dr− C = 0.

�� C(r → ∞) = 0�Ǳ

C(r) = const × e−r

r.

��

��

� r ∼ n−1/3�� r�� r/n−1/3 → r��

d2C

dr2+

2

r

dC

dr− g2/3

(4π)1/31

r2dC

dr− (4π)2/3g2/3C = 0.

�� g2/3��

C(r) = const × 1

r.

� r ∼ e2/T �� r�� r/(e2/T ) → r��

d2C

dr2+

2

r

dC

dr− 1

r2dC

dr− g2C = 0.

314 ��

�� g2��

C = const + const × e−1/r.

��

��Ǳ��

��

��

��Ǳ

C(r) =

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

−1 + exp(−e2/Tr) +O(g2), r � e2

T,

− e2

Tr+O(g2/3), r ∼ n−1/3,

− e2

Trexp(−r/λD) +O(g), r � λD.

(11.65)

��(11.65)��

C(r) = −1 + exp

[− e2

Tre−r/λD

]. (11.66)

��O(g)��(11.66)�� (11.65)��(11.66)��Ǳ

G2(r) = exp

[− e2

Tre−r/λD

]. (11.67)

��(11.1)��(11.67)�� g = 10−4��

�� 0� 1�

��Ǳ�� C(r)��

��O(g)��

��

�� g2�� g��

�� g��

��(11.65)�(11.66)�� r ∼ n−1/3 � ��(11.65)�� O(g2/3)��

�� g��Ǳ�

11.5.2 ��

��Ǳ��

��

11.5 �� 315

10−4

10−2

100

0

0.2

0.4

0.6

0.8

1

r/λD

G2(r

)

� 11.1: ��g = 10−4��

��(11.31)�� g��

��

G2 � 1− e2

Trexp(−r/λD). (11.68)

��

P = nT − n2

6

∫r∂φ

∂rG2(r)d

3r

= nT − T

24πλ3D.

��P

nT= 1− 1

24πnλ3D= 1− g

6. (11.69)

��

�� g � 1��

��

�� g��Ǳ�

��

��Ǳ��

��Ǳ

E =3

2NT

(1− g

3

), (11.70)

316 ��

�Ǳ

S = N

[(ln

V

λ3TN+

5

2

)+

5

6g

], (11.71)

��Ǳ

F = −NT[(

lnV

λ3TN+ 1

)+

1

3g

]. (11.72)

11.6 ��

� ��

�� g � 1��

��

��

��

Ǳ��

��ǱZe��Ǳ a�� a��

�� Ǳ

4

3πa3 =

V

Ni

=1

ni

(11.73)

��ni ��

4π

3a3ne = Z,

��Ǳ

ne =3Z

4πa3.

��ǱZe��Ǳ a� ��

��Ǳ

φe(r) = − 3Ze

4πa3

∫r′<a

d3r′

|r− r′| = −3Ze

2a+Ze

2a

(ra

)2.

��Ǳ

−1

2

∫r<a

3Ze

4πa3φe(r)d

3r =3

5

(Ze)2

a.

��Ǳ

−∫r<a

Ze

rned

3r = −3

2

(Ze)2

a

11.7 �� 317

��Ǳ

U =3

5

(Ze)2

a− 3

2

(Ze)2

a= −0.9

(Ze)2

a. (11.74)

�Ǳ��

��(11.74)��

11.7 ��

1. �� V ��

2Ekin +

N∑i=1

ri · Fi = 0,

�� ri� i�� Fi� i�� Ekin��

Ekin =

N∑i=1

1

2mv2i ,

� (· · · )��

(· · · ) = limτ→∞

∫ τ

0

(· · · )dt.

2. ��1�� [��⟨

N∑i=1

ri · Fi

⟩= −

⟨N∑i=1

ri ·∂ΦN

∂ri

⟩− P

∮∂V

r · df ,

�� ΦN ��]

3. ��Ǳ��Ǳ

φ(r) =

{+∞ for r < r0,

0 for r > r0.

��

318 ��

4. ��Ǳ

ε =3

2nT(1− g

3

).

5. ��Ǳ

F = −NT[ln

V

λ3TN+ 1 +

g

3

].

