Beta decay
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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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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

      • 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?



  • 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

The neutrino

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



  • 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

Spin in beta decay

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

Spin in beta decay1

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)

Q value calculation review

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



  • 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

Weak interaction model of beta decay

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;

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

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

Weak interaction1

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

Weak interaction2

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

Coulomb correction

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

Kurie plot

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

Fermi golden rule

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

Comparative half lives

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)

Comparative half lives1

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

Log ft calculation

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

Log ft data

Log ft data

  • What drives changes in log ft values for 205Hg?

    • Examine spin and parity changes between parent and daughter state

Extranuclear effects of ec

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

Selection rules

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

Selection rules1

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

Forbidden transitions

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

Other beta decay

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

Topic review

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

Homework questions

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.

Pop quiz

Pop Quiz

  • Calculate the logft for the decay of 241Pu, 162Eu, 44Ti, and 45Ti. Provide the transition for each?

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