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Let us review some principles of physics needed for the study of the release of nuclear energy and its conversion into thermal and electrical forms.. To appreciate the relationship of st

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Nuclear Energy

Seventh Edition

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Nuclear Energy

An Introduction to the Concepts,

Systems, and Applications of

Nuclear Processes

Seventh Edition

Raymond L Murray Keith E Holbert

AMSTERDAM • BOSTON • HEIDELBERG • LONDON

NEW YORK • OXFORD • PARIS • SAN DIEGO SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO

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Joe Hayton

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Library of Congress Cataloging-in-Publication Data

Murray, Raymond LeRoy,

1920-Nuclear energy : an introduction to the concepts, systems, and applications of nuclear processes / Raymond L Murray, Keith E Holbert – Seventh edition.

pages cm

Includes bibliographical references and index.

ISBN 978-0-12-416654-7 (alk paper)

1 Nuclear engineering 2 Nuclear energy I Holbert, Keith E II Title.

TK9145.M87 2014

British Library Cataloguing-in-Publication Data

A catalogue record for this book is available from the British Library.

ISBN: 978-0-12-416654-7

For information on all Butterworth–Heinemann publications visit

our website at store elsevier.com

Printed and bound in United States of America

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About the Authors

Raymond L Murray (Ph.D., University of Tennessee, 1950) was a long-time faculty member in theDepartment of Nuclear Engineering of North Carolina State University Professor Murray studiedunder J Robert Oppenheimer at the University of California at Berkeley In the Manhattan Project

of World War II, he contributed to the uranium isotope separation process at Berkeley and Oak Ridge

In the early 1950s, he helped found the first university nuclear engineering program and the firstuniversity nuclear reactor During his 30 years of teaching and research in reactor analysis at NorthCarolina State, he taught many of our leaders in universities and industry throughout the world

He was the author of textbooks in physics and nuclear technology and the recipient of a number ofawards, including the Eugene P Wigner Reactor Physicist Award of the American Nuclear Society

in 1994 He was a Fellow of the American Physical Society, a Fellow of the American NuclearSociety, and a member of several honorary, scientific, and engineering societies After retirement fromthe university, Dr Murray was a consultant on criticality for the Three Mile Island Recovery Program,served as chairman of the North Carolina Radiation Protection Commission, and served as chairman ofthe North Carolina Low-Level Radioactive Waste Management Authority He provided an annual lec-ture at MIT for the Institute of Nuclear Power Operations

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Keith E Holbert (Ph.D., University of Tennessee, 1989) is presently an Associate Professor in theSchool of Electrical, Computer and Energy Engineering at Arizona State University His researchexpertise is in the area of instrumentation and system diagnostics including radiation effects on sensors.

Dr Holbert has performed tests on safety-related systems in more than a dozen nuclear power plants inthe United States He has published more than 100 journal and conference papers, a textbook, and holdsone patent Dr Holbert is a registered professional (nuclear) engineer He is a member of the AmericanNuclear Society and a Senior Member of the IEEE Dr Holbert holds a Guest Scientist affiliation withLos Alamos National Laboratory As the Director of the Nuclear Power Generation Program at ASU,Professor Holbert teaches undergraduate and graduate engineering courses on electric power genera-tion (from all forms of energy), nuclear reactor theory and design, nuclear power plant controls anddiagnostics, reactor safety analysis, and health physics and radiation measurements Dr Holbert hasbeen the recipient of multiple teaching awards Keith is a Christian, who ascribes to the doctrine thatGod has entrusted humanity with good stewardship of His creation

xvi About the Authors

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Besides nuclear power generation, associated technologies are utilized in a variety of applicationsincluding nuclear medicine and smoke detectors Furthermore, since the terrorist attacks of 2001, radi-ation detectors have been installed at ports of entry worldwide to intercept illicit shipments ofnuclear materials.

Like politics and religion, the subject of nuclear energy generates heated debate in certain circles.Hence, a purpose of this book must be to bring factual information to the discussion Topics that seem

to generate the most concern inevitably include the persistent nuclear waste issue, nuclear power plantsafety, radiation, and atomic weapons Therefore, the authors are compelled to devote coverage to these(sometimes controversial) areas

Those familiar with earlier editions will quickly realize that the ordering of chapters in the last thirds of the textbook has changed noticeably Part I retains its focus on foundational nuclear concepts.Part II is now devoted to topics concerning radiation and its generation, effects, and utilization; whereasPart III is aligned to nuclear power generation Besides changes to the organizational structure, signif-icant amounts of up-to-date nuclear data have been added (e.g., see Appendix A), thereby increasingthe utility of this book as a reference

two-Student learning is enhanced by performing calculations and analyses on nuclear quantities Thisedition provides Exercises, solvable by handheld calculator, with final answers given in Appendix B

