Two neutron transfer in Sn isotopes

Transcription

1 Kyoto, October 24th, 2011 slide 1/21 Two neutron transfer in Sn isotopes Grégory Potel Aguilar (Sevilla) Andrea Idini (Milano) Francisco Barranco Paulano (Sevilla) Enrico Vigezzi (Milano) Francesco Marini (Milano) Ricardo A. Broglia (Milano) Kyoto, October 24th, 2011

2 Introduction and Outline Kyoto, October 24th, 2011 slide 2/21 This talk will be devoted to two particle transfer reactions as the specific probe to study pairing correlations. Emphasis will be made in the connection between structure aspects and the resulting two particle transfer cross sections. Outline: Reaction mechanism : two particle transfer in second order DWBA A Sn(p, t) A 2 Sn reactions: transition between pairing vibrational (closed shell) to pairing rotational (superfluid) regimes in the tin isotopic chain.

3 Two nucleon transfer reactions Kyoto, October 24th, 2011 slide 3/21 Two valence nucleons go from core b of nucleus a to core A of nucleus B Probing two particle correlations. Investigating structure properties such as pairing and superfluidity in a finite fermion system (the atomic nucleus). Get absolute values as well as the angular distribution for the cross sections in second order DWBA. A(a, b)b Examples: 112 Sn(p,t) 110 Sn 1 H( 11 Li, 9 Li) 3 H

4 First Part Kyoto, October 24th, 2011 slide 4/21 Reaction mechanism: second order DWBA

5 Elements of the calculation yoto, October 24th, 2011 slide 5/21 Ψ a ( r 1, r 2 ), Ψ B ( r 1, r 2 ): internal wave functions of the transferred nucleons in each nucleus χ(r): distorted wave describing the ( relative motion ) in the optical P 2 potential U(R) = V (R) + iw (R) R 2µ + U(R) χ(r) = E CM χ(r) V A,V a : mean field potentials of the two nuclei

6 Elements of the calculation yoto, October 24th, 2011 slide 5/21 Ψ a ( r 1, r 2 ), Ψ B ( r 1, r 2 ): internal wave functions of the transferred nucleons in each nucleus χ(r): distorted wave describing the ( relative motion ) in the optical P 2 potential U(R) = V (R) + iw (R) R 2µ + U(R) χ(r) = E CM χ(r) V A,V a : mean field potentials of the two nuclei V A (V a ) is the interaction potential that transfers the nucleons from one nucleus to the other in the prior (post) representation

7 Elements of the calculation Kyoto, October 24th, 2011 slide 5/21 Ψ a ( r 1, r 2 ), Ψ B ( r 1, r 2 ): internal wave functions of the transferred nucleons in each nucleus χ(r): distorted wave describing the ( relative motion ) in the optical P 2 potential U(R) = V (R) + iw (R) R 2µ + U(R) χ(r) = E CM χ(r) V A,V a : mean field potentials of the two nuclei V A (V a ) is the interaction potential that transfers the nucleons from one nucleus to the other in the prior (post) representation it is a single particle potential!!

8 Simultaneous and Successive contributions α = φ a (ξ b, r 1, r 2 ) φ A (ξ A )χ aa (r aa ) β = φ b (ξ b )φ B (ξ A, r 1, r 2 ) χ bb (r bb ) H a φ a = E a φ a H A φ A = E A φ A (T aa + U aa )χ aa = E aa χ aa H b φ b = E b φ b H B φ B = E B φ B (T bb + U bb )χ bb = E bb χ bb. yoto, October 24th, 2011 slide 6/21

9 Simultaneous and Successive contributions α = φ a (ξ b, r 1, r 2 ) φ A (ξ A )χ aa (r aa ) β = φ b (ξ b )φ B (ξ A, r 1, r 2 ) χ bb (r bb ) H a φ a = E a φ a H A φ A = E A φ A (T aa + U aa )χ aa = E aa χ aa H b φ b = E b φ b H B φ B = E B φ B (T bb + U bb )χ bb = E bb χ bb. yoto, October 24th, 2011 slide 6/21

