fundamentals of nuclear science & engineering

Although nuclearand atomic radiation had been used during the first half of the twentieth century,mainly for medical purposes, nuclear technology as a distinct engineering disciplinebega

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

D E K K E R

ISBN: 0-8247-0834-2

This book is printed on acid-free paper.

Headquarters

Marcel Dekker, Inc.

270 Madison Avenue, New York, NY 10016 tel: 212-696-9000; fax: 212-685-4540

Eastern Hemisphere Distribution

Marcel Dekker AG Hutgasse 4, Postfach 812, CH-4001 Basel, Switzerland tel: 41-61-261-8482; fax: 41-61-261-8896

World Wide Web

http://www.dekker.com The publisher offers discounts on this book when ordered in bulk quantities For more information, write to Special Sales/Professional Marketing at the headquarters address above.

Copyright © 2002 by Marcel Dekker, Inc All Rights Reserved.

Neither this book nor any part may be reproduced or transmitted in any form or by any means, tronic or mechanical, including photocopying, microfilming, and recording, or by any information storage and retrieval system, without permission in writing from the publisher.

elec-Current printing (last digit):

1 0 9 8 7 6 5 4 3 2 1

PRINTED IN THE UNITED STATES OF AMERICA

Nuclear engineering and the technology developed by this discipline began andreached an amazing level of maturity within the past 60 years Although nuclearand atomic radiation had been used during the first half of the twentieth century,mainly for medical purposes, nuclear technology as a distinct engineering disciplinebegan after World War II with the first efforts at harnessing nuclear energy forelectrical power production and propulsion of ships During the second half of thetwentieth century, many innovative uses of nuclear radiation were introduced in thephysical 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 forelectrical 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 yearsthan 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 clear technology to its present level of maturity are too many to credit here Onlytheir important results are presented in this book For their efforts, which havegreatly benefited all nuclear engineers, not least ourselves, we extend our deepestappreciation As university professors we have enjoyed learning of the work of ourcolleagues We hope our present and future students also will appreciate these pastaccomplishments and will build on them to achieve even more useful applications

nu-of nuclear technology We believe the uses nu-of nuclear science and engineering willcontinue to play an important role in the betterment of human life

At a more practical level, this book evolved from an effort at introducing anuclear engineering option into a much larger mechanical engineering program atKansas State University This book was designed to serve both as an introduction

to the students in the nuclear engineering option and as a text for other engineeringstudents who want to obtain an overview of nuclear science and engineering We

believe that all modern engineering students need to understand the basic aspects

of nuclear science engineering such as radioactivity and radiation doses and theirhazards

Many people have contributed to this book First and foremost we thank ourcolleagues Dean Eckhoff and Fred Merklin, whose initial collection of notes for anintroductory course in nuclear engineering motivated our present book intendedfor a larger purpose and audience We thank Professor Gale Simons, who helpedprepare an early draft of the chapter on radiation detection Finally, many revisionshave been made in response to comments and suggestions made by our students onwhom we have experimented with earlier versions of the manuscript Finally, thecamera 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

J Kenneth Shultis and Richard E Faw

1 Fundamental Concepts

1.1 Modern Units1.1.1 Special Nuclear Units1.1.2 Physical Constants1.2 The Atom

1.2.1 Atomic and Nuclear Nomenclature1.2.2 Atomic and Molecular Weights1.2.3 Avogadro's Number

1.2.4 Mass of an Atom1.2.5 Atomic Number Density1.2.6 Size of an Atom

1.2.7 Atomic and Isotopic Abundances1.2.8 Nuclear Dimensions

1.3 Chart of the Nuclides1.3.1 Other Sources of Atomic/Nuclear Information

2 Modern Physics Concepts

2.1 The Special Theory of Relativity2.1.1 Principle of Relativity2.1.2 Results of the Special Theory of Relativity2.2 Radiation as Waves and Particles

2.2.1 The Photoelectric Effect2.2.2 Compton Scattering2.2.3 Electromagnetic Radiation: Wave-Particle Duality2.2.4 Electron Scattering

2.2.5 Wave-Particle Duality2.3 Quantum Mechanics

2.3.1 Schrodinger's Wave Equation2.3.2 The Wave Function

2.3.3 The Uncertainty Principle2.3.4 Success of Quantum Mechanics2.4 Addendum 1: Derivation of Some Special Relativity Results2.4.1 Time Dilation

2.4.2 Length Contraction2.4.3 Mass Increase2.5 Addendum 2: Solutions to Schrodinger's Wave Equation2.5.1 The Particle in a Box

2.5.2 The Hydrogen Atom2.5.3 Energy Levels for Multielectron Atoms

Atomic/Nuclear Models

3.1 Development of the Modern Atom Model3.1.1 Discovery of Radioactivity3.1.2 Thomson's Atomic Model: The Plum Pudding Model3.1.3 The Rutherford Atomic Model