In addition, MATLAB programs and Excel spreadsheets for the solution of computer exercises in thetext can be downloaded fromhttp://booksite.elsevier.com/9780124166547/

Persons providing valuable ideas and information are recognized at appropriate places in the book.The author welcomes any constructive comments and corrections to the text (holbert@asu.edu)

Keith E HolbertTempe, Arizona, 2013

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Since all processes involve interactions of particles, it is necessary to develop a backgroundunderstanding of the basic physical facts and principles that govern such interactions In Part I

we shall examine the concept of energy, describe the models of atomic and nuclear structure,discuss radioactivity and nuclear reactions in general, review the ways radiation reacts withmatter, and concentrate on two important nuclear processes: fission and fusion

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Energy

1 CHAPTER OUTLINE

1.1 Forces and Energy 3

1.2 Units of Measure 5

1.3 Thermal Energy 6

1.4 Radiant Energy 7

1.5 The Equivalence of Matter and Energy 9

1.6 Energy and the World 11

1.7 Summary 11

1.8 Exercises 12

1.9 Computer Exercise 13

References 13

Further Reading 14

Our material world is composed of many substances distinguished by their chemical, mechanical, and electrical properties They are found in nature in various physical states—the familiar solid, liquid, and gas, along with the ionic plasma However, the apparent diversity of kinds and forms of material is reduced

by the knowledge that there are only a little more than 100 distinct chemical elements and that the chemical and physical features of substances depend merely on the strength of force bonds between atoms

In turn, the distinctions between the elements of nature arise from the number and arrangement of basic particles: electrons, protons, and neutrons At both the atomic and nuclear levels, the structure of elements is determined by internal forces and energy

A limited number of basic forces exist: gravitational, electrostatic, electromagnetic, and nuclear Asso-ciated with each of these is the ability to do work Thus, energy in different forms may be stored, released, transformed, transferred, and “used” in both natural processes and man-made devices It is often convenient to view nature in terms of only two basic entities: particles and energy Even this distinction can be removed, because we know that matter can be converted into energy and vice versa Let us review some principles of physics needed for the study of the release of nuclear energy and its conversion into thermal and electrical forms We recall that if a constant forceF is applied to an object

to move it a distances, the amount of work W done is the product W¼Fs As a simple example, we pick

up a book from the floor and place it on a table Our muscles provide the means to lift against the force

of gravity on the book We have done work on the object, which now possesses stored energy (potential energy), because it could do work if allowed to fall back to the original level Now a forceF acting on a

3

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massm provides an acceleration a, given by Newton’s law F¼ma Starting from rest, the object gains aspeedv, and at any instant has energy of motion (kinetic energy) in amount

EK¼1

For objects falling under the force of gravity, we find that the potential energy is reduced as the kineticenergy increases, but the sum of the two energy types remains constant This is an example of the prin-ciple of conservation of energy Let us apply this principle to a practical situation and perform someillustrative calculations

As we know, falling water provides one primary source for generating electrical energy In a electric plant, river water is collected by a dam and allowed to fall through a considerable heighth,known as the head The potential energy of water is thus converted into kinetic energy The water

hydro-is directed to strike the blades of a hydraulic turbine, which turns an electric generator

The potential energy of a massm located at the top of a dam is EP¼Fh, being the work done to place

it there The force is the weightF¼mg, where g is the acceleration of gravity Thus, the potentialenergy is

EXAMPLE 1.1

Find the velocity of water descending through a dam with a 50 m head Ignoring friction, thepotential energy in kinetic form would appear at the bottom, that is,EP¼EK Using gravitationalacceleration at the Earth’s surface*g0¼9.81 m/s2, the water speed would be

v¼

ffiffiffiffiffiffiffiffi2EK

m

r

¼

ffiffiffiffiffiffiffiffi2EP

m

r

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffi2mg0hm

r

¼pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 9ð :81 m=s2Þ 50 mð Þ¼ 31:3 m=s

Energy takes on various forms, classified according to the type of force that is acting The water inthe hydroelectric plant experiences the force of gravity, and thus gravitational energy is involved It istransformed into mechanical energy of rotation in the turbine, which is then converted to electricalenergy by the generator At the terminals of the generator, there is an electrical potential difference,which provides the force to move charged particles (electrons) through the network of the electricalsupply system The electrical energy may then be converted into mechanical energy as in motors, intolight energy as in light bulbs, into thermal energy as in electrically heated homes, or into chemicalenergy as in a storage battery

The automobile also provides familiar examples of energy transformations The burning of gasolinereleases the chemical energy of the fuel in the form of heat, part of which is converted to energy of

rounded off to a few significant digits Only when accuracy is important will more figures or decimals be used The principal

( Haynes et al., 2011 ).