10 Simultaneous and Successive contributions α = φ a (ξ b, r 1, r 2 ) φ A (ξ A )χ aa (r aa ) β = φ b (ξ b )φ B (ξ A, r 1, r 2 ) χ bb (r bb ) H a φ a = E a φ a H A φ A = E A φ A (T aa + U aa )χ aa = E aa χ aa H b φ b = E b φ b H B φ B = E B φ B (T bb + U bb )χ bb = E bb χ bb. yoto, October 24th, 2011 slide 6/21

11 Simultaneous and Successive contributions α = φ a (ξ b, r 1, r 2 ) φ A (ξ A )χ aa (r aa ) β = φ b (ξ b )φ B (ξ A, r 1, r 2 ) χ bb (r bb ) H a φ a = E a φ a H A φ A = E A φ A (T aa + U aa )χ aa = E aa χ aa H b φ b = E b φ b H B φ B = E B φ B (T bb + U bb )χ bb = E bb χ bb. yoto, October 24th, 2011 slide 6/21

12 Simultaneous and Successive contributions α = φ a (ξ b, r 1, r 2 ) φ A (ξ A )χ aa (r aa ) β = φ b (ξ b )φ B (ξ A, r 1, r 2 ) χ bb (r bb ) H a φ a = E a φ a H A φ A = E A φ A (T aa + U aa )χ aa = E aa χ aa H b φ b = E b φ b H B φ B = E B φ B (T bb + U bb )χ bb = E bb χ bb. yoto, October 24th, 2011 slide 6/21

13 Simultaneous and Successive contributions α = φ a (ξ b, r 1, r 2 ) φ A (ξ A )χ aa (r aa ) β = φ b (ξ b )φ B (ξ A, r 1, r 2 ) χ bb (r bb ) H a φ a = E a φ a H A φ A = E A φ A (T aa + U aa )χ aa = E aa χ aa H b φ b = E b φ b H B φ B = E B φ B (T bb + U bb )χ bb = E bb χ bb. yoto, October 24th, 2011 slide 6/21

14 Simultaneous and Successive contributions α = φ a (ξ b, r 1, r 2 ) φ A (ξ A )χ aa (r aa ) β = φ b (ξ b )φ B (ξ A, r 1, r 2 ) χ bb (r bb ) H a φ a = E a φ a H A φ A = E A φ A (T aa + U aa )χ aa = E aa χ aa H b φ b = E b φ b H B φ B = E B φ B (T bb + U bb )χ bb = E bb χ bb. yoto, October 24th, 2011 slide 6/21

15 Simultaneous and Successive contributions α = φ a (ξ b, r 1, r 2 ) φ A (ξ A )χ aa (r aa ) β = φ b (ξ b )φ B (ξ A, r 1, r 2 ) χ bb (r bb ) H a φ a = E a φ a H A φ A = E A φ A (T aa + U aa )χ aa = E aa χ aa H b φ b = E b φ b H B φ B = E B φ B (T bb + U bb )χ bb = E bb χ bb. yoto, October 24th, 2011 slide 6/21

16 Two particle transfer in second order DWBA Kyoto, October 24th, 2011 slide 7/21 Some details of the calculation of the differential cross section for two nucleon transfer reactions T 2NT = ( ) B jf B ji T (1) (j i, j f ) + T succ(j (2) i, j f ) T (2) NO (j i, j f ) j f j i dσ dω = µ iµ f (4π 2 ) 2 k f k i T 2NT 2 Simultaneous transfer T (1) (j i, j f ) = 2 σ 1 σ 2 dr ff dr b1 dr A2 [Ψ j f (r A1, σ 1 )Ψ j f (r A2, σ 2 )] 0 0 χ ( ) bb (r bb) v(r b1 )[Ψ j i (r b1, σ 1 )Ψ j i (r b2, σ 2 )] Λ µχ (+) aa (r aa)

