The Babcock & Wilcox Company Steam 41 / Fundamentals of Nuclear Energy 47-1 Chapter 47 Fundamentals of Nuclear Energy Fundamental particles and structure of the atom 1 In the fifth century B.C. Greek philosophers pos- tulated that all matter is composed of indivisible par- ticles called atoms. Over the next 24 centuries there were many speculations on the basic structure of matter but all theories lacked any experimental basis. It was not until late in the nineteenth and early twenti- eth centuries that the existence of fundamental particles, the proton, neutron and electron, was confirmed. The discovery of nuclear fission, credited to Otto Hahn, Lise Meitner and Fritz Strassman, was made possible through an accumulation of knowledge on the structure of matter beginning with Becquerel’s 1896 detection of radioactivity. As shown in Fig. 1, the structure of an atom is pic- tured as a dense, positively charged nucleus sur- rounded by an array of negatively charged electrons. Each proton, the equivalent of a hydrogen atom nucleus, carries an elemental positive charge of elec- tricity. The number of protons determines the type of chemical element of the atom. Each neutron is an elec- trically neutral particle with a mass slightly greater than the proton. Because of their association with the nucleus of the atom, protons and neutrons are also referred to as nucleons. Each electron shown orbiting around the nucleus has an elemental negative charge and a mass about 1/2000 that of a proton. The nucleus of an atom with a characteristic number of neutrons and protons is called a nuclide. Despite the minute size of the atom, there is a rela- tively great distance between the nucleus and the or- biting electrons. This distance, approximately 10 5 times the dimension of the nucleus, accounts for the ability of various radiations to pass through appar- ently dense materials. Fig. 1 also indicates the relationship of the positively charged protons in the nucleus and the negatively charged electrons. In the un-ionized state, the num- ber of protons is balanced by an equal number of elec- trons. An atom becomes ionized by gaining or losing one or more electrons. A gain of electrons yields nega- tive ions and a loss results in positive ions. An atom in an ionized state can interact with other elements to form various compounds. When an atom has more than two electrons, their orbits are located in a series of separate and distinct groupings of energy levels or shells. Each shell is ca- pable of containing a specific number of electrons. In general, an inner shell fills to its maximum number of electrons before electrons begin to form in the next shell. An x-ray is a quantum of electromagnetic en- ergy that is emitted when an electron transitions from an outer shell to an inner shell or between energy levels within the same shell. The number of electrons in the outermost shell de- termines certain chemical properties of the elements. The properties are similar for elements which have similar electron distributions in the outer shell regard- less of the number of inner shells. This accounts for the repetition of chemical properties in the Periodic Table of the Elements. (See Appendix 1.) Nuclides and isotopes A nuclide is characterized by its atomic number Z (number of protons in the nucleus) and the mass num- ber A (total number of protons plus neutrons). When nuclides are described by chemical symbol, the atomic Fig. 1 Structure of the atom (Z = atomic number; A = mass number).
Transcript
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Steam 41 / Fundamentals of Nuclear Energy 47-1
Chapter 47Fundamentals of Nuclear Energy
Fundamental particles and structureof the atom1
In the fifth century B.C. Greek philosophers pos-tulated that all matter is composed of indivisible par-ticles called atoms. Over the next 24 centuries therewere many speculations on the basic structure ofmatter but all theories lacked any experimental basis.It was not until late in the nineteenth and early twenti-eth centuries that the existence of fundamental particles,the proton, neutron and electron, was confirmed.
The discovery of nuclear fission, credited to OttoHahn, Lise Meitner and Fritz Strassman, was madepossible through an accumulation of knowledge on thestructure of matter beginning with Becquerel’s 1896detection of radioactivity.
As shown in Fig. 1, the structure of an atom is pic-tured as a dense, positively charged nucleus sur-rounded by an array of negatively charged electrons.Each proton, the equivalent of a hydrogen atomnucleus, carries an elemental positive charge of elec-tricity. The number of protons determines the type ofchemical element of the atom. Each neutron is an elec-trically neutral particle with a mass slightly greaterthan the proton. Because of their association with thenucleus of the atom, protons and neutrons are alsoreferred to as nucleons. Each electron shown orbitingaround the nucleus has an elemental negative chargeand a mass about 1/2000 that of a proton. The nucleusof an atom with a characteristic number of neutronsand protons is called a nuclide.
Despite the minute size of the atom, there is a rela-tively great distance between the nucleus and the or-biting electrons. This distance, approximately 105
times the dimension of the nucleus, accounts for theability of various radiations to pass through appar-ently dense materials.
Fig. 1 also indicates the relationship of the positivelycharged protons in the nucleus and the negativelycharged electrons. In the un-ionized state, the num-ber of protons is balanced by an equal number of elec-trons. An atom becomes ionized by gaining or losingone or more electrons. A gain of electrons yields nega-tive ions and a loss results in positive ions. An atom inan ionized state can interact with other elements toform various compounds.
When an atom has more than two electrons, theirorbits are located in a series of separate and distinctgroupings of energy levels or shells. Each shell is ca-pable of containing a specific number of electrons. In
general, an inner shell fills to its maximum numberof electrons before electrons begin to form in the nextshell. An x-ray is a quantum of electromagnetic en-ergy that is emitted when an electron transitions froman outer shell to an inner shell or between energylevels within the same shell.
The number of electrons in the outermost shell de-termines certain chemical properties of the elements.The properties are similar for elements which havesimilar electron distributions in the outer shell regard-less of the number of inner shells. This accounts forthe repetition of chemical properties in the PeriodicTable of the Elements. (See Appendix 1.)
Nuclides and isotopesA nuclide is characterized by its atomic number Z
(number of protons in the nucleus) and the mass num-ber A (total number of protons plus neutrons). Whennuclides are described by chemical symbol, the atomic
Fig. 1 Structure of the atom (Z = atomic number ; A = mass number).
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number is the left hand subscript and the mass num-ber the right hand superscript. For example, the αparticle can be represented as α, 2α4, or 2He4. With mostchemical elements there are several types of atomshaving different mass numbers, i.e., different numbersof neutrons in the nucleus but the same number ofprotons. An isotope is one of two or more nuclides of thesame chemical element having different mass numbers.
Two isotopes of hydrogen and two isotopes of oxy-gen are depicted in Fig. 1. An ordinary hydrogen atom1H1, contains one proton and no neutrons. Its atomicnumber Z and mass number A are 1. It combines withoxygen to form H2O or regular or light water. The deu-terium atom, 1H2 or 1D2, has one proton and one neu-tron; Z is 1 and A is 2. It combines with oxygen to formD2O or heavy water. A third isotope of hydrogen, tri-tium, 1H3 or 1T3, has one proton and two neutrons.
Heavier nuclides are also identified by chemicalname and mass number. Oxygen-16 and oxygen-17(Fig. 1) have 16 and 17 nucleons respectively, althougheach has eight protons. For example, symbols for the twoisotopes of oxygen described above are 8O16 and 8O17.
Although either the atomic number or the chemi-cal symbol could identify the chemical element, sub-scripts sometimes are useful in accounting for the to-tal number of charges in an equation.
MassIt is customary to list the mass of atoms and funda-
mental particles in atomic mass units (AMU). This isa relative scale in which the nuclide 6C12 (carbon-12)is assigned the exact mass of 12 AMU by agreementat the 1962 International Union of Chemists andPhysicists. One AMU is the equivalent of approxi-mately 1.66 × 10-24 grams, or the reciprocal of the pres-ently accepted Avogadro number, 0.602214 × 1024 at-oms per gram-atom. A gram-atom of an element is aquantity having a mass in grams numerically equalto the atomic weight of the element.