3.1.4 The Bohr Atomic Model3.1.5 Extension of the Bohr Theory: Elliptic Orbits3.1.6 The Quantum Mechanical Model of the Atom3.2 Models of the Nucleus

3.2.1 Fundamental Properties of the Nucleus3.2.2 The Proton-Electron Model

3.2.3 The Proton-Neutron Model3.2.4 Stability of Nuclei

3.2.5 The Liquid Drop Model of the Nucleus3.2.6 The Nuclear Shell Model

3.2.7 Other Nuclear Models

Nuclear Energetics

4.1 Binding Energy4.1.1 Nuclear and Atomic Masses4.1.2 Binding Energy of the Nucleus4.1.3 Average Nuclear Binding Energies4.2 Niicleon Separation Energy

4.3 Nuclear Reactions4.4 Examples of Binary Nuclear Reactions4.4.1 Multiple Reaction Outcomes4.5 Q-Value for a Reaction

4.5.1 Binary Reactions4.5.2 Radioactive Decay Reactions4.6 Conservation of Charge and the Calculation of Q-Values4.6.1 Special Case for Changes in the Proton Number4.7 Q-Value for Reactions Producing Excited Nulcei

Radioactivity

5.1 Overview5.2 Types of Radioactive Decay5.3 Energetics of Radioactive Decay5.3.1 Gamma Decay

5.3.2 Alpha-Particle Decay5.3.3 Beta-Particle Decay

5.3.4 Positron Decay5.3.5 Electron Capture5.3.6 Neutron Decay5.3.7 Proton Decay5.3.8 Internal Conversion5.3.9 Examples of Energy-Level Diagrams5.4 Characteristics of Radioactive Decay

5.4.1 The Decay Constant5.4.2 Exponential Decay5.4.3 The Half-Life5.4.4 Decay Probability for a Finite Time Interval5.4.5 Mean Lifetime

5.4.6 Activity5.4.7 Half-Life Measurement5.4.8 Decay by Competing Processes5.5 Decay Dynamics

5.5.1 Decay with Production5.5.2 Three Component Decay Chains5.5.3 General Decay Chain

5.6 Naturally Occurring Radionuclides5.6.1 Cosmogenic Radionuclides5.6.2 Singly Occurring Primordial Radionuclides5.6.3 Decay Series of Primordial Origin

5.6.4 Secular Equilibrium5.7 Radiodating

5.7.1 Measuring the Decay of a Parent5.7.2 Measuring the Buildup of a Stable Daughter

6 Binary Nuclear Reactions6.1 Types of Binary Reactions6.1.1 The Compound Nucleus6.2 Kinematics of Binary Two-Product Nuclear Reactions6.2.1 Energy/Mass Conservation

6.2.2 Conservation of Energy and Linear Momentum6.3 Reaction Threshold Energy

6.3.1 Kinematic Threshold6.3.2 Coulomb Barrier Threshold6.3.3 Overall Threshold Energy6.4 Applications of Binary Kinematics6.4.1 A Neutron Detection Reaction6.4.2 A Neutron Production Reaction6.4.3 Heavy Particle Scattering from an Electron6.5 Reactions Involving Neutrons

6.5.1 Neutron Scattering6.5.2 Neutron Capture Reactions6.5.3 Fission Reactions

6.6 Characteristics of the Fission Reaction

6.6.1 Fission Products6.6.2 Neutron Emission in Fission6.6.3 Energy Released in Fission6.7 Fusion Reactions

6.7.1 Thermonuclear Fusion6.7.2 Energy Production in Stars6.7.3 Nucleogenesis

7 Radiation Interactions with Matter

7.1 Attenuation of Neutral Particle Beams7.1.1 The Linear Interaction Coefficient7.1.2 Attenuation of Uncollided Radiation7.1.3 Average Travel Distance Before an Interaction7.1.4 Half-Thickness

7.1.5 Scattered Radiation7.1.6 Microscopic Cross Sections7.2 Calculation of Radiation Interaction Rates7.2.1 Flux Density

7.2.2 Reaction-Rate Density7.2.3 Generalization to Energy- and Time-Dependent Situations7.2.4 Radiation Fluence

7.2.5 Uncollided Flux Density from an Isotropic Point Source7.3 Photon Interactions

7.3.1 Photoelectric Effect7.3.2 Compton Scattering7.3.3 Pair Production7.3.4 Photon Attenuation Coefficients7.4 Neutron Interactions

7.4.1 Classification of Types of Interactions7.4.2 Fission Cross Sections

7.5 Attenuation of Charged Particles7.5.1 Interaction Mechanisms7.5.2 Particle Range

7.5.3 Stopping Power7.5.4 Estimating Charged-Particle Ranges

8 Detection and Measurement of Radiation

8.1 Gas-Filled Radiation Detectors8.1.1 lonization Chambers8.1.2 Proportional Counters8.1.3 Geiger-Mueller Counters8.2 Scintillation Detectors