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motion of mechanical parts, while the rest is transferred to the atmosphere and highway The vehicle’salternator provides electricity for control and lighting In each of these examples, energy is changedfrom one form to another but is not destroyed The conversion of heat to other forms of energy is gov-erned by two laws, the first and second laws of thermodynamics The first law states that energy isconserved; the second specifies inherent limits on the efficiency of the energy conversion.

Energy can be classified according to the primary source We have already noted two sources ofenergy: falling water and the burning of the chemical fuel gasoline, which is derived from petroleum,one of the main fossil fuels To these we can add solar energy; the energy from winds, tides, or thesea motion; and heat from within the Earth Finally, we have energy from nuclear reactions (i.e.,the “burning” of nuclear fuel)

For many purposes, we use the metric system of units, more precisely designated as SI or Syste`meInternationale In this system (seeNISTin the chapter’s references), the base units are the kilogram(kg) for mass, the meter (m) for length, the second (s) for time, the mole (mol) for amount of substance,the ampere (A) for electric current, the kelvin (K) for thermodynamic temperature, and the candela (cd)for luminous intensity.Table 1.1 summarizes these SI base units and important derived quantities

Table 1.1 SI Base and Derived Quantities and Units

5 1.2 Units of measure

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In addition, the liter (L) and metric ton (tonne) are in common use (1 L¼10–3m3; 1 tonne¼1000 kg).However, for understanding earlier literature, one requires knowledge of other systems Table A.3 inAppendix A lists useful conversions from British units to SI units.

The transition in the United States from British units to SI units has been much slower than expected

To ease understanding by the typical reader, a dual display of numbers and their units are frequently given

in this book Familiar and widely used units such as the centimeter, the barn, the curie, and the rem aremaintained

In dealing with forces and energy at the level of molecules, atoms, and nuclei, it is conventional touse another energy unit, theelectronvolt (eV) Its origin is electrical in character, being the amount ofkinetic energy that would be imparted to an electron (charge 1.60210–19

coulombs) if it were erated through a potential difference of 1 volt Because the work done on 1 coulomb would be 1 J, wesee that 1 eV¼1.60210–19

accel-J The unit is of convenient size for describing atomic reactions Forinstance, to remove one electron from the hydrogen atom requires 13.6 eV of energy However, whendealing with nuclear forces, which are very much larger than atomic forces, it is preferable to use themillion-electronvolt unit (MeV) To separate the neutron from the proton in the nucleus of heavyhydrogen, for example, requires an energy of about 2.2 MeV (i.e., 2.2106eV)

E¼3

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wherek is Boltzmann’s constant, 1.3810–23J/K (Recall that the Kelvin scale has the same spacing ofdegrees as the Celsius scale, but its zero is at273.15C.)

To illustrate, let the mass flow rate _m of water in the hydropower plant ofExample 1.1be 2106kg/s

As power is the time rate of change of energy, the power available is

W) Such multiples of units are used because of the enormous range of magnitudes ofquantities in nature, from the submicroscopic to the astronomical.Table 1.2 gives the standard set

of prefixes for the system of units

Another form of energy is electromagnetic or radiant energy We recall that this energy may be released

by heating of solids, as in the wire of an incandescent light bulb; by electrical oscillations, as in radio ortelevision transmitters; or by atomic interactions, as in the sun The radiation can be viewed in either of

7 1.4 Radiant energy

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two ways—as a wave or as a particle—depending on the process under study In the wave view, it is acombination of electric and magnetic vibrations moving through space In the particle view, it is a com-pact moving uncharged object, the photon, which is a bundle of pure energy, having mass only by virtue

of its motion Regardless of its origin, all radiation can be characterized by its frequency, which isrelated to speed and wavelength Lettingc be the speed of light,l its wavelength, and n its frequency,

of the components are not fixed Of particular interest are the ionizing wavelengths, which begin withinthe ultraviolet (UV) region (10 eV) Later, we surmise that this is the reason some UV rays causeskin cancer

To appreciate the relationship of states of matter, atomic and nuclear interactions, and energy, let usvisualize an experiment in which we supply energy to a sample of water from a source of energy that is

Table 1.2 Prefixes for Numbers and Abbreviations

(nu) The Greek alphabet is compiled in Table A.1 for reference The reader must be alert to symbols used for more than one quantity.

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as large and as sophisticated as we wish Thus, we increase the degree of internal motion and eventuallydissociate the material into its most elementary components Suppose inFigure 1.2that the water isinitially ice at nearly absolute zero temperature, where water (H2O) molecules are essentially at rest.

