-decay theory

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 Vo E 0.0 2a

8. Vo >> E ; a is “large” e4aimportant!!

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 assumedV(x) >> E • Where in x is this approximation less good? Z1 daughter Z2 = 2 ()

12. Coulomb barrier penetration If E << V(R)  Low Energy -particle  b >> 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 vin Frequency of hits on Coulomb barrier Velocity of -particle inside daughter Remember: Vo is negative! -particle decay constant - probability per unit time for decay

16. -particle decay lifetime

17. -particle decay lifetime Qis measured for -decay m is calculated for  and daughter; Z1 & Z2 are known R is assumed to be RoA1/3 Calculate B Vo 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.,