Generalized pairing models, Saclay, June 2005 Generalized models of pairing in non-degenerate orbits J. Dukelsky, IEM, Madrid, Spain D.D. Warner, Daresbury,

Generalized pairing models, Saclay, June 2005

Generalized models of pairingin 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


The nuclear shell model

• Mean field plus residual interaction (between valence nucleons).

• Assume a simple mean-field potential:

• Contains– Harmonic-oscillator potential with constant .

– Spin-orbit term with strength ls.

– Orbit-orbit term with strength ll.

€

ˆ H =pk

2

2m+

1

2mω2rk

2 −ζ ls lk ⋅sk −ζ ll lk2

⎡

⎣ ⎢

⎤

⎦ ⎥

k=1

A

∑ + ˆ V RI k,l( )1≤k<l

A

∑


Shell model for complex nuclei

• Solve the eigenvalue problem associated with the matrix (n active nucleons):

• Methods of solution:– Diagonalization (Lanczos): d~109.– Monte-Carlo shell model: d~1015.– Density Matrix Renormalization Group: d~10120?

€

′ i 1K ′ i n ˆ V RI k, l( )1≤k<l

n

∑ i1K in


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:

€

ˆ H =pk

2

2m+

1

2mω2rk

2 −ζ ls lk ⋅sk −ζ ll lk2

⎡

⎣ ⎢

⎤

⎦ ⎥

k=1

A

∑

+ ˆ V RI k, l( )1≤k<l

A

∑


Racah’s SU(2) pairing model

• Assume pairing interaction in a single-j shell:

• Spectrum 210Pb:

€

j 2JMJˆ V pairing j 2JMJ =

− 1

22 j +1( )g0, J = 0

0, J ≠ 0

⎧ ⎨ ⎩


Solution of the pairing hamiltonian

• Analytic solution of pairing hamiltonian for identical nucleons in a single-j shell:

• Seniority (number of nucleons not in pairs coupled to J=0) is a good quantum number.

• Correlated ground-state solution (cf. BCS). €

j nυ J ˆ V pairing

1≤k<l

n

∑ j nυJ = −g01

4n −υ( ) 2 j − n −υ + 3( )

G. Racah, Phys. Rev. 63 (1943) 367


Nuclear “superfluidity”

• Ground states of pairing hamiltonian have the following correlated character:– Even-even nucleus (=0):– Odd-mass nucleus (=1):

• 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.

€

ˆ S +( )n / 2

o

a j+ ˆ S +( )

n / 2o


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.


Integrability of pairing hamiltonian

A.K. Kerman, Ann. Phys. (NY) 12 (1961) 300

• Pair operators (several shells):

• The pairing hamiltonian for degenerate shells

• … is solvable by virtue of an underlying SU(2) “quasi-spin” symmetry:

€

ˆ S + = ˆ S +j , ˆ S − = ˆ S +( )

+

j

∑

€

ˆ V pairing = −g0ˆ S + ˆ S − = −g0

ˆ S 2 − ˆ S z2 + ˆ S z( )

€

ˆ S +, ˆ S −[ ] = 1

22 ˆ n − Ω( ) ≡ 2 ˆ S z, ˆ S z, ˆ S ±[ ] = ± ˆ S ±, Ω = 2 j +1

j∑


Generalized pairing model

• Hamiltonian for pairing interaction in non-degenerate shells:

• Is the pairing model with non-degenerate orbits integrable?€

ˆ H = ε jˆ n j

j

∑ − g0ˆ S +

j ˆ S −j '

jj '

∑


Richardson-Gaudin models

R.W. Richardson, Phys. Lett. 5 (1963) 82M. Gaudin, J. Phys. (Paris) 37 (1976) 1087.

• Algebraic structure:

• The Gaudin operators

• …commute if Xij and Yij are antisymmetric and satisfy the equations

Any combination of Ri is integrable.

€

ˆ S +i , ˆ S −

j[ ] = 2δij

ˆ S zj , ˆ S z

i, ˆ S ±j

[ ] = ±δ ijˆ S +

j

€

ˆ R j = ˆ S zj − 4g0

1

2X ij

ˆ S +j ˆ S −

i + ˆ S −j ˆ S +

i( ) + Yij

ˆ S zj ˆ S z

i{ }

i ≠ j( )

∑

€

Yij X jk + YkiX jk + XkiX ij = 0 ⇔ ˆ R i, ˆ R j[ ] = 0


Pairing with non-degenerate orbits

€

1

2ε j − eαj

∑ S+j

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

α =1

n / 2

∏ o

1− 2g0

2 j +1

2ε j − eα

− 8g0

1

eα − eβ

= 0, α =1,2,K ,n /2β ≠α( )

∑j

∑

J. Dukelsky et al., Phys. Rev. Lett. 87 (2001) 066403

• If we choose

A hamiltonian for pairing in non-degenerate shells is integrable! Solution:

