Nov 2006, Lectures &9 Nuclear Physics Lectures, Dr. Armin Reichold 1 Lectures 8 & 9  decay theory

Nov 2006, Lectures &9Nuclear Physics Lectures, Dr. Armin Reichold Overview 8.1 QM tunnelling and  decays 8.2 Fermi theory of  decay and electron capture 8.3 The Cowan and Reines Experiment 8.4 The Wu experiment 8.5  decays (very brief)

Nov 2006, Lectures &9Nuclear Physics Lectures, Dr. Armin Reichold 3  Decay Theory Previously looked at kinematics and energetics now study the dynamics i.e. the interesting bit. Will need this to calculate life times Will get to understand variations in lifetimes

Nov 2006, Lectures &9Nuclear Physics Lectures, Dr. Armin Reichold  Decay Theory Consider 232 Th, Z=90, with radius of R=7.6 fm It alpha decays with E a =4.08 MeV at r=  But at R=7.6 fm the potential energy of the alpha would be E ,pot =34 MeV if we believe: Question: How does the  escape from the Th nucleus? Answer: by QM tunnelling which we really should!

Nov 2006, Lectures &9Nuclear Physics Lectures, Dr. Armin Reichold 5 r nucleusinside barrier (negative KE)small flux of real α 8.1  Decay Theory I IIIII potential energy of  total energy of  Exponential decay of  radial wave function in alpha decay in 3 regions oscillatory  r=t r=R see also Williams, p.85 to 89

Nov 2006, Lectures &9Nuclear Physics Lectures, Dr. Armin Reichold QM Tunnelling through a square well (the easy bit) Boundary condition for  and d  /dx at r=0 and r=t give 4 equations for times such that Kt>>1 and approximating k≈K we get transmission probability: T=|D| 2 ~ exp(-2Kt) [Williams, p.85] in regions I and III in region II unit incoming oscillatory wave reflected wave of amplitude A two exponential decaying waves of amplitude B and C transmitted oscillatory wave of amplitude D Wave vector Ansatz: Stationary Wavefunction Ansatz: E tot Potential :V r=0 r=t V=V 0 IIIIII r V=0 4 unknowns !

Nov 2006, Lectures &9Nuclear Physics Lectures, Dr. Armin Reichold  -decay Neutrons Protons AlphasAlphas  E bind ( 4 2  )=28.3 MeV > 4*6MeV  E sep ≈6MeV per nucleon for heavy nuclei

Nov 2006, Lectures &9Nuclear Physics Lectures, Dr. Armin Reichold Tunnelling in  -decay Assume there is no recoil in the remnant nucleus Assume we can approximate the Coulomb potential by sequence of many square wells of thickness  r with variable height V i Transmission probability is then product of many T factors where the K inside T is a function of the potential: The region between R and R exit is defined via: V(r)>E kin Inserting K into the above gives: We call G the Gamov factor

Nov 2006, Lectures &9Nuclear Physics Lectures, Dr. Armin Reichold Tunnelling in  -decay Use the Coulomb potential for an a particle of charge Z 1 and a nucleus of charge Z 2 for V(r) the latter defines the relation between the exit radius and the alpha particles kinetic energy inserted into: and Z 1 =2 gives

Nov 2006, Lectures &9Nuclear Physics Lectures, Dr. Armin Reichold Tunnelling in  -decay How can we simplify this ? for nuclei that actually do a-decay we know typical decay energies and sizes R typ ≈10 fm, E typ ≈ 5 MeV, Z typ ≈ 80  R exit,typ ≈ 60 fm >>R typ since Inserting all this into G gives: And further expressing R exit via E kin gives:

Nov 2006, Lectures &9Nuclear Physics Lectures, Dr. Armin Reichold  -decay Rates How can we turn the tunnelling probability into a decay rate? We need to estimate the “number of hits” that an  makes onto the inside surface of a nucleus. Assume: the a already exists in the nucleus it has a velocity v 0 =(2E kin /m) 1/2 it will cross the nucleus in  t=2R/v 0  it will hit the surface with a rate of  0 =v 0 /2R Decay rate  is then “rate of hits” x tunnelling probability Note:  0 is a very rough plausibility estimate! Williams tells you how to do it better but he can’t do it either!

