i. bentley and s. frauendorf department of physics university of notre dame, usa
DESCRIPTION
Calculation of the Wigner Term in the Binding Energies by Diagonalization of the Isovector Pairing Hamiltonian. I. Bentley and S. Frauendorf Department of Physics University of Notre Dame, USA. Important for p/rp process near the N=Z line. Quantify X. Subtract the Coulomb energy. A=44. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: I. Bentley and S. Frauendorf Department of Physics University of Notre Dame, USA](https://reader036.vdocument.in/reader036/viewer/2022062500/56815734550346895dc4d33f/html5/thumbnails/1.jpg)
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
![Page 2: I. Bentley and S. Frauendorf Department of Physics University of Notre Dame, USA](https://reader036.vdocument.in/reader036/viewer/2022062500/56815734550346895dc4d33f/html5/thumbnails/2.jpg)
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
![Page 3: I. Bentley and S. Frauendorf Department of Physics University of Notre Dame, USA](https://reader036.vdocument.in/reader036/viewer/2022062500/56815734550346895dc4d33f/html5/thumbnails/3.jpg)
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
![Page 4: I. Bentley and S. Frauendorf Department of Physics University of Notre Dame, USA](https://reader036.vdocument.in/reader036/viewer/2022062500/56815734550346895dc4d33f/html5/thumbnails/4.jpg)
![Page 5: I. Bentley and S. Frauendorf Department of Physics University of Notre Dame, USA](https://reader036.vdocument.in/reader036/viewer/2022062500/56815734550346895dc4d33f/html5/thumbnails/5.jpg)
![Page 6: I. Bentley and S. Frauendorf Department of Physics University of Notre Dame, USA](https://reader036.vdocument.in/reader036/viewer/2022062500/56815734550346895dc4d33f/html5/thumbnails/6.jpg)
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
![Page 7: I. Bentley and S. Frauendorf Department of Physics University of Notre Dame, USA](https://reader036.vdocument.in/reader036/viewer/2022062500/56815734550346895dc4d33f/html5/thumbnails/7.jpg)
“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.
![Page 8: I. Bentley and S. Frauendorf Department of Physics University of Notre Dame, USA](https://reader036.vdocument.in/reader036/viewer/2022062500/56815734550346895dc4d33f/html5/thumbnails/8.jpg)
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
![Page 9: I. Bentley and S. Frauendorf Department of Physics University of Notre Dame, USA](https://reader036.vdocument.in/reader036/viewer/2022062500/56815734550346895dc4d33f/html5/thumbnails/9.jpg)
![Page 10: I. Bentley and S. Frauendorf Department of Physics University of Notre Dame, USA](https://reader036.vdocument.in/reader036/viewer/2022062500/56815734550346895dc4d33f/html5/thumbnails/10.jpg)
Micro-Macro with Nilsson potential
![Page 11: I. Bentley and S. Frauendorf Department of Physics University of Notre Dame, USA](https://reader036.vdocument.in/reader036/viewer/2022062500/56815734550346895dc4d33f/html5/thumbnails/11.jpg)
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)
![Page 12: I. Bentley and S. Frauendorf Department of Physics University of Notre Dame, USA](https://reader036.vdocument.in/reader036/viewer/2022062500/56815734550346895dc4d33f/html5/thumbnails/12.jpg)
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:
![Page 13: I. Bentley and S. Frauendorf Department of Physics University of Notre Dame, USA](https://reader036.vdocument.in/reader036/viewer/2022062500/56815734550346895dc4d33f/html5/thumbnails/13.jpg)
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
![Page 14: I. Bentley and S. Frauendorf Department of Physics University of Notre Dame, USA](https://reader036.vdocument.in/reader036/viewer/2022062500/56815734550346895dc4d33f/html5/thumbnails/14.jpg)
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)
![Page 15: I. Bentley and S. Frauendorf Department of Physics University of Notre Dame, USA](https://reader036.vdocument.in/reader036/viewer/2022062500/56815734550346895dc4d33f/html5/thumbnails/15.jpg)
02
ˆ
0
ˆ
np
ppnn
np
ppnn
y
z
N Z p n : E(r ) E()
All directions of the pair field are equivalent.