Table 1 lists the masses of the fundamental particlesand the atoms of hydrogen, deuterium and helium inatomic mass units.
Mass defectThe mass of the hydrogen atom 1H1 listed in Table
1 is almost but not quite equal to the sum of the massesof its individual particles, one proton and one electron.However, the mass of a deuterium atom 1H2 is notice-ably less than the sum of its constituents – a proton,neutron and electron. Measurements show that themass of a nuclide is always less than the sum of themasses of its protons, neutrons and electrons. Thisdifference, the mass defect (MD), is customarily cal-culated as:
MD Zm A Z m mh n e= + −( ) − (1)
where
MD = mass defect, AMUZ = number of protons in the nucleus of the
nuclidemh = mass of the hydrogen atom, AMUA – Z = number of neutrons in the nucleus
mn = mass of neutron, AMUme = mass of the nuclide including its Z elec-
trons, AMU
Binding energyAlthough most nuclei contain a plurality of protons
with mutually repulsive positive charges, the nucleusremains tightly bound together, and it takes consid-erable energy to cause disintegration. This energy,called binding energy, is equivalent to the mass de-fect. From the equivalence of mass and energy, asdefined by Einstein’s equation E = mc2, one AMUequals 931 million electron volts (MeV ). An electronvolt is the energy gained by a unit electrical chargewhen it passes, without resistance, through a poten-tial difference of 1 volt. Therefore,
Binding energy MeV( )= + −( ) −{ } ×Zm A Z m mh n e 931 (2)
This represents the amount of radiant or heat en-ergy released when an atom is formed from neutronsand hydrogen atoms. It also represents the energywhich must be added to fission an atom into its basicnucleons, i.e., neutrons and protons.
Dividing Equation 2 by A, the number of nucleonsin the nucleus, yields the binding energy per nucleon.This in turn can be plotted as a function of A, the massnumber as shown in Fig. 2. The result shows that thebinding energy per nucleon rapidly increases for lowmass numbers, reaches a maximum for mass numbersin the range of 40 to 80, and then drops off with anincreasing slope. On the rising portion of this curve,fusion or joining of nucleons to atoms of higher massnumber means that there is an increased binding en-ergy per nucleon and consequently a release of energy.On the falling portion of this curve, the fission process orsplitting of an atom results in nuclides of lesser mass num-bers and greater binding energy per nucleon. Again, en-ergy is released because of the increased mass defect.
Radioactivity and decay
Nuclear radiationsNuclides that occur in nature are stable in most
cases. However, a few are unstable, especially thoseof atomic number 84 and above. The unstable nuclides
Table 1Masses of Particles and Light Atoms
Isotope or Particle Mass, AMU
Electron 0.000549
Proton, 1p1 1.007277
Neutron, 0n1 1.008665
Hydrogen, 1H1 1.007825
Deuterium, 1H2 2.01410
Helium, 2He4 4.00260
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undergo spontaneous change at specific rates by ra-dioactive disintegration or decay. Many nuclides de-cay into other unstable nuclides, resulting in a decaychain that continues until a stable isotope is formed.
There are generally three types of radiation commonlyarising out of the decay of specific nuclides: alpha (α)particles, beta (β) particles and gamma (γ) rays.
Alpha particle The α particle is equivalent to thenucleus of a helium atom comprising two neutronsand two protons. It results from radioactive decay ofan unstable nuclide and, with very few exceptions, isobserved only in the decay of heavy nuclides.
Beta particle The β particle results from radioactivedecay and has the same mass and charge as an elec-tron. It is believed that the nucleus of an atom doesnot contain electrons and, in radioactive β decay, theβ radiation arises from conversion of a neutron into aproton and β particle; therefore:
neutron proton + + energy→ β
In some instances a positive β particle (called apositron) is produced from the conversion of a protonto a neutron.
Gamma ray Gamma rays are electromagnetic ra-diation resulting from a nuclear reaction. They canbe treated like particles in many nuclear reactions andare included among the fundamental particles. Al-though γ rays have physical characteristics similar to x-rays, their energy is greater and wave length shorter.The only other difference between the two rays is that γrays originate from within the nucleus while x-rays origi-nate from within the shell structure of the atom.
Biological effects of radiationThe organs of the human body are composed of tis-
sue which is composed of atoms. When electrons areknocked out of or added to atoms (ionized), the chemi-cal bonds which bind atoms together to make moleculesare broken. This process is called ionization. The re-combination of these broken molecules can result inchanges in the molecular structure of a cell which mayaffect the way the cell functions, its growth charac-teristics and its interaction with other cells. In cases
of high radiation exposure, cancer may result. Theterm used to describe the effects of radiation on hu-mans is biological damage.
Ionization is a direct result of alpha and beta par-ticles interacting with human tissue. These particlesproduce a continuous path of ionization as they travelthroughout the body. Gamma radiation produces ion-ization indirectly in matter by the photoelectric effect,the Compton effect and pair production. All three pro-cesses yield electrons which in turn produce most ofthe ionization which occurs within the body.2
In the photoelectric effect, the γ ray transfers all itsenergy to the electron that it strikes. The electron thencauses ionization in the medium. In the Compton ef-fect, only part of the γ ray’s energy is transferred tothe electron as kinetic energy. The remaining energygives rise to a lower energy γ ray. Pair production oc-curs when the γ ray has an energy greater than 1.02MeV. All of its energy is given up and two particles,an electron and a positron, are produced. (All threeprocesses produce electrons which then ionize theabsorbing matter.)
Although ionization causes biological damage, theseriousness of the damage is determined by manyfactors such as the types of cells effected (how radi-osensitive they are), the age of the person receivingthe exposure (young cells are more radiosensitive) andwhether the dose is received over a short (acute expo-sure) or long (chronic exposure) period of time.
Acute exposure is more detrimental because a largeamount of damage is incurred rapidly and the cellsdo not have an adequate amount of time to repairthemselves. Under chronic exposure, cells are veryeffective in repairing injury. The effects of chronicexposure and the amount required to produce dam-age have been extrapolated primarily from knowncases of acute radiation exposure. More recent radio-biology laboratory and other studies have developeddata which assess the effects of chronic exposure. Asthe nuclear industry continues, the database for lowlevel occupational exposure grows and it becomes moreevident that the associated risks are low. The dataavailable suggest that the body can tolerate low dosesof radiation received over long periods of time withlittle risk. This is supported by the exposure andhealth history of x-ray technicians, physicians andradiation workers. In many laboratory and historicexposure studies, beneficial effects of low level radia-tion are demonstrated. These studies certainly counterthe argument that no level is low enough and thatallowable levels must be endlessly reduced in the pur-suit of safety.
Radiation protectionAlpha particles, due to their large mass and double
positive charge, travel very short distances. Theirrange in air is only a few centimeters. Fig. 3 illustratesthe penetration ranges of various types of radiation.Alpha particles are seldom harmful when the sourceis located outside the human body because an α par-ticle of the highest energy will barely penetrate theouter layer of skin. If emitted inside the body, how-ever, α radiation can be serious. It is important, there-Fig. 2 Binding energy per nucleon versus mass number.
