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