I. Bentley and S. Frauendorf Department of Physics University of Notre Dame, USA Calculation of the...
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Transcript of I. Bentley and S. Frauendorf Department of Physics University of Notre Dame, USA Calculation of the...
I. Bentley and S. FrauendorfDepartment of Physics
University of Notre Dame, USA
Calculation of the Wigner Term in the Binding Energies by Diagonalization of
the Isovector Pairing Hamiltonian
Eground bvolume A bsurface A2 / 3
E shell (N,Z) bCoulomb
Z 2
A1/ 3 cCoulombZZ
A
1/ 3
bsymmetry
| Tz | (| Tz |X)
A
Tz N Z
2
bsymmetry 92MeV , A 100
linear term : ~ 1MeV| Tz |
Important for p/rp process near the N=Z line
Quantify X
Subtract the Coulomb energy
ECoulomb bCoulomb
Z 2
A1/ 3 cCoulombZ
Z
A
1/ 3
ECoulomb
Z A
2bCoulomb
Z
A1/ 3 cCoulomb
1
2
1/ 3
E
Z mirror nuclei
Estrong Eground ECoulomb
A=44
A=56
A=68
2
)1()1(
||2/
||
|)(|
)|(|||2
:chain isobarican Along
zstrongzstrong
z
shellz
z
strong
zshell
zzsymmetrystrong
TETE
T
EXTc
T
E
constTEA
XTTbE
“Experimental” Wigner X
Substantial scatter caused by shell effectsMean value ~1 for A<70Mean value ~4 for 80<A<90
Contains shell effects! Separation is problematic.
Eground bvolumeA bsurfaceA2 / 3
E shell (N,Z) bCoulomb
Z 2
A1/ 3 cCoulombZZ
A
1/ 3
bsymmetry(A)| Tz | (| Tz |X(A))
A
Tz N Z
2P. Moeller et al. Atomic Data and
Nuclear Data Tables 59, 185 (1995):
Phenomenological expression for symmertry energy,
which corresponds to X 1
Phenomenological treatment: Micro-Macro
Density functionals
Skyrme–Hartree–Fock–Bogoliubov mass formula by N. Chamel, S. Goriely, J.M. Pearson, Nuclear Physics A 812 (2008) 72–98:
Skyrme HFB give parameter dependent values of X, substantially smaller than 1, sensitive to effective mass (Satula, Wyss, Rep. Prog. Phys. 68, 131 (05)
Unsatisfactory!
Relativistic Mean Field gives X approximately 1(Ban et al., Phys. Lett. B 633, 231 (06)
What is the origin of X?
Isovector Proton-Neutron Pairing.
Strength is fixed by isospin invariance of stronginteraction. It gives X approximately 1 by symmetry.(Frauendorf, Sheikh, Nucl. Phys. A 645, 509, (99)
1) Fixing the isovector pairing strength to the standard value for pp, nn pairing, obtained from even-odd mass differences, we quantitatively reproduce the experimental X.
2) Possibilities for implementation into density functional approaches (ongoing)
There is a well founded mechanism, which has to be there:
Isovector Pairing Hamiltonian
Generate all configurations by lifting pp, nn, pn pairs and diagonalize.6 or 7 levels around the Fermi level -> dimension ~ 10000 Few cases with 8 levels -> no significant change if G is scaled.
H (k, )k, ˆ N k, G ˆ P k,
ˆ P k,
ˆ P k, 1 ˆ p k
ˆ p k ˆ P k,1
ˆ n k ˆ n
k
ˆ P k,0
1
2ˆ n k ˆ p
k ˆ p k
ˆ n k
[H,r T ]0 : Isospin is conserved.
Solve the pairing problem by diagonalization:-Isospin is good-No problems with instabilities of the pair field
Why X=1? Strong pairing limit Spontaneous breaking of isorotational symmetry
hmf (k, )k, ˆ N k
ˆ P k, ˆ P k,
k,
quasiparticles mixed from proton and neutron particles and holes
ˆ P k,
k
Pair field is a vector, which spontaneously breaks the isorotational symmetry.
