### FUNDAMENTALS OF NUCLEAR

### SCIENCE AND ENGINEERING

### J. KENNETH SHULTIS

### RICHARD E. FAW

*Kansas State University*
*Manhattan, Kansas, U.S.A.*

M A R C E L

### MARCEL DEKKER, INC. NEW YORK • BASEL

**ISBN: 0-8247-0834-2**

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**Preface**

Nuclear engineering and the technology developed by this discipline began and reached an amazing level of maturity within the past 60 years. Although nuclear and atomic radiation had been used during the first half of the twentieth century, mainly for medical purposes, nuclear technology as a distinct engineering discipline began after World War II with the first efforts at harnessing nuclear energy for electrical power production and propulsion of ships. During the second half of the twentieth century, many innovative uses of nuclear radiation were introduced in the physical and life sciences, in industry and agriculture, and in space exploration.

The purpose of this book is two-fold as is apparent from the table of contents. The first half of the book is intended to serve as a review of the important results of "modern" physics and as an introduction to the basic nuclear science needed by a student embarking on the study of nuclear engineering and technology. Later in this book, we introduce the theory of nuclear reactors and its applications for electrical power production and propulsion. We also survey many other applications of nuclear technology encountered in space research, industry, and medicine.

The subjects presented in this book were conceived and developed by others. Our role is that of reporters who have taught nuclear engineering for more years than we care to admit. Our teaching and research have benefited from the efforts of many people. The host of researchers and technicians who have brought nu-clear technology to its present level of maturity are too many to credit here. Only their important results are presented in this book. For their efforts, which have greatly benefited all nuclear engineers, not least ourselves, we extend our deepest appreciation. As university professors we have enjoyed learning of the work of our colleagues. We hope our present and future students also will appreciate these past accomplishments and will build on them to achieve even more useful applications of nuclear technology. We believe the uses of nuclear science and engineering will continue to play an important role in the betterment of human life.

believe that all modern engineering students need to understand the basic aspects of nuclear science engineering such as radioactivity and radiation doses and their hazards.

Many people have contributed to this book. First and foremost we thank our colleagues Dean Eckhoff and Fred Merklin, whose initial collection of notes for an introductory course in nuclear engineering motivated our present book intended for a larger purpose and audience. We thank Professor Gale Simons, who helped prepare an early draft of the chapter on radiation detection. Finally, many revisions have been made in response to comments and suggestions made by our students on whom we have experimented with earlier versions of the manuscript. Finally, the camera copy given the publisher has been prepared by us using I^TEX, and, thus, we must accept responsibility for all errors, typographical and other, that appear in this book.

**Contents**

**1 Fundamental Concepts**

1.1 Modern Units

1.1.1 Special Nuclear Units 1.1.2 Physical Constants 1.2 The Atom

1.2.1 Atomic and Nuclear Nomenclature 1.2.2 Atomic and Molecular Weights 1.2.3 Avogadro's Number

1.2.4 Mass of an Atom 1.2.5 Atomic Number Density 1.2.6 Size of an Atom

1.2.7 Atomic and Isotopic Abundances 1.2.8 Nuclear Dimensions

1.3 Chart of the Nuclides

1.3.1 Other Sources of Atomic/Nuclear Information

**2 Modern Physics Concepts**

2.1 The Special Theory of Relativity 2.1.1 Principle of Relativity

2.1.2 Results of the Special Theory of Relativity 2.2 Radiation as Waves and Particles

2.2.1 The Photoelectric Effect 2.2.2 Compton Scattering

2.2.3 Electromagnetic Radiation: Wave-Particle Duality 2.2.4 Electron Scattering

2.2.5 Wave-Particle Duality 2.3 Quantum Mechanics

2.3.1 Schrodinger's Wave Equation 2.3.2 The Wave Function

2.3.3 The Uncertainty Principle 2.3.4 Success of Quantum Mechanics

2.4.2 Length Contraction 2.4.3 Mass Increase

2.5 Addendum 2: Solutions to Schrodinger's Wave Equation 2.5.1 The Particle in a Box

2.5.2 The Hydrogen Atom

2.5.3 Energy Levels for Multielectron Atoms

**Atomic/Nuclear Models**

3.1 Development of the Modern Atom Model 3.1.1 Discovery of Radioactivity

3.1.2 Thomson's Atomic Model: The Plum Pudding Model 3.1.3 The Rutherford Atomic Model

3.1.4 The Bohr Atomic Model

3.1.5 Extension of the Bohr Theory: Elliptic Orbits 3.1.6 The Quantum Mechanical Model of the Atom 3.2 Models of the Nucleus

3.2.1 Fundamental Properties of the Nucleus 3.2.2 The Proton-Electron Model

3.2.3 The Proton-Neutron Model 3.2.4 Stability of Nuclei

3.2.5 The Liquid Drop Model of the Nucleus 3.2.6 The Nuclear Shell Model

3.2.7 Other Nuclear Models

**Nuclear Energetics**

4.1 Binding Energy

4.1.1 Nuclear and Atomic Masses 4.1.2 Binding Energy of the Nucleus 4.1.3 Average Nuclear Binding Energies 4.2 Niicleon Separation Energy

4.3 Nuclear Reactions

4.4 Examples of Binary Nuclear Reactions 4.4.1 Multiple Reaction Outcomes 4.5 Q-Value for a Reaction

4.5.1 Binary Reactions

4.5.2 Radioactive Decay Reactions

4.6 Conservation of Charge and the Calculation of Q-Values 4.6.1 Special Case for Changes in the Proton Number 4.7 Q-Value for Reactions Producing Excited Nulcei

**Radioactivity**

5.1 Overview

5.2 Types of Radioactive Decay 5.3 Energetics of Radioactive Decay

5.3.4 Positron Decay 5.3.5 Electron Capture 5.3.6 Neutron Decay 5.3.7 Proton Decay 5.3.8 Internal Conversion

5.3.9 Examples of Energy-Level Diagrams 5.4 Characteristics of Radioactive Decay

5.4.1 The Decay Constant 5.4.2 Exponential Decay 5.4.3 The Half-Life

5.4.4 Decay Probability for a Finite Time Interval 5.4.5 Mean Lifetime

5.4.6 Activity

5.4.7 Half-Life Measurement

5.4.8 Decay by Competing Processes 5.5 Decay Dynamics

5.5.1 Decay with Production

5.5.2 Three Component Decay Chains 5.5.3 General Decay Chain

5.6 Naturally Occurring Radionuclides 5.6.1 Cosmogenic Radionuclides

5.6.2 Singly Occurring Primordial Radionuclides 5.6.3 Decay Series of Primordial Origin

5.6.4 Secular Equilibrium 5.7 Radiodating

5.7.1 Measuring the Decay of a Parent

5.7.2 Measuring the Buildup of a Stable Daughter

6 Binary Nuclear Reactions 6.1 Types of Binary Reactions

6.1.1 The Compound Nucleus

6.2 Kinematics of Binary Two-Product Nuclear Reactions 6.2.1 Energy/Mass Conservation

6.2.2 Conservation of Energy and Linear Momentum 6.3 Reaction Threshold Energy

6.3.1 Kinematic Threshold 6.3.2 Coulomb Barrier Threshold 6.3.3 Overall Threshold Energy 6.4 Applications of Binary Kinematics

6.4.1 A Neutron Detection Reaction 6.4.2 A Neutron Production Reaction

6.4.3 Heavy Particle Scattering from an Electron 6.5 Reactions Involving Neutrons

6.5.1 Neutron Scattering

6.5.2 Neutron Capture Reactions 6.5.3 Fission Reactions

6.6.1 Fission Products

6.6.2 Neutron Emission in Fission 6.6.3 Energy Released in Fission 6.7 Fusion Reactions

6.7.1 Thermonuclear Fusion 6.7.2 Energy Production in Stars 6.7.3 Nucleogenesis

**7 Radiation Interactions with Matter**

7.1 Attenuation of Neutral Particle Beams 7.1.1 The Linear Interaction Coefficient 7.1.2 Attenuation of Uncollided Radiation

