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P H G N 4 2 2 : N U C L E A R P H Y S I C S
PHGN 422: Nuclear PhysicsLecture 17: γ Decay
Prof. Kyle Leach
October 29, 2019
Slide 1
P H G N 4 2 2 : N U C L E A R P H Y S I C S
Last Class...
• β decay converts a proton to a neutron (or vice versa) within abound nucleus.
• That process is mediated by the weak interaction.
• β± are three body decays, and EC is a two-body decay.
• Fermi theory of beta decay allowed us to estimate the decayrates from Fermi’s golden rule.
• Assignment 3 is now online!• Topic selection on my door (and online to read...)
Slide 2 — Prof. Kyle Leach — PHGN 422: Nuclear Physics
P H G N 4 2 2 : N U C L E A R P H Y S I C S
γ Decay• γ decay is an electromagnetic process where the nucleus
decreases in excitation energy, but does not change proton orneutron numbers
• This decay process only involves the emission of photons
Slide 3 — Prof. Kyle Leach — PHGN 422: Nuclear Physics
P H G N 4 2 2 : N U C L E A R P H Y S I C S
The Spectrum of Electromagnetic Radiation
Slide 4 — Prof. Kyle Leach — PHGN 422: Nuclear Physics
P H G N 4 2 2 : N U C L E A R P H Y S I C S
γ DecayThere are only two-bodies in the final state for γ decay:
γ Decay
AZX∗N →A
Z X(∗)N + γ
Slide 5 — Prof. Kyle Leach — PHGN 422: Nuclear Physics
P H G N 4 2 2 : N U C L E A R P H Y S I C S
γ Decay in a Nutshell• The photon emission of the nucleus essentially results from a
re-ordering of nucleons within the shells:
Source: Krane, Fig. 5.11
Slide 6 — Prof. Kyle Leach — PHGN 422: Nuclear Physics
P H G N 4 2 2 : N U C L E A R P H Y S I C S
γ Decay in a Nutshell• This re-ordering often follows α or β decay, and moves the
system into a more energetically favourable state:
Slide 7 — Prof. Kyle Leach — PHGN 422: Nuclear Physics
P H G N 4 2 2 : N U C L E A R P H Y S I C S
Classical ElectrodynamicsFirst let’s think of this in broader terms:• The nucleus is a collection of moving charges, which can induce
magnetic/electric fields
• The power radiated into a small area element is proportional tosin2(θ)
• The average power radiated for an electric dipole is:
P =1
12πε0
ω4
c3 d2
• For a magnetic dipole is:
P =1
12πε0
ω4
c5 µ2
Slide 8 — Prof. Kyle Leach — PHGN 422: Nuclear Physics
P H G N 4 2 2 : N U C L E A R P H Y S I C S
Classical ElectrodynamicsFirst let’s think of this in broader terms:• The nucleus is a collection of moving charges, which can induce
magnetic/electric fields
• The power radiated into a small area element is proportional tosin2(θ)
• The average power radiated for an electric dipole is:
P =1
12πε0
ω4
c3 d2
• For a magnetic dipole is:
P =1
12πε0
ω4
c5 µ2
Slide 8 — Prof. Kyle Leach — PHGN 422: Nuclear Physics
P H G N 4 2 2 : N U C L E A R P H Y S I C S
Classical ElectrodynamicsFirst let’s think of this in broader terms:• The nucleus is a collection of moving charges, which can induce
magnetic/electric fields
• The power radiated into a small area element is proportional tosin2(θ)
• The average power radiated for an electric dipole is:
P =1
12πε0
ω4
c3 d2
• For a magnetic dipole is:
P =1
12πε0
ω4
c5 µ2
Slide 8 — Prof. Kyle Leach — PHGN 422: Nuclear Physics
P H G N 4 2 2 : N U C L E A R P H Y S I C S
Classical ElectrodynamicsFirst let’s think of this in broader terms:• The nucleus is a collection of moving charges, which can induce
magnetic/electric fields
• The power radiated into a small area element is proportional tosin2(θ)
• The average power radiated for an electric dipole is:
P =1
12πε0
ω4
c3 d2
• For a magnetic dipole is:
P =1
12πε0
ω4
c5 µ2
Slide 8 — Prof. Kyle Leach — PHGN 422: Nuclear Physics
P H G N 4 2 2 : N U C L E A R P H Y S I C S
Classical ElectrodynamicsFirst let’s think of this in broader terms:• The nucleus is a collection of moving charges, which can induce
magnetic/electric fields
• The power radiated into a small area element is proportional tosin2(θ)
• The average power radiated for an electric dipole is:
P =1
12πε0
ω4
c3 d2
• For a magnetic dipole is:
P =1
12πε0
ω4
c5 µ2
Slide 8 — Prof. Kyle Leach — PHGN 422: Nuclear Physics
P H G N 4 2 2 : N U C L E A R P H Y S I C S
Classical ElectrodynamicsFirst let’s think of this in broader terms:• The nucleus is a collection of moving charges, which can induce
magnetic/electric fields
• The power radiated into a small area element is proportional tosin2(θ)
• The average power radiated for an electric dipole is:
P =1
12πε0
ω4
c3 d2
• For a magnetic dipole is:
P =1
12πε0
ω4
c5 µ2
Slide 8 — Prof. Kyle Leach — PHGN 422: Nuclear Physics
P H G N 4 2 2 : N U C L E A R P H Y S I C S
Electric/Magnetic DipolesElectric and magnetic dipole fields have opposite parity: Magneticdipoles have even parity, and electric dipole fields have odd parity.