Since [Tx ,Ty ] iTz (SU2) strong breaking generates an isorotational band
E(T)T(T 1)
"strong" means >> level distance
Frauendorf SG, Sheikh JACranked shell model and isospin symmetry near N=Z NUCLEAR PHYSICS A 645, 509 (1999)
Isorotations (strong symmetry breaking)Bayman, Bes, Broglia, PRL 23 (1969) 1299 ( 2 particle transfer)Frauendorf, Sheikh, NPA 645, 509 (1999) Frauendorf, Sheikh, Physica Scripta T88, 162 (2000)
intrinsic state : | Described by common
quasi proton and quasi neutron excitations without an pn - pair field
isorotational state : DTz
T
0( ,,0) |
isorotational bands : E(T,Tz) hmf Tz T(T 1)
2Spectra of deformed N Z nuclei organanize into
spatio - iso - rotational bands.
Afanasjev AV, Frauendorf S, PRC 71, 064318 (2005)Afanasjev AV, Frauendorf S, NPA 746, 575C (2004 )Kelsall NS, Svensson CE, Fischer S, et al. EURO. PHYS. J. A 20, 131 (2004)….
Wigner X with AutoTAC Deformations
• Not perfect, but promising.• Two problems :
44≤A≤58 too strong scatter74≤A≤88 Xc~1 Xe~4
• Why?• Calculated deformations • not good enough
smallmediumlarge
Rotational response
Optimize the deformation
• Nilsson calculated• Woods Saxon calculated•Folded Yukawa calculated•Experimental (BE2(2->0)•Experimental yrast energies
“adopted deformations”
• Isovector proton neutron pairing with the strength fixed by isospin conservation gives the correct X•Mean field treatment (HFB) is insufficient – violates isospin conservation•In devising approximations beyond mean field it is decisive to incorporate restoration of isospin
Isovector and isoscalar pairing
H (k, )k, ˆ N k, GV
ˆ P k, ˆ P k,
,k
GSˆ Q k ˆ Q k
k
ˆ P k, 1 ˆ p k
ˆ p k ˆ P k,1
ˆ n k ˆ n
k
ˆ P k,0 1
2ˆ n k ˆ p
k ˆ p k
ˆ n k
ˆ Q k
1
2ˆ n k ˆ p
k ˆ p k
ˆ n k
• Isoscalar pairing attenuates the staggering between the even-even and odd-odd N=Z nuclei: some indication from experiment•Small isoscalar pair correlation would only slightly increase the X values: within the tolerance range of the isovector scenario•What is GS/GV ?
Implementation into mean field approaches
•8 levels around the Fermi level is not enough-> dimensions explode->approximations.•Iso-cranking approximation•HFB + RPA•HFB + SCRPA•T-,N-,Z- projected HFB•BCS-truncation
Iso-cranking Frauendorf, Sheikh, NPA 645, 509 (1999)
For spatial rotations of well deformed nuclei do HFB with:
H 'H Jx, with constraint | Jx | >= I(I 1)
E | H | >I(I 1)
2
In analogy do HBF with:
H 'H Tz ˆ A ,
with constraints , | Tz |, >= T(T 1), , | ˆ A |, >= A
or equivalentely :
H 'H 1ˆ N 1
ˆ Z
with constraints , | ˆ N |, >= N , , | ˆ Z |, >= Z
T(T 1) T 1/2 and N Z
Problem: It works only for a sufficiently strong pair field.
HFB+Lipkin-Nogami may mend the problem.
HFB+QRPA
H (k, )k, ˆ N k, G ˆ P k,
ˆ P k,
2
r T
r T
ˆ P k, 1 ˆ p k
ˆ p k ˆ P k,1
ˆ n k ˆ n
k
ˆ P k,0
1
2ˆ n k ˆ p
k ˆ p k
ˆ n k
[H,r T ]0 : Isospin is conserved.
K. Neergard PLB 537, 287 (2002); PLB 572, 159 (2003); PRC 80, 044313 (2009)
T-, N-, Z-, projected HFB
| T,Tz,N,Z sinddT ,TT ()e iTy d
0
2
d0
2
e i(N ˆ N ) i(Z ˆ Z )
0
(uk vkPk ) | 0
k,1
vk
T,Tz,N,Z | H | T,Tz,N,Z T,Tz,N,Z | T,Tz,N,Z
0, uk2 uk
2 1
Simplified version: projected BCS
uk
vk
1
21
ek
ek 2 2
1/ 2
, 2 vk2 ( , )
Z, 1
N, 1k
E(1, 1) T,Tz,N,Z | H | T,Tz,N,Z T,Tz,N,Z | T,Tz,N,Z
minimum
Only nn pairing
In BCS state a certain configuration has the weight : w fk with fk uk if no pair
vk if pair
k
All configuarations with w taken.
BCS
BCS