Transcript Nuclear Structure Models

Generalized models of pairing
in non-degenerate orbits
J. Dukelsky, IEM, Madrid, Spain
D.D. Warner, Daresbury, United Kingdom
A. Frank, UNAM, Mexico
P. Van Isacker, GANIL, France
Symmetries of pairing models
Generalized pairing models
Deuteron transfer
Generalized pairing models, Saclay, June 2005

The nuclear shell model
• Mean field plus residual interaction (between
valence nucleons).
• Assume a simple mean-field potential:
A
pk2 1

Hˆ    m 2rk2   ls lk  sk   ll lk2 VˆRI k,l
2
 1kl
k1 2m
A
• Contains
– Harmonic-oscillator potential with constant .
– Spin-orbit term with strength ls.
– Orbit-orbit term with strength ll.
Generalized pairing models, Saclay, June 2005

Shell model for complex nuclei
• Solve the eigenvalue problem associated with
the matrix (n active nucleons):
i1 in
n
Vˆ k,l i
RI
1
in
1kl
• Methods of solution:
– Diagonalization (Lanczos): d~109.
– Monte-Carlo shell model: d~1015.
– Density Matrix Renormalization Group: d~10120?
Generalized pairing models, Saclay, June 2005
Symmetries of the shell model
• Three bench-mark solutions:
– No residual interaction  IP shell model.
– Pairing (in jj coupling)  Racah’s SU(2).
– Quadrupole (in LS coupling)  Elliott’s SU(3).
• Symmetry triangle:
pk2 1

2
2
2
Hˆ    m rk   ls l k  sk   ll lk 
2m 2

k1 
A

A
Vˆ k,l
RI
1kl
Generalized pairing models, Saclay, June 2005
Racah’s SU(2) pairing model
• Assume pairing interaction in a single-j shell:
j 2JMJ Vˆpairing
1


2 j 1g0, J  0
2
2
j JMJ  
0,
J 0

• Spectrum 210Pb:

Generalized pairing models, Saclay, June 2005

Solution of the pairing hamiltonian
• Analytic solution of pairing hamiltonian for
identical nucleons in a single-j shell:
j nJ
n
1
n
ˆ
V
j

J

g
 pairing
0 4 n   2 j  n    3
1kl
• Seniority  (number of nucleons not in pairs
coupled to J=0) is a good quantum number.
• Correlated ground-state solution (cf. BCS).
G. Racah, Phys. Rev. 63 (1943) 367
Generalized pairing models, Saclay, June 2005
Nuclear “superfluidity”
• Ground states of pairing hamiltonian have the
following correlated character:
– Even-even nucleus (=0):
– Odd-mass nucleus (=1):
  o
a Sˆ  o
Sˆ 

j
n /2
n /2

• Nuclear superfluidity leads to
– Constant energy of first 2+ in even-even nuclei.

– Odd-even staggering in masses.
– Smooth variation of two-nucleon separation
energies with nucleon number.
– Two-particle (2n or 2p) transfer enhancement.
Generalized pairing models, Saclay, June 2005
Two-nucleon separation energies
• Two-nucleon separation energies S2n:
(a) Shell splitting dominates over interaction.
(b) Interaction dominates over shell splitting.
(c) S2n in tin isotopes.
Generalized pairing models, Saclay, June 2005



Integrability of pairing hamiltonian
• Pair operators (several shells):
Sˆ    Sˆ j ,
 
Sˆ  Sˆ 

j
• The pairing hamiltonian for degenerate shells

Vˆpairing  g0 Sˆ  Sˆ  g0 Sˆ 2  Sˆ z2  Sˆ z

• … is solvable by virtue of an underlying SU(2)
“quasi-spin” symmetry:
ˆ , Sˆ  1 2nˆ    2Sˆ , Sˆ , Sˆ   Sˆ ,    2 j  1
S
   2
 z  
z
j
A.K. Kerman, Ann. Phys. (NY) 12 (1961) 300
Generalized pairing models, Saclay, June 2005

Generalized pairing model
• Hamiltonian for pairing interaction in nondegenerate shells:
Hˆ   j nˆ j  g0  Sˆj Sˆj'
j
jj'
• Is the pairing model with non-degenerate
orbits integrable?
Generalized pairing models, Saclay, June 2005



Richardson-Gaudin models
• Algebraic structure:



Sˆ i , Sˆj  2ij Sˆ zj ,

Sˆ zi , Sˆ j  ij Sˆ j
• The Gaudin operators
Rˆ j  Sˆ zj  4g0 
 
i j 
1
2

X ij Sˆ j Sˆi  Sˆj Sˆ i  Yij Sˆ zj Sˆ zi

• …commute if Xij and Yij are antisymmetric and
satisfy the equations


Yij X jk  Yki X jk  X ki X ij  0  Rˆ i , Rˆ j  0
• Any combination of Ri is integrable.
R.W. Richardson, Phys. Lett. 5 (1963) 82
M. Gaudin, J. Phys. (Paris) 37 (1976) 1087.
Generalized pairing models, Saclay, June 2005
Pairing with non-degenerate orbits
• If we choose
1
X ij  Yij 
 Hˆ  2 j Rˆ j   j nˆ j  g0  Sˆ j Sˆj'
i   j
j
j
jj'
•  A hamiltonian for pairing in non-degenerate
shells is integrable! Solution:




1
j

S
 2  e  o
j

 1  j

2 j 1
1
1 2g0 
 8g0 
 0,   1,2, ,n /2
2 j  e
j
   e  e
n /2
J. Dukelsky et al., Phys. Rev. Lett. 87 (2001) 066403
Generalized pairing models, Saclay, June 2005
Pairing with neutrons and protons
• For neutrons and protons two pairs and hence
two pairing interactions are possible:
– Isoscalar (S=1,T=0):
010


ˆ10 , Sˆ10  2l  1a 1 1  a 1 1  , Sˆ10  Sˆ10
Sˆ10

S





l 
 l 2 2
22
– Isovector (S=0,T=1):

 
001


Sˆ 01  Sˆ01, Sˆ 01  2l  1al1 1  al1 1  , Sˆ01  Sˆ 01
 2 2
2 2 

Generalized pairing models, Saclay, June 2005

 


Neutron-proton pairing hamiltonian
• A hamiltonian with two pairing interactions
ˆ10  gSˆ 01  Sˆ 01
Vˆpairing  g0Sˆ10

S


0 

• …has an SO(8) algebraic structure.
• Vpairing is integrable and solvable (dynamical
symmetries) for g0=0, g0=0 and g0=g0.
Generalized pairing models, Saclay, June 2005

SO(8) “quasi-spin” formalism
• A closed algebra is obtained with the pair
operators S± with in addition
000
011




nˆ  2 2l  1al1 1  al1 1  , Yˆ  2l  1al1 1  al1 1 
 2 2
 2 2
2 2 
2 2 
000
0 
001
010




Sˆ   2l  1al1 1  al1 1  , Tˆ  2l  1al1 1  al1 1 
 2 2
 2 2
2 2 
2 2 
00 
0 0
• This set of 28 operators forms the Lie algebra
SO(8) with subalgebras
SO 6  SU 4  Sˆ, Tˆ , Yˆ , SO S 5  nˆ, Sˆ, Sˆ 01,
SO T 5  nˆ, Tˆ , Sˆ10
, SO S 3  Sˆ, SO T 3  Tˆ 
 
B.H. Flowers & S. Szpikowski, Proc. Phys. Soc. 84 (1964) 673
Generalized pairing models, Saclay, June 2005
Solvable limits of the SO(8) model
• Pairing interactions can expressed as follows:
Sˆ  Sˆ  Cˆ SO 5 Cˆ SO 3 (2l  nˆ  1)(2l  nˆ  7)
Sˆ  Sˆ  Sˆ  Sˆ  Cˆ SO 8 Cˆ SO 6 (2l  nˆ  1)(2l  nˆ  13)
Sˆ  Sˆ  Cˆ SO 5 Cˆ SO 3 (2l  nˆ  1)(2l  nˆ  7)


01

01

01

01

10

10

1
2
2
10

1
2
T
1
2
10

2
1
2
S
2
T
1
2
2
1
2
2
1
8
S
1
8
2
1
8
• Symmetry lattice of the SO(8) model:
SO S 5  SO T 3


SO(8)   SO(6)  SU 4  SO S 3  SO T 3
SO 5  SO 3 
 T  
S  
• Analytic solutions for g0=0, g0=0 & g0=g0.
Generalized pairing models, Saclay, June 2005

Quartetting in N=Z nuclei
• T=0 and T=1 pairing has a quartet structure
with SO(8) symmetry. Pairing ground state of
an N=Z nucleus:
cos Sˆ
10

 Sˆ  sin Sˆ  Sˆ
10

01

01


n /4
o
•  Condensate of “-like” objects.
• Observations:
– Isoscalar component in condensate survives only
in N~Z nuclei, if anywhere at all.
– Spin-orbit term reduces isoscalar component.
Generalized pairing models, Saclay, June 2005