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fore, to prevent the ingestion or inhalation of α emit-ting nuclides. Particular care is required to keep theair in working spaces free of dust containing α emit-ters. Fabrication of α emitters such as plutonium nor-mally takes place inside gloveboxes that remain un-der a slight negative pressure. The boxes dischargeair effluent through a filter designed to prevent αbearing dust from entering the working spaces andthe atmosphere in general. The air is continuouslymonitored and analyzed.
Beta particles penetrate up to an inch of wood orplastic material and travel several yards in air. Theskin and the lenses of the eye are most vulnerable toexternal β radiation. However, clothing and safetyglasses provide adequate protection for external ex-posure to β radiation. The β particle is not as great aninternal hazard as the α particle. The β particle, dueto its smaller mass and lower charge, will travel far-ther than the α particle through tissue and will de-posit less energy in a localized area.
Gamma rays penetrate deeply and deposit theirenergy throughout the entire body. They have greatranges in air and may present a hazard at large dis-tances from the source; however, they do not presentas large an internal hazard as α particles.
Neutrons, like gamma radiation, are an externalhazard and their damage extends throughout thebody. In addition, the effectiveness of neutrons to pro-duce biological damage is 2.5 to 10 times greater than γrays. The concrete and water shielding provide the pri-mary protection against neutrons and gamma rays.
Decay rateThe rate at which a radioactive nuclide emits ra-
diation is a characteristic of the nuclide and is unaf-fected by temperature, pressure or the presence ofother elements that may dilute the radioactive sub-stance. Each nucleus of a specific radioactive nuclidehas the same probability of decaying in a definiteperiod of time at a rate characterized by its radioac-tive decay constant. The rate of decay at any time, t,always remains proportional to the number of radio-active atoms existing at that particular instant. Thedecay is calculated by:
N t N e t( ) = ( ) −0 λ (3)
where
N(t) = the number of atoms per cm3 at time tN(0) = the number of atoms per cm3 at time zeroλ = radioactive decay constant
Half lifeDecay is usually expressed in terms of a unit of time
called radioactive half life, T1/2. This represents a measur-able period of time, the period it takes for a quantity of radio-active material to decay to one half of its original amount.The relationship between half life and decay constant canbe determined by substituting (T1/2) for t and N(0)/2 forN(t) in Equation 3 and solving for λ. The result is:
λ = 0 693
1 2
.
/T(4)
Decay constants for radioactivity isotopes are eas-ily obtainable with a listing of measured half lives.3
If N(t) represents the number of radioactive atomspresent at time t, then λN(t) becomes the number ofradioactive nuclei that decay per unit of time at timet. This is referred to as the radioactivity, or more sim-ply the activity, of the atoms and is expressed in cu-ries. A curie is 3.70 × 1010 disintegrations per second.N(t) can be converted directly to curies by the rela-tion N(t)/(3.70 × 1010). Fig. 4 shows on a relative scalehow the activity decreases during several half lives.This curve applies to all radioactive substances.
It is sometimes difficult to distinguish betweenstable and radioactive nuclides. All nuclides heavierthan bismuth-209 (Bi-209) are unstable. However, thespecific nuclides thorium-232 (Th-232), uranium-235(U-235) and U-238 have half lives of 108 to 1010 yearsand can be considered stable. In fact, they are gener-ally referred to as the stable isotopes of the heavierchemical elements. Table 2 shows half lives of somenuclides of high atomic number.
Induced nuclear reactionsBy providing sufficient energy, all of the fundamen-
tal particles can be made to react with various nuclei.In this procedure, the particle strikes or enters anatomic nucleus causing a transfiguration or changein structure of the nucleus and the release of a quan-tity of energy. Particles which activate these reactions in-clude neutrons, deuterons, α, β, protons, electrons and γ.Many reactions produce artificial radioactive nuclides.
Charged particles such as alphas, betas and pro-tons do not have great penetration power in matterbecause the interaction with the existing electricalfield within the atoms either slows or stops them. Col-lisions between like charged particles require exceed-ingly high kinetic energy. Electrical fields, however,can not deflect electrically neutral neutrons; therefore,they collide with nuclei of the material on a statisticalbasis. The neutron is therefore the most effective par-ticle for inducing nuclear reactions including fission.
Reactions of principal interest in the design ofnuclear reactors are those that involve neutrons andthose that involve the interaction of the particles pro-Fig. 3 Ranges of various types of radiation in materials.
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duced by fission and other nuclear processes withinthe reactor and surrounding materials.
The expression:
ZA
Z
AX P P X11
1 2 2
2( ) (5)
is generally used to denote a nuclear reaction and isthe short form for:
ZA
ZA
ZAX P X X P1
11 2
22+ → → +*
The X* notation indicates formation of a compoundnucleus which is generally unstable. Terms P1 and P2
represent incident and resultant particles respectively.The individual sum of the Zs (atomic numbers) andthe sum of the As (mass numbers) on the left side ofthe equation always equals, respectively, the sum ofthe Zs and the sum of the As on the right side of theequation. (See Table 3).
Nuclear reactions always conserve total mass-energyon the two sides of the equation in accordance withEinstein’s equation for the equivalence of mass and en-ergy. However, there is usually an energy difference and
a mass difference. A good statistical probability for thereaction to occur exists when energy is released as a resultof reaction. The more probable reactions generally canbe initiated by lower energy particles. The less probableones, those that require significant energy addition, canbe initiated only by high energy particles. Table 3 givessome typical induced nuclear reactions.
In a nuclear reactor, many heavy nuclides withmass numbers greater than 238 are produced by aseries of neutron captures accompanied by β and/or αdecay. These artificially produced nuclides have manyuses – some are readily fissionable, some are fertilebecause they absorb a neutron and then transmuteto a fissionable nuclide, some have high energy α de-cay modes that can be used in neutron sources, andsome are used as sources in medical procedures.
Probability of nuclear reactions –cross-sections
The probability of a nuclear reaction between a neu-tron, or other fundamental particle, and a particularnuclide is expressed as the cross-section for that reac-tion. The probability is dependent on the energy of theinteracting particle. Because neutron interactions aremost important in a nuclear reactor, the following dis-cussion is directed toward neutron cross-sections; how-ever, the concepts can be directly applied to all par-ticle interactions.
Two types of nuclear cross-sections are defined:
1. The microscopic cross-section or interaction probabilityis an intrinsic characteristic of the nuclei of the mate-rial. It has dimensions of an area and is normally ex-pressed in barns where one barn equals 10-24 cm2.
Table 2Half Lives of Heavy Elements
Decay Mode Half Life
Naturally occurring nuclides: Thorium-232 α 1.39 x 1010 yr Uranium-238 α 4.51 x 109 yr Uranium-235 α 7.13 x 108 yr
Artificial nuclides: Thorium-233 β 22.1 min Protactinium-233 β 27.4 d Uranium-233 α 1.62 x 105 yr Uranium-239 β 23.5 min Neptunium-239 β 2.35 d Plutonium-239 α 2.44 x 104 yr
* Indicates formation of a compound nucleus which is generally unstable.
Fig. 4 Exponential decay of radioactive nuclides.
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2. The macroscopic cross-section is a probability ofinteraction per centimeter of neutron path andtakes into account the density of the material. Ithas the dimensions cm-1.