��

��

��

��

��

12.1 ��

��

��

��

��N�� N(t; r1,p1; . . . rN ,pN )

��

∂�N∂t

+

N∑i=1

{∂�N∂ri

· ∂HN

∂pi− ∂�N∂pi

· ∂HN

∂ri

}= 0, (12.1)

��

HN =N∑i=1

p2i2m

+∑

1�i<j�N

φ(|ri − rj|), (12.2)

��(12.2)��(12.1)��

∂�N∂t

+

N∑i=1

vi ·∂�N∂ri

−N∑i=1

∂

∂ri

[ ∑1�k<j�N

φ(|rk − rj|)]· ∂�N∂pi

= 0. (12.3)

�� s��Fs(t; r1,p1; · · · ; rs,ps)�

�� N ��Ǳ

Fs(t; r1,p1; · · · ; rs,ps) = V s

∫Ω

· · ·∫Ω

�N(t; r1,p1; · · · ; rN ,pN)N∏

i=s+1

d3rid3pi, (12.4)

12.2 �� 321

��

∂F2

∂t+ v1 ·

∂F2

∂r1+ v2 ·

∂F2

∂r2− ∂φ(|r1 − r2|)

∂r1· ∂F2

∂p1

− ∂φ(|r1 − r2|)∂r2

· ∂F2

∂p2

= 0.

��ǱdF2(t; r1,p1; r2,p2)

dt= 0, (12.8)

�� (r1,p1)� (r2,p2)��

H2 =p21

2m+

p22

2m+ φ(|r1 − r2|).

��(12.8)��Ǳ

F2[t; r1(t),p1(t); r2(t),p2(t)] = F2[t0; r10,p10; r20,p20].

�� {r1(t),p1(t); r2(t),p2(t)}��Ǳ t�� {r10,p10; r20,p20}��Ǳ t0 ��Ǳ��Ǳ��

��Ǳ t0��

��

F2(t0; r10,p10; r20,p20) = F1(t0; r10,p10)F1(t0; r20,p20). (12.9)

��(12.6)��

∂F1

∂t+ v1 ·

∂F1

∂r1= C(F ), (12.10)

�� C(f)Ǳ��

C(F ) = n

∫∂φ(|r1 − r2|)

∂r1· ∂

∂p1{F1(t0; r10,p10)F1(t0; r20,p20)}d3r2d3p2. (12.11)

�� Æ��Ǳ�� r0��

� 10−10��

��(12.11)��Ǳ

C(F ) = n

∫∂φ(|r1 − r2|)

∂r1· ∂

∂p1{F1(t0;p10)F1(t0;p20)}d3r2d3p2. (12.12)

�� F1(t0;p10)F1(t0;p20)��Ǳ��

��

322 ��

�� (12.8)�

d

dtF1(t0;p10)F1(t0;p20)

=d

dtF1[t0;p10(r1,p1; r2,p2)]F1[t0;p20(r1,p1; r2,p2)]

=

(v1 ·

∂

∂r1+ v2 ·

∂

∂r2− ∂φ(r12)

∂r1· ∂

∂p1

− ∂φ(r12)

∂r2· ∂

∂p2

)F1(t0;p10)F1(t0;p20)

= 0.

��(12.12)�� ∂/∂p2 ��Ǳ�� p10 �

p20�� (r1 − r2)��

C(F ) =

∫vrel ·

∂

∂r{F1(t0;p10)F1(t0;p20)}d3rd3p2,

�� vrel = v1 − v2��Ǳ z−��

vrel ·∂

∂r= vrel

∂

∂z,

� z��

C(F ) =

∫{F1(t0;p10)F1(t0;p20)}z=∞

z=−∞vrelρdρdφd3p2.

� z = −∞��

p10 = p1, p20 = p2,

� z = ∞�� (p10,p20)��

��Ǳ (p1,p2)�� (p10,p20)��Ǳ�� ρ��

p10 = p′1(ρ), p20 = p′

2(ρ),

�� Ǳ

ρdρdφ = dσ,

r0v

� t− t0 �L

v,

�� L�� t0� t��

��

C(F ) =

∫[F1(t,p

′1)F1(t,p

′2)− F1(t,p1)F1(t,p2)] vreldσd

3p2. (12.13)

12.3 �� 323

��

∂F1(t, r1,p1)

∂t+ v1 ·

∂F1(t, r1,p1)

∂r1=

∫[F1(t,p

′1)F1(t,p

′2)− F1(t,p1)F1(t,p2)] vreldσd

3p2

(12.14)

12.3 ��

��

��

F αβ2 (t; ra,pa; rβ,pβ) = fα(t; r,p)fβ(t; r

′,p′) + P αβ2 (t; r,p; r′,p′), (12.15)

�� fα(t, r,p)� α��

�� g = 1/4πnλ3D ��

�� Ǳ�� g��

��

∂fα∂t

+ v · ∂fα∂r

− ∂

∂r

∑β

nβ

[∫φαβ(|r− r′|)fβ(t; r′,p′)d3r′d3p′

]· ∂fα∂p

= 0.