7.1.3 Average Travel Distance Before an Interaction 7.1.4 Half-Thickness

7.1.5 Scattered Radiation 7.1.6 Microscopic Cross Sections 7.2 Calculation of Radiation Interaction Rates

7.2.1 Flux Density

7.2.2 Reaction-Rate Density

7.2.3 Generalization to Energy- and Time-Dependent Situations 7.2.4 Radiation Fluence

7.2.5 Uncollided Flux Density from an Isotropic Point Source 7.3 Photon Interactions

7.3.1 Photoelectric Effect 7.3.2 Compton Scattering 7.3.3 Pair Production

7.3.4 Photon Attenuation Coefficients 7.4 Neutron Interactions

7.4.1 Classification of Types of Interactions 7.4.2 Fission Cross Sections

7.5 Attenuation of Charged Particles 7.5.1 Interaction Mechanisms 7.5.2 Particle Range

7.5.3 Stopping Power

7.5.4 Estimating Charged-Particle Ranges

**8 Detection and Measurement of Radiation**

8.1 Gas-Filled Radiation Detectors 8.1.1 lonization Chambers 8.1.2 Proportional Counters 8.1.3 Geiger-Mueller Counters 8.2 Scintillation Detectors

8.3 Semiconductor lonizing-Radiation Detectors 8.4 Personal Dosimeters

8.4.1 The Pocket Ion Chamber 8.4.2 The Film Badge

8.5 Measurement Theory

8.5.1 Types of Measurement Uncertainties

8.5.2 Uncertainty Assignment Based Upon Counting Statistics 8.5.3 Dead Time

8.5.4 Energy Resolution

**9 Radiation Doses and Hazard Assessment**

9.1 Historical Roots 9.2 Dosimetric Quantities

9.2.1 Energy Imparted to the Medium 9.2.2 Absorbed Dose

9.2.3 Kerma

9.2.4 Calculating Kerma and Absorbed Doses 9.2.5 Exposure

9.2.6 Relative Biological Effectiveness 9.2.7 Dose Equivalent

9.2.8 Quality Factor

9.2.9 Effective Dose Equivalent 9.2.10 Effective Dose

9.3 Natural Exposures for Humans

9.4 Health Effects from Large Acute Doses 9.4.1 Effects on Individual Cells

9.4.2 Deterministic Effects in Organs and Tissues 9.4.3 Potentially Lethal Exposure to Low-LET Radiation 9.5 Hereditary Effects

9.5.1 Classification of Genetic Effects 9.5.2 Summary of Risk Estimates

9.5.3 Estimating Gonad Doses and Genetic Risks 9.6 Cancer Risks from Radiation Exposures

9.6.1 Dose-Response Models for Cancer

9.6.2 Average Cancer Risks for Exposed Populations 9.7 Radon and Lung Cancer Risks

9.7.1 Radon Activity Concentrations 9.7.2 Lung Cancer Risks

9.8 Radiation Protection Standards 9.8.1 Risk-Related Dose Limits

9.8.2 The 1987 NCRP Exposure Limits

**10 Principles of Nuclear Reactors**

10.1 Neutron Moderation

10.2 Thermal-Neutron Properties of Fuels

10.3 The Neutron Life Cycle in a Thermal Reactor 10.3.1 Quantification of the Neutron Cycle 10.3.2 Effective Multiplication Factor 10.4 Homogeneous and Heterogeneous Cores 10.5 Reflectors

10.6.1 A Simple Reactor Kinetics Model 10.6.2 Delayed Neutrons

10.6.3 Reactivity and Delta-k

10.6.4 Revised Simplified Reactor Kinetics Models 10.6.5 Power Transients Following a Reactivity Insertion 10.7 Reactivity Feedback

10.7.1 Feedback Caused by Isotopic Changes 10.7.2 Feedback Caused by Temperature Changes 10.8 Fission Product Poisons

10.8.1 Xenon Poisoning 10.8.2 Samarium Poisoning

10.9 Addendum 1: The Diffusion Equation 10.9.1 An Example Fixed-Source Problem 10.9.2 An Example Criticality Problem

10.9.3 More Detailed Neutron-Field Descriptions 10.10 Addendum 2: Kinetic Model with Delayed Neutrons 10.11 Addendum 3: Solution for a Step Reactivity Insertion

**11 Nuclear Power**

11.1 Nuclear Electric Power

11.1.1 Electricity from Thermal Energy 11.1.2 Conversion Efficiency

11.1.3 Some Typical Power Reactors 11.1.4 Coolant Limitations

11.2 Pressurized Water Reactors

11.2.1 The Steam Cycle of a PWR 11.2.2 Major Components of a PWR 11.3 Boiling Water Reactors

11.3.1 The Steam Cycle of a BWR 11.3.2 Major Components of a BWR 11.4 New Designs for Central-Station Power

11.4.1 Certified Evolutionary Designs 11.4.2 Certified Passive Design

11.4.3 Other Evolutionary LWR Designs 11.4.4 Gas Reactor Technology

11.5 The Nuclear Fuel Cycle

11.5.1 Uranium Requirements and Availability 11.5.2 Enrichment Techniques

11.5.3 Radioactive Waste 11.5.4 Spent Fuel

11.6 Nuclear Propulsion

11.6.1 Naval Applications

11.6.2 Other Marine Applications 11.6.3 Nuclear Propulsion in Space

**12 Other Methods for Converting Nuclear Energy to Electricity**

12.1 Thermoelectric Generators

12.2 Thermionic Electrical Generators 12.2.1 Conversion Efficiency

12.2.2 In-Pile Thermionic Generator 12.3 AMTEC Conversion

12.4 Stirling Converters

12.5 Direct Conversion of Nuclear Radiation

12.5.1 Types of Nuclear Radiation Conversion Devices 12.5.2 Betavoltaic Batteries

12.6 Radioisotopes for Thermal Power Sources 12.7 Space Reactors

12.7.1 The U.S. Space Reactor Program 12.7.2 The Russian Space Reactor Program

**13 Nuclear Technology in Industry and Research**

13.1 Production of Radioisotopes

13.2 Industrial and Research Uses of Radioisotopes and Radiation 13.3 Tracer Applications

13.3.1 Leak Detection 13.3.2 Pipeline Interfaces 13.3.3 Flow Patterns

13.3.4 Flow Rate Measurements 13.3.5 Labeled Reagents

13.3.6 Tracer Dilution 13.3.7 Wear Analyses 13.3.8 Mixing Times 13.3.9 Residence Times 13.3.10 Frequency Response

13.3.11 Surface Temperature Measurements 13.3.12 Radiodating

13.4 Materials Affect Radiation 13.4.1 Radiography 13.4.2 Thickness Gauging 13.4.3 Density Gauges 13.4.4 Level Gauges

13.4.5 Radiation Absorptiometry 13.4.6 Oil-Well Logging

13.4.7 Neutron Activation Analysis

13.4.8 Neutron Capture-Gamma Ray Analysis 13.4.9 Molecular Structure Determination 13.4.10 Smoke Detectors

**14 Medical Applications of Nuclear Technology**
14.1 Diagnostic Imaging

14.1.1 X-Ray Projection Imaging 14.1.2 Fluoroscopy

14.1.3 Mammography 14.1.4 Bone Densitometry

14.1.5 X-Ray Computed Tomography (CT)

14.1.6 Single Photon Emission Computed Tomography (SPECT) 14.1.7 Positron Emission Tomography (PET)

14.1.8 Magnetic Resonance Imaging (MRI) 14.2 Radioimmunoassay

14.3 Diagnostic Radiotracers 14.4 Radioimmunoscintigraphy 14.5 Radiation Therapy

14.5.1 Early Applications 14.5.2 Teletherapy

14.5.3 Radionuclide Therapy 14.5.4 Clinical Brachytherapy

14.5.5 Boron Neutron Capture Therapy

**Appendic A: Fundamental Atomic Data**

**Appendix B: Atomic Mass Table**

**Appendix C: Cross Sections and Related Data**

### Chapter 1

**Fundamental Concepts**

The last half of the twentieth century was a time in which tremendous advances in science and technology revolutionized our entire way of life. Many new technolo-gies were invented and developed in this time period from basic laboratory research to widespread commercial application. Communication technology, genetic engi-neering, personal computers, medical diagnostics and therapy, bioengiengi-neering, and material sciences are just a few areas that were greatly affected.

Nuclear science and engineering is another technology that has been transformed in less than fifty years from laboratory research into practical applications encoun-tered in almost all aspects of our modern technological society. Nuclear power, from the first experimental reactor built in 1942, has become an important source of electrical power in many countries. Nuclear technology is widely used in medical imaging, diagnostics and therapy. Agriculture and many other industries make wide use of radioisotopes and other radiation sources. Finally, nuclear applications are found in a wide range of research endeavors such as archaeology, biology, physics, chemistry, cosmology and, of course, engineering.