=⇒ π(ML) = (−1)L+1 and π(EL) = (−1)L
Slide 9 — Prof. Kyle Leach — PHGN 422: Nuclear Physics
P H G N 4 2 2 : N U C L E A R P H Y S I C S
Higher Order MultipolesIt is also possible to describe the angular distribution of the radiationfield as a function of the multipole order using Legendre polynomials.So, as a refresher:
• L: The index of radiation2L: The multipole order of the radiation
• L = 1→ DipoleL = 2→ QuadrupoleL = 3→ Octopole....
• The associated Legendre polynomials P2L(cos(θ)) are:For L = 1: P2 = 1
2 (3cos2(θ)− 1)
For L = 2: P4 = 18 (35cos4(θ)− 30cos2(θ) + 3)
etc....
Slide 10 — Prof. Kyle Leach — PHGN 422: Nuclear Physics
P H G N 4 2 2 : N U C L E A R P H Y S I C S
Classical to Quantum Electrodynamics
Nuclei are (of course) quantum mechanical object. So, we need toadjust our theory to properly describe γ emission from nuclear decay.How do we do this?
• First, we need to quantize the electromagnetic field. We firststart by replacing our electromagnetic moments by the relevantelectromagnetic operators:
m(σL)→ mfi(σL)
Where mfi(σL) =∫ψ∗
f m(σL) ψi dV
• But we’ve done this before! Let’s head to the chalkboard again...
Slide 11 — Prof. Kyle Leach — PHGN 422: Nuclear Physics
P H G N 4 2 2 : N U C L E A R P H Y S I C S
Electric Transitions
Electric Transition Probability
λ(EL) = 8π(L+1)L[(2L+1)!!]2
e2
4πε0~c
(E~c
)2L+1 ( 3L+3
)2 cR2L
Where R = r0a1/3, as usual. From here, we can now make someestimates for the various multipoles:
λ(E1) = 1.0× 1014A2/3E3 (1)
λ(E2) = 7.3× 107A4/3E5 (2)
λ(E3) = 3.4× 101A2E7 (3)
λ(E4) = 1.1× 10−5A8/3E9 (4)
where λ is in s−1 and E is in MeV.
Slide 12 — Prof. Kyle Leach — PHGN 422: Nuclear Physics
P H G N 4 2 2 : N U C L E A R P H Y S I C S
Magnetic Transitions
Magnetic Transition Probability
λ(ML) = 80π(L+1)L[(2L+1)!!]2
(~
mpc
)2e2
4πε0~c
(E~c
)2L+1 ( 3L+2
)2 cR2L−2
Therefore, the magnetic multipole estimates are:
λ(M1) = 5.6× 1013E3 (5)
λ(M2) = 3.5× 107A2/3E5 (6)
λ(M3) = 1.6× 101A4/3E7 (7)
λ(M4) = 4.5× 10−6A2E9 (8)
where λ is in s−1 and E is in MeV.
Slide 13 — Prof. Kyle Leach — PHGN 422: Nuclear Physics
P H G N 4 2 2 : N U C L E A R P H Y S I C S
The Weisskopf Estimates
We have made the assumption here that only a single particle(proton) changing states in the shell-model is responsible for the thede-excitation via γ decay.
Electric Transitions
λ(E1) = 1.0× 1014A2/3E3
λ(E2) = 7.3× 107A4/3E5
λ(E3) = 3.4× 101A2E7
λ(E4) = 1.1× 10−5A8/3E9
Magnetic Transitions
λ(M1) = 5.6× 1013E3
λ(M2) = 3.5× 107A2/3E5
λ(M3) = 1.6× 101A4/3E7
λ(M4) = 4.5× 10−6A2E9
These estimates (for both ML and EL) are known as the WeisskopfEstimates. What can we learn from these?
