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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