€

X ij = Yij =1

ε i −ε j

⇒ ˆ H = 2 ε jˆ R j ≡ ε j

ˆ n j − g0ˆ S +

j ˆ S −j '

jj '

∑j

∑j

∑


Pairing with neutrons and protons

• For neutrons and protons two pairs and hence two pairing interactions are possible:– Isoscalar (S=1,T=0):

– Isovector (S=0,T=1):

€

ˆ S +01 ⋅ ˆ S −

01, ˆ S +01 = 2l +1 a

l 1

2

1

2

+ × al 1

2

1

2

+ ⎛ ⎝ ⎜ ⎞

⎠ ⎟001( )

, ˆ S −01 = ˆ S +

01( )

+

€

ˆ S +10 ⋅ ˆ S −

10, ˆ S +10 = 2l +1 a

l 1

2

1

2

+ × al 1

2

1

2

+ ⎛ ⎝ ⎜ ⎞

⎠ ⎟010( )

, ˆ S −10 = ˆ S +

10( )

+


Neutron-proton pairing hamiltonian

• A hamiltonian with two pairing interactions

• …has an SO(8) algebraic structure.

• Vpairing is integrable and solvable (dynamical symmetries) for g0=0, g′0=0 and g0=g′0.

€

ˆ V pairing = −g0ˆ S +

10 ⋅ ˆ S −10 − ′ g 0 ˆ S +

01 ⋅ ˆ S −01


SO(8) “quasi-spin” formalism

• A closed algebra is obtained with the pair operators S± with in addition

• This set of 28 operators forms the Lie algebra SO(8) with subalgebras

€

ˆ n = 2 2l +1 al 1

2

1

2

+ × al 1

2

1

2

+ ⎛ ⎝ ⎜ ⎞

⎠ ⎟000

000( )

, ˆ Y μν = 2l +1 al 1

2

1

2

+ × al 1

2

1

2

+ ⎛ ⎝ ⎜ ⎞

⎠ ⎟0μν

011( )

ˆ S μ = 2l +1 al 1

2

1

2

+ × al 1

2

1

2

+ ⎛ ⎝ ⎜ ⎞

⎠ ⎟00μ

001( )

, ˆ T ν = 2l +1 al 1

2

1

2

+ × al 1

2

1

2

+ ⎛ ⎝ ⎜ ⎞

⎠ ⎟0ν 0

010( )

B.H. Flowers & S. Szpikowski, Proc. Phys. Soc. 84 (1964) 673

€

SO 6( ) ≈ SU 4( ) = ˆ S , ˆ T , ˆ Y { }, SOS 5( ) = ˆ n , ˆ S , ˆ S ±01

{ },

SOT 5( ) = ˆ n , ˆ T , ˆ S ±10

{ }, SOS 3( ) = ˆ S { }, SOT 3( ) = ˆ T { }


Solvable limits of the SO(8) model• Pairing interactions can expressed as follows:

• Symmetry lattice of the SO(8) model:

Analytic solutions for g0=0, g′0=0 & g0=g′0.

€

ˆ S +01 ⋅ ˆ S −

01 = 1

2ˆ C 2 SOT 5( )[ ] − 1

2ˆ C 2 SOT 3( )[ ] − 1

8(2l − ˆ n +1)(2l − ˆ n + 7)

ˆ S +01 ⋅ ˆ S −

01 + ˆ S +10 ⋅ ˆ S −

10 = 1

2ˆ C 2 SO 8( )[ ] − 1

2ˆ C 2 SO 6( )[ ] − 1

8(2l − ˆ n +1)(2l − ˆ n +13)

ˆ S +10 ⋅ ˆ S −

10 = 1

2ˆ C 2 SOS 5( )[ ] − 1

2ˆ C 2 SOS 3( )[ ] − 1

8(2l − ˆ n +1)(2l − ˆ n + 7)

€

SO(8)⊃

SOS 5( )⊗SOT 3( )

SO(6) ≈ SU 4( )

SOT 5( )⊗SOS 3( )

⎧

⎨ ⎪

⎩ ⎪

⎫

⎬ ⎪

⎭ ⎪⊃SOS 3( )⊗SOT 3( )


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:

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.

€

cosθ ˆ S +10 ⋅ ˆ S +

10 − sinθ ˆ S +01 ⋅ ˆ S +

01( )

n / 4o


Generalized neutron-proton pairing

• Hamiltonian for pairing interactions in non-degenerate shells:

• Solution techniques:– Richardson-Gaudin for SO(8) model.– Boson mappings:

• requiring same commutation relations;

• associating state vectors.

€

ˆ H = ε jˆ n j

j

∑ − g0ˆ S +

10 ⋅ ˆ S −10 − ′ g 0 ˆ S +

01 ⋅ ˆ S −01


Generalized pairing models

J. Dukelsky et al., to be published

• Pairing in degenerate orbits between identical particles has SU(2) symmetry.

