james e. martin · 2013. 7. 23. · problems and solutions 1999 isbn: 978-0-471-29711-6 bevelacqua,...

James E. Martin

Physics for Radiation Protection

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James E. Martin

Physics for Radiation Protection

Third Completely Updated Edition

The Author

James E. Martin2604 Bedford RoadAnn Arbour MI 48104USA

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To the memory of

Frank A. and Virginia E. Martinand JoAnn Martin Burkhart.

VII

Preface XVII

1 Structure of Atoms 1

1.1 Atom Constituents 2

1.2 Structure, Identity, and Stability of Atoms 5

1.3 Chart of the Nuclides 6

1.4 Nuclear Models 8

Problems – Chapter 1 9

2 Atoms and Energy 11

2.1 Atom Measures 12

2.2 Energy Concepts for Atoms 14

2.2.1 Mass-energy 15

2.2.2 Binding Energy of Nuclei 16

2.3 Summary 18

Other Suggested Sources 18


3 Radioactive Transformation 21

3.1 Processes of Radioactive Transformation 21

3.1.1 Transformation of Neutron-rich Radioactive Nuclei 23

3.1.2 Double Beta (bb) Transformation 27

3.1.3 Transformation of Proton-rich Nuclei 27

3.1.4 Positron Emission 29

3.1.5 Average Energy of Negatron and Positron Emitters 32

3.1.6 Electron Capture (EC) 33

3.1.7 Radioactive Transformation of Heavy Nuclei by Alpha ParticleEmission 35

3.1.8 Theory of Alpha Particle Transformation 38

3.1.9 Transuranic (TRU) Radionuclides 40

3.1.10 Gamma Emission 41

3.1.11 Internal Transition (Metastable or Isomeric States) 42

3.1.12 Internal Conversion 43

Contents

Physics for Radiation Protection, Third Edition. James E. MartinCopyright � 2013 WILEY-VCH Verlag & Co. KGaA. Published 2013 by WILEY-VCH Verlag GmbH & Co. KGaA.