8.3 Semiconductor lonizing-Radiation Detectors8.4 Personal Dosimeters

8.4.1 The Pocket Ion Chamber8.4.2 The Film Badge

8.4.3 The Thermoluminescent Dosimeter

8.5 Measurement Theory8.5.1 Types of Measurement Uncertainties8.5.2 Uncertainty Assignment Based Upon Counting Statistics8.5.3 Dead Time

8.5.4 Energy Resolution

9 Radiation Doses and Hazard Assessment

9.1 Historical Roots9.2 Dosimetric Quantities9.2.1 Energy Imparted to the Medium9.2.2 Absorbed Dose

9.2.3 Kerma9.2.4 Calculating Kerma and Absorbed Doses9.2.5 Exposure

9.2.6 Relative Biological Effectiveness9.2.7 Dose Equivalent

9.2.8 Quality Factor9.2.9 Effective Dose Equivalent9.2.10 Effective Dose

9.3 Natural Exposures for Humans9.4 Health Effects from Large Acute Doses9.4.1 Effects on Individual Cells9.4.2 Deterministic Effects in Organs and Tissues9.4.3 Potentially Lethal Exposure to Low-LET Radiation9.5 Hereditary Effects

9.5.1 Classification of Genetic Effects9.5.2 Summary of Risk Estimates9.5.3 Estimating Gonad Doses and Genetic Risks9.6 Cancer Risks from Radiation Exposures

9.6.1 Dose-Response Models for Cancer9.6.2 Average Cancer Risks for Exposed Populations9.7 Radon and Lung Cancer Risks

9.7.1 Radon Activity Concentrations9.7.2 Lung Cancer Risks

9.8 Radiation Protection Standards9.8.1 Risk-Related Dose Limits9.8.2 The 1987 NCRP Exposure Limits

10 Principles of Nuclear Reactors

10.1 Neutron Moderation10.2 Thermal-Neutron Properties of Fuels10.3 The Neutron Life Cycle in a Thermal Reactor10.3.1 Quantification of the Neutron Cycle10.3.2 Effective Multiplication Factor10.4 Homogeneous and Heterogeneous Cores10.5 Reflectors

10.6 Reactor Kinetics

10.6.1 A Simple Reactor Kinetics Model10.6.2 Delayed Neutrons

10.6.3 Reactivity and Delta-k10.6.4 Revised Simplified Reactor Kinetics Models10.6.5 Power Transients Following a Reactivity Insertion10.7 Reactivity Feedback

10.7.1 Feedback Caused by Isotopic Changes10.7.2 Feedback Caused by Temperature Changes10.8 Fission Product Poisons

10.8.1 Xenon Poisoning10.8.2 Samarium Poisoning10.9 Addendum 1: The Diffusion Equation10.9.1 An Example Fixed-Source Problem10.9.2 An Example Criticality Problem10.9.3 More Detailed Neutron-Field Descriptions10.10 Addendum 2: Kinetic Model with Delayed Neutrons10.11 Addendum 3: Solution for a Step Reactivity Insertion

11 Nuclear Power

11.1 Nuclear Electric Power11.1.1 Electricity from Thermal Energy11.1.2 Conversion Efficiency

11.1.3 Some Typical Power Reactors11.1.4 Coolant Limitations

11.2 Pressurized Water Reactors11.2.1 The Steam Cycle of a PWR11.2.2 Major Components of a PWR11.3 Boiling Water Reactors

11.3.1 The Steam Cycle of a BWR11.3.2 Major Components of a BWR11.4 New Designs for Central-Station Power11.4.1 Certified Evolutionary Designs11.4.2 Certified Passive Design11.4.3 Other Evolutionary LWR Designs11.4.4 Gas Reactor Technology

11.5 The Nuclear Fuel Cycle11.5.1 Uranium Requirements and Availability11.5.2 Enrichment Techniques

11.5.3 Radioactive Waste11.5.4 Spent Fuel

11.6 Nuclear Propulsion11.6.1 Naval Applications11.6.2 Other Marine Applications11.6.3 Nuclear Propulsion in Space

12 Other Methods for Converting Nuclear Energy to Electricity

12.1 Thermoelectric Generators12.1.1 Radionuclide Thermoelectric Generators

12.2 Thermionic Electrical Generators12.2.1 Conversion Efficiency12.2.2 In-Pile Thermionic Generator12.3 AMTEC Conversion

12.4 Stirling Converters12.5 Direct Conversion of Nuclear Radiation12.5.1 Types of Nuclear Radiation Conversion Devices12.5.2 Betavoltaic Batteries