As we add thermal energy to increase the temperature to 0C or 32F, molecular movement increases

to the point at which the ice melts to become liquid water, which can flow rather freely To cause achange from the solid state to the liquid state, a definite amount of energy (termed the heat of fusion)

is required In the case of water, this latent heat is 334 J/g In the temperature range in which water isliquid, thermal agitation of the molecules permits some evaporation from the surface At the boilingpoint, 100C or 212F at atmospheric pressure, the liquid turns into the gaseous form as steam Again,

energy is required to cause the change of state, with a heat of vaporization of 2258 J/g Further heating,

by use of special high temperature equipment, causes dissociation of water into atoms of hydrogen (H)and oxygen (O) By electrical means, electrons can be removed from hydrogen and oxygen atoms,leaving a mixture of charged ions and electrons Through nuclear bombardment, the oxygen nucleuscan be broken into smaller nuclei, and in the limit of temperatures in the billions of degrees, thematerial can be decomposed into an assembly of electrons, protons, and neutrons

The connection between energy and matter is provided by Einstein’s theory of special relativity It dicts that the mass of any object increases with its speed Letting the mass when the object is stationary

pre-bem0, therest mass; letting m be the mass when it is at speed v; and noting that the speed of light in avacuum isc, then the relativistic mass is

FIGURE 1.1

Electromagnetic spectrum (ROYGBIV: red, orange, yellow, green, blue, indigo, violet)

9 1.5 The equivalence of matter and energy

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For low speeds,v c, EKis approximately1

2mv2, the classical relation (seeExercise 1.7)

is more than a million times that from chemical fuel To prove this startling statement, we firstfind the result of the complete transformation of 1 kilogram of matter into energy, namely,(1 kg)(3.0108m/s)2¼91016J The nuclear fission process, as one method of converting mass intoenergy, is relatively inefficient, because the burning of 1 kg of uranium involves the conversion of only0.87 g of matter into energy This corresponds to approximately 7.81013J/kg of the uranium con-sumed The enormous magnitude of this energy release can be appreciated only by comparison withthe energy of combustion of a familiar fuel such as gasoline, 5107J/kg The ratio of these numbers,1.5106, reveals the tremendous difference between nuclear and chemical energies

Calculations involving Einstein’s theory of relativity are readily accomplished using a MATLABprogram ALBERT, described inComputer Exercise 1.A

All of the activities of human beings depend on energy, as we realize when we consider the dimensions

of the world’s energy problem The efficient production of food requires machines, fertilizer, andwater, each making use of energy in a different way Energy is vital to transportation, protection againstthe weather, and the manufacturing of all goods An adequate long-term supply of energy is thereforeessential for humanity’s survival The world energy problem has many dimensions: the increasing cost

to acquire fuels as they become more scarce; the potential for global climate change resulting fromburning fossil fuels; the effects on safety and health of the by-products of energy consumption; theinequitable distribution of energy resources among regions and nations; and the discrepancies betweencurrent energy use and human expectations throughout the world

Associated with each basic type of force is an energy, which may be transformed to another form forpractical use The addition of thermal energy to a substance causes an increase in temperature, themeasure of particle motion Electromagnetic radiation arising from electrical devices, atoms, or nuclei

11 1.7 Summary

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may be considered to be composed of waves or of photons Matter can be converted into energyand vice versa according to Einstein’s formulaE¼mc2 The energy of nuclear fission is millions oftimes as large as that from chemical reactions Energy is fundamental to all human endeavors and,indeed, survival.

1.6 Find the frequency of a gamma-ray photon of wavelength 1.510–12m

1.7 (a) For very small velocities compared with the velocity of light, show that the relativistic formulafor kinetic energy is ½m0v2.Hint: Use the series expansion (1þx)n¼1þnxþ (b) Find theapproximate relativistic mass increase of a car with rest mass 1000 kg moving at 20 m/s.1.8 Noting that the electronvolt is 1.6010–19J, how many joules are released in the fission of oneuranium nucleus, which yields 190 MeV?

1.9 Applying Einstein’s formula for the equivalence of mass and energy, E¼mc2, how manykilograms of matter are converted into energy in Exercise 1.8?

1.10 If the atom of uranium-235 has mass of (235)(1.6610–27

) kg, what amount of equivalent energydoes it have?

1.11 Using the results ofExercises 1.8,1.9, and1.10, what fraction of the mass of a U-235 nucleus isconverted into energy when fission takes place?

1.12 Show that to obtain a power of 1 W from fission of uranium, it is necessary to cause 3.31010

fission events per second Assume that each fission releases 190 MeV of useful energy.1.13 Using the rest mass of each, compute the rest mass energy in MeV for (a) a proton and (b) aneutron Compare to values given in Table A.2

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1.14 Using the mass of 1.660538910–27kg for one atomic unit, calculate the equivalent energy(in MeV) to 5 significant digits.

1.15 (a) If the fractional mass increase caused by relativity is DE/E0, show that

v=c ¼qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 1 þ DE=Eð 0Þ2(b) At what fraction of the speed of light does a particle have a mass that is 1% higher than the restmass? 10%? 100%?

