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Fundamentals of Nuclear Engineering
Module 2: Radioactive Decay
Dr. John H. Bickel
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Objectives:1. Explain physical rate laws for radioactive
decay
2. Explain concept of half-life
3. Explain concept of branching reactions
4. Describe natural radioactive decay chains
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Radioactive Decay• Why we need to understand this:
• Delayed neutrons emitted in fission arise from radioactive decay processes
• Certain delayed fission products: 54Xe135, 62Sa149 are very strong neutron absorbers and impact reactor control
• Decay heat from power reactors persists according to radioactive decay processes.
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Rate of radioactive decay• Radioactive decay follows simple 1st order rate law• Rate of disintegration is proportional to quantity present:
dN /dt = -λtot N • N is measure of quantity present.• N could be expressed in: grams, moles, # of atoms• λtot is rate constant – with units of sec.-1, hr.-1, yr.-1
• λtot is based upon unique internal physics and energy levels of disintegrating nucleus
• λtot can be theoretically derived from physics, but in general is just experimentally measured
• λtot can be expressed as linear combination of rates of different decay processes (α, β… decay)
• λtot = λα + λβ + …
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Rate of radioactive decay
• Solution of: dN/dt = -λtot N via integration yields:• N(t) = Noexp(-λtot t) -where No is the initial quantity
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Concept of Half-life
• Rate of radioactive decay can be characterized by time required to decrease by factor of ½.
• T1/2 is obtained by solving: • N(T1/2)/No = Noexp(-λtot T1/2)/No = exp(-λtot T1/2) =½• -λtot T1/2 = ln(½) thus: T1/2 = -ln(½)/λtot = 0.693/λtot
• Or: λtot = 0.693/T1/2
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Decay, half-life, data from Isotope Table
Source: http://ie.lbl.gov/schematics.html
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Activity of radioactive sources• Amount of material No based upon measured count rates or
“activity”.
• Activity: A(t) = -dN/dt = λtot N• Historical unit for activity: curie (Ci)• 1 curie (Ci) is activity equivalent of 1 gram: 88Ra226
• Activity is from: 88Ra226 86Rn222 + 2He4
• 1 curie (Ci) = 3.7x1010 dps (disintegrations/sec.)• S.I. unit for activity is: Bequerel (Bq)• 1 Bq = 1 dps (disintegrations/sec.)
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Example of Radioactive Decay
• Someone wants to ship a 6,000 Ci 27Co60 source.
• Physically how big is this?
• The reaction is: 27Co60
28Ni60 + -1β0
• With: T1/2 = 5.27 yrs.
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Example of Radioactive Decay• A(t) = λtot N(t) thus: N(t) = A(t)/λtot
• A(t) = (6x103Ci)(3.7x1010dps/Ci) = 2.22x1014dps
• λtot = 0.693/T1/2 thus: N(t) = A(t)T1/2/0.693
• T1/2 = (5.27 yrs.)(8760hrs/yr)(3600sec/hr) = 1.659x108 sec.
• N(t) = (2.22x1014dps)(1.659x108 sec.)/0.693 = 5.315x1022
• MCo = (59.93 AMU)(1.6604x10-27kg/AMU)(5.315x1022)= 5.28 g
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Radioactive decay chains
• Involve changes from A B C … X
• Buildup/decay also follow first order rate law.
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Waterfall Analogy: Growth and Decay Chains
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Radioactive decay chains
• General formulation involves system of linear differential equations:
• dX0/dt = [material added] – [material which decays]
• dX1/dt = - λ1X1 -X1 decays, no replenishment
• dX2/dt = λ1X1 – λ2X2 -X2 decays, replenished by λ1X1
• dX3/dt = λ2X2 – λ3X3 -X3 decays, replenished by λ2X2
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Radioactive decay chains• General solution for 2-stage growth/decay process:• NA(t) = NA(0)exp(-λAt)• NB(t) = λANA(0)[exp(-λAt)- exp(-λAt)]/(λB –λA) + NB(0)exp(-λBt)• When: λB >>λA daughter product builds up and matches
parent• When: λB <<λA daughter product builds up and lingers long
after parent has died off
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Equilibrium in radioactive decay chains• As in “waterfall analogy” long term production/decay will lead to constant
inventories.