The symbol σ represents the microscopic cross-sec-tion. In the case of a neutron approaching a fission-able atom it becomes possible to consider a total cross-section σT in which:
σ σ σ σ σ σT c s f a s= + + = + (6)
where
σc = the capture cross-section, a measure of theprobability for absorption without fission
σs = the scattering cross-section, the probabilitythat the nucleus will scatter the neutron
σf = the fission cross-section (present in only a fewof the many nuclei), the probability for a neu-tron to strike and cause a fission to occur
σa = the absorption cross-section, the sum of theprobabilities for capture and fission
The macroscopic cross-section can be obtained from:
Experimental measurements determine microscopiccross-sections for each element. Total cross-sectionmeasurements are generally made by transmissiontechniques. For example, placing a material of knowndensity and thickness in front of a neutron source per-mits measuring the intensity of neutrons at a particu-lar energy on each side of the material. The differencerepresents the loss or attenuation of neutrons by thematerial. The cross-section required to obtain this at-tenuation is derived by calculation.
Isotopes of the same element can have very differ-ent cross-sections. For example, the isotope xenon-134(Xe-134) has a microscopic cross-section for neutronabsorption of about 0.2 barn, yet Xe-135 has a micro-scopic cross-section of 2.7 × 106 barns. U-235 is fission-able at any energy, and U-238 only at high energy.
Cross-sections of some nuclides also contain abruptpeaks called resonances at certain energy bands (Fig.5). Because a cross-section is really a function of rela-tive energy of the neutron and the nucleus, an effec-tive change in the cross-section results when the en-ergies of the neutron and the nucleus increase or de-crease as a result of temperature changes.
Where the curve of cross-section versus energy isfairly smooth, the effect of a temperature change ofthe target nucleus is relatively small. However, theeffect of a temperature change is large and importantin the vicinity of a resonance. An increase in tempera-ture results in increased vibration of the nucleus with
a corresponding increase in the number of probablecollisions between the nucleus and neutron occurringat energies in the vicinity of the resonance. Therefore,an increase in the temperature of the nucleus resultsin an apparent broadening of the energy width of theresonance. This in turn results in a very effective in-crease in resonance neutron absorption, i.e., captureand fission. Conversely, a decrease in temperature ofthe nucleus results in narrowing the resonance widthand decreasing resonance absorption. The change inresonance energy width with temperature, known asthe Doppler effect, is important in reactor control.
Fig. 5 illustrates a typical cross-section curve show-ing the neutron capture cross-section of U-238 as afunction of neutron energy. Below 102 eV the heightof the resonance peaks is about 103 barns. Between102 and 4 × 103 eV the great number of resonancesmakes it impractical to show the cross-section as acurve, although the resonance parameter data areavailable for use on a computer. Between 4 × 103 and105 eV the curve represents a statistical average of themeasurements which have been made.
The fission processNuclear fission is the splitting of a nucleus into two
or more separate nuclei accompanied by release of alarge amount of energy. In the fission of an atom dueto a neutron, the mass of the neutron plus its energymust be equal to or greater than the mass defect as-sociated with the two fission products. U-235 is theonly naturally occurring nuclide that is capable of un-dergoing fission by interaction with low energy or slowneutrons. Some of the artificially produced heavy nu-clides are also fissionable with slow neutrons, includ-ing plutonium-239 (Pu-239) and U-233. Other nu-clides such as U-238 and Pu-242 require higher en-ergy neutrons to cause fission.
This difference in fission capability occurs becausethe binding energy of a nucleus is not only determinedby its mass number but also by whether the numberof protons and neutrons is even or odd. A nuclide withan even number of neutrons and protons, such as U-238, has the highest binding energy per nucleon andrequires the most added energy to fission. Fission onlyoccurs with high energy or fast neutrons (> 1MeV). Anuclide with an odd number of protons and an evennumber of neutrons or an even number of protons andan odd number of neutrons, such as U-235, Pu-239and U-233, has a lower binding energy per nucleon.Nucleons with both an odd number of protons andneutrons have the lowest binding energy per nucleonfor essentially equivalent mass numbers. U-235 andPu-239 are fissionable with slow or thermal neutrons(~0.025 eV). The term thermal neutron refers to a neu-tron energy distribution that is in thermal equilibriumwith the temperature of the surrounding materials.
Fission occurs when the fissionable nucleus absorbsa neutron. In the case of U-235 the reaction is:
92235
01
92236
92236
11
22
012 43
U n U
U X Y nZA
ZA
+ →
→ + + +
*
* . Energy
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As previously noted, the asterisk in 92U236* indicatesan unstable nuclide. The value 2.43 applies to U-235fission by a thermal neutron and is the statistical av-erage of the number of neutrons produced per reac-tion. X and Y represent the fission products which aredistributed as shown in Fig. 6.
Energy from fissionFission also produces γ rays, neutrons, β particles
and other particles. The energy release per fissionamounts to about 204 MeV for U-235 and is distrib-uted as shown in Table 4. Release of approximatelythis amount of energy per fission can be predicted byexamination of Fig. 2 for binding energy, consideringthat two fission products are formed as shown in Fig.6. Table 4 also includes data for Pu-239.
Neutrons from fissionThe fact that additional neutrons are born or gen-
erated by a fission event makes it possible to estab-lish a chain reaction, as depicted in Fig. 7, that cansustain itself as long as sufficient fissionable materialand neutrons are present. The neutrons produced perfission event, υ, and the average number of neutronsproduced per neutron absorbed in the fuel, η, varywith the different fissionable isotopes and with theenergy of the neutron producing the fission. Statisti-cal averages for these quantities are given in Table 5.There are individual fission reactions that produce onlyone neutron, possibly none, or as many as five. Be-cause some of the neutrons are absorbed without pro-ducing fission, η becomes a more meaningful quan-tity in reactor design than υ. The values in the tableare for fission by low energy or thermal neutrons atroom temperature [0.025 eV neutron energy or 7218ft/s (220 m/s) neutron velocity]. To maintain a chainreaction, the average number of neutrons per absorp-tion must be significantly greater than 1 because someof the neutrons will be lost to absorption in the mod-erator and structural materials, to absorption in con-trol materials, and to leakage from the core. Controlmaterials such as boron and silver-indium-cadmiumare used to maintain a steady-state chain reaction by
absorbing any additional neutrons over that neededfor steady-state operation.
Fission neutron energy distributionNeutrons released from fission vary in initial energy
over a wide range up to 15 MeV and above. The distri-bution of neutrons produced by fission as a function ofenergy has been determined from several experimentalmeasurements, and typical results for U-235 and Pu-239are shown in Fig. 8. Based on these measurements, theaverage energy of fission neutrons is about 2 MeV; thepeak of the energy distribution occurs at about 0.8 MeV.
Fig. 6 Mass distribution of fission products from fission of U-235.
Fig. 5 Capture cross-section of uranium-238.
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BurnupAs a mass of uranium undergoes fission it produces
energy. The term burnup is used to represent theamount of energy produced per unit mass of the ma-terial. The units of burnup are megawatt days permetric ton (MWd/tm) of initial heavy metal, i.e., ura-nium. One megawatt day represents about 2.6 × 1021
fissions. A nuclear fuel assembly is typically dischargedfrom the reactor when it has achieved a burnup ofabout 50,000 MWd/tm. In commercial power reactorsthat are fueled with uranium comprising mainly theisotope U-238, with 4% or less U-235, most of the en-ergy produced comes from the fissioning of U-235. Atburnups of 50,000 MWd/tm only about 5% of the ini-tial uranium content of the fuel assembly has fis-sioned. Even when the other nuclear reactions thatcause loss of the original uranium atoms are considered,there is still more than 90% of the initial uranium re-maining in the fuel assembly when it is discharged.
