Lectures%208%20

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Lectures 8 9

• abg decay theory

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8.0 Overview

• 8.1 QM tunnelling and a decays
• 8.2 Fermi theory of b decay and electron capture
• 8.3 The Cowan and Reines Experiment
• 8.4 The Wu experiment
• 8.5 g decays (very brief)

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abg 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

4
8.1 a Decay Theory

• Consider 232Th, Z90, with radius of R7.6 fm
• It alpha decays with Ea4.08 MeV at r?
• But at R7.6 fm the potential energy of the alpha
  would be Ea,pot34 MeV if we believe
• Question How does the a escape from the Th
  nucleus?
• Answer by QM tunnelling

which we really should!
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8.1 a Decay Theory
I
II
III
r
nucleus
inside barrier (negative KE)
small flux of real a
rt
rR

• see also Williams, p.85 to 89

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8.1 QM Tunnelling through a square well (the
easy bit)
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
4 unknowns !

• Boundary condition for Y and dY/dx at r0 and rt
  give 4 equations
• for times such that Ktgtgt1 and approximating kK
  we get transmission probability TD2exp(-2Kt)
  Williams, p.85

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8.1 a-decay
Protons
Alphas
Neutrons
8
8.1 Tunnelling in a-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 Dr
  with variable height Vi
• 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 Rexit is defined via
  V(r)gtEkin
• Inserting K into the above gives
• We call G the Gamov factor

9
8.1 Tunnelling in a-decay

• Use the Coulomb potential for an a particle of
  charge Z1 and a nucleus of charge Z2 for V(r)

the latter defines the relation between the exit
radius and the alpha particles kinetic energy
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8.1 Tunnelling in a-decay

• How can we simplify this ?
• for nuclei that actually do a-decay we know
  typical decay energies and sizes
• Rtyp10 fm, Etyp 5 MeV, Ztyp 80
• ?Rexit,typ 60 fm gtgtRtyp
• since
• Inserting all this into G gives
• And further expressing Rexit via Ekin gives

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8.1 a-decay Rates

• How can we turn the tunnelling probability into a
  decay rate?
• We need to estimate the number of hits that an
  a makes onto the inside surface of a nucleus.
• Assume
• the a already exists in the nucleus
• it has a velocity v0(2Ekin/m)1/2
• it will cross the nucleus in Dt2R/v0
• ? it will hit the surface with a rate of w0v0/2R
• Decay rate w is then rate of hits x tunnelling
  probability
• Note w0 is a very rough plausibility estimate!
  Williams tells you how to do it better but he
  cant do it either!

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8.1 a-decay experimental tests

• Predict exponential decay rate proportional to
  (Ekin)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)
• El is small compared to ECoulomb
• E.g. l1, R15 fm ? El0.05 MeV compared to
• Z90 ? Ecoulomb17 MeV.
• but still generates noticeable extra exponential
  suppression.
• Spin (DJ) and parity (DP) change from parent to
  daughter
• DJLa DP(-1)L

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8.1 a-decay experimental tests

• We expect

ln(decay rate)
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8.2 Fermi b Decay Theory

• Consider simplest case of b-decay, i.e. n decay
• At quark level d?uW followed by decay of
  virtual W to electron anti-neutrino
• this section is close to Cottingham Greenwood
  p.166 - ff
• but also check that you understand Williams p.
  292 - ff

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8.2 Fermi Theory

• 4 point interaction
• Energy of virtual W ltlt mW ? life time is
  negligible
• assume interaction is described by only a single
  number
• we call this number the Fermi constant of beta
  decay Gb
• also assume that p is heavy and does not recoil
  (it is often bound into an even heavier nucleus
  for other b-decays)
• We ignore parity non-conservation

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8.2 Fermi Theory

• as we neglect nuclear recoil energy
• electron energy distribution is determined by
  density of states

• but pe and pn or Ee and En are correlated to
  conserve energy ? we can not leave them both
  variable

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8.2 Fermi Theory ? Kurie Plot

• FGR to get a decay rate and insert previous
  results

lets plot that from real data
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8.2 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

Simple Spectrum
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8.2 Kurie Plot

• A plot of should be linear
• but it does not! Why?
• because thats off syllabus!
• But if you really must know
• Electron notices Coulomb field of nucleus ?
• Ye gets enhanced near to proton (nucleus)
• The lower Ee the bigger this effect
• We compensate with a Fudge Factor
  scientifically aka Fermi Function K(Z,pe)
• Can be calculated but we dont have means to
  do so ?
• We cant integrate I(pe) to give a total rate