VIII

3.1.13 Multiple Modes of Radioactive Transformation 49

3.1.14 Transformation by Delayed Neutron Emission 51

3.1.15 Transformation by Spontaneous Fission 51

3.1.16 Proton Emission 53

3.2 Decay Schemes 54

3.3 Rate of Radioactive Transformation 57

3.3.1 Activity 58

3.3.2 Units of Radioactive Transformation 58

3.3.3 Mathematics of Radioactive Transformation 60

3.3.4 Half-Life 62

3.3.5 Mean Life 63

3.3.6 Effective Half-life 64

3.4 Radioactivity Calculations 65

3.4.1 Half-life Determination 68

3.5 Activity–mass Relationships 70

3.5.1 Specific Activity 70

3.6 Radioactive Series Transformation 73

3.6.1 Series Decay Calculations 73

3.6.2 Recursive Kinetics: the Bateman Equations 76

3.7 Radioactive Equilibrium 77

3.7.1 Secular Equilibrium 78

3.7.2 Transient Equilibrium 80

3.7.3 Radionuclide Generators 81

3.8 Total Number of Transformations (Uses of s and kEff) 84

3.9 Discovery of the Neutrino 86

Acknowledgments 87



4 Interactions 91

4.1 Production of X-rays 91

4.2 Characteristic X-rays 93

4.2.1 X-rays and Atomic Structure 95

4.2.2 Auger Electrons 96

4.3 Nuclear Interactions 98

4.3.1 Cross-Section 100

4.3.2 Q-values for Nuclear Reactions 102

4.4 Alpha Particle Interactions 104

4.4.1 Alpha–Neutron Reactions 105

4.5 Transmutation by Protons and Deuterons 106

4.5.1 Proton–Alpha Particle (p,a) Reactions 108

4.5.2 Proton–Neutron (p,n) Reactions 109

4.5.3 Proton–Gamma (p,c) Reactions 110

4.5.4 Proton–Deuteron Reactions 110

4.5.5 Deuteron–Alpha (d,a) Reactions 111

Contents

IX

4.5.6 Deuteron–Proton (d,p) and Deuteron–Neutron (d,n) Reactions 111

4.6 Neutron Interactions 114

4.6.1 Radiative Capture (n,c) Reactions 114

4.6.2 Charged Particle Emission (CPE) 115

4.6.3 Neutron–Proton (n,p) Reactions 116

4.6.4 Neutron–Neutron (n,2n) Reactions 116

4.7 Activation Product Calculations 117

4.7.1 Neutron Activation Product Calculations 119

4.7.2 Charged Particles Calculations 124

4.8 Medical Isotope Reactions 126

4.9 Transuranium Elements 128

4.10 Photon Interactions 130

4.10.1 Activation by Photons 130

4.11 Fission and Fusion Reactions 133

4.11.1 Fission 133

4.11.2 Fusion 134

4.12 Summary 138



5 Nuclear Fission and its Products 143

5.1 Fission Energy 145

5.2 Physics of Sustained Nuclear Fission 147

5.3 Neutron Economy and Reactivity 152

5.4 Nuclear Power Reactors 154

5.4.1 Reactor Design: Basic Systems 155

5.5 Light Water Reactors (LWRs) 157

5.5.1 Pressurized Water Reactor (PWR) 157

5.5.2 Boiling Water Reactor (BWR) 159

5.5.3 Inherent Safety Features of LWRs 161

5.5.4 Decay Heat in Power Reactors 163

5.5.5 Uranium Enrichment 164

5.6 Heavy Water Reactors (HWRs) 165

5.6.1 HWR Safety Systems 168

5.7 Breeder Reactors 169

5.7.1 Liquid Metal Fast Breeder Reactor (LMFBR) 171

5.8 Gas-cooled Reactors 174

5.8.1 High-temperature Gas Reactor (HTGR) 175

5.9 Reactor Radioactivity 176

5.9.1 Fuel Cladding 177

5.9.2 Radioactive Products of Fission 178

5.9.3 Production of Individual Fission Products 182

5.9.4 Fission Products in Spent Fuel 184

5.9.5 Fission Product Poisons 185

5.10 Radioactivity in Reactors 188

Contents

5.10.1 Activation Products in Nuclear Reactors 188

5.10.2 Tritium Production in Reactors 191

5.10.3 Low-level Radioactive Waste 192

5.11 Summary 193

Acknowledgments 194



6 Naturally Occurring Radiation and Radioactivity 197

6.1 Discovery and Interpretation 197

6.2 Background Radiation 199

6.3 Cosmic Radiation 200

6.4 Cosmogenic Radionuclides 203

6.5 Naturally Radioacitve Series 207

6.5.1 Neptunium Series Radionuclides 214

6.6 Singly Occurring Primordial Radionuclides 214

6.7 Radioactive Ores and Byproducts 216

6.7.1 Resource Recovery 218

6.7.2 Uranium Ores 218

6.7.3 Water Treatment Sludge 219

6.7.4 Phosphate Industry Wastes 219

6.7.5 Elemental Phosphorus 220

6.7.6 Manhattan Project Wastes 221

6.7.7 Thorium Ores 223

6.8 Radioactivity Dating 224

6.8.1 Carbon Dating 224

6.8.2 Dating by Primordial Radionuclides 225

6.8.3 Potassium–Argon Dating 226

6.8.4 Ionium (230Th) Method 227

6.8.5 Lead-210 Dating 227

6.9 Radon and its Progeny 228

6.9.1 Radon Subseries 229

6.9.2 Working Level for Radon Progeny 232

6.9.3 Measurement of Radon 236

6.10 Summary 240

Acknowledgements 241



7 Interactions of Radiation with Matter 245

7.1 Radiation Dose and Units 245

7.1.1 Radiation Absorbed Dose 246

7.1.2 Radiation Dose Equivalent 246

7.1.3 Radiation Exposure 247

7.2 Radiation Dose Calculations 249

ContentsX

7.2.1 Inverse Square Law 249

7.3 Interaction Processes 250

7.4 Interactions of Alpha Particles and Heavy Nuclei 252

7.4.1 Recoil Nuclei and Fission Fragments 254

7.4.2 Range of Alpha Particles 254

7.5 Beta Particle Interactions and Dose 257

7.5.1 Energy Loss by Ionization 258

7.5.2 Energy Losses by Bremsstrahlung 258

7.5.3 Cerenkov Radiation 259

7.5.4 Attenuation of Beta Particles 261

7.5.5 Range Versus Energy of Beta Particles 262