Fission productsWhen a nucleus undergoes fission, experimental
measurements have shown that predominantly two fis-sion products or fragments are generated. The distribu-tion of these fission products for U-235 fissions has alsobeen measured and is shown in Fig. 6. In fission, as inany nuclear reaction, there is always conservation of
total mass-energy so that one of the two fission productswill come from each hump of the distribution.
Examination of Fig. 6 reveals that mass number ofthe fission products ranges from about 70 to 170 withtwo plateaus at approximately 95 and 135. Curves areshown for fission caused by thermal energy neutronsand by 14 MeV neutrons. The higher the neutron en-ergy causing fission the more uniform the fission prod-uct distribution.
Many fission products interact with neutrons, ab-sorbing them so that they are not available to thechain reaction. As these fission products build up (seeFig. 9), they act as absorbers to retard the chain reac-tion. All long lived fission products except samariumbuild up as the core is operated, reaching a maximumeffect at the end of core life.
Table 5Neutrons from Fission
Avg. Neutrons per Avg. Neutrons per Fuel Fission, υ Absorption in Fuel, η
Instantaneous Kinetic energy of fission products 169 175 γ ray energy 8 8 Kinetic energy of fission neutrons 5 6 182 189
Delayed β particles from fission products 8 8 γ rays from fission products 7 6
Neutron-capture γs* 7 10 22 24
Total 204 213
* Energy produced depends on reactor composition.
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After discharge from the reactor and sitting in stor-age for a few years, most of the fission products willhave decayed due to their relative short half lives.However, some of the long half life fission products willcontribute significantly to the γ ray source that mustbe shielded against in handling the fuel during its ul-timate disposal. These isotopes include strontium-90(Sr-90), ruthenium-106 (Ru-106), cesium-134 (Cs-134) and 137 (Cs-137), cerium-144 (Ce-144) and eu-ropium-154 (Eu-154). These isotopes will either bebound in the fuel pellets or on the inside surface ofthe cladding. Other isotopes important to fuel handlingare the artificially produced nuclides in the fuel pel-let that decay by spontaneously fissioning. These iso-topes include Pu-238, americium-241 (Am-241), andcurium-242 (Cm-242) and 244 (Cm-244). In addition,the γ emitting isotope cobalt-60 (Co-60) is produced inthe structural steels and Inconel® through neutron ac-tivation and must be shielded against.
Fission product behavior with timeCertain fission products, specifically Xe-135 and
samarium-149 (Sm-149), both of which have veryhigh cross-sections for absorption of thermal neutrons,are not only produced directly from fission but also arethe decay products of other fission products.
Essentially all initial products of fission are highlyradioactive and decay rapidly to less active isotopeswith somewhat longer half lives. There are usuallyseveral isotopes in the chain before a stable end prod-uct is reached. The most significant such decay chainis the following:
β β β β
52135
53135
54135
55135
56135
1 0 6 7 9 2
Te I Xe Cs Ba→ → → →
< . . .min. h h 22 106× yr
In this chain, tellurium-135 (Te-135) with a oneminute half life decays to iodine-135 (I-135) with a6.7 hour half life and then to Xe-135. This xenon iso-tope, which fortunately has only a 9.2 hour half life,has a macroscopic cross-section for thermal neutronabsorption approximately 100,000 cm-1, as great as allthe long lived fission products together. Unfortunately,the nuclides of this chain occur abundantly as fissionproducts – a predictable happening, because the massnumber 135 occurs at a peak in the fission product dis-tribution (Fig. 6). The Xe-135 absorbs an appreciable
fraction of available neutrons as long as the reactor isoperating. This changes some of the Xe-135 to Xe-136(which has negligible neutron absorption). As a result,when a water reactor operates at constant power level,the Xe-135 builds up to its equilibrium value in 36 to 48hours.
When reactor power lessens and, particularly, whenthe reactor is shut down, the I-135 formed at the origi-nal power level continues for a time to generate Xe-135 at a rate corresponding to the original power level.Therefore, the Xe-135 builds up rapidly after shut-down because fewer neutrons are available for con-version to Xe-136. The buildup reaches a peak 4 to12 hours after shutdown and then slowly decays. Thetime behavior of Xe-135 must be addressed in reactorcontrol.
Sm-149, the second important fission product in thereactor core during operation, is generated as follows:
60149
61149
62149Nd Pm Sm→ → (stable)
In the chain, neodymium-149 (Nd-149) decays (1.7hour half life) into promethium-149 (Pm-149), whichin turn decays (47 hour half life) into Sm-149. Al-though Sm-149 is a stable isotope it is destroyed sorapidly by neutron absorption that it reaches an equi-librium value when the reactor operates at constantpower. Typically, Sm-149 reaches equilibrium in apressurized water reactor after 50 to 100 days.Buildup after shutdown is slower and less extensivethan for Xe-135.
Decay after shutdown has little consequence for allother fission fragments because of their long lives andcomparable capture cross-sections between parent anddaughter nuclide.
Nuclear reactor compositionA reactor core is composed of fuel, structural and
control materials, and a moderator and/or coolant.Typical pressurized water reactor (PWR) fuel assem-bly and control component designs are shown in Fig.10. A depiction of the interactions that take placewithin a fuel assembly is shown in Fig. 11. Neutronsreact not only with uranium, but also with the nucleiof most other elements present. Therefore, many ma-terials which have high neutron absorption proper-ties should not be used for structural purposes. For-tunately, a few materials such as aluminum and zir-conium have low neutron absorption cross-sections.Inconel, steel and stainless steel also can be used to alimited extent.
The choice of coolants includes water, helium andliquid sodium. Thermal reactor design, where the neu-tron energy distribution is essentially in thermal equi-librium with its surroundings, requires a moderator(a material containing atoms of a light element suchas hydrogen, deuterium or carbon) in the core to re-duce the kinetic energy of the neutrons. Hydrogen isthe ideal moderator because its nucleus is as light asa neutron and, therefore, it can absorb the energy ofthe neutron in a single direct collision. Hydrogen isnormally present in the form of water. The heavier theFig. 9 Fission product chain.
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atom the less energy it absorbs per collision, and theless slowing effect on the neutron. Carbon, in the formof graphite, is about the heaviest atom that can beused practically as a moderator. Carbon has the addi-tional advantage of absorbing very few neutrons, evenless than hydrogen. Other low weight atoms can beused, but they are less practical because of excessneutron absorption or high material cost. Because thehelium nucleus absorbs essentially no neutrons, itwould be an ideal moderator except that helium is agas at reactor operating temperatures. In this state itwould be impossible to provide the core with a suffi-cient number of atoms to be an effective moderator.
All thermal reactors contain some fast (high energy)neutrons because only fast neutrons result from fission.The amount of moderator provided determines the de-gree of thermalization (energy reduction) or the per-centage of thermal neutrons present in the reactor. Per-haps the greatest advantage of the thermal reactor isits compatibility with the water coolant. Water hasproven to be the most practical and economical coolant.
The low neutrons per fission available with natu-ral uranium makes it difficult to design a natural ura-nium (0.71% U-235) reactor. However, it can be ac-complished if all materials are of especially high pu-rity and if discrete sections of fuel are placed in a het-erogeneous array such that the neutrons can slowdown in the moderator and then re-enter the fuelmaterial as slow neutrons. Under these conditions,there is a relative high probability of causing fission.The normal moderators used in natural uranium re-
actors are either graphite or deuterium oxide (heavywater). The absorption cross-section of normal hydro-gen makes it less desirable than the other moderators.Also, minute quantities of any high neutron absorb-ers can prevent a chain reaction. Nevertheless, allearly reactors used natural uranium and many stilloperate, notably the large plutonium producing reac-tors and the Canadian CANDU pressurized heavywater reactors. (See Chapter 46.)