12.6 Radioisotopes for Thermal Power Sources12.7 Space Reactors

12.7.1 The U.S Space Reactor Program12.7.2 The Russian Space Reactor Program

13 Nuclear Technology in Industry and Research

13.1 Production of Radioisotopes13.2 Industrial and Research Uses of Radioisotopes and Radiation13.3 Tracer Applications

13.3.1 Leak Detection13.3.2 Pipeline Interfaces13.3.3 Flow Patterns13.3.4 Flow Rate Measurements13.3.5 Labeled Reagents

13.3.6 Tracer Dilution13.3.7 Wear Analyses13.3.8 Mixing Times13.3.9 Residence Times13.3.10 Frequency Response13.3.11 Surface Temperature Measurements13.3.12 Radiodating

13.4 Materials Affect Radiation13.4.1 Radiography13.4.2 Thickness Gauging13.4.3 Density Gauges13.4.4 Level Gauges13.4.5 Radiation Absorptiometry13.4.6 Oil-Well Logging

13.4.7 Neutron Activation Analysis13.4.8 Neutron Capture-Gamma Ray Analysis13.4.9 Molecular Structure Determination13.4.10 Smoke Detectors

13.5 Radiation Affects Materials13.5.1 Food Preservation13.5.2 Sterilization13.5.3 Insect Control13.5.4 Polymer Modification13.5.5 Biological Mutation Studies13.5.6 Chemonuclear Processing

14 Medical Applications of Nuclear Technology

14.1 Diagnostic Imaging14.1.1 X-Ray Projection Imaging14.1.2 Fluoroscopy

14.1.3 Mammography14.1.4 Bone Densitometry14.1.5 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 Radiotracers14.4 Radioimmunoscintigraphy14.5 Radiation Therapy

14.5.1 Early Applications14.5.2 Teletherapy14.5.3 Radionuclide Therapy14.5.4 Clinical Brachytherapy14.5.5 Boron Neutron Capture Therapy

Appendic A: Fundamental Atomic Data Appendix B: Atomic Mass Table

Appendix C: Cross Sections and Related Data Appendix D: Decay Characteristics of Selected Radionuclides

Chapter 1

Fundamental Concepts

The last half of the twentieth century was a time in which tremendous advances inscience 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 neering, personal computers, medical diagnostics and therapy, bioengineering, andmaterial sciences are just a few areas that were greatly affected

engi-Nuclear science and engineering is another technology that has been transformed

in less than fifty years from laboratory research into practical applications 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

encoun-of electrical power in many countries Nuclear technology is widely used in medicalimaging, diagnostics and therapy Agriculture and many other industries make wideuse of radioisotopes and other radiation sources Finally, nuclear applications arefound 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 ing how various types of radiation interact with matter and how these interactionsaffect matter In this book, we will describe sources of radiation, radiation inter-actions, and the results of such interactions As the word "nuclear" suggests, wewill address phenomena at a microscopic level, involving individual atoms and theirconstituent nuclei and electrons The radiation we are concerned with is generallyvery penetrating and arises from physical processes at the atomic level

quantify-However, before we begin our exploration of the atomic world, it is necessary tointroduce some basic fundamental atomic concepts, properties, nomenclature andunits used to quantify the phenomena we will encounter Such is the purpose ofthis introductory chapter

1.1 Modern Units

With only a few exceptions, units used in nuclear science and engineering are thosedefined by the SI system of metric units This system is known as the "InternationalSystem of Units" with the abbreviation SI taken from the French "Le SystemeInternational d'Unites." In this system, there are four categories of units: (1) baseunits of which there are seven, (2) derived units which are combinations of the baseunits, (3) supplementary units, and (4) temporary units which are in widespread

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

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 WcVftWb T Hz Bq Pa

Symbol racl sr

Symbol

A

ha bar atm b Ci R Gy Sv

N m-' 2

in s" 1

kg m~^

o m

3.7 x 10 H) Bq 2.58 x 10~ 4 C kg" 1

1 J kg- 1

Source: NBS Special Publication 330, National Bureau of Standards, Washington, DC, 1977.

use for special applications These units are shown in Table 1.1 To accommodatevery small and large quantities, the SI units and their symbols are scaled by usingthe 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 convertsome 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 SIgrammar is presented in Table 1.4

Table 1.2 SI prefixes Table 1.3 Conversion factors.

