WHY ARE NUCLEI PROLATE: Deformation is a collective effect Pavel Stránský Alejandro Frank Roelof Bijker Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México XXXIV Symposium on Nuclear Physics, Cocoyoc, Mexico, 2011 7th January 2011

WHY ARE NUCLEI PROLATE: Deformation is a collective effect Pictures with deformation 1. Single particle x collective approaches 2. Deformed liquid drop model Binding energy (Mass formula) Quadrupole deformation Shape stabilization: Shell corrections Results Prolate-oblate energy difference for experimental data of electric quadrupole moments and B(E2) transitions

Single-particle x collective approaches Microscopic – Nilsson model (1 slide) Chocolate (chocolate – nahuatl) box (1 slide) Adiabatic approximation (1 slide) Deformation – collective effect Collective approaches – too small??? (1 slide)

Single-particle description Collective excitations 1. Single-particle models Single-particle description Collective excitations Nillson-like models deformed liquid drop models 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

Chocolate-box model y x = y x z Spectrum: Volume conservation: 1. Single-particle models Chocolate-box model a demonstration of the single-particle approach y x = y x z Spectrum: Volume conservation: xyz = const. deformation parameter

Chocolate-box model – Nilsson diagram 1. Single-particle models Chocolate-box model – Nilsson diagram E oblate d prolate

Each level is occupied by 1 particle only 1. Single-particle models Chocolate-box model – Level occupation N = 8 Each level is occupied by 1 particle only Total energy E d

Chocolate-box model – Level occupation 1. Single-particle models Chocolate-box model – Level occupation N = 8 Total energy E d

Chocolate-box model – Level occupation 1. Single-particle models Chocolate-box model – Level occupation N = 8 Total energy E d

Chocolate-box model – Level occupation 1. Single-particle models Chocolate-box model – Level occupation N = 8 Total energy E d

Chocolate-box model – Level occupation 1. Single-particle models Chocolate-box model – Level occupation N = 8 Total energy E d

Chocolate-box model – Level occupation 1. Single-particle models Chocolate-box model – Level occupation N = 8 Total energy E d

Chocolate-box model – Level occupation 1. Single-particle models Chocolate-box model – Level occupation N = 8 Total energy E d

Chocolate-box model – Level occupation 1. Single-particle models Chocolate-box model – Level occupation N = 8 Total energy E d

Chocolate-box model – Level occupation 1. Single-particle models Chocolate-box model – Level occupation N = 8 Total energy E d

Chocolate-box model – Level occupation 1. Single-particle models Chocolate-box model – Level occupation N = 8 Total energy E d

Chocolate-box model – Level occupation 1. Single-particle models Chocolate-box model – Level occupation N = 8 Total energy E d

Chocolate-box model – Adiabatic approximation 1. Single-particle models Chocolate-box model – Adiabatic approximation This approximation allows us to determine the shape for a given particle number N uniquely Procedure: Lowest single-particle levels are occupied for small deformation |d| Deformation is then changed “adiabatically”, making the particles stay on the same levels as in the beginning, no matter if there happens to appear another level with lower energy Total energy under this approximation has always 1 (spherical) or 2 (deformed) minima; the deeper minimum determines the shape

Chocolate-box model – Adiabatic approximation 1. Single-particle models Chocolate-box model – Adiabatic approximation N = 8 E d

Chocolate-box model – Adiabatic approximation 1. Single-particle models Chocolate-box model – Adiabatic approximation N = 8 E d

Chocolate-box model – Adiabatic approximation 1. Single-particle models Chocolate-box model – Adiabatic approximation N = 8 E d

Chocolate-box model – Adiabatic approximation 1. Single-particle models Chocolate-box model – Adiabatic approximation N = 8 E d

Chocolate-box model – Adiabatic approximation 1. Single-particle models Chocolate-box model – Adiabatic approximation N = 8 E d

Chocolate-box model – Adiabatic approximation 1. Single-particle models Chocolate-box model – Adiabatic approximation N = 8 E d