7.5.6 Radiation Dose from Beta Particles 264

7.5.7 Beta Dose from Contaminated Surfaces 267

7.5.8 Beta Contamination on Skin or Clothing 268

7.5.9 Beta Dose from Hot Particles 269

7.6 Photon Interactions 270

7.6.1 Photoelectric Interactions 271

7.6.2 Compton Interactions 272

7.6.3 Pair Production 274

7.6.4 Photodisintegration 276

7.7 Photon Attenuation and Absorption 277

7.7.1 Attenuation (l) and Energy Absorption (lEn) Coefficients 280

7.7.2 Effect of E and Z on Photon Attenuation/Absorption 284

7.7.3 Absorption Edges 286

Checkpoints 288

7.8 Energy Transfer and Absorption by Photons 288

7.8.1 Electronic Equilibrium 293

7.8.2 Bragg–Gray Theory 295

7.9 Exposure/Dose Calculations 296

7.9.1 Point Sources 297

7.9.2 Gamma Ray Constant, C 298

7.9.3 Exposure and Absorbed Dose 300

7.9.4 Exposure, Kerma, and Absorbed Dose 301

7.10 Summary 303

Acknowledgments 303



8 Radiation Shielding 307

8.1 Shielding of Alpha-Emitting Sources 307

8.2 Shielding of Beta-Emitting Sources 308

8.2.1 Attenuation of Beta Particles 308

8.2.2 Bremsstrahlung Effects for Beta Shielding 311

8.3 Shielding of Photon Sources 314

8.3.1 Shielding of Good Geometry Photon Sources 315

Contents XI

8.3.2 Half-Value and Tenth-Value Layers 322

8.3.3 Shielding of Poor Geometry Photon Sources 324

8.3.4 Use of Buildup Factors 330

8.3.5 Effect of Buildup on Shield Thickness 331

8.3.6 Mathematical Formulations of the Buildup Factor 333

8.4 Gamma Flux for Distributed Sources 338

8.4.1 Line Sources 339

8.4.2 Ring Sources 341

8.4.3 Disc and Planar Sources 342

8.4.4 Shield Designs for Area Sources 343

8.4.5 Gamma Exposure from Thick Slabs 350

8.4.6 Volume Sources 355

8.4.7 Buildup Factors for Layered Absorbers 356

8.5 Shielding of Protons and Light Ions 357

8.6 Summary 360

Acknowledgments 360



9 Internal Radiation Dose 365

9.1 Absorbed Dose in Tissue 365

9.2 Accumulated Dose 366

9.2.1 Internal Dose: Medical Uses 369

Checkpoints 369

9.3 Factors In The Internal Dose Equation 370

9.3.1 The Dose Reciprocity Theorem 377

9.3.2 Deposition and Clearance Data 378

9.3.3 Multicompartment Retention 378

9.4 Radiation Dose from Radionuclide Intakes 383

9.4.1 Risk-Based Radiation Standards 384

9.4.2 Committed Effective Dose Equivalent (CEDE) 385

9.4.3 Biokinetic Models: Risk-Based Internal Dosimetry 386

9.4.4 Radiation Doses Due to Inhaled Radionuclides 388

9.4.5 Radiation Doses Due to Ingested Radionuclides 398

9.5 Operational Determinations of Internal Dose 405

9.5.1 Submersion Dose 406

Checkpoints 406

9.6 Tritium: a Special Case 408

9.6.1 Bioassay of Tritium: a Special Case 410

9.7 Summary 411



10 Environmental Dispersion 415

10.1 Atmospheric Dispersion 417

ContentsXII

10.1.1 Atmospheric Stability Effects on Dispersion 420

10.1.2 Atmospheric Stability Classes 422

10.1.3 Calculational Procedure: Uniform Stability Conditions 424

10.1.4 Distance xmax of Maximum Concentration (vmax) 426

10.1.5 Stack Effects 427

Checkpoints 429

10.2 Nonuniform turbulence: Fumigation, Building Effects 429

10.2.1 Fumigation 429

10.2.2 Dispersion for an Elevated Receptor 431

10.2.3 Building Wake Effects: Mechanical Turbulence 432

10.2.4 Concentrations of Effluents in Building Wakes 433

10.2.5 Ground-level Area Sources 435

10.2.6 Effect of Mechanical Turbulence on Far-field Diffusion 436

10.3 Puff Releases 438

10.4 Sector-Averaged v/Q Values 439

10.5 Deposition/Depletion: Guassian Plumes 443

10.5.1 Dry Deposition 443

10.5.2 Air Concentration Due to Resuspension 447

10.5.3 Wet Deposition 449

10.6 Summary 452



11 Nuclear Criticality 455

11.1 Nuclear Reactors and Criticality 456

11.1.1 Three Mile Island Accident 456

11.1.2 Chernobyl Accident 458

11.1.3 NRX Reactor: Chalk River, Ontario, December 1952 461

11.1.4 SL-1 Accident 461

11.1.5 K-reactor, Savannah River Site, 1988 462

11.1.6 Fukushima-Daichi Plant—Japan, March 11, 2011 463

11.2 Nuclear Explosions 464

11.2.1 Fission Weapons 464

11.2.2 Fusion Weapons 465

11.2.3 Products of Nuclear Explosions 466

11.2.4 Fission Product Activity and Exposure 467

Checkpoints 469

11.3 Criticality Accidents 470

11.3.1 Y-12 Plant, Oak Ridge National Laboratory, TN: June 16, 1958 470

11.3.2 Los Alamos Scientific Laboratory, NM: December 30, 1958 471

11.3.3 Idaho Chemical Processing Plant: October 16, 1959,January 25, 1961, and October 17, 1978 472

11.3.4 Hanford Recuplex Plant: April 7, 1962 473

11.3.5 Wood River Junction RI: July 24, 1964 473

11.3.6 UKAEA Windscale Works, UK: August 24, 1970 474

Contents XIII

11.3.7 Bare and Reflected Metal Assemblies 474

11.4 Radiation Exposures in Criticality Events 475

11.5 Criticality Safety 476

11.5.1 Criticality Safety Parameters 478

11.6 Fission Product Release in Criticality Events 482

11.6.1 Fast Fission in Criticality Events 483

11.7 Summary 485

Acknowledgments 486



12 Radiation Detection and Measurement 489

12.1 Gas-Filled Detectors 489