��

Φ(t, r) =1

eα

∑β

nβ

∫φαβ(|r− r′|)fβ(t; r′,p′)d3r′d3p′,

=∑β

nβ

∫eβ

|r− r′|fβ(t; r′,p′)d3r′d3p′ (12.16)

��

��∂fα∂t

+ v · ∂fα∂r

− eα∂Φ

∂r· ∂fα∂p

= 0, (12.17)

��(12.16)��∇2� ��

∇2

(1

r

)= −4πδ(r),

��

∇2Φ = −4π∑β

eβnβ

∫fβ(t; r,p)d

3p. (12.18)

324 ��

��(12.17)��—��

��

��

�� Ǳ

∂fα∂t

+ v · ∂fα∂r

− eα

(E+

1

cv×B

)· ∂fα∂p

= 0, (12.19)

�� E� B��

∇×B =1

c

∂E

∂t+

4π

cj, ∇ · E = 4πρ, (12.20a)

∇× E= −1

c

∂B

∂t, ∇ ·B = 0. (12.20b)

��

j =∑α

eαnα

∫vfα(t, r,p)d

3p, ρ =∑α

eαnα

∫fα(t, r,p)d

3p. (12.21)

�� Te � mec2��

��Ǳ�

∂fα∂t

+ v · ∂fα∂r

− eαmα

(E+

1

cv×B

)· ∂fα∂v

= 0, (12.22)

�� fα�� t�� r�� v��

��Ǳ

j =∑α

eαnα

∫vfα(t, r,v)d

3v, ρ =∑α

eαnα

∫fα(t, r,v)d

3v. (12.23)

�� BBGKY��Ǳ��Ǳ��

��

��

12.4 ��

��(12.19)�(12.20)�(12.21)��

12.4 �� 325

�� εl(ω, k)��

��Ǳ��

��Ǳ

fa(t, r,v) = fαM(v) + δfα(t, r,v), (12.24)

��

fαM =1

(2π)3/2v3αexp

(− v2

2v2α

), (12.25)

Ǳ�� vα = (Tα/mα)1/2 ��

�� δfα��

|δfα/fαM | � 1. (12.26)

��

∂δfα∂t

+ v · ∂δfα∂r

+eαmα

E · ∂fαM∂v

= 0, (12.27)

∇ · E = 4π∑α

eαnα

∫δfαd

3v. (12.28)

��(12.27)�(12.28)��

δfα,E ∝ eik·r−iωt.

��(12.27)��

δfα = − eαmα

E · ∂fαM/∂vi(k · v − ω)

, (12.29)

��

δρ =∑α

eαnα

∫δfαd

3p = −[∑

α

e2αnα

mα

∫∂fαM/∂v

i(k · v − ω)d3v

]· E.

��∇ · P = −δρ�� ik · P = −δρ��

D(ω,k) = E(ω,k) + 4πP = εl(ω, k)E(ω,k),

��Ǳ

εl(ω, k) = 1−∑α

ω2pα

k2

∫k · ∂fαM/∂vk · v − ω

d3v, (12.30)

326 ��

�� ω2pα = 4πnαe

2α/mα � α ��(12.30)�

�� k ��Ǳ z−��(12.25)��(12.30)�� k��

εl(ω, k) = 1−∑α

ω2pα

k

∫ ∞

−∞

∂FαM/∂vzkvz − ω

dvz, (12.31)

��

FaM (vz) =

∫fαM(v)dvxdvy =

1

(2π)1/2vαexp

(− v2z2v2Tα

), (12.32)

��

��(12.31)�� vz ��Ǳ��

��

ω = kvz = k · v. (12.33)

�� Ǳ��

ω → ω + i0. (12.34)

��

��

E ∝ e−iωt+δt → 0 as t→ −∞, .

�� δ → +0��

�� eδt �� t → +∞�� Ǳ��

��Ǳ

εl = 1−∑α

ω2pα

k

∫ ∞

−∞

∂FαM/∂vzkvz − ω − i0

dvz. (12.35)

�(12.32)��

εl(ω, k) = 1 +∑α

1

(kλDα)2

[1 +

ω√2kvα

1√π

∫ ∞

−∞

exp(−z2)z − ω/

√2kvα − i0

dz

](12.36)

�� λDα = (Tα/4πnαe2α)� α��(12.36)��

��

Z(ξ) =1√π

∫ ∞

−∞

exp(−z2)z − ξ − i0

dz. (12.37)

12.5 �� 327

��Ǳ

εl(ω, k) = 1 +∑α

1

(kλDα)2[1 + ξαZ(ξα)] , (12.38)