1.16 The heat of combustion of hydrogen by the reaction 2HþO¼H2O is quoted as

34.18 kilogram calories per gram of hydrogen (a) Find how many Btu per pound this is withthe conversions 1 Btu¼0.252 kcal, 1 lb¼454 grams (b) Find how many joules per gram this isnoting 1 cal¼4.184 J (c) Calculate the heat of combustion in eV per H2molecule

Note: Recall the number of particles per gram of molecular weight, Avogadro’s number,

1.A Properties of particles moving at high velocities are related in a complicated way according

to Einstein’s theory of special relativity To obtain answers easily, the MATLAB programALBERT (after Dr Einstein) can be used to treat the following quantities: velocity,

momentum, total energy, kinetic energy, and ratio of mass to rest mass Given one of these, for aselected particle, ALBERT calculates the others Test the program with various inputs,including (a) an electron with velocity 0.5c, (b) a proton with 1000 MeV total energy, (c) aneutron with 0.025 eV kinetic energy, (d) deuteron withm/m0¼1.01, and (e) alpha withmomentum 10–19kgm/s

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Further Reading 30

A complete understanding of the microscopic structure of matter and the exact nature of the forces acting on that matter is yet to be realized However, excellent models have been developed to predict behavior to an adequate degree of accuracy for most practical purposes These models are descriptive

or mathematical, often based on analogy with large-scale processes, on experimental data, or on advanced theory

The most elementary concept is that matter is composed of individual particles—atoms—that retain their identity as elements in ordinary physical and chemical interactions Thus, a collection of helium atoms that forms a gas has a total weight that is the sum of the weights of the individual atoms Also, when two elements combine to form a compound (e.g., if carbon atoms combine with oxygen atoms to form carbon monoxide molecules), the total weight of the new substance is the sum of the weights of the original elements

There are more than 100 known elements Most are found in nature; some are artificially produced Each is given a specific number in the periodic table of the elements; examples are hydrogen (H) 1, helium (He) 2, oxygen (O) 8, and uranium (U) 92 The symbolZ is assigned to the atomic number, which is also the number of electrons in the atom and determines its chemical properties The periodic table is shown inFigure 2.1

Generally, the further an element is in the periodic table, the heavier its atoms Theatomic weight M is the weight in grams of a definite number of atoms, 6.021023, which is Avogadro’s number, NA

15

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Although we often use the termsatomic weight and atomic mass interchangeably, atomic mass describesthe mass of a single atom of a particular isotope, whereas atomic weight provides a weighted averagemass for an element based on the abundance of its constituent isotopes For the elements just mentioned,the values ofM are approximately H, 1.008; He, 4.003; O, 16.00; and U, 238.0 Atomic weight isexpressed using grams/mole or atomic mass units (u), and atomic mass is quantified using atomic massunits (u) Accurate values of atomic weights of all the elements are given in Table A.4 in Appendix A.

If an element has a nonnatural abundance of its isotopes (i.e., the elemental material is eitherenriched or depleted), it is necessary to compute the atomic weight of the element (M) from theweighted sum of all the atomic masses of the isotopes (Mj) rather than use the tabulatedM value found

in a reference In such cases, the isotopic abundance may be expressed either as an atom abundance orfraction (gj), or as a weight or mass fraction (oj) This distinction leads to two formulas for determiningthe elemental atomic weight:

Periodic table of the elements

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We can easily find the number of atoms per cubic centimeter in a substance if its density r (rho) ingrams per cubic centimeter is known This procedure can be expressed as a convenient formula forfindingN, the atomic number density for any material

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wheren is the number of particles and k is Boltzmann’s constant An increase in the temperature of thegas as a result of heating causes greater molecular motion, which results in an increase of particle bom-bardment of a container wall and thus of pressure on the wall.

The gas particles, each of mass m, have a variety of speeds v in accord with Maxwell’sgas theory, as shown in Figure 2.2 Maxwell’s formula for the number of molecules per unitspeed is

v¼

ffiffiffiffiffiffiffiffi8kTpm

Maxwellian distribution of molecular speeds

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2.3 THE ATOM AND LIGHT

Until the twentieth century, the internal structure of atoms was unknown, but it was believed that electriccharge and mass were uniform Rutherford supervised some crucial experiments in which gold atomswere bombarded by charged particles He deduced in 1911 that most of the mass and positive charge

of an atom were concentrated in anucleus of radius only approximately 10–5times that of the atom,and thus occupying a volume of approximately 10–15 times that of the atom (seeExercises 2.2 and

2.12) The new view of atoms paved the way for Bohr to find an explanation for the production of light

It is well known that the color of a heated solid or gas changes as the temperature is increased,tending to go from the red end of the visible region toward the blue end (i.e., from long wavelengths

to short wavelengths) The measured distribution of light among the different wavelengths at acertain temperature can be explained by the assumption that light is in the form of photons Theseare absorbed and emitted with definite amounts of energyE that are proportional to the frequency