Physical description of nuclearchain reactions4-7
To use fission as a continuous process for power pro-duction, it is necessary to initiate and maintain a fis-sion chain reaction at a controlled rate or level whichcan be varied with power demand. Obtaining a steady-state chain reaction requires the availability of morethan one neutron for each fission produced becausenon-fission reactions absorb some neutrons and oth-ers leak from the reactor. For a reactor as a whole, theneutrons born in each instant of time constitute onegeneration of neutrons. The term effective multipli-cation factor, keff, is defined as the ratio of the num-ber of neutrons in one generation to that in the pre-vious generation. With a multiplication factor of lessthan one, the system is a decaying one and will neverbe self-sustaining. With a multiplication factor greaterthan one, a nuclear system produces more neutronsthan it uses, and power increases. For a steady-statechain reaction, keff = 1.
The steady-state equation for neutron balance ina chain reaction can also be written as:
Production Absorption + Leakage= (8)
When this condition is obtained, the reactor hasgone critical meaning that the necessary amount ofneutron production has been achieved to balance theleakage and the absorption of neutrons. The mass offissionable material required to achieve this conditionis called the critical mass.
Production, absorption and leakage all dependheavily on interactions of neutrons with nuclei of thevarious materials in the reactor. Production dependsFig. 10 Typical PWR fuel assembly and control components.
primarily on those interactions with U-235 nucleiwhich result in fission. Absorption depends on inter-actions of neutrons with any nuclei in the core thatresult in absorption of neutrons with or without fis-sion. Leakage depends on the scattering effect of col-lisions between neutrons, nuclei and other particles,which results in transport of neutrons toward theboundaries of the chain reacting system and ultimateescape from the system.
Table 5 indicates that each fission of a U-235 atomby a thermal neutron makes available an average of2.43 neutrons. However, only 2.07 neutrons are pro-duced per thermal neutron absorbed in U-235 as alsoindicated in Table 5. The other 0.36 neutrons are ab-sorbed in non-fission reactions. If natural uranium isused, the number of neutrons produced per thermalneutron absorbed is reduced to 1.34 because of theneutron absorption in U-238. Most of this absorptionresults in the ultimate production of fissionable Pu-239. However, neutrons used in this manner are nolonger available to help maintain the chain reaction.
A neutron chain reaction is possible because eachfission event on the average produces more than oneneutron. If the process is controlled such that just oneof the neutrons produced from fission causes anotherfission reaction then a steady state process is main-tained and a constant power level is achieved. To bestunderstand the process, a general description of thelifetime of a neutron follows. It must be realized thatfor a power reactor the number of neutrons crossinga 1 cm2 surface (neutron flux) near the center of thereactor is on the order of 1013 neutrons/cm2/s. The life-time of a neutron is on the order of 10-9 seconds. There-fore, this description is only representative of whathappens in an instant of time.
As described previously, a neutron produced by thefission process has an average energy of about 2 MeVand is termed a fast neutron. At this energy, elastic orinelastic scattering reaction with the moderator, thefuel or with the structural materials are most prob-able. However, about 2% of the fast neutrons cause afission in U-238 and produce an even higher numberof neutrons than produced from thermal fission.
In an elastic scattering reaction some of the neu-tron energy is transferred to the nucleus with whichit collides in the form of kinetic energy; this can causethe nucleus to be displaced from its normal lattice po-sition in the material. In an inelastic collision a com-pound nucleus is formed in an excited state and itreduces its energy by releasing a lower energy neu-tron and a γ ray. Some of the original neutron’s en-ergy is also deposited as the kinetic energy on thenucleus. The net impact of the scattering reactions isto cause the neutron to lose energy or slow down intothe energy range where resonance interactions withvarious materials (especially U-238) can take place.In the resonance range, neutrons will have a highprobability of being captured if their energy coincideswith any of the resonances associated with the ura-nium fuel or the fission products and will not be avail-able to react with the U-235 to help sustain the chainreaction. The resonance escape probability is the frac-tion of neutrons that escape capture while slowing
down to the thermal energy range. The term resonanceescape refers particularly to U-238 because in the in-termediate range of neutron energy there are severalresonance peaks of absorption cross-section for this iso-tope. These resonances are useful in the production ofplutonium; however, to maintain criticality in a ther-mal reactor, sufficient neutrons must escape absorptionin the resonance region.
In contrast, collisions with hydrogen atoms in themoderator can slow down the neutron very rapidly,and therefore the resonance escape probability is highfor water moderated systems. Once past the resonancerange the neutron is said to become thermalized andbecomes part of a Maxwellian energy distribution withan average energy of about 0.2 eV. As the fission cross-sections for U-235 and Pu-239 are the highest in thethermal energy range, most fission reactions takeplace in this range.
The probability of having a fission reaction is re-ferred to as the thermal utilization and is the ratio ofthe fission cross-section to the absorption cross-section(capture plus fission) averaged over the moderatorplus the fuel and structural materials.
The other mechanism for loss of neutrons is leak-age from the system. This is dependent on the size ofthe system and the length of the path neutrons travelbetween source and capture. For a thermal reactor thislength is approximately 2.95 in. (75 mm). Today’spower reactors are on the order of 13.1 ft (4 m) in di-ameter and therefore only neutrons born in the fuelassemblies on the periphery of the core have a signifi-cant probability of being lost from the system.
Control of the chain reactionMore than 99% of neutrons produced in fission are
prompt neutrons; that is, they are produced almostinstantaneously. About 0.73%, in the case of U-235,are released by the decay of fission products ratherthan directly from fission. The average half life forthese delayed neutrons from fissioning of U-235 isabout 13 seconds. This provides time for the reactoroperator to respond to small changes in either powerdemand by the system or reactor system parameters.
The chain reaction can be regulated by placing ma-terials with high neutron absorption capability in thereactor and providing a means of varying theiramounts. These materials tend to stop the chain reac-tion and include boron, cadmium-silver-indium, haf-nium and gadolinium. One or more of these materialsusually goes into the reactor in the form of control rodsthat can be withdrawn to start up the reactor or to increasepower level and reinserted to reduce power or to shut downthe unit. To assure accurate control of power, at least someof the rods must be capable of fine regulation.
ReactivityThe first step in establishing a chain reaction is to
bring the reactor to critical condition at essentially nothermal power or zero power level. The reactor poweris gradually increased up to the desired level by theremoval of a control material and maintained there.The objective in each step of this procedure is to ob-tain and hold a constant value of keff = 1.0. In view of
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the changes that continually occur in a nuclear sys-tem, it is never possible to keep keff = 1.0 for more thana short period of time without adjustments to compen-sate for variations. Operating a reactor at any con-stant power level at an effective multiplication factorof unity corresponds to steering a ship on a compasscourse. It takes a continual effort, either automatic ormanual, to hold the ship on the exact course.
If a reactor operates at a specific power level withkeff = 1.0, and if anything changes to increase or de-crease the multiplication factor, a reactivity change issaid to have occurred. It may be of positive or nega-tive change depending upon the direction of changein keff. Reactivity, represented by the symbol ρ, is de-fined as the ratio:
ρ = −( )k keff eff1 / (9)
Reactivity has been given units of dollars ($) andcents (¢) or inhours. It is usually expressed in theunits of pcm (per cent milli) which corresponds to a ρof 1 × 10-5. As will be shown, most processes in the re-actor, except for control rod insertion, generate reac-tivities of 1 to 10 pcm which constitute a very smallchange from a keff of 1.