12.2 Crystalline Detectors/Spectrometers 493

12.3 Semiconducting Detectors 494

12.4 Gamma Spectroscopy 495

12.4.1 Gamma-Ray Spectra: hm £ 1.022 MeV 495

12.4.2 Gamma-Ray Spectra: hm ‡ 1.022 MeV 500

12.4.3 Escape Peaks and Sum Peaks 502

12.4.4 Gamma Spectroscopy of Positron Emitters 503

12.5 Portable Field Instruments 504

12.5.1 Geiger Counters 504

12.5.2 Ion Chambers 505

12.5.3 Microrem Meters 506

12.5.4 Alpha Radiation Monitoring 506

12.5.5 Beta Radiation Surveys 507

12.5.6 Removable Radioactive Surface Contamination 508

12.5.7 Instrument Calibration 509

12.6 Personnel Dosimeters 509

12.6.1 Film Badges 509

12.6.2 Thermoluminescence Dosimeters (TLDs) 510

12.6.3 Pocket Dosimeters 511

12.7 Laboratory Instruments 511

12.7.1 Liquid Scintillation Analysis 511

12.7.2 Proportional Counters 515

12.7.3 End-window GM Counters 517

12.7.4 Surface Barrier Detectors 518

12.7.5 Range Versus Energy of Beta Particles 519



ContentsXIV

13 Statistics in Radiation Physics 523

13.1 Nature of Counting Distributions 523

13.1.1 Binomial Distribution 525

13.1.2 Poisson Distribution 525

13.1.3 Normal Distribution 527

13.1.4 Mean and Standard Deviation of a Set of Measurements 530

13.1.5 Uncertainty in the Activity of a Radioactive Source 531

13.1.6 Uncertainty in a Single Measurement 533

Checkpoints 533

13.2 Propagation of Error 534

13.2.1 Statistical Subtraction of a Background Count or Count Rate 535

13.2.2 Error Propagation of Several Uncertain Parameters 537

13.3 Comparison of Data Sets 538

13.3.1 Are Two Measurements Different? 538

13.4 Statistics for the Counting Laboratory 541

13.4.1 Uncertainty of a Radioactivity Measurement 541

13.4.2 Determining a Count Time 542

13.4.3 Efficient Distribution of Counting Time 544

13.4.4 Detection and Uncertainty for Gamma Spectroscopy 545

13.4.5 Testing the Distribution of a Series of Counts (the Chi-squareStatistic) 547

13.4.6 Weighted Sample Mean 548

13.4.7 Rejection of Data 549

13.5 Levels of Detection 551

13.5.1 Critical Level 552

13.5.2 Detection Limit (Ld) or Lower Level of Detection (LLD) 554

13.6 Minimum Detectable Concentration or Contamination 558

13.6.1 Minimum Detectable Concentration (MDConc.) 558

13.6.2 Minimum Detectable Contamination (MDCont.) 560

13.6.3 Less-than Level (Lt) 561

13.6.4 Interpretations and Restrictions 561

13.7 Log Normal Data Distributions 562

13.7.1 Particle Size Analysis 565

Acknowledgment 569


Chapter 13 – Problems 569

14 Neutrons 571

14.1 Neutron Sources 571

14.2 Neutron Parameters 573

14.3 Neutron Interactions 575

14.3.1 Neutron Attenuation and Absorption 576

14.4 Neutron Dosimetry 578

14.4.1 Dosimetry for Fast Neutrons 581

14.4.2 Dose from Thermal Neutrons 583

Contents XV

14.4.3 Monte Carlo Calculations of Neutron Dose 585

14.4.4 Kerma for Neutrons 588

14.4.5 Dose Equivalent Versus Neutron Flux 588

14.4.6 Boron Neutron Capture Therapy (BNCT) 591

14.5 Neutron Shielding 591

14.5.1 Neutron Shielding Materials 591

14.5.2 Neutron Shielding Calculations 593

14.5.3 Neutron Removal Coefficients 594

14.5.4 Neutron Attenuation in Concrete 597

14.6 Neutron Detection 598

14.6.1 Measurement of Thermal Neutrons 599

14.6.2 Measurement of Intermediate and Fast Neutrons 600

14.6.3 Neutron Foils 602

14.6.4 Albedo Dosimeters 604

14.6.5 Flux Depression of Neutrons 604

14.7 Summary 605

Acknowledgment 605



Answers to Selected Problems 607

Appendix A 613

Appendix B 615

Appendix C 625

Appendix D 629

Index 657

ContentsXVI

XVII

This book is the outcome of teaching radiation physics to students beginning acourse of study in radiation protection, or health physics. This 3rd editionattempts as the first two did to provide in one place a comprehensive treatise ofthe major physics concepts required of radiation protection professionals. Numer-ous real-world examples and practice problems are provided to demonstrate con-cepts and hone skills, and even though its limited uses are thoroughly developedand explained, some familiarity with calculus would be helpful in grasping someof the subjects.

The materials in this compendium can be used in a variety of ways, both forinstruction and reference. The first two chapters describe the atom as an energysystem, and as such they may be of most use for those with minimal science back-ground. Chapter 3 addresses the special condition of radioactive transformation(or disintegration) of atoms with excess energy, regardless of how acquired. Chap-ters 4 and 5 describe activation and fission processes and the amount of energygained or lost due to atom changes; these define many of the sources that areaddressed in radiation protection. Chapter 6 develops natural sources of radiationand radioactive materials primarily as reference material; however, the sections onradioactive dating and radon could be used as supplemental, though specialized,material to Chapter 3.

