Why Does Uranium Alpha Decay?

Transcription

1 Why Does Uranium Alpha Decay? Consider the alpha decay shown below where a uranium nucleus spontaneously breaks apart into a 4 He or alpha particle and Th U 4 He Th E( 4 He) = 4.2 MeV To study this reaction we first map out the 4 He Th potential energy. We reverse the decay above and use a beam of 4 He nuclei striking a Th target. The 4 He beam comes from the radioactive decay of another nucleus Po and E(4 He) = MeV. 1 What is the distance of closest approach of the 4 He to the Th target if the Coulomb force is the only one that matters? 2 Is the Coulomb force the only one that matters? 3 What is the lifetime of the U? Jerry Gilfoyle Alpha Decay 1 / 26

2 What Do We Know? The Th α Potential V = Z 1Z 2 e 2 r V(r) (MeV) attractive nuclear part r (fm) Jerry Gilfoyle Alpha Decay 2 / 26

3 Mapping the Potential Energy Rutherford Scattering What is the distance of closest approach of the 4 He to the Th target if only the Coulomb force is active? Is the Coulomb force the only one active? The energy of the 4 He emitted by the Po to make the beam is E( 4 He) = MeV. Collimator Alpha source Po He + Pb ZnS Microscope Scattered helium Alpha beam Thorium target Jerry Gilfoyle Alpha Decay 3 / 26

4 Rutherford Trajectories Rutherford trajectories for different impact parameters y/b x Jerry Gilfoyle Alpha Decay 4 / 26

5 Mapping the Potential Energy The Differential Cross Section y z 4 He trajectory Pb target z axis Jerry Gilfoyle Alpha Decay 5 / 26

6 Mapping the Potential Energy The Differential Cross Section y z 4 He trajectory Pb target z axis dσ dω = ( Z1 Z 2 e 2 ) 2 1 4E cm sin 4 ( θ 2 ) Jerry Gilfoyle Alpha Decay 5 / 26

7 Actual Rutherford Scattering Results (θ/2) 4 Counts/sin H.Geiger and E.Marsden, Phil. Mag., 25, p. 604 (1913) alphas on gold θ (deg) :38:14 Jerry Gilfoyle Alpha Decay 6 / 26

8 Actual Rutherford Scattering Results (θ/2) 4 Counts/sin dσ dω = ( Z1 Z 2 e 2 ) 2 1 4E cm sin 4 ( θ 2 ) H.Geiger and E.Marsden, Phil. Mag., 25, p. 604 (1913) alphas on gold θ (deg) :38:14 Jerry Gilfoyle Alpha Decay 6 / 26

9 Interpretation of Rutherford Scattering Results Measured/Predicted Scattering Angle (deg) What does this say about the 4 2He Th potential energy? Jerry Gilfoyle Alpha Decay 7 / 26

10 Measuring the Size of the Nucleus θ cm (deg) Jerry Gilfoyle Alpha Decay 8 / 26

11 Measuring the Size of the Nucleus PRL 109, (2012) E cm = 23.1 MeV E cm = 28.3 MeV θ cm (deg) Jerry Gilfoyle Alpha Decay 8 / 26

12 The 4 He Th Potential α-th Potential Blue - known Red - a guess V(MeV) r(fm) Jerry Gilfoyle Alpha Decay 9 / 26

13 The Paradox of Alpha Decay 1 We have probed the Th potential into an internuclear distance of r DOCA = 48 fm using a 4 He beam of E( 4 He) = MeV. 2 The data are consistent with the Coulomb force and no others. 3 The radioactive decay U Th + 4 He emits an α (or 4 He) with energy E α = 4.2 MeV. 4 For a classical decay the emitted α should have an energy of at least E min = MeV. 5 It appears the decay α starts out at a distance r emit = 62 fm. 6 How do we explain this? Jerry Gilfoyle Alpha Decay 10 / 26

14 The Paradox of Alpha Decay 1 We have probed the Th potential into an internuclear distance of r DOCA = 48 fm using a 4 He beam of E( 4 He) = MeV. 2 The data are consistent with the Coulomb force and no others. 3 The radioactive decay U Th + 4 He emits an α (or 4 He) with energy E α = 4.2 MeV. 4 For a classical decay the emitted α should have an energy of at least E min = MeV. 5 It appears the decay α starts out at a distance r emit = 62 fm. 6 How do we explain this? Quantum Tunneling! Jerry Gilfoyle Alpha Decay 10 / 26

