pavel stránský 29 th august 2011 w hat drives nuclei to be prolate? instituto de ciencias...
Post on 16-Dec-2015
212 Views
Preview:
TRANSCRIPT
Pavel Stránský
29th August 2011
WHAT DRIVES NUCLEI TO BE PROLATE?
Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México
Alejandro FrankRoelof Bijker
CGS14, University of Guelph, Ontario, Canada, 2011
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
WHAT DRIVES NUCLEI TO BE PROLATE?
Surface tensionCoulomb energy…
Shell structureSpin-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
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
1s
1p
1d2s
2p
1f
1g
2d1h 2s
012
3
4
01
23
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
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
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
surface
Coulomb
surface
Coulomb
Almost the same contribution (despite the different functional form)
Coulomb and surface contribution
2. Deformed liquid drop model
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
Shape stabilizationPure liquid drop model is not able
to explain the ground state deformation (spherical shape is
always preferred)
Necessity of introducing shell correctionsShell corrections (Strutinsky)
N
E
Exact cumulative level density
Smooth cumulative level
densityspherical
deformed deformation decreases the size of the corrections
2. Deformed liquid drop model
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
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 (1st approximation)
2. Deformed liquid drop model
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
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
top related