1. PHL424: Nuclear surface vibration

2. 2+ Systematics 2+ 0+ Excitation energy (keV)
Ground state Configuration.
Spin/parity Iπ=0+ ; Ex = 0 keV

3. 4+/2+ Energy ratio: mirrors 2+ systematics
Excitation energy (keV) 4+ 2+ 0+
Ground state Configuration.
Spin/parity Iπ=0+ ; Ex = 0 keV

4. Evolution of nuclear structure as a function of nucleon number

5. Collective vibration In general, λ=0 λ=1 λ=2 λ=3 OK...
forbidden – density change!
λ=0
forbidden – CM moves!
λ=1
time
OK...
λ=2
λ=3

6. Collective vibration In general, λ=2: quadrupole vibration
harmonic vibration
λ=2: quadrupole vibration
λ=3: octupole vibration

7. Collective vibration classical Hamiltonian constants:
Binding energy of a nucleus:
aV = MeV aS = MeV aC = MeV aA = MeV aP = MeV

8. Collective vibration classical Hamiltonian quantization
energy eigenvalue
wave function
Hermite polynomials

9. Second quantization Hamilton operator rule for boson operators:
𝛼 𝜆𝜇 = ℏ 2 𝐵 𝜆 𝜔 𝜆 ∙ 𝛽 𝜆𝜇 + + − 𝜇 𝛽 𝜆𝜇
𝜋 𝛼 𝜆𝜇 = ℏ 𝐵 𝜆 𝜔 𝜆 2 ∙ − 𝜇 𝛽 𝜆−𝜇 + − 𝛽 𝜆𝜇
rule for boson operators:

10. Second quantization Hamilton operator rule for boson operators:
creation & annihilation operators
increase or decrease the number of phonons in a wave function
ground state (vacuum)
1-phonon state
2-phonon state

11. Second quantization Hamilton operator rule for boson operators:
1-phonon energy:

12. Second quantization Hamilton operator rule for boson operators:
2-phonon energy:
etc.
2 𝐻 2 ~2∙ℏ∙𝜔∙
normalization of wave function !!!

13. Second quantization Hamilton operator rule for boson operators:
2-phonon state:
normalization (approximation):

14. Second quantization Hamilton operator rule for boson operators:
2-phonon state:
normalization:

15. Collective vibration 𝐸 𝑁 =ℏ𝜔∙ 𝑁+ 5 2 𝐸 3 =ℏ𝜔∙ 3+ 5 2 𝐸 2 =ℏ𝜔∙ 2+ 5 2
𝐸 1 =ℏ𝜔∙
𝐸 0 =ℏ𝜔∙ 5 2

16. Example of vibrational excitations
(E-E ground state)
n=1,  = 2, 2+ phonon
n = 2,  = 2, J = 0+, 2+, 4+
ħ2
2ħ2
3ħ2
3- state?
multiple  = 2 phonon
states, ideally degenerate

17. Second quantization Hamilton operator rule for boson operators:
2-phonon state:
reduced transition probability:

18. Reduced transition probabilities
2-phonon state
3-phonon state

19. Reduced transition probabilities
1-phonon state
2-phonon state
3-phonon state