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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
- Beta decay half-life
- few milliseconds to ~ 1016 years
- How does this compare to alpha decay?

-Decay

- Class includes any 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
- + decay needs to exceed 1.02 MeV
- Below 1.02 MeV EC dominates
- + increases with increasing energy

- Decay energies of -unstable nuclei rather systematically with distance from stability
- Predicted 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
- Range of particle 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

- 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
- 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

- High A

- Fermi (F) (S=0)
- A source can produce a mixture of F and GT spins

Spin in Beta Decay

- Decay of even-even nuclei with N=Z (mirror nuclei)
- neutron and protons are in the same orbitals
- 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

Energetics (Review)

- Beta decay
- electron can be combined with the positive ion to create a neutral atom
- release of very small binding energy
- use neutral atoms to calculate the Q value
- assuming that the mass of the antineutrino is very small

- electron can be combined with the positive ion to create a neutral atom
- Consider beta decay of 14C
- 14C14N+ + β- +antineutrino + energy
- Energy = mass 14C – mass 14N

- 14C14N+ + β- +antineutrino + energy
- Positron decay
- 2 extra electrons (daughter less Z, emission of positron)

- Electron Capture

Q value calculation (Review)

- 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?

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
- Interaction of electron into a whole in sea of electrons of negative energy
- simultaneous emission of corresponding amount of energy in form of radiation
- Responsible for short lifetime of positrons
- No positron capture decay

- Interaction of electron into a whole in sea of electrons of negative energy
- 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
- Electromagnetic interaction characterized by electron charge
- Needs to be replaced for beta decay
- Fermi constant (g)
- Value determined by experiment
- 10-3of the electromagnetic force constant

- Electromagnetic interaction characterized by electron charge
- 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

Mifis matrix element characterizing the transition from the initial to the final nuclear state

Mif2 a measure of the amount of overlap between the wave functions of initial and final nuclear states

dn/dEo is the density of final states with the electron in the specified momentum interval

number of states of the final system per unit decay energy

Fermi constant (g) governs other interactions in addition to beta decay

m-meson decay, p-meson decay, neutrino-electron scattering

Weak interactions

Weak InteractionWeak Interaction

- Integration over all electron momenta from zero to maximum possible to evaluate spectrum should provide transition probabilities or lifetimes
- Variations in number of electrons at a given energy

- 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 nuclear 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 division of energy 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
- At higher Z need a correction factor to account for coulomb interaction
- Coulomb interaction between nucleus and the 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 the 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 against beta-particle energy

- Linear relationship designates allowed transition

Fermi Golden Rule

- Treat beta decay as transition that depends upon strength of coupling between the initial and final states
- Decay constant given by Fermi's Golden Rule
- matrix element couples initial and final states
- phase space factor which describes volume of phase space available for the outgoing leptons
- Electron is charged lepton
- electron, muon, and tau

- Neutral lepton is neutrino

- Electron is charged lepton
- Small system perturbation
- Contained within M

- E is Q value
- Rate proportional to strength of coupling between initial and final states factored by the density of final states available to system

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

- Z is daughter and Eo is maximum energy in MeV (Q value)
- Log ft = log f + log t1/2
- t1/2 in seconds

- 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 the changes in the log ft values for 24Na and 205Hg?

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 own 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 the 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 the same
- 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 the allowed spectrum shape

Not all these transitions have similar fot values

transitions with low fot values are “favored” or “superallowed”

found among emitters of low Z and between mirror nuclei (one contains n neutrons and n+1 protons, the 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 the 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 must be the 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 the 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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