The discipline of nuclear science and engineering is concerned with quantify-ing how various types of radiation interact with matter and how these interactions affect matter. In this book, we will describe sources of radiation, radiation inter-actions, and the results of such interactions. As the word "nuclear" suggests, we will address phenomena at a microscopic level, involving individual atoms and their constituent nuclei and electrons. The radiation we are concerned with is generally very penetrating and arises from physical processes at the atomic level.

However, before we begin our exploration of the atomic world, it is necessary to introduce some basic fundamental atomic concepts, properties, nomenclature and units used to quantify the phenomena we will encounter. Such is the purpose of this introductory chapter.

**1.1 Modern Units**

**Table 1.1. The SI system of units arid their four categories.**

Base SI units: Physical quantity length mass time electric current thermodynamic temperature luminous intensity

quantity of substance

Examples of Derived SI Physical quantity

force

work, energy, quantity of heat power

electric charge

electric potential difference electric resistance

magnetic flux magnetic flux density frequency

radioactive decay rate pressure velocity mass density area volume molar energy

electric charge density

Supplementary Units: Physical quantity plane angle solid angle Temporary Units: Physical quantity length velocity length area pressure pressure area radioactive activity radiation exposure absorbed radiation dose radiation dose equivalent

Unit name meter kilogram second ampere kelvin candela mole units: Unit name ricwton joule watt coulomb volt ohm weber tesla hertz bequerel pascal Unit name radian steradian Unit name nautical mile knot angstrom hectare bar standard atmosphere barn curie roentgen gray sievert Symbol m kg s A K cd mol Symbol N J W

### c

V### ft

Wb T Hz Bq Pa Symbol racl sr Symbol A ha bar atm b Ci R Gy Sv Formula kg m s N m J s-1 A s W A'1 V A-1 V s Wb m"2 s-1 s-1 N m-'2 in s"1 kg m~^o m in3 J mor1 C m-3

Value in SI unit 1852 m

*1852/3600 rn s~[*

0.1 nm = ICT10 rn
1 hm2* = 10*4 m2
0.1 MPa
0.101325 MPa
10~24 cm2
3.7 x 10H) Bq
2.58 x 10~4 C kg"1
1 J kg-1

use for special applications. These units are shown in Table 1.1. To accommodate very small and large quantities, the SI units and their symbols are scaled by using the SI prefixes given in Table 1.2.

There are several units outside the SI which are in wide use. These include the
*time units day (d), hour (h) and minute (min); the liter (L or I); plane angle degree*
(°), minute ('), and second ("); and, of great use in nuclear and atomic physics,
the electron volt (eV) and the atomic mass unit (u). Conversion factors to convert
some non-Si units to their SI equivalent are given in Table 1.3.

Finally it should be noted that correct use of SI units requires some "grammar" on how to properly combine different units and the prefixes. A summary of the SI grammar is presented in Table 1.4.

**Table 1.2. SI prefixes.** **Table 1.3. Conversion factors.**

Factor 1024 1021 1018 1015 1012 109 106 103 102 101 lo-1 io-2 10~3 10~6 io-9 10~12 io-15 10-18 io-21 io-24 Prefix yotta zetta exa peta tera giga mega kilo hecto deca deci centi milli micro nano pico femto atto zepto yocto Symbol Y Z E P T G M k h da d c m M n P f a z y Property Length Area Volume Mass Force Pressure Energy Unit in. ft mile (int'l) in2 ft2 acre

square mile (int'l) hectare

oz (U.S. liquid) in3 gallon (U.S.) ft3 oz (avdp.) Ib ton (short) kgf lbf ton

lbf/in2 (psi) lbf/ft2 atm (standard) in. H2O (@ 4 °C) in. Hg (© 0 °C) mm Hg (@ 0 °C) bar eV cal Btu kWh MWd SI equivalent

2.54 x 1CT2 ma
3.048 x 10~* 1* ma
1.609344 X 103 ma
6.4516 x 10~4 m2 a
9.290304 X 10~2 m2 a
4.046873 X 103 m2
2.589988 X 106 m2
1 x 104 m2

2.957353 X 10~5 m3 1.638706 X 10~5 m3 3.785412 X 10~3 m3 2.831685 x 10~2 m3 2.834952 x 10~2 kg 4.535924 X lO^1 kg 9.071 847 x 102 kg 9.806650 N a 4.448222 N 8.896444 X 103 N 6.894757 x 103 Pa 4.788026 x 101 Pa 1.013250 x 105 Paa 2.49082 x 102 Pa 3.38639 x 103 Pa 1.33322 x 102 Pa 1 x 105 Paa 1.60219 x 10~19 J 4.184 Ja

1.054350 X 103 J 3.6 x 106 Ja 8.64 x 1010 Ja "Exact converson factor.

**Table 1.4. Summary of SI grammar.**

Grammar Comments

capitalization

space plural raised dots

solidis

mixing units/names

prefix

double vowels

hyphens

numbers

A unit name is never capitalized even if it is a person's name. Thus curie, not Curie. However, the symbol or abbreviation of a unit named after a person is capitalized. Thus Sv, not sv.

Use 58 rn, not 58m .

A symbol is never pluralized. Thus 8 N, not 8 Ns or 8 Ns.

Sometimes a raised dot is used when combining units such as N-m2-s; however, a single space between unit symbols is preferred as in N m2 s.

For simple unit combinations use g/cm3 or g cm~3. However, for more complex expressions, N m~2 s""1 is much clearer than N/m2/s. Never mix unit names and symbols. Thus kg/s, not kg/second or kilogram/s.

Never use double prefixes such as ^g; use pg. Also put prefixes in the numerator. Thus km/s, not m/ms.

When spelling out prefixes with names that begin with a vowel, su-press the ending vowel on the prefix. Thus megohm and kilohm, not megaohm and kiloohm.

Do not put hyphens between unit names. Thus newton meter, not newton-meter. Also never use a hyphen with a prefix. Hence, write microgram not micro-gram.

For numbers less than one, use 0.532 not .532. Use prefixes to avoid large numbers; thus 12.345 kg, not 12345 g. For numbers with more than 5 adjacent numerals, spaces are often used to group numerals into triplets; thus 123456789.12345633, not 123456789.12345633.

**1.1.1 Special Nuclear Units**

When treating atomic and nuclear phenomena, physical quantities such as energies and masses are extremely small in SI units, and special units are almost always used. Two such units are of particular importance.

**The Electron Volt**

The energy released or absorbed in a chemical reaction (arising from changes in electron bonds in the affected molecules) is typically of the order of 10~19 J. It

*is much more convenient to use a special energy unit called the electron volt. By*
*definition, the electron volt is the kinetic energy gained by an electron (mass me*

*and charge —e) that is accelerated through a potential difference AV of one volt*
= 1 W/A = 1 (J s~1)/(C s-1) = 1 J/C. The work done by the electric field is
*-e&V = (1.60217646 x 1(T*19 C)(l J/C) = 1.60217646 x 10~19 J = 1 eV. Thus

*If the electron (mass me) starts at rest, then the kinetic energy T of the electron*

after being accelerated through a potential of 1 V must equal the work done on the electron, i.e.,

*T = \m^ = -eAV = I eV. (1.1)*

*Zi*

*The speed of the electron is thus v = ^/2T/me* ~ 5.93 x 105 m/s, fast by our

*everyday experience but slow compared to the speed of light (c ~ 3 x 10*8 m/s).

**The Atomic Mass Unit**

Because the mass of an atom is so much less than 1 kg, a mass unit more appropriate
to measuring the mass of atoms has been defined independent of the SI kilogram
*mass standard (a platinum cylinder in Paris). The atomic mass unit (abbreviated*
*as amu, or just u) is defined to be 1/12 the mass of a neutral ground-state atom*
of 12*C. Equivalently, the mass of Na 12*C atoms (Avogadro's number = 1 mole) is

0.012 kg. Thus, 1 amu equals (1/12)(0.012 kg/JVa) = 1.6605387 x 10~27 kg.

**1.1.2 Physical Constants**

Although science depends on a vast number of empirically measured constants to
make quantitative predictions, there are some very fundamental constants which
*specify the scale and physics of our universe. These physical constants, such as the*
*speed of light in vacuum c, the mass of the neutron me, Avogadro's number 7V*a,

etc., are indeed true constants of our physical world, and can be viewed as auxiliary units. Thus, we can measure speed as a fraction of the speed of light or mass as a multiple of the neutron mass. Some of the important physical constants, which we use extensively, are given in Table 1.5.

