PHL424: Nuclear surface vibration

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

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

Evolution of nuclear structure as a function of nucleon number

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

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

Collective vibration classical Hamiltonian constants: Binding energy of a nucleus: aV = 15.560 MeV aS = 17.230MeV aC = 0.6970 MeV aA = 23.385 MeV aP = 12.000 MeV

Collective vibration classical Hamiltonian quantization energy eigenvalue wave function Hermite polynomials

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:

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

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

Second quantization Hamilton operator rule for boson operators: 2-phonon energy: etc. 2 𝐻 2 ~2∙ℏ∙𝜔∙ 2+ 1 2 normalization of wave function !!!

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

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

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

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

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

Reduced transition probabilities 2-phonon state 3-phonon state

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