![Page 16: I. Bentley and S. Frauendorf Department of Physics University of Notre Dame, USA](https://reader036.vdocument.in/reader036/viewer/2022062500/56815734550346895dc4d33f/html5/thumbnails/16.jpg)
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)….
![Page 17: I. Bentley and S. Frauendorf Department of Physics University of Notre Dame, USA](https://reader036.vdocument.in/reader036/viewer/2022062500/56815734550346895dc4d33f/html5/thumbnails/17.jpg)
level spacing dominates
pair fielddominates 1 3
T
![Page 18: I. Bentley and S. Frauendorf Department of Physics University of Notre Dame, USA](https://reader036.vdocument.in/reader036/viewer/2022062500/56815734550346895dc4d33f/html5/thumbnails/18.jpg)
![Page 19: I. Bentley and S. Frauendorf Department of Physics University of Notre Dame, USA](https://reader036.vdocument.in/reader036/viewer/2022062500/56815734550346895dc4d33f/html5/thumbnails/19.jpg)
![Page 20: I. Bentley and S. Frauendorf Department of Physics University of Notre Dame, USA](https://reader036.vdocument.in/reader036/viewer/2022062500/56815734550346895dc4d33f/html5/thumbnails/20.jpg)
![Page 21: I. Bentley and S. Frauendorf Department of Physics University of Notre Dame, USA](https://reader036.vdocument.in/reader036/viewer/2022062500/56815734550346895dc4d33f/html5/thumbnails/21.jpg)
spherical sphericaldeformed
![Page 22: I. Bentley and S. Frauendorf Department of Physics University of Notre Dame, USA](https://reader036.vdocument.in/reader036/viewer/2022062500/56815734550346895dc4d33f/html5/thumbnails/22.jpg)
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
![Page 23: I. Bentley and S. Frauendorf Department of Physics University of Notre Dame, USA](https://reader036.vdocument.in/reader036/viewer/2022062500/56815734550346895dc4d33f/html5/thumbnails/23.jpg)
smallmediumlarge
Rotational response
Optimize the deformation
• Nilsson calculated• Woods Saxon calculated•Folded Yukawa calculated•Experimental (BE2(2->0)•Experimental yrast energies
“adopted deformations”
![Page 24: I. Bentley and S. Frauendorf Department of Physics University of Notre Dame, USA](https://reader036.vdocument.in/reader036/viewer/2022062500/56815734550346895dc4d33f/html5/thumbnails/24.jpg)
Adjusted deformations
![Page 25: I. Bentley and S. Frauendorf Department of Physics University of Notre Dame, USA](https://reader036.vdocument.in/reader036/viewer/2022062500/56815734550346895dc4d33f/html5/thumbnails/25.jpg)
![Page 26: I. Bentley and S. Frauendorf Department of Physics University of Notre Dame, USA](https://reader036.vdocument.in/reader036/viewer/2022062500/56815734550346895dc4d33f/html5/thumbnails/26.jpg)
![Page 27: I. Bentley and S. Frauendorf Department of Physics University of Notre Dame, USA](https://reader036.vdocument.in/reader036/viewer/2022062500/56815734550346895dc4d33f/html5/thumbnails/27.jpg)
2 1
2E(N 1,Z 1) 2E(N,Z) E(N 1,Z 1) , N,Z even
Tz 12
(N Z)
![Page 28: I. Bentley and S. Frauendorf Department of Physics University of Notre Dame, USA](https://reader036.vdocument.in/reader036/viewer/2022062500/56815734550346895dc4d33f/html5/thumbnails/28.jpg)
• 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
![Page 29: I. Bentley and S. Frauendorf Department of Physics University of Notre Dame, USA](https://reader036.vdocument.in/reader036/viewer/2022062500/56815734550346895dc4d33f/html5/thumbnails/29.jpg)
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
![Page 30: I. Bentley and S. Frauendorf Department of Physics University of Notre Dame, USA](https://reader036.vdocument.in/reader036/viewer/2022062500/56815734550346895dc4d33f/html5/thumbnails/30.jpg)
![Page 31: I. Bentley and S. Frauendorf Department of Physics University of Notre Dame, USA](https://reader036.vdocument.in/reader036/viewer/2022062500/56815734550346895dc4d33f/html5/thumbnails/31.jpg)
![Page 32: I. Bentley and S. Frauendorf Department of Physics University of Notre Dame, USA](https://reader036.vdocument.in/reader036/viewer/2022062500/56815734550346895dc4d33f/html5/thumbnails/32.jpg)
1
4E(N 1,Z 1) 2E(N,Z) E(N 1,Z 1) , N Z even
Indication for weak isoscalar pairing correlations?
