14

18 June 2005
International Conference on Finite Fermi
Systems: Nilsson Model 50 years, Lund, Sweden
Stuart Pittel
Bartol Research Institute, University of Delaware, Newark, Delaware
19716, USA
*
Work carried out in collaboration with J. Dukelsky (CSIC, Madrid),
G.G. Dussel (CNEA, Buenos Aires) and C. Esebbag (Alcala).
Exactly

solvable Richardson

Gaudin models and
their applications
*
14

18 June 2005
International Conference on Finite Fermi
Systems: Nilsson Model 50 years, Lund, Sweden
Shown by Richardson in the 60s that the pure pairing model with
constant
g
and non

degenerate single

particle energies is exactly
solvable.
Recently, a revival of work on exactly

solvable pairing models
building on work of Richardson and related work of Gaudin.

Will summarize recent advances

Will then discuss one particular example of relevance to nuclear
structure.
Introductory Remarks and Outline
14

18 June 2005
International Conference on Finite Fermi
Systems: Nilsson Model 50 years, Lund, Sweden
2000

Richardson’s exact solution revived and used to provide new
insight into transition from superconducting regime to fluctuation

dominated regime in small metallic grains. [J. Dukelsky and G.
Sierra, Phys. Rev. B61 (2000) 12302].
2001

Richardson’s solution of pure pairing model generalized to a
much wider variety of exactly

solvable pairing hamiltonians,
relevant to both fermion and bosons systems. [J. Dukelsky, C.
Esebbag and P. Schuck, PRL 87 (2001) 066403]
2001

Extended models applied to system of bosons confined to an
oscillator trap and interacting via a repulsive interaction. Showed
that fragmentation of the ground condensate possible. [J. Dukelsky
and P. Schuck, PRL 86 (2001) 4207]
2001

Models used to identify a new mechanism for enhancing s

d
boson dominance in interacting boson models of nuclei, arising
from repulsive interaction due to Pauli exchange of constituent
nucleons. [J. Dukelsky and S. Pittel, PRL 86 (2001) 4791]
Summary of Recent Developments
14

18 June 2005
International Conference on Finite Fermi
Systems: Nilsson Model 50 years, Lund, Sweden
2002

Exactly

solvable nature of pure pairing model used to find
a new pictorial representation of how superconductivity arises in a
finite fermi system like the nucleus. [J. Dukelsky, C. Esebbag and
S. Pittel, PRL 88 (2002) 062501.]
2004

Review article on Richardson

Gaudin exactly

solvable
models. [J. Dukelsky, S. Pittel and G. Sierra, RMP 76 (2004)
643.]
2004

Exactly

solvable models extended to describe coupling
between an atomic system governed by pairing correlations and
another bosonic mode. Used to model a system of bosonic atoms
coupled to a molecular dimer. [ J. Dukelsky, G. G. Dussel, S.
Pittel and C. Esebbag, PRL 93 (2004) 050403.]
14

18 June 2005
International Conference on Finite Fermi
Systems: Nilsson Model 50 years, Lund, Sweden
m
l
a
m
lm
a
l
A
lm
a
m
lm
a
l
N
l
A
ll
l
A
g
l
l
N
l
H
and
ˆ
'
'
2
ˆ
Standard pure pairing hamiltonian:
Richardson’s solution of pure pairing model
(fermions)
Richardson ansatz for ground state (N pairs):
l
l
l
N
i
A
e
B
B
2
1
,
0


1

Ψ
> is an exact eigenstate of H if
pair energies
e
α
satisfy
2
/
1
,
0
4
2
2
1
l
e
e
g
e
g
l
l
l
l
14

18 June 2005
International Conference on Finite Fermi
Systems: Nilsson Model 50 years, Lund, Sweden
These coupled eqns., one for each Cooper pair, called the
Richardson equations.
The ground state energy is a sum of the resulting pair energies,
E
α
=
Σ
α
e
α
Method can be used to get all eigenstates of H and all eigen

energies.
14

18 June 2005
International Conference on Finite Fermi
Systems: Nilsson Model 50 years, Lund, Sweden
Electrostatic analogy for pairing models
There is an electrostatic analogy for such pairing models that
emerges from the Richardson equations. Will focus on pure
pairing for fermion systems.
In this case,
ground state solution governed by pair energies
obtained from set of coupled Richardson equations:
0
4
2
2
1
e
e
g
e
g
l
l
l
14