This is seen to be a very minute amount of energy

Bohr (1913)first explained the emission and absorption of light from incandescent hydrogen gaswith a novel model of the hydrogen atom He assumed that the atom consists of a single electron mov-ing at constant speed in a circular orbit about a nucleus—the proton—as shown inFigure 2.3 Eachparticle has an electric charge of 1.610–19coulombs (C), but the positively charged proton has a mass

FIGURE 2.3

Hydrogen atom

19 2.3 The atom and light

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that is 1836 times that of the negatively charged electron The radius of the orbit is set by the equality ofelectrostatic force, attracting the two charges toward each other, to centripetal force, required to keepthe electron on a circular path If sufficient energy is supplied to the hydrogen atom from the outside,the electron is caused to jump to a larger orbit of definite radius At some later time, the electron fallsback spontaneously to the original orbit, and energy is released in the form of a photon of light Thephoton energyhn is equal to the difference between energies in the two orbits The smallest orbit has aradiusR1¼0.5310–10m, whereas the others have radii increasing as the square of integers,n, whichare called principalquantum numbers Thus if n is 1, 2, 3, ,7, the radius of the n-th orbit is

Rn¼ n2

Figure 2.4shows the allowed electron orbits in hydrogen The energy of the atom system when theelectron is in the first orbit isE1¼ 13.6 eV, where the negative sign means that energy must be sup-plied to remove the electron to a great distance and leave the hydrogen as a positive ion The energywhen the electron is in thenth orbit is

The various discrete levels are shown inFigure 2.5

The electronic structure of the other elements is described by the shell model, in which a limitednumber of electrons can occupy a given orbit or shell The atomic numberZ is unique for each chemicalelement and represents both the number of positive charges on the central massive nucleus of the atomand the number of electrons in orbits around the nucleus The maximum allowed numbers of electrons

in orbits asZ increases for the first few shells are 2, 8, and 18 The number of electrons in the outermost,

or valence, shell determines the chemical behavior of elements For example, oxygen withZ¼8 hastwo electrons in the inner shell, six in the outer Thus, oxygen has an affinity for elements with twoelectrons in the valence shell The formation of molecules from atoms by electron sharing is illustrated

byFigure 2.6, which shows the water molecule

Electron “jump”

Possible orbits

Electron orbits in hydrogen (Bohr theory)

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The Bohr model of atoms is useful for visualization, but quantum mechanics provides a more orous view There, the location of the electron in the H atom is described by a probability expression Akey feature of quantum mechanics is Heisenberg’s uncertainty principle It states that the precise values

rig-of both a particle’s position and momentum cannot be known

up inside by a combination of reflection and stimulation An avalanche of photons is produced thatmakes a very intense beam Light moving in directions other than the long axis of the laser is lostthrough the sides, so that the beam that escapes from the end proceeds in only one direction The reflec-tion between the two end mirrors assures a coherent beam (i.e., the waves are in phase)

Lasers can be constructed from several materials The original one (1960) was the crystalline gemruby Others use gases such as a helium-neon mixture, liquids with dye in them, or semiconductors.The external supply of energy can be chemical reactions, a discharge produced by accelerated electrons,energetic particles from nuclear reactions, or another laser Some lasers operate continuously, whereasothers produce pulses of energy as short as a fraction of a nanosecond (10–9sec) with the power of a tera-watt (1012watts) Because of the high intensity, laser light, if viewed directly, can be hazardous to the eyes.Lasers are widely used where an intense well-directed beam is required, as in metal cutting andwelding, eye surgery and other medical applications, and accurate surveying and range finding Newerapplications are noise-free phonographs, holograms (3D images), and communication between air-plane and submarine

In later chapters, we will describe some nuclear applications: isotope separation (Section 15.5) andthermonuclear fusion (Section 26.4)

Most elements are composed of atoms of different mass, calledisotopes For instance, hydrogen hasthree isotopes of weights in proportion 1, 2, and 3—ordinary hydrogen, heavy hydrogen (deuterium),and tritium Each has atomic numberZ¼1 and the same chemical properties, but they differ in thecomposition of the central nucleus, where most of the mass resides The nucleus of ordinary hydrogen

is the positively charged proton; the deuteron consists of a proton plus a neutron; the triton contains aproton plus two neutrons The neutron is a neutral particle of mass very close to that of the proton Todistinguish isotopes, we identify theatomic mass number A as the total number of nucleons, which arethe heavy particles in the nucleus The atomic weight, a real number, is approximated by the mass num-ber, which is an integer,M ﬃ A The complete shorthand notation for an isotope is given by the