Chocolate-box model – Adiabatic approximation 1. Single-particle models Chocolate-box model – Adiabatic approximation N = 8 E d

Chocolate-box model – Adiabatic approximation 1. Single-particle models Chocolate-box model – Adiabatic approximation N = 8 E d

Chocolate-box model – Adiabatic approximation 1. Single-particle models Chocolate-box model – Adiabatic approximation N = 8 E d

Adiabatic approximation 1. Single-particle models Chocolate-box model – shapes particle number N Adiabatic approximation d deformation

Chocolate-box model – prolate-oblate assymetry 1. Single-particle models Chocolate-box model – prolate-oblate assymetry Oblate: beginnings of the “shells” Prolate: ends of the “shells”

Chocolate-box model Spheroidal cavity d 1. Single-particle models Chocolate-box model Spheroidal cavity d I. Hamamoto and B.R. Mottelson, Phys. Rev. C 79, 034317 (2009)

2. Deformed liquid drop model Principal terms of the spherical LDM + improvements (1 slide) (curvature as a surface term of higher order) Deformation, quadrupole deformation (1 slide) Prolate – oblate energy difference Prolate – always favored. Shape stabilization by shell corrections (basics of microscopic structure included) (1 slide) 2. Deformed liquid drop model

Total mass/energy (Weizsäcker formula) 2. Liquid drop model Total mass/energy (Weizsäcker formula) microscopic corrections (shell effects, pairing) binding energy A = N + Z curvature energy, surface and volume redistribution energy… volume energy surface energy Coulomb energy Adjustable constants: Shape functions: W.D. Myers, W.J. Swiatecki, Nucl. Phys. 81, 1 (1966)

Fixed by a condition of volume conservation 2. Liquid drop model Quadrupole deformation (axially symmetric) Fixed by a condition of volume conservation a2 = 0 a2 > 0 a2 < 0 spherical oblate prolate Surface shape functions: Coulomb

Quadrupole deformation – shape functions 2. Liquid drop model Quadrupole deformation – shape functions Surface Coulomb shape functions:

Negative for a2 < 0 – prolate shape has always lower energy 2. Liquid drop model Quadrupole deformation – shape functions Symmetric with respect to the sign of a2 Surface Coulomb shape functions: Negative for a2 < 0 – prolate shape has always lower energy Deformation parameter Values of the coefficients

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

Shell corrections Mid-shell correction < 3MeV 40 80 120 2. Liquid drop model Shell corrections Mid-shell correction < 3MeV 40 80 120 Shell corrections are highly important near closed shells, but less for deformed nuclei in mid-shells W.D. Myers, W.J. Swiatecki, Nucl. Phys. 81, 1 (1966)

3. Prolate-oblate energy difference from experiments

Electric quadrupole moment 3. Numerical results Electric quadrupole moment Deformation parameter: measured intrinsic where rare-earth region is a typical value for well-deformed nuclei N.J. Stone, At. Data Nucl. Data Tables 90, 75 (2005)

Prolate-oblate energy difference 3. Numerical results Prolate-oblate energy difference

Prolate-oblate energy difference 3. Numerical results Prolate-oblate energy difference rare-earth region

Almost the same contribution 3. Numerical results Prolate-oblate energy difference surface surface Almost the same contribution Coulomb Coulomb

Distribution of DB values 3. Numerical results Distribution of DB values 495 nuclei totally

B(E2) transition probabilities 3. Numerical results 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)

Thank you for your attention Last slide Conclusions Predominance of prolate states can be explained by a simple deformed liquid drop model. This approach is robust with a transparent physical understanding, in contrast with single particle studies that require fine-tunning procedures and are strongly model dependent. Prolate-oblate energy difference of the order of DB = 500keV is high enough to be considered as non-negligible (for comparison, first 2+ excited state for well-deformed even-even nuclei is typically of the order of 100keV). Microscopic shell effects are necessary to stabilize deformed shape, but in most cases the prolate-oblate asymmetry in energy they give is not strong enough to compete with collective effects. Thank you for your attention