1 Passage of Charged Particles Through Matter ................ 1
1.1 Various Types of Processes ................................. 1
1.2 Kinematics ................................................. 1
1.2.1 Laboratory (Lab) System (LS) and Centre of Mass
System (CM) ................................................ 1
1.2.2 Total Linear Momentum in the CM System Is Zero ...... 2
1.2.3 Relation Between Velocities in the LS and CMS ....... 3
1.2.4 Relation Between the Angles in LS and CMS ........... 4
1.2.5 Recoil Angle ........................................ 5
1.2.6 Limits on the Scattering Angle θ .................... 6
1.2.7 Limits on the Recoil Angle φ ........................ 7
1.2.8 Scattering in Three Dimensions ...................... 8
1.2.9 Scattering Cross-Section ............................ 9
1.2.10 Relation Between Differential Scattering Cross-
Sections ........................................... 10
1.2.11 Kinematics of Elastic Collisions ................... 11
1.2.12 To Derive an Expression for the Recoil Velocity ν2
as a Function of φ ................................. 12
1.2.13 Available Energy in the Lab System and CM System ... 12
1.3 Rutherford Scattering ..................................... 20
1.3.1 Derivation of Scattering Formula ................... 20
1.3.2 Darwin's Formula ................................... 27
1.3.3 Mott's Formula ..................................... 28
1.3.4 Cross-Section for Scattering in the Angular
Interval θ' and θ" ................................. 28
1.3.5 Probability of Scattering .......................... 28
1.3.6 Rutherford Scattering in the LS and CM System ...... 29
1.3.7 Validity of Classical Description of Scattering .... 31
1.3.8 Coulomb Scattering with a Shielded Potential Under
Born's Approximation ............................... 32
1.3.9 Discussion of Rutherford's Formula ................. 33
1.3.10 The Scattering of a Particles and the Nuclear
Theory of Atom ..................................... 34
1.4 Multiple Scattering ....................................... 40
1.4.1 Mean Scattering Angle .............................. 40
1.4.2 Choice of b(max) and b(min) ........................ 43
1.4.3 Mean Square Projected Angle and the Mean Square
Displacement ....................................... 44
1.5 Theory of Ionization ...................................... 46
1.5.1 Bohr's Formula ..................................... 46
1.5.2 Range-Energy-Relation .............................. 50
1.5.3 Energy Loss to Electrons and Nuclei ................ 58
1.5.4 Energy Loss of Heavy Fragments ..................... 58
1.5.5 Energy Loss of Electrons ........................... 60
1.6 Delta Rays ................................................ 60
1.6.1 Energy Spectrum .................................... 60
1.6.2 Angular Distribution ............................... 61
1.6.3 Delta Ray Density .................................. 62
1.7 Straggling ................................................ 63
1.7.1 Theory ............................................. 63
1.7.2 Energy Straggling .................................. 64
1.7.3 Range Straggling ................................... 65
1.8 Cerenkov Radiation ........................................ 68
1.9 Identification of Charged Particles ....................... 72
1.9.1 (a) Momentum and Velocity .......................... 72
1.9.2 (b) Momentum Times Velocity (pβ) and Velocity ...... 72
1.9.3 Energy and Velocity ................................ 72
1.9.4 Simultaneous Measurement of dE/dx and E ............ 72
1.9.5 Energy and Emission Angle .......................... 73
1.10 Bremsstrahlung ............................................ 73
1.11 Questions ................................................. 74
1.12 Problems .................................................. 76
References ................................................ 81
2 Passage of Radiation Through Matter ....................... 83
2.1 Kinds of Interaction ...................................... 83
2.2 The Compton Effect ........................................ 84
2.2.1 Shift in Wavelength ................................ 84
2.2.2 Shift in Frequency ................................. 85
2.2.3 Angular Relation ................................... 86
2.2.4 Differential Cross-Section ......................... 86
2.2.5 Spectrum of Scattered Radiation .................... 87
2.2.6 Compton Attenuation Coefficients ................... 88
2.3 Photoelectric Effect ...................................... 90
2.3.1 Measurement of Photon Energy ....................... 94
2.4 Pair-Production ........................................... 94
2.4.1 Angular Distribution of Pair Electrons ............. 96
2.4.2 Energy Distribution of Pair Electrons .............. 96
2.4.3 Total Pair-Production Cross-Section per Nucleus .... 96
2.5 Nuclear Resonance Fluorescence ............................ 99