17 Two particle transfer in second order DWBA Some details of the calculation of the differential cross section for two nucleon transfer reactions T (2) T 2NT = ( ) B jf B ji T (1) (j i, j f ) + T succ(j (2) i, j f ) T (2) NO (j i, j f ) j f j i succ(j i, j f ) = 2 K,M σ 1 σ 2 σ 1 σ 2 Successive transfer dr ff dr b1 dr A2 [Ψ j f (r A1, σ 1 )Ψ j f (r A2, σ 2 )] 0 0 χ ( ) bb (r bb)v(r b1 )[Ψ j f (r A2, σ 2 )Ψ j i (r b1, σ 1 )] K M dr ff dr b1 dr A2 G(r ff, r ff )[Ψj f (r A2, σ 2)Ψ j i (r b1, σ 1)] K M 2µ ff 2 v(r f 2 )[Ψj i (r b2, σ 2)Ψ j i (r b1, σ 1)] Λ µχ (+) aa (r aa ) Kyoto, October 24th, 2011 slide 7/21

18 Two particle transfer in second order DWBA Kyoto, October 24th, 2011 slide 7/21 Some details of the calculation of the differential cross section for two nucleon transfer reactions T (2) T 2NT = ( ) B jf B ji T (1) (j i, j f ) + T succ(j (2) i, j f ) T (2) NO (j i, j f ) j f j i NO (j i, j f ) = 2 K,M Non orthogonality term σ 1 σ 2 σ 1 σ 2 dr ff dr b1 dr A2 [Ψ j f (r A1, σ 1 )Ψ j f (r A2, σ 2 )] 0 0 χ ( ) bb (r bb)v(r b1 )[Ψ j f (r A2, σ 2 )Ψ j i (r b1, σ 1 )] K M dr b1 dr A2 [Ψj f (r A2, σ 2)Ψ j i (r b1, σ 1)] K M [Ψ j i (r b2, σ 2)Ψ j i (r b1, σ 1)] Λ µχ (+) aa (r aa )

19 Cancellation of simultaneous and non-orthogonal contributions Kyoto, October 24th, 2011 slide 8/21 very schematically, the first order (simultaneous) contribution is T (1) = β V α, while the second order contribution can be separated in a successive and a non-orthogonality term T (2) = T (2) succ + T (2) NO = γ β V γ G γ V α γ β γ γ V α. If we sum over a complete basis of intermediate states γ, we can apply the closure condition and T (2) NO exactly cancels T (1) the transition potential being single particle, two-nucleon transfer is a second order process.

20 Ingredients of the calculation Kyoto, October 24th, 2011 slide 9/21 Structure input for, e.g., the 112 Sn(p,t) 110 Sn reaction: E (MeV) real part imaginary part d 5/2 3s 1/2 1g 7/2 2d 3/2 1h 11/2 single particle wave functions E (MeV) proton neutron potential optical potential r (fm) r (fm) r (fm) plus the B j spectroscopic amplitudes needed to define the two neutron wavefunction: Φ(r 1, σ 1, r 2, σ 2 ) = j B j [ ψ j (r 1, σ 1 )ψ j (r 2, σ 2 ) ] 0 0 (1)

21 Standard procedure : first order DWBA Kyoto, October 24th, 2011 slide 10/ Sn(p,t) 110 Sn reaction, E p = 26 MeV (Guazzoni et al. PRC (2006)) with first order DWBA one obtains the angular distribution of the angular differential cross section

22 Standard procedure : first order DWBA Kyoto, October 24th, 2011 slide 10/ Sn(p,t) 110 Sn reaction, E p = 26 MeV (Guazzoni et al. PRC (2006)) with first order DWBA one obtains the angular distribution of the angular differential cross section absolute normalization relative cross sections

23 Standard procedure : first order DWBA Kyoto, October 24th, 2011 slide 10/ Sn(p,t) 110 Sn reaction, E p = 26 MeV (Guazzoni et al. PRC (2006)) with first order DWBA one obtains the angular distribution of the angular differential cross section absolute normalization relative cross sections Give up absolute cross sections!!

