Structure of exotic nuclei from relativistic Hartree Bogoliubov model (II)

Shan-Gui Zhou

Email: sgzhou@itp.ac.cn; 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