2.5.1 Restoring Mechanisms .............................. 101
2.5.2 Mechanical Motion (Ultra Centrifuge) .............. 101
2.5.3 Thermal Motion .................................... 102
2.5.4 Preceding β or γ Emission ......................... 102
2.6 Mossbauer Effect ......................................... 103
2.6.1 Elementary Theory ................................. 105
2.6.2 Importance of Mossbauer Effect .................... 107
2.6.3 Applications ...................................... 108
2.7 Questions ................................................ 119
2.8 Problems ................................................. 120
References ............................................... 122
3 Radioactivity ............................................ 125
3.1 Natural Radioactivity .................................... 125
3.1.1 The Radioactive Decay Law ......................... 125
3.1.2 Mean Life and Half-Life ........................... 126
3.1.3 Activity .......................................... 126
3.1.4 Units of Radioactivity ............................ 127
3.1.5 Unit of Exposure and Unit of Dose ................. 127
3.1.6 Determination of Half-Life Time ................... 132
3.1.7 Law of Successive Disintegration .................. 133
3.1.8 Age of the Earth .................................. 138
3.1.9 Radiocarbon Dating ................................ 141
3.2 β-Decay .................................................. 145
3.2.1 Potential Barrier Problem ......................... 146
3.2.2 Barrier of an Arbitrary Shape ..................... 149
3.2.3 Determination of Nuclear Radius ................... 151
3.2.4 Geiger Nuttall Law ................................ 151
3.2.5 Success of Gamow's Theory ......................... 152
3.2.6 Fine Structure of a Spectrum ...................... 152
3.2.7 Angular Momentum and Parity in a-Decay ............ 153
3.3 β-Decay .................................................. 156
3.3.1 Fermi's Theory .................................... 158
3.3.2 Selection Rules ................................... 166
3.4 Range-Energy Relation .................................... 170
3.4.1 Double β Decay .................................... 170
3.5 Electron Capture ......................................... 178
3.5.1 Decay Constant .................................... 178
3.5.2 Detection ......................................... 181
3.6 Gamma Decay .............................................. 181
3.6.1 Multipole Order of Radiation ...................... 181
3.6.2 Selection Rules for γ-Emission (or Absorption) .... 182
3.6.3 γ-Ray Emission Probability ........................ 182
3.6.4 Internal Conversion ............................... 183
3.6.5 Isomers ........................................... 186
3.6.6 Angular Correlation of Successive Radiation ....... 186
3.7 Questions ................................................ 189
3.8 Problems ................................................. 192
References ............................................... 196
4 General Properties of Nuclei ............................. 197
4.1 Nuclear Sizes ............................................ 197
4.1.1 Scattering of a Particles ......................... 197
4.1.2 Coulomb Energy Term in Weisacker's Mass Formula ... 199
4.1.3 β Transition Energies in Mirror Nuclei ............ 199
4.1.4 High Energy Electron Scattering ................... 200
4.1.5 Mesic Atoms ....................................... 215
4.1.6 Half Lifetimes of α Emitters ...................... 221
4.1.7 High Energy Neutron Scattering .................... 222
4.2 Constituents of the Atomic Nucleus ....................... 222
4.3 Definitions .............................................. 223
4.4 Atomic Mass Unit ......................................... 224
4.5 Nuclear Force ............................................ 224
4.6 Mass Defect, Packing Fraction and Binding Energy ......... 224
4.7 Mass and Energy Equivalence .............................. 226
4.8 Nuclear Instability ...................................... 227
4.9 Stability Against β Decay ................................ 229
4.10 Stability Against Neutron and α Decay and Fission ....... 230
4.11 Charge Independence of Nuclear Forces .................... 230
4.11.1 Iso-spin .......................................... 231
4.12 Ground and Excited States of Nuclei ...................... 234
4.12.1 Nuclear Spin ...................................... 235
4.12.2 Nuclear Parity .................................... 236
4.13 Determination of Nuclear Spin ............................ 237
4.13.1 Nuclear Spin from Statistics ...................... 237
4.13.2 Nuclear Spin from Hyperfine Structure ............. 239
4.13.3 Nuclear Spin from Zeeman Effect ................... 242
4.14 Nuclear Magnetic Dipole Moment ........................... 242
4.14.1 Magnetic Moment of the Neutron ................... 247