�� ξα = ω/√2kvα�

�� εt(ω, k)Ǳ��

��

εt(ω, k) = 1 +∑α

ω2pα

ω2ξαZ(ξα). (12.39)

12.5 ��

�� Z(ξ)��

��

��(12.37)��Z(ξ)��

��Ǳ

Z(ξ) =1√π

∫C

exp(−z2)z − ξ

dz, (12.40)

�� C �� z−�� ξ ��

12.1�� (12.34)��Ǳ��C �� ξ ��

�� C ��Ǳ

Z(ξ) =

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

1√π

∫ ∞

−∞

exp(−z2)z − ξ

dz, if Im ξ > 0,

1√πP

∫ ∞

−∞

exp(−z2)z − ξ

dz + i√πe−ξ2, if Im ξ = 0,

1√π

∫ ∞

−∞

exp(−z2)z − ξ

dz + i2√πe−ξ2 , if Im ξ < 0,

(12.41)

��(12.41)� ξ��

��Ǳ��Ǳ��

�� ξ−��Ǳ��(��)�

Z(ξ) = i√π exp(−ξ2)− 2e−ξ2

∫ ξ

0

ep2

dp. (12.42)

�� ξǱ�� 12.2��

328 ��

Re z

Im z

ξ

Im ξ > 0

C

(a)

Re z

Im z

ξ

Im ξ = 0

C

(b)

Re z

Im z

ξ

Im ξ < 0

C

(c)

12.1: ��Z(ξ)��

0 1 2 3 4-0.4

0

0.4

0.8

1.2

ξ

Re[Z(ξ)]Im[Z(ξ)]

12.2: �Ǳ��

12.6 �� 329

12.5.1 ��

��(12.36)�� Ǳ��mi → ∞��Ǳ

εl(ω, k) = 1 +1

(kλDe)2

[1 +

ω√2kve

Z

(ω√2kve

)].

��ω√2kve

→ xe, (kλDe)−1 → α,

��Ǳ

εl = 1 + α2 [1 + xeZ(xe)] . (12.43)

�� ω �� xe = iy� ��(12.48)�(12.43)��Ǳ

εl = 1 + α2{1−√πy exp(y2)[1− erf(y)]}.

�� y = 0��

εl(0, k) = 1 + α2.

� y 1��

erf(y) ∼ 1− exp(−y2)πy

∞∑k=0

(−1)kΓ(k + 1/2)

y2k,

��

εl ∼ 1 +α2

2y2+ · · · .

�� y → +∞�� εl → 1�� y ��

1 + α2�� 1�� 12.3�� ω��

ω → ±∞��Ǳ� 1�

12.6 ��

��

��Ǳ

εl = 1 + α2

[1− 2e−x2

e

∫ xe

0

ep2

dp

]+ iπ1/2α2xe exp(−x2e),

330 ��

12.3: ��εl��

�� α = (kλDe)−1� xe = ω/

√2kvTe��Ǳ��

��

ε′′l = π1/2α2xe exp(−x2e) > 0, when ω > 0.

��Ǳ��

�� Ǳ

Q =ω

8πε′′l |E|2 =

ω

8π1/2|E|2α2xe exp(−x2e).

��

��Ǳ��

��Ǳ�� 1946��Ǳ��

��

�� Ǳ

E(x, t) = ReE0ei(kz−ωt)eδt,

�� δ�� t = −∞��

w = w0 + δw, z = z0 + δz,

�� w�� z�� w0� z0� �� δw

� δz0��

12.6 �� 331

��Ǳ

medδw

dt= −eE(t, z0) = −eReE0e

ikt(w0−ω/k)etδ,

��

δw = − e

meRe

E(t, z0)

ik(w0 − ω/k) + δ,

δz = − e

me

ReE(t, z0)

[ik(w0 − ω/k) + δ]2.

�� Ǳ

q = −ewE(t, z) = −e(w0 + δw)E(t, z0 + δz)

= −ew0(∂E/∂z0)δz − eδwE(t, x0)

=e2

2me|E|2Re

{−iω + δ

[ik(w0 − ω/k) + δ]2

}

=e2

2me|E|2 δ(δ2 + ω2 − k2w2

0)

[k2(w0 − ω/k)2 + δ2]2

=e2

2m|E|2 d

dw0

w0δ

δ2 + k2(w0 − ω/k)2.

