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

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

4. W HAT DRIVES NUCLEI TO BE PROLATE? Surface tension Coulomb energy … Shell structure Spin-orbit and l2 interaction … Macroscopic effects: Microscopic effects: ? 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 Stable ground-state configuration

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

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

7. 1s 1p 1d 2s 2p 1f 1g 2d 1h 2s 0 1 2 3 4 0 1 2 3 4 Level dynamics – Spheroid infinite well E (a.u.)  Projection of the angular momentum 1. Single-particle models I.Hamamoto, B.R. Mottelson, Phys. Rev. C 79, 034317 (2009) 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

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

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

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

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

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

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

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

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

16. Shortcomings of the pure LDM

17. 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 E Exact cumulative level density Smooth cumulative level density spherical deformed deformation decreases the size of the corrections 2. Deformed liquid drop model

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

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

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

21. 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  B = 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 Last slide Thank you very much for your attention