**1.2 The Atom**

Crucial to an understanding of nuclear technology is the concept that all matter is
*composed of many small discrete units of mass called atoms. Atoms, while often*
viewed as the fundamental constituents of matter, are themselves composed of other
*particles. A simplistic view of an atom is a very small dense nucleus, composed of*
*protons and neutrons (collectively called nucleons), that is surrounded by a swarm of*
negatively-charged electrons equal in number to the number of positively-charged
protons in the nucleus. In later chapters, more detailed models of the atom are
introduced.

It is often said that atoms are so small that they cannot been seen. Certainly, they cannot with the naked human eye or even with the best light microscope. However, so-called tunneling electron microscopes can produce electrical signals, which, when plotted, can produce images of individual atoms. In fact, the same instrument can also move individual atoms. An example is shown in Fig. 1.1. In this figure, iron atoms (the dark circular dots) on a copper surface are shown being moved to form a ring which causes electrons inside the ring and on the copper surface to form standing waves. This and other pictures of atoms can be found on the web at http://www.ibm.com/vis/stm/gallery.html.

**Table 1.5. Values of some important physical constants as internationally **

recom-mended in 1998.

Constant Symbol Value

Speed of light (in vacuum) Electron charge

Atomic mass unit

Electron rest mass

Proton rest mass

Neutron rest mass

Planck's constant

Avogadro's constant Boltzmann constant

Ideal gas constant (STP) Electric constant

2.99792458 x 108* m s~l*

1.60217646 x 10'19 C 1.6605387 x 10~27 kg (931.494013 MeV/c2) 9.1093819 x 10~31 kg (0.51099890 MeV/c2) (5.48579911 x 10~4 u) 1.6726216 x 10~27 kg (938.27200 MeV/c2) (1.0072764669 u) 1.6749272 x 10~27 kg (939.56533 MeV/c2) (1.0086649158 u) 6.6260688 x 10~34 J s 4.1356673 x 10~15 eV s 6.0221420 x 1023 mol"1 1.3806503 x 10~23 J K ~] (8.617342 x 10~5 eV K"1) 8.314472 J mor1 K"1 8.854187817 x 10~12 F m"1

*Source: P.J. Mohy and B.N. Taylor, "CODATA*

*Fundamental Physical Constants," Rev. Modern*

Recommended Values of the

*Physics, 72, No. 2, 2000.*

*together by yet other particles called gluons. Whether quarks arid gluons are *
them-selves fundamental particles or are composites of even smaller entities is unknown.
*Particles composed of different types of quarks are called baryons. The electron and*
*its other lepton kin (such as positrons, neutrinos, and muons) are still thought, by*
current theory, to be indivisible entities.

However, in our study of nuclear science and engineering, we can viewr the

elec-tron, neutron and proton as fundamental indivisible particles, since the composite nature of nucleons becomes apparent only under extreme conditions, such as those encountered during the first minute after the creation of the universe (the "big bang") or in high-energy particle accelerators. We will not deal with such gigantic energies. Rather, the energy of radiation we consider is sufficient only to rearrange or remove the electrons in an atom or the neutrons and protons in a nucleus.

**1.2.1 Atomic and Nuclear Nomenclature**

**Figure 1.1. Pictures of iron atoms on a copper surface being moved to**

form a ring inside of which surface copper electrons are confined and form standing waves. Source: IBM Corp.

However, atoms of the same element may have different numbers of neutrons in
the nucleus. Atoms of the same element, but with different numbers of neutrons,
*are called isotopes. The symbol used to denote a particular isotope is*

*where X is the chemical symbol and A = Z + TV, which is called the mass number.*
For example, two uranium isotopes, which will be discussed extensively later, are
2g|U and 2*g2U. The use of both Z and X is redundant because one specifies the*
*other. Consequently, the subscript Z is often omitted, so that we may write, for*
example, simply 235U and 238U.1

*1To avoid superscripts, which were hard to make on old-fashioned typewriters, the simpler form*

Because isotopes of the same element have the same number and arrangement of electrons around the nucleus, the chemical properties of such isotopes are nearly identical. Only for the lightest isotopes (e.g., 1H, deuterium 2H, and tritium 3H)

are small differences noted. For example, light water 1H2O freezes at 0 °C while

heavy water 2H2O (or D2O since deuterium is often given the chemical symbol D)

freezes at 3.82 °C.

A discussion of different isotopes arid elements often involves the following basic nuclear jargon.

**nuclide: a term used to refer to a particular atom or nucleus with a specific neutron**
*number N and atomic (proton) number Z. Nuclides are either stable (i.e.,*
unchanging in time unless perturbed) or radioactive (i.e., they spontaneously
*change to another nuclide with a different Z and/or N by emitting one or*
*more particles). Such radioactive nuclides are termed rachonuclides.*

**isobar: nuclides with the same mass number A = N + Z but with different number***of neutrons N and protons Z. Nuclides in the same isobar have nearly equal*
masses. For example, isotopes which have nearly the same isobaric mass of
14 u include ^B. ^C, ^N, and ^O.

**isotone: nuclides with the same number of neutrons A**r but different number of

*protons Z. For example, nuclides in the isotone with 8 neutrons include ^B.*
^C. Jf N and *f O.

* isorner: the same nuclide (same Z and A") in which the nucleus is in different *
long-lived excited states. For example, an isomer of "Te is 99mTe where the m

denotes the longest-lived excited state (i.e., a state in which the nucleons in the nucleus are not in the lowest energy state).

**1.2.2 Atomic and Molecular Weights**

*The atomic weight A of an atom is the ratio of the atom's mass to that of one neutral*
atom of 12*C in its ground state. Similarly the molecular weight of a molecule is the*

ratio of its molecular mass to one atom of 12C. As ratios, the atomic and molecular

weights are dimensionless numbers.

*Closely related to the concept of atomic weight is the atomic mass unit, which*
we introduced in Section 1.1.1 as a special mass unit. Recall that the atomic mass
unit is denned such that the mass of a 12C atom is 12 u. It then follows that the

*mass M of an atom measured in atomic mass units numerically equals the atom's*
*atomic weight A. From *Table 1.5 we see 1 u ~ 1.6605 x 10~27 kg. A detailed

listing of the atomic masses of the known nuclides is given in Appendix B. From
this appendix, we see that the atomic mass (in u) and. hence, the atomic weight of
*a nuclide almost equals (within less than one percent) the atomic mass number A*
*of the nuclide. Thus for approximate calculations, we can usually assume A — A.*

Most naturally occurring elements are composed of two or more isotopes. The

*isotopic abundance 7, of the /-th isotope in a given element is the fraction of the*

of all naturally occurring isotopes of the element, weighted by the isotopic abun-dance of each isotope, i.e.,

where the summation is over all the isotopic species comprising the element. Ele-mental atomic weights are listed in Appendix Tables A. 2 and A. 3.

**Example 1.1: What is the atomic weight of boron? From **Table A.4 we find
that naturally occurring boron consists of two stable isotopes 10B and nB with
isotopic abundances of 19.1 and 80.1 atom-percent, respectively. From Appendix
B the atomic weight of 10B and UB are found to be 10.012937 and 11.009306,
respectively. Then from Eq. (1.2) we find

*AB = (7io-4io +7n./4ii)/100*

= (0.199 x 10.012937) + (0.801 x 11.009306) = 10.81103.

*This value agrees with the tabulated value AB = 10.811 as listed in Tables A.2*

and A.3.

**1.2.3 Avogadro's Number**

Avogadro's constant is the key to the atomic world since it relates the number of microscopic entities in a sample to a macroscopic measure of the sample. Specif-ically, Avogadro's constant 7Va ~ 6.022 x 1023 equals the number of atoms in 12 grams of 12C. Few fundamental constants need be memorized, but an approximate value of Avogadro's constant should be.

The importance of Avogadro's constant lies in the concept of the mole. A
*mole (abbreviated mol) of a substance is denned to contain as many "elementary*
particles" as there are atoms in 12 g of 12C. In older texts, the mole was often
called a "gram-mole" but is now called simply a mole. The "elementary particles"
can refer to any identifiable unit that can be unambiguously counted. We can, for
example, speak of a mole of stars, persons, molecules or atoms.

Since the atomic weight of a nuclide is the atomic mass divided by the mass of
one atom of 12C, the mass of a sample, in grams, numerically equal to the atomic
weight of an atomic species must contain as many atoms of the species as there
are in 12 grams (or 1 mole) of 12C. The mass in grams of a substance that equals
*the dimensionless atomic or molecular weight is sometimes called the gram atomic*
*weight or gram molecular weight. Thus, one gram atomic or molecular weight of*
any substance represents one mole of the substance and contains as many atoms or
molecules as there are atoms in one mole of 12*C, namely Na* atoms or molecules.

