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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 (4He2+) based on observed emission bands Enegetic

Alpha decay energies 4-9 MeV

• AZ(A-4)(Z-2) + 4He + Q

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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 tightly 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 (Qα is positive) However alpha emission is dominant decay process only for

heaviest nuclei, A≥210 Energy ranges 1.8 MeV (144Nd) to 11.6 MeV (212mPo) half-life of 144Nd is 5x1029 times longer then 212mPo 212

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Alpha separation energy

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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 249Cf goes to 9th excited state of 245Cm 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 212mPo

2.922 MeV above 212Po ground state Decays to ground state of 208Pb with emission of 11.65 MeV alpha

particle

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Energetics• Calculation of Q value from mass excess

238U234Th + + Q Isotope Δ (MeV)

238U 47.3070 234Th 40.6124He 2.4249

Q=47.3070 – (40.612 + 2.4249) = 4.270 MeV Q energy divided between the α particle and the heavy recoiling

daughter kinetic 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

p2=2mT where m= mass and T=kinetic energy• 238U alpha decay energy =4.270 (234/238)=4.198 MeV

)mm

m(QT

)mm

(

Q

)m

m(TQ

m

TmTQ

d

d

d

d

d

1

1

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Energetics

• Q values generally increase with A variation due to shell effects can impact trend increase Peaks at N=126 shell Stable end daught 208Pb is doubly magic α decay of 211Pb and 213Po will not lead to this daughter

• 82 neutron closed shell in the rare earth region increase in Qα, α-decay for nuclei with N=84 as it decays to N=82 daughter

• short-lived α-emitters near doubly magic 100Sn 107Te, 108Te, 111Xe

• alpha emitters have been identified by the proton dripline above A=100

• For isotopes the decay energy generally decreases with increasing mass

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Q value for different A

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Energetics

• Alpha decay energies are small compared to the required energy for the reverse reaction

• Systematics result from Coulomb potential

Higher mass accelerates products larger mass

daughter 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

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Mass parabolas

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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

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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

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Alpha Decay Theory• Geiger Nuttall law of alpha

decay Log t1/2=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 105 in the half-life

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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 f is frequency factor P 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 uncertainties 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 twice the radius lower limit for velocity could be obtained from

the kinetic energy of emitted alpha particle However particle is moving inside a potential

energy well and its velocity should be larger and correspond to the well depth plus the external energy

211 22 R

h

R

vf

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Alpha Decay Theory

• Determination of decay constant from potential information

• Using the square-well potential, integrating and substituting

R

R

MM

MM

2

1

2/12/121

))(()2(4

exp2

R

R

drTrUhR

h

2

2

2

2

1v

R

ZzeT

1

2

R

ZzeB

2/12/12/12

21

1arccos8

exp2 B

T

B

T

B

T

hv

Zze

R

h

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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 10-1

• even-even nuclei undergoing l=0 decay average preformation factor is ~

10-2

neglects effects of angular momentum Assumes 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

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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 brokenless 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.

if 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

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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 208Pb from 220Ra for 12C emission Actually found 14C from 223Ra 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 14C residue inside the heavy nucleus

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Proton Decay• For proton-rich nuclei, the Q value

for proton emission can be positive Line where Qp 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