15 The Plan For Calculating Nuclear Lifetimes 1 The α particle ( 4 He) is confined by the nuclear potential and bounces back and forth between the walls of the nucleus. Assume its energy is the same as the emitted nucleon. 2 Each time it bounces off the nuclear wall it has a finite probability of tunneling through the barrier. 3 The decay rate will the product of the rate of collisions with a wall and the probability of transmission. 4 The lifetime is the inverse of the decay rate. 5 The radius of a nucleus has been found to be described by r nuke = 1.2A 1/3 where A is the mass number of the nucleus. 6 We are liberally copying the work of Gamow, Condon, and Gurney. Like them we will assume V = 0 inside the nucleus and V = 0 from the classical turning point to infinity. Jerry Gilfoyle Alpha Decay 11 / 26

16 The 4 He Th Potential 30 Gamow, Condon and Gurney 20 V(MeV) MeV r(fm) Jerry Gilfoyle Alpha Decay 12 / 26

17 The Transfer-Matrix Solution V(MeV) MeV r(fm) Jerry Gilfoyle Alpha Decay 13 / 26

18 Recall For a Single Barrier ζ 1 = tζ 3 = d 12 p 2 d 21 p 1 2 ζ 3 = t 11 = 1 4 d 12 = 1 2 ( 1 + k 2 k 1 1 k 2 k 1 1 k 2 k k 2 k 1 ) ( p 1 e ik 2 2a 0 1 = 0 e ik 22a k 1 = ( t11 t 12 t 21 t 22 ) d 21 = 1 2 p 2 = ) ζ 3 T = 1 t 11 2 ( 1 + k 1 k 2 1 k 1 k 2 1 k 1 k k 1 k 2 ) ( e ik 2 2a 0 0 e ik 22a 2mE 2m(E V ) 2 k 2 = 2 [( 1 + k ) ( 2 e ik 22a 1 + k ) ( k ) ( 2 e ik 22a 1 k )] 1 k 1 k 2 k 1 k 2 ) Jerry Gilfoyle Alpha Decay 14 / 26

19 The Transfer-Matrix Solution V(MeV) MeV r(fm) Jerry Gilfoyle Alpha Decay 15 / 26

20 The Transfer-Matrix Solution V(MeV) MeV r(fm) Jerry Gilfoyle Alpha Decay 16 / 26

21 Difference Between Lecture and Readings There are some differences between the formula for Rutherford scattering in the reading (go here) that are discussed below. The lecture formula is dσ dω = ( Z1 Z 2 e 2 ) 2 1 4E cm while the expression in the reading is the following. dσ dcos θ = π 2 z2 Z 2 α 2 [ c KE sin 4 ( θ 2 To go from Eq 1 to Eq 2 you need to make the following changes. ) (1) ] 2 1 (1 cos θ) 2 (2) 1 Change some variable names so Z 1 = z, Z 2 = Z, E cm = KE. 2 Use dω = sin θdθdφ = dcos θdφ and integrate over all φ or φ = 0 2π. This gives you a factor of 2π in front of Eq 1. 3 Make the following substitutions and you get Eq 2. dσ d cos θ = 2π 0 dσ dσ dφ = 2π dω dω e 2 = α c and sin 2 θ 2 = 1 (1 cos θ) (4) 2 Jerry Gilfoyle Alpha Decay 17 / 26 (3)

22 Results Points->Data, Line->Theory Lifetime (s) E α (MeV) Jerry Gilfoyle Alpha Decay 18 / 26

23 Additional slides. Jerry Gilfoyle Alpha Decay 19 / 26

24 What is an Angle? Jerry Gilfoyle Alpha Decay 20 / 26

25 What is an Angle? y dθ θ r ds x dθ = ds r Jerry Gilfoyle Alpha Decay 20 / 26

26 Solid Angle Jerry Gilfoyle Alpha Decay 21 / 26

27 Solid Angle Jerry Gilfoyle Alpha Decay 22 / 26

28 Solid Angle Jerry Gilfoyle Alpha Decay 23 / 26

29 Solid Angle Jerry Gilfoyle Alpha Decay 24 / 26

30 Solid Angle Jerry Gilfoyle Alpha Decay 25 / 26

31 Solid Angle Jerry Gilfoyle Alpha Decay 26 / 26

32 Solid Angle da = rdθ r sin θdφ = r 2 sin θdθdφ Jerry Gilfoyle Alpha Decay 26 / 26

33 Solid Angle da = rdθ r sin θdφ = r 2 sin θdθdφ dω = da r 2 = sin θdθdφ Jerry Gilfoyle Alpha Decay 26 / 26