1

Alpha Decay• Readings

§ Nuclear and Radiochemistry: Chapter 3§ Modern Nuclear Chemistry: Chapter 7

• 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§ Energetics

à Alpha decay energies 4-9 MeVà Originally thought to be monoenergetic, fine structure discovered

• AZ(A-4)(Z-2) + 4He + Qa

2

Fine Structure and Energetics

• Different alpha decay energies for same isotope§ Relative intensities vary§ Coupled with gamma decay

• Over 350 artificially produced alpha emitting nuclei§ Alpha energy variations used

to develop decay schemes • 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) à half-life of 144Nd is 5x1029

times longer then 212mPo

3

Energetics• Q values generally increase with

A§ variation due to shell

effects can impact trend increase

§ Peaks at N=126 shell• For isotopes decay energy

generally decreases with increasing mass

• 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 proton dripline above A=100

4

Alpha Decay Energetics• Q value positive for alpha decay

§ Q value exceeds alpha decay energy§ maTa = mdTd

§ md and Td represent daughter• From semiempirical mass equation

§ emission of an α-particle lowers Coulomb energy of nucleus

§ increases stability of heavy nuclei while not affecting the overall binding energy per nucleonà tightly bound α-particle has

approximately same binding energy/nucleon as the original nucleus* Emitted particle must have

reasonable energy/nucleon* Energetic reason for alpha rather

than proton• Energies of alpha particles generally increase

with atomic number of parent

)()1(

)1(Q

Q

d

d

d

d

d

mm

mQT

mmQ

m

mT

m

TmT

5

Energetics• Calculation of Q value from mass excess

§ 238U234Th + a + Qà Isotope Δ (MeV)

238U 47.3070 234Th 40.6124He 2.4249

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

daughter à kinetic energy of the alpha particle will be slightly less than Q

value• Conservation of momentum in decay, daughter and alpha are equal rd=r

§ 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

)(d

d

mm

mQT

MeVT 198.4)

2344

234(270.4

6

Energetics• Kinetic energy of emitted particle is less than Coulomb

barrier α-particle and daughter nucleus§ Equation specific of alpha

§ For 238 U decay

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

• Alpha particle carries as much energy as possible from Q value

• For even-even nuclei, alpha decay leads to the ground state of the daughter nucleus§ as little angular momentum as possible§ ground state spins of even-even parents, daughters and

alpha particle are l=0

fmMeVR

Ze

R

Z

oc 44.1

2

4

2V

2

MeVfm

fmMeVfmMeV

fmc 283.9

25944.1

)4234(2.1

)90(2V

3/13/1

7

Energetics• Some decays of odd-A heavy nuclei populate low-

lying daughter excited states that match spin of parent § Leads to fine structure of alpha decay

energy• Orbital angular momentum of α particle can be

zero§ 83% of alpha decay of 249Cf goes to 9th

excited state of 245Cm § lowest lying state with same spin and

parity as parent• Long range alpha decay

§ Decay from excited state of parent nucleus to ground state of the daughter

§ 212mPoà 2.922 MeV above 212Po ground

stateà Decays to ground state of 208Pb

with emission of 11.65 MeV alpha particle

• Systematics result from§ Coulomb potential

à Higher mass accelerates products§ larger mass

à 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 decay chain

Beta Decay to Energy minimum, then Alpha decay to different A

8

Alpha decay theory• Distance of closest approach for

scattering of a 4.2 MeV alpha particle is ~62 fm§ Distance at which alpha particle

stops moving towards daughter § Repulsion from Coulomb

barrier§ Alpha particle should not get

near the nucleus from outside• Alpha particle should be trapped behind

a potential energy barrier• 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 barrierà Related to decay

probability

Alpha decay energy

Vc

9

Alpha Decay Theory• 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, more frequently the particle will encounter barrier§ Increase probability of

barrier penetration• Geiger Nuttall law of alpha decay

§ Log t1/2=A+B/(Qa)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

10

Expanded Alpha Half Life Calculation

• More accurate determination of half life from Hatsukawa, Nakahara and Hoffman

• Theoretical description of alpha emission based on calculating 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