The interaction of radiation with matter and the resulting deposition of energyis covered in Chapter 7 along with the corollary subjects of radiation exposure anddose. Radiation shielding, also related to interaction processes, is described inChapter 8 for various source geometries. Chapters 9 and 10, on internal radiationdose and environmental dispersion of radioactive materials, are also fundamentalfor understanding how such materials produce radiation dose inside the body andhow they become available for intakes by humans. These are followed by specialtychapters on nuclear criticality (Chapter 11); radiation detection and measurement(Chapter 12); applied statistics (Chapter 13); and finally (Chapter 14) neutronsources and interactions. A course in radiation physics would likely include thematerial in Chapters 3, 4, 5, 7, and 8 with selections from the other chapters, allor in part, to develop needed background and to address specialty areas of interestto instructor and student. In anticipation of such uses, attempts have been made

Preface


XVIII

to provide comprehensive and current coverage of the material in each chapterand relevant data sets.

Health physics problems require resource data. To this end, decay schemes andassociated radiation emissions are included for about 100 of the most commonradionuclides encountered in radiation protection. These are developed in thedetail needed for health physics uses and cross referenced to standard compen-diums for straightforward use when these more in-depth listings need to be con-sulted. Resources are also provided on activation cross sections, fission yields, fis-sion-product chains, photon absorption coefficients, nuclear masses, and abbre-viated excerpts of the Chart of the Nuclides. These are current from the NationalNuclear Data Center at Brookhaven National Laboratory; the Center and its staffare a national resource.

The units used in radiation protection have evolved over the hundred years orso that encompass the field. They continue to do so with a fairly recent, but notentirely accepted, emphasis on System Internationale (SI) Units while U.S. stan-dards and regulations have continued to use conventional units. To the degreepossible, this book uses fundamental quantities such as eV, transformations,time, distance, and the numbers of atoms or emitted particles and radiations todescribe nuclear processes, primarily because they are basic to concepts beingdescribed but partially to avoid conflict between SI units and conventional ones.Both sets of units are defined as they apply to radiation protection, but in generalthe more fundamental parameters are used. For the specific units of radiation pro-tection such as exposure, absorbed dose, dose equivalent, and activity, text mate-rial and examples are generally presented in conventional units because the fieldis very much an applied one; however, the respective SI unit is also includedwhere feasible. By doing so, it is believed presentations are clearer and relevant tothe current conditions, but it is recognized that this quandary is likely to continue.

This endeavor has been possible because of the many contributions of myresearch associates and students whose feedback shaped the teacher on the extentand depth of the physics materials necessary to function as a professional healthphysicist. I am particularly indebted to Chul Lee who began this process with mewith skill and patience and to Rachael Nelson who provided invaluable help incapping off this 3rd edition. I hope it helps all who undertake study in this excitingfield to appreciate how physics underpins it.

In an undertaking of this scope, it is inevitable that undetected mistakes creepin and remain despite the best efforts of preparers and editors; thus, reports([email protected]) of errors found would be appreciated.

James E. Martin, Ph.D., CHPAssociate Professor (emeritus) of

Radiological HealthThe University of Michigan, 2012

Preface

1

1Structure of Atoms

“I have discovered somethingvery interesting.”

W. C. Roentgen (Nov. 8, 1895)

The fifty years following Roentgen’s discovery of x-rays saw remarkable changesin physics that literally changed the world forever, culminating in a host of newproducts from nuclear fission. Discovery of the electron (1897) and radioactivity(1898) focused attention on the makeup of atoms and their structure as did otherdiscoveries. For example, in 1900, Planck introduced the concept that the emis-sion (or absorption) of electromagnetic radiation occurs only in a discrete amountwhere the energy is proportional to the frequency, v, of the radiation, or

E = hv

where h is a constant (Planck’s constant) of nature; its value is

h= 6.1260693 � 10–34 J s or 4.13566743 � 10–15 eV s

Planck presumed that he had merely found an ad hoc solution for blackbodyradiation, but in fact he had discovered a basic law of nature: any physical systemcapable of emitting or absorbing electromagnetic radiation is limited to a discreteset of possible energy values or levels; energies intermediate between them simplydo not occur. Planck’s theory was revolutionary because it states that the emissionand absorption of radiation must be discontinuous processes, i.e. only as a transi-tion from one particular energy state to another where the energy difference is anintegral multiple of hv. This revolutionary theory extends over 22 orders of magni-tude from very long wavelength radiation such as radio waves up to and includinghigh energy gamma rays. It includes the energy states of particles in atoms andgreatly influences their structure.

Einstein (in 1905) used Planck’s discrete emissions (or quanta) to explain whylight of a certain frequency (wavelength) causes the emission of electrons fromthe surface of various metals (the photoelectric effect). Light photons clearly haveno rest mass, but behaving like a particle, a photon can hit a bound electron and“knock” it out of the atom.


Quanta

The kinetic energy of the ejected electrons is KE = hv – f.

Example 1–1. Light with a wavelength of 5893 � produces electrons from a potas-sium surface that are stopped by 0.36 volts. Determine: a) the maximum energyof the photoelectron, and b) the work function.Solution. a) The maximum energy KEmax of the ejected photoelectrons is equal tothe stopping potential of 0.36 eV.

b) the work function is the energy of the incident photon minus the energy giv-en to the ejected electron, or

f = [4.13566743 � 10–15 eV s) (3 � 108 m/s) / 5.893 � 10–10 m] – 0.36 eV= 1.7454 eV.