• Richardson-Gaudin models can be generalized to higher-rank algebras:

€

ˆ R i = ˆ H is + g0

ˆ X iμ gμν

ˆ X jν

2ε i − 2ε jμ ,ν

∑j ≠ i( )

L

∑

g0

Λia

eaα − 2ε i

− g0

Aba

eaα − ebβ

= δas

β =1

M b

∑b=1

r

∑i=1

L

∑


Example: SO(5) pairing• Hamiltonian:

• “Quasi-spin” algebra is SO(5) (rank 2).

• Example: 64Ge in pfg9/2 shell (d~91014).

€

ˆ H = ε jˆ n j

j

∑ − g0ˆ S +

10 ⋅ ˆ S −10

S. Dimitrova, unpublished


Model with L=0 vector bosons

• Correspondence:

• Algebraic structure is U(6).

• Symmetry lattice of U(6):

• Boson mapping is exact in the symmetry limits [for fully paired states of the SO(8)].

€

ˆ S +10 → b10

+ ≡ s+ ˆ S +01 → b01

+ ≡ p+

€

U(6)⊃US 3( )⊗UT 3( )

SU 4( )

⎧ ⎨ ⎩

⎫ ⎬ ⎭⊃SOS 3( )⊗SOT 3( )

P. Van Isacker et al., J. Phys. G 24 (1998) 1261


Masses of N~Z nuclei

• Neutron-proton pairing hamiltonian in non-degenerate shells:

• HF maps into the boson hamiltonian:

• HB describes masses of N~Z nuclei.

€

ˆ H B = a ˆ C 2 SU 4( )[ ] + b ˆ C 1 US 3( )[ ]

+ c1ˆ C 1 U 6( )[ ] + c2

ˆ C 2 U 6( )[ ] + d ˆ C 2 SOT 3( )[ ]

E. Baldini-Neto et al., Phys. Rev. C 65 (2002) 064303

€

ˆ H F = ε jˆ n j

j

∑ − g0ˆ S +

10 ⋅ ˆ S −10 − ′ g 0 ˆ S +

01 ⋅ ˆ S −01


Two-nucleon transfer

• Amplitude for two-nucleon transfer in the reaction A+aB+b:

• Nuclear-structure information contained in GN(L,S,J) which for L=0 transfer reduces to

€

ℑ→ =λ GN L,S,J( )KNLM Lkα ,kβ( )

N

∑

N.K. Glendenning, Direct Nuclear Reactions

€

GN L = 0,S = J( ) = 00N0;0 nl nl;0nl

∑ β nlTS

β nlTS = ΦB anl1/ 21/ 2

+ × anl1/ 21/ 2+

( )0TS( )

ΦA


Deuteron transfer• Overlap of uncorrelated pair:

• Bosons correspond to correlated pairs:

• Scale property:

P. Van Isacker et al., Phys. Rev. Lett. 94 (2005) 162502

€

nlTS = ΦB anl1/ 21/ 2

+ × anl1/ 21/ 2+

( )0TS( )

ΦA

€

nl anl1/ 21/ 2+ × anl1/ 21/ 2

+( )

0TS( )

nl

∑ ≡ ˆ S +TS → bTS

+

€

nlTS =

2l +1

α n' l ' 2l'+1n' l '

∑ΦB

ˆ S +TS ΦA


Deuteron transfer with bosons• Correspondence does not take

account of Pauli principle.• The following correspondence is shown to be

exact [in the Wigner limit]:– Even-even odd-odd

– Odd-odd even-even

€

ˆ S +TS → bTS

+

€

ΦBˆ S +

TS ΦA = 1

22Ω − Nb +1( ) Nb +1[ ]φB bTS

+ Nb[ ]φA

€

ΦBˆ S +

TS ΦA = 1

22Ω − Nb + 6( ) Nb +1[ ]φB bTS

+ Nb[ ]φA


Masses of pf-shell nuclei• Boson hamiltonian:

• Rms deviation is 306 (or 254) keV.• Parameter ratio: b/a5. €

ˆ H B = a ˆ C 2 SU 4( )[ ] + b ˆ C 1 US 3( )[ ] + c1ˆ C 1 U 6( )[ ] + c2

ˆ C 2 U 6( )[ ] + d ˆ C 2 SOT 3( )[ ]


Deuteron transfer in N=Z nuclei• Deuteron-transfer

intensity cT2 calculated

in sp-boson IBM based on SO(8).

• Ratio b/a fixed from masses in lower half of 28-50 shell.

€

cT2 = Nb +1[ ]φB bTS

+ Nb[ ]φA

2

Generalized pairing models, Saclay, June 2005 Generalized models of pairing in non-degenerate orbits J. Dukelsky, IEM, Madrid, Spain D.D. Warner, Daresbury,

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