18 June 2005
International Conference on Finite Fermi
Systems: Nilsson Model 50 years, Lund, Sweden
Consider
energy functional
If we differentiate
U
with respect to
the
e
α
and equate to zero,
we recover precisely the Richardson equations.
Question:
What is the physical meaning of
U
?

2
2

ln
8
1


ln
2
1

2

ln
)
(
2
1
]
[
4
1
j
j
j
i
j
j
j
j
j
j
e
e
e
e
g
U


ln
)
,
(
2
1
2
1
2
1
r
r
q
q
r
r
v
14

18 June 2005
International Conference on Finite Fermi
Systems: Nilsson Model 50 years, Lund, Sweden
–
A number of fixed charges (one for each active orbit) located at 2
ε
i
and with charges
Ω
i
/2. Called
orbitons.
–
N free charges located at
e
α
and with unit charge. Called
pairons
.
–
A Coulomb interaction between all charges.
–
A uniform electric field with strength 1/4g.
Reminder:
The Coulomb interaction between two point charges in 2D is:
Thus: U
represent
s
the
physics
of a
classical 2D electrostatic
problem with the following ingredients:

2
2

ln
8
1


ln
2
1

2

ln
)
(
2
1
]
[
4
1
j
j
j
i
j
j
j
j
j
j
e
e
e
e
g
U


ln
)
,
(
2
1
2
1
2
1
r
r
q
q
r
r
v
14

18 June 2005
International Conference on Finite Fermi
Systems: Nilsson Model 50 years, Lund, Sweden
For fermion systems, can show that

Orbitons are c
onstrained to real axis, since s.p. energies all
real.

Pairons
lie either on real axis or in complex conjugate pairs.
14

18 June 2005
International Conference on Finite Fermi
Systems: Nilsson Model 50 years, Lund, Sweden
Application to Nuclear Pairing
Will use electrostatic analogy to obtain pictorial representation of
how “superconductivity” develops in nuclei.
Typically hard to see effects of transition to superconductivity
because of limited number of nucleons involved.
Will use info on classical positions of pairons (from analogous 2D
problem) to provide insight into quantum problem which otherwise
was not readily evident.
Will focus on even Sn isotopes, with closed Z=50 proton shell and
N

50 active (valence) neutrons. Will do calculations as function of
g.
14

18 June 2005
International Conference on Finite Fermi
Systems: Nilsson Model 50 years, Lund, Sweden
The tin isotopes
Orbiton
Position
Charge
d
5/2
0

1.5
g
7/2
0.44

2.0
s
1/2
3.8

0.5
d
3/2
4.4

1.0
h
11/2
5.6

3.0
14

18 June 2005
International Conference on Finite Fermi
Systems: Nilsson Model 50 years, Lund, Sweden
114
Sn , 7 pairons
Lines drawn to connect each
pairon with its nearest
neighbor.
For weak pairing, pairons
organize themselves as
artificial atoms
around
associated orbitons, subject to
Pauli principle.
Note:
Physical g
≈

0.095 MeV
14

18 June 2005
International Conference on Finite Fermi
Systems: Nilsson Model 50 years, Lund, Sweden
114
Sn, Evolution with
g
As pairing strength grows, a
transition takes place from a
set of isolated “atoms” to a
“cluster”, in which pairons
have lost memory of the
orbitons from which they
came.
14

18 June 2005
International Conference on Finite Fermi
Systems: Nilsson Model 50 years, Lund, Sweden
116
Sn , 8 pairons
14

18 June 2005
International Conference on Finite Fermi
Systems: Nilsson Model 50 years, Lund, Sweden
116
Sn , stronger pairing
Two

stage transition to full superconductivity.
Comments 0
Log in to post a comment