Structure of exotic nuclei from relativistic Hartree Bogoliubov model (II)
37
Shan-Gui Zhou Email: [email protected] ; URL: http://www.itp.ac.cn/~sgzhou 1. Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 2. Center of Theoretical Nuclear Physics, National Laboratory of Heavy Ion Accelerator, Lanzhou Structure of exotic nuclei from relativistic Hartree Bogoliubov model (II) HISS-NTAA 2007 Dubna, Aug. 7-17
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Structure of exotic nuclei from relativistic Hartree Bogoliubov model (II). Shan-Gui Zhou Email: [email protected] ; URL: http://www.itp.ac.cn/~sgzhou Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing - PowerPoint PPT Presentation
Transcript of Structure of exotic nuclei from relativistic Hartree Bogoliubov model (II)
Shan-Gui Zhou
Email: [email protected]; URL: http://www.itp.ac.cn/~sgzhou
1. Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing
2. Center of Theoretical Nuclear Physics, National Laboratory of Heavy Ion Accelerator, Lanzhou
Structure of exotic nuclei from relativistic Hartree
Bogoliubov model (II)
HISS-NTAA 2007
Dubna, Aug. 7-17
23/4/19 2
Magic numbers in super heavy nuclei
Zhang et al. NPA753(2005)106
23/4/19 3
Contents Introduction to Relativistic mean field model
Basics: formalism and advantages Pseudospin and spin symmetries in atomic nuclei Pairing correlations in exotic nuclei
Contribution of the continuum BCS and Bogoliubov transformation
Spherical relativistic Hartree Bogoliubov theory Formalism and results
Summary I Deformed relativistic Hartree Bogoliubov theory in a Woods-
Saxon basis Why Woods-Saxon basis Formalism, results and discussions
Single particle resonances Analytical continuation in coupling constant approach Real stabilization method
Summary II
23/4/19 4
Deformed Halo? Deformed core?
Decoupling of the core and valence nucleons？
11,14BeNe isotopes…
Nunes, NPA757(05)349
Misu, Nazarewicz, Aberg, NPA614(97)44
Bennaceur et al., PLB296(00)154
Hamamoto & Mottelson, PRC68(03)034312
Hamamoto & Mottelson, PRC69(04)064302
Poschl et al., PRL79(97)3841
Pei, Xu & Stevenson, NPA765(06)29
23/4/19 5
Hartree-Fock Bogoliubov theoryDeformed non-relativistic HFB in r space
Deformed relativistic Hartree-Bogoliubov or Hartree-Fock-Bogoliubov theory in harmonic oscillator basis
No deformed relativistic Hartree-Bogoliubov or Hartree-Fock-Bogoliubov theory in r space available yet
Terasaki, Flocard, Heenen & Bonche, NPA 621, 706 (1996)
Stoitsov, Dobaczewski, Ring & Pittel, PRC61, 034311 (2000)
Terán, Oberacker & Umar, PRC67, 064314 (2003)
Vretenar, Lalazissis & Ring, PRL82, 4595 (1999)
23/4/19 6
Harmonic oscillator basis and r-space
Average potential in atomic nucleus Woods-Saxon potential: no analytic solution harmonic oscillator potential: a good approx. for stable nuclei; matrix
diagonalization
Drip line nuclei: large space distribution, contribution of continuum HO basis: localization r-space: complicated and time-consuming (deformation and pairing) Woods-Saxon basis: a reconciler of r-space & HO basis?
