1. 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

2. 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

3. 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)

4. 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

5. 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

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

7. 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

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

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

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

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

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

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

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

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

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

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

18. 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

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

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

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

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

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

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

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

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

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

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

29. 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”

30. 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, (2009)

31. 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

32. 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)

33. 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

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

35. 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

36. 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)

37. 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)

38. 3. Prolate-oblate energy difference from experiments

39. 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)

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

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

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

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

44. 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)

45. 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