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Beta Decay

Beta Decay. Readings: Nuclear and Radiochemistry: Chapter 3, Modern Nuclear Chemistry: Chapter 8 Neutrino Hypothesis Derivation of Spectral Shape Kurie Plots Beta Decay Rate Constant Selection Rules Transitions Majority of radioactive nuclei are outside range of alpha decay Beta decay

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Beta Decay

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  1. Beta Decay • Readings: Nuclear and Radiochemistry: Chapter 3, Modern Nuclear Chemistry: Chapter 8 • Neutrino Hypothesis • Derivation of Spectral Shape • Kurie Plots • Beta Decay Rate Constant • Selection Rules • Transitions • Majority of radioactive nuclei are outside range of alpha decay • Beta decay • Second particle found from U decay • Negative particle • Distribution of energies • Need another particle to balance spin • Parent, daughter, and electron • Need to account for half integer spin • Radioactive decay process in which A remains unchanged, but Z changes • - decay, electron capture, + decay • energetic conditions for decay: • - decay: MZ  MZ+1 • Electron capture: MZMZ-1, • + decay: MZ  MZ-1+2me • Beta decay half-life • few milliseconds to ~ 1016 years • How does this compare to alpha decay?

  2. -Decay • Decay energies of  -unstable nuclei vary systematically with distance from stability • Shown by mass parabolas • Energy-lifetime relations are not nearly so simple as alpha decay •  -decay half lives depend strongly on spin and parity changes as well as energy • For odd A, one -stable nuclide; for even A, at most three -stable nuclides • Information available from mass parabolas • Odd-odd nuclei near the stability valley (e.g., 64Cu) can decay in both directions • Form even-even nuclei • Beta particle energy not discrete • Continuous energy to maximum

  3. The Neutrino • Solved problems associated with -decay • Continuum of electron emission energies • Zero charge • neutron -> proton + electron • Small mass • Electron goes up to Q value • Anti-particle • Account for creation of electron particle • spin of ½ and obeys Fermi statistics • couple the total final angular momentum to initial spin of ½ ħ, • np+ + e- is not spin balanced, need another fermion

  4. Neutrino • Carries away appropriate amount of energy and momentum in each  process for conservation • Nearly undetectable due to small rest mass and magnetic moment • observed by inverse  processes • 37Cl+37Ar+e-: Detection of 37Ar • 71Ga+71Ge+e-: Detection of 71Ge • Antineutrinos emitted in - decay, neutrinos emitted in + decay • indistinguishable properties, except in capture reactions • Neutrinos created at moment of emission • n  p + - + n • p  n + + +  • Spin of created particles are key in assigning decay • Spin up and spin down

  5. Spin in Beta Decay • Spins of created particles can be combined in two ways • Electron and neutrino spin both 1/2 • S=1 in a parallel alignment • S= 0 in an anti-parallel alignment • two possible relative alignments of "created" spins • Fermi (F) (S=0) • Low A • Gamow-Teller (GT) (S =1) • High A • Spin change since neutron number tends to be larger than proton • A source can produce a mixture of F and GT spins • Can be used to define decay

  6. Spin in Beta Decay • Decay of even-even nuclei with N=Z (mirror nuclei) • neutron and protons are in the same orbitals • shell model, Nuclear Structure and Models lecture • 0+ to 0+ decay can only take place by a Fermi transition • Heavy nuclei with protons and neutrons in very different orbitals (from shell model) • GT is main mode, need to account for spin difference • Complex nuclei • rate of decay depends on overlap of wave functions of ground state of parent and state of the daughter • final state in daughter depends on decay mode • spin and parity state changes from parent to daughter • Half life information can be used to understand nuclear states • Decay constant can be calculated if wave functions are known • Observed rate indicates quantum mechanical overlap of initial and final state wave functions • Basis of model to calculate decay constant • Fermi golden rule (slide 15)

  7. Q value calculation (Review) Beta decay • Find Q value for the Beta decay of 24Na • 1 amu = 931.5 MeV • M (24Na)-M(24Mg) • 23.990962782-23.985041699 • 0.005921 amu • 5.5154 MeV • From mass excess • -8.4181 - -13.9336 • 5.5155 MeV • Q value for the EC of 22Na • M (22Na)-M(22Ne) • 21.994436425-21.991385113 • 0.003051 amu • 2.842297 MeV • From mass excess • -5.1824 - -8.0247 • 2.8432 MeV • Q- are ~0.5 – 2 MeV, Q + ~2-4 MeV and QEC ~ 0.2 – 2 MeV • What about positron capture instead of EC? Positron decay Electron Capture