Basic idea Numerical solutions for spherical WS potential in r space Large-box boundary condition to discretize the continuum WS wave functions used as a complete basis matrix diagonalization
problem
23/4/19 7
Schroedinger Woods-Saxon basis
max
max
1/)(0
WS
001)(
Rr
RreVrV
aRr
)()()( ll lmnlnlm YrRr
)()()()1(1
2
12
22
rRErRrVr
ll
rr
rrM nlnlnlWS
nlnrRnl ,,2,1,0;,2,1,0);(
r
VWS(r)
0 Rmax
Shooting Method
23/4/19 8
)(
)()(
)()(
)(~
t
Yr
rF
Yr
rGi
t
ljm
ljm
m
r
Spherical RMF in Schroedinger WS basis
max
max
~
0
~~~
0
)()(
)()(
n
nlnn
n
nnln
rrRfirF
rrRgirG
n
n
n
n
nmnm
nmmn
f
g
f
g
~~~~~
~
max
max
max
max
0~~~~
2~~
0~~
2~
0~~
2~
0
2
)()()()(
)(1
)(
)(1
)(
)()()()(
R
lnlmnm
R
nllmnm
R
lnmlnm
R
nlmlmn
rRMrSrVrdrRr
rRr
k
rrdrRr
rRr
k
rrdrRr
rRMrSrVrdrRr
1~maxmax nn
)()()()()(
)()()()()(
rFrVrSMrGrr
rF
rGrVrSMrFrr
rG
23/4/19 9
Dirac Woods-Saxon basis
,2,1;,1,0;0)];,,(,[ 000 nts mnmnmn r
1,),,(),,( maxmaxmax0
0max
nnntsctsn
nmnnm rr
max0
0 ,,2,1,max
nmcHcc m
n
nmnnmm
max
max
0
000
0
0
000
0
000
0
)()()()()()(
)()()()()()(
)()()()()()(
R
nm
R
nm
nmmn
rFrSrSrVrVrdrF
rGrSrSrVrVrdrG
SSVVH rrrrrr
mmSMV )()( rrp
000WSWS )()( mnmnmnSMV rrp
mmSMV )()( rrp
23/4/19 10
Dirac-WS: negative energy states
V0 [MeV] E/A [MeV] Rrms [fm]
54 8.013 | 8.547 2.568 | 2.385
72 8.015 | 8.117 2.567 | 2.531
90 8.012 | 8.427 2.567 | 2.610
Completeness of the basis (no contradiction with no-sea)
Underbound without inclusion of n.e. states
Results independent of basis parameters
23/4/19 11
Basis: Dirac-WS versus Schroedinger-WS
1
1
1
1
1
n
n
n
n
n
F
F
F
F
G
G
G
1
1
1
1
1
n
n
n
n
n
1
1
1
n
n
Schroedinger WS
nFmax = nGmax + 1
Dirac WS
nmax < n+max
Smaller Basis!
23/4/19 12
Neutron density distribution: 48Ca
23/4/19 13
Spherical Rela. Hartree calc.: 72Ca
SGZ, Meng & Ring, PRC68,034323(03)
Woods-Saxon basis reproduces r space
23/4/19 14
RMF in a Woods-Saxon basis: progress
Shape Model Schrödinger
W-S basis
DiracW-S basis
Spherical
Rela. Hartree SRH SWS SRH DWS √
Axially deforme
d
Rela. Hartree + BCS DRH DWS
√
Axially deforme
d
Rela. Hartree-Bogoliubov DRHB DWS
√
Triaxially
deformed
Rela. Hartree-Bogoliubov TRHB DWS
SGZ, Meng & Ring,PRC68,034323(03)
SGZ, Meng & Ring, in preparation
Woods-Saxon basis might be a reconciler between the HO basis and r space
SGZ, Meng & Ring, AIP Conf. Proc. 865, 90 (06)
23/4/19 15
Deformed RHB in a Woods-Saxon basis
)()(
)()(1)(
mi
mimi YrF
YriG
rpr
i mimik
mimik
mk
mk
pv
pu
pV
pU
r
r
r
r~
~,
,
pV
pUE
pV
pU
pphpp
pppphd
E
E
E
E
p
r
r
r
r
rrrr
rrrrr
'''
'''
''';''';
''';''';' *
3
miku ,
mikv ~,
Axially deformed nuclei
imi
mikmi
mikkm avau ~
~,,
23/4/19 16
, even or odd, 0 or 1
DRHB matrix elements
mArSrVrFrFrSrVrGrGdrh iiiim
ii ,',,'''''',
S
S
SSM
SMppSM rY,
21;2121r
21
21
2
22
1
1122112211 ;;,~, 2
1
pp
SMpp
pi
pi
SMpp
SMM
mii rrRrdrR SS
S
S
I m
iih '',
I m
iih '~',~
m
ii '~',
m
iim
ii '~','',~
YrVV r
YrSS r, even, 0
23/4/19 17
Pairing interaction
Phenomenological pairing interaction with parameters: V0, 0, and ( = 1)
21210
10 1
4
11
rr
rVVpp
Smooth cutoff
Soft cutoff