A. H. Compton used a similar approach to explain x-ray scattering as interac-tions between “particle-like” photons and loosely bound (or “free”) electrons of car-bon (now known as the Compton effect). Energy and momentum are conservedand the calculated wavelength changes agreed with experimental observations.

1.1Atom Constituents

Atoms consist of protons and neutrons (discovered in 1932 by Chadwick) boundtogether to form a nucleus which is surrounded by electrons that counterbalanceeach proton in the nucleus to form an electrically neutral atom. Its componentsare: a) protons which have a reference mass of about 1.0 and an electrical charge of+1; b) electrons which have a mass about 1/1840 of the proton and a (–1) electricalcharge; and c) neutrons which are electrically neutral and slightly heavier than theproton. The number of protons (or Z) establishes the identity of the atom and itsmass number (A) is the sum of protons and neutrons (or N) in its nucleus. Elec-trons do not, and, according to the uncertainty principle, cannot exist in thenucleus, although they can be manufactured and ejected during radioactive trans-formation. Modern theory has shown that protons and neutrons are made up ofquarks, leptons, and bosons (recently discovered), but these are not necessary forunderstanding atoms or how they produce radiant energy.

Four forces of nature determine the array of atom constituents. The electromag-netic force between charged particles is attractive if the charges (q1 and q2) are ofopposite signs (i.e. positive or negative); if of the same sign, the force F will be

1 Structure of Atoms2

repulsive and quite strong for the small distances between protons in the nucleusof an atom. This repulsion is overcome by the nuclear force (or strong force) whichis about 100 times stronger; it only exists in the nucleus and only between protonsand neutrons (there is no center point towards which nucleons are attracted). Theweak force (relatively speaking) has been shown to be a form of the electromag-netic force; it influences radioactive transformation; and the gravitational force,though present, is negligible in atoms.

The nucleus of an atom containing Z protons is essentially a charged particle(with charge Ze) that attracts an equal number of electrons that orbit the nucleussome distance away. Thomson theorized that each negatively charged electronwas offset by a positively charged proton and that these were arrayed somewhatlike a plum pudding to form an electrically neutral atom. This model proved un-satisfactory for explaining the large-angle scattering of alpha particles by gold foilsas observed by Rutherford and Geiger-Marsden. Such large deflections were duethe electromagnetic force between a positively charged nucleus (Ze) at the centerof the atom and that of the alpha particle (2e).

αα

α

α

The force for such deflections is inversely proportional to the distance r betweenthem, or

F ¼ kq1q2

r2¼ k

ð2eÞðZeÞr2

;

which yields a value of r of about 10–15 m which Rutherford proposed as theradius of a small positively-charged nucleus surrounded by electrons in orbitsabout 10–10 m in size. This model had a fatal flaw: according to classical physicsthe electrons would experience acceleration, m2/r, causing them to continuouslyemit radiation and to quickly (in about 10–8 s) spiral into the nucleus.

In 1913, Niels Bohr explained Rutherford’s conundrum by simply declaring (postu-late I) that atoms are stable and that an electron in its orbit does not radiate energy, butonly does so when it experiences one of Planck’s quantum changes to an orbit oflower potential energy (postulate III) with the emission of a photon of energy

hm = E2 – E1

31.1 Atom Constituents

1 Structure of Atoms

And, that the allowed stationary states for orbiting electrons (postulate II) arethose for which the orbital angular momentum, L, is an integral multiple of h/2p,or:

L ¼ nh

2p

where n = 1, 2 ,3, 4, ..., represents the principal quantum number for discrete,quantized energy states. Since L = mvr, the calculated radius of the first (orwhen n = 1) electron orbit for hydrogen (the simplest atom) was found to be0.529 � 10–8 cm, which agreed with experiment. Postulate III is apparently basedon Planck’s quantum hypothesis, but postulate II appeared to be arbitrary eventhough it worked , at least for hydrogen. It would only be explained by de Broglie’shypothesis some 13 years later (see below).

Bohr assumed that electrons orbiting a nucleus moved in circular orbits underthe influence of two force fields: the coulomb attraction (a centripetal force) pro-vided by the positively charged nucleus and the centrifugal force of each electronin orbital motion at a radius, rn, and velocity, mn. These forces are equal and oppo-site each other, or:

mv2n

rn¼ k

q1q2

r2n

where rn can be calculated from postulate II. And, since q1 and q2 are unity forhydrogen

rn ¼ðnhÞ2

ð2pÞ2kmq2¼ n2r1

where n = 1, 2, 3, 4, ... and r1 is the radius of the first orbit of the electron in thehydrogen atom, the so-called Bohr orbit. And since the quantum hypothesis limitsvalues of n to integral values, the electron can only be in those orbits which aregiven by:

rn = r1, 4r1, 9r1, 16r1, ...