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8.2 Selection Rules

• Fermi Transitions
• en couple to give spin Sen0
• Allowed transitions Len0 ? DJn?p0.
• Gamow-Teller transitions
• en couple to give spin Sen1
• Allowed transitions Len0 ? DJn?p0 or 1
• Forbidden transitions
• See arguments on slide 15
• Higher order terms correspond to non-zero DL.
  Therefore suppressed depending on (q.r)2L
• Usual QM rules give DJn?pLenSen

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8.2 Electron Capture

• capture atomic electron
• Can compete with b decay.
• Use FGR again and first look at matrix element

• For allowed transitions we consider Ye and Yn
  const.

• Only le0 has non vanishing Ye(r0) and for ne1
  this is largest.

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8.2 Electron Capture

• Density of states easier now
• only a 2-body final state (n,n)
• n is assumed approximately stationary ? only n
  matters
• ? final state energy En

• apply Fermis Golden Rule AGAIN

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8.3 Anti-neutrino Discovery

• Inverse Beta Decay
• Assume again no recoil on n
• But have to treat positron fully relativistic
• Same matrix elements as b-decay because all wave
  functions assume to be plane waves
• Fermis Golden Rule (only positron moves in final
  state!)

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8.3 Anti-neutrino Discovery

• Phase space factor
• Neglect neutron recoil
• Combine with FGR

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8.3 The Cowan Reines Experiment

• for inverse b-decay _at_ En 1MeV ? s 10-47 cm2
• Paulis prediction verified by Cowan and Reines.

Liquid Scint.
PMT
1 GW Nuclear Reactor
H20CdCl2
PMT
Liquid Scint.
original proposal wanted to use a bomb instead!
all this well under ground to reduce cosmic rays!
Shielding
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8.4 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,P0 ? 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!

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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 hg -1) can be
  defined wrt. vacuum
• For other particles we can define relative
  parity. e.g. arbitrarily define hp1, hn1 then
  we can determine parity of other nuclei wrt. this
  definition
• parity of anti-particle is opposite particles
  parity
• Parity is a hermitian operator as it has real
  eigenvalues!
• If parity is conserved ltpseudo-scalargt0 (see
  next transparency).
• Nuclei are Eigenstates of parity

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Parity Conservation

• Let Op be an observable pseudo scalar operator,
  i.e. H, Op0
• Let parity be conserved H, P0 ? P, Op0
• Let Y be Eigenfunctions of P and H with intrinsic
  parity hp

insert Unity
as POp-OpP since P, Op0
use E.V. of Y under parity

• ltOpgt - ltOpgt 0 QED
• it is often useful to think of parity violation
  as a non vanishing expectation value of a pseudo
  scalar operator

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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?
• cant 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

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Mme. Wus Cool Experiment

• Adiabatic demagnetisation to get T 10 mK
• Align spins of 60Co with magnetic field.
• Measure angular distribution of electrons and
  photons relative to B field.
• Clear forward-backward asymmetry of the electron
  direction (forwarddirection 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

100
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The Wu Experiment
gs from late cascade decays of Ni measure
degree of polarisation of Ni and thus of Co
gamma det. signals summed over both B
orientations!
electron signal shows asymmetry of the electron
distribution
scintillator signal
sample warms up ? asymmetry disappears
see also Burcham Jobes, P.370
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Interpreting the Wu Experiment

• Lets make an observable pseudo scalar Op
• OpJCo pe Polarisation (axial vector dot real
  vector)
• If parity were conserved this would have a
  vanishing expectation value
• But we see that pe prefers to be anti-parallel to
  B and thus to JCo
• Thus parity is violated

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Improved Wu-Experiment

• Polar diagram of angular dependence of electron
  intensity
• q 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

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8.5 g decays

• When do they occur?
• Nuclei have excited states similar to atoms.
  Dont worry about details E,JP (need a proper
  shell model to understand).
• EM interaction less strong then the strong
  (nuclear) interaction
• Low energy excited states Elt6 MeV above ground
  state cant usually decay by nuclear interaction
  ? g-decays
• g-decays important in cascade decays following a
  and b decays.
• Practical consequences
• Fission. Significant energy released in g decays
  (see later lectures)
• Radiotherapy g from Co60 decays
• Medical imaging eg Tc (see next slide)

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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 g-rays
• dont need to eat the detector as g s
  penetrate the body
• call this substance a tracer

MeV
interesting meta stable state
both b decay leaves Tc in excited state.
MeV