Calculation of reactor physicsparameters4-7
The calculation of a chain reaction (or the designof a nuclear reactor) requires solution of the steady-state equation:
Production = Absorption + Leakage
or its counterpart for nonsteady-state (transient) con-ditions:
Production Absorption Leakage− − = dn dt/ (10)
where dn/dt is the variation of the neutron densitywith time.
The production rate is evaluated as:
Production = υ Σf nv (11)
where
υ = neutrons per fissionΣf = macroscopic fission cross-sections, cm-1
n = neutron density, neutrons/cm3
v = neutron velocity, cm/s
If the neutron flux is defined as the product of the neu-tron density and neutron velocity, then the produc-tion rate can be defined as:
Production = υ φΣf (12)
where
φ = nv = neutron flux
Similarly, the absorption rate is defined as:
Absorption = Σa φ (13)
where
Σa = macroscopic absorption cross-section, cm-1
Leakage is a complicated function of the gradientsin the neutron flux at boundaries of the region underconsideration.
In performing reactor calculations, the core is di-vided into discrete spatial regions or nodes, and thecontinuous neutron energy range is divided into anumber of groups. The above equations then becomea coupled set of partial differential equations that aresolved in both the space and energy domains to findthe neutron flux and reaction rates in each node.
In the design of the early reactors, most calculationsof reactor physics parameters were done by hand us-ing homogeneous models of the core components. Neu-tron cross-sections were obtained from experimentalplots of total, absorption, scattering and fission cross-sections. Critical experiments were performed mock-ing up the fuel designs and modeling both the fuel andcontrol rods. From these experiments, modeling ad-justments were developed to ensure high accuracy pre-dictions of reactor performance parameters. As withmost other disciplines, the advent of more powerfulcomputers has permitted the designer to perform de-tailed calculations in both space and energy. The fol-lowing sections describe some techniques that havebeen used in calculations for commercial PWR fuelcycle design and licensing.
Cross-sectionsBasic neutron and gamma cross-sections are com-
piled and verified as part of the industry effort to main-tain the cross-section libraries. These libraries providethe basic data needed to calculate the cross-section ofeach isotope as a function of incident neutron energy.Typically two major energy groups, each containingsubgroups, are used by the designer. The first, the fastand epithermal group, spans energies from 1.85 eVto 20 MeV and is composed of 40 or more subgroups.The second, the thermal energy group, covers ener-gies from 0.00001 eV to 1.85 eV and is composed of50 or more subgroups. In the fast and epithermal en-ergy range, only scattering reactions that decreaseneutron energy are used in the calculations. In thethermal energy range, the neutron energies are inthermal equilibrium with the moderator, and energycan be imparted to the neutron during scattering colli-sions. Therefore, scattering reactions, which can in-crease or decrease neutron energy, are permitted in thecalculations for this group.
When designing a reactor core, specific cross-sec-tions are calculated using the total multigroup cross-section set (40 epithermal plus 50 thermal groups) foreach material present. Each cell type in the fuel as-sembly can then be modeled (Fig. 12). This calcula-tion determines the neutron flux distribution in eachenergy group and cell. The resulting neutron flux isthen used to average the cross-sections over the twomajor energy groups and over various regions of thefuel assembly. A full core representation, indicating theposition of fuel assemblies and control components, isshown in Fig. 13.
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The important quantities that describe the reactorcore and its performance include critical boron concen-tration, core power generation distribution, control rodreactivity worths, and the reactivity coefficients asso-ciated with changes in reactor conditions.
Critical boron concentrationThere is sufficient U-235 in a commercial power
reactor to power the unit for 12 to 24 months beforerefueling. Without additional absorbers present in thesystem, the keff would be greater than 1. It is imprac-tical to provide the absorption using control rods.Therefore, for long-term control keff = 1 is maintainedby soluble boron in the moderator (boron dissolved inwater) and/or burnable absorbers. Burnable absorb-ers contain a limited concentration of absorber atomssuch that, as neutrons are absorbed, the effectivenessof absorption decreases and essentially burns out withtime. Burnable absorbers may be in the fuel rods orin fixed absorber rods that are placed in fuel assem-bly guide tubes. The boron concentration in the cool-ant can be near 1800 ppm at the beginning of a fuelcycle. This concentration can decrease to near zero atthe end of the cycle, when a significant portion of the
uranium is depleted and fission products have builtup. The critical boron concentration is the boron levelrequired to maintain steady-state reactor power lev-els. The burnable absorber limits the amount ofsoluble boron that is used at the beginning of the cycle.
Power distributionThe primary factor governing acceptable reactor
operation is the energy production rate at every pointin each fuel rod. If the rate is too high, the fuel canmelt and cause rod failure. Excessive energy rates canalso cause coolant steaming, which results in poor heattransfer and can lead to cladding burnout.
The energy production rate, or power, at each fuelrod location is referred to as the core power distribu-tion. This has typically been calculated by forming athree-dimensional model of the reactor and represent-ing each fuel assembly as a homogenized region overan axial interval. A more recent method for power cal-culations involves reconstructing the power produc-tion in each fuel rod based on detailed fuel assemblycalculations. This method is more accurate because itprovides a better fuel assembly model that can includewater, burnable absorbers or guide tube control rods.With proper fuel assembly placement, the core powerdistribution can be optimized to limit power peaks infuel rod segments. As part of this primary analysis,initial enrichments of the fuel and burnable absorb-ers can be determined, as well as the necessary solubleboron concentration in the coolant over the cycle.
Control rod worthsControl rods are used in a pressurized water reac-
tor to change power levels and to shut down the reac-tor. Approximately 48 rods, divided into banks of 8,are commonly used and core reactivity is changed bysequentially moving the banks. The control rod banksare grouped into shutdown and control (or regulat-ing) banks. The shutdown banks provide negative re-activity to bring the reactor from hot to cold conditions.During a shutdown, the core temperature decreasesFig. 12 Cross-sectional view of fuel assembly.
Fig. 13 Cross-sectional computer modeling of PWR core.
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and its reactivity increases. The shutdown control rodbanks counteract this increase. The controlling banksare used in conjunction with variations in the solubleboron concentration to take the reactor from hot zeropower to full power. These banks also provide the reac-tivity to handle rapid power changes. The negative reac-tivity caused by the buildup of xenon in the core follow-ing a shutdown is offset by a combination of control rodremoval and decreases in the coolant boron concentration.
The reactivity worth of a control rod is a measureof its ability to reduce core reactivity. Control rodworths are calculated by first calculating the core withthe control rod inserted to determine its keff. The coreis then similarly calculated with the rod withdrawn.Control rod worth is defined as:
k k
k keff i eff w
eff w eff i
( ) ( )
( ) ( )
−(14)
where
keff(w) = multiplication factor, control rod withdrawnkeff(i) = multiplication factor, control rod inserted
Bank worths are similarly calculated.
Reactivity coefficientsReactivity coefficients define the rate of core reac-
tivity change associated with the rate of change inreactor conditions. These coefficients indicate the rela-tive sensitivity of the core operation to changes in op-erating parameters.