  8. Positrons • Postulated in 1931 • Relativistic equations could be solved for electrons with positive energy states • Require energies greater than electron mass • Creation of positive hole with electron properties • Pair production process involves creation of a positron-electron pair by a photon in nuclear field • Nucleus carries off some momentum and energy • Positron-electron annihilation • Conversion of mass to energy when positron and electron interact • simultaneous emission of corresponding amount of energy in form of radiation • Responsible for short lifetime of positrons • No positron capture decay • Annihilation radiation • energy carried off by two  quanta of opposite momentum • Annihilation conserves momentum • Exploited in Positron Emission Tomography

  9. Weak Interaction: Model of Beta Decay • Fermi's theory of beta decay based on electromagnetic theory for light emission • Fermions interact during reaction • Degree of interaction from Fermi constant(g) • Value determined by experiment • 10-3of the electromagnetic force constant • Used to determine emitted electron momentum range per unit time P(pe) dpe;

  10. P(pe)dpe probability electron with momentum pe+dpe e electron wave function n neutrino wave function e(0)2 and n(0)2 probability of finding electron and neutrino at nucleus Mifmatrix element characterizes transition from initial to final nuclear state Mif2 a measure of overlap amount between wave functions of initial and final nuclear states dn/dEo is density of final states with electron in specified momentum interval number of states of final system per unit decay energy Weak Interaction

  11. Weak Interaction • Integration over all electron momenta from zero to maximum should provide transition probabilities or lifetimes • Variations in number of electrons at a given energy • Derivation of emission spectrum • Calculation of decay constant • Classically allowed transitions both have electron and neutrino emitted with zero orbital angular momentum • Allowed have s orbital angular momentum • Relatively high probabilities for location of electron and neutrino at nucleus for s wave compared to higher l • p,d,f, etc. • 2 of allowed transitions  2of forbidden transitions • Magnitudes of (0) and Mif are independent of energy division between electron and neutrino

  12. Weak Interaction • Spectrum shape determined entirely by e(0) and dn/dEo • dn/dEo density of final states with electron momentum • Coulomb interaction between nucleus and emitted electron (e(0)) neglected • Reasonable for low Z • Density of final states determined from total energy W • W is total (kinetic plus rest) electron energy • Wo is maximum W value • dn/dEo goes to zero at W = 1 and W = Wo • Yields characteristic bell shape beta spectra

  13. Coulomb Correction • Agreement of experiment and modeling at low Z • Minimized charge on nucleus • At higher Z need a correction factor to account for coulomb interaction • Coulomb interaction between nucleus and emitted electron • decelerate electrons and accelerate positrons • Electron spectra has more low-energy particles • Positron spectra has fewer low-energy particles • Treat as perturbation on electron wave function e(0) • Called Fermi function • Defined as ratio of e(0)2Coul /e(0)2free • perturbation on e(0) and spectrum multiplied by Fermi function • Z daughter nucleus • v beta velocity • + for electrons • - for positron

  14. Kurie Plot • Comparison of theory and experiment for momentum measurements • Square root of number of beta particles within a certain range divided by Fermi function plotted against beta-particle energy (W) • x axis intercept is Q value • Linear relationship designates allowed transition

  15. Fermi Golden Rule • Used for transition probability • Treat beta decay as transition that depends upon strength of coupling between initial and final states • Decay constant given by Fermi's Golden Rule • matrix element couples initial and final states • density of states that are available to system after transition • Wave function of initial and final state • Operator which coupled initial and final state • Rate proportional to strength of coupling between initial and final states factored by density of final states available to system • final state can be composed of several states with the same energy • Degenerate states

  16. Comparative Half Lives • Based on probability of electron energy emission coupled with spectrum and Coulomb correction fot1/2 • comparative half life of a transition • Assumes matrix element is independent of energy • true for allowed transitions • Yields ft (or fot1/2), comparative half-life • may be thought of as half life corrected for differences in Z and W • W is total kinetic energy • fo can be determine when Fermi function is 1 (low Z) • Rapid estimation connecting ft and energy • Simplified route to determine ft (comparative half-life)