Generalized neutron-proton pairing
• Hamiltonian for pairing interactions in nondegenerate shells:
ˆ10  gSˆ 01  Sˆ 01
Hˆ   j nˆ j  g0Sˆ10

S


0 

j
• Solution techniques:
– Richardson-Gaudin for SO(8) model.
– Boson mappings:
• requiring same commutation relations;
• associating state vectors.
Generalized pairing models, Saclay, June 2005

Generalized pairing models
• Pairing in degenerate orbits between identical
particles has SU(2) symmetry.
• Richardson-Gaudin models can be generalized
to higher-rank algebras:
L
Rˆ i  Hˆ is  g0  
j i ,
Xˆ i g Xˆ j
2i  2 j

Aba
g0 
 g0  
 as
i1 ea  2i
b1  1 ea  eb
L
a
i
r Mb
J. Dukelsky et al., to be published
Generalized pairing models, Saclay, June 2005

Example: SO(5) pairing
• Hamiltonian:
ˆ10
Hˆ   j nˆ j  g0Sˆ10

S


j
• “Quasi-spin” algebra is
SO(5) (rank 2).
• Example: 64Ge in pfg9/2
shell (d~91014).
S. Dimitrova, unpublished
Generalized pairing models, Saclay, June 2005

Model with L=0 vector bosons
• Correspondence: Sˆ10  b10  s
• Algebraic structure is U(6).
• Symmetry lattice of U(6):

Sˆ01  b01
 p

U S 3  UT 3
U(6)  
 SO S 3  SO T 3
 SU 4

• Boson mapping is exact in the symmetry limits
[for fully paired states of the SO(8)].
P. Van Isacker et al., J. Phys. G 24 (1998) 1261
Generalized pairing models, Saclay, June 2005
Masses of N~Z nuclei
• Neutron-proton pairing hamiltonian in nondegenerate shells:
ˆ10  gSˆ 01  Sˆ 01
Hˆ F   j nˆ j  g0Sˆ10

S


0 

j

• HF maps into the boson hamiltonian:
Hˆ B  aCˆ 2 SU 4 bCˆ1U S 3
 c1Cˆ1U6 c 2Cˆ 2 U6 dCˆ 2 SO T 3
• HB describes masses of N~Z nuclei.

E. Baldini-Neto et al., Phys. Rev. C 65 (2002) 064303
Generalized pairing models, Saclay, June 2005


Two-nucleon transfer
• Amplitude for two-nucleon transfer in the
reaction A+aB+b:
   GN L,S,J KNLM k ,k 
L
N
• Nuclear-structure information contained in
GN(L,S,J) which for L=0 transfer reduces to
GN L  0,S  J    00N0;0 nlnl;0  nlTS
nl
  B a
TS
nl

nl1/ 21/ 2

nl1/ 21/ 2
a
0TS 

A
N.K. Glendenning, Direct Nuclear Reactions
Generalized pairing models, Saclay, June 2005


Deuteron transfer
• Overlap of uncorrelated pair:
  B a

nl1/ 21/ 2
TS
nl

nl1/ 21/ 2
a
0TS 

A
• Bosons correspond to correlated pairs:
 nl a

nl1/ 21/ 2

nl1/ 21/ 2
a
0TS 

 Sˆ TS  bTS
nl
• Scale property:
 
TS
nl

n'l'
2l  1
B Sˆ TS A
 n' l' 2l'1
P. Van Isacker et al., Phys. Rev. Lett. 94 (2005) 162502
Generalized pairing models, Saclay, June 2005
Deuteron transfer with bosons
• Correspondence SˆTS  bTS does not take account
of Pauli principle.
• The following correspondence is shown to be
exact [in 
the Wigner limit]:
– Even-even  odd-odd
B Sˆ TS A 
1
2
2  N b  1 N b  1B bTS N b A
– Odd-odd  even-even
B Sˆ TS A 
1
2
2  N b  6 N b  1B bTS N b A


Generalized pairing models, Saclay, June 2005

Masses of pf-shell nuclei
• Boson hamiltonian:
ˆ  aCˆ SU 4 bCˆ U 3 c Cˆ U6 c Cˆ U6 dCˆ SO 3
H
B
2
1
S
1 1
2 2
2
T
• Rms deviation is 306 (or 254) keV.
• Parameter ratio: b/a5.
Generalized pairing models, Saclay, June 2005

Deuteron transfer in N=Z nuclei
• Deuteron-transfer
intensity cT2 calculated
in sp-boson IBM based
on SO(8).
c  N b  1 B b
2
T

TS
N b  A
2
• Ratio b/a fixed from
masses in lower half of
28-50 shell.
Generalized pairing models, Saclay, June 2005