In determining a coefficient, the core is first modeledwith a given power and/or temperature distribution,and keff is calculated. The reactor is then similarly mod-eled at a different power or temperature distribution.The coefficient is defined as the reactivity change perunit of power or temperature. Fig. 14 shows the impactof various core parameter changes on core reactivity.
There are three basic reactivity coefficients that areimportant to reactor operation: 1) the moderator tem-perature, 2) Doppler, and 3) power coefficients. Themoderator temperature coefficient is defined as thechange in reactivity associated with a change in mod-erator temperature. It includes the effects of modera-tor density changes and the changes in nuclear cross-sections. The Doppler coefficient is defined as thechange in reactivity associated with changes in fueltemperature that occurs primarily because of fuelnuclides with large resonances. The power coefficientis defined as the change in reactivity associated witha change in power level. It is a combination of themoderator and Doppler coefficients and is based on thechange in temperature as power changes.
When the coolant temperature increases, the asso-ciated density reduction normally decreases its mod-erating capability. A net decline in core reactivity, i.e.,a negative moderator temperature coefficient, occurs.Similarly, when the fuel temperature increases, corereactivity decreases. Both mechanisms provide an in-herent safety response for the system. However, if thesoluble boron concentration is too high, a positivemoderator temperature coefficient can occur, and core
reactivity could increase. For this reason boron con-centration is limited to ensure a negative moderatorcoefficient at full power.
Neutron detectorsWhen operating a nuclear chain reaction system,
neutron detectors are used to measure the intensityof the neutron radiation (flux). This radiation is a di-rect indication of nuclear reaction intensity. Neutrondetectors can be counters or ionization chambers.Counters, which detect neutrons by sensing the indi-vidual ionizations they produce, are most useful whenthe neutron flux is low. Ionization chambers are moreuseful at high neutron fluxes. They measure the elec-trical current that flows when neutrons ionize gas ina chamber. These two types of detectors, placed on theexterior of the pressure vessel, only measure neutronsleaking from the reactor.
Two other types of detectors, a self-powered neu-tron detector and a miniature fission chamber, can beused to measure the neutrons inside the instrumenttube of operating fuel assemblies. Self-powered detec-tors are usually arranged in strings and are positionedto continuously measure the neutron reactions at upto seven axial locations in the fuel assembly. The min-iature fission chamber contains uranium, and fissionevents are detected by the electronics. The fission cham-ber is moved into and out of the core on a periodic ba-sis, normally monthly, and measures the neutron levelalong the entire length of the fuel assembly.
Neutron sourceDuring the start of the chain reaction, it is essential
that the operator monitor nuclear instrumentation thatis counting neutrons and not gamma rays. The massof fuel in the core is much greater than the critical massrequired to sustain a chain reaction. Control rods andsoluble boron dissolved in the reactor coolant keep thecore subcritical when no power output is required.
When the core is to be brought to criticality, controlrods are withdrawn to initiate the chain reaction.However, if the control rods are withdrawn before ameasurable neutron flux is available, a reaction couldbe initiated by a stray neutron and build up to a highpower level before the control rods could be reinsertedto maintain the desired core output.
Fig. 14 Core reactivity as a function of operating parameters.
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To control the rate of buildup of the reaction, it isnecessary to have a neutron source present in the corefor startup. With a neutron source available, it is pos-sible to measure neutron flux before moving controlrods and help ensure a safe reactor startup.
Several types of neutron sources are available. Oneis an americium-beryllium-curium source rod. The ra-dioactive isotopes americium and curium emit α particleswhich react with the beryllium to produce neutrons.
Fast reactorsIt is possible to design for a chain reaction to occur
predominantly with either fast (high energy) neutronsor slow (thermal energy) neutrons. In fast reactors thechain reaction is maintained primarily by fast neutrons.Thermal reactors are those in which the chain reactionoccurs primarily from thermal neutrons. Today’s com-mercial water reactors are thermal reactors.
Fertile material such as U-238 can be used as a fuelby first converting it to plutonium in a reactor. Fastreactors and particularly fast breeder reactors accom-plish this conversion most effectively. Liquid sodium,which has essentially no moderating effect, is the cool-ant most often considered for fast breeders. Helium isused as a coolant in high temperature gas-cooled re-actor designs.
Conversion and breedingMost important of the conversion reactions is the
capture of a neutron by a U-238 nucleus, resulting ina nucleus of fissionable Pu-239:
92238
01
92239
00
92239
93239
94239
23
U n U
U Np Pu
+ → +
→ →
γβ β
min. 2.3 d
The U-238 nucleus absorbs a neutron and becomesU-239, which decays with a half life of 23 minutes,into neptunium-239 (Np-239). Again this nuclide, witha half life of 2.3 days, is transmuted to Pu-239 by βdecay. This is the nuclear process by which the mostuseful isotope of plutonium is formed.
There are other reactions during operation whichform different isotopes of plutonium. Some of these iso-topes, including Pu-241, are fissionable, and others,including Pu-240, are converted to fissionable isotopesby neutron absorption. These isotopes can not be sepa-rated economically from Pu-239, and therefore must betaken into account when plutonium is used in a reactor.
The large commercial PWRs in the United Statesoperate on slightly enriched uranium fuel. These re-actors convert U-238 to plutonium and produce about50% as much plutonium as the U-235 consumed. A re-actor is considered to be a converter when the amountof fissionable material produced, e.g., plutonium, is lessthan the amount of fissionable material consumed, e.g.,U-235. A breeder reactor is one in which more fission-able material is produced than consumed.
By definition, a breeder reactor must have thevalue of η (neutrons produced per neutron absorbedin fissionable fuel, Table 5) greater than 2.0. Oneneutron is required to maintain the chain reaction,one or more for the breeding, and an additional frac-tion for absorption in nonfuel materials and leakage.
It is not possible to make a breeder reactor with natu-ral uranium fuel, because η is less than 2.0. Table 5shows that η does not greatly exceed 2.0 with any com-mon fissionable isotopes for fissions produced by ther-mal neutrons. Consequently, it becomes difficult andimpractical to make a breeder with a thermal reactor.
Fortunately, at high neutron energies, η has agreater value than at thermal energies, particularlywith Pu-239. For this reason, and because the absorp-tion cross-sections of most materials are less than atthermal energy, fast reactors breed plutonium moreeffectively than thermal reactors. In addition, fast re-actors operate best with plutonium fuel.
Inconel is a trademark of the Special Metals Corporation group ofcompanies.
1. Evans, R.D., The Atomic Nucleus, R.E. Krieger Pub-lishing Company, New York, New York, 1982.2. Moe, H.J., et al., Radiation Safety Training TechnicianTraining Course, ANL-7291, Rev. 1, Argonne NationalLaboratory, Argonne, Illinois, 1972.3. Firestone, R.B., and Shirley, V.S., Eds., Table of Iso-topes, Sixth Ed., Wiley-Interscience, Hobocan, New Jer-sey, 1998.4. Lamarsh, J.R., Introduction to Nuclear ReactorTheory, American Nuclear Society, LaGrange Park, Illi-nois, 2002.
References5. Lamarsh, J.R. and Baratta, A., Introduction to NuclearEngineering, Third Ed., Prentice-Hall, Reading, Massa-chusetts, 2001.6. Duderstadt, J., and Hamilton, L., Nuclear ReactorAnalysis, Wiley Publishers, Hobocan, New Jersey, 1976.7. Bell, G., Nuclear Reactor Theory, R.E. Krieger Pub-lishing Company, New York, New York, 1979.
