Critical Safety

8/3/2019 Critical Safety

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BNL47594INFORMAL REPORT

_.,.....-..-. . ae, ,4 LECTURE NOTES-- FOR

CR ITlCALlTY SAFETY

byRalph Fullwood

March 1992

DEPARTMENTOF NUCLEAR ENERGY, BROMHAVEN NATIONAL LABORATORYUPTON, NEW YORK 11973

Prepared for the U.S. Department of EnergyUnder Contract No. DE-AC02-76CH00016WIRIBUTION OF T i l l s DOCUMENT 1s UNLiMiTqo


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NOTICE I

This report was preparedasan accountof work sponsored by an agency of the UnitedStates Government. Neither the United States Government nor any agency thereof, or any oftheir employees, makes any warranty, expressed or implied, or assumes any legal liability orresponsibilityforany hi id party's use, or the results of such use, of any information, apparatus,product or process disclosed in this report, or represents that its use by such third party wouldnot infringe privately owned rights.The views expressed in this report are not necessarily those of the U.S. Departmentof Energy.


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DISCLAIMER

This report was prepared as an account of work sponsored by anagency of the United States Government. Neither the United StatesGovernment nor any agency Thereof, nor any of their employees,makes any warranty, express or implied, or assumes any legalliability or responsibility for the accuracy, completeness, orusefulness of any information, apparatus, product, or processdisclosed, or represents that its use would not infringe privatelyowned rights. Reference herein to any specific commercial product,process, or service by trade name, trademark, manufacturer, orotherwise does not necessarily constitute or imply its endorsement,recommendation, or favoring by the United States Government or anyagency thereof. The views and opinions of authors expressed hereindo not necessarily state or reflect those of the United StatesGovernment or any agency thereof.


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DISCLAIMER

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LECTURENOTES FORCRITICALITY SAFETY

Prepared by:Ralph Fullwood

March 1992

BNL--47594DE92 016009

Engineering Technology Division-Department of Nuclear EnergyBrookhaven National Laboratory/Associated Universities, Inc.Upton, New York 11973

Prepared for:U.S. epartment of EnergyWashington, D.C. 20585

DISTRIBUTIONOF THIS DOCUMENT IS UNLIMIT-


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ABSTRACTThese lecture notes for criticality safety are prepared for the training of Department ofEnergy supervisory, project managem ent, an d administrative staff. Technical training andbasic mathematics a re assumed. Th e notes are designed for a two-day course, taught by twolecturers. Video tape s may be used a t the options of the instructors. Th e notes provide allthe materials that are necessary but outside reading will assist in the fullest understanding.Th e course begins with a n uclear physics overview. Th e reade r is led from the macroscopicworld into the microscopic world of atoms and the elementary particles that constituteatoms. Th e particles, their masses and sizes and prop erties associated with radioactivedecay and fission ar e introduc ed along with Einsteins mass-energy equivalence. Radioactivedecay, nuclear reactions, radiation penetration, shielding and health-effects are discussedto understand protection in case of a criticality accident. Fission, the fission products,particles and energy released are presented t o app reciate the dang ers of criticality. Nuclearcross sections are introdu ced to understand the effectiveness of slow neutrons to producefission.Ch ain reactors a re presented as an economy; effective use of the neu tron s from fission leadsto mo re fission resulting in a pow er reactor or a criticality excursion. T he six-factor formulais presented for managing the neutron budget. This leads to concepts of material andgeom etric buckling which are used in simple calculations to assure safety from criticality.Experimen tal measurem ents and computer code calculations of criticality are discussed.To emph asize th e reality, historical criticality accidents are presen ted in a table with m ajorone s discussed to provide lessons-learned. Finally, stand ards, N R C guides and regulations,and D O E ord ers relating to criticality protection are presented.

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C ONTENTS PageAB S TR AC T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .L IS T O F F IG U R E S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .LIST OF TAB LES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .I N T R O D U C T I O N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .AC KNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1 NUC LEA R P HYS ICS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.1 Physics Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.2 Nuclear Particles and Natural Radioactivity . . . . . . . . . . . . . . . .1.3 Nuclear Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.4 Radiation Protection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.5 Dose. Dose Rate and Health Protection . . . . . . . . . . . . . . . . . . .1.6 Nuclear Resonances and Nuclear Fission . . . . . . . . . . . . . . . . . .

2. NUC LEAR C HAIN R EAC TIONS AND C R ITICALITY . . . . . . . . . . .2.12.22.32.42.52.62.72.8

T he Principle of the Chain Reaction . . . . . . . . . . . . . . . . . . . . .Th e Six-Factor Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Neutron Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Lethargy and the Average Logarithmic Energy Dec remen t . . . . . .Nuclear Reactor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Time Response of a Nuclear Chain Reactor . . . . . . . . . . . . . . . .Rules for Avoiding Criticality . . . . . . . . . . . . . . . . . . . . . . . . . .

Neutron Economy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3. DETER MINING C R ITIC ALITY . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.1 Buckling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.2 Measurement of Criticality . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.3 Subcritical Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.4 Calculation of Criticality without a Com pute r . . . . . . . . . . . . . . .3.5 Computer Calculation of Criticality . . . . . . . . . . . . . . . . . . . . . .

4. C R ITIC ALITY AC C IDENTS AND TH EIR P R EVENTION . . . . . . . .4.1 Som e Characteristics of Criticality Acc idents . . . . . . . . . . . . . . . .4.2 Selected Criticality Accidents . . . . . . . . . . . . . . . . . . . . . . . . . . .4.3 Summ ary Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

...111viiviiiixX1-11-11-41-81-121-161-202-1 2-1 2-12-2 2-3 2-4 2-7 2-92-113-13-1 3-5 3-8 3-113-174-14-1 4-24-9

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CONTENTS (Continued)Page

5. S TANDAR DS . GUIDES AND OR DER S . . . . . . . . . . . . . . . . . . . . . . 5-15.1 ANSIIANS Standards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-1 5.3 Depar tment of Energy Orders . . . . . . . . . . . . . . . . . . . . . . . . . . 5-105.2 Nuclear Regulatiory Comm ission Regulations and Guides . . . . . . 5-8

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L IS T O F F I G U R E SPage

1-11-21-31-41-51-62- 12-22-32-42-53-13-23-33-43-53-63-73-85-1

A Cross Section Targe t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-8 Fission Cross Section of U-235 from 0.01 to 5 eV . . . . . . . . . . . . . . . . . 1-20Five Stages in Liquid-Drop Breakup . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-21Fission Product Mass Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-21Fission Neutron Energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-22Isotropic Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-12

How a Reactor Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-1 Six factors abo ut the Critical Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-3 Neutron-Proton Collision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-4Thermal Energy Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-4 Simplified Reactor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-7....lux Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Critical Mass and Radius for a B are and a Reflected Plutonium-Multi-Parameter U-235 .........ub-criticality Limits . . . . . . . . . . . . . . . .Multi-Parameter U-235 ....ub-criticality Limits . . . . . . . . . . . . . . . . . . . .Extrapolation Lengths ...ylinders Con taining U 0.F. . . . . . . . . . . . . . 3-1.xtrapolation Lengths for Cylinders of Uranium Metal and &phasePlutonium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Figure ... odified .........ass-Concentration Equation . . . . . . . . . . . .Birdcage ........ontainer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-14. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Th e Ring of A N S V A N S Standard Protection . . . . . . . . . . . . . . . . . . . . 5-2

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LIST OF TABLESPage

1-11-21-31-42-12-23-13-23-34-14-24-34-45-15-25-3

Everyday Nuclear Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-5Q-factors for V arious Radiations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Q-factors for N eutrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-17Some Maximum Doses from N CRP-91 . . . . . . . . . . . . . . . . . . . . . . . . .1-171-18

Mod erating Prope rties of Several Materials . . . . . . . . . . . . . . . . . . . . . 2-5Delayed Neutrons from Uranium-235 Gro ups . . . . . . . . . . . . . . . . . . . . 2-10Bucklings for Regular Geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-3 Single Par am eter Lim its for Uniform N itrate Aque ous Solutions ofFissile Nuclides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-7 Single Par am eter L imits for Metal A ssemblies . . . . . . . . . . . . . . . . . . . 3-7 Process Accidents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-11Bare and Reflected Metal System Accidents . . . . . . . . . . . . . . . . . . . . . 4-12Mo derate d Me tal and Oxide System Accidents . . . . . . . . . . . . . . . . . . . 4-13Fissile Solutions and Miscellaneous System Accidents . . . . . . . . . . . . . . 4-14Some 10CFR Titles Applicable to the NR C . . . . . . . . . . . . . . . . . . . . . 5-8 Some Pa rts of D O E O rde r 5480 Relevant to Criticality . . . . . . . . . . . . .Som e Division 3 Regulatory G uides Relevant to Criticality . . . . . . . . . . 5-9 5-10

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ACKNOWLEDGMENTSI gratefully acknowledge the assistance of L uth er Lowry of Law rence Livermore La boratorywho kindly provided video tapes and many booklets used in criticality training at hisLaboratory.Dr. Joh n Carew, the BNL Criticality S afety Officer, very kindly and ungrud gingly providedhis time an d advise in th e preparation of this course.I also wish to thank: Jerry C. Cadwell for his many helpful suggestions in p reparing thiscourse; D r. Avril Woo dhead for the effective technical editing and help in radiation effects;and Susan Monteleone for the document preparation.

