PHGN 422: Nuclear Physics Lecture 15: Introduction to β Decay

Transcription

1 PHGN 422: NUCLEAR PHYSICS PHGN 422: Nuclear Physics Lecture 15: Introduction to β Decay Prof. Kyle Leach October 9, 2018 Slide 1

2 Last Week... We learned α decay is a result of having a Coulomb term in the nuclear potential (ie. strong + electromagnetic interactions) α decay occurs because of quantum tunnelling through the Coulomb barrier The α decay T 1/2 is heavily dependent on the Q value... However, the height of the barrier can change depending on the nuclear structure. Slide 2 Prof. Kyle Leach PHGN 422: Nuclear Physics

3 β Decay β decay is a weak interaction process where a bound proton (neutron) is converted into a neutron (proton) inside of the nucleus This decay process involves quarks, leptons (positrons, electrons, and neutrinos), and the weak-interaction force carriers β decay occurs in three modes: Slide 3 Prof. Kyle Leach PHGN 422: Nuclear Physics

4 Three Types of β Decay β Decay A Z X N A Z+1 Y N 1 + e + ν e β + Decay A Z X N A Z 1 W N+1 + e + + ν e Electron Capture (EC) A Z X N + e A Z 1 W N+1 + ν e Slide 4 Prof. Kyle Leach PHGN 422: Nuclear Physics

5 Moving Through the Nuclear Chart Phil Walker, New Scientist Magazine, October 2011 Slide 5 Prof. Kyle Leach PHGN 422: Nuclear Physics

6 Electron Capture (EC) In EC decay, there are only two components to the final state: the daughter nucleus and the neutrino. Therefore we call this a two-body decay. Orbital Electron Capture Signature of EC Slide 6 Prof. Kyle Leach PHGN 422: Nuclear Physics

7 Three Body Decay For β ± decay, there are three components to the final state: the daughter nucleus, the electron (positron), and the anti-neutrino (neutrino). This a three-body decay. Slide 7 Prof. Kyle Leach PHGN 422: Nuclear Physics

8 Three Body Decay Source: Slide 8 Prof. Kyle Leach PHGN 422: Nuclear Physics

9 What Does This Actually Mean? Up (u) m = 2.4 MeV/c 2 q = +2/3 u u u d d Proton u,u,d Down (d) m = 4.8 MeV/c 2 q = 1/3 d Neutron u,d,d Slide 9 Prof. Kyle Leach PHGN 422: Nuclear Physics

10 A Taste of Feynman Diagrams Slide 10 Prof. Kyle Leach PHGN 422: Nuclear Physics

11 Feynman Diagrams: A Simple Way to Illustrate Complicated Mathematics Slide 11 Prof. Kyle Leach PHGN 422: Nuclear Physics

12 β Decay For β decay: A Z X N Z+1 A Y N 1 + e + ν e u d d ν e W e Neutron Fermi s golden rule: λ = 2π M fi 2 dn de M fi : β-decay transition matrix element connecting the initial and final states u d Proton u Slide 12 Prof. Kyle Leach PHGN 422: Nuclear Physics

13 β + Decay u d u For β + decay: A Z X N Z 1 A W N+1 + e + + ν e W + ν e e + Proton Fermi s golden rule: λ = 2π M fi 2 dn de M fi : β-decay transition matrix element connecting the initial and final states u d d Neutron Slide 13 Prof. Kyle Leach PHGN 422: Nuclear Physics

14 Fermi s Golden Rule In 1934, Enrico Fermi developed the first theory of β decay. Let s recall again Fermi s golden rule: λ = 2π V fi 2 ρ(e f ) Slide 14 Prof. Kyle Leach PHGN 422: Nuclear Physics

15 Fermi s Golden Rule In 1934, Enrico Fermi developed the first theory of β decay. Let s recall again Fermi s golden rule: λ = 2π V fi 2 ρ(e f ) The transition matrix element that connects the initial and final states: V fi = f V i. Slide 14 Prof. Kyle Leach PHGN 422: Nuclear Physics

