Pavel Str ánský

1. WHAT DRIVES NUCLEI TO BE PROLATE? Pavel Stránský Alejandro Frank Roelof Bijker Institutode Ciencias Nucleares, Universidad Nacional Autónoma de México 29th August 2011 CGS14, University of Guelph, Ontario, Canada, 2011

2. Experimental deformation of nuclei is a typical value for well-deformed nuclei rare-earth region Deformation parameter (from measured quadrupole moments): where measured intrinsic N.J. Stone, At. Data Nucl. Data Tables 90, 75 (2005)

4. WHAT DRIVES NUCLEI TO BE PROLATE? Macroscopic effects: Microscopic effects: Surface tension Coulomb energy … Shell structure Spin-orbit and l2 interaction … ? Stable ground-state configuration Minimization of the total sum of the lowest-lying occupied one-particle energies with respect to the size of the potential deformation Minimization of the equilibrium energy with respect to the size of the shape deformation

5. 1. Single-particle models Microscopic single-particle models (a short discussion)

6. 1. Single-particle models 3D spheroid potential (axially symmetric elipsoid) V = const Noninteracting fermions (only 1 type of particles) Pure harmonic potential Infinite potential well Equal number of prolate and oblate configurations • Volume saturation of the nuclear force • Sharp surface N N

7. 1. Single-particle models Level dynamics – Spheroid infinite well E (a.u.) Projection of the angular momentum 4 2s 1h 2d 3 1 3 0 1g 2 2 2p 1 4 0 1f 2s 1d 1p 1s b Sharp surface pushes down shells with higher orbital momentum l, containing additional downsloping states with low projection m on the prolate side; the predominance of these low-m states, together with their mutual repulsion, causes the prolate-oblate deformation asymmetry I.Hamamoto, B.R. Mottelson, Phys. Rev. C 79, 034317 (2009)

8. Deformed liquid drop model (A little of the theory and results)

9. 2. Deformed liquid drop model Total mass/energy (Weizsäcker formula) microscopic corrections (asymmetry energy, shell effects, pairing) binding (bulk) energy A = N + Z curvature energy, surface and volume redistribution energy… volume energy surface energy Coulomb energy Adjustable constants: Shape functions:

10. 2. Deformed liquid drop model Quadrupole deformation (axially symmetric) a2 < 0 a2 > 0 oblate a2 = 0 prolate Fixed by a condition of volume conservation spherical Symmetric with respect to the sign of a2 Negative for a2 < 0 – prolate shape has always lower energy Surface Numerically shape functions: Deformation parameter Coulomb W.J. Swiatecki, Phys. Rev. 104, 993 (1956)

11. 2. Deformed liquid drop model Prolate-oblate energy difference keV

12. 2. Deformed liquid drop model Prolate-oblate energy difference keV rare-earth region

13. 2. Deformed liquid drop model Coulomb and surface contribution surface surface Almost the same contribution (despite the different functional form) Coulomb Coulomb

14. 2. Deformed liquid drop model DB from the B(E2) transition probabilities • Only absolute value of the deformation • Only even-even nuclei S. Raman, C.W. Nestor, and P. Tikkanen, At. Data Nucl. Data Tables 78, 1 (2001)

15. 2. Deformed liquid drop model Distribution of DB values 495 nuclei totally

16. Shortcomings of the pure LDM

17. 2. Deformed liquid drop model Shape stabilization Pure liquid drop model is not able to explain the ground state deformation (spherical shape is always preferred) Necessity of introducing shell corrections Shell corrections (Strutinsky) N deformed deformation decreases the size of the corrections Smooth cumulative level density spherical Exact cumulative level density E

18. 2. Deformed liquid drop model Shape stabilization Pure liquid drop model is not able to explain the ground state deformation (spherical shape is always preferred) Necessity of introducing shell corrections

19. 2. Deformed liquid drop model Shape stabilization Pure liquid drop model is not able to explain the ground state deformation (spherical shape is always preferred) Necessity of introducing shell corrections Shell effects (1st approximation) Symmetric with respect to the sign of the deformation W.D. Myers, W.J. Swiatecki, Nucl. Phys. 81, 1 (1966)

20. 2. Deformed liquid drop model Size of the shell corrections Positive corrections: Create the oblate and prolate minima S (N,Z) Mid-shell correction < 3MeV Negative corrections: deepen the spherical minimum 40 80 120 Shell corrections are highly important near closed shells, but less for deformed nuclei in mid-shells

21. Last slide Thank you very much for your attention Conclusions & Outlook • Collective effects (surface and Coulomb energy of the quadrupole deformed simple liquid drop model) give a significant amount of the prolate-oblate energy difference up to DB = 800keV (for comparison, the first 2+ excited state for well-deformed even-even nuclei is typically of the order of 100keV) • This model is not capable of explaining the origin of the deformation: In order to stabilize a deformed shape, microscopic corrections (that may lower the prolate minimum, however) must be included • Microscopic pure single-particle models explain the prolate preponderance as a consequence of the sharp surface and saturation of the nuclear matter. Complex calculations (such as the self-consistent the HF+BCS or the shell model with random interactions) favor the prolate shape, but the underlying responsible physics is hidden • In the future: To find a link between the microscopic shell structure (i.g. the ordering of levels) and the exact shape of a nucleus