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