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INTR ODUC TIONAn accident resulting in an uncontrolled nuclear chain reaction is called a "criticalityaccident." W e will learn to recognize and calculate what materials, in what qua ntities,combinations, and shapes can result in a chain reaction'. We will learn how to protectourselves and o the rs if such a reaction takes place. We will discuss the few serious and fewfatal accidents th at have occurred to show the reality of th e criticality, and to gain lessons-learned to pre ven t criticality. Finally, we will review the ANSI/ANS standard, NRCRegulatory Guides and D O E O rders for preventing criticality accidents.Th e following are the chapters to be covered:

1. Nuclear Physics,2. Nuclear Chain Reactions and Criticality,3. Determ ining Criticality4. Criticality Accidents and their Prevention5 . Standards, Guides, and Orders

Chapter 1 Nuclear Physics provides an unde rstanding of the particles involved, radioactivedecay, reactions, cross sections, and radiation protection by distance, shielding, and reducedexposure time. Measures of radiation and the effects of radiation on hum ans are presented,followed by discussions of nuclear resonances, fission, and the energy distribution of thefission neutrons.Chapter 2 - Nuclear Chain Reactions and Criticality covers the principle of the chainreaction, the six-factor formula for calculating the neutron multiplication (k-factor), how tocontrol criticality, the mode ration and reflection of neutrons, how a reactor works, and endswith a discussion of the time response of a reactor.Chapter 3 - Determining Criticality begins by showing that four of t h e factors in the sixfactor formula are equal to k, - the neutron multiplication factor for an infinite reactor.The o ther two factors are related to the Fermi age, the thermal diffusion length, andbuckling. Buckling, related to the size of the reactor, can be used to deter mi ne if anassembly or process is critical. Next, there is a discussion of the experimental measurementof criticality and critical facilities. Single pa ram ete r and m ultiple-p aram eter limits tha t havebeen deter mine d a re presented , and used for the hand-calculation of criticality by fourmethods. The chapter ends with a discussion of the two types of computer codes forcalculating criticality.

A chain reaction is a succession of fissions occurring one a fter the oth er like a chain - th epreceding event causing the following.X


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Chapter 4 - Criticality Acc idents and their Prev ention begins by discussing the explosive andlethal effects of criticality accidents. Sixteen accidents are discussed, one of which was achem ical explosion. O n e criticality acciden t, Chern obyl, killed mo re people than all of the4 1 criticality accidents that are summarized in four tables.Chapter 5 - Standards, Guides, and Orders end the course with the rules that have beendeveloped, based on ex perience and theory, to prevent criticality accidents.

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1. NUCLEAR PHYSICS1.1 Physics ReviewT h e following is a review of definitions and principles, presen ted in a lpha betical order, tha tare commonly used in criticality safety.Atomic Mass Unit - A M UO n e A M U = 1/12 the m ass of a carbon 12 neutral atom. This value is the standard forcomparison of al l of th e o ther atomic masses. 1A M U = 931.494 M eV = 1.66054E-27 kg.Atomic Number - 2Th e charge of th e nucleus in m ultiples of the elem ental charge; the n umb er of protons inthe nucleus.Atomic Weight - ATh e weight of an atom in multiples of an atomic mass unit (AM U)AtomsAtoms are constituents of molecules. They ar e compo sed of a central nucleus wherepractically all of th e atom 's m ass is con centrated an d planetary-like electron s circling th enucleus. Atoms exchange electrons with other atoms to p roduce th e force binding atomsto form molecules.ElectronT he electron is the nuclear particle that flows in wires all abo ut us to power homes andindustry. Electron s ar e negatively charged, very light-weight particles, th at circle the posi-tively cha rged ato mic nucleus to neutralize th e overall atomic charge. If there a re more orless electrons than are needed to balance the nuclear charge, the atom is said to be"ionized."Electron VoltCharged particles acquire energy when they pass between two electrodes of differentpotential according to the formula: E = e*V, where E is the energy, e is the elementalcharg e (discussed below), and V is th e accelerating voltage. Thus, energy is proportionalto, an d may be m easure d by, voltage instead of i ts fundam ental and very large unit, Joules.Th is unit, called electron-volts is related t o the Jo ule as:

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1 Joule = 1.602E-19 electron-volt (e V)Elemental Charge

(1-1)

Electrical charge com es in multiples of the elemental charge which is:e = 1.602E-19 Coulombs (C). (1-2)

Electrons a re negatively charged; protons ar e positively charged, but the am oun t of chargecarried by e ithe r particle is the same, regard less of sign.Electric charge must b e conservedTh e sum of the electric charges before a react ion must be the sam e as the sum of thecharges after a reaction.Energy-Mass. EquivalenceEinstein discovered th at mass and energy are related throug h the famou s equation:

E = m*c*, (1-3)where m is the mass, and c is the velocity of light. T ha t is, if an am ou nt of mass mvanishes, an amount of energy given by equation 1-3 must be released. This relationshipis used to express nuclear mass in units of energy.IsotopeNuclei ar e called isotopes if they have the sam e nu mb er of protons but different numb ersof neutrons. For example, ='U and ='U ar e isotopes. Both have 92 protons but the formerhas 235-92 = 143 neutrons, while th e latter h as 146 neutrons. Chemically they are identical,because chemical properties are th e result of the num ber of protons - however, nuclearproperties may be quite different.Neut ronsOne of the constituents of a nucleus is the neutron, a particle having n o charge andweighing 1836 times th e electro n weight.

The other nuclear constituent is the proton, having the same amount of charge as theelectron bu t of opposite sign. It also weighs 1836 times the electron weight.

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Review Questions:1) What particles are nuclear constituents?2) W hat do proton s and electrons have in common?3) W hat do protons and n eutrons have in common?4) Wh at is an isotope?5 ) Wh at units are used to express the energy of molecules, atom s and particles?6) How can the mass of a nu cleus be expressed in units of energy?

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1.2 Nuc lear Particles a nd Natural Radioactivity1.2.1 Sum mary of ParticlesScientists in th e DOE complex working with very high energy accelerators produce exotic,rapidly decaying particles that ar e no t relevant to criticality safety. Th is discussion is limitedto what I call the "everyday" particles.Th e 1903 Nobel prize was awarded to Henri Becquerel and the Curies for the discovery ofradioactivity in certain uranium an d radium bearing ores. In a mag netic field, the radiationwas found to be composed of three types of particles which they identified by the eq uivalentof A, B and C except they used G ree k letters:

A. Highly ionizing heavy particles having a positive charge called the "alpha" (a)particle, late r identified as the nuclei of helium.

B. Lightly ionizing light weight particles with a negative ch arge called t he "betall ("B")particle, late r identified as an electron.C. Lightly ionizing particles not affected by the magnetic field called the "gamma"

( y ) particle, and later identified with electromagnetic, x-radiation.Before this and subsequently, other particles of concern t o us were discovered. Table 1-1summarizes the everyday particles, classifymg them by mass and charge; to simplify ourdiscussion, they have bee n num bered.1. Electron (e-)Electrons are the particles circling the atomic nucleus, having a mass of 9.108E-31 kg, anda charge of -1.602E-19 C.2. Positron (e+)Th e positron is the mirror image of the electron, with a mass of 9.108E-31 kg, and a ch argeof +1.602E-19 C. The positron is the anti-particle of the electron according to Dirac'srelativistic quan tum m echanics. Every particle ha s its anti-particle with which it mayannihilate e.g., B++ B-+2* y. T he energy of annihilation is con tained in two dentical gam marays each having 0.511 M eV emitted in opposite directions to carry off and conserve theenergy and momentum.3. Proton (PITh e proton is a hydrogen atom without the electron i.e. the nucleus of the hydrogen atom .T he pro ton ha s a mass of 1.672E-27 kg, which is 1836 times heavier than a n electron. Th e

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charge is the same as the electron but of opposite sign.4. Neutron (n)Similar to the p roton in that both constitute the atomic nucleus. Its mass is 1.6725E-27kg,and i t has no charge.

I1 Table 1-1 Evewdav Nuclear Particles

1.2.

Nam e Mass Chargeelectron, B, eta minus 0.51099 neg.

(MeV)

positron, B+, beta plus 0.51099 pos.

3. I proton, p I 938.27 I POS.4.5.6.7.

neutron, n 939.56 nonegamma, y, photon none noneneutrino, v, none nonedeuteron, d 1875.6 pos.

Decay

9.

stable

alpha, a 3727.4 Ppos . s tab le

anti-electron henc eann ihilates with elec-tronsstable888 s when freestable

stable8. triton, t 12809. 1 12.3 y

H3+He3+13-

5 . G a m m a (Y)T he gamma or ph oton is an electromagnetic wave particle with n o mass.6. Neutrino (u)T he "lit tle neutron" has no charge and n o mass but i t is different from a y-ray. It wasoriginally postulated to exist to conserve energy, charge, an d some oth er q uantities in th ebeta decay of radioactive material . It is very hard to detect because it can pene trate greatthicknesses of material without interacting. Its existence has been established in manyexperime nts since it s existence was postulated by Pa uli in t he early 1930s. Th ere ar e severaltypes of neutrinos: the one listed in Table 1-1is the on e that accompanies beta decay.