Nov 2006, Lectures &9Nuclear Physics Lectures, Dr. Armin Reichold  -decay experimental tests Predict exponential decay rate proportional to (E kin ) 1/2 Agrees approximately with data for even-even nuclei. But angular momentum effects complicate the picture: Additional angular momentum barrier (as in atomic physics) E l is small compared to E Coulomb E.g. l=1, R=15 fm  E l ~0.05 MeV compared to Z=90  E coulomb ~17 MeV. but still generates noticeable extra exponential suppression. Spin (  J ) and parity (  P ) change from parent to daughter  J=L   P=(-1) L

Nov 2006, Lectures &9Nuclear Physics Lectures, Dr. Armin Reichold  -decay experimental tests We expect: ln(decay rate)

Nov 2006, Lectures &9Nuclear Physics Lectures, Dr. Armin Reichold Fermi  Decay Theory Consider simplest case: of  -decay, i.e. n decay At quark level: d  u+W followed by decay of virtual W to electron + anti- neutrino this section is close to Cottingham & Greenwood p ff but also check that you understand Williams p ff W - e - ( ) e d u u d u d n p

Nov 2006, Lectures &9Nuclear Physics Lectures, Dr. Armin Reichold Fermi Theory 4 point interaction Energy of virtual W << m W  life time is negligible assume interaction is described by only a single number we call this number the Fermi constant of beta decay G  also assume that p is heavy and does not recoil (it is often bound into an even heavier nucleus for other  -decays) We ignore parity non-conservation e - ( ) e d u u d u d n p

Nov 2006, Lectures & Fermi Theory as we neglect nuclear recoil energy electron energy distribution is determined by density of states but p e and p or E e and E are correlated to conserve energy  we can not leave them both variable

Nov 2006, Lectures &9Nuclear Physics Lectures, Dr. Armin Reichold Fermi Theory  Kurie Plot FGR to get a decay rate and insert previous results: A let’s plot that from real data

Nov 2006, Lectures &9Nuclear Physics Lectures, Dr. Armin Reichold Electron Spectrum Observe electron kinetic energy spectrum in tritium decay Implant tritium directly into a biased silicon detector Observe internal ionisation (electron hole pairs) generated from the emerging electron as current pulse in the detector number of pairs proportional to electron energy Observe continuous spectrum  neutrino has to carrie the rest of the energy End point of this spectrum is function of neutrino mass But this form of spectrum is bad for determining the endpoint accurately E kin,e (keV) Relative Intensity Simple Spectrum

Nov 2006, Lectures &9Nuclear Physics Lectures, Dr. Armin Reichold Kurie Plot A plot of: should be linear …but it does not! Why? …because that’s off syllabus! But if you really must know … Electron notices Coulomb field of nucleus   e gets enhanced near to proton (nucleus) The lower E e the bigger this effect We compensate with a “Fudge Factor” scientifically aka “Fermi Function” K(Z,p e ) Can be calculated but we don’t have means to do so  We can’t integrate I(p e ) to give a total rate (I(p)/p 2 K(Z,p)) 1/2 E kin,e (keV) Kurie-Plot

Nov 2006, Lectures &9Nuclear Physics Lectures, Dr. Armin Reichold Selection Rules Fermi Transitions: e couple to give spin S e =0 “Allowed transitions” L e =0   J n  p =0. Gamow-Teller transitions: e couple to give spin S e =1 “Allowed transitions” L e =0   J n  p =0 or ±1 “Forbidden” transitions See arguments on slide 15 Higher order terms correspond to non-zero  L. Therefore suppressed depending on (q.r) 2L Usual QM rules give:  J n  p =L e +S e

Nov 2006, Lectures &9Nuclear Physics Lectures, Dr. Armin Reichold Electron Capture capture atomic electron Can compete with  + decay. Use FGR again and first look at matrix element For “allowed” transitions we consider  e and  const. Only l e =0 has non vanishing  e (r=0) and for n e =1 this is largest.

Nov 2006, Lectures &9Nuclear Physics Lectures, Dr. Armin Reichold Electron Capture Density of states easier now only a 2-body final state ( n, ) n is assumed approximately stationary  only matters  final state energy = E apply Fermi’s Golden Rule AGAIN:

Nov 2006, Lectures &9Nuclear Physics Lectures, Dr. Armin Reichold Anti-neutrino Discovery Inverse Beta Decay Assume again no recoil on n But have to treat positron fully relativistic Same matrix elements as  -decay because all wave functions assume to be plane waves Fermi’s Golden Rule (only positron moves in final state!)

Nov 2006, Lectures &9Nuclear Physics Lectures, Dr. Armin Reichold Anti-neutrino Discovery Phase space factor: Neglect neutron recoil: Combine with FGR

Nov 2006, Lectures &9Nuclear Physics Lectures, Dr. Armin Reichold The Cowan & Reines Experiment for inverse  E ~ 1MeV   ~ cm 2 Pauli’s prediction verified by Cowan and Reines. 1 GW Nuclear Reactor PMT H 2 0+CdCl 2 Liquid Scint. Shielding original proposal wanted to use a bomb instead! Liquid Scint. PMT  -beam all this well under ground to reduce cosmic rays!