Factor

10 24 1021 1018 1015 1012

io-9 10~12

io-15

10 -18io-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

ft mile (int'l)

in 2

ft 2 acre square mile (int'l) hectare

oz (U.S liquid)

in 3 gallon (U.S.)

ft 3

oz (avdp.) Ib ton (short) kgf

lb f ton lbf/in 2 (psi)

lb f /ft 2 atm (standard)

in H 2 O (@ 4 °C)

in Hg (© 0 °C)

mm Hg (@ 0 °C) bar

eV cal Btu kWh MWd

SI equivalent 2.54 x 1CT2 m a 3.048 x 10~ 1 m a 1.609344 X 10 3 m a 6.4516 x 10~ 4 m 2 a 9.290304 X 10~ 2 m 2 a 4.046873 X 10 3 m 2 2.589988 X 10 6 m 2

1 x 10 4 m 2 2.957353 X 10~ 5 m 3 1.638706 X 10~ 5 m 3 3.785412 X 10~ 3 m 3 2.831685 x 10~ 2 m 3 2.834952 x 10~ 2 kg 4.535924 X lO^ 1 kg 9.071 847 x 10 2 kg 9.806650 N a 4.448222 N 8.896444 X 10 3 N 6.894757 x 10 3 Pa 4.788026 x 10 1 Pa 1.013250 x 10 5 Pa a 2.49082 x 10 2 Pa 3.38639 x 10 3 Pa 1.33322 x 10 2 Pa

1 x 10 5 Pa a 1.60219 x 10~ 19 J 4.184 J a

1.054350 X 10 3 J 3.6 x 10 6 J a 8.64 x 10 10 J a

"Exact converson factor.

Source: Standards for Metric Practice, ANSI/ASTM

E380-76, American National Standards Institute, New York, 1976.

Table 1.4 Summary of SI grammar.

Use 58 rn, not 58m

A symbol is never pluralized Thus 8 N, not 8 Ns or 8 N s Sometimes a raised dot is used when combining units such as N-m 2 -s; however, a single space between unit symbols is preferred as in

N m 2 s.

For simple unit combinations use g/cm 3 or g cm~ 3 However, for more complex expressions, N m~ 2 s"" 1 is much clearer than N/m 2 /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, press the ending vowel on the prefix Thus megohm and kilohm, not megaohm and kiloohm.

su-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 energiesand masses are extremely small in SI units, and special units are almost alwaysused Two such units are of particular importance

The Electron Volt

The energy released or absorbed in a chemical reaction (arising from changes inelectron 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 m e

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(T19 C)(l J/C) = 1.60217646 x 10~19 J = 1 eV Thus

1 e V = 1.602 176 46 x 10~19 J

If the electron (mass m e ) 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 theelectron, i.e.,

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

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

everyday experience but slow compared to the speed of light (c ~ 3 x 108 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 12C Equivalently, the mass of N a 12C atoms (Avogadro's number = 1 mole) is0.012 kg Thus, 1 amu equals (1/12)(0.012 kg/JVa) = 1.6605387 x 10~27 kg

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-chargedprotons in the nucleus In later chapters, more detailed models of the atom areintroduced

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 sameinstrument can also move individual atoms An example is shown in Fig 1.1 Inthis figure, iron atoms (the dark circular dots) on a copper surface are shown beingmoved to form a ring which causes electrons inside the ring and on the coppersurface to form standing waves This and other pictures of atoms can be found onthe web at http://www.ibm.com/vis/stm/gallery.html

Although neutrons and protons are often considered as "fundamental" particles,

we now know that they are composed of other smaller particles called quarks held

Table 1.5 Values of some important physical constants as internationally

recom-mended in 1998.

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 10 8 m s~ l

1.60217646 x 10' 19 C 1.6605387 x 10~ 27 kg (931.494013 MeV/c 2 ) 9.1093819 x 10~ 31 kg (0.51099890 MeV/c 2 ) (5.48579911 x 10~ 4 u) 1.6726216 x 10~ 27 kg (938.27200 MeV/c 2 ) (1.0072764669 u) 1.6749272 x 10~ 27 kg (939.56533 MeV/c 2 ) (1.0086649158 u) 6.6260688 x 10~ 34 J s 4.1356673 x 10~ 15 eV s 6.0221420 x 10 23 mol" 1

1.3806503 x 10~ 23 J K ~ ]

(8.617342 x 10~ 5 eV K" 1 ) 8.314472 J mor 1 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 tron, neutron and proton as fundamental indivisible particles, since the compositenature of nucleons becomes apparent only under extreme conditions, such as thoseencountered during the first minute after the creation of the universe (the "bigbang") or in high-energy particle accelerators We will not deal with such giganticenergies Rather, the energy of radiation we consider is sufficient only to rearrange

elec-or remove the electrons in an atom elec-or the neutrons and protons in a nucleus

1.2.1 Atomic and Nuclear Nomenclature

The identity of an atom is uniquely specified by the number of neutrons N and protons Z in its nucleus For an electrically neutral atom, the number of electrons equals the number of protons Z, which is called the atomic number All atoms of the same element have the same atomic number Thus, all oxygen atoms have 8

protons in the nucleus while all uranium atoms have 92 protons

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 inthe 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 2g2U 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

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

U-235 and U-238 was often employed However, with modern word processing, this form should

no longer be used.