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7. Deuteron (d)Th e deuteron is the nucleus of deu terium w hich com bines with oxygen to make heavy water.It is not an elementary particle because it is composed of a proton and a neutron. Itsweight is not th e sum of the neutron and proton weights because of the binding energy. Ithas a mass of 3.344E-27 kg, and one uni t of positive charge.8. Triton - tThis nucleus of tritium is not an elementary particle becaus e it is composed of twoneutronsand one proton. Its mass is 5.0058E-27 kg, with one elem ental positive charge.9. Alpha - a!This nucleus of tritium is not an elem entary particle because it is composed of two neutronsand two protons. Its mass is 1.10367E-26 kg, with two elem ental positive charges.1.2.2 Radioactive DecayRadioactive isotopes ar e distinguished by the rate w ith which they decay and by t he type ofparticles emitted. Some isotopes decay much faster than others. Even th e same isotopemay em it one type of particle faster than another.Experimentally, it is observed that th e rate, dN/dt, a t which particles ar e being emitted isproportional to the number, N, of nuclei that can emit particles. Writing this state m ent asan equat ion:

dN/dt = - l * N , t1-41where A is the proportionality decay constant, and the negative sign means that N isdiminishing with time, t. Eq ua tion 1-4 is easily integ rated t o give:N = N,*exp(-l*t), (1-5)

where No is the number of radionuclides a t time t=O.If we ask, how long doe s it take for half of th e mate rial to decay, we set the ratio N/No =?A and solve for time; th e result is called th e half-life.

t,h = ln(%)/ l = 0.693/31. t1-61The ra te of radioactive decay is expressed in un its of halflife. and may be fo und in the C ha rt

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of the Nuclides2. To use such data in equation 1-5, it is necessary to determine the decayconstant (1) sing equation 1-6.Whv E mr ess the D ecav Con stant in Half-Lives?Equation 1-5 is easily calculated with a hand calculator if the decay constant is known.However, calculating the decay using half lives can be don e by simple multiplication anddivision. Suppose a radionuclide h as a 2-year half-life an d you wa nt to know how m uch isleft after 8 years. Th en 8/2 = 4 half-lives ar e of concern, so %*%*%*% = (%) ln = 1/16,and only 1/16-th of the m aterial is left after 8 years.Review Questions

1. What is the mass of an electron in kilograms? W ha t is its weight in pound s?2. W hat do gammas and neutrinos have in common?3. W ha t is the energy of each gamma if an electron and a positron annihilate?4. Tritium has a half-life of 12.26 years. If a nu clear b om b is filled with 4 grams of

W hat is the mass of a positron?

tritium, how much is left after 24.52 years if there are no eaks and no refilling?

' "Nuclidesand Isotopes", General Electric Company, Nuclear Energy Operations 175 Curtner Ave. M /C 684,San Jose CA 95 125.

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1.3 Nuclear ReactionsCollision Mean Free Path LengthIf a particle reacts with a nucleus, the particle may be deflected, it may lose energy byexciting the nucleus, or th e nucleus may be transformed. T h e likelihood of a particle doingany of thes e things is expressed in the unit s of are a and is called the cross-section. Supposewe have a beam , like a flashlight beam , of particles h itting a target, how m any reaction s willtake place per second? Th e converse problem is, what fraction of the in cident particles passthrough the target unaffected? These are the subjects of this sub-section.Figure 1-2 shows a slab of material that is the target for our beam of particles. Th e targethas dimensions of L1 by L2, giving an a rea A, the thickness is dx,which is so thin that eachnucleus is exposed t o the beam. Th e figure shows an incident particle and a target nucleusin grazing incidence i.e. the centers of th e two particles must come within a distance d orelse ther e is a miss. This is tru e for all angles ab ou t the collision circle tha t has a circulararea:

I

Other nuclei

IL2

a L 1 \.Figure 1-1A Cross Section Target

\d X

where d = r+R with r being theradius of the projectile and R beingthe radius of the target particle ornucleus.T he units used for cross-sections a rethe barn defined to be:

1 barn = 1E-24 cm2. (1-8)Th e name is said t o have arisen fromth e expression "big as a barn." It islarge in terms of nuclear sizes, but itis a convenient area for expressingnuclear cross-sections.Th e probability, p of a single particlecolliding is just th e ra tio of the area

presented by nuclei to the total area, hence:p = Area of nuclei and particlesflarget area = n*o*dx, (1-9)

where n is the number of nuclei per unit volume (number density).

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Th e probability th at N particles penetrate to depth x of material, where dN collide in thethickness dx equals the ratio dN/N as N becomes large. The n equation 1-9 becomes:dN/N = -n*o*dx. (1-10)

T h e ratio, dN/N, may be called the differential transmission, dT. Eq uati on 1-10 is solvedby integration to give the total attenuation t o a depth x as:T = exp(-n*o*x), (1-11)

where expo is the base of natural logarithms raised to the power of the contents of th eparentheses.Notice that if x=O, then T = l to indicate complete transmission.Conversely we may define atten uation as one-minus-transmission:

A = l - T .Using equation 1-11:

A = 1 - exp(-n*u*x),which is zero for z ero thickness as you might suspect.

(1-12)

(1-13)

To use equations 1-12 or 1-13, the nu mb er of nuclei per u nit volume, n is needed.The num ber of atoms in a gram-atomic weight is Avogadro's num ber A, = 6.02E23. T h eatomic weight, A is the mass in grams per gram-atom so AJA is the number of atoms pergram. Th e density p is the mass grams per cm3. Combining this information, the nuc leardensity is:

n = p*AJA. (1-14)MicroscoDic Neutron Cross-SectionsTh e cross-section w e have be en discussing is called the micro scotic cross-section, u. I t isthe a rea presented by a n ucleus to a neu tron of a certain energy for producing a given typeof reaction.A primary collection of data on the m icroscopic cross section of neu tron projectiles incidenton isotopic targets is the "barn bo ok f (BNL-325). To find th e attenuation of neutro ns bymaterial of thickness x, multiply the cross-section in barns by n, given by equation 1-14.Since Avogadro's number is 6.02E23 and a barn is 1E-24, the result of the multiplication

1-9


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is 0.602 to give the attenuation as:A = l-exp[-0.602*p*a(barns)*x(cm)/A]. (1- 15)

Be su re th e cross-section selected is that for the co rrect type of reaction, neutron energy,and isotopic compo sition.MacroscoDic C ross-SectionTh e microscop ic cross-section usually occurs in combination with n, hence is convenient todefine the macroscopic cross-section as the produ ct of n times u or :

2 = n*o = n*p*A,,/A. (1-16)This has the units of l/length and is convenient because with it, equa tion 1-11becomes:

T = exp(-X*x). (1-17)Th e values of Z for elem ents, including uraniu m, a re tabulate d fo r various energy groups3.Mean-Free Path LengthAttenu ation is also expressed in terms of the "m ean-fre e path length", symbolized by e x > -the average pen etration distance which is:

e x > = l /(n*u) (1-18)substituting into equ ation 1-11gives:

T = exp( -d) . - (1-19)Thus, the mean-free path length is the thickness of material that reduces the transmittedradiation by " one-e th" (l/e ), where e = 2.71828, or 36.8%.Half-ThicknessTh e thickness to a ttenu ate the radiation by one-half is called the half-thickness. It is relatedto the mean free path length as:

x,/, = 0 . 6 9 3 * < ~ > . ( 1 - 2 0 )

For example, see H. Etherington, editor, Nuclear Engineering Handbook, McGraw-Hill, NY , 1958.1-10


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Th e Advantage of Using the Half-ThicknessThe use of th e half-thickness, like using half-lives, simplifies calculations. Ra diation tha tpenetrates one half-thickness of target is % of the original amount . After ano ther half-thickness that % is reduced by anoth er % and so on.Num erical Example of a Calculation involving Half-Thick nessSuppose water of thickness 2 cm will attenuate neu trons by 2. How much will 6 cm ofwaterat tenuate neutrons? Th e 6 cm thick ness is 3 half-thicknessesso the at tenuation is %*%*?A= 1/8.Review Questions

1. What is a cross-section?2. How big is a bar n?3. If a n eutro n cross-section is 10barns, th e ma terial density is 1gm/cm3, he atomic4. If th e ma croscop ic cross-section is O.S/cm an d th e thick ness is 2 cm, what is the5. What is a half-thickness?6. If a m aterial is 10 cm thick an d th e half-thickness for radiation is 2 cm, what isthe at tenuat ion?

numb er is 10, an d th e thickness is 3 cm, what is the atten uation ?at tenuat ion?

1-11


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1.4 Radiation ProtectionRadiation protection is the use of materials, distance, and limited exposure times to protec tpersonnel, equipment, and the environment from radiation.1.4.1 G eom etric ShieldingI Z Point Source

Dose is related to the amount ofradiation absorbed by people orequipment. Figure 1-2 shows apoin t radioactive s ourc e radiating Sparticles per second equally in alldirections: it is called an isotropicsource and hence, is suited tospherical geometry. If all S par-ticles pass through this sphere inone second, the surface flux density(SD), the number of particles inci-dent upon an area, is:

Figure 1-2 Isotropic R adiation

SD = S*f/4*7r*r2, (1-21)where t is the duration of the radiation. T he total radiation incident on the area is:

D = SD*area = area* S*f/4*7r*r2 (1-22)T he difficulty comes from the fac t that th e .areal surface in equation 1-21 is curved on asphere of radius r, cast by the outline of the object being irradiated. If this area is smallcompared with the radius of the sph ere, it may be approximated as the flat area , in whichcase, the solid angle is:

s1 = area/r2. (1-23)Substituting equation 1-23 into 1-22:

D = s1*S*t/(4*7r). (1-24)This equation shows tha t the radiation incident is proportional t o the solid angle, the sourcestrength, and the time duration of exposure.If th e a rea nearly eq uals r2, it may be calculatedin Figure 1-2:

1-12

by integrating the very small area shown


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Area = jr2*sin+*dO*d+. (1-25)Substituting equation 1-25 into equation 1-23, we find the solid angle can, in general, becalculated as:

= Jsin+*dO*d+/(4*~). (1-26)An Example of Calculating the Interception of RadiationSuppose you are 1.8 m tall and 0.4 m wide, your area is 1.8*0.4 = 0.072 m. If you arestanding 10 m from a source emitting lElO n/sec, then the flux density is: SD =lE10 / (4*~*100)= 8E 6 n/m2*sec. Becau se your are a is 0.072 m2, the n um be r of neutronshitting your body is 0.072*8E6 = 5.7E5 n/sec.Doubling your distance from a point source reduces the exposure by 75%.Geometric Shielding - Line SourceThe effect of a line of radiation sources can be easily calculated if the radiation from apoint source is known by simply adding the effects of a number of point sources along aline. In doing this the poin t source solution is called a Green s function and th e combinedeffects is an integral over the G reens function. W ithout giving the full mathem aticaldetails, the attenuation from a line source is:

SD(a) = SD(b)*(b/a) (1-27)This equation indicates that the flux density from a line source is inversely related to thedistance from the source, where a is the distance to the object being irradiated, an d b is arefere nce distance, with SD (b) being the flux density at b. Dou bling the distance from aline source reduces th e exposure by 50%.Geometric Shielding: - Plane SourceIf the source of radiation is uniform over th e plane, the surfac e density is:

SD(a) = SD(b)*ln(b/a). (1-28)Thus the flux density from a plane source is attenuated as the logarithm of the distancefrom the plane - doubling the distance only cuts the dose by 31%.1.4.2 Ma terial ShieldingT he discussion of geometric shielding ignored the effects of materials between the objectand the source of radiation. Equation 1-11 showed that:

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transmission = 1 - attenuation = exp(-x/) (1-29)where x is the thickness of the shielding and is the mean-free path length of theshielding material for the type and energy of the radiation. Now, we put this into equation1-22 to get:

SD = S* exp(-x/ < > /(4 *T* 2). (1-30)which shows tha t radiation is attenuated bo th by the distance from th e source, an d by theshielding material.1.4.3 Buildup FactorEquation (1-30) is based on the assumption that if a radiation particle impacts a nucleus,it disappears. T he collision processes depe nd very much on the type of particles involvedin the collision. Heavily ionizing particles such as, alpha particles o r protons, ar e very easilystopped by a small amount of material because they leave a dense trail of ions. They ar eno t generally remo ved by a single collision but slowed with e nergy going into th e ionizingprocess. On the other hand, electrons scatter off other electrons, and in this process, loseenergy and produce a gamm a. Subsequently, the gamm a may react with an othe r electronto produce an electron and gamma. This process is called a gamma cascade which iscomp licated to calculate. Ne utro ns may suffer many elastic collisions with nuclei, in asimilar cascade process, until their energy is too low for this gamma cascade to occur. Inaddition, neutrons may inelastically scatter from nuclei to produce gamma rays or theneutrons may be captured. If they a re captured to produc e fission, on the average 2.5 ne whigh energy neutrons will be produced.The se complex processes a re simply treated as an a ttenuatio n process with a build-up factorin equation 1-30 to correct for the effect of these secondary effects. Th is buildup factor,B(E,r) corrects equation 1-30 for the cascades as:

SD = S*B(E,r) *exp( r/ < > /(4* IT* 2). (1-31)Large com puters calculate theoretical models of the secondary processes to produce tablesof build-up factors. The se tables are prepared for neutrons and gamm as of various energiesin many geometries and material combinations. Tabulated buildup factors depend on thetype of primary radiation, the energy, E, of the primary radiation, the charge, Z, atomicnumber, A, and thickness of the shielding material.

A Numerical Example of a Build-Up FactorFor example, a 1 Me V point isotropic source of gamma-radiation has a buildup factor of

1-14


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2.1 when penetrating a mean -free thickness of water. If the build-up factor is ignored (setto on e), equation 1-33 shows that exp(-1) = 0.36, or 36% of the radiation passes throughthe shield. But when the buildup factor is included, 2.1*0.36 = 76% of the radiationpenetrates the shield.Summary of ShieldingTh e calculation of shielding and the atten uation it provides is a complex problem involvingtransport theory. Th e complexity includes a geom etric effect, exponential attenuation, anda buildup factor. Th e attenuation may be approximated, bu t usually requires recourse totables of buildup factors for the type of radiation, energy, shielding material, andpen etration depth . Shielding geom etries may be complex, such as labyrinth corridors orducts, so there'w ill be n o straight-line paths. Shadow shielding may be used such thatworkers are in the shadow of the shield. Mo nte Carlo compu ter codes are required tocalculate such complex problems accurately.Review Questions

1. What is the solid angle of a 1 m* area, 10 meters from a point?2. If a n isotropic neutro n so urce is emitting lElO neutron&, and you are 1.8 m tall3. Why is a build-up factor n'eeded for shielding calculations?4. How does th e radiation vary with distance from p oint; line an d plane sources of

and about 0.6 m wide, how many neutrons are hitting you per second.

radiation?

1-15


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1.5 Dose, Dose Rate and H ealth ProtectionIn this section, we will discuss how radioactivity, dose, dose rate, and health effects aredefined. Th en we will discuss how dose can be redu ced by reducing exposure time, distancefrom the source, and by shielding materials between you and the source of radiation.Units of RadioactivityTh e activity of a sam ple, source or contam inated material is the r ate a t which radioactivedisintegrations are taking place. Th e earliest term for this is the curie, defined to be:

One Curie is the amount of radioactive material that undergoes 3.7EIO drjintegrationsper second.Th e more m odern unit is the Becquerel defined to be:

One Becquerel is the amoun t of radioactive material that undergoes I dkintegration persecond.thus

One Curie (C i) equals 3.7ElO Becquerels (Bq )Dose and Dose RateT he underlying item of interest is the effects of radiation on the human body. The se effectsare related t o the term "dose", which is a measure of the amount of energy deposited in anorgan from the entry or passage of radiation. The rate at which dose is deposited in theorgan is called the "dose rate". T he usual situation is tha t radiation is being emitted atsome rate (say x neutro ns per second), and is producing a dose rat e on organs or the w holebody. T he major effect on health is not the dose rate bu t the total dose received:

dose = dose rate * time of exposure.Thu s, dose may be re duce d by a short 'exposure time even though th e dose rate m ay behigh.Units of DoseDose com es from the energy deposited in tissue d ue t o the trail of ions left by a radiationparticle a s it loses energy in traveling th roug h ma terial. In a gas, the charge from ionizationcan be collected as a measure of the energy deposited. Thus, the Roentgen was defined interms of charge deposited:

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One Roentgen is that amount of radiation that willdeposit 1 esu of charge in one cubic centimeterofairadiation QGamma rays I at standard temperature and pressure.X-rays 1Beta rays and electrons 1NeutronsProtons 10 tissue thu s, giving rise to:Alpha particles 20Heavy ions 20

This definition was not very useful because the primaryinterest is in th e deposition of radiation energy in human

One Rem (Roentgen equivalent man) defined to 0.01Joules per kilogram of tissue.1-2 Q-factors for Recently, this quantity was redefined to be the "gray1'which is:arious Radiations

One gray (Gy) s that amount of radiation that will deposit 1 Joule of energy in 1 kg ofmass of tissue.Around the beginning of this century, canc er and illness was discovered to b e associatedwith excessive us e of xrays. Mo uth can cer in watch painte rs was associated with th e use ofradium in the paint. It soon came to be realized that radiation can have negative healtheffects.Health EffectsNeutron Energy(MeV) Q2.5 X IO-* (thermal) 21x o-' 2

1x 10-6 21x10-5 21x io-* 21x 10-3 21x10-2 2.5

Neutron Energy(MeV) Q5 87 7

10 6.514 7.520 a40 760 5 . 5

The concepts introduced as the rad (gray) dono t adequately describe the impact of energy ontissue because ab sorption of a given amount ofenergy in a given mass does not describe thedose effect. Tissue damage increases with thelinear energy transfer (LE T) i.e. t he density ofthe ionization along the track. Th e dose isestimated by correcting the gray by the "relativebiological effectiveness" (RBE) or "Q" actor ofvarious radiations relative to 200 keV x-rays.Table 1-2 gives th e Q-factors f or sev eral types ofradiation; note tha t Q increases with LET. Neu-trons and protons have similar LET because

i ; ; 5 , 2 !:;3 X l d

2.5 4 X l d 3.5 neu tron s collide with proton s.1-3 Q-factorsfor Neutrons Table 1-3 shows Q-factors for n eutron s of var-ious energies.

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Table 1-4 Some Maximum Doses from NCRlType of Exposure

C. Education a nd training (annual total)1. Effective dose2. Dos e limit for lens of ey e, skin, and ex tremitiesD. Embryo-fetus exposures1. Total dose equivalent limit2. Dos e equivalent limit in a mo nth

A. Annual occupational exposure1.Equivalent stochastic effects2. Dose equivalents for tissues and organ (non-stochastic)a. Lens of eyeb. All othe rs (e.g. bone m arrow, breast, lung, gonads, skin etc.)3. Guidance: cumulative exposure

15050.5

B. Public (annual)1. Effective dose: continuous, o r frequent2. Effective dose: infrequent3. Remed ial action recom mended when:a. Effective dose equivalentb. Exposure to radon and its daughters4. Dos e equivalent limits for lens of eye, skin and extremities

-91Dose (mSv)

5015050010*age15>5>0.007 Jhm"50

Dose(rem)

51550l *age0.10.5> O S>2 WLM5

0.15

~

0.50.0050.001

He alth Effects: Dose Equivalent - the SievertUsing the quality factor in conjunction with the energy deposited (the dose), we canestimate the he alth effects (H). This is expressed in th e equ ation:

H = D*Q (1-36)wh ere th e dose equivalent is expressed in Sieverts and th e dose is in grays. (100 Sieverts= 1 rem). It is important to note that the determination of dose equivalent requiresknowledge of the absorbed dose, the composition of the radiation an d it s energy distribu-tion.Table 1-4 summarizes so me g uideline limits for radiological protection4.

From the National Council on Radiation Protection and Measurements, "Recomm endationson imits forExposure to Ionizing Radiation," NCR P report No. 91, 7910 Woo dmont Ave., B ethesda M D, Jun e 1987.1-18


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Review Questions1. What is the difference between a Curie an d a Becquerel?2. W hat is th e difference between Sievert and a rem?3. Suppose thermal n eutrons deposit a Joule in you body, and you weigh 70 kg.What is your dose in rem, and in Sieverts? Is this a significant dose ?4. If the dose in problem 3 was received in on e hour, what was the dose rate?