*That one mole of any substance contains Na entities is known as Avogadro's law*

**Example 1.2: How many atoms of** 10B are there in 5 grams of boron? From
Table A. 3, the atomic weight of elemental boron AB = 10.811. The 5-g sample
*of boron equals m/ AB moles of boron, and since each mole contains Na* atoms,

the number of boron atoms is

*Na = = (5 g)(0.6022 x 10" atoms/mo.) = y*

*AB (10.811 g/mol)*

From Table A. 4, the isotopic abundance of 10B in elemental boron is found to
*be 19.9%. The number Nw of* 10B atoms in the sample is, therefore, A/io =
(0.199)(2.785 x 1023) = 5.542 x 1022 atoms.

**1.2.4 Mass of an Atom**

With Avogadro's number many basic properties of atoms can be inferred. For
example, the mass of an individual atom can be found. Since a mole of a group
*of identical atoms (with a mass of A grams) contains 7V*a atoms, the mass of an

individual atom is

*M (g/atom) = A/Na ~ A/Na. (1.3)*

*The approximation of A by A is usually quite acceptable for all but the most precise*
calculations. This approximation will be used often throughout this book.

In Appendix B. a comprehensive listing is provided for the masses of the known atom. As will soon become apparent, atomic masses are central to quantifying the energetics of various nuclear reactions.

**Example 1.3: Estimate the mass on an atom of** 238U. From Eq. (1.3) we find
238 (g/mol)

6.022 x 1023 atoms/mol = 3.952 x 10 g/atom.

From Appendix B, the mass of 238U is found to be 238.050782 u which numerically
*equals its gram atomic weight A. A more precise value for the mass of an atom*
of 238U is, therefore,

,238in ___ 238.050782 (g/mol)

M(238U) = I,w* ' = 3.952925 x IQ~" g/atom.*
v ; 6.022142 x 1023 atoms/mol &/
*Notice that approximating A by A leads to a negligible error.*

**1.2.5 Atomic Number Density**

In many calculations, we will need to know the number of atoms in 1 cm3 of a

*contains p/A moles of the substance and pNa/A atoms. The atom density N is*

thus

*N (atoms/cm*3) - (1.4)

*To find the atom density Ni of isotope i of an element with atom density N simply*
*multiply N by the fractional isotopic abundance 7^/100 for the isotope, i.e., Ni —*

Equation 1.4 also applies to substances composed of identical molecules. In this
*case, N is the molecular density and A the gram molecular weight. The number of*
atoms of a particular type, per unit volume, is found by multiplying the molecular
density by the number of the same atoms per molecule. This is illustrated in the
following example.

**Example 1.4: What is the hydrogen atom density in water? The molecular**

*weight of water AH Q = 1An + 2Ao — 2A# + AO = 18. The molecular density*
of EbO is thus

A r / T T* _ pH2°Na (I g cm"*3) x (6.022 x 1023 molecules/mol)
7 V ( r i 2 O ) = : = ; :

V* ' ^H*20 18g/mol

= 3.35 x 1022 molecules/cm3.

The hydrogen density 7V(H) = 27V(H2O) = 2(3.35xlO22) = 6.69xlO22 atoms/cm3.

The composition of a mixture such as concrete is often specified by the mass
*fraction Wi of each constituent. If the mixture has a mass density p, the mass*
*density of the iih constituent is pi — Wip. The density Ni of the iih component is*
thus

*PiNa wlPNa*

*1 =*

* ~A~*

**S\i S^-i**=* ~A~'*

( }
If the composition of a substance is specified by a chemical formula, such as
XnYm*, the molecular weight of the mixture is A = nAx + mAy and the mass*
fraction of component X is

*/- -,*
(1.6)

*t •*

*nAx + mAy*

Finally, as a general rule of thumb, it should be remembered that atom densities in solids and liquids are usually between 1021 and 1023 /cm~3. Gases at standard temperature and pressure are typically less by a factor of 1000.

**1.2.6 Size of an Atom**

of the size of atoms reveals a diffuse electron cloud about the nucleus. Although
there is no sharp edge to an atom, an effective radius can be defined such that
outside this radius an electron is very unlikely to be found. Except for hydrogen,
atoms have radii of about 2 to 2.5 x 10~8* cm. As Z increases, i.e., as more electrons*

and protons are added, the size of the electron cloud changes little, but simply becomes more dense. Hydrogen, the lightest element, is also the smallest with a radius of about 0.5 x 10~8 cm.

**1.2.7 Atomic and Isotopic Abundances**

During the first few minutes after the big bang only the lightest elements (hydrogen, helium and lithium) were created. All the others were created inside stars either during their normal aging process or during supernova explosions. In both processes, nuclei are combined or fused to form heavier nuclei. Our earth with all the naturally occurring elements was formed from debris of dead stars. The abundances of the elements for our solar system is a consequence of the history of stellar formation and death in our corner of the universe. Elemental abundances are listed in Table A. 3. For a given element, the different stable isotopes also have a natural relative abundance unique to our solar system. These isotopic abundances are listed in

Table A. 4.

**1.2.8 Nuclear Dimensions**
Size of a Nucleus

If each proton and neutron in the nucleus has the same volume, the volume of a
*nu-cleus should be proportional to A. This has been confirmed by many measurements*
that have explored the shape and size of nuclei. Nuclei, to a first approximation, are
spherical or very slightly ellipsoidal with a somewhat diffuse surface, In particular,
it is found that an effective spherical nuclear radius is

*R = R0Al/3, with R0* ~ 1.25 x 1CT13 cm. (1.7)

The associated volume is

*Vicious = ^ - 7.25 X W~39A Cm3*. (1.8)

Since the atomic radius of about 2 x 10~8 cm is 105 times greater than the

nuclear radius, the nucleus occupies only about 10~15 of the volume of a atom. If

an atom were to be scaled to the size of a large concert hall, then the nucleus would be the size of a very small gnat!

**Nuclear Density**

Since the mass of a nucleon (neutron or proton) is much greater than the mass of
*electrons in an atom (mn* = 1837 me), the mass density of a nucleus is

m*nucleus A/Na 14* , 3

*^nucleus = T7 - = ~, \ r> ~ 2A X 1U* S/cm '
^nucleus

**1.3 Chart of the Nuclides**

*The number of known different atoms, each with a distinct combination of Z and*
A, is large, numbering over 3200 nuclides. Of these, 266 are stable (i.e.,
non-radioactive) and are found in nature. There are also 65 long-lived radioisotopes
found in nature. The remaining nuclides have been made by humans and are
ra-dioactive with lifetimes much shorter than the age of the solar system. The lightest
*atom (A = 1) is ordinary hydrogen JH, while the mass of the heaviest is *
contin-ually increasing as heavier and heavier nuclides are produced in nuclear research
*laboratories. One of the heaviest (A = 269) is meitnerium logMt.*

A very compact way to portray this panoply of atoms and some of their
*proper-ties is known as the Chart of the Nuclides. This chart is a two-dimensional matrix of*
*squares (one for each known nuclide) arranged by atomic number Z (y-axis) versus*
*neutron number N (x-axis). Each square contains information about the nuclide.*
The type and amount of information provided for each nuclide is limited only by
the physical size of the chart. Several versions of the chart are available on the
internet (see web addresses given in the next section and in Appendix A).

Perhaps, the most detailed Chart of the Nuclides is that provided by General Electric Co. (GE). This chart (like many other information resources) is not avail-able on the web; rather, it can be purchased from GE ($15 for students) and is highly recommended as a basic data resource for any nuclear analysis. It is available as a 32" x55" chart or as a 64-page book. Information for ordering this chart can be found on the web at http://www.ssts.lmsg.lmco.com/nuclides/index.html.

**1.3.1 Other Sources of Atomic/Nuclear Information**

A vast amount of atomic and nuclear data is available on the world-wide web. However, it often takes considerable effort to find exactly what you need. The sites listed below contain many links to data sources, and you should explore these to become familiar with them and what data can be obtained through them.