24 Early results with second order DWBA Kyoto, October 24th, 2011 slide 11/21 Götz, Ichimura, Broglia and Winther, Phys. Rep. 16 (1975) Igarashi, Kubo and Yagi, Phys. Rep. 199(1991)1 Bayman and Chen, PRC 26 (1982)1509, respectively

25 Examples of calculations yoto, October 24th, 2011 slide 12/ H( 11 Li, 9 Li(3/2 ;gs)) 3 H 1 H( 11 Li, 9 Li(1/2 ;2.69 MeV)) 3 H E lab =33MeV dσ/dω (mb/sr) dσ/dω (mb/sr) 10 1 E lab =33MeV dσ/dω(µ b/sr) θ CM Sn(p,t) 120 Sn, Elab =26MeV θ CM 0.4 dσ/dω (µ b/sr) θ CM Sn(p,t) 110 Sn, Elab =26 MeV θ CM 10 3 Pb( Pb( O, O, O) O) Pb Pb good results obtained for halo nuclei, population of excited states, superfluid nuclei, normal nuclei (pairing vibrations), heavy ion reactions... Potel et al., arxiv: Pb(t,p) 208 Pb, Elab =12 MeV dσ/dω (mb/sr) dσ/dω (µ b/sr) E lab =86 MeV E lab =86 MeV θ CM θ CM

26 Second Part Kyoto, October 24th, 2011 slide 13/21 From pairing vibrations to pairing rotations: Tin isotopic chain

27 112 Sn(p,t) 110 Sn, reaction mechanism Kyoto, October 24th, 2011 slide 14/21

28 112 Sn(p,t) 110 Sn, results Kyoto, October 24th, 2011 slide 15/ experiment pure (d 5/2 ) 2 configuration Shell Model BCS dσ/dω (µ b/sr) Sn(p,t) 110 Sn, Elab =26 MeV enhancement factor with respect to the transfer of uncorrelated neutrons: ε = θ CM Experimental data and shell model wavefunction from Guazzoni et al. PRC (2006)

29 Pairing correlations and two particle transfer cross sections Kyoto, October 24th, 2011 slide 16/21 two particle transfer transition strength: Ψ A 2 P Ψ A 2 In superfluid nuclei (open shell): Ψ A 2 P Ψ A BCS P BCS = α 0 = ν>0 U ν V ν = /G dσ/dω(a, g.s A + 2, g.s.) α 2 0 In normal nuclei (closed shell), = α 0 = 0: [ ] dσ/dω (α α 0 ) 2 = Ψ A P P Ψ A Ψ A PP Ψ A /2

30 A Sn(p,t) A 2 Sn, results 116 Sn(p,t) Sn(p,t) 110 Sn, Elab =26MeV Sn, Elab =26MeV 10 4 dσ/dω (µ b/sr) dσ/dω (µ b/sr) θ CM 118 Sn(p,t) 116 Sn, Elab =24.6MeV θ CM Sn(p,t) 118 Sn, Elab =21MeV dσ/dω (µ b/sr) θ CM 122 Sn(p,t) 120 Sn, Elab =26MeV dσ/dω (µ b/sr) θ CM 124 Sn(p,t) 122 Sn, Elab =25 MeV Comparison with the experimental data available so far for superfluid tin isotopes Potel et al., PRL 107, (2011) dσ/dω (µ b/sr) dσ/dω (µ b/sr) θ CM θ CM yoto, October 24th, 2011 slide 17/21

31 A Sn(p,t) A 2 Sn, superfluid isotopic chain Kyoto, October 24th, 2011 slide 18/ Integrated cross section for tin isotopes 2500 σ(µ b) experimental result theoretical prediction A

32 134,132 Sn(p,t) 132,130 Sn, pairing vibrations yoto, October 24th, 2011 slide 19/ Sn(p,t) 130 Sn and 134 Sn(p,t) 132 Sn reactions can probe the predicted pairing vibrations of the exotic double magic nucleus 132 Sn Potel et al., PRL 107, (2011)

33 Conclusions Kyoto, October 24th, 2011 slide 20/21 Second order DWBA has proven to be a valuable reaction formalism to obtain reliable absolute values,along with angular distributions, for the two particle transfer nuclear reactions angular differential cross sections. Two nucleon transfer reactions are an ideal tool to probe two neutrons correlations in nuclei. We have studied the transition between pairing vibrational (closed shell) to pairing rotational (superfluid) regimes in the tin isotopic chain. We hope that the predictions made for reactions with exotic beams such as 132 Sn(p,t) 130 Sn will stimulate future experiments!.

34 Thank You! Kyoto, October 24th, 2011 slide 21/21