1Nuclear decayNuclear decayRadioactivityRadioactivity2Nuclear decayNuclear decay• Radioactivity• Curie, Becquerel• N radioactive nuclei• dN decay in a time dt( )teNtNNdtdNNdNλλ−=−=∝03λ = decay constant = probability of a nucleus decaying per secondHalf-life = time for half the nuclei to decayLifetime (average) λ2ln21=tλτ1=Nt4• Units: 1 Becquerel (Bq) = 1 decay/second• 1 Curie (Ci) = 3.7 x 1010decays/sec (1g of radium)• 3 types of radiation emitted spontaneouslyα = 4He nucleus (2 p and 2 n)β = e–or e+ γ = high-energy photons (keV, MeV)5Effect of a magnetic fieldEffect of a magnetic fieldB into pagegammaalpha (+)beta –XX6• Decay rate (“Activity”)• Half-life e.g. ( )tNeNdtdNRtλλλ===−0hrtNo1&2021==etc52101200Nt (hr)7α α decaydecay8ααdecaydecayα-particle is a 4He nucleus (2p & 2n)• Z ↓ by 2• N ↓ by 2• A ↓ by 4α+→−−− 242 NAZNAZYX“parent”“daughter”α+→ ThU2349023892yrst91047.421×=9Disintegration energyDisintegration energy()2cmMMQDPα−−=MeVcmQumMMumuMuMThUThU275.4502.931004589.0004589.0002602.4043593.234050784.2382=×=⋅∆==−−===αα10Disintegration energyDisintegration energy• Most of the energy (Q) is KE of α• Decay occurs if Q > 0 (energy released)• Spontaneous decay does not occur if Q < 0 Conservation of momentum (daughter + alpha) and Energy gives …( )−⋅=AAQKE4α( )MeVKEUge 2.4238234275.4..238=×=α())(daughterKEKEQ+=α11Theory of Theory of αα--decaydecay• Questions to answer:(1) How do α-particles with KE ~ 4 MeV escape the nucleus while incoming α-particles with KE ~ 10 MeV are scattered ?(2) KE(α) range is ~ 4 to 9 MeV but half-life varies over 24 orders of magnitude ! (nanoseconds to billions of years)12Theory of Theory of αα--decaydecay• 1911: Geiger & Nuttall noticed that large Q ↔ short half-life• 1928: Gamow, Gurney, Condon et al. QM tunnelling through a nuclear potential barrier by α• The tunnelling model works well, even though it supposes the α to be preformed inside the nucleus.Krane 8.313Theory of Theory of αα--decaydecayα experiences a Coulombic potential barrierZ-2r{}3131)4(42.1~)()()2(88.2)(4)2(2)(02−+−=−×=AfmRfmrZMeVVreZrVπε14Barrier heightBarrier height{}MeVMeVfmfmR9.273.99088.2~)(heightBarrier3.923442.1~)(3131=×∴=+α+→ ThU234902389227.9 MeV4.2 MeV9.3 fm15Barrier widthBarrier widthfmwidthfmrMeVrqqV4.523.97.617.612.44021=−=∴=∴==πε28 MeV4.2 MeV9.3 fm61.7 fm16Theory of Theory of αα--decaydecay• An early triumph of QM was the Geiger-Nuttall relation.loglog21ConstKnt=⋅+αconstantLivesey 9.44)log(21t)log(αK17Theory of Theory of αα--decaydecay122020302211)(sinh)(41)()(2)(2)(12211−∗∗−−−+===−=+==+=RkEVEVAAFFTFexEVmkDeCexmEkBeAexxikxkxkxikxikφφφhh12318Theory of Theory of αα--decaydecay2121)(98.248)(97.322exp~0202−====+−fmmebMeVmeaZRbEZaTπεεhhE = α energy in MeVR = radius of ‘daughter’ in fmZ = atomic number of parent19Theory of Theory of αα--decaydecay39106)88exp(3.99098.22.49097.3exp~3.9~&90,2.4−×=−=×+−==TfmRZMeVEDThus, the probability of an α particletunnelling out of the nucleus when it‘hits’ the potential barrier is VERY SMALL.BUT ----- how many times does the α ‘hit’the barrier per second ? i.e. How many escape attempts ?