H.O. [202] 3 2 (2) (2) H.O. 4.0 [200] 1 2 [202] 5 2 (2) (4) (2) 3.5 [211] 1 2 (2) (6) [211] 3 2 (2) 3.0 (2) [220] ε

Transcription

1 E/ħω H r 0 r Y0 0 l s l l N + l + l s [0] 3 H.O. ε = H.O. ε = 0 + l s + l [00] n z = 0 d 3/ 4 [0] N = s / N n z d 5/ 6 [] n z = N lj [] ε [0] n z =

2 interaction of two levels with the same Ω π quantum numbers from different j shells avoided crossings

3 Nilsson diagrams deformed shell gaps minima in PES 3

4 Nilsson Strutinsky method: microscopic macroscopic approach E sh > 0 E sh < 0 E E SH E LDM A i E i SH ~ e E SH combined with the Nilsson model 4

5 second minimum: fission isomer fission isomers superdeformed shape 5

6 let s look at odd nuclei: ground-state spins of deformed mid-shell nuclei: 67 Er 99 7/ + 7 Yb 0 / 73 Yb 03 5/ 77 Hf 05 7/ 79 Hf 07 9/ + 6

7 odd particle coupled to deformed even-even core particle rotor model axial symmetry R perpendicular to z Ω = K we expand the wave function of the particle in the basis NljΩ : φ ν = a ν NljΩ NljΩ Nj the rotating core is described by the Wigner D functions: D MK 3,, K symmetry for 80 rotation about the x axis: D D MK M K where: φ ν = j Ω a ν NljΩ Nlj Ω Nj H R rot j j 3 j j with the ladder operators j j j i i j j j j Coriolis term coupling of particle and collective rotation j K j / K K K / j j j

8 j K j / K K K / j j j the lowering and raising operators change ΔK = ± diagonal matrix elements of the Coriolis term are only non-zero for K = ±/ exercise: find + = + + = + j j j = j + j + j j = j + Coriolis term j j contributes only for K = /: j j a K it was: DMK DM K φ ν = a ν NljΩ NljΩ Nj φ ν = j Ω a ν NljΩ Nlj Ω Nj j with a j j j Nj a Nlj a has a fixed value for each Ω = / state 8

9 we assume for now that is small j j e H sp H sp H rot H full Hamiltonian single-particle energies and wave functions 9 odd particle coupled to deformed even-even core particle rotor model 3 rot j j j j H K e E K for K / with the single-particle energies measured relative to the Fermi level a K e E K K

10 rotational bands on all available single-particle states also hole states the energies of the band heads are approximately what one would expect from the Nilsson diagram for =0.3 the bands with =/ look funny we suspect that is related to the Coriolis term E for K==/ we get an additional term: K e K K a a is called the decoupling parameter for a: /,3/; 5/,7/... degenerate for the [4]/ band: a0.8 for the [54]/ band: a+3 ordering! 0

11 band heads only

12

13 so far we have considered the case where the particle is aligned with the deformation axis: deformation alignement this is also called strong coupling if the coupling is weak: rotation alignment: the Coriolis term aligns the particle angular momentum with the rotational axis H = H sp + H rot = H sp + ħ + j j j term tries to align intrinsic and total spin this is especially important for large j and large in particular: intruder orbitals, e.g. h /, i 3/ large j unique parity in shell pure wave functions 3

14 rotation alignment favored for high spin high particle spin j lower deformation we can have a situation between the two extreme cases need to diagonalize the full particle rotor Hamiltonian 4

15 fully rotation-aligned case: α = j R = α E rot = ħ = ħ + + j j + j j j + + j = ħ R R + + const. symmetry for 80 rotation restored like even-even nucleus expect rotational band with = j, j +, j + 4 same energies as even-even neighbor plus constant parabola with minimum at j 5

16 fully rotation-aligned case: α = j R = α E rot = ħ = ħ + + j j + j j j + + j = ħ R R + + const. symmetry for 80 rotation restored like even-even nucleus expect rotational band with = j, j +, j + 4 same energies as even-even neighbor plus constant only favored states observed: = j, j +, j + 4, unfavored states = j +, j + 3, shifted high up in energy decoupled bands 6

17 57 protons: Fermi level near [550]/ high j low Ω only favored states observed: = j, j +, j + 4, unfavored states = j +, j + 3, shifted high up in energy decoupled bands 7

18 in between gradual transition from: rotation alignment to deformation alignment N= weakly deformed more deformed low- larger decoupled strongly coupled 8

19 gradual transition from: rotation alignment to deformation alignment N= weakly deformed more deformed low- larger decoupled strongly coupled 9

20 M E M 9/ + 5/ + 66 P. Nieminen et al 9/ + / + A. Hürstel et al. Phys. Rev. C 69, Eur. Phys. J. A, / / / + 9/ / + 5/ / / + / / / / + 3/ + 3/ / / - 9/ - 9/ - 9/ - 93 Bi Bi 89 Bi 87 Bi strongly coupled bands deformation aligned large rapid transition decoupled bands rotation aligned small

21 so far: macroscopic description of collective rotational motion now: microscopic description: cranking model z independent particles moving in an average potential that is rotating with axial symmetry ;0,, ; t r V t r V transformation: rotation of angle t : t i U x exp U r wavefunction of a particle in the rotated potential: h t i with t U i t U i t i r r x x h U i i h U we can then write: r r r x h h for A particles: A i x h i H H, the energy in the rotating frame is called Routhian i x time independent

22 x H H energy in lab system: x H H E we expand: 0 E E x d de 0 E E in a somewhat sloppy way we have used the angular momentum remember: we should use: 0 E E

23 kinematic vs. dynamic moment of inertia E de d E kinematic moment of inertia related to the overall motion of the nucleus dynamic moment of inertia describe its response to a torque the first derivative is related to the kinematic moment of inertia de d de d E depends on absolute angular momentum the second derivative is related to d E the dynamic moment of inertia d d E d d d E independent of absolute angular momentum measures the variation of d d rigid rotor:

24 Coriolis and centrifugal forces break time-reversal symmetry different energy depending on motion with or against rotation of core cranked Hamiltonian invariant under rotation of 80 signature quantum number R x exp i j eigenvalues r x i levels with large j and small have strong dependence on signature splitting strong mixing x calculate potential energy using Nilsson-Strutinsky for different rotational frequancy total Routhian surface 68 Hf