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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
- 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

- Second particle found from U decay

- Beta decay
- 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?

-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

- 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

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

- observed by inverse processes
- 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

- 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

- Electron and neutrino spin both 1/2
- 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

- High A

- Fermi (F) (S=0)
- A source can produce a mixture of F and GT spins
- Can be used to define decay

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

- neutron and protons are in the same orbitals
- 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)

- Basis of model to calculate decay constant

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

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

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;

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

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

Weak InteractionWeak 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 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

- Coulomb interaction between nucleus and emitted electron (e(0)) neglected

- dn/dEo density of final states with electron momentum
- 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

- 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

- 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

- 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

- final state can be composed of several states with the same energy

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

- may be thought of as half life corrected for differences in Z and W
- 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-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

- 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

- What drives changes in log ft values for 205Hg?
- Examine spin and parity changes between parent and daughter state

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

- important in measuring disintegration rates of EC nuclides

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

- 0 or 1
- 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

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

Selection RulesForbidden 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

- 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

- Very long half-life

- 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

- 1 % of neutrons delayed
- decay populates some high energy states in Kr daughter
- 51 neutrons, neutron emission to form 86Kr

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

- What influences shape
- Kurie Plots
- Beta Decay Rate Constant
- Calculations
- Selection rules
- Log ft
- How do values compare and relate to spin and parity

- Log ft

- Other types of beta decay

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

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