1.  -decay theory

2.  -particle penetration through Coulomb Barrier Goal: estimate the parent lifetime for  -decay Assume an  -particle is formed in the parent nucleus  Parent nucleus =  -particle + daughter nucleus  -decay   -particle must “tunnel” through the Coulomb barrier from R (nuclear matter radius) to b (called the “turning point”).

4.  -particle penetration through Coulomb Barrier Goal: estimate the parent lifetime for  -decay Consider the  -particle moving in the potential of the daughter nucleus. For r > R, this is a central force problem with Motion of a reduced mass  -particle relative to center of daughter nucleus

5.  -particle penetration through Coulomb Barrier Goal: estimate the parent lifetime for  -decay Assume that the rate of  -decay per nucleus ( ) can be obtained as = (probability for  -particle to penetrate the barrier)  (number of hits on the boundary per sec.) (probability for  -particle to penetrate the barrier) = “transmission coefficient” T

6.  -particle penetration through Coulomb Barrier (probability for  -particle to penetrate the barrier) = “transmission coefficient” T To calculate T for the Coulomb barrier -- consider the  -particle transmission through a one-dimensional rectangular barrier. From chapter 2, we have the transmission coefficient T for (simple) case --

7. 1-dimensional rectangular barrier 0.0 VoVo E 2a

8. V o >> E ;  a is “large” e  4  a important!!

9. Coulomb barrier penetration Consider penetration through Coulomb barrier as a sequence of rectangular barriers Total transmission coefficient is

10. Coulomb barrier penetration Ln term small

11. Coulomb barrier penetration We have assumed V(x) >> E Where in x is this approximation less good? Z 1 daughter Z 2 = 2 (  )

12. Coulomb barrier penetration If E > R

13. Coulomb barrier penetration You can easily show this ratio is true.

14. Coulomb barrier penetration

15. Barrier hit frequency For r < R,  -particle moves with v in Frequency of hits on Coulomb barrier Velocity of  -particle inside daughter Remember: V o is negative!  -particle decay constant - probability per unit time for decay

16.  -particle decay lifetime

17. Q is measured for  -decay m is calculated for  and daughter; Z 1 & Z 2 are known R is assumed to be R o A 1/3  Calculate B V o is taken to be ~35 MeV (from other studies) A is for parent nucleus; may not give best radius!

18.  -particle decay lifetime Assumptions:  -particle preformation in nucleus; did not estimate the rate for this from Fermi GR -- involves Final nuclear state of  -particle + daughter nucleus Initial nuclear state of parent nucleus V  causes transition from  i to  f

19.  -particle decay lifetime Assumptions: Assumed nucleus is a sphere; many large-A nuclei are deformed (ellipsoids) Assumed the  -particle takes off zero angular momentum, i.e.,