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1.6 Nuclear Resonances and Nuclear Fission1.6.1 Nuclear ResonancesFission is the splitting of a nucleus into two rarely three) fragm ents of the original nucleuswith the release of energy and particles. Fission occurs spontaneou sly in som e isotopesheavier than uranium, but it usually occurs when a neutron is captured in a fissionablenucleus. The most common nuclei capable of fission are , UZ5,UD3,nd PuZ9(notice theseare even in Z but odd in A).

1 4 0 0 .1 , i i l l l l l l l l l l I l , I I I I I , i , j1000.

500.-nb

A 84 OR L G w X 71 GE L D e60 KUR MO * 70 MOL CeQ 77 LR L Cz Y 66 LR L BoA 73 BUC Bo x 66 ORL De+ 73 GE L De ENDp/B-vY 72 BUC Mi

to . so. 100. 300. 1000E, (mv)

200.

100.

5 0 .

i o .nUb 5.

a 64 ORL Gw80 KUR Mov 79 GE L Wa0 78 QRL Mo.A 71 GEL De+ 66 HAR Br

v 66 KU R M oX 66 KUR Mo* 66 LRL BoA 66 ORL DeENDF/B-V

Figure 1-3 Fission Cross Section of U-235 from 0.01 to 5 e VThe cross-section for a neutron being captured in a nucleus depends on the neutronsenergy and the target nucleus. Nuclei are very complicated and ar e subject to resonances,shown by very large ch ang es in the cross-section corresponding with small ch ang es of neu-tron energy.Figure 1-3(left), from the barn book, shows the fission cross-section of Uu5 for incidentneutrons from 0.01 to 1 ev; Figure 1-3(right) shows the fission cross-section for incidentneutrons from 1 to 5 eV. Figure 1-3(left) clearly shows a general trend of neutron cross-sections changing in proportion to the reciprocal neutron velocity (l /v, where v is theneutron velocity). This, an d the fact that neutrons accum ulate in the thermal region wherethey enter in to thermal equilibrium with materials, a re the rea sons that most reactors usethermal neutrons to produce fission.Th e right part of Figure 1-3shows the resonance structure which c auses variations in thecross-section by nearly a factor of 100with small energy changes. Th ese resonance s greatlycomplicate detailed calculations of criticality.

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1.6.2 Nuc lear FissionA Liquid-Drop Model of Fission .Th e particular of reaction in which we are interested is the fission reaction. If a neutronis captured in a nucleus, the co mpou nd nucleus acts like0 0 ~ 0 00

Figure 1-4 Five Stages in Liquid-Drop Breaku p

a liauid drop in which the liquidbegins to slosh and break-u p in totwo or more large fragments -called fission fragme nts. Fivestages from t he initial capture ofa neutron in a nucleus to th edeform ations becoming so severethat the drop breaks u p are shown in Figure 1-4.

Nuclear Fissiona

Th e compo und nucleus does not breakinto halves, but breaks into light frag-ment having mass about 95 A M U a ndheavy fragments having mass about140 AMU. Figure 1-5 shows the dis-tribution of the masses of the fissionproducts.Neutrons and other radiation areemitted during fission. T he averagenumber of neutrons emitted (callednu-bar) is 2.55, 2.47, and 2.91 fo rthermal neutron fission of Uu3,Uu5 ,and PuD9, espectively. T he kinet-ic energy budget for particles fromfission is: 165 MeV for fission frag-ments; 5 MeV for Bs from fissionfragments; 7 MeV for prompt ys; 6MeV for ys from fission fragm ents; 10MeV for neutrinos; and 5 MeV forneutrons, for a total of about 200MeV.

Figure 1-5 Fission Product Mass Distribution Neutrons emitted from fission aredistributed around 0.75 MeV, asshown in Figure 1-6. From Figure 1-3, it is apparent that the cross-section is lower athigher energies than at lower energies; therefore, fission occurs more readily at lowerneutron energies.1-21


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0 4 2 3 4 5E, M E VFigure 1-6 Fission Neutron Energies

T he distribution of the neutron ener-gies coming from fission is shown inFigure 1-6. It should be noted thatthe peak of the distribution (mostprobable) occurs at abo ut 0.75 M eV(this corresponds to a speed of 27million mph).Review Questions

1. What is the total amountof energy released in fission?2. How much of the energyfrom fission is useful for the produc-tion of heat?3. W ha t is the fission cross section of U-235 at 25 me V (milli-eV - this energy is theaverage energy of room-temperature thermal neutrons).4. Indicate the correct answer: Neutron cross-sections: a) a re no t affected by neutro nenergy, b) increase with increasing neutron energy, c) decrease with neutron energy inproportion to the reciprocal of the velocity.5. W hat relatively plentiful isotopes ar e capable of low energy ne utron fission.6. Are the Zs of the isotopes of problem 5 , even or odd (parity)? W ha t is the parityof the As of problem 2?

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Summary of Chapter 1The nucleus of an atom is an extremely small central core containing 99.95% of the massof the atom. Th e nucleus contains protons and neutrons. Th e chemical character thatidentifies it as an element is specified hy the number of protons in the nucleus. Th enumber of neutrons may vary to form different isotopic species of an element that arechemically the same but have different nuclear properties. One isotope may spontaneouslydecay into another or may de-excite by the emission of alpha, beta, gamma, neutron,neu trino particles, o r it m ay und ergo fission.The ra te of emission of particles is proportional to the amount of material decaying, hence,the decay is exponential with time. Radiation is attenu ated by distance from the sourc e ofradiation; for a point source, it decreases inversely with the square of the distance.Radiation is attenu ated by the presence of material between th e object an d the source; thisatte nu atio n is exponential with the thickness of material except for a correction facto r calledthe buildup factor. Th e radiation is proportional to the source's strength (rate of particleemission).Nuclear cross-sections express the area presented by nuclei as a target for being hit byelemen tary particles. T he a rea is expressed in a unit called a "barn", defined to be 10E-24cm2. Cross-sections expressed in these units are tabulated for neutrons impacting variousisotopes. Pen etrating ability also is expressed in half-thickness for ease of calculation.Neutron cross-sections vary greatly according to the isotopic target and with the neutronenergy. Th ere is a gen eral "one-over-velocity" depend ence, but a t neutron energies above1 eV, th ere are "resonances" in which the cross-section varies as much as 100 with smallenergy changes.If neutrons a re cap tured in certain odd-Z, odd-N nuclei specifically, Uu3,UD5 nd P u ' ~ ~ ,henucleus un derg oes fission with the release of a large am oun t of energy and nu clear particles.This chapter has laid the ground work for the study of the combinations and shapes ofmaterials necessary to produce a chain-reaction - called a criticality event.

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2. NUCLEAR CHA IN REACTIONS AND CRITICALITY2.1 Th e Principle of th e Ch ain ReactionFigure 2-1illustrates the feedb ack process th at occurs in a sustained chain reaction. A t thetop of the Figure, neutron s are born in th e fissioning of a nucleus at an energy of about 0.75MeV. At this energy, the fission cross-section is quite small but it increasesgenerally as l/v. A criticality requiresf issionable mater ial , neutrons, andsurrounding material to slow down theneutrons and t o retain neutrons to continuecaus ing fissions.2.2 Neutron EconomvW het her an economy is growing, shrinking,or staying constant dep ends on the balancebetween expenditure and collection. Th esame is true of a reactor'. A fission pro-duces about 2.5 neutrons, on average. Ifonly one of these neutrons is used toproduce an ot he r fission, the reaction is self-sustaining and will c ontin ue until th e fissilematerial becomes depleted.Th e neutron multiplication abou t a fissioncycle is designated by the parameter k,defined t o be:

Firsion Neutron Spectrum

Group 1

Thermal (Maxxellian)hermal (Maxxellian)neutron spectrumnCroup 2

Figure 2-1 How a Reactor W orks

k = Number of neutrons in on e generationNumber of neutrons in th e previous generationor: k = Neutron production rateNeutron loss rateWhen: k = 1, a rea ctor is critical. It is neither gaining nor losing neutrons.

k < 1, it is losing neutron sk > 1, it is supercritical.

' In this and su bsequent chapters, the word "reactor"means any configuration of materials capable of a chainreaction - not necessarily a device intended to becom e critical and generate power.2-1


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If k = 1, the reactor will continue a t constant power; if k > 1 he power incre ases with eachneutron generation. The bigger it is the f aster th e increase. If k < 1, the power decreases.2.3 T he Six-Factor FormulaT he n eutr on m ultiplication is described by t he six-factor formu la which identifies th e term scontrolling the n eutron economy:

(2-1)* * *k = 77 E P f Pfd*Pbdwhere,77 (eta) - the number of ne utro ns produced in a fission to the number absorbed,E - fast fission fa ctor: ratio of the total number of neu trons prod uced by all fissionsto the number produced by slow neutro n fission ( E = 1.03 for natural uranium),p - reson ance esca pe probability: the probability that neu tron s will esca pe cap turean d reach therm al ener gies wh ere they may cause fission,f - therm al utilization: th e ratio of thermal neu trons absorbed in the fue l to the totalnumber of thermal n eutron s absorbed,pfnr fast neutr on non leaka ge probability, a ndphl therm al neu tron non leaka ge probability.

The first term, 7, of equation 2-1 is a fact of nature concerning th e fissionable materialsselected. Similarly; the fast fission factor , E is a function of geometry and the material. T heremaining two factors may be varied by introducing neutron absorbing material. If theabsorber abso rbs neutrons at e nergies above thermal energies, then p, the resonance escapeprobability will increase . If it primarily absorbs neu tron s a t therm al ene rgies, th e effect willreducep,nl, he therm al utilization; if absorption occurs a t high energy,pfnlwill decrease. T heuse of m oderators to slow and contain neutron s may affect either pfn, r pln1.The six-factor formula is illustrated in Figure 2-2. At the bot tom, 405 neutrons causefission. As 7 = 2.47, 1000 neutrons are produced, some of which produc e fa st fissions (E= 1.04). In the slowing down process, 140 neutro ns ar e lost from fast leakage, an d 300neutro ns ar e lost from resonant capture. O ne hundred neu trons are lost by thermalleakage, 50 by non -fuel material absorption, 45 are lost by captu re processes in th e fuel tha tdo not p rodu ce fission, and finally, 405 prod uce fission to star t the cycle over.

