1. PHL424: γ-decay
γ-decay is an electromagnetic process where the nucleus decreases in excitation energy, but does not change proton or neutron numbers
This decay process only involves the emission of photons

2. Electromagnetic spectrum

3. γ-decay
Gamma-ray emission is usually the dominant decay mode
Measurements of γ-rays let us deduce:
Energy, Spin (angular distr. / correl.), Parity (polarization), magnetic moment,
lifetime (recoil distance, Doppler shift), …
of the involved nuclear levels.
137Cs detected in red: NaI scintillator
blue: HPGe (high purity Ge semiconductor)
𝑍 𝐴 𝑋 𝑁 ∗ → 𝑍 𝐴 𝑌 𝑁 ∗

4. γ-decay in a Nutshell
The photon emission of the nucleus essentially results from a re-ordering of nucleons within the shells.
This re-ordering often follows α or β decay, and moves the system into a more energetically favorable state.

5. γ-decay
γ-ray spectrum of natU

6. γ-decay
Most β-decay transitions are followed by γ-decay.

7. Classical Electrodynamics
The nucleus is a collection of moving charges, which can induce magnetic/electric fields
The power radiated into a small area element is proportional to 𝑠𝑖𝑛 2 𝜃
The average power radiated for an electric dipole is:
𝑃= 1 12𝜋 𝜖 0 𝜔 4 𝑐 3 𝑑 2
For a magnetic dipole is
𝑃= 1 12𝜋 𝜖 0 𝜔 4 𝑐 5 𝜇 2

8. Electric/Magnetic Dipoles
Electric and magnetic dipole fields have opposite parity:
Magnetic dipoles have even parity and electric dipole fields have odd parity.
⇒ 𝜋 𝑀ℓ = −1 ℓ+1 𝑎𝑛𝑑 𝜋 𝐸ℓ = −1 ℓ

9. Higher Order Multipoles
It is possible to describe the angular distribution of the radiation field as a function of the multipole order using Legendre polynomials.
ℓ: The index of radiation
2 ℓ : The multipole order of the radiation
ℓ=1 → 𝐷𝑖𝑝𝑜𝑙𝑒
ℓ=2 → 𝑄𝑢𝑎𝑑𝑟𝑢𝑝𝑜𝑙𝑒
ℓ=3 → 𝑂𝑐𝑡𝑢𝑝𝑜𝑙𝑒
The associated Legendre polynomials 𝑃 2ℓ 𝑐𝑜𝑠 𝜃 are:
For ℓ=1: 𝑃 2 = ∙ 𝑐𝑜𝑠 2 𝜃 −1
For ℓ=2: 𝑃 4 = 𝑐𝑜𝑠 4 𝜃 −30 𝑐𝑜𝑠 2 𝜃 +3

10. Angular Momentum in γ-Decay
The photon is a spin-1 boson
Like α-decay and β-decay the emitted γ-ray can carry away units of angular momentum ℓ, which has given us different multipolarities for transitions.
For orbital angular momentum, we can have values ℓ=0,1,2,3,⋯ that correspond to our multipolarity.
Therefore, our selection rule is:
𝐽 𝑖 − 𝐽 𝑓 ≤ℓ≤ 𝐽 𝑖 + 𝐽 𝑓

11. Characteristics of multipolarity
π(Eℓ) / π(Mℓ)
angular distribution 1
dipole
-1 / +1 2
quadrupole
+1 / -1 3
octupole 4
hexadecapole ⁞
ℓ = 1
ℓ =2
𝐸 𝛾 = 𝐸 𝑖 − 𝐸 𝑓
𝐼 𝑖 − 𝐼 𝑓 ≤ℓ≤ 𝐼 𝑖 + 𝐼 𝑓
∆𝜋 𝐸ℓ = −1 ℓ
∆𝜋 𝑀ℓ = −1 ℓ+1

12. Characteristics of multipolarity
π(Eℓ) / π(Mℓ)
angular distribution 1
dipole
-1 / +1 2
quadrupole
+1 / -1 3
octupole 4
hexadecapole ⁞
ℓ = 1
ℓ =2
parity: electric multipoles π(Eℓ) = (-1)ℓ, magnetic multipoles π(Mℓ) = (-1)ℓ+1
The power radiated is proportional to:
where σ means either E or M and ℳ 𝜎ℓ is the E or M multipole moment of the appropriate kind.
𝑃 𝜎ℓ ∝ 2 ℓ+1 ∙𝑐 𝜀 0 ∙ℓ ∙ 2ℓ+1 ‼ 𝜔 𝑐 2ℓ+2 ℳ 𝜎ℓ 2