α+→ ThU234902389220Theory of Theory of αα--decaydecayαvRt2=Time to ‘cross’ the nucleus isAttempt frequency (“knocking rate”) isRvtf21α==Alpha particle speed = ?smmEvMeVmMeVE/104.1~2)4.3727(2.4~7×=∴=αα120157105.7~103.92104.1−−××××=∴sf21HalfHalf--life of life of αα--decaydecay)1046.4expt(109.41054.12ln105.4105.7106991711820392121yryrstts×=×=×=∴==×=×××−−−λα+→ ThU234902389222HalfHalf--life of life of αα--decaydecay)310expt(2104.31014.1/1006.2103~9.170.99.26widthBarrier2.26heightBarrier02.978.82118121713nsnstsfTsfsmvTfmfmMeVfmRMeVE==∴=×=⋅×=×=×=−====−−−λααα+→ PbPo208822128423β β decaydecay24β β decaydecayβ-particles are electrons or positrons (anti-electrons)• Z changes by +1 or –1• N changes by –1 or +1• A is unchanged−−++→β11 NAZNAZYX−+→βNC147146++−+→β11 NAZNAZYX(Not the whole story – see neutrinos)25• Mechanismsβ β decaydecay−+→ epn++→ enpnep →+−β– β+or “Electron Capture”an ‘atomic’ electronThe e–or e+is NOT present in the nucleus prior to decay – it is created during the decay process.Uncertainty calculation26ββdecaydecayMeVQdtePoBi16.15212108421083==++→−LEisberg & Resnick 16.10A range of energies !27ββdecaydecay• The masses of the parent and daughter nuclei are both fixed so why is there a range of KE (β) ?• Problems with ‘Energy Conservation’ and ‘Momentum Conservation’• 1930: Pauli – there must be another particle involved !• 1934: Fermi – “neutrino”• VERY (!) weakly interacting0=q0≈m28Neutrino inNeutrino inββdecaydecayMomentum conservation probleme–cfe–ν29NeutrinoNeutrino• Three fermions: all have spin = ½• Spin angular momentum is not conserved in the above reaction• The ‘other’ particle must have spin = ½++→ enp½ ½ ½eenpνβ++→++:eepnνβ++→−−:Electron-anti-neutrinoElectron-neutrinoCannot occur for a FREE proton ….. ?Decay of a free neutron. ‘Lifetime’ ~ 886 s30Neutrino detectionNeutrino detection• 1956: Reines & Cowan++→+ enpeνγ+→→+ CdCdCdn109*109108γγ+→+−+eeDetect the reaction by detecting the 3 gamma-rays31An application of An application of ββ--decaydecay“Carbon Dating”“Carbon Dating”• The ratio of 14C to 12C is ~ 1.3 x 10–12• Living organisms exchange CO2with surroundings so ratio is ~ stable.• Exchange stops at death so ratio ↓ because the 14C decays.eeNCν++→−147146yrst 573021=3225 g charcoal14C activity = 250 decays/minDecay constant1121083.32ln21−−×== stλNumber of 12C nuclei in 25 g 24231026.112251002.6 ×=××=()12122414106.1103.11026.1 ×=×××=−CNoInitial activity (at death)min/370sec/13.6 decaysdecaysNRoo===λyrsteeNRtt32393702500=→==−−λλλ33Decay energyDecay energy()( )( )222:2::cMMQECcmMMQcMMQDPeDPDP−=−−=−=+−ββIf the decay energy Q were shared ONLYbetween the Daughter atom and the beta particle, there would NOT be a range of beta energies !Conservation of energy and momentum would be enough to define the beta energy34Theory of Theory