Because isotopes of the same element have the same number and arrangement

of electrons around the nucleus, the chemical properties of such isotopes are nearlyidentical 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 whileheavy 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 basicnuclear 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 Ar 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 mdenotes the longest-lived excited state (i.e., a state in which the nucleons inthe 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 12C 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 molecularweights 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 massunit 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 detailedlisting of the atomic masses of the known nuclides is given in Appendix B Fromthis 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

atoms in the element that are that isotope Isotopic abundances are usually pressed in atom percent and are given in Appendix Table A.4 For a specified

ex-element, the elemental atomic weight is the weighted average of the atomic weights

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

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

Ele-Example 1.1: What is the atomic weight of boron? From Table A.4 we findthat naturally occurring boron consists of two stable isotopes 10B and nB withisotopic 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

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 oftencalled 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, forexample, 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 ofone atom of 12C, the mass of a sample, in grams, numerically equal to the atomicweight of an atomic species must contain as many atoms of the species as thereare 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 ormolecules as there are atoms in one mole of 12C, namely N a atoms or molecules

That one mole of any substance contains N a entities is known as Avogadro's law

and is the fundamental principle that relates the microscopic world to the everydaymacroscopic world

Example 1.2: How many atoms of 10 B 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 N a 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 10 B in elemental boron is found to

be 19.9% The number Nw of 10 B atoms in the sample is, therefore, A/io = (0.199)(2.785 x 10 23 ) = 5.542 x 10 22 atoms.

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 knownatom As will soon become apparent, atomic masses are central to quantifying theenergetics of various nuclear reactions

Example 1.3: Estimate the mass on an atom of 238 U From Eq (1.3) we find

238 (g/mol) 6.022 x 10 23 atoms/mol = 3.952 x 10 g/atom.

From Appendix B, the mass of 238 U 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 238 U is, therefore,

, 238in _ 238.050782 (g/mol) M( 238 U) = I, w ' = 3.952925 x IQ~" g/atom.

v ; 6.022142 x 10 23 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 asubstance Again, Avogadro's number is the key to finding the atom density For a

homogeneous substance of a single species and with mass density p g/cm3, 1 cm3

contains p/A moles of the substance and pN a /A atoms The atom density N is

thus

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 moleculardensity by the number of the same atoms per molecule This is illustrated in thefollowing 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

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

Pi N a w lP N a

1 = ~A~ = ~A~' S\i S^-i ( }

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

/- -,

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 standardtemperature and pressure are typically less by a factor of 1000

1.2.6 Size of an Atom

For a substance with an atom density of TV atoms/cm3, each atom has an associated

volume of V = I/A7" cm3 If this volume is considered a cube, the cube width is F1/3.For 238U, the cubical size of an atom is thus I/A7"1/3 = 2.7 x 10~8 cm Measurements

of the size of atoms reveals a diffuse electron cloud about the nucleus Althoughthere is no sharp edge to an atom, an effective radius can be defined such thatoutside 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 simplybecomes more dense Hydrogen, the lightest element, is also the smallest with aradius 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 eitherduring 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 naturallyoccurring elements was formed from debris of dead stars The abundances of theelements for our solar system is a consequence of the history of stellar formationand 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 relativeabundance 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, arespherical or very slightly ellipsoidal with a somewhat diffuse surface, In particular,

it is found that an effective spherical nuclear radius is

R = R 0 A l/3 , with R 0 ~ 1.25 x 1CT13 cm (1.7)The associated volume is

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

Since the atomic radius of about 2 x 10~8 cm is 105 times greater than thenuclear 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 (m n = 1837 me), the mass density of a nucleus is

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., radioactive) and are found in nature There are also 65 long-lived radioisotopesfound 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

non-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 bythe physical size of the chart Several versions of the chart are available on theinternet (see web addresses given in the next section and in Appendix A)

Perhaps, the most detailed Chart of the Nuclides is that provided by GeneralElectric 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 highlyrecommended 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 befound 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 siteslisted below contain many links to data sources, and you should explore these tobecome 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 itories in the world

These sites contain much information about nuclear technology and other relatedtopics Many are home pages for various governmental agencies and some are sitesoffering 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.htmlhttp://www.iaea.org/worldatom/

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 isreported 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 gramequivalent 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 tions you make

assump-12 Dry air at normal temperature and pressure has a mass density of 0.0012 g/cm3with a mass fraction of oxygen of 0.23 WThat is the atom density (atom/cm3)

of 180?

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 remainderbeing 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 ofsodium in the crystal?

16 A concrete with a density of 2.35 g/cm3 has a hydrogen content of 0.0085weight 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 moststable isotopes?

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

20 The earth has a radius of about 6.35 x 106 m and a mass of 5.98 x 1024 kg.What would be the radius if the earth had the same mass density as matter in

a nucleus?