4.15 Electric Quadrupole Moment ............................... 248
4.16 Questions ................................................ 258
4.17 Problems ................................................. 259
References ............................................... 261
5 The Nuclear Two-Body Problem ............................. 263
5.1 Deuteron ................................................. 263
5.1.1 Binding Energy of Deuteron ........................ 263
5.1.2 The Ground State of Deuteron ...................... 264
5.1.3 The Probability that the Neutron and Proton Are
Found Outside the Range of Nuclear Forces ......... 268
5.1.4 Excited States of Deuteron ........................ 270
5.1.5 Root Mean Square Radius ........................... 271
5.1.6 The Inclusion of Hard Core Potential in the
Square Well ....................................... 272
5.1.7 Use of the Exponential Wave Function in the
Solution of a Square Well Potential Problem ....... 275
5.1.8 Magnetic Dipole Moment of Deuteron ................ 277
5.1.9 Tensor Force ...................................... 279
5.1.10 Constants of Motion for the Two-Body System ....... 282
5.1.11 Quadrupole Moment ................................. 286
5.2 Nucleon-Nucleon Scattering: Phase Shift Analysis ......... 291
5.2.1 Introduction ...................................... 291
5.2.2 Neutron-Proton Scattering ......................... 292
5.2.3 Phase-Shift Analysis .............................. 293
5.2.4 Physical Interpretation of Partial Waves and
Phase-Shifts ...................................... 298
5.2.5 Integral Expression for Phase Shift ............... 300
5.2.6 Angular Distribution of Scattered Neutrons at
Low Energies ...................................... 303
5.2.7 Optical Theorem ................................... 305
5.2.8 Total Cross Section ............................... 306
5.2.9 Comparison of Experimental Cross-Sections with
the Theory and Evidence for Spin Dependence
of Nuclear Forces ................................. 308
5.2.10 Finite Range Correction ........................... 310
5.2.11 Evidence for Neutron Spin (1/2) ................... 311
5.2.12 Scattering Length ................................. 312
5.3 Effective Range Theory ................................... 313
5.3.1 Triplet Scattering ................................ 317
5.3.2 Singlet Scattering ................................ 318
5.3.3 Nature of the Singlet and Triplet States .......... 321
5.3.4 Cross-Sections for Protons Bound in Molecules ..... 322
5.4 Proton-Proton Scattering; Low Energy ..................... 323
5.5 High Energy Nucleon-Nucleon Scattering ................... 330
5.5.1 Polarization ...................................... 331
5.5.2 Mechanism of Polarization ......................... 334
5.6 Properties of the Nucleon-Nucleon Force .................. 336
5.6.1 Exchange Forces ................................... 336
5.6.2 Effect of Exchange Forces ......................... 337
5.6.3 Exchange Forces and Saturation .................... 339
5.7 Yukawa's Theory ..................................... 341
5.8 Questions ................................................ 348
5.9 Problems ................................................. 350
References ............................................... 351
6 Nuclear Models ........................................... 353
6.1 Need for a Model ......................................... 353
6.2 Type of Nuclear Models ................................... 353
6.3 Fermi Gas Model .......................................... 354
6.3.1 Fermi Energy ...................................... 356
6.3.2 Asymmetric Term (δ) in the Mass Formula ........... 358
6.3.3 Odd-Even Term in the Mass Formula ................. 359
6.3.4 Threshold for Particle Production in Complex
Nuclei ............................................ 359
6.3.5 Application to Neutron Stars ...................... 360
6.3.6 Energy Levels of Individual Nucleons .............. 361
6.4 Shell Model .............................................. 362
6.4.1 Magic Numbers ..................................... 364
6.4.2 Theory ............................................ 365
6.4.3 LS Coupling ....................................... 368
6.4.4 Predictions of the Shell Model .................... 371
6.4.5 Magnetic Moments .................................. 373
6.4.6 Schmidt Lines ..................................... 375
6.4.7 Parity of Nuclei .................................. 377
6.4.8 Nuclear Isomerism ................................. 377
6.4.9 Criticism of the Shell Model ...................... 379
6.5 The Liquid Drop Model .................................... 380
6.5.1 Semi-Empirical Mass Formula ....................... 381
6.5.2 Nuclear Instability Against β Emission ............ 383
6.5.3 Instability Against Neutron Decay ................. 385
6.5.4 Instability Against Alpha Decay ................... 385