These relationships can be used to calculate the total energy En of an electron inthe nth orbit where the sum of its kinetic and potential energy is

En ¼mv2

n

2þ � ke2

rn

� �

En ¼ � 1n2

ð2pÞ2k2q4m2h2

¼ � 1n2

· 13:58 eV

4

1.2 Structure, Identity, and Stability of Atoms

which is the binding energy of the electron in hydrogen and is in perfect agree-ment with the measured value of the energy required to ionize hydrogen. Forother values of n, the allowed energy levels of hydrogen are:

En ¼ � E1

4; � E1

9; � E1

16; :::

where E1 = –13.58 eV. These predicted energy levels can be used to calculate thepossible emissions (or absorption) of electromagnetic radiation and their wave-lengths for hydrogen. When Bohr did so for n = 3 he obtained the series of wave-lengths measured by Balmer, and since the theory holds for n = 3, Bohr postulatedthat it should also hold for other values of n, and the corresponding wavelengthswere soon found providing dramatic proof of the theory.

In 1926, Louis de Broglie postulated that if Einstein’s and Compton’s assign-ment of particle properties to waves was correct, why shouldn’t the converse betrue; i.e., that particles have wave properties such that an electron (or a car for thatmatter) has a wavelength,associated with its motion, or

k ¼ hp¼ h

mv

and that as a wave it has momentum, p, with the value:

p ¼ hk

This simple but far-reaching concept was later proved by Davisson and Germerwho observed diffraction (a wave phenomenon) of electrons (clearly particles)from a nickel crystal. De Broglie’s wave/particle behavior of electrons also openedthe door to description of the dynamics of particles by wave mechanics, perhapsthe most revolutionary development in physics since Einstein’s special theory ofrelativity.

Simple though it appears, de Broglie’s hypothesis has consequences as signifi-cant as Einstein’s equivalence of mass and energy (E = mc2) which are related toeach other through c2, a large constant of proportionality; in de Broglie’s equa-tions, the wavelength,and the momentum p of a particle are related to each otherthrough Planck’s constant, a very small one.

1.2Structure, Identity, and Stability of Atoms

The identity of an atom is determined by the number and array of protons andneutrons in its nucleus. An atom with one proton is defined as hydrogen; it hasone orbital electron for electrical neutrality. Deuterium (or hydrogen-2) also con-

5


tains one proton and one electron but also a neutron and is quite stable; tritium(hydrogen-3) with one proton and one electron has a second neutron whichcauses it to be unstable, or radioactive. These three are isotopes of hydrogen.

Two protons cannot be joined to form an atom because the repulsive electro-magnetic force between them is so great that it even overcomes the stronglyattractive nuclear force. If, however, a neutron is present, the distribution of forcesis such that a stable nucleus is formed and two electrons will then join up to bal-ance the two plus (+) charges of the protons to create a stable, electrically neutralatom of helium so defined because it has two protons. Its mass number (A) is 3(2 protons plus 1 neutron) and is written as helium-3 or 3He. Because neutronsprovide a cozy effect, yet another neutron can be added to obtain 4He which stillhas two electrons to balance the two positive charges. This atom is the predomi-nant form (or isotope) of helium on earth, and it is very stable (this same atom,minus the two orbital electrons, is ejected from some radioactive atoms as analpha particle, i.e., a charged helium nucleus). Helium-5 (5He) cannot be formedbecause the extra neutron creates a very unstable atom that breaks apart very fast(in 10–21 s or so). But, for many atoms an extra neutron(s) is easily accommodatedto yield one or more isotopes of the same element, and for some elements addingan extra neutron (or proton) to a nucleus only destabilizes it; i.e., it will often existas an unstable, or radioactive, atom. Such is the case for hydrogen-3 (3H, or tri-tium) and carbon-14 (14C). Elements are often identified by name and mass num-ber, e.g., hydrogen-3 (3H) or carbon-14 (14C).

Three protons can be assembled with three neutrons to form lithium-6 (6Li) orwith four neutrons, lithium-7 (7Li). Since lithium contains three protons, it mustalso have three orbital electrons, but because the first orbit can only hold two elec-trons (there is an important reason for this which is explained by quantum theory)the third electron occupies another orbit further away.

1.3Chart of the Nuclides

As shown in Figure 1-1, a plot of the number of protons versus the number ofneutrons increases steadily for heavier atoms because extra neutrons are neces-sary to distribute the nuclear force and moderate the repulsive electrostatic forcebetween protons. The heaviest element in nature is 238U with 92 protons and 146neutrons; it is radioactive, but very long-lived. The heaviest stable element in na-ture is 209Bi with 83 protons and 126 neutrons. Lead with 82 protons is muchmore common in nature than bismuth and for a long time was thought to be theheaviest of the stable elements; it is also the stable endpoint of the radioactivetransformation of uranium and thorium, two primordial naturally occurringradioactive elements (see Chapter 6).

The chart of the nuclides contains basic information on each element, how manyisotopes it has (atoms on the horizontal lines) and which ones are stable (shaded)or unstable (unshaded). A good example of such information is shown in

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1.3 Chart of the Nuclides

Fig. 1-1 Part of the chart of the nuclides.(Nuclides and Isotopes, 16th Edition, KAPL, Inc, 2002.)