  17. Comparative half-lives • Log ft = log f + log t1/2 • t1/2 in seconds • Z is daughter • Eo is maximum energy in MeV (Q value) • 14 O to 14N • positron decay • Q=1.81 MeV • T1/2 =70.6 s • Log fb+ = 1.83, log t = 1.84 • Log ft=3.67

  18. Log ft calculation • 212Bi beta decay • Q = 2.254 MeV • T1/2 = 3600 seconds • 64 % beta branch • lb=1.22E-4 s-1 • T1/2Beta =5625 seconds • Log f=3.73; log t=3.75 • Log ft=7.48

  19. Log ft data • What drives changes in log ft values for 205Hg? • Examine spin and parity changes between parent and daughter state

  20. Extranuclear Effects of EC • If K-shell vacancy is filled by L electron, difference in binding energies emitted as x-ray or used in internal photoelectric process • Auger electrons are additional extranuclear electrons from atomic shells emitted with kinetic energy equal to characteristic x-ray energy minus its binding energy • Fluorescence yield is fraction of vacancies in shell that is filled with accompanying x-ray emission • important in measuring disintegration rates of EC nuclides • radiations most frequently detected are x-rays

  21. Selection Rules • Allowed transitions are ones in which electron and neutrino carry away no orbital angular momentum • largest transition probability for given energy release • If electron and neutrino do not carry off angular momentum, spins of initial and final nucleus differ by no more than h/2 and parities must be same • 0 or 1 • Fermi or Gamow-Teller transitions • If electron and neutrino emitted with intrinsic spins antiparallel, nuclear spin change (I )is zero • singlet • If electron and neutrino spins are parallel, I may be +1, 0, -1 • triplet

  22. All transitions between states of I=0 or 1 with no change in parity have allowed spectrum shape I is nuclear spin Not all these transitions have similar fot values transitions with low fot values are “favored” or “superallowed”  emitters of low Z between mirror nuclei one contains n neutrons and n+1 protons, other n+1 neutrons and n protons Assumption of approximately equal Mif2 values for all transitions with I=0, 1 without parity change was erroneous Selection Rules

  23. Forbidden Transitions • When transition from initial to final nucleus cannot take place by emission of s-wave electron and neutrino • orbital angular momenta other than zero • l value associated with given transition deduced from indirect evidence • ft values, spectrum shapes • If l is odd, initial and final nucleus have opposite parities • If l is even, parities are same • Emission of electron and nucleus in singlet state requires I  l • Triple-state emission allows I  l+1

  24. Other Beta Decay • Double beta decay • Very long half-life • 130Te and 82Se as examples • Can occur through beta stable isotope • 76Ge to 76Se by double beta • 76Ge to 76As • Q= -73.2130- (-72.2895) • Q= -0.9235 MeV • Possible to have neutrinoless double beta decay • two neutrinos annihilate each other • Neutrino absorbed by nucleon • Beta delayed decay • Nuclei far from stability can populate unbound states and lead to direct nucleon emission • First recognized during fission • 1 % of neutrons delayed • 87Br is produced in nuclear fission and decays to 87Kr • decay populates some high energy states in Kr daughter • 51 neutrons, neutron emission to form 86Kr

  25. Topic Review • Fundamentals of beta decay • Electron, positron, electron capture • Neutrino Hypothesis • What are trends and data leading to neutrino hypothesis • Derivation of Spectral Shape • What influences shape • Particles, potentials • Kurie Plots • Beta Decay Rate Constant • Calculations • Selection rules • Log ft • How do values compare and relate to spin and parity • Other types of beta decay

  26. Homework questions • For beta decay, what is the correlation between decay energy and half life? • What is the basis for the theory of the neutrino emission in beta decay. •  In beta decay what are the two possible arrangements of spin? • What is the basis for the difference in positron and electron emission spectra? • What log ft value should we expect for the -decay to the 1- state of 144Pr? • Why is there no  decay to the 2+ level? • Calculate and compare the logft values for EC, positron and electron decay for Sm isotopes.

  27. Pop Quiz • Calculate the logft for the decay of 241Pu, 162Eu, 44Ti, and 45Ti. Provide the transition for each? Put comments on blog Bring homework to class on Tuesday 1 October

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