• a=P*fà f is frequency factorà P is transmission coefficient

),(446.20]1([arccos))(()(log 2/12/110 NZCXXX

QA

AZAt

p

d

)126(067.0)82(105.042.1[),(

)126(070.0)82(020.094.1[),(

0),(

NZNZC

NZNZC

NZC

Outside of closed shells

78Z82; 100N12682Z90; 100N126

)2

)(4(2249.12

3/13/1

eZ

QAX

d

11

Alpha Decay Theory• Alternate expression includes additional factor that describes probability of

preformation of alpha particle inside 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

à uncertainties in the theoretical estimates that contribute to the differences

• Frequency for alpha particle to reach edge of a nucleus § estimated as velocity divided by the distance across the nucleus

à twice the radius à lower limit for velocity could be obtained from the kinetic energy of

emitted alpha particleà 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

§ On the order of 1021 s-1

R

QV

R

vf o

2

/)(2

2

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Alpha Decay Calculations

• Alpha particle barrier penetration from Gamow§ T=e-2G

• Determination of decay constant from potential information

• Using the square-well potential, integrating and substituting§ Z daughter, z alpha

R

R

MM

MM

2

1

2/12/12

1

))(()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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Gamow calculations

• From Gamow

§ Log t1/2=A+B/(Qa)0.5

• 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

G

o

eQVfP

t 22/12/1 )/)(2(

2ln2ln2ln

14

Alpha Decay Theory• Even-even nuclei undergoing l=0 decay

§ average preformation factor is ~ 10-2

§ neglects effects of angular momentum à 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• Previous derivation only holds for even-even nuclei

§ odd-odd, even-odd, and odd-even nuclei have longer half-lives than predicted due to hindrance factors

• Assumes existence of pre-formed -particles§ ground-state transition from a nucleus containing an odd

nucleon in highest filled state can take place only if that nucleon becomes part of the -particle à another nucleon pair is brokenà less favorable situation than formation of an -particle from

existing pairs in an even-even nucleus * observed hindrance.

à if -particle is assembled from existing pairs in such a nucleus, the product nucleus will be in an excited state, * explain the “favored” transitions to excited states

•Hindrance factor determine by ratio of measured alpha decay half life over calculated alpha decay half life

§ Calculations underpredict half life§ Hindrance factors between 1 and 3E4

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Hindrance Factors

• Transition of 241Am (5/2-) to 237Np § states of 237Np (5/2+) ground state and (7/2+) 1st excited state have

hindrance factors of about 500§ Main transition to 60 keV above ground state is 5/2-, almost

unhindered• 5 classes of hindrance factors (half live measure/half life calculated)

§ Between 1 and 4, the transition is called a “favored”à emitted alpha particle is assembled from two low lying pairs of

nucleons in the parent nucleus, leaving the odd nucleon in its initial orbital

§ Hindrance factor of 4-10 indicates a mixing or favorable overlap between the initial and final nuclear states involved in the transition

§ Factors of 10-100 indicate that spin projections of the initial and final states are parallel, but the wave function overlap is not favorable

§ Factors of 100-1000 indicate transitions with a change in parity but with projections of initial and final states being parallel

§ Hindrance factors of >1000 indicate that the transition involves a parity change and a spin flip

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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 much smaller than the alpha decay

• simple barrier penetration estimate can be attributed to the very small probability to preform 14C residue inside the heavy nucleus

18

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

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Topic Review

• Understand and utilize systematics and energetics involved in alpha decay

• Calculate Q values for alpha decay§ Relate to alpha energy and fine structure

• Correlate Q value and half-life• Models for alpha decay constant

§ Tunneling and potentials• Hindered of alpha decay • Understand proton and other charged particle

emission

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Homework Questions

• Calculate the alpha decay Q value and Coulomb barrier potential for the following, compare the values§ 212Bi, 210Po, 238Pu, 239Pu, 240Am, 241Am

• What is the basis for daughter recoil during alpha decay?• What is the relationship between Qa and the alpha decay energy (Ta)

• What are some general trends observed in alpha decay?• Compare the calculated and experimental alpha decay half life for

the following isotopes§ 238Pu, 239Pu, 241Pu, 245Pu§ Determine the hindrance values for the odd A Pu isotopes above

• What are the hindrance factor trends?• How would one predict the half-life of an alpha decay from

experimental data?

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Pop Quiz• Calculate the alpha decay energy for 252Cf and 254Cf from the

mass excess data below.• Which is expected to have the shorter alpha decay half-life and

why? • Calculate the alpha decay half-life for 252Cf and 254Cf from the

data below.