![Page 33: I. Bentley and S. Frauendorf Department of Physics University of Notre Dame, USA](https://reader036.vdocument.in/reader036/viewer/2022062500/56815734550346895dc4d33f/html5/thumbnails/33.jpg)
• 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 ?
![Page 34: I. Bentley and S. Frauendorf Department of Physics University of Notre Dame, USA](https://reader036.vdocument.in/reader036/viewer/2022062500/56815734550346895dc4d33f/html5/thumbnails/34.jpg)
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
![Page 35: I. Bentley and S. Frauendorf Department of Physics University of Notre Dame, USA](https://reader036.vdocument.in/reader036/viewer/2022062500/56815734550346895dc4d33f/html5/thumbnails/35.jpg)
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
![Page 36: I. Bentley and S. Frauendorf Department of Physics University of Notre Dame, USA](https://reader036.vdocument.in/reader036/viewer/2022062500/56815734550346895dc4d33f/html5/thumbnails/36.jpg)
Problem: It works only for a sufficiently strong pair field.
HFB+Lipkin-Nogami may mend the problem.
![Page 37: I. Bentley and S. Frauendorf Department of Physics University of Notre Dame, USA](https://reader036.vdocument.in/reader036/viewer/2022062500/56815734550346895dc4d33f/html5/thumbnails/37.jpg)
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)
![Page 38: I. Bentley and S. Frauendorf Department of Physics University of Notre Dame, USA](https://reader036.vdocument.in/reader036/viewer/2022062500/56815734550346895dc4d33f/html5/thumbnails/38.jpg)
T
TdT
dE )(0
ji
ji eeTTEETETE1
220 )(
2)0()(
2)1(
)(
TT
TE
Equidistant levels
Iso-cranking
![Page 39: I. Bentley and S. Frauendorf Department of Physics University of Notre Dame, USA](https://reader036.vdocument.in/reader036/viewer/2022062500/56815734550346895dc4d33f/html5/thumbnails/39.jpg)
HFB+QRPA unreliable near the critical G. We are close by.
![Page 40: I. Bentley and S. Frauendorf Department of Physics University of Notre Dame, USA](https://reader036.vdocument.in/reader036/viewer/2022062500/56815734550346895dc4d33f/html5/thumbnails/40.jpg)
SCQRPA is not worked out for full isovector pairing. Hung & Dang RIKEN working on it.
nn pairing
![Page 41: I. Bentley and S. Frauendorf Department of Physics University of Notre Dame, USA](https://reader036.vdocument.in/reader036/viewer/2022062500/56815734550346895dc4d33f/html5/thumbnails/41.jpg)
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
![Page 42: I. Bentley and S. Frauendorf Department of Physics University of Notre Dame, USA](https://reader036.vdocument.in/reader036/viewer/2022062500/56815734550346895dc4d33f/html5/thumbnails/42.jpg)
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
![Page 43: I. Bentley and S. Frauendorf Department of Physics University of Notre Dame, USA](https://reader036.vdocument.in/reader036/viewer/2022062500/56815734550346895dc4d33f/html5/thumbnails/43.jpg)
• Generalization to full isovector pairing OK• Not implemented yet• ?