��

Q = ne

∫ ∞

−∞qf(w0)dw0

= − ne

2me

e2|E|2∫ ∞

−∞

w0δ

δ2 + k2(w0 − ω/k)2df(w0)

dw0

dw0,

��

limδ→0

δ

δ2 + x2= πδ(x),

�

Q = − ne

2mee2|E|2π

∫1

kδ(w0 − ω/k)w0

df(w0)

dw0dw0

= −ω2pe

8|E|2 1

kw0df(w0)

dw0

∣∣∣∣w0=ω/k

=ω

8π1/2|E|2α2xe exp(−x2e).

��

�� me

4πnee2

∫ ∞

0

ωε′′l (ω, k)dω =π

2.

332 ��

12.7 ��

��Ǳ�

��Ǳ��Ǳ

εl(ω, k) = 1 + α2 [1 + xeZ(xe)]

��

εl(ω, k) = 0.

�� ω ��Ǳ�� εl ��

z = ρ exp(−iπ/4)��(��)� ��

Im εl =

√π

2α2ρ

{sin(ρ2)[1 + 2S(ρ)] + cos(ρ2)[1 + 2C(ρ)]

}= 0,

Re εl = 1 + α2{1 +

√π/2ρ[cos ρ2(1 + 2S(ρ))− sin2 ρ(1 + 2C(ρ))]

}= 0.

��

S(x) ∼ 1

2− 1√

2πxcos x2 +O(1/x2),

C(x) ∼ 1

2+

1√2πx

sin x2 +O(1/x2),

��Ǳ

Im εl ∼ 2√πα2ρ sin(ρ2 + π/4) = 0.

��

ρ2 = 3π/4 + 2mπ,

��Ǳ 0 (��Ǳ��)��

(kλD)2 = 2π(2m+ 3/4)1/2 − 1,

��Ǳ��m��

�� 12.4�� k��

�� α2 1�(��)��

| Imω/Reω| � 1,

12.7 �� 333

12.4: ��εl(ω, k) = 0��

��

ω = ωr + iγ, |γ/ω| � 1.

��

εl(ω, k) = Re εl(ωr, k) + iγ∂ Re εl∂ωr

+ i Im εl = 0,

��Ǳ

Re εl(ωr, k) = 0,

γ = − Im εl(ωr, k)

(∂ Re εl/∂ωr).

��

ωr/kvTe 1.

��

Re εl = 1 + α2

[− 1

(ωr/kvTe)2− 3

(ωr/kvTe)4

]= 0,

��Ǳ

ω2r = ω2

pe + 3k2v2Te = ω2pe(1 + 3k2λ2De).

334 ��

��Ǳγ

|ωr|= −

√π

8α3 exp

[−1

2α2 − 3

2

].

�� |γ/ωγ| � 1�� α 1��

12.8 ��

��

� � ��Ǳ

εl = 1 + α2 [1 + xeZ(xe)] + α2ZTeTi

[1 + xiZ(xi)] ,

��

xe =ω√2kvTe

, xi =ω√2kvT i

.

��

vT i �ω

k� vTe,

��

xe � 1, xi 1.

��

εl = 1 + α2(1 + iπ1/2xe) + α2ZTeTi

(− 1

2x2i− 3

4x4i+ iπ1/2xie

−x2i

).

��Ǳ��

1 + α2 + α2ZTeTi

(− 1

2x2i− 3

4x4i

)= 0.

��Ǳ (ωr

kvi

)2

=α2

1 + α2

ZTeTi

,

�� (ωr

kvi

)2

=ZTeTi

��Ǳ

γ

ωr= −

[√π

8

Zme

mi+

√π

2

(ZTeTi

)3/2

e−ZTe/2Ti

].

12.9 �� 335

��

ZTeTi

1.

��ω

k=

√ZTemi

��ZTe/Ti 1��

γ

ωr= −

√π

8

Zme

mi.

12.9 ��

��

�� S(k, ω)��Ǳ

S(k, ω) =1

2πN

∫ ∞

−∞dτ〈n∗

k(t)nk(t+ τ)〉eiωτ . (12.44)

��

��

�� n(r, t)�� k = 0� ω = 0�

��Ǳ��

��

��Ǳ��

��

� ��

��

��N ��N/Z �� i��

��Ǳ

∇ ·D = −4πeδ(r− ri0 − vit),

��

��D = εlE = −εl∇φ�� ε(ω,k)��

ε exp(ik · r− iωt) = ε(ω,k) exp(ik · r− iωt)��

∇ · (ε∇φ) = 4πeδ(r− ri0 − vit),

336 ��

��

φik(t) =

4πe

εl(k · vi,k)k2e−ik·(ri0+vit).

�� φik(t)��Ǳ�� Ǳ��

��Ǳ��

∂δfe∂t

+ v · ∂δfe∂r

− eE · ∂fM∂p

= 0.