Bonche et al., NPA443,39 (1985)
23/4/19 18
RHB in Woods-Saxon basis for axially deformed nuclei (-force in pp channel)
)()(
)()(1)(
mi
mimi YrF
YriG
rpr
i mimik
mimik
mk
mk
pv
pu
pV
pU
r
r
r
r~
~,
,
miku ,
mikv ~,
I m
iih '',
I m
iih '~',~
m
ii '~',
m
ii '',~
cos2121 Prpppp r
cos2121 Prpppp r
cosPrr
cosPrVV r
cosPrSS r2,1p
23/4/19 19
How to fix the pairing strength and the pairing window
Zero pairing energy for the neutron
23/4/19 20
Convergence with E+cut and compared to
spherical RCHB results
E+cut: 100 MeV
~16 main shellsE ~ 0.1 MeVr ~ 0.002 fm
23/4/19 21
Routines checks: comparison with available programs
Compare with spherical RCHB model Spherical, Bogoliubov
Compare with deformed RMF in a WS basis Deformed, no pairing
Compare with deformed RMF+BCS in a WS basis
Deformed, BCS for pairing
23/4/19 22
Compare with spherical RCHB model
23/4/19 23
Properties of 44Mg
23/4/19 24
Density distributions in 44Mg
cosPrr
23/4/19 25
Density distributions in 44Mg
cosPrr
23/4/19 26
Density distributions in 44Mg
23/4/19 27
Pairing tensor in 44Mg
cos2121 Prpppp r
23/4/19 28
Canonical single neutron states in 44Mg
23/4/19 29
Contents Introduction to Relativistic mean field model
Basics: formalism and advantages Pseudospin and spin symmetries in atomic nuclei Pairing correlations in exotic nuclei
Contribution of the continuum BCS and Bogoliubov transformation
Spherical relativistic Hartree Bogoliubov theory Formalism and results
Summary I Deformed relativistic Hartree Bogoliubov theory in a Woods-
Saxon basis Why Woods-Saxon basis Formalism, results and discussions
Single particle resonances Analytical continuation in coupling constant approach Real stabilization method
Summary II
23/4/19 30
Analytical continuation in coupling constant
Padé approximant
Kukulin et al., 1989
23/4/19 31
Analytical continuation in coupling constant
Zhang, Meng, SGZ, & Hillhouse,
PRC70 (2004) 034308
23/4/19 32
Analytical continuation in coupling constant
Zhang, Meng, SGZ, & Hillhouse,
PRC70 (2004) 034308
23/4/19 33
Real stabilization method
Box boundary condition
Stable against changing of box size: resonance
Stable behavior: width
0
Hazi & Taylor, PRA1(1970)1109
23/4/19 34
Real stabilization method
Zhang, SGZ, Meng, & Zhao, 2007
RMF (PK1)
23/4/19 35
Real stabilization method
Zhang, SGZ, Meng, & Zhao, 2007
RMF (PK1)
23/4/19 36
ComparisonsACCC: analytical continuation in coupling constantS: scattering phase shift RSM: real stabilization method
RMF (NL3)Zhang, SGZ, Meng, & Zhao,
2007
23/4/19 37
Summary II
Deformed exotic nuclei, particularly halo Weakly bound and large spatial extension Continuum contributing
Deformed relativistic Hartree Bogoliubov model in a Woods-Saxon basis for exotic nuclei W-S basis as a reconciler of the r space and the oscillator basis Preliminary results for 44Mg Halo in deformed nucleus tends to be spherical
Single particle resonances: bound state like methods Analytical continuation in the coupling constant approach Real stabilization method
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