6.5.5 Fission and Fusion ................................ 385
6.5.6 Defects of Liquid Drop Model ...................... 395
6.5.7 Criticism of Liquid Drop Model .................... 397
6.6 The Collective Model or Unified Model .................... 398
6.6.1 Rotational States ................................. 398
6.6.2 Vibrational States ................................ 399
6.6.3 Electric Quadrupole Moments ....................... 399
6.6.4 Shortcomings of the Shell Model ................... 400
6.6.5 General Theory of Deformed Nuclei ................. 401
6.6.6 Rotational Model .................................. 403
6.6.7 Vibrational Model ................................. 408
6.6.8 Collective Oscillations ........................... 409
6.6.9 Giant Resonances .................................. 412
6.6.10 Nilsson Model ..................................... 413
6.7 Questions ................................................ 420
6.8 Problems ................................................. 421
References ............................................... 423
7 Nuclear Reactions ........................................ 425
7.1 Types of Reactions ....................................... 425
7.2 Energy and Mass Balance .................................. 427
7.3 Conservation Laws for Nuclear Reactions .................. 428
7.3.1 Quantities that Are not Conserved ................. 429
7.4 Cross-Sections ........................................... 429
7.5 Exoergic and Endoergic Reactions ......................... 431
7.5.1 Exoergic Reactions (Q-Value is Positive) .......... 433
7.5.2 Endoergic Reactions (Q-Value is Negative) ......... 434
7.6 Behaviour of Cross-Sections near Threshold ............... 436
7.7 Inverse Reaction ......................................... 438
7.8 Qualitative Features of Nuclear Reactions ................ 440
7.9 Reaction Mechanisms ...................................... 441
7.10 Nuclear Reactions via Compound Nucleus Formation ......... 442
7.10.1 Resonances in the Formation of the Compound
Nucleus ........................................... 444
7.10.2 Width of Resonance Levels ......................... 445
7.10.3 Experimental Verification of the Compound
Nucleus Hypothesis ................................ 446
7.10.4 Energy Level Density .............................. 446
7.11 Partial Wave Analysis of Nuclear Reactions ............... 448
7.12 Slow Neutron Resonances and the Breit-Wigner Theory ...... 452
7.12.1 Resonance Absorption and the 1/ν Law .............. 456
7.12.2 Elastic Scattering ................................ 459
7.13 Optical Model ............................................ 460
7.14 Direct Reactions ......................................... 465
7.14.1 Inelastic Scattering .............................. 466
7.14.2 Charge-Exchange Reactions ......................... 467
7.14.3 Nucleon Transfer Reactions ........................ 468
7.14.4 Break-up Reactions ................................ 472
7.14.5 Knock-out Reactions ............................... 473
7.15 Comparison of Compound Nucleus Reactions and Direct
Reactions ................................................ 474
7.16 Pre-equilibrium Reactions ................................ 474
7.17 Heavy-Ion Reactions ...................................... 477
7.17.1 Characteristics of Heavy Ion Reactions ............ 477
7.17.2 Types of Interactions ............................. 478
7.17.3 Distant Collisions ................................ 479
7.17.4 Deep Inelastic Collisions ......................... 482
7.18 Fusion ................................................... 483
7.18.1 Quark-Gluon Plasma ................................ 488
7.19 Questions ................................................ 497
7.20 Problems ................................................. 498
References ............................................... 501
8 Nuclear Power ............................................ 503
8.1 Nuclear Fission Reactor .................................. 503
8.2 The Thermal Reactor ...................................... 504
8.2.1 Moderation of Neutrons ............................ 504
8.2.2 The Average Energy Decrement ...................... 507
8.2.3 Forward Scattering ................................ 509
8.3 Thermal Neutrons ......................................... 510
8.4 Scattering Mean Free Path (M.F.P.) ....................... 513
8.4.1 Transport Mean Free Path .......................... 514
8.4.2 The Mean Square Distance of Scattering ............ 515
8.5 Slowing-Down Density ..................................... 515
8.5.1 Slowing-Down Time ................................. 517
8.6 Resonance Escape Probability ............................. 518
8.6.1 The Effective Resonance Integral .................. 520
8.7 Diffusion of Neutrons .................................... 521
8.7.1 Leakage of Neutrons ............................... 523