Chapter 2

Modern Physics Concepts

During the first three decades of the twentieth century, our understanding of thephysical universe underwent tremendous changes The classical physics of Newtonand the other scientists of the eighteenth and nineteenth centuries was shown to beinadequate to describe completely our universe The results of this revolution inphysics 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 ativity, which extended Newtonian mechanics; (2) wave-particle duality, which saysthat both electromagnetic waves and atomic particles have dual wave and particleproperties; and (3) quantum mechanics, which revealed that the microscopic atomicworld is far different from our everyday macroscopic world The results and insightsprovided by these three advances in physics are fundamental to an understanding

rel-of nuclear science and technology This chapter is devoted to describing their basicideas and results

2.1 The Special Theory of Relativity

The classical laws of dynamics as developed by Newton were believed, for over 200years, to describe all motion in nature Students still spend considerable effortmastering the use of these laws of motion For example, Newton's second law, inthe 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)

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 necessarycorrection In his theory of special relativity,1 Einstein showed that Eq (2.1) is stillcorrect, 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

l ln 1915 Einstein published the general theory of relativity, in which he generalized his special

theory to include gravitation We will not need this extension in our study of the microscopic world.

varies with the body's speed as

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 108 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 m 0 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 — v 2 /c 2 = 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 — m 0

Here (v/c) 2 = (7.5 x 103 /2.998 x 10 8 ) 2 = 6.258 x 1CT 10 With this value of v 2 /c 2

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 v 2 /c 2

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 + -±— — L t + — - ^ - '-C + • • • ~ 1 + ne for |e| « 1.

Thus, with e = —v 2 /c 2 and n — —1/2 we find

so that the fractional mass increase is

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 spaceare the same amongst themselves, whether the space is at rest or moves uniformlyforward 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 (x 1 , 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 ( mertia l) coordinate systems?

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,

F x , acting on some mass m Then the second law in the S' moving system is

F x = md 2 x'/dt/ 2 Now transform this law to the stationary S system We find (for

In the 1870s Maxwell introduced his famous laws of electromagnetism Theselaws explained all observed behavior of electricity, magnetism, and light in a uni-form system However, when Eqs (2.3) are used to transform Maxwell's equations

to another inertial system, they assume a different form Thus from optical 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 20years of research only continued to reconfirm them Eventually, some scientistsbegan to wonder if the problem lay in the Galilean transformation of Eqs (2.3).Indeed, Lorentz observed in 1904 that if the transformation

experi-/ X VL i i A.I VXIC

is used, Maxwell's equations become the same in all inertial coordinate systems

Poincare, about this time, even conjectured that all laws of physics should

re-main unchanged under the peculiar looking Lorentz transformation The Lorentztransformation 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 indeedthe correct transformation relating all inertial coordinate systems He also showedhow Newton's laws of motion must be modified to make them invariant under thistransformation

Einstein based his analysis on two postulates:

• The laws of physics are expressed by equations that have the same form in allcoordinate 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 ofour 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.,

V ' ^ l - v2/ 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 (m 0 > 0) cannot even travel at the speed of light;

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

2 The length of a moving object in the direction of its motion appears smaller

to an observer at rest, namely

where L 0 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 t 0 for the same phenomenon to occur in the

stationary system The relation between t and t 0 is

'

4 Perhaps the most famous result from special relativity is the demonstration

of the equivalence of mass and energy by the well-known equation

Consider a particle with rest mass m 0 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 = m 0

Figure 2.2 A force F accelerates

a particle along a path of length s.

or finally

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

Equivalently, we can write this result as me2 = m 0 c 2 + 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 theequivalence 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 beingconverted into energy

E — me 2

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 = meC 2 = (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 MeVWhen dealing with masses on the atomic scale, it is often easier to use massesmeasured in atomic mass units (u) and the conversion factor of 931.49 MeV/u.With this important conversion factor we obtain

E = m e c 2 = (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 8c4the kinetic energy of a slowly moving particle is

Thus the relativistic kinetic energy reduces to the classical expression for kinetic

energy if v « c, a reassuring result since the validity of classical mechanics is well

established in the macroscopic world

Relation Between Kinetic Energy and Momentum

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

For relativistic particles, the relationship between momentum and kinetic energy

is not as simple Square Eq (2.5) to obtain

expres-effects, the particle's speed v = 0.045c (see Problem 2).

