• • • • • •

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 ( 4 He 2+ ) based on observed emission bands



Energetics



Alpha decay energies 4-9 MeV



Originally thought to be monoenergetic, fine structure discovered A Z



(A-4) (Z-2) + 4 He + Q

a 1

• • •

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 All nuclei with mass numbers is against alpha emission (Q positive)



α However alpha emission is dominant decay process



only for heaviest nuclei, A≥210 Energy ranges 1.8 MeV ( ( 144 Nd) to 11.6 MeV



half-life of then 144 Nd is

2

• • • • •

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 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 proton dripline above A=100

3

Alpha Decay Energetics

• • •

Q value positive for alpha decay



Q value exceeds alpha decay energy

 

m

a

T

a

= m d T d m d and T d represent daughter From semiempirical mass equation



emission of an α-particle lowers Coulomb energy of nucleus

 Q

increases stability of heavy nuclei while not affecting the overall binding energy per nucleon



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

Q ( 1   

T T Q

a a

m

a

m d

 )

m

( 1   a

m d m m T T

a a a

d

) 

Energies of alpha particles generally increase with atomic number of parent

Q

(

m

a

m d



m d

4 )

• • •

Energetics

Calculation of Q value from mass excess



238 U



234 Th +

a

+ Q



Isotope Δ (MeV) 238 234 U Th 47.3070 40.612

 

4 He 2.4249

Q

a

=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

r

d =

r a 

recoil momentum and the

a

-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

T

a  4 .

270 ( 4 234  234 )  4 .

198

MeV T

a 

Q

(

m

a

m d

5 

m d

)

•

Energetics

Kinetic energy of emitted particle is less than Coulomb barrier α-particle and daughter nucleus

 

Equation specific of alpha For 238 U decay

V

c

 2

Z R

4

e

2 

o

 2

Z R

1 .

44

MeV fm

V

c

 1 .

2 ( 2 ( 90 ) 234 1 / 3  4 1 / 3 )

fm

1 .

44

MeV fm

 259

MeV

9 .

3

fm fm

 28

MeV

• • •

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

6

Energetics

• • • • •

low-lying daughter excited states that match



Leads to fine structure of alpha decay energy Orbital angular momentum of α particle can be zero

 

83% of alpha decay of parity as parent 245 249 Cm Cf goes to lowest lying state with same spin and Long range alpha decay



Decay from excited state of parent



daughter 212m



Po 2.922 MeV above 212 Po ground

 

Decays to ground state of 208 Pb with emission of 11.65 MeV 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 decay chain

Beta Decay to Energy minimum, then Alpha decay to different A 7

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

 

Tunneling



classically trapped particle has component of wave function outside the potential barrier



main component inside the barrier finite piece outside barrier Some probability to go through barrier



Related to decay probability

V c Alpha decay energy 8

• • • •

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 t 1/2 =A+B/(Q

a

) 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 of 10 5

a

-decay energy results in a change in the half-life

9

Expanded Alpha Half Life Calculation

• •

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

log 10 (

t

1 / 2 ) 

A

(

Z

)(

A d A p Q

a ) 1 / 2 [arccos

X



X

( 1

C

(

Z

,

N

)  0 Outside of closed shells

C

(

Z

,

N

)  [ 1 .

94  0 .

020 ( 82 

Z

)  0 .

070 ( 126 

N

)

C

(

Z

,

N

)  [ 1 .

42  0 .

105 (

Z

 82 )  0 .

067 ( 126 

N

) 

X

]  20 .

446 

C

(

Z

,

N

) 78  Z  82; 100  N  126 82  Z  90; 100  N  126

X

 1 .

2249 (

A

1 / 3  4 1 / 3

Q

)( 2

Z d

a

e

2 )

Theoretical description of alpha emission based on calculating



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*f

 

f is frequency factor P is transmission coefficient

10

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



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



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 On the order of 10 21 s -1

f



v

2

R

 2 (

V o



Q

) /  2

R

11

Alpha Decay Calculations

• • •

Alpha particle barrier penetration from Gamow



T=e

-2G



Determination of decay constant from potential information



h

2 

R

1 2 exp     4 

h

( 2  ) 1 / 2

R R

1 2  (

U

(

r

) 

T

) 1 / 2

dr

  

Using the square-well potential, integrating and substituting



Z daughter, z alpha

T



Zze

2

R

2  1 2 

v

2  

h

2 

R

1 2 exp  8 

Zze

2

hv

   arccos

T B

1 / 2 

T B

1 / 2 1

T B

 1 / 2     

M M

a a 

M M R R B



Zze

2

R

1 12

Gamow calculations

• •

t

1 / 2  ln  2  ln 2

fP

 ( 2 (

V o

ln 2 

Q

a ) /  ) 1 / 2

e

 2

G

From Gamow



Log t

1/2

=A+B/(Q

a

)

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

13

•

Alpha Decay Theory

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

14

Hindered

a

-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

a

-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

a

-particle



another nucleon pair is broken



less favorable situation than formation of an

a

-particle from existing pairs in an even-even nucleus

*

observed hindrance.

• 

if

* a

-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

15

Hindrance Factors

• •

Transition of 241 Am (5/2-) to 237 Np



states of 237 Np (5/2+) ground state and (7/2+) 1 hindrance factors of about 500 st excited state have



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”



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

16

• • • •

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 much smaller than the alpha decay simple barrier penetration estimate can be attributed to the very small probability to preform 14 C residue inside the heavy nucleus

17

• • •

Proton Decay

For proton-rich nuclei, the Q value for proton emission can be positive



Line where Q p proton drip line is positive,



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

18

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

19

Homework Questions

• • • • • • •

Calculate the alpha decay Q value and Coulomb barrier potential for the following, compare the values



212 Bi, 210 Po, 238 Pu, 239 Pu, 240 Am, 241 Am What is the basis for daughter recoil during alpha decay?

What is the relationship between Q a and the alpha decay energy (T a ) What are some general trends observed in alpha decay?

Compare the calculated and experimental alpha decay half life for the following isotopes



238 Pu, 239 Pu, 241 Pu, 245 Pu



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?

20

• • •

Pop Quiz

Calculate the alpha decay energy for 252 Cf and 254 Cf 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 252 Cf and 254 Cf from the data below.

21