13. Emission of electromagnetic radiation
where Eγ = Ei – Ef is the energy of the emitted γ quantum in MeV (Ei, Ef are the nuclear level energies, respectively), and the reduced transition probabilities B(Eℓ) in units of e2(barn)ℓ and B(Mℓ) in units of 𝜇 𝑁 2 = 𝑒ℏ 2 𝑚 𝑁 𝑐 𝑓𝑚 2ℓ−2

14. Single particle transition (Weisskopf estimate)
For the first few values of λ, the Weisskopf estimates are
gamma energy Eγ [keV]
transition probability λ [s-1]

15. Conversion electrons Energetics of CE-decay (i=K, L, M,….)
Ei = Ef + Ece,i + EBE,i
γ- and CE-decays are independent; transition probability (λ ~ Intensity)
λT = λγ + λCE = λγ + λK + λL + λM……
Conversion coefficient
ai = λCE,i / λγ

16. Conversion electrons Te = Eγ - Be
For an electromagnetic transition internal conversion can occur instead of emission of gamma radiation. In this case the transition energy Q = Eγ will be transferred to an electron of the atomic shell.
Te: kinetic energy of the electron
Be: binding energy of the electron
Te = Eγ - Be
internal conversion is important for:
heavy nuclei ~ Z4
high multipolarities Eℓ or Mℓ
small transition energies

17. Comparison of α-decay, β-decay and γ-decay
𝜆= ℎ 𝑝 = ℎ∙𝑐 𝐸 𝑘𝑖𝑛 ∙ 𝐸 𝑘𝑖𝑛 +2𝑚 𝑐 2 = 𝑀𝑒𝑉 𝑓𝑚 𝐸 𝑘𝑖𝑛 ∙ 𝐸 𝑘𝑖𝑛 +2𝑚 𝑐 2
de Broglie wavelength:
decay
Energy [MeV]
de Broglie λ [fm]
α-particle, mα = 3727 MeV/c2 5
6.42
β-particle, me = MeV/c2 1
871.92
γ-photon
𝜆= ℎ∙𝑐 𝐸 = 1240 𝐸
For α-particles this dimension is somewhat smaller than the nucleus and this is why a semi-classical treatment of α-decay is successful.
The typical β-particle has a large wavelength λ in comparison to the nuclear size and a quantum mechanical is dictated and wave analysis is called for.
For γ-decay the wavelength λ ranges from – 1240 fm (0.1 – 1 MeV). Clearly, only a quantum mechanical approach has a chance of success.

18. γ-decay
γ-spectroscopy yields some of the most precise knowledge of nuclear structure, as spin, parity and ΔE are all measurable.
Transition rates between initial Ψ 𝑁 ∗ and final Ψ 𝑁 ´ nuclear states, resulting from electromagnetic decay producing a photon with energy 𝐸 𝛾 can be described by Fermi´s Golden rule:
where ℳ 𝑒𝑚 is the electromagnetic transition operator and 𝑑 𝑛 𝛾 𝑑 𝐸 𝛾 is the density of final states. The photon wave function 𝜓 𝛾 and ℳ 𝑒𝑚 are well known, therefore measurements of λ provide detailed knowledge of nuclear structure.
𝜆= 2𝜋 ℏ Ψ 𝑁 ´ 𝜓 𝛾 ℳ 𝑒𝑚 Ψ 𝑁 ∗ 𝑑 𝑛 𝛾 𝑑 𝐸 𝛾
A γ-decay lifetime is typically [s] and sometimes even as short as [s]. However, this time span is an eternity in the life of an excited nucleon. It takes about 4·10-22 [s] for a nucleon to cross the nucleus.

19. Electromagnetic excitation
208Pb beam
238U target
energy of gamma radiation (keV)
log (intensity)

20. Electric field lines of two interacting charges

21. Electric fields of multipoles
In general the electric potential due to
an arbitrary charge distribution is
𝑈 𝑟 = 𝜌 𝑝 𝑟 ′ 𝑟 − 𝑟 ′ 𝑑𝜏′
expansion
1 𝑟−𝑟′ = ℓ=0 ∞ 𝑟′ ℓ 𝑟 ℓ+1 4𝜋 2ℓ+1 𝑚=−ℓ ℓ 𝑌 ℓ𝑚 𝜗,𝜑 𝑌 ℓ𝑚 ∗ 𝜗 ′ ,𝜑′
ℓ=𝑚= 𝑌 00 𝜗,𝜑 = 𝑌 00 𝜗 ′ ,𝜑′ = 1 4𝜋
special case: electric monopole
→ 𝑟−𝑟′ = 1 𝑟