These two sites have links to the some of the major nuclear and atomic data repos-itories in the world.

http://www.nndc.bnl.gov/wallet/yellows.htm

http://www.nndc.bnl.gov/usndp/usndp-subject.html

The following sites have links to many sources of fundamental nuclear and atomic data.

http://www.nndc.bnl.gov/

http://physics.nist.gov/cuu/index.htm

http://isotopes.Ibl.gov/isotopes/toi.html

http://wwwndc.tokai.jaeri.go.jp/index.html

http://wwwndc.tokai.j aeri.go.jp/nucldata/index.html

These sites contain much information about nuclear technology and other related topics. Many are home pages for various governmental agencies and some are sites offering useful links, software, reports, and other pertinent information.

http://physics.nist.gov/ h t t p : / / w w w . n i s t . g o v / http://www.energy.gov/ h t t p : / / w w w . n r c . g o v / h t t p : / / w w w . d o e . g o v / h t t p : / / w w w . e p a . g o v / o a r / h t t p : / / w w w . n r p b . o r g . u k /

http://www-rsicc.ornl.gov/rsic.html http://www.iaea.org/worldatom/ h t t p : / / w w w . n e a . f r /

**PROBLEMS**

1. Both the hertz and the curie have dimensions of s"1. Explain the difference between these two units.

2. Explain the SI errors (if any) in and give the correct equivalent units for the following units: (a) m-grams/pL, (b) megaohms/nm, (c) N-m/s/s, (d) gram cm/(s~1/mL). and (e) Bq/milli-Curie.

3. In vacuum, how far does light move in 1 ps?

4. In a medical test for a certain molecule, the concentration in the blood is reported as 123 mcg/dL. What is the concentration in proper SI notation? 5. How many neutrons and protons are there in each of the following riuclides:

(a) 10B. (b) 24Na, (c) 59Co, (d) 208Pb. and (e) 235U?

6. What are the molecular weights of (a) H2 gas, (b) H2O, and (c) HDO? 7. What is the mass in kg of a molecule of uranyl sulfate UC^SCV/

8. Show by argument that the reciprocal of Avogadro's constant is the gram equivalent of 1 atomic mass unit.

9. How many atoms of 234U are there in 1 kg of natural uranium? 10. How many atoms of deuterium are there in 2 kg of water?

11. Estimate the number of atoms in a 3000 pound automobile. State any assump-tions you make.

13. A reactor is fueled with 4 kg uranium enriched to 20 atom-percent in 235U. The remainder of the fuel is 238U. The fuel has a mass density of 19.2 g/cm3. (a) What is the mass of 235U in the reactor? (b) What are the atom densities of 235U and 238U in the fuel?

14. A sample of uranium is enriched to 3.2 atom-percent in 235U with the remainder being 238U. What is the enrichment of 235U in weight-percent?

15. A crystal of Nal has a density of 2.17 g/cm3. What is the atom density of sodium in the crystal?

16. A concrete with a density of 2.35 g/cm3 has a hydrogen content of 0.0085 weight fraction. What is the atom density of hydrogen in the concrete?

17. How much larger in diameter is a uranium atom compared to an iron atom? 18. By inspecting the chart of the nuclides, determine which element has the most

stable isotopes?

**19. Find an internet site where the isotopic abundances of mercury may be found.**

### Chapter 2

**Modern Physics Concepts**

During the first three decades of the twentieth century, our understanding of the physical universe underwent tremendous changes. The classical physics of Newton and the other scientists of the eighteenth and nineteenth centuries was shown to be inadequate to describe completely our universe. The results of this revolution in physics are now called "modern" physics, although they are now almost a century old.

Three of these modern physical concepts are (1) Einstein's theory of special rel-ativity, which extended Newtonian mechanics; (2) wave-particle duality, which says that both electromagnetic waves and atomic particles have dual wave and particle properties; and (3) quantum mechanics, which revealed that the microscopic atomic world is far different from our everyday macroscopic world. The results and insights provided by these three advances in physics are fundamental to an understanding of nuclear science and technology. This chapter is devoted to describing their basic ideas and results.

**2.1 The Special Theory of Relativity**

The classical laws of dynamics as developed by Newton were believed, for over 200 years, to describe all motion in nature. Students still spend considerable effort mastering the use of these laws of motion. For example, Newton's second law, in the form originally stated by Newton, says the rate of change of a body's momentum p equals the force F applied to it, i.e.,

*dp d(mv)*

*F*

*-^-^r~*

### (2

### -

### 1}

For a constant mass m, as assumed by Newton, this equation immediately reduces
*to the modem form of the second law, F = ma, where a = dv/dt, the acceleration*
of the body.

In 1905 Einstein discovered an error in classical mechanics and also the necessary correction. In his theory of special relativity,1 Einstein showed that Eq. (2.1) is still

correct, but that the mass of a body is not constant, but increases with the body's
*speed v. The form F = ma is thus incorrect. Specifically, Einstein showed that m*

*lln 1915 Einstein published the general theory of relativity, in which he generalized his special*

varies with the body's speed as

m = , m° =, (2.2)

V l - ^2/ c2

where m0 is the body's "rest mass," i.e., the body's mass when it is at rest, and

*c is the speed of light (~ 3 x 10*8 m/s). The validity of Einstein's correction was
immediately confirmed by observing that the electron's mass did indeed increase as
its speed increased in precisely the manner predicted by Eq. (2.2).

Most fundamental changes in physics arise in response to experimental results
that reveal an old theory to be inadequate. However, Einstein's correction to the
laws of motion was produced theoretically before being discovered experimentally.
This is perhaps not too surprising since in our everyday world the difference between
*ra and m0* is incredibly small. For example, a satellite in a circular earth orbit of

7100 km radius, moves with a speed of 7.5 km/s. As shown in Example 2.1, the mass
*correction factor ^/l — v2/c2 = I — 0.31 x 10~*9, i.e., relativistic effects change the
satellite's mass only in the ninth significant figure or by less than one part a billion!
Thus for practical engineering problems in our macroscopic world, relativistic effects
can safely be ignored. However, at the atomic and nuclear level, these effects can
be very important.

**Example 2.1: What is the fractional increase in mass of a satellite traveling at**

a speed of 7.5 km/s? Prom Eq. (2.2) find the fractional mass increase to be

*m — m0*

*Here (v/c)2 = (7.5 x 10*3/2.998 x 108)2 = 6.258 x 1CT10*. With this value of v2/c2*

most calculators will return a value of 0 for the fractional mass increase. Here's
*a trick for evaluating relativistically expressions for such small values of v2/c2 .*

*The expression (1 + e)n* can be expanded in a Taylor series as

*/•, \ n -, n ( n + l )* 2 n ( n — l)(n — 2) 3 ., . . .

(1 + e)n* = 1 + ne + -±— — Lt + — - ^ - '-C + • • • ~ 1 + ne for |e| « 1.*

*Thus, with e = —v2/c2 and n — —1/2 we find*

so that the fractional mass increase is

*m0* c

*s*

**2.1.1 Principle of Relativity**

The principle of relativity is older than Newton's laws of motion. In Newton's words
(actually translated from Latin) "The motions of bodies included in a given space
are the same amongst themselves, whether the space is at rest or moves uniformly
forward in a straight line." This means that experiments made in a laboratory in
uniform motion (e.g.. in an non-accelerating
*y y' train) produce the same results as when the*
<v laboratory is at rest. Indeed this principle
a of relativity is widely used to solve problems
*(x, y, z, t) in mechanics by shifting to moving frames of*
*(x1, y , z ' , t ' ) reference to simplify the equations of motion.*

*v The relativity principle is a simple *
*intu-x' itive and appealing idea. But do all the laws*
of physics indeed remain the same in all

**non-Figure 2.1. Two inertial coordinate** accelerati* (mertial) coordinate systems?*

systems. ° v* ' J*

Consider the two coordinate systems shown
*in Fig. 2.1. System S is at rest, while system S' is moving uniformly to the right*
*with speed v. At t — 0, the origin of S' is at the origin of S. The coordinates*
*of some point P are ( x , y , z ) in S and (x'.y'.z') in S'. Clearly, the primed and*
unprimed coordinates are related by

*x' = x - vt\ y' = y; z = z; and t' = t.* (2.3)
If these coordinate transformations are substituted into Newton's laws of motion,
we find they remain the same. For example, consider a force in the x-direction,
*Fx, acting on some mass m. Then the second law in the S' moving system is*

*Fx = md2x'/dt/2. Now transform this law to the stationary S system. We find (for*

v constant)

*d2x' d2(x-vt) d2x*

**r ~~~ TY1**** — TY1 — TD**

*x dt'2 ~ d(t}2 dt* '*

Thus the second law has the same form in both systems. Since the laws of motion are the same in all inertial coordinate systems, it follows that it is impossible to tell, from results of mechanical experiments, whether or not the system is moving. In the 1870s. Maxwell introduced his famous laws of electromagnetism. These laws explained all observed behavior of electricity, magnetism, and light in a uni-form system. However, when Eqs. (2.3) are used to transuni-form Maxwell's equations to another inertial system, they assume a different form. Thus from optical experi-ments in a moving system, one should be able to determine the speed of the system. For many years Maxwell's equations were thought to be somehow incorrect, but 20 years of research only continued to reconfirm them. Eventually, some scientists began to wonder if the problem lay in the Galilean transformation of Eqs. (2.3). Indeed, Lorentz observed in 1904 that if the transformation

*/ X VL i i A.I VXIC . .*

*x*

* ~~ /- o / o' y*

*=*

* y> z = z] t — , == i^-*

4### j

re-main unchanged under the peculiar looking Lorentz transformation. The Lorentz
transformation is indeed strange, since it indicates that space and time are not
*in-dependent quantities. Time in the S' system, as measured by an observer in the S*
system, is different from the time in the observer's system.