• In this case all derivatives vanish: ~0
• dX1/dt = ~0 = P∞ - λ1X1
• dX2/dt = ~0 = λ1X1 – λ2X2
• dX3/dt = ~0 =λ2X2 – λ3X3
• X1 = P∞/λ1
• X2 = λ1X1 / λ2 = λ1(P∞/λ1) / λ2 = P∞/λ2
• X3 = λ2X2 / λ3 = λ2(P∞/λ2) / λ3 = P∞/λ3
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Radioactive decay with branching
• d[Ac227]/dt = λPa [Pa231] – λAc [Ac227]
• d[Fr223]/dt = αoλAc [Ac227] – λFr [Fr223] - αo, branching ratio: λα / λtot
• d[Th227]/dt = (1- αo)λAc [Ac227] – λTh [Th227]
• d[Ra223]/dt = (1- βo)λFr [Fr223] + λTh [Th227] – λRa [Ra223]
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Radioactive decay with branching• Th231 buildup limited by U235 which has T1/2 =7.1x108 yrs.• Treat decay chain via assuming equilibrium rate: P∞ ~ λU [U235]• Then relative amounts fall out simply as:
[Th231] = P∞/λTh
[Pa231] = P∞/λPa
[Ac227] = P∞/λAc
[Th227] = (1-α)P∞/λTh
[Fr223] = αP∞/λFr
[Ra223] = [(1-α)+(1-β)α]P∞/λRa= (1- αβ)P∞/λRa
[At219] = αβP∞/λAt
[Rn219] = [(1-γ)αβ+(1-α)+(1-β)α]P∞/λRn = (1- αβγ)P∞/λRn
[Bi215] = αβγP∞/λBi
[Po215] = [αβγ+(1-γ)αβ+(1-α)+(1-β)α]P∞/λPo = P∞/λPo
• Thus: given just branching coefficients and half-lives one can determine relative abundances of an entire decay chain.
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Radioactive Decay Found in Nature
• Dominant radioactive decay sources in environment are: U238,U235, Th232
• Uranium, Thorium are more abundant in earth’s crust than Gold or Platinum
• Uranium, Thorium found in granite, soil, limestone, and coal at ~4ppm as relatively insoluble oxide forms.
• Primary health hazards are from various Radon gas isotopes given off in closed spaces
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U238 Natural Decay Series• Natural U238 decays to Pb206
• Decay chain controlled by 4.468x109yr U238 half-life
• All others, with exception of: U234 Th230 +α decay occur with relatively short half-lives
• Ra226 Rn222 +α is principle source of Radon
• Rn222 is inert noble gas and inhalation hazard
• Rn218 also produced
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U235 Natural Decay Series
• Natural U235 eventually decays to Pb207
• Decay chain timing controlled by 7.038x108yr half-life of: U235 Th231 + α
• All others decay occur with relatively short half-lives
• Shorter U235 half-life compared to U238 explains 0.72% relative abundance
• U238 abundance is 99.27%
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Th232 Natural Decay Series
• Natural Th232 eventually decays to Pb208
• Decay chain timing controlled by 1.405x1010yr half-life of: Th232 Ra228 + α
• Ra224 Rn220 +α is second major environmental source of Radon gas
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Man Made Radioisotopes
• Neutron activation in reactor creates fission products and artificial “heavy elements” not found in nature
• Some “heavy elements” are fissionable and valuable as fuel sources, or as thermionic heat sources.
• Example artificial heavy elements used as nuclear fuels: Pu239, U233
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Pu239 Production
• U238 + n U239 + γ via (n, γ) reaction
• U239 Np239 + β- t1/2 = 23.45 minutes
• Np239 Pu239 + β- t1/2 = 2.356 days
• Pu239 U235 + α+ t1/2 = 24,110 years
• Pu239 is fissionable
• Decay of U235 follows previously described natural radioactive decay series leading to Pb207
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U233 Production
• Th232 + n Th233 + γ via (n, γ) reaction
• Th233 Pa233 + β- t1/2 = 22.3 minutes
• Pa233 U233 + β- t1/2 = 26.967 days
• U233 Th229 + α+ t1/2 = 1.592 x 105 years
• U233 is fissionable
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Summary• Radioactive decay process governed by simple first order
rate law: - dN/dt = λtot N(t)• Decay rate is inversely proportional to half-life: λtot = 0.693/T1/2
• Decay rates for one isotope are additiveadditive: λtot = λα + λβ + • Half-lives are not additivenot additive• Decay chains can be modeled like a series of waterfallsseries of waterfalls• Relative abundances of all isotopes can be related to decay decay
ratesrates and decay branching ratiosdecay branching ratios• Man-made isotopes can be made by neutron bombardment
followed by decay processes
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