1 Alpha Decay Energetics of Alpha Decay Theory of Alpha Decay Hindrance Factors Heavy Particle Radioactivity Proton Radioactivity Identified at positively charged particle by Rutherford §Helium nucleus ( 4 He 2+ ) based on observed emission bands §Enegetic àAlpha decay energies 4-9 MeV A Z  (A-4) (Z-2) + 4 He + Q 

2 Energetics Q value positive for alpha decay From semiempirical mass equation §emission of an α-particle lowers the Coulomb energy of nucleus §increases stability of heavy nuclei while not affecting the overall binding energy per nucleon because àtightly bound α-particle has approximately same binding energy/nucleon as the original nucleus *Emitted particle must have reasonable energy/nucleon Energies of the alpha particles generally increase with the atomic number of parent §kinetic energy of the emitted particle is less than Coulomb barrier α-particle and daughter nucleus All nuclei with mass numbers greater than A of 150 are thermodynamically unstable against alpha emission (Q α is positive) §However alpha emission is dominant decay process only for heaviest nuclei, A≥210 §Energy ranges 1.8 MeV ( 144 Nd) to 11.6 MeV ( 212m Po) §half-life of 144 Nd is 5x10 29 times longer then 212m Po 212

3 Alpha separation energy

4 Energetics Alpha particle carries as much energy as possible from Q value, alpha decay leads to the ground state of the daughter nucleus so that the §as little angular momentum as possible §ground state spins of even-even parents, daughters and alpha particle are l=0 Some decays of odd-A heavy nuclei populate low-lying daughter excited states that match spin of the parent orbital angular momentum of the α particle can be zero §83% of alpha decay of 249 Cf goes to 9 th excited state of 245 Cm §lowest lying state with the same spin and parity as the parent Fine structure alpha decay §decay to several different excited states of a daughter nucleus Long range alpha decay §Decay from excited state of parent nucleus to ground state of the daughter § 212m Po à2.922 MeV above 212 Po ground state àDecays to ground state of 208 Pb with emission of MeV alpha particle

5 Energetics Calculation of Q value from mass excess  238 U  234 Th +  + Q àIsotope Δ (MeV) 238 U Th He  Q  = – ( ) = MeV §Q energy divided between the α particle and the heavy recoiling daughter àkinetic energy of the alpha particle will be slightly less than Q value Conservation of momentum in decay, daughter and alpha are equal  d =   §recoil momentum and the  -particle momentum are equal in magnitude and opposite in direction §p 2 =2mT where m= mass and T=kinetic energy 238 U alpha decay energy =4.270 (234/238)=4.198 MeV

6 Energetics Q values generally increase with A §variation due to shell effects can impact trend increase §Peaks at N=126 shell §Stable end daught 208 Pb is doubly magic §α decay of 211 Pb and 213 Po will not lead to this daughter 82 neutron closed shell in the rare earth region §increase in Q α, §α-decay for nuclei with N=84 as it decays to N=82 daughter short-lived α-emitters near doubly magic 100 Sn § 107 Te, 108 Te, 111 Xe alpha emitters have been identified by the proton dripline above A=100 For isotopes the decay energy generally decreases with increasing mass

7 Q value for different A

8 Energetics Alpha decay energies are small compared to the required energy for the reverse reaction Systematics result from §Coulomb potential àHigher mass accelerates products §larger mass àdaughter and alpha particle start further apart mass parabolas from semiempirical mass equation §cut through the nuclear mass surface at constant A §Explains beta decay in chain

9 Mass parabolas

10 Alpha decay theory Distance of closest approach for scattering of a 4.2 MeV alpha particle is ~62 fm. §Distance at which the alpha particle stops moving towards the daughter §Repulsion from Coulomb barrier An alpha particle should not get near the nucleus or For decay §alpha particle should be trapped behind a potential energy barrier

11 Alpha decay theory Wave functions are only completely confined by potential energy barriers that are infinitely high §With finite size barrier wave function has different behavior §main component inside the barrier §finite piece outside barrier Tunneling §classically trapped particle has component of wave function outside the potential barrier §Some probability to go through barrier Closer the energy of the particle to the top of the barrier more likely the particle will penetrate barrier More energetic the particle is relative to a given barrier height, the more frequently the particle will encounter barrier §Increase probability of barrier penetration

12 Alpha Decay Theory Geiger Nuttall law of alpha decay  Log t 1/2 =A+B(Q  )0.5 §constants A and B have a Z dependence. simple relationship describes the data on α-decay §over 20 orders of magnitude in decay constant or half-life  1 MeV change in  - decay energy results in a change of 10 5 in the half-life

13 Alpha Decay Theory Theoretical description of alpha emission based on calculating the rate in terms of two factors §rate at which an alpha particle appears at the inside wall of the nucleus §probability that the alpha particle tunnels through the barrier =P*f àf is frequency factor àP is transmission coefficient Some investigators suggest expression should be multiplied by an additional factor that describes probability of preformation of alpha particle inside the parent nucleus no clear way to calculate such a factor §empirical estimates have been made §theoretical estimates of the emission rates are higher than observed rates §preformation factor can be estimated for each measured case àuncertainties in the theoretical estimates that contribute to the differences frequency for an alpha particle to reach edge of a nucleus §estimated as velocity divided by the distance across the nucleus àtwice the radius àlower limit for velocity could be obtained from the kinetic energy of emitted alpha particle àHowever particle is moving inside a potential energy well and its velocity should be larger and correspond to the well depth plus the external energy

14 Alpha Decay Theory Determination of decay constant from potential information Using the square-well potential, integrating and substituting

15 Alpha Decay Theory calculated emission rate typically one order of magnitude larger than observed rate §observed half-lives are longer than predicted §Observation suggest probability to find a ‘preformed’ alpha particle on order of even-even nuclei undergoing l=0 decay §average preformation factor is ~ §neglects effects of angular momentum àAssumes α-particle carries off no orbital angular momentum (ℓ = 0) §If α decay takes place to or from excited state some angular momentum may be carried off by the α-particle §Results in change in the decay constant when compared to calculated

16 Hindered  -Decay The previous derivation only holds for even-even nuclei §odd-odd, even-odd, and odd-even nuclei have longer half-lives than predicted by this formula, due to hindrance factors assumes the existence of pre-formed  -particles §a ground-state transition from a nucleus containing an odd nucleon in the highest filled state can take place only if that nucleon becomes part of the  -particle and therefore if another nucleon pair is broken àless favorable situation than the formation of an  -particle from already existing pairs in an even-even nucleus and may give rise to the observed hindrance. àif the  -particle is assembled from existing pairs in such a nucleus, the product nucleus will be in an excited state, and this may explain the “favored” transitions to excited states

17 Heavy Particle Decay Possible to calculate Q values for the emission of heavier nuclei §Is energetically possible for a large range of heavy nuclei to emit other light nuclei. Q-values for carbon ion emission by a large range of nuclei §calculated with the smooth liquid drop mass equation without shell corrections Decay to doubly magic 208 Pb from 220 Ra for 12 C emission §Actually found 14 C from 223 Ra §large neutron excess favors the emission of neutron-rich light products §emission probability is so much smaller than the simple barrier penetration estimate can be attributed to the very small probability to preform 14 C residue inside the heavy nucleus

18 Proton Decay For proton-rich nuclei, the Q value for proton emission can be positive §Line where Q p is positive, proton drip line §Describes forces holding nuclei together Similar theory to alpha decay §no preformation factor for the proton §proton energies, even for the heavier nuclei, are low (Ep~1 to 2 MeV) barriers are large (80 fm) §Long half life