5-1

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 electronØ 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: MZMZ-1,à + decay: MZ MZ-1+2me

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

EnergyNeNa 2210

2211

EnergyMgeAl 2612

2613

EnergyXeI 13154

13153

5-2

-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

5-3

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

5-4

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 37Cl+37Ar+e-: Detection of 37Arà 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

5-5

Spin in Beta Decay• Spins of created particles can be combined in two

ways§ Electron and neutrino spin both 1/2

à S=1 in a parallel alignment à S= 0 in an anti-parallel alignment

• two possible relative alignments of "created" spins§ Fermi (F) (S=0)

à Low A§ Gamow-Teller (GT) (S =1)

à 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

5-6

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 lectureà 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 functionsà Basis of model to calculate decay constant

* Fermi golden rule (slide 15)

5-7

Q value calculation (Review)• Find Q value for the Beta decay of 24Na

§ 1 amu = 931.5 MeV § M (24Na)-M(24Mg)

à 23.990962782-23.985041699 à 0.005921 amu

* 5.5154 MeV§ From mass excess

à -8.4181 - -13.9336 à 5.5155 MeV

• Q value for the EC of 22Na§ M (22Na)-M(22Ne)§ 21.994436425-21.991385113 § 0.003051 amu

à 2.842297 MeV§ From mass excess

à -5.1824 - -8.0247 à 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?

QZZ )1(

QZZ )1(

)e2)1Z(M()Z(MQ

)1Z(M)Z(MQ

Q)1Z(Z

)1Z(M)Z(MQEC

Beta decay

Positron decay

Electron Capture

5-8

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 positronsà No positron capture decay

• Annihilation radiation§ energy carried off by two

quanta of opposite momentum§ Annihilation conserves

momentum§ Exploited in Positron Emission

Tomography

5-9

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 experimentà 10-3

of the electromagnetic force constant

• Used to determine emitted electron momentum range per unit time P(pe) dpe;

0

22222

)0()0(4

)(dE

dngM

hdppP ifeee

5-10

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

0

22222

)0()0(4

)(dE

dngM

hdppP ifeee

5-11

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 là 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

0

22222

)0()0(4

)(dE

dngM

hdppP ifeee

5-12

Weak Interaction• Spectrum shape determined entirely

by e(0) and dn/dEo

§ dn/dEo density of final states with electron momentumàCoulomb 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

dWWWWWh

cm

dE

dno

o

o

22/126

452

)()1(16

5-13

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 particlesà 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)2

free

§ perturbation on e(0) and spectrum multiplied by Fermi function

à Z daughter nucleusà v beta velocityà + for electronsà - for positron

v

Zex

x

xWZF

2

;)2exp(1

2),(

5-14

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

5-15

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

energyà Degenerate states

fM 22

dvVM if

5-16

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 Wà 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)

oif fMKt

2

2/1

2ln

oW

oo

o

dWWWWWWZFf

hgcmK

1

22/12

72454

)()1(),(

/64

5-17

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)

)1log(5.36.5log0.2log

3log)1(009.0007.079.0log0.4log

log)1(005.002.078.0log0.4log

2

ZEf

EZZEf

EZZEf

oEC

oo

oo

• 14 O to 14N§ positron decay§ Q=1.81 MeV

§ T1/2 =70.6 s

• Log f + b = 1.83, log t = 1.84

• Log ft=3.67

2

2

3

81.1log)17(009.0)7(007.079.081.1log0.4log

3log)1(009.0007.079.0log0.4log

f

EZZEf o

o

5-18

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

254.2log)184(005.0)84(02.078.0254.2log0.4log

log)1(005.002.078.0log0.4log

f

EZZEf oo

5-19

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

§ Examine spin and parity changes between parent and daughter state

5-20

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 nuclidesà radiations most

frequently detected are x-rays

5-21

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

5-22

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 Mif2 values for all transitions with I=0, 1 without parity change was erroneous

5-23

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

5-24

Other Beta Decay• Double beta decay

§ Very long half-life à 130Te and 82Se as

examples§ Can occur through beta

stable isotope§ 76Ge to 76Se by double beta

à 76Ge to 76Asà Q= -73.2130- (-72.2895) à Q= -0.9235 MeV

§ Possible to have neutrinoless double beta decayà two neutrinos annihilate

each otherà Neutrino absorbed by

nucleon

• Beta delayed decay

§ Nuclei far from stability can populate unbound states and lead to direct nucleon emission

§ First recognized during fissionà 1 % of neutrons delayed

* 87Br is produced in nuclear fission and decays to 87Kr

§ decay populates some high energy states in Kr daughterà 51 neutrons, neutron emission to form

86Kr

5-25

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

5-26

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.

5-27

Pop Quiz

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

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