**2.1.2 Results of the Special Theory of Relativity**

It was Einstein who, in 1905, showed that the Lorentz transformation was indeed the correct transformation relating all inertial coordinate systems. He also showed how Newton's laws of motion must be modified to make them invariant under this transformation.

Einstein based his analysis on two postulates:

• The laws of physics are expressed by equations that have the same form in all coordinate systems moving at constant velocities relative to each other.

• The speed of light in free space is the same for all observers and is independent of the relative velocity between the source and the observer.

The first postulate is simply the principle of relativity, while the second states that
*we observe light to move with speed c even if the light source is moving with respect*
to us. From these postulates, Einstein demonstrated several amazing properties of
our universe.

1. The laws of motion are correct, as stated by Newton, if the mass of an object
*is made a function of the object's speed v, i.e.,*

*m(v) = , m°* (2.5)

V* ' ^ l - v*2/ c2

This result also shows that no material object can travel faster than the speed
*of light since the relativistic mass m(v) must always be real. Further, an*
*object with a rest mass (m0 > 0) cannot even travel at the speed of light;*

otherwise its relativistic mass would become infinite and give it an infinite kinetic energy.

2. The length of a moving object in the direction of its motion appears smaller to an observer at rest, namely

*L = L0^/l-v2/c2. (2.6)*

*where L0* is the "proper length" or length of the object when at rest.

The passage of time appears to slow in a system moving with respect to a
*stationary observer. The time t required for some physical phenomenon (e.g.,*
the interval between two heart beats) in a moving inertial system appears to
*be longer (dilated) than the time t0* for the same phenomenon to occur in the

*stationary system. The relation between t and t0* is

*t= , l° =.* (2.7)

4. Perhaps the most famous result from special relativity is the demonstration of the equivalence of mass and energy by the well-known equation

*E = me2. (2.8)*

This result says energy and mass can be converted to each other. Indeed, all changes in energy of a system results in a corresponding change in the mass of the system. This equivalence of mass and energy plays a critical role in the understanding of nuclear technology.

The first three of these results are derived in the Addendum 1 to this chapter. The last result, however, is so important that it is derived below.

**Derivation of E = me**

*Consider a particle with rest mass m0* initially

*at rest. At time t = 0 a force F begins to*
act on the particle accelerating the mass
*un-til at time t it has acquired a velocity v (see*
Fig. 2.2). From the conservation-of-energy
prin-ciple, the work done on this particle as it moves
*along the path of length s must equal the *
*ki-netic energy T of the particle at the end of the*
path. The path along which the particle moves
is arbitrary, depending on how F varies in time.
The work done by F (a vector) on the
*parti-cle as it moves through a displacement ds (also*
a vector) is F»ds. The total work done on the
particle over the whole path of length s is

*d(mv]*

t=0

*v = 0, m. = m0*

**Figure 2.2. A force F accelerates**

*a particle along a path of length s.*

= / F«ds = *•ds =* '** d(mv]*

*dt*

*ds ,*
*• — dt.*

*dt*

*But, ds/dt* *v, a vector parallel to d(mv}/dt; thus,*
"** d(mv*

*T =* *-v dt —*

*Substitution of Eq. (2.5) for m yields*

*= m*

or finally

*T = me2 -- m0c2.* (2.10)

Thus we see that the kinetic energy is associated with the increase in the mass of the particle.

*Equivalently, we can write this result as me2 = m0c2 + T. We can interpret me2*
*as the particle's "total energy" E, which equals its rest-mass energy plus its kinetic*
energy. If the particle was also in some potential field, for example, an electric field,
the total energy would also include the potential energy. Thus we have

(2.11)

This well known equation is the cornerstone of nuclear energy analyses. It shows the
equivalence of energy and mass. One can be converted into the other in precisely
*the amount specified by E = me2. When we later study various nuclear reactions,*
we will see many examples of energy being converted into mass and mass being
converted into energy.

*E* *— me2.*

**Example 2.2: What is the energy equivalent in MeV of the electron rest mass?**

From data in Table 1.5 and Eq. 1.1 we find

*E = meC2 = (9.109 x 10~*31 kg) x (2.998 x 108 m/s)2

x(l J/(kg m2 s~2)/(1.602 x 10~13 J/MeV) = 0.5110 MeV

When dealing with masses on the atomic scale, it is often easier to use masses measured in atomic mass units (u) and the conversion factor of 931.49 MeV/u. With this important conversion factor we obtain

*E = mec2 = (5.486 x 10~*4 u) x (931.49 MeV/u) = 0.5110 MeV.

**Reduction to Classical Mechanics**

*For slowly moving particles, that is, v « c, Eq. (2.10) yields the usual classical*
result. Since,

(2-12) 1

0 _ V2/C2 ~ ^ " '" ' ' 2c2 8c4 the kinetic energy of a slowly moving particle is

*T = m*0c2

*v*
*Zc2'*

*- 1 | ~ ~m0v2.*
*£*

(2.13)

**Relation Between Kinetic Energy and Momentum**

*Both classically and relativistically the momentum p of a particle is given by,*

*p = mv. (2-14)*

*In classical physics, a particle's kinetic energy T is given by,*

*mv2 p2*
*~~ 2 " 2m'*

which yields

*p = \llmT. (2.15)*

For relativistic particles, the relationship between momentum and kinetic energy is not as simple. Square Eq. (2.5) to obtain

*-^-=mo'*

or, upon rearrangement.

*p2 = (mv}2* - (me)2* - (m0c)2 = ^[(mc*2)2 - (m0c2)2].

Then combine this result with Eq. (2.10) to obtain

*P2 = ^ [(T + m0c2)2 - (m*0c2)2*] = ~ [T2 + 2Tm0c2} . (2.16)*

Thus for relativistic particles

1

*P= -* (2.17;

**Particles**

For most moving objects encountered in engineering analyses, the classical
expres-sion for kinetic energy can be used. Only if an object has a speed near c must we
use relativistic expressions. From Eq. (2.10) one can readily calculate the kinetic
energies required for a particle to have a given relativistic mass change. Listed in
Table 2.1 for several important atomic particles are the rest mass energies and the
kinetic energies required for a 0.1% mass change. At this threshold for relativistic
*effects, the particle's speed v = 0.045c (see Problem 2).*

**2.2 Radiation as Waves and Particles**

**Table 2.1. Rest mass energies and kinetic energies for a 0.1%**

relativistic mass increase for four particles.

Particle electron proton neutron a-particle

rest mass
*energy m0c2*

0.511 MeV 938 MeV 940 MeV 3751 MeV

kinetic energy for a 0.1% increase in mass 511 eV

938 keV 940 keV 3.8 MeV

~ 0.5 keV ~ 1 MeV ~ 1 MeV ~ 4 MeV

However, near the beginning of the twentieth century, several experiments
in-volving light and X rays were performed that indicated that radiation also possessed
particle-like properties. Today we understand through quantum theory that matter
(e.g., electrons) and radiation (e.g., x rays) both have wave-like and particle
*proper-ties. This dichotomy, known as the wave-particle duality principle, is a cornerstone*
of modern physics. For some phenomena, a wave description works best; for others,
a particle model is appropriate. In this section, three pioneering experiments are
reviewed that helped to establish the wave-particle nature of matter.

**2.2.1 The Photoelectric Effect**

In 1887, Hertz discovered that, when metal surfaces were irradiated with light,
"electricity" was emitted. J.J. Thomson in 1898 showed that these emissions were
*electrons (thus the term photoelectrons). According to a classical (wave theory)*
description of light, the light energy was absorbed by the metal surface, and when
sufficient energy was absorbed to free a bound electron, a photoelectron would
"boil" off the surface. If light were truly a wave, we would expect the following
observations:

• Photoelectrons should be produced by light of all frequencies.

