1. 16.451 Lecture 174/11/2003 1

2. Analysis: (solutions will be posted on the web site under “homework”) basic recipe: about 40% of the marks for knowing the course material, 20% for being able to solve a familiar homework problem (slightly modified), 40% for being able to solve a new problem applying what you have learned in class most people did not know what the neutron charge form factor looks like 2 b) was marked “generously” – the  does not travel at the speed of light! a few of you were obviously having trouble getting started on question 2 b)... next time, please ask for help if you are stuck!!! 2

3. Recall the problem with beta decay spectra (lecture 16): N(K) Too many low energy e - Too few low energy e + 3 N(K e ) Ideal case: N(K) is proportional to: electron neutrino recoil 0

4. Discrepancy: neglect of Coulomb effects in the final state. Key point: Coulomb distortions of the energy spectra occur AFTER the electron/positron are emitted in the weak decay process. Modified density of electron/positron states: original result “Fermi function”, depends on the charge Z’ of the “daughter nucleus” (final state) and the electron/positron momentum Approximate correction factor for  decay: Modified electron/positron spectrum prediction: 4

5. Fermi-Kurie Plot (Krane, fig. 9.4): Idea: for “allowed decays”, corresponding to our approximation: inside the nucleus, the electron energy spectrum can be “linearized” if one accounts for the Coulomb distortion via the Fermi function F(Z’,p): linear function, endpoint Q 66 Ga  66 Zn decay, Phys. Rev. 129, 1782 ++ 5

6. Neutrino Mass effect: Idea: shape of the electron energy spectrum near the endpoint (Q) is sensitive to the mass of the electron antineutrino: electron neutrino recoil When K e  Q, K R  K  0. if m  0, then in this limit, mass effects are most pronounced. recall: 6

7. figure from “KATRIN” proposal, 2001: (Karlsruhe Tritium Neutrino expt.) Best direct upper limit: m < 2.5 eV from Sudbury neutrino observatory and other experiments, we have convincing indirect evidence of nonzero neutrino mass that is much smaller than this (special lecture...) 7

8. Implications for the universe ( http:/www.astro.queensu.ca/~dursi/dm-tutorial/dm0.html ) we can only “see” about 10% of the matter distribution that is required to be present to explain astronomical observations, e.g. rotational curves of galaxies etc. direct upper limit on neutrino mass is too small to account for the discrepancy 8

9. Total decay rate: Our formalism determines if, which is the rate (s -1 ) to a particular final state electron (or positron) momentum p: The total decay rate is obtained by integrating if over all allowed e  momenta p: (  refers to   decay modes: the Fermi function is different the two cases) “Fermi integral”, f (Z’,Q) (Note: literature, and Krane Ch. 9, use the symbol E o for Q in this expression) Key point: by integrating over the momentum distribution, which is specific only to the kinematics of the case under consideration, we can relate rates for various decays that differ only in the nuclear matrix element... 9

10. Krane, Figure 9.8 : Dimensionless Fermi integral By convention: log 10 f(Z’,E o ) 10

11. Comparative half lives: By convention, the half-life, is used as a comparison standard for different nuclear beta decays: Using our formalism and, we can write: Notice: The only difference in the “ft” value between different nuclear beta decays is the value of the nuclear matrix element. If |M nucl | 2 = 1 (“superallowed” case in nuclei), the ft values can be used to determine the weak coupling constants G = (G V, G A ) Special case: “superallowed” decays in nuclei with initial and final nuclear states 0+  0+, e.g. must have S = 0 for the leptons  pure Fermi decay... 11

12. Superallowed Fermi 0+  0+ decays: world’s best data for light nuclei all have the same ft value ~ 3100 sec determines the weak coupling constant for Fermi decays: (And G A /G V = -1.25, more later....) 12

13. Can we understand beta decay rates in general? first page of Krane, Appendix C: (symbol  stands for electron capture/  + decay)  27 isotopes: 8  - decays, 6  + decays, spanning 16 orders of magnitude in rate! 13

14. Some anomalies: According to our theory, the very slow decay: (1.6 x 10 6 yrs) should not occur at all, because angular momentum does not add up, i.e.: Another example: (16.1 hr) This should not occur because the wavefunctions in the nuclear matrix element have opposite parity, so the integrand is odd and should vanish: 14

15. Forbidden Decays: These are two examples of forbidden decays – they cannot proceed under the allowed approximation, ie. electron neutrino recoil Reconsider the electron and antineutrino wave function as a multipole expansion: 15

16. Effect on beta transitions: angular momentum coupling for the multipole order L, together with S and nuclear angular momentum allows previously impossible reactions to proceed multipole term has parity (-1) L, which allows nuclear states of opposite parity to be “connected” by the beta decay operator momentum dependence of the matrix element varies as... since this is small, the lowest multipole order L that satisfies the conservation laws will dominate the transition rate ~ |M| 2 ~ (p R r/ħ) 2L  (0.01) 2L  dramatically smaller for large L momentum dependence also affects the shape of the spectrum; Kurie plots are not linear unless “shape factors” are taken into account.... naming convention: 16

17. Classification: all known decays.... 17