��

∂δfek(t)

∂t+ ik · vδfek(t) + ikeφi

k(t) ·∂fM∂p

= 0

�Ǳ φik(t) ∝ exp(−ik · vit)��Ǳ

δfek(t) =4πe2

εl(k · vi,k)k2k · ∂fM/∂pk · (vi − v)

e−ik·(ri0+vit).

�� i��Ǳ

δnk(t) =

[1 +

∫d3p

4πe2


]e−ik·(ri0+vit)

�� j��Ǳ

δnk(t) = −Z∫d3p

4πe2


e−ik·(Ri0+Vit).

��Ǳ

δnk(t) =N∑i=1

[1 +

∫d3p

4πe2


]e−ik·(ri0+vit)

− Z

N/Z∑j=1

∫d3p

4πe2


e−ik·(Ri0+Vit)

12.9 �� 337

��Ǳ

S(ω,k) =1

2πN

∫ ∞

−∞dτ〈n∗

k(t)nk(t+ τ)〉eiωτ

=1

2πN

∫ ∞

−∞dτeiωτ

⟨{N∑i=1

[1 +

∫d3p

4πe2

ε∗l (k · vi,k)k2k · ∂fM/∂pk · (vi − v)

]eik·(ri0+vit)

−ZN/Z∑j=1

∫d3p

4πe2


e−ik·(Ri0+Vit)

⎫⎬⎭

×{

N∑i=1

[1 +

∫d3p

4πe2


]e−ik·(ri0+vit+viτ)

−ZN/Z∑j=1

∫d3p

4πe2


e−ik·(Ri0+Vit+Viτ)

⎫⎬⎭⟩.

��Ǳ��

〈eik·ri0e−ik·ri′0〉 = δi,i′, 〈eik·Rj0e−ik·Rj′0〉 = δj,j′, 〈eik·rj0e−ik·Rj0〉 = 0.

��

S(ω,k) =1

2πN

∫ ∞

−∞dτeiωτ

⟨N∑i=1

∣∣∣∣1 +∫d3p

4πe2


∣∣∣∣ e−ik·viτ

⟩

+Z2

2πN

∫ ∞

−∞dτeiωτ

⟨N/Z∑i=1

∣∣∣∣∫d3p

4πe2


∣∣∣∣ e−ik·Viτ

⟩

=1

2πN

N

(2π)3/2v3e

∫ ∞

−∞dτeiωτ

∫d3v′

∣∣∣∣1 +∫d3p

4πe2

εl(k · v′,k)k2k · ∂fM/∂pk · (v′ − v)

∣∣∣∣ e−v′2/2v2ee−ik·v′τ

+Z2

2πN

N/Z

(2π)3/2v3i

∫ ∞

−∞dτeiωt

∫d3v′

∣∣∣∣∫d3p

4πe2

εl(k · v′,k)k2k · ∂fM/∂pk · (v′ − v)

∣∣∣∣ e−v′2/2v2i e−ik·v′τ .

�� τ �� ∫ ∞

−∞dτei(ω−k·v′)τ = 2πδ(ω − k · v′),

��

S(ω,k) =1

(2π)3/2v3e

∫d3v′

∣∣∣∣1 +∫d3p

4πe2

εl(k · v′,k)k2k · ∂fM/∂pk · (v′ − v)

∣∣∣∣ e−v′2/2v2eδ(ω − k · v′)

+Z

(2π)3/2v3i

∫d3v′

∣∣∣∣∫d3p

4πe2

εl(k · v′,k)k2k · ∂fM/∂pk · (v′ − v)

∣∣∣∣ e−v′2/2v2i δ(ω − k · v′).

338 ��

��Ǳ�� k�� k��

��

S(ω,k) =1

(2π)1/2ve

∫dv′

�

∣∣∣∣1 +∫d3p

4πe2

εl(kv′�,k)k2

k · ∂fM/∂pkv′� − k · v

∣∣∣∣ e−v′2�/2v2e δ(ω − kv′

�)

+Z

(2π)1/2vi

∫dv′

�

∣∣∣∣∫d3p

4πe2

εl(kv′�,k)k2

k · ∂fM/∂pkv′� − k · v

∣∣∣∣ e−v′2�/2v2eδ(ω − kv′

�).

��

S(ω,k) =1

(2π)1/2kve

∣∣∣∣1 +∫d3p

4πe2

εl(ω,k)k2k · ∂fM/∂pω − k · v

∣∣∣∣2

exp

[− ω2

2k2v2e

]

+Z

(2π)1/2kvi

∣∣∣∣∫d3p

4πe2

εl(ω,k)k2k · ∂fM/∂pω − k · v

∣∣∣∣2

exp

[− ω2

2k2v2i

].