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chemical symbol X with leading superscriptA and subscript Z values, that is,ZAX.Figure 2.7shows thenuclear and atomic structure of the three hydrogen isotopes (i.e.,1H,1H, and1H) Each has one electron

in the outer shell, in accord with the Bohr theory described earlier

The structure of some of the lighter elements and isotopes is shown inFigure 2.8 In each case,the atom is neutral, because the negative charge of theZ electrons in the outside shell balancesthe positive charge of the Z protons in the nucleus The symbols for the isotopes shown in

The dimensions of nuclei are found to be very much smaller than those of atoms Whereas the hydrogenatom has a radius of approximately 510–9cm, its nucleus has a radius of only approximately

10–13cm Because the proton mass is much larger than the electron mass, the nucleus is extremelydense The nuclei of other isotopes may be viewed as closely packed particles of matter—neutronsand protons—forming a sphere whose volume,43pR3, depends onA, the number of nucleons A usefulrule of thumb to calculate radii of nuclei is

R cm½ ¼ 1:25 1013A1 =3 ð2:10Þ

Hydrogen +

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BecauseA ranges from 1 to approximately 250, we see that all nuclei are smaller than 10–12cm Theradius of an entire atom is much larger, on the order of 10–8cm.

The masses of atoms, labeledM, are compared on a scale in which an isotope of carbon126C has amass of exactly 12 For1H, the atomic mass isMH-1¼1.007825, for1H,MH-2¼2.014102, and so on.The atomic mass of the proton is 1.007276, of the neutron 1.008665, the difference being only about0.1% The mass of the electron on this scale is 0.000549 A list of atomic masses appears inTable A.5

+ +

−

+ +

−

+ +

FIGURE 2.8

Atomic and nuclear structure

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The atomic mass unit (amu or simply “u”), as one-twelfth the mass of612C, corresponds to an actualmass of 1.66053910–24g (ﬃ1.6610–24g) To verify this approximation merely divide 1 g by Avo-gadro’s number 6.0221023 We can calculate the actual masses of atoms and nuclei by multiplyingtheir mass in atomic mass units by the mass of 1 amu.

EXAMPLE 2.5

The rest mass of the neutron is

1:008665 amu

ð Þ 1:660539 10 24g=amu¼ 1:674928 1024g

By use of Einstein’s E¼mc2 with constants in Table A.2, we found in Exercise 1.14 that

1 amu¼931.494 MeV (ﬃ 931.5 MeV) This mass–energy equivalence factor will prove useful

cm; the distance of separation of centers is about twice that The nuclearforce acts only when the nucleons are very close to each other and binds them into a compact structure.Associated with the net force is a potential energy of binding

To disrupt a nucleus and separate it into its component nucleons, energy must be supplied from theoutside Recalling Einstein’s relation between mass and energy, this is the same as saying that a givennucleus is lighter than the sum of its separate nucleons, the difference being the mass defect Let themass of an atom including nucleus and external electrons beM, and let mnandMHbe the respectivemasses of the neutron and the proton plus matching electron Then themass defect is

Dm ¼ Nm|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}nþ ZMH

total mass ofseparate particles

25 2.7 Binding energy

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Neglected in this relation is a small energy of atomic or chemical binding The corresponding nuclearbinding energy is simply

If the binding energy per nucleon (BE/A) can be increased, then energy will be liberated in the process.TheBE/A curve shown in Figure 2.9reveals a maximum around56Fe, but 62Ni is the most boundnucleus The release of binding energy is exploited in both nuclear fission and fusion

pro-DmT ¼ NTmnþ ZTMH MT

¼ 2 1:008665ð Þ þ 1 1:007825ð Þ 3:016049 ¼ 0:009106 amuConverting by use of the relation 1 amu¼931.5 MeV, the binding energy is BE¼8.48 MeV

Example 2.6shows the necessity of utilizing masses known to 6 or more significant digits in thecomputation of the mass defect Calculations such as these are required for several purposes: to com-pare the stability of one nucleus with that of another, to find the energy release in a nuclear reaction, and

to predict the possibility of fission of a nucleus

FIGURE 2.9

Binding energy per nucleon

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We can speak of the binding energy associated with one particle such as a neutron Suppose thatM1

is the mass of an atom andM2is its mass after absorbing a neutron The binding energy of the additionalneutron of massmnis then

BEn¼ M½ð 1þ mnÞ M2c2 ð2:13ÞExplanations of binding energy effects by means of physical logic and measured atomic masseshave led to what are called semiempirical formulas for binding energy The BE for any nuclidemay be approximated using a liquid drop model that accounts for (1) attraction of nucleons for eachother due to strong nuclear force, (2) electrostatic (Coulombic) repulsion, (3) surface tension effects,and (4) the imbalance of neutrons and protons in the nucleus The Bethe-Weizsa¨cker formula is onesuch expression to calculate the binding energy:

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The binding energy per nucleon is readily seen as BE/A¼(1786.8 MeV)/(235)¼7.60MeV/nucleon, which compares favorably to the known value of 7.59 MeV/nucleon (see

Figure 2.9) Finally, combining Equations(2.11)and(2.12)provides a relation for the atomic mass

to dissociate a nucleus into its components

2.1 Find the number of carbon (12

6C) atoms in 1 cm3of graphite, density 1.65 g/cm3.2.2 Estimate the radius and volume of the gold atom, using the metal density of 19.3 g/cm3

Assumethat atoms are located at corners of cubes and that the atomic radius is that of a sphere withvolume equal to that of a cube

2.3 Calculate the most probable speed of a “neutron gas” at temperature 20C (293 K), noting that

the mass of a neutron is 1.67510–27kg

2.4 Prove that the specific heat at constant volume of an atomic gas is given by cV¼(3/2)(k/m), by use

of the formula for average energy of a molecule, i.e., Equation(1.4), and that heat added (Q)causes a temperature rise (DT) according to Q¼mcVDT

2.5 Use the formula derived in the previous exercise to determine the specific heat of helium, acommon coolant for high-temperature gas reactors

2.6 Calculate the energy in electronvolts of a photon of yellow light (seeSection 2.3)

2.7 What frequency of light is emitted when an electron jumps into the smallest orbit of hydrogen,coming from a very large radius (assume infinity)?

2.8 Calculate the energy in electronvolts of the electron orbit in hydrogen for which n¼3, and findthe radius in centimeters How much energy would be needed to cause an electron to go from theinnermost orbit to this one? If the electron jumped back, what frequency of light would

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2.10 If A nucleons are visualized as spheres of radius r that can be deformed and packed tightly in anucleus of radiusR, show that r¼1.2510–13cm.

2.11 What is the radius of the nucleus of uranium-238 viewed as a sphere? What is the area of thenucleus, seen from a distance as a circle?

2.12 Find the fraction of the volume that is occupied by the nucleus in the gold-197 atom, by use of therelationship of radiusR to mass number A Recall fromExercise 2.2that the radius of the atom is1.5910–8cm

2.13 Find the mass defect in amu and binding energy in MeV of ordinary helium,2He

2.14 How much energy (in MeV) would be required to completely dissociate the uranium-235 nucleusinto its component protons and neutrons?

2.15 Find the mass density of the nucleus, the electrons, and the atom of U-235, assuming sphericalshapes and the following data:

Discuss the results

2.16 (a) Use the Maxwellian distribution of Equation(2.4)to verify by differentiation that the peak ofthe curve occurs atvpgiven by Equation(2.5) (b) Confirm by integration that the average speed

is given by Equation(2.6).Hint: Let mv2/2kT¼x

2.17 The temperature of the surface of the sun is approximately 5800 K To what light frequency andwavelength does that correspond?

2.18 The division between ionizing and nonionizing electromagnetic radiation is sometimes quoted as

10 eV Determine the corresponding frequency and wavelength

2.19 Use the Bethe-Weizsa¨cker formula to determine the atomic mass of (a)16O, (b)17O, (c)92Rb, (d)

140Cs, and (e)238U Compare the resulting values to those in Table A.5

2.20 Using Equations(2.8)and(2.9), make a log-log plot of |En| versusRnforn¼1, 2, 6 What isthe shape of the resulting curve?

2.21 Using the liquid drop model, create an approximation of the binding energy per nucleon graph of

Figure 2.9 The atomic mass number of stable isotopes can be approximated as a power lawfunction of the atomic number according toA¼1.47Z1.123

(Paar et al., 2002) where the resultingvalue must be rounded to an integer for use in Equation(2.14)

2.A Use the program ALBERT (seeChapter 1) to complete the following table comparing basicquantities of the electron, proton, and neutron at 1 MeV

29 2.10 Computer exercises

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Particle Total Energy(MeV) Velocity(m/s) Mass to RestMass Ratio Momentum(kg m/s)

of the first four terms of the Bethe-Weizsa¨cker formula, BINDING constructs an area graph of thecontribution to the binding energy per nucleon (BE/A) for the volume, surface, Coulomb, andasymmetry energy terms such that the netBE/A is clearly distinguishable as a function of A

References

Bohr, N., 1913 On the constitution of atoms and molecules Philosophical Magazine Series 6, 26 (151), 1–25.Paar, V., Pavin, N., Rubcˇic´, A., Rubcˇic´, J., 2002 Power laws and fractal behavior in nuclear stability, atomicweights and molecular weights Chaos, Solitons and Fractals 14, 901–916

Rutherford, E., 1911 LXXIX The scattering ofa and b particles by matter and the structure of the atom.Philosophical Magazine Series 6, 21 (125), 669–688

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