A critical condition can be preven ted or stopped by breaking this cycle.We will now discuss in m ore detail the ele me nts of th e c ritical cycle.

2-2


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2.4 Neutron ControlNeu tron absorbers2 may b e liquids o r solidmaterials. f - 0.90 loo llamms last iPd- 0.833Solid absorbers used primarily to stop areactor are called scram rods; absorbersused to control a reactor are called"control rods". In both cases, the abso rbermay no t be rod shaped3. Examples ofmaterials used as th e neu tron absorber incontrol rods are: hafnium, europium,cadmium, and boron.Solid absorbers may be built into a newcore as "burnable poison" to help holddown th e excess reactivity. As the core isused, th e poison is designed to bum out tocompensate for the reduction in fissilematerial.Liquid reactor control is achieved bydissolving a n eutron-absorbing materia l e.g.boric acid, in th e cooling water. Th is is a

_I p-0.667Pu 0.865

45 nentroaslost

I....................:::;:A>:....... . ...................................................................................

ma.........................................................................Figure 2-2 Six-factors ab ou t th e CriticalCycle

slow con trol mechanism used in conjunction with control rods. Wh en a co re is new, anddesigned to last a long time, it has extra fuel. Liquid abso rber is used to "hold down" th ereactivity t o a conv enien t level for auto matic control by t he control rods.Ga s absorbers are n ot in cu rrent use b ut th ere have been discussions a t the Savannah RiverPlant and O ak Ridge National Laboratory on the use of the gas He3as an emergency shut-down mechanism.If you have ever played pool, you will know that if the cue ball hits ano ther ball dead-on,the cue ball may come to a dead stop and th e othe r ball will go on. If the angle is a littledifferent, both balls will continue moving but more slowly, and if the cue ball hits at aglancing angle, it is hardly affected. This is wh at happ ens if elastic balls of equal masscollide, and also hap pens if a neu tron collides with a proton, such as in water.

' A neutron absorber is an isotopic m aterialwith a large cross-section for absorbing neutrons without resultingin fission.Fermi used cadmium sheets that dropped like a guillotine into the reactor if th e "SCRAM"rope were cut.

In current reactors, scram rods are moved in and out by the "Control Rod Drive Mechanism" (CR DM ).2-3


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As a result of multiple collisions of thetype shown in Figure 2-3, the neutronsloose energy and enter into thermalequilibrium like a 'he utr on gas". Inmany respects, the neutro ns behave likehydrogen gas inside of the reactor Protonmaterials. The rma l neutro ns i.e.neutrons in thermal equilibrium with areactor have a speed corresponding tothe temperature of the reactor. T heenergy of these neutrons has a Maxwell-Boltzmann distribution as shown inFigure 2-4. Th e average energy of neu-trons in this inverted "bell curve with atail" is:

After

Before

Neutron\eutronFigure 2-3 Neutron-P roton Collision

E(ev) = 8.61E-5*T("K). (2-2)A t room tem perature, 20C (68"F), th e neutron energy is 0.025 eV and i ts speed is 2200 m/s(5,000 mph) so the left part of Figure 1.3 encompasses the thermal distnbutior(except for a low en ergy tail).

The fact that the average energy ofthermal neutrons- s proportionalto the temperature and the factthat neutron cross sectionsincrease as neutron energydecreases explains why somesolutions and reactors go criticalwhen they cool-down. EFigure 2-4 Therm al Energy Spectrum2.5 Lethargv. nd the A verageLogarithmic Energv Decrement

If you average t he en ergy loss of a neutro n ov er th e various possible scattering angles, aquantity called the average logarithmic energy decrement, E , may be calculated:E =


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Review Ouestions1. What moderator can cause in a chain reaction in natural, unenriched uranium?2. On he average, how many collisions must a fission-energy neutron make with3. State the six-factor formula and explain its terms.4. How much does the average thermal energy change when a reactor cools from

hydrogen nu clei to reach therma l energy?

250C to 28C. (Remember to use Kelvin temperature.)

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2.6 Nuclear ReactorFigure 2-5 is a highly simplified diagram of a reactor such as a Savannah River Siteproduction reactor. N eut ron s Droduce fissions in th e core which consists of fissionable

/ control rodt;;;M;lechanism,,/at 'changer

I

uranium metal encased inaluminum tubes. Heavy water(D,O) flows in the spacebetween tubes to serve twopurposes: remove the heat fromthe fissioning, and to moderate

~ ~ ~ f i ~ r yhe high-energy neutrons fromP E P fission. No t all of the neutronsw-toheatsink

will be contained in the coreregion. To reduce neutron loss,the cor e region is surrounded byan additional D,O moderatorcalled the reflector. Its purposeis to reflect back, into the core,as many neutrons as possible -hence to keep the p*f factors inthe six-factor formu la a s close to1 as possible.

pressure vesselL

Figure2-5 Simplified Reactor

isolationvalves The reactor is controlled byintroducing neutron absorbingrods to reduce k by reducing thep*f facto rs. W hich facto r is pri-marily affected will depend onthe energy region in whichabsorption takes place. Th e control rods are moved up and down by the control rod drivemechanisms. Som e types of reactors dissolve a neutron abso rber (such a s boric acid) in theprimary coolantlmoderator.

The primary envelope consists of the pressure vessel (also called the reactor vessel), thepumps, a nd the piping th at connects with the hea t exchanger which is used to transfer theheat from the primary loop into a secondary cooling loop so that primary water which isslightly con tam inated with radioactivity is not exposed to the environment.A reactor is designed to go critical and o pera te with a power producing chain reaction. Itis standard practice to enclose the chain-reacting core in an enclosure called thecontainme nt as shown in Figure 2-5 (although the Savannah River reactors were designedbefore this was the practice). Th e containmen t is the last line of the defense-in-de pth policyof multiple barriers protecting the public.

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Review Q uestions1. What is the purpose of the reflector in a reactor? Th e fuel? Th e control rods?2. The k factor of a reactor is slightly temperature dependen t. Which way should3. The coolant of a reactor serves two purposes. What are they?it change for safety?

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2.7 Time ResDonse of a Nuclear Chain ReactionPrompt PeriodThe rate of disappearance of neutrons, N, from a reactor is proportional to the number ofneutrons in it:

dN/dt = N E (2-6)where T is called the reactor period, which is the average time from fission to disappearanceof the neutron either by escape, capture, or capture-fission; T will depend on the averageneutron velocity and the size of the reactor. A fast reactor is characterized by the wholeprocess from fission to fission capture taking place at high energy - in the range betweena few MeV and a few keV. A fast reactor will have a T of, sa y 10 psec, an atomic bombhas a T of about 0.01 psec. A thermal reactor, such as a production reactor, has a T ofabout 0.001 sec. The number of neutrons, N, is related to the power, p, being produced bythe reactor. With this substitution, equation 2-6 is easily integrated to give:

This equation shows that the power will increase by 2.71828 in a time t=T. For example,a thermal reactor will increase its power by 10,000 times in 1 second (t=lO*T).From this, it would seem that a reactor is difficult to control. Luckily, this is not the wholestory, because some neutrons are delayed.Delaved PeriodBut nature is not so perverse as to leave us in this situation. Instead, about 0.75% of theneutrons are delayed because they are associated with beta emission. As we saw in Figure1-5,when a nucleus undergoes fission, it breaks into two parts centered about A = 90 andA = 140. For example, consider the fission reaction:

9235U + $ - z 7 L a + g B r + 2*#EBr - 5.6 sec. + B- + ZKr

ZKr-ZKr+$

(2-8) .

The first equation shows the fissioning of uranium-235 into lanthanum 147 and bromine-87with two prompt neutrons (as required to balance the nucleons). Bromine-87 emits anelectron with a decay constant of 55.6 sec. changing to krypton-87, which decays to krypton-86 by emitting a neutron. This is just one of the reactions that produce delayed neutrons.Table 2-2 summarizes the delayed emission as 5 groups.

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It may seem th at because th e delayed neu-trons are only 0.75% of the neutronsemitted that they are not significant. How-ever if k=99.25% on the prompt neutrons,the delayed neutrons make the differencebetween being critical an d subcritical. Th uscontrolling the reactor on the delayedneutro ns raises the time constant to tens ofseconds instead of milliseconds.PromDt and Delayed CriticalAn imp ortant distinction is ma de b etweenk 2 1 without including the delayedneutro ns or including the delayed neutrons.Th e former is called prom pt critical. It isa run-away condition tha t usually is unstoppab le. T h e reaction must go to completion,which usually means that so much heat and pressure is generated that the reactordisassembles itself. For example, it blows the liquid out of solution criticality o r d istortsmetal parts to increase the neutron leakag e sufficiently to stop. This whole process happ ensvery fast. A solution, assembly, or reactor that is delaved critical means that the delayedneutrons' which con stitute 0.79% of all of the neutrons, cause k 2 1. A small amoun t ofabsorber will shutdown the reaction and shu t it down with a time constant correspondingwith the delayed neutron times. All controlled nuclear reactors operate in the delayedcritical region.Review q uestions

1. What causes the neutrons to be delayed in being emitted?2. Why ar e the re five delayed n eutron groups?3. Thirty seconds after fission, what fraction of the delayed neutrons have beenemitted?