16 Fermi s Golden Rule In 1934, Enrico Fermi developed the first theory of β decay. Let s recall again Fermi s golden rule: λ = 2π V fi 2 ρ(e f ) The transition matrix element that connects the initial and final states: V fi = f V i. The density of final states. How do we calculate this? Slide 14 Prof. Kyle Leach PHGN 422: Nuclear Physics

17 Fermi s Golden Rule In 1934, Enrico Fermi developed the first theory of β decay. Let s recall again Fermi s golden rule: λ = 2π V fi 2 ρ(e f ) The transition matrix element that connects the initial and final states: V fi = f V i. The density of final states. How do we calculate this? Similar to α decay, V fi 2 is what we need to find a way to understand. How do we do that? First, let s slightly rewrite this... Slide 14 Prof. Kyle Leach PHGN 422: Nuclear Physics

18 Fermi s Golden Rule λ = 2π M fi 2 dn de (e, ν e ) Slide 15 Prof. Kyle Leach PHGN 422: Nuclear Physics

19 Fermi s Golden Rule λ = 2π M fi 2 dn de (e, ν e ) The hard stuff to calculate. This depends on our knowledge of the weak interaction. Slide 15 Prof. Kyle Leach PHGN 422: Nuclear Physics

20 Fermi s Golden Rule λ = 2π M fi 2 dn de (e, ν e ) The hard stuff to calculate. This depends on our knowledge of the weak interaction. The β and ν are emitted as free particles, so we can normalize their wavefunctions in a closed box of volume V. Slide 15 Prof. Kyle Leach PHGN 422: Nuclear Physics

21 Fermi s Golden Rule λ = 2π M fi 2 dn de (e, ν e ) The hard stuff to calculate. This depends on our knowledge of the weak interaction. The β and ν are emitted as free particles, so we can normalize their wavefunctions in a closed box of volume V. So, let s head to the chalkboard and take a look at all of this... Slide 15 Prof. Kyle Leach PHGN 422: Nuclear Physics

22 Density of Final States We can, however, make a simplification in our approximation to only two particles. So, starting with our result for a single particle in a volume V: dn de = V dp 4π p2 (2π ) 3 de We obtain: dn de (e, ν e ) = V (2π ) 6 d de p 2 edp e dω e p 2 νdp ν dω ν = V (2π ) 6 dω edω ν p 2 edp e p 2 dp ν νdp ν de Slide 16 Prof. Kyle Leach PHGN 422: Nuclear Physics

23 Density of Final States So, integrating over dω e and dω ν (ie. over all possibilities): dn de (e, ν e ) = V2 p 2 4π 4 e(e E 6 e ) 2 1 m2 ν c4 (E E e) dp 2 e So, our differential decay rate from Fermi s golden rule becomes: dλ(p e ) = 2π M fi 2 dn de (e, ν e ) = 2π M fi 2 V 2 4π 4 6 p2 e(e E e ) 2 1 m2 νc 4 (E E e ) 2 dp e Slide 17 Prof. Kyle Leach PHGN 422: Nuclear Physics

24 Approximating m ν 0 Considering that m ν 0, we can simplify this to: dn de (e, ν e ) = V2 4π 4 6 p 2 e(e E e ) 2 dp e Which, in turn, gives us something more managable: dλ(p e ) = 2π M fi 2 dn de (e, ν e ) = 2π M fi 2 V 2 4π 4 6 p2 e(e E e ) 2 dp e So this is good...right? Well, we ve neglected the hardest part of this equation so far, M fi 2. Slide 18 Prof. Kyle Leach PHGN 422: Nuclear Physics

25 Upcoming Weeks... Assignment #2 Due on Friday Oct 12 No Class Thursday Oct 11 or Tuesday Oct 16 (Fall Break) Midterm Thursday Oct 18 More on the Fermi theory of β decay β decay selection rules β decay ft values Slide 19 Prof. Kyle Leach PHGN 422: Nuclear Physics