Beta Decay Readings Nuclear and Radiochemistry Chapter 3

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 à Second particle found from U decay * Negative particle * Distribution of energies * Need another particle to balance spin Ø Parent, daughter, and electron Ø 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: à - decay: MZ MZ+1 à Electron capture: MZ MZ-1, à + decay: MZ MZ-1+2 me Beta decay half-life § few milliseconds to ~ 1016 years § How does this compare to alpha decay? • • 5 -1

-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. , 64 Cu) can decay in both directions § Form even-even nuclei • Beta particle energy not discrete § Continuous energy to maximum 5 -2

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 5 -3

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 à 37 Cl+ 37 Ar+e-: Detection of 37 Ar à 71 Ga+ 71 Ge+e-: Detection of 71 Ge • Antineutrinos emitted in - decay, neutrinos emitted in + decay § indistinguishable properties, except in capture reactions • Neutrinos created at moment of emission § n p + - + § p n + + + • Spin of created particles are key in assigning decay § Spin up and spin down 5 -4

Spin in Beta Decay • Spins of created particles can be combined in two ways § Electron and neutrino spin both 1/2 àS =1 in a parallel alignment à S = 0 in an anti-parallel alignment • two possible relative alignments of "created" spins § Fermi (F) (S =0) àLow A § Gamow-Teller (GT) (S =1) àHigh 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 5 -5

Spin in Beta Decay • Decay of even-even nuclei with N=Z (mirror nuclei) § neutron and protons are in the same orbitals à shell model, Nuclear Structure and Models lecture à 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 à 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 à Basis of model to calculate decay constant * Fermi golden rule (slide 15) 5 -6

Q value calculation (Review) • • Find Q value for the Beta decay of 24 Na § 1 amu = 931. 5 Me. V § M (24 Na)-M(24 Mg) à 23. 990962782 -23. 985041699 à 0. 005921 amu * 5. 5154 Me. V § From mass excess à -8. 4181 - -13. 9336 à 5. 5155 Me. V Q value for the EC of 22 Na § M (22 Na)-M(22 Ne) § 21. 994436425 -21. 991385113 § 0. 003051 amu à 2. 842297 Me. V § From mass excess à -5. 1824 - -8. 0247 à 2. 8432 Me. V Beta decay Positron decay Electron Capture Q - are ~0. 5 – 2 Me. V, Q + ~2 -4 Me. V and QEC ~ 0. 2 – 2 Me. V What about positron capture instead of EC? 5 -7

• • • 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 à No positron capture decay Positrons • Annihilation radiation § energy carried off by two quanta of opposite momentum § Annihilation conserves momentum § Exploited in Positron Emission Tomography 5 -8

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) à Value determined by experiment à 10 -3 of the electromagnetic force constant • Used to determine emitted electron momentum range per unit time P(pe) dpe; 5 -9

Weak Interaction • • P(pe)dpe probability electron with momentum pe+dpe e electron wave function neutrino wave function e(0) 2 and (0) 2 probability of finding electron and neutrino at nucleus • Mif matrix 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/d. Eo is density of final states with electron in specified momentum interval § number of states of final system per unit decay energy 5 -10

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 à p, d, f, etc. à 2 of allowed transitions 2 of forbidden transitions • Magnitudes of (0) and Mif are independent of energy division between electron and neutrino 5 -11

Weak Interaction • Spectrum shape determined entirely by e(0) and dn/d. Eo § dn/d. Eo density of final states with electron momentum àCoulomb 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/d. Eo goes to zero at W = 1 and W = Wo § Yields characteristic bell shape beta spectra 5 -12

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 à Electron spectra has more low-energy particles à 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) 2 Coul / e(0) 2 free § perturbation on e(0) and spectrum multiplied by Fermi function à Z daughter nucleus à v beta velocity à + for electrons à - for positron 5 -13

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 5 -14

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 à Degenerate states 5 -15

Comparative Half Lives • Based on probability of electron energy emission coupled with spectrum and Coulomb correction fot 1/2 § comparative half life of a transition • Assumes matrix element is independent of energy § true for allowed transitions • Yields ft (or fot 1/2), comparative half-life § may be thought of as half life corrected for differences in Z and W àW 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) 5 -16

• Log ft = log f + log t 1/2 Comparative half-lives § t 1/2 in seconds • Z is daughter • Eo is maximum energy in Me. V (Q value) • • • 14 O to 14 N § positron decay § Q=1. 81 Me. V § T 1/2 =70. 6 s Log f + = 1. 83, log t = 1. 84 Log ft=3. 67 5 -17

Log ft calculation • 212 Bi beta decay • Q = 2. 254 Me. V • T 1/2 = 3600 seconds § 64 % beta branch § l =1. 22 E-4 s-1 § T 1/2 Beta =5625 seconds • Log f=3. 73; log t=3. 75 • Log ft=7. 48 5 -18

Log ft data • What drives changes in log ft values for 205 Hg? § Examine spin and parity changes between parent and daughter state 5 -19

Extranuclear Effects of EC • If K-shell vacancy is filled by L electron, difference in binding energies emitted as xray 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 xray emission § important in measuring disintegration rates of EC nuclides àradiations most frequently detected are x-rays 5 -20

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 à 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 5 -21

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” à emitters of low Z àbetween 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 5 -22

• 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 Forbidden Transitions § 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 5 -23

Other Beta Decay • Double beta decay § Very long half-life à 130 Te and 82 Se as examples § Can occur through beta stable isotope § 76 Ge to 76 Se by double beta à 76 Ge to 76 As à Q= -73. 2130 - (-72. 2895) • à Q= -0. 9235 Me. V § Possible to have neutrinoless double beta decay à two neutrinos annihilate each other à 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 à 1 % of neutrons delayed * 87 Br is produced in nuclear fission and decays to 87 Kr § decay populates some high energy states in Kr daughter à 51 neutrons, neutron emission to form 86 Kr 5 -24

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 à Particles, potentials • Kurie Plots • Beta Decay Rate Constant § Calculations § Selection rules à Log ft * How do values compare and relate to spin and parity • Other types of beta decay 5 -25

Homework questions • For beta decay, what is the correlation between decay energy and half life? • What is the basis for 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 144 Pr? • Why is there no decay to the 2+ level? • Calculate and compare the logft values for EC, positron and electron decay for Sm isotopes. 5 -26

Pop Quiz • Calculate the logft for the decay of 241 Pu, 162 Eu, 44 Ti, and 45 Ti. Provide the transition for each? Put comments on blog Bring homework to class on Tuesday 1 October 5 -27