22. Electric fields of multipoles
In general the electric potential due to
an arbitrary charge distribution is
𝑈 𝑟 = 𝜌 𝑝 𝑟 ′ 𝑟 𝑑𝜏′
homogenous charge distribution
𝜌 𝑝 𝑟′ = 3∙𝑍𝑒 4𝜋∙ 𝑅 0 3
𝑈 𝑟 = 3∙𝑍𝑒 4𝜋∙ 𝑅 𝑟 𝑟′ 2 𝑑 𝑟 ′ 𝑠𝑖𝑛 𝜗 ′ 𝑑 𝜗 ′ 𝑑𝜑′
special case: electric monopole
𝑈 𝑟 = 3∙𝑍𝑒 4𝜋∙ 𝑅 0 3 ∙ 1 𝑟 ∙ 𝑅 ∙4𝜋
= 𝒁𝒆 𝒓

23. Electric fields of multipoles
In general the electric potential due to
an arbitrary charge distribution is
𝑈 𝑟 = 𝜌 𝑝 𝑟 ′ 𝑟 − 𝑟 ′ 𝑑𝜏′
1 𝑟−𝑟′ = 1 𝑟 𝑓𝑜𝑟 𝑟>𝑟´ 1 𝑟´ 𝑓𝑜𝑟 𝑟<𝑟´
𝑈 𝑟 =4𝜋∙ 3∙𝑍𝑒 4𝜋∙ 𝑅 𝑟 1 𝑟 ∙ 𝑟´ 2 𝑑𝑟´ + 𝑟 𝑅 𝑟´ ∙ 𝑟´ 2 𝑑𝑟´
𝑈 𝑟 = 3∙𝑍𝑒 𝑅 𝑟 𝑅 − 𝑟 2 2
special case: electric monopole
𝑈 𝑟 = 𝑍𝑒 2∙ 𝑅 0 ∙ 3− 𝑟 𝑅

24. Monopole, dipole, quadrupole, …
octupole
𝑈 𝑟 ∝ 1 𝑟
𝑈 𝑟 ∝ 1 𝑟 2
𝑈 𝑟 ∝ 1 𝑟 3
𝑈 𝑟 ∝ 1 𝑟 4

25. Electric fields of multipoles
In general the electric potential due to
an arbitrary charge distribution is
𝑈 𝑟 = 𝜌 𝑝 𝑟 ′ 𝑟 − 𝑟 ′ 𝑑𝜏′
expansion
1 𝑟−𝑟′ = ℓ=0 ∞ 𝑟′ ℓ 𝑟 ℓ+1 4𝜋 2ℓ+1 𝑚=−ℓ ℓ 𝑌 ℓ𝑚 𝜗,𝜑 𝑌 ℓ𝑚 ∗ 𝜗 ′ ,𝜑′
multipole moment
𝑀 ∗ ℓ,𝑚 = 𝜌 𝑝 𝑟´ ∙ 𝑟´ ℓ ∙ 𝑌 ℓ𝑚 ∗ 𝜗´,𝜑´ 𝑑𝜏´
𝑀 ∗ ℓ=2,𝑚 = 3∙𝑍𝑒∙ 𝑅 𝜋 ∙ 𝛽 2
special case: electric quadrupole matrixelement
𝐵 𝐸ℓ=2; 𝐼 𝑖 → 𝐼 𝑓 = 𝑀 𝑓 𝑚 𝐼 𝑓 𝑀 𝑓 𝐾 𝑓 𝑀 ℓ=2,𝑚 𝐼 𝑖 𝑀 𝑖 𝐾 𝑖 2
B(E2) – value:
𝑈 𝑟 = 𝑚=−2 𝑚=2 4𝜋 𝑟 3 𝑌 ℓ=2,𝑚 𝜗,𝜑 ∙ 𝑀 ∗ ℓ=2,𝑚
electric quadrupole potential:

26. Electric multipoles 𝜋 𝐸ℓ = −1 ℓ
𝑉 𝑟 = 1 4𝜋 𝜀 0 ∙ 𝑄 0 𝑟 + 𝑄 1 𝑟 𝑄 2 𝑟 3 ⋯
𝑄 0 = 𝑑𝑟´𝜌 𝑟´
𝑄 1 = 𝑑𝑟´𝑧´𝜌 𝑟´
𝑄 2 = 𝑑𝑟´ 3 𝑧´ 2 − 𝑟´ 2 ∙𝜌 𝑟´
Under the parity operation, 𝑟 →− 𝑟 , the parity of the electric dipole radiation is 𝜋 𝐸 1 =−1.
For an electric quadrupole radiation the parity is 𝜋 𝐸 2 =+1.
In general, the parity of an electric multipole is:
𝜋 𝐸ℓ = −1 ℓ