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PHY20004NuclearPhysicsClass4:RadioactiveDecayLaw
Inthisclasswewillstudythemathematicallawwhichdescribesthestatisticsofradioactivedecays
withtime,anditspowerfulapplications
Thestatisticsofradioactivity
• Radioactivedecayisarandomprocessgovernedbyprobability
• Wecannot predictwhenanindividualnucleusdecays,butwecan predicttheaveragenumberofdecaysofalargenumber
Wecannotpredicttheresultofasinglediceroll,butwecanpredictthenumberof6’swhenalargenumberofdicearerolled
Credit:http://ch302.cm.utexas.edu
Mathematicsofdecay
• Theprobabilitythatanindividualnucleuswilldecayinatimeinterval𝑑𝑡 is𝑑𝑃 = 𝜆𝑑𝑡,where𝜆 isaparametercalledthedecayconstant
• Therefore,givenalargenumberofnuclei𝑁,thenumberofnucleithatdecayintime𝑑𝑡 is𝑑𝑁()*+, = 𝑁𝜆𝑑𝑡
• Thisreduces thenumber,so𝑑𝑁 = −𝑑𝑁()*+, = −𝑁𝜆𝑑𝑡
Thedecayrateishencegivenby:
𝑑𝑁𝑑𝑡 = −𝜆𝑁
Solvingthisequationforthenumberremainingattime𝑡:
𝑁 𝑡 = 𝑁 0 𝑒012
• Thisisthefamouslawofexponentialdecay
Mathematicsofdecay
• Theradioactive half-life 𝑡3/5 isthetimerequiredforhalfthenucleiinagivensampletodecay
𝑁 𝑡3/5 =12𝑁(0)
12𝑁 0 = 𝑁 0 𝑒012:/; 𝑡3/5 =
ln 2𝜆 =
0.693𝜆
Credit:https://courses.lumenlearning.com
Populationdropsby3
5ineachtime
interval𝑡3/5
Mathematicsofdecay
• Theradioactive half-life 𝑡3/5 isthetimerequiredforhalfthenucleiinagivensampletodecay
𝑁 𝑡3/5 =12𝑁(0)
12𝑁 0 = 𝑁 0 𝑒012:/; 𝑡3/5 =
ln 2𝜆 =
0.693𝜆
• Themeanlifetime𝑡̅ istheaveragetimetakenforanucleustodecay,givenprobabilityofsurvivingtotime𝑡 isC(2)
C(D)= 𝑒012
𝑡̅ =∫ 𝑡𝑃(𝑡)𝑑𝑡FD
∫ 𝑃(𝑡)𝑑𝑡FD
=∫ 𝑡𝑒012𝑑𝑡FD
∫ 𝑒012𝑑𝑡FD
𝑡̅ =1𝜆
Unitsofradioactivity
• Thelevelofradioactivityofasampleismeasuredasthenumberofdecayspersecond(
GCHIJKLG2
= 𝜆𝑁)
• 1Becquerel (Bq)sampleradioactivityisequivalentto1decaypersecond(thisisaverysmallnumber)
• AmorepracticalunitistheCurie (Ci),definedsuchthat1Ci = 3.7×103DBq
MarieandPierreCurie(NobelPrizes1903,1911)
Radioactivedecaychains
• Theproductsofradioactivedecaymaythemselvesbeunstabletodecay– creatingadecaychain
Credit:http://hyperphysics.phy-astr.gsu.edu/
𝛼-decaycreatesanucleuswith(𝑍 − 2,𝑁 − 2)
𝛽-decaycreatesanucleuswith(𝑍 + 1,𝑁 − 1)
Radioactivedecaychains
• Tomodelthepopulationofadaughternuclideformingfromaparent(withdecayconstant𝜆3),whilstitselfdecaying(withdecayconstant𝜆5),weneedtousecoupledequations…
ModellingadecaychainLet𝑁3(𝑡) and𝑁5(𝑡) bethenumberofparentanddaughternucleiattime𝑡,respectively.We’llassume𝑁3 0 = 𝑁D and𝑁5 0 = 0.Fromthedecaylaws:
𝑑𝑁3 = −𝜆3𝑁3(𝑡)𝑑𝑡𝑑𝑁5 = +𝜆3𝑁3(𝑡)𝑑𝑡 − 𝜆5𝑁5(𝑡)𝑑𝑡
Solvingthe1st equation:𝑁3 𝑡 = 𝑁D𝑒01:2
Tryasolutiontothe2nd equation:𝑁5 𝑡 = 𝐴𝑒01:2 + 𝐵𝑒01;2.Usingthecondition𝑁5 0 = 0,wefind𝐴 + 𝐵 = 0.Substitutingin,wefind𝑁5 𝑡 = 1:CZ
1;01:𝑒01:2 − 𝑒01;2
Moreonhalf-lives
• Thehalf-life ofnuclidesvariesdramatically,fromafractionofasecondtotheageoftheUniverse!
Credit:http://ne.phys.kyushu-u.ac.jp
• Examplefrom𝛼-decay–rangesover30ordersofmagnitude!
• Thisisbecause𝛼-decaydependsonquantumtunneling,whichisexponentiallysensitivetotheheightoftheenergybarrier
• Differenthalf-livescanbeusedindifferentapplications
Carbondating
• Theisotope C3[ iscontinuouslyproducedintheatmospherebycosmicraybombardment,mixingwiththedominantisotope C35
• Livingorganismscontinuallyexchangecarbonwiththeatmosphere,andhencehaveafixedisotopicabundanceratio C3[ / C35 ≈ 1.3×10035
• Afterdeath, C3[ 𝛽-decaysinto N3[ withahalf-lifeof5,730yr,whereasC35 isstable.Hencethe C3[ / C35 ratiodeclines,tracingtheobject’sage
Credit:http://hyperphysics.phy-astr.gsu.edu/hbase/Nuclear/cardat.html
Geologicaldating
• Radioactivedatingusingnuclideswithlongerhalf-livescanbeusedtoestimatetheageofrocks,oreventheEarthitself
• Thereareseveralversions,butingeneralweneedtocomparetherelativeabundanceofparent anddaughter nuclides
• Uranium-lead datingusesthedecaychainsof U5bc → Pb5Dg
(half-life7.1×10hyr)andU5bh → Pb5Di (half-life
4.47×10kyr)Credit:http://geologylearn.blogspot.com,http://www.astronomy.ohio-state.edu
Keytake-aways
• Radioactivedecayisaspontaneous,randomprocessgovernedbythelawsofprobability
• Aradioactivepopulationofnucleideclinesas𝑁 𝑡 =𝑁 0 𝑒012withdecayconstant𝜆
• Apopulationhasradioactivehalf-life𝑡3/5 = 0.693/𝜆andmeanlifetime𝑡̅ = 1/𝜆
• TheradioactivityofasampleismeasuredinCurieswhere1Ci = 3.7×103Ddecays03
• Radioactivenuclidescanbeusedasclocks,inapplicationssuchascarbonorgeologicaldating