Slide 14 — Prof. Kyle Leach — PHGN 422: Nuclear Physics
P H G N 4 2 2 : N U C L E A R P H Y S I C S
Interpreting the Weisskopf Estimates
The Weisskopf estimates are only intended to be a comparativeguideline, and are not meant to be exact representations of all cases(remember, we have removed our dependance on the nuclearstructure). So, how do we actually interpret a comparison betweenthese estimates, and what we would observed experimentally (thetransition rate):
λ(σL)Exp. � λ(σL)Weiss.:A poor overlap between ψi and ψf which decreases the probability forthe transition to occur.
λ(σL)Exp. � λ(σL)Weiss.:More than one single particle contributes to the decay, thusincreasing the probability for the transition.
Slide 15 — Prof. Kyle Leach — PHGN 422: Nuclear Physics
P H G N 4 2 2 : N U C L E A R P H Y S I C S
Angular Momentum in γ Decay
• The photon is a spin-1 Boson.
• Like α decay and β decay the emitted γ can carry away units ofangular momentum L, which has given us the differentmultipolarities for the transitions described aboveFor orbital angular momentum, we can have valuesL = 0, 1, 2, 3... that correspond to our multipolarity
• Therefore, our selection rule is:
Slide 16 — Prof. Kyle Leach — PHGN 422: Nuclear Physics
P H G N 4 2 2 : N U C L E A R P H Y S I C S
Angular Momentum in γ Decay
• The photon is a spin-1 Boson.
• Like α decay and β decay the emitted γ can carry away units ofangular momentum L, which has given us the differentmultipolarities for the transitions described above
For orbital angular momentum, we can have valuesL = 0, 1, 2, 3... that correspond to our multipolarity
• Therefore, our selection rule is:
Slide 16 — Prof. Kyle Leach — PHGN 422: Nuclear Physics
P H G N 4 2 2 : N U C L E A R P H Y S I C S
Angular Momentum in γ Decay
• The photon is a spin-1 Boson.
• Like α decay and β decay the emitted γ can carry away units ofangular momentum L, which has given us the differentmultipolarities for the transitions described aboveFor orbital angular momentum, we can have valuesL = 0, 1, 2, 3... that correspond to our multipolarity
• Therefore, our selection rule is:
Slide 16 — Prof. Kyle Leach — PHGN 422: Nuclear Physics
P H G N 4 2 2 : N U C L E A R P H Y S I C S
Angular Momentum in γ Decay
• The photon is a spin-1 Boson.
• Like α decay and β decay the emitted γ can carry away units ofangular momentum L, which has given us the differentmultipolarities for the transitions described aboveFor orbital angular momentum, we can have valuesL = 0, 1, 2, 3... that correspond to our multipolarity
• Therefore, our selection rule is:
|Ji − Jf | ≤ L ≤ |Ji + Jf |
Let’s head to the chalkboard for an example...
Slide 16 — Prof. Kyle Leach — PHGN 422: Nuclear Physics
P H G N 4 2 2 : N U C L E A R P H Y S I C S
Internal Conversion
Each of the above cases were for Ji 6= Jf , but what about Ji = Jf ?
Slide 17 — Prof. Kyle Leach — PHGN 422: Nuclear Physics
P H G N 4 2 2 : N U C L E A R P H Y S I C S
Internal Conversion
Each of the above cases were for Ji 6= Jf , but what about Ji = Jf ?
There are no L = 0 photons! So what about 0→ 0 transitions withinthe nucleus? Do they occur?
Slide 17 — Prof. Kyle Leach — PHGN 422: Nuclear Physics
P H G N 4 2 2 : N U C L E A R P H Y S I C S
Internal Conversion
Each of the above cases were for Ji 6= Jf , but what about Ji = Jf ?
There are no L = 0 photons! So what about 0→ 0 transitions withinthe nucleus? Do they occur?Yes!, but they cannot decay by photon emission....
Slide 17 — Prof. Kyle Leach — PHGN 422: Nuclear Physics
P H G N 4 2 2 : N U C L E A R P H Y S I C S
Internal ConversionEach of the above cases were for Ji 6= Jf , but what about Ji = Jf ?
There are no L = 0 photons! So what about 0→ 0 transitions withinthe nucleus? Do they occur?Yes!, but they cannot decay by photon emission....
• The decrease in energy fromstate i→ f can instead ”kick”an electron out of the atom
• This process is called internalconversion
• In general, this processcompetes with γ decay..
• However, for 0→ 0transitions, it is the onlypossible decay mode
We won’t cover IC any further, butsee Krane 10.6 for a detailed
discussionSlide 17 — Prof. Kyle Leach — PHGN 422: Nuclear Physics
P H G N 4 2 2 : N U C L E A R P H Y S I C S
Next Week...
Reading Before Next Class
• Chapter 7.1 in Krane
Next Class Topics
• Rutherford scattering and introduction to nuclear reactions• Radiation interaction with matter
Slide 18 — Prof. Kyle Leach — PHGN 422: Nuclear Physics