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RFSS: Lecture 5 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: MZMZ-1, + decay: MZ MZ-1+2me Beta decay half-life few milliseconds to ~ 1016 years How does this compare to alpha decay?
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Q value calculation (Review)
Beta decay Find Q value for the Beta decay of 24Na 1 amu = MeV M (24Na)-M(24Mg) amu MeV From mass excess MeV Q value for the EC of 22Na M (22Na)-M(22Ne) amu MeV 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
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-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
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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, np+ + e- is not spin balanced, need another fermion
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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 involved in assigning decay Spin up and spin down
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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
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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)
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Positrons Annihilation radiation
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 Positrons Annihilation radiation energy carried off by two quanta of opposite momentum Annihilation conserves momentum Exploited in Positron Emission Tomography
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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 10-3 of the electromagnetic force constant Used to determine emitted electron momentum range per unit time P(pe) dpe;
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Weak Interaction 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 Mif matrix element characterizes transition from initial to final nuclear state Mif2 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
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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 locating electron and neutrino at nucleus for s wave compared to higher l p,d,f, etc. 2 of allowed transitions 2 of forbidden transitions Magnitudes of (0) and Mif are independent of energy division between electron and neutrino
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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
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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
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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
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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
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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)
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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=4.121 MeV T1/2 =70.6 s Log fb+ = 3.30, log t = 1.85 Log ft=5.15
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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
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Log ft data What drives changes in log ft values for 205Hg?
Examine spin and parity changes between parent and daughter state
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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
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Selection Rules 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 Mif2 values for all transitions with I=0, 1 without parity change was erroneous
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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 Forbidden Transitions
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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
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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= ( ) Q= 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
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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
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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.
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