Parity Definitions Parity transforms from a left to a right handed co-ordinate system and vice versa Eigenvalues of parity are +/- 1. If parity is conserved: [H,P]=0  eigenstates of H are eigenstates of parity  all observables have a defined parity If Parity is conserved all result of an experiment should be unchanged by parity operation If parity is violated we can measure observables with mixed parity, i.e. not eigenstates of parity best read Bowler, Nuclear Physics, chapter 2.3 on parity!

Nov 2006, Lectures &9Nuclear Physics Lectures, Dr. Armin Reichold 27 Parity Conservation If parity is conserved for reaction a + b  c + d. Absolute parity of states that can be singly produced from vacuum (e.g. photons   = -1 ) can be defined wrt. vacuum For other particles we can define relative parity. e.g. arbitrarily define  p =+1,  n =+1 then we can determine parity of other nuclei wrt. this definition parity of anti-particle is opposite particle’s parity Parity is a hermitian operator as it has real eigenvalues! If parity is conserved =0 (see next transparency). Nuclei are Eigenstates of parity

Nov 2006, Lectures &9Nuclear Physics Lectures, Dr. Armin Reichold 28 Parity Conservation Let O p be an observable pseudo scalar operator, i.e. [H, O p ]=0 Let parity be conserved [H, P]=0  [P, O p ]=0 Let  be Eigenfunctions of P and H with intrinsic parity  p = - = 0 QED it is often useful to think of parity violation as a non vanishing expectation value of a pseudo scalar operator insert Unity as PO p =-O p P since [P, O p ]=0 use E.V. of  under parity

Nov 2006, Lectures &9Nuclear Physics Lectures, Dr. Armin Reichold 29 Q: Is Parity Conserved In Nature? A1: Yes for all electromagnetic and strong interactions. Feynman lost his 100$ bet that parity was conserved everywhere. In 1956 that was a lot of money! A2: Big surprise was that parity is violated in weak interactions. How was this found out? can’t find this by just looking at nuclei. They are parity eigenstates (defined via their nuclear and EM interactions) must look at properties of leptons in beta decay which are born in the weak interaction see Bowler, Nuclear Physics, chapter 3.13

Nov 2006, Lectures &9Nuclear Physics Lectures, Dr. Armin Reichold 30 Mme. Wu’s “Cool” Experiment Adiabatic demagnetisation to get T ~ 10 mK Align spins of 60 Co with magnetic field. Measure angular distribution of electrons and photons relative to B field. Clear forward-backward asymmetry of the electron direction (forward=direction of B)  Parity violation. Note: Spin S= axial vector Magnetic field B = axial vector Momentum p = real vector  Parity will only flip p not B and S  - allowed Gamov Teller decay  J= MeV 1.33 MeV 0 MeV Excitation Energy 60 Ni 60 Co ~100%

Nov 2006, Lectures &9 31 The Wu Experiment  ’s from late cascade decays of Ni* measure degree of polarisation of Ni* and thus of Co gamma det. signals summed over both B orientations! scintillator signal electron signal shows asymmetry of the electron distribution see also Burcham & Jobes, P.370 sample warms up  asymmetry disappears

Nov 2006, Lectures &9Nuclear Physics Lectures, Dr. Armin Reichold 32 Interpreting the Wu Experiment Let’s make an observable pseudo scalar O p : O p =J Co * p e = Polarisation (axial vector dot real vector) If parity were conserved this would have a vanishing expectation value But we see that p e prefers to be anti-parallel to B and thus to J Co Thus: parity is violated

Nov 2006, Lectures &9Nuclear Physics Lectures, Dr. Armin Reichold 33 Improved Wu-Experiment Polar diagram of angular dependence of electron intensity  is angle of electron momentum wrt spin of 60Co or B using many detectors at many angles points indicate measurements if P conserved this would have been a circle centred on the origin

Nov 2006, Lectures &9Nuclear Physics Lectures, Dr. Armin Reichold  decays When do they occur? Nuclei have excited states similar to atoms. Don’t worry about details E,J P (need a proper shell model to understand). EM interaction less strong then the strong (nuclear) interaction Low energy excited states E<6 MeV above ground state can’t usually decay by nuclear interaction   -decays  -decays important in cascade decays following  and  decays. Practical consequences Fission. Significant energy released in  decays (see later lectures) Radiotherapy:  from Co60 decays Medical imaging eg Tc (see next slide)

Nov 2006, Lectures &9Nuclear Physics Lectures, Dr. Armin Reichold 35 Energy Levels for Mo and Tc Make Mo-99 in an accelerator attach it to a bio-compatible molecule inject that into a patient and observe where the patient emits  -rays don’t need to “eat” the detector as  ’s penetrate the body call this substance a tracer both  decay leaves Tc in excited state. MeV interesting meta stable state