��Ǳ

χe(ω, k) =4πe2

k2

∫d3p

k · ∂fM/∂pω − k · v ,

��Ǳ

S(ω,k) =1

(2π)1/2kve

∣∣∣∣1 + χe(ω, k)

εl(ω, k)

∣∣∣∣2

exp

[− ω2

2k2v2e

]+

Z

(2π)1/2kvi

∣∣∣∣χe(ω, k)

εl(ω, k)

∣∣∣∣2

exp

[− ω2

2k2v2i

].

12.10 ��

12.10.1 ��

�� Im ξ > 0��

1

z − ξ= i

∫ ∞

0

dte−i(z−ξ)t,

��(12.41)��

Z(ξ) =i√π

∫ ∞

−∞exp(−z2)dz

∫ ∞

0

exp[−i(z − ξ)t]dt

=i√π

∫ ∞

0

dt exp(iξt− t2/4

) ∫ ∞

−∞dz exp

[−(z + it/2)2

]

= i exp(−ξ2)∫ ∞

0

dt exp[−(t/2− iξ)2]

= i√π exp(−ξ2)− 2e−ξ2

∫ ξ

0

ep2

dp.

12.10 �� 339

Im z < 0� 1/(x− z)��Ǳ��

1

z − ξ= −i

∫ ∞

0

dtei(z−ξ)t.

��

Z(ξ) = i√π exp(−ξ2)− 2e−ξ2

∫ ξ

0

ep2

dp. (12.45)

�� ξ−��

12.10.2 ��Fresnel��

��ξ = ρ exp(−iπ/4)��ǱFresnel��(12.45)��ξ = ρ exp(−iπ/4)��

Z(ρe−iπ/4) = iπ1/2eiρ2 − 2eiρ

2

e−iπ/4

∫ ρ

0

e−it2dt

= iπ1/2eiρ2

{1 +

2

π1/2eiπ/4

[∫ ρ

0

cos(t2)dt− i

∫ ρ

0

sin(t2)dt

]},

��Fresnel��

S(x) =

√2

π

∫ x

0

sin(t2)dt,

C(x) =

√2

π

∫ x

0

cos(t2)dt,

��

Z(ρe−iπ/4) = i√πeiρ

2{1 + (2i)1/2[C(ρ)− iS(ρ)]}. (12.46)

�ρ 1�� Fresnel��

C(x) ≈ 1

2, S(x) ≈ 1

2,

��

Z(ρe−iπ/4) ≈ 2π1/2ei(ρ2+π/2) when ρ 1. (12.47)

340 ��

12.10.3 ��

�� ξ = iy�� y��

Z(iy) = i√π exp(y2)[1− erf(y)], (12.48)

�� erf(x)��Ǳ

erf(y) =2√π

∫ y

0

e−p2dp.

12.10.4 ��

�� (12.45)��

−2e−ξ2∫ ξ

0

ep2

dp = −2

∫ ξ

0

e−(ξ2−p2)dp = −2∞∑n=0

(−)nξ2n+1

n!

∫ 1

0

(1− t2)ndt

= −√πξ

∞∑n=0

(−)nξ2n

Γ(n+ 3/2).

��

Z(ξ) = i√πe−ξ2 −

√πξ

∞∑n=0

(−)nξ2n

Γ(n+ 3/2). (12.49)

�|ξ| 1��

��Ǳ��Ǳ

Z(ξ) =1√πP

∫ ∞

−∞

exp(−z2)z − ξ

dz + i√πe−ξ2 .

�ξ 1��Ǳ

1√πP

∫ ∞

−∞

exp(−z2)z − ξ

dz ∼ − 1√π

1

ξP

∫ ∞

−∞

exp(−z2)1− (z/ξ)

dz

= − 1√πξ

∫ ∞

−∞

∞∑n=0

(z

ξ

)n

e−z2dz

= − 1√πξ

∫ ∞

−∞

∞∑n=0

(z

ξ

)2n

e−z2dz

= −∞∑n=0

Γ(n + 1/2)√πξ2n+1

.

12.11 �� 341

��

Z(ξ) ∼ i√πe−ξ2 −

∞∑n=0

Γ(n + 1/2)√πξ2n+1

, when Im ξ = 0.

��Ǳ

Z(ξ) ∼ i√πσe−ξ2 −

∞∑k=0

Γ(k + 1/2)√πξ2k+1

, (12.50)

��

σ =

⎧⎪⎪⎨⎪⎪⎩

0, when Im ξ > 0,

1, when Im ξ = 0,

2, when Im ξ < 0.