2.2 Radiation as Waves and Particles

For many phenomena, radiant energy can be considered as electromagnetic waves.Indeed Maxwell's equations, which describe very accurately interactions of longwave-length radiation, readily yield a wave equation for the electric and magneticfields of radiant energy Phenomena such as diffraction, interference, and otherrelated optical effects can be described only by a wave model for radiation

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

in-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 arereviewed 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 whensufficient 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 followingobservations:

• Photoelectrons should be produced by light of all frequencies

• At low intensities a time lag would be expected between the start of irradiationand the emission of a photoelectron since it takes time for the surface to absorbsufficient energy to eject an electron

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

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

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

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

photoelec-• There is no time lag between the start of irradiation and the emission ofphotoelectrons, no matter how low 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 thelight and riot on its intensity The higher the frequency, the more energetic isthe photoelectron

In 1905 Einstein introduced a new light model which explained all theseobservations.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 aphotoelectron 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 /V^ \ x/ / Xngh t plain qualitatively the observed char-

ex-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 theexperiment shown schematically in Fig 2.3 Photoelectrons emitted from freshlypolished metallic surfaces were absorbed by a collector causing a current to flowbetween the collector and the irradiated metallic surface As an increasing negativevoltage 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 V 0 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 V 0 e the photoelectron

must overcome, i.e.,

V 0 e = hv - A,

or

e e where e is the electron charge In 1912 Hughes showed that, for a given metallic surface, V 0 was a linear function of the light frequency v In 1916 Milliken who had previously measured the electron charge e verified that plots of V 0 versus v for different metallic surface had a slope of h/e, from which h could be evaluated.

2 It is an interesting historical fact that Einstein received the Nobel prize for his photoelectric research and not for his theory of relativity, which he produced in the same year.

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 clearlydemonstrated the quantum nature of radiant energy Although the wave theory oflight clearly explained diffraction and interference phenomena, scientists were forced

to accept that the energy of electromagnetic radiation could somehow come togetherinto 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 energyable to free a valence electron)? At the minimum frequency, a photon has justenough energy to free an electron From Eq (2.18) the minimum frequency toyield a photon with zero kinetic energy (E=0) is

z,min = A/h = 2.35 eV/4.136 x 1(T15 eV/s = 5.68 x 1014 s"1

The wavelength of such radiation isAmax = 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 greenportion of the visible electromagnetic spectrum

2.2.2 Compton Scattering

Other experimental observations showed that light, besides having quantized 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 Comptonobserved that x rays scattered from electrons had a decrease in the wavelength

en-AA = A' — A proportional to (1 — cos9 s ) where 9 S was the scattering angle (seeFig 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 ing must equal the energy and momentum after scattering Conservation of linearmomentum requires the initial momentum of the incident photon (the electron isassumed 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

scatter-of Fig 2.5 to be closed, i.e.,

or from the law of cosines

Pe = P 2X + PX ~ 2% c o sft- (2-21)

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 + m e c 2 — p, c + me 2

where m e 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 m e = m0), can be rewritten as

where h/(m e c) = 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 tion which could not be obtained with a wave model

predic-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^

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, crowaves, visible light, X rays, and gamma rays Many properties are described

mi-by a wave model in which the wave travels at the speed of light c and has a length A and frequency v, which are related by the wave speed formula

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 photoelectricinteractions described above)

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

Photon Properties

Some particles must always be treated relativistically For example, photons, by

definition, travel with the speed of light c From Eq (2.5), one might think that

photons have an infinite relativistic mass, and hence, from Eq (2.17), infinite mentum This is obviously not true since objects, when irradiated with light, arenot observed to jump violently This apparent paradox can easily be resolved if weinsist that the rest mass of the photon be exactly zero, although its relativistic mass

mo-is finite In fact, the total energy of a photon, E = hv, mo-is due strictly to its motion Equation (2.17) immediately gives the momentum of a photon (with m 0 = 0) as,

cause photons had a discrete energy E = hv arid momentum p = h/\ de Broglie

suggested that a particle, because of its momentum, should have an associated

wavelength A = h/p.

incident electrons

reflected electrons

crystal plane

Figure 2.6 Electrons scattering from atoms on a

crystalline plane, interfere constructively if the

dis-tance AB is a multiple of the electron's de Broglie

wavelength.

N(0)

Figure 2.7 Observed number of

electrons N(0) scattered into a fixed cone or directions about an angle 9

by the atoms in a nickel crystal.

Davisson and Germer in 1927 confirmed that electrons did indeed behave likewaves with de Broglie's predicted wavelength In their experiment, shown schemat-ically in Fig 2.6, Davisson and Germer illuminated the surface of a Ni crystal by

a perpendicular beam of 54-eV electrons and measured the number of electrons

N(9} reflected at different angles 0 from the incident beam According to the

par-ticle model, electrons should be scattered by individual atoms isotropically and

N(9) should exhibit no structure However N(9) was observed to have a peak

near 50° (see Fig 2.7) This observation could only be explained by recognizingthe peak as a constructive interference peak — a wave phenomenon Specifically,two reflected electron waves are in phase (constructively interfere) if the difference

in their path lengths AB in Fig 2.6 is an integral number of wavelengths, i.e if

dsmO = nX, n = 1 2 , where d is the distance between atoms of the crystal This

experiment and many similar ones clearly demonstrated that electrons (and otherparticles such as atoms) have wave-like properties