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2.8 Rules for Avoiding Criticalitv AccidentsTh e six-factor formula (eq uation 2-1) gives us gu idan ce for avoiding criticality. Anythingthat b reaks the critical cycle will prevent criticality. O ne m ethod is to mak e i t easy for theneu trons to escape by controlling the:

Geometry.For an assembly, or solution t o g o critical, it must be likely th at a n eutro n born from fissionwill cause another fission. Neutrons from fission are prevented from escaping beingsurrounded by either more fissionable material or by material that reflects the neutronsback to fissionable materials for anoth er try a t fission. Th e geometry that d oes this best isthe sphere4. Spherical geom etries will go critical with less fissionable mate rial th an anyothe r shape. Conversely, this is why fissionable ma terial is stored in rod -like cylinders - tomake it easier for the n eutro n to escape without causing fission.A n assembly whose shape is such that it cannotpossibly go critical is called "geometrically afe."An other meth od for controlling fission is:

Neutron absorption.Abso rbing ne utro ns in a non-fission reaction is essentially the s ame a s letting them escape.Criticality may be avoided by the use of such materials as boron (especially the isotopeboro n-lo), cadmium or oth er materials used for scram or control rods.Process streams may contain dissolved neutron absorbers (called "poisons") or absorbersmay be build into the walls of the process containers - called rashig rings.Th e speed o n th e neu trons, hence the fission cross section d epends o n:

TemperatureRaising the tem perature of an assembly, or process raises the average thermalized neu tronenergy reducing the fission neutron cross section, thereby increasing the possibility thatneutrons will escape or be captured in a non-fission reaction.Neutrons may be slowed by collisions with light-weight nuclei which increase the fissioncross section. T he slowing-down process is called:

' A sphere has the least surface area for a given volume of any figure.2-11


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ModerationLight nuclei, including the hum an body are effective moderators. Th e presence of suchmaterial causes neutrons to be reflected back into the assembly and slows the neutrons.Both of these processes increases the reactivity and may make a subcritical assembly gocritical.Th e process of returning neu trons for another try a t fission is called:

ReflectionMaterials that are good moderators a re often good reflectors even thou gh th e principles ofmoderation and reflection are different. The purpose of a mo derator is to slow neutrons;the purpose of a reflector is to cause n eutro ns to change direction.Review Questions

1. Name 5 things affecting c riticality.2. W hat is the meaning of "geometrically safe"?3. W hat is a good sha pe for geometric safety?

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Summary of Chapter 2Whether or not a reactor, assembly or solution goes or exceeds criticality depends on thek-factor.

If k= 1, the reactor is iustcritical i.e. as many neutrons are being generated as arebeing lost.If k = kprompt 2 1, the reaction is prompt critical and must stop itself;If k = kprompt neutrons + belayed neutrons 2 1 but kprompt neutrons < 1, the reaction is delavedcritical and is con trollable by neutron absorption or leakage if done promptly;If kprompt + kdelayed < 1, the reaction is sub-critical.

The 6-factor formula, used to calculate the k-factor, expresses the neutron budget about thechain reaction cycle. The six factors account for the number of neutrons produced byfission, additional neutrons from fast fission, the probability that neutrons are lost in slowingdown from high energy, through the resonance region to reach and enter into thermalequilibrium. The six-factor formula accounts for neutron loss in the thermal region, non-fission capture, and for the number of neutrons born in fission.This neutron slowing down process is natural and increases the likelihood of fission as slowneutrons are more likely to cause fission than fast neutrons. Light nuclei (small Z numbers)are used to slow down (moderate) the neutron energy to enhance the fission process.The chain reaction takes place very rapidly, but some neutrons are released by beta-decayprocesses, having half-lives of the order of seconds. These delayed neutrons give the timeneeded to control a reactor.Criticality accidents have occurred. Criticality may be prevented by interrupting the chainreaction cycle i.e., modifying terms in the six-factor formula.

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k,*exp(-B2*T)/( 1+L2*Bz) = 1, (3-3)B, is the material buckling tha t can be related to the six-factor form ula by identifymg th efast-neutron nonleakage probability to be:

and the thermal neutron nonleakage probabili ty to be:ptnr= 1/(1+L2*Bm2) (3-5)

In the limit of large buckling, exp(-Bm2*T)= 1 - B,~*T) = 1/(1+Bm2*7) nd equation 3-3becomeskJ(l+Bm2(T+LZ) = kJ( l+Bm2*M2) = 1, (3-6)

where M = ~ ( T + L * )s called the migration length. This is called the m igration ar eamethod.Ge ome tric Buckling,After the neutrons enter into thermal equilibrium with the material, they act like a gasleaking from an enclosure - very similar to the problem of the temperature distribution ina h ot block of material in which hea t is flowing from the bound ary to the surroundings. Forneutron diffusion, the equation is:

V%P + B,2*+ = 0, (3-7)where 4 (phi) is the n eutron flux defined a s the num ber of neutrons per unit volume timesthe ir velocity, B, is th e geo metr ic buckling, an d V (del) is a partial differential operatorused to m athematically describe diffusion from an enclosure.For a slab, one dimensional reactor, equation 3-7 is solved as:

4 = ~,*cos(x*B,), (3-6)where $, is the flux in th e ce nter of the slab. Notice 4 is a maximum in the center (x =0) and goes to zero when x = ,+(a/2+S). Generally, the flux goes to zero a t an additiondistance 8 , called the extrapolation distance.

It is clear that equation 3-4 can be written as B; = -T+/+.Th e quantity on the right is the curvatureor buckling in mathematics, hence, the reason John Wheeler, a d eveloper of fission theory and reactors, gaveit the name buckling.

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Fig.3-1 Slab Flux ProfileTable 3-1 includes a factor S in addi-tion to the physical length to accountfor the extrapolation length or thesum of the extrapolation length andthe reflector savings.

Figure 3-1 illustrates the fact that the geometricbuckling is the distance from the peak flux to theextrapolated vanishing point. Th is is a very simplifiedexposition of buckling to illustrate the concept. Th ebuckling of a complex reactor such as a processsystem criticality accident is much m ore complex.

Cylind er, radiu s r, [2.405/(r+S)]2+Cub e, dimen sions [.rr/(a+2*S)I2+[T / ( +2 *a)] +

height h [?r/(h+2* a)]*a, b, c [r/(+2*S)]2

Buckling in Regular GeometriesAn infinite slab geom etry is no t realistic. Ta ble 3-1provides the geometric bucklings for a sphere, an d acube.

Table3-1 Bucklings for Regular GeometriesI

II Geometry

Critical ConditionTh e reactor becom es critical when the geom etric and m aterial bucklings are eq ual:

B2 = B,2 = Bm2, (3-7)the refo re, the task of estimating criticality condition s is on e of estimating the m aterial andgeom etric bucklings. A t critical, the two bucklings are th e same; hence, they ar e designatedby B2.A reactor is subcritical when:

B,2 < Bm2,and supercritical when: B,2 > B,?

(3-8)

(3-9)

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3.2 Measurement of CriticalityTheories alone, especially those formulated before modern computers, lacked sufficientprecision for risking health and safetyon the calculations. These theories were supportedby experimental measurement of the conditions for criticality. In fact, experimental resultswere used to calibrate calculations and provide a basis for their refinements.Critical FacilitiesA critical facility is designed for determining the conditions for criticality. Because of thedangers involved, these facilities are remote and can be operated from a safe distance.Criticality measurements were performed before the Los Alamos Pajarito Site beganoperation in 1948, but this oldest of criticality facilities has regularly performed measure-ments on all three of the fissile species.Criticality facilities exist or existed at most of the National Laboratories, and a t someof theindustrial reactor designers and nuclear material processors but usually focus on criticalitydeterminations for their mission geometries and compositions. For example ArgonneNational Laboratory performed Zero-Power Reactor and Zero-Power Plutonium Reactor(ZPPR) experiments in support of the design of the breeder reactor.Mechanicallv Separable FacilitiesExperimentsperformed with solid fuels and moderators use mechanical methods to stop thereaction before damage is done. For a large assembly such as a reactor lattice, this takesthe form of a horizontal or vertical split bed. The assembly is divided in half; one half isstationary and the other is movable. The two halves are slowly moved together, andneutron multiplication is measured. If multiplication becomes excessive, during theapproach of the two halves, indicating the onset of criticality, the halves are quicklyseparated.The process is similar for highly enriched, spherical geometries, such as those investigatedat Los Alamos or Livermore Laboratories. The two halves, which may be bare spheres ofuranium or plutonium can move together to determine the neutron multiplication andrapidly separated. In some experiments, super-critical assemblies are brought together toseparate themselves by the large energy release from the chain reaction.Liquid Dump FacilitiesExperiments with solutions containing fissile materials are performed similarly. Theso h ions, contained in geometrically safe containers and pipes are slowly transferred intoa container with the shape being investigated. If multiplication becomes excessive, valvesrapidly open to discharge the solution into a safe geometry.

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From many critical experiments, a criticality databas e has b een constructed for special andstandard geometries, com binations of fuel, moderator amounts, and types.Measurement of MultiplicationSuppose we wish to m easure the k-factor of a potentially critical reactor e.g. a sphericalstorage vessel which contains plutonium a t different concentrations. T he object of th eme asurem ents is to determ ine t he crit ical concentration.A neutron source is located in the vessel and one or mo re neutron detectors, such as boron-trifluoride counters, a re located abou t the container. Th e count ing rate of the .neutrondetectors are measured when the solution contains no plutonium; call this R(0).Adding plutonium increases k, and multiplies the number of neutrons in the reactor,resulting in a detected neutron counting rate R(k), and a neutron multiplication of

M = R(k)/R(O) (3-12)T he n eutr on m ultiplication is:

M = R(O)*k + R(O)*k* + R(0)*k3 + .... /R(O)where the definit ion o f the k-factor is:

k = number of neutron s in o ne generation.num ber of n eutrons in th e previous generationEquation 3-13 is an infinite series that can b e written as:

and M = l / ( l -k) .k = (M-1)/M

(3-13)

(3-14)(3-15)

Numerical ExampleW hat is the k-factor of an unreflected sphere 10 cm radius which gives a ne utron countingrate of 100 cpm when filled with a solution without plutonium, and 10,000 cpm when theplutonium concentration is 5 kglt? T he multiplication is 100. Substituting into equation3-15: k = 99/100 = 0.99.Extrapolated CriticalityPlotting the k-factors obtained from many ne utron multiplication experiments as a functionof a reactor param eter (such as plutonium concentration) enables the extrapolation of the

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point at which the parameter wouldresult in criticality. Graphs of parame-ters that result in criticality are pre-pared from a multitude of experimentsto provide guidance for safe plant oper-ations.