�Ǳ��

Z(ξ) = i√π(1− ξ2)− ξ(2− 4ξ2/3) + · · · , (12.51a)

Z(ξ) ∼ i√πσe−ξ2 −

(1

ξ+

1

2ξ3+

3

4ξ5+ · · ·

)when |ξ| 1. (12.51b)

12.10.5 ��

��(12.45)��

dZ

dξ= −2(1 + ξZ),

Z(ξ = 0) = iπ1/2.

��Ǳ��

��Ǳ��

12.11 ��

1. ��Ǳ

εt(ω, k) = 1 +∑α

ω2pα

ω2ξαZ(ξα).

2. ��Imx < 0��Ǳ

Z(x) = i√π exp(−x2)− 2e−x2

∫ x

0

ep2

dp.

342 ��

0 1 2 3 4-0.4

0

0.4

0.8

1.2

ξ

Re[Z(ξ)]Im[Z(ξ)]

3. ��|ω| → ∞��

εl(ω, k) ∼ 1−ω2pe

ω2.

��

Braginskii��, Braginskii equations, 153

Kruskal-Shafranov stable criterion, 181

Yvon-Born-Green��Yvon-Born-Greenhierarchy, 307

��Alfven wave, 185��Attwood number, 146��Airy equation, 282

��surface plasma wave, 243

��Boltzmann distribution, 11��wave zone, 96��wave breaking, 257�� Poisson’s equation, 11��trapped particle, 75

��magnetic mirror, 68��mirror ratio, 69��magnetohydrodynamics, 129��magnetic flux surface, 74��magnetoacoustic wave, 186

��one component plasma,17

��one-component plasma,312

��guiding center, 47, 58��Debye length, 12

��Debye screening, 11��plasma parameter, 18��plasma permittivity,

325��plasma dispersion func-

tion, 326��plasma coupling pa-

rameter, 17��lower hybrid wave, 240��lower-hybrid frequency, 230��electromagnetic wave, 225��electric conductivity, 197��electric polarization, 197��current flux function, 169��electric induction, 197��electron plasma wave,

222��classical electron radius,

114

��dynamic form factor, 117

��Faraday rotation, 235��Fermi energy, 16��radiation intensity, 98��Vlasov equation, 323

��upper-hybrid wave, 240��upper-hybrid frequency, 230

344 ��

��resonance absorption, 284Æ�Æ�pinch angle, 68��proper time, 27��generalized Ohm’s law, 154��—��Hamilton-Jacobi equa-

tion, 33

��Helmholtz free energy,300

��chemical equilibrium con-stant, 21

��toroidal magnetic field, 73��cyclotron radiation, 108

��shock wave, 146��laser wakefield, 286��laser wakefield accel-

erator, 287��cluster expansion, 312

Æ��poloidal magnetic field, 73Æ��poloidal magnetic flux func-

tion, 168��permittivity, 198

��electrostatic ion cyclotronwave, 241

��local thermal equilibrium, 129��adiabatic equation, 134

��spatial dispersion, 198��Coulomb logarithm, 151

��Lagrangian, 27��,sausage instability, 179� ��Landau damping, 330

��Langmiur frequency, 14��Langmiur oscillation, 14��ion-sphere model, 316��ion-acoustic wave, 223��ideal gas, 300

��continuity equation, 130��critical density, 278��Liuoville theorem, 319

��Rutherford’s formula, 39��Loerentz invariance, 27

��—��Navier-Stokes equa-tion, 136

��kink instability, 179��—��Euler—Lagrange equa-

tion, 31��Euler equation, 132��dipole radiation, 100��partition function, 299

��drift wave, 246��drift approximation, 58��frequency dispersion, 198

��thermal conductivity, 137��—� ��Rayleigh-Taylor in-

stability, 144

��Saha’s equation, 22��whistler wave, 234

��time dispersion, 198��stimulated Brillouin scat-

tering, 294

�� 345

��stimulated Raman scatter-ing, 294

��Thomson scattering, 113

��passing particle, 75��tokamak, 72

��virial coefficient, 311��configuration integral, 300

�� banana orbit, 75

�� cyclic coordinate, 33

��ponderomotive force, 57��Hugoniot relation, 148

��reduced distribution func-tion, 301

��viscous stress tensor, 135��canonical distribution, 299

��law of mass action, 21

��equation of state, 134��self-consistent field, 323�� stopping power, 212��the principle of least ac-

tion, 31��priciple of least action,

27��action, 27

��,bremsstralung radiation, 101

final - ustc

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