• At low intensities a time lag would be expected between the start of irradiation and the emission of a photoelectron since it takes time for the surface to absorb sufficient energy to eject an electron.

• As the light intensity (i.e., wave amplitude) increases, more energy is absorbed per unit time and, hence, the photoelectron emission rate should increase.

• The kinetic energy of the photoelectron should increase with the light intensity since more energy is absorbed by the surface.

However, experimental results differed dramatically with these results. It was ob-served:

• For each metal there is a minimum light frequency below which no photoelec-trons are emitted no matter how high the intensity.

*• The intensity of the light affects only the emission rate of photoelectrons.*
• The kinetic energy of the photoelectron depends only on the frequency of the

light and riot on its intensity. The higher the frequency, the more energetic is the photoelectron.

In 1905 Einstein introduced a new light model which explained all these
observations.2* Einstein assumed that light energy consists of photons or "quanta of*
*energy," each with an energy E = hv^ where h is Planck's constant (6.62 x 10~*34
*J s) and v is the light frequency. He further assumed that the energy associated*
with each photon interacts as a whole, i.e., either all the energy is absorbed by
an atom or none is. With this "particle" model, the maximum kinetic energy of a
photoelectron would be

*E = hv-A, (2.18)*

*where A is the amount of energy (the so-called work function) required to free an*
*electron from the metal. Thus if hv < A, no photoelectrons are produced. *
Increas-ing the light intensity only increases
I \ the number of photons hitting the
.Xcurrent \ collector metal surface per unit time and, thus.
*Wmeter \/ the rate of photoelectron emission.*

X' photoelectron . 1 1 -V 0 Although Einstein was able to ex-/V^ \ x/ / Xngh t plain qualitatively the observed char-I •^^•(B acteristics of the photoelectric effect, it was several years later before

**Ein-Figure 2.3. A schematic illustration of stein's prediction of the maximum **

en-the experimental arrangement used to ver- ergy of a photoelectron, Eq. (2.18),
ify photoelectric effect. ^ verified quajltitatively using the
experiment shown schematically in Fig. 2.3. Photoelectrons emitted from freshly
polished metallic surfaces were absorbed by a collector causing a current to flow
between the collector and the irradiated metallic surface. As an increasing negative
voltage was applied to the collector, fewer photoelectrons had sufficient kinetic
en-ergy to overcome this potential difference and the photoelectric current decreased to
*zero at a critical voltage V0* at which no photoelectrons had sufficient kinetic energy

to overcome the opposing potential. At this voltage, the maximum kinetic energy
*of a photoelectron, Eq. (2.18), equals the potential energy V0e the photoelectron*

must overcome, i.e.,

*V0e = hv - A,*
or

*V0 = HO. - I, (2.19)*

*e e*

*where e is the electron charge. In 1912 Hughes showed that, for a given metallic*
*surface, V0 was a linear function of the light frequency v. In 1916 Milliken. who*

*had previously measured the electron charge e. verified that plots of V0 versus v*

*for different metallic surface had a slope of h/e, from which h could be evaluated.*

*Milliken's value of h was in excellent agreement with the value determined from*
measurements of black-body radiation, in whose theoretical description Planck first
*introduced the constant h.*

The prediction by Einstein and its subsequent experimental verification clearly demonstrated the quantum nature of radiant energy. Although the wave theory of light clearly explained diffraction and interference phenomena, scientists were forced to accept that the energy of electromagnetic radiation could somehow come together into individual quanta, which could enter an individual atom and be transferred to a single electron. This quantization occurs no matter how weak the radiant energy.

**Example 2.3: What is the maximum wavelength of light required to liberate**

photoelectrons from a metallic surface with a work function of 2.35 eV (the energy able to free a valence electron)? At the minimum frequency, a photon has just enough energy to free an electron. From Eq. (2.18) the minimum frequency to yield a photon with zero kinetic energy (E=0) is

z,min* = A/h = 2.35 eV/4.136 x 1(T*15 eV/s = 5.68 x 1014 s"1.
The wavelength of such radiation is

Amax = c/i/min = 2.998 x 108 m s~V5.68 x 1014 s~x = 5.28 x 1(T7 m. This corresponds to light with a wavelength of 528 nm which is in the green portion of the visible electromagnetic spectrum.

**2.2.2 Compton Scattering**

Other experimental observations showed that light, besides having quantized
en-ergy characteristics, must have another particle-like property, namely momentum.
According to the wave model of electromagnetic radiation, radiation should be
scat-tered from an electron with no change in wavelength. However, in 1922 Compton
observed that x rays scattered from electrons had a decrease in the wavelength
*AA = A' — A proportional to (1 — cos9s) where 9S* was the scattering angle (see

Fig. 2.4). To explain this observation, it was necessary to treat x rays as particles
*with a linear momentum p = h/X and energy E — hv = pc.*

In an x-ray scattering interaction, the energy and momentum before scatter-ing must equal the energy and momentum after scatterscatter-ing. Conservation of linear momentum requires the initial momentum of the incident photon (the electron is assumed to be initially at rest) to equal the vector sum of the momenta of the scat-tered photon and the recoil electron. This requires the momentum vector triangle of Fig. 2.5 to be closed, i.e.,

**PA= PA, + Pe** (2-20)

or from the law of cosines

**scattered**
**photon**

**incident**
**photon •>**

**recoil**
**electron**

**Figure 2.4. A photon with wavelength A**

is scattered by an electron. After scattering,
the photon has a longer wavelength A' and
*the electron recoils with an energy T(- and*

**momentum p****e****.**

**Figure 2.5. Conservation of momentum **

*re-quires the initial momentum of the photon p\*
equal the vector sum of the momenta of the
scattered photon and recoil electron.

The conservation of energy requires

*p c + mec2 — p, c + me2*

*X A* (2.22)

*where me* is the rest-mass of the electron before the collision when it has negligible

*kinetic energy, and m is its relativistic mass after scattering the photon. This result,*
*combined with Eq. (2.16) (in which me* = m0), can be rewritten as

mec (2.23)

*Substitute for pe* from Eq. (2.21) into Eq. (2.23), square the result, and simplify to

obtain

(2.24)

*Then since A = h/p. this result gives the decrease in the scattered wavelength as*

*h*
A ' - A =

*mec*

•(1 (2.25)

*where h/(mec) = 2.431 x 10 6/j,m. Thus, Compton was able to predict the *

wave-length change of scattered x rays by using a particle model for the x rays, a predic-tion which could not be obtained with a wave model.

This result can be expressed in terms of the incident and scattered photon
*energies, E and'E", respectively. With the photon relations A = cji> and E = hv^*
Eq. (2.25) gives

(2.26)

### 1

*E'*

1

*E* ~* l ( 1 -*2 V1

*mec*

**Example 2.4: What is the recoil kinetic energy of the electron that scatters**

a 3-MeV photon by 45 degrees? In such a Compton scattering event, we first
*calculate the energy of the scattered photon. From Eq. (2.26) the energy E' of*
the scattered photon is found to be

### =

### 1

### -

### 10 MeV

*-Because energy is conserved, the kinetic energy Te* of the recoil election must
*equal the energy lost by the photon, i.e., Te = E - E' = 3 - 1.10 = 1.90 MeV.*

**2.2.3 Electromagnetic Radiation: Wave-Particle Duality**

Electromagnetic radiation assumes many forms encompassing radio waves,
mi-crowaves, visible light, X rays, and gamma rays. Many properties are described
*by a wave model in which the wave travels at the speed of light c and has a *
*wave-length A and frequency v, which are related by the wave speed formula*

*c = \v. (2.27)*

The wave properties account for many phenomena involving light such as diffraction and interference effects.

However, as Einstein and Compton showed, electromagnetic radiation also has
particle-like properties, namely, the light energy being carried by discrete quanta
*or packets of energy called photons. Each photon has an energy E = hv and*
interacts with matter (atoms) in particle-like interactions (e.g., in the photoelectric
interactions described above).

Thus, light has both wave-like and particle-like properties. The properties or model we use depend on the wavelength of the radiation being considered. For example, if the wavelength of the electromagnetic radiation is much longer than the dimensions of atoms ~ 10~10 m (e.g., visible light, infrared radiation, radar and radio waves), the wave model is usually most useful. However, for short wavelength electromagnetic radiation < 10~12 m (e.g., ultraviolet, x rays, gamma rays), the corpuscular or photon model is usually used. This is the model we will use in our study of nuclear science and technology, which deals primarily with penetrating short-wavelength electromagnetic radiation.

**Photon Properties**