Figure 1-2 for four isotopes of carbon (actually there are 8 measured isotopes ofcarbon but these 4 are the most important). They are all carbon because each con-tains 6 protons, but each has a different number of neutrons, hence they are dis-tinct isotopes with different weights. 12C and 13C are shaded and thus are stable,as are the two shaded blocks for boron (5 protons) and nitrogen (7 protons) alsoshown in Figure 1-2. The nuclides in the unshaded blocks (e.g., 11C and 14C) areunstable simply because they don’t have the right array of protons and neutrons to bestable (we will use these properties later to discuss radioactive transformation). Thedark band at the top of the block for 14C denotes that it is a naturally-occurring radio-active isotope, a convention used for several other such radionuclides. The block tothe far left contains information on naturally abundant carbon: it contains the chemi-cal symbol, C, the name of the element, and the atomic weight of natural carbon, or12.0107 grams/mole, weighted according to the percent abundance of the twonaturally occurring stable isotopes. The shaded blocks contain the atom percentabundance of 12C and 13C in natural carbon at 98.90 and 1.10 atom percent,respectively; these are listed just below the chemical symbol. Similar informationis provided for all of the elements in the chart of the nuclides.

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Fig. 1-2 Excerpt from the chart of the nuclides for the twostable isotopes of carbon (Z = 6) and its two primary radio-active isotopes, in relation to primary isotopes of nitrogen(Z = 7) and boron (Z = 5). (Adapted from KAPL, 2002.)

1.4Nuclear Models

The array of protons and neutrons in each element is unique because natureforces these constituents toward the lowest potential energy possible; when theyattain it they are stable, and until they do they have excess energy and are thusunstable, or radioactive; e.g. tritium or carbon-14. Descriptions of the dynamicsand changes in energy states of nuclear constituents often use a shell model; how-ever, descriptions of fission and other phenomena are best done with a liquiddrop model. The exact form of the nuclear force in the nucleus is not yet knownnor the structure of potential energy states of its constituents, but a shell modelcorresponds nicely with the emission of gamma rays from excited nuclei. Theseemissions are similar to those that occur when orbital electrons change to one oflower potential energy.

The nucleus exhibits periodicities that suggest energy shells not unlike thoseobserved for electron shells. Atoms that have 2, 8, 20, 28, 50, and 82 neutrons orprotons and 126 neutrons are particularly stable. These values of N and Z arecalled magic numbers, and elements with them have many more stable isotopesthan their immediate neighbors. For example, Sn (Z = 50) has 10 stable isotopes,while In (Z = 49) and Sb (Z = 51) each have only 2. Similarly, for N = 20, thereare 5 stable isotones (differing elements with the same N number), while forN = 19, there is none, and for N = 21, there is only one. The same pattern holdsfor other magic numbers.

8

1.4 Nuclear Models

A nebular model of the atom, as shown in Figure 1-3, provides an overall descrip-tion of the atom. In it the electrons are spread as waves of probability over thewhole volume of the atom, a direct consequence of de Broglie’s discovery of thewave characteristics of electrons and other particles. Electrons are distributedaround a nucleus in energy states that are an equal number of de Broglie wave-lengths, or nk, where n, the principal quantum number, corresponds to energyshells, K, L, M, etc. for n = 1, 2, 3, ...; changes between electron states are quan-tized with discrete energies. The outer radius of the nebular cloud of electrons isabout 10–10 m which is some 4 to 5 orders of magnitude greater than the nuclearradius at about one femtometer (10–15 m), commonly called one fermi in honor ofthe great Italian physicist and nuclear navigator, Enrico Fermi. The radius of thenucleus is proportional to A1/3 or

r = roA1/3

where A is the atomic mass number of the atom in question and the constant ro

has an average value of about 1.3 � 10–15 m, or 1.3 fermi.

Fig. 1–3 Very simplified model of a nebular atom consistingof an array of protons and neutrons with shell-like stateswithin a nucleus surrounded by a cloud of electrons withthree dimensional wave patterns and also with shell-likeenergy states.

Problems – Chapter 1

1–1. How many neutrons and how many protons are there in: a) 14C, b) 27Al,c) 133Xe, and d) 209Bi?1–2. Calculate the radius of the nucleus of 27Al in meters and fermis.1–3. When light of wavelength 3132 � falls on a caesium surface, a photoelectronis emitted for which the stopping potential is 1.98 volts. Calculate the maximumenergy of the photoelectron, the work function, and the threshold frequency.

9


1–4. The work function of potassium is 2.20 eV. What should be the wavelengthof the incident electromagnetic radiation so that the photoelectrons emitted frompotassium will have a maximum kinetic energy of 4 eV? Also calculate the thresh-old frequency.1–5. Calculate the de Broglie wavelength associated with the following:a) an electron with a kinetic energy of 1 eVb) an electron with a kinetic energy of 510 keVc) a thermal neutron (2200 m/s)d) a 1500 kg automobile at a speed of 100 km/h.1–6. Calculate the de Broglie wavelength associated with: a) a proton with 15 MeVof kinetic energy, and b) a neutron of the same energy.

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james e. martin · 2013. 7. 23. · problems and solutions 1999 isbn: 978-0-471-29711-6 bevelacqua,...

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