1 u-235

Review Ouestions

1 0.78

1. Describe a critical facility fora solid assembly.2. Describe a critical facility forsoh ions.3. What is the neutron multipli-cation factor of process container witha k-factor of 0.95?

I1 6.2

Table 3-2 Single Param eter Limits for U niformNitrate Aaueou s Solutions of Fissile Nuclides

I 11.6

ParameterFissile mass (kg)Cylinder dia. (cm)Slab thickness (cm)Solution volume (e)Fissile conc entration(de)Area l den sity (g/cm*)Hydrogen/fissileatomic ratio

Parameter U-235 U-233 Pu-238Fissile mass (kg)Cylinder dia. (cm)

( 4Slab thicknessEnrichment

14.4

20.1 6.0 5.07.3 4.5 4.41.3 0.38 0.65

5%

4.9

Maximum density 18.81 18.65 19.52

0.43250

U-233 Pu-239*10.8

0.35 0.25

2390 I 3630

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3.3 Subcritical LimitsEvery process situation or measurementthat may be encountered cannot be im-mediately evaluated experimentally -o rtheoretically to determine if it is danger-ous. For practicality, guidelines contain-ing safety margins must be available.Single-parameter limits are the mostrestrictive; multi-parameter limits are lessrestrictive and allow a closer. approach tothe critical condition.Single Parameter LimitsTable 3-2 provides subcritical limits foruniform aqueous solutions of the threefissile nuclides in which one single pa-rameter, fissile mass, is controlled to pre-vent criticality. Aslong as the controllingparameter is not exceeded, criticality isnot possible. The values assume thatthere is unlimited water reflection. Thelinear limits refers to the diameter of theinfinite cylinder, or to the thickness ofinfinite slab. The limit on atomic ratio is

0 5 10 15 20 25C R I T I C A L RADIUS .cmFigure 3-2 Critical Mass and Radius for aBare and a Reflected Plutonium-SolutionSphere

equivalent to the limit on solution's concentration, but may be applied to non-aqueoussolutions regardless of chemical composition.Table 3-3 presents the single-parameter limits on fully water-reflected metal assemblies.The mass limits assume that there are no re-entrant holes filled with moderator.Multiple-Parameter LimitsConformance with the single-parameter limits would prevent the use of the 100,000 kginventories of UO, that must be fabricated for an initial reactor-core loading. Multi-pa-rameter limits relax the single-parameter restrictions through administrative constraintsonoperations. Control of solution concentration for specified geometries allows the use ofmu1 i-parame ter limits.Figure 3-2 shows the multiparameter limits for a spherical aqueous solution of plutonium.The ordinate is the critical mass and the abscissa is the critical radius. The upper curvepresents data for a bare (unreflected) spherical container of the solution; the lower curvepresents data for a fully water-reflected sphere. The lines at 45" angles indicate the

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plutonium concentration a t the intersection of these lines with the curves. Th ese dat a showthat a fully reflected plutonium metal sphere, as small as 4 cm radius, density 19.6 g/cm3,is critical if fully reflected an d would co ntain 5.3 kg of plutonium. If the sphere is bare, alarger quantity, 9.8 kg, of plutonium is needed fo r criticality in which case t he r adius is 5cm .402

S

2

3 40m

VII$ 5I24a2 2uanYVI coo

5

2

1 6 10-2 2 5 40-1 2 5 too 2 5 40 2URANIUM CONCENTRATION ( h p U / l )

Figure 3-3 Multi-Parameter U-235 Spherical Sub-criticality Limits

Figure 3-3 presents multiparameter l imits for a sphere of U-235 aqueous solution as afunction of uranium concentration and sphere mass. T he lower curve applies to 30 cm ofwater reflection; the upper curve applies to 2.5 cm of wa ter reflection. It is see n tha t theminimum uranium mass to go critical is 0.6 kg in a solution 0.05 kg Ul t , or about 500hydrogen atoms for every uranium atom.Figure 3-4 shows multiparameter limits for U-235 solution in slab geometry. T h e leftportion of th e curve applies to a uranyl dioxide difluoride solution an d th e right to uraniumme tal mixed with water. T h e upp er curve is fo r 2.5 cm thick moderation; the lower curveis for 30 cm thick moderation. It may be no ted that slabs as thin as 1.3 cm a re critical iffully reflected and in concentrations as high as 2 kglt.

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2

to

InIn3Y 2

2

Review Questions1. W hat is the purpose of criticality lim its?2. W h at is the single param eter limit for spherical fissile mass in nitrate aqu eou s3. W hat is the single param eter limit for spherical fissile mass in a metal assembly?4. Why ar e multiparameter l imits needed?5. W h at is the sm allest amo unt of plutonium th at can b e critical?6. W hat is the smallest amo unt of U-235 tha t can be critical?7. W h at is the radius of a fully reflected sphere of Pu-239 solution that will result

solution mass?In a solution mass?

in criticality?

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3.4 Calculation of Criticalitv without a Com puter

2 -

Unfortunately, there are many complex geometries and material combinations that occurin process an d experimental situations tha n can be accurately represen ted by multiparam etercriticality grap hs. T h e criticality asp ects of these complex geometries can b e calculated withcomputers, but several methods have been developed that are simple enough for handcalculations.

-- Ii

-+I m- o I ^ ! - --UNREFLECTEDI I

BucklindS ape C onversion

i-- I NF I N I TE SLAB

The buckling/shape conversion method determines the equivalent geometry for whichcriticality limits are available to the geometry being assessed for criticality by relating thebuckling of one to the o ther . If the buckling is greater than the subcriticality limited

I INFINITE CYL INDER

7 Ir I

6

A , H / u ~ ~ =3200 , S P H E R ESOLUTI ONS I N /qs-IN - T H I C K A L U M I N U M C O N T AI N E RSa I3 I I 1 I I II I I I

Fig. 3-5 Extrapolation L engths for Cylinders Containing UO,F,(h=height ; d=diameter)buckling shap e, the unknow n is deem ed to be subcritical; it is deeme d to b e critical if itbuckling is less.However, to use these formulae, the extrapolation distance, 8 , is need ed. While theextrapolation distance is important, it need not be known precisely because it is usuallysmall compared with the bucklings provided in Tab le 3-1for several regular geometricshapes.Figures 3-5 and 3-6 provide extrapolation length data for uran ium and plutonium cylinders,specifically for uran ium enriched to 93.2% in the isotope U-235 in Figure 3-5, while Figure

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3-6 is for enrichm ent to 93.5%.

4 5

4 0

3 52uU

3 0

\R- RE FLE CTE D U(93 51t-r 11 /-e r A-

I /

.C,,a4*---.- -I I

URANIUM(48 8 G OF U R A N I U M / C M 3 )35 65 G OF PCUTONIUM/CM3)

o-n*4-= AII

lN F l NIT CYLl N D E RI I ,

0 , SPHERE A , S P HE RE0 , CYLINDER 0 , CYLINDER

2 0 )

d 5 '

I 1- I NF I N I T E S C A BG O t 0 2 0 3 0 4 0 5 0 6 0 7 O B 0 9 ( 0

( h / d ) / [ ! + (h /d i ]Fig. 3-6 Extrapolation Lengths for Cylinders of Uranium M etal and &ph ase Plutonium(h=height; d=diameter)

Example of BucklindShaDe ConversionSuppo se we have a fully reflected, cylindrical process con taine r tha t is 30-cm high a nd 20-cm in diameter. The rat io (h/d)/[l+(h/d)] = 1.5/(1+1.5) = 0.6. Figure 3-5 gives theextrapolation length 8 = 5 .9 cm . To find the spherical equivalent to the cylinder, we equatetheir bucklings from Table 3-1:

ET/( r+a)]' = 12.4054 +a)]'+ [ r / ( h +2*S)I2[T/( +5.9) 2 = [2.405/(10+5.9)]*+ [T / ( 0+2 *5.9)]2,

(3-14)(3-15)

where the same extrapolation length is used for the sphere as for th e cylinder. T he resultis that the equivalent sphere is 12.7 cm radius, which is equivalent t o a spherical volume of8.58 e . The concentration is C = m/V, where m is the mass and V is the volume.Substituting for V, m = 8.58*C. This is the equation of a straight line on a log-log plot witha slope of 45" which has been affixed to Figure 3-5, as shown in Figure 3-7. T h eintersection of this straigh t line with th e fully reflected (lower) curve occ urs at a mass of 7kg of ura nium which also corresponds with density of 0.815 kgle. Thus, a concentration inexcess of 0.815 kgle will go critical. T he volume of the cylinder is 9.724 e, hence, themaximum subcritical uranium con tent is 7.68 kg of U-235.

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In this example, we found th e sp here equivalent to a cylinder. It also would have bee ncorrect to relate a finite cylinder to the criticality limits for a n infinite cylinder. Howevershape conversion between quite different shapes, such as spheres and infinite slabs, maylead to error.Surface Dcnsitv Method

10-2 2 5 40-' 2 5 IO0 2 5 to' 2U R A N I U M C O N C E N T R A T I O N [ k g U / l l

Fig. 3-7 Figure 3-3 Modified to Include Mass-Concentration Equa tion

Consider a fissile inventory storeroom stacked with shipping containers like the " b ir d ~ a g e " ~con tainers shown in Figure 3-8. If a catastrophe occurred and th e material were depositedon the floor, forming a slab, and this slab became flooded with

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