8.7.2 The Diffusion Equation for Thermal Neutrons ....... 524
8.7.3 Extrapolation Distance ............................ 525
8.7.4 Diffusion Length .................................. 529
8.7.5 Relationship Between (r2) and L2 for a Point
Source ............................................ 529
8.7.6 Experimental Measurement of Diffusion Length ...... 531
8.7.7 The Albedo ........................................ 531
8.7.8 Determination of Diffusion Length from the
'Exponential' Pile ................................ 532
8.8 Elementary Theory of the Chain-Reacting Pile ............. 534
8.8.1 Life History of Neutrons and Four-Factor Formula .. 535
8.8.2 Fast-Fission Factor (ε) ........................... 538
8.8.3 Resonance Absorption .............................. 539
8.9 Neutron Leakage and Critical Size ........................ 542
8.10 The Critical Dimension of a Reactor ...................... 543
8.10.1 One Group Theory .................................. 543
8.11 Reactor with a Reflector ................................. 548
8.12 Multigroup Theory ........................................ 549
8.12.1 Experimental Measurement of Critical Size ......... 550
8.13 Fast Neutron Diffusion and the Fermi Age Equation ........ 551
8.13.1 Correction for Neutron Capture .................... 554
8.13.2 Application of Diffusion Equation to a Thermal
Reactor ........................................... 555
8.13.3 Critical Equation and Reactor Buckling ............ 557
8.13.4 The Non-leakage Factors ........................... 558
8.13.5 Criticality of Large Thermal Reactors ............. 560
8.13.6 The Diffusion Length for a Fuel-Moderator
Mixture ........................................... 560
8.13.7 koo for a Heterogeneous Reactor ................... 561
8.13.8 Power ............................................. 565
8.14 The Chain Reaction Requirements .......................... 566
8.15 The Reactor Period ....................................... 569
8.15.1 Thermal Lifetime and Generation Time ............. 569
8.16 Effect of Delayed Neutrons ............................... 571
8.17 Classification of Reactors ............................... 572
8.17.1 Homogeneous Reactor ............................... 572
8.17.2 Heterogeneous Reactors ............................ 573
8.17.3 Fast Reactors ..................................... 575
8.17.4 Breeder Reactors .................................. 576
8.17.5 Thermal Breeders .................................. 577
8.17.6 Fast Breeders ..................................... 578
8.17.7 Doubling Time ..................................... 578
8.18 Other Types of Reactors .................................. 578
8.18.1 Power Reactors .................................... 578
8.18.2 The Pressurized Water Reactor ..................... 579
8.18.3 The Boiling Water Reactor ......................... 579
8.18.4 The Gas-Cooled Natural-Uranium-Graphite Reactor ... 579
8.18.5 The Homogeneous Reactor ........................... 579
8.18.6 Research Reactors ................................. 580
8.18.7 The Water Boiler .................................. 580
8.18.8 The Swimming Pool Reactor ......................... 581
8.18.9 The Tank-Type Reactor ............................. 581
8.18.10 The Graphite-Moderated Natural Uranium Reactor ... 581
8.19 Variation of Reactivity .................................. 581
8.19.1 Fuel Depletion and Fuel Production ................ 581
8.19.2 Effect of Fission Products Accumulation ........... 582
8.19.3 Temperature Effects ............................... 582
8.19.4 Temperature Coefficient and Reactor Stability ..... 583
8.20 Nuclear Fusion ........................................... 584
8.20.1 Fusion Reactions .................................. 584
8.20.2 Three Important Fusion Reactions .................. 586
8.20.3 Reaction Rate ..................................... 587
8.20.4 Power Density ..................................... 588
8.20.5 Thermonuclear Reactions in the Laboratory ......... 590
8.20.6 The D-T Reaction .................................. 591
8.20.7 Energy Losses ..................................... 591
8.20.8 Lawson Criterion .................................. 592
8.20.9 Ignition Temperature .............................. 593
8.20.10 Controlled Fusion Reactions ...................... 593
8.20.11 Confinement ...................................... 594
8.20.12 Containment of a Plasma .......................... 594
8.20.13 Plasma Diagnostics ............................... 597
8.21 Questions ................................................ 599
8.22 Problems ................................................. 600
References ............................................... 604
References ............................................... 605
Index ......................................................... 609
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