Milano University and INFN The Niels Bohr Institute, Copenhagen · 2018. 11. 14. · t Present...

R.A. Broglia, A. Idini, E. Vigezzi

Milano University and INFN

The Niels Bohr Institute, Copenhagen

F. Barranco, G. Potel

Sevilla University

Two-nucleon transfer reactions

G. Baym, Fi+y years of BCS theory , Denver, March 2007

Nuclear Physics 33 (1962) 685--692; ~ North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

N O T E O N T H E T W O - N U C L E O N S T R I P P I N G R E A C T I O N

SHIRO YOSHIDA t Radiation Laboratory, University o/ Pittsburgh, Pittsburgh, Pennsylvania tt

Received 9 February 1962

A b s t r a c t : The magnitude of the two-nucleon stripping reactions is calculated using the pairing interaction model. The calculation also is applied to final states of collective type. For some types of reaction a collective enhancement of the reaction cross section is predicted.

Recently much attention has been directed to reactions like (6, d), (He 8, n), (t, p) (or their reverse reactions), for which the two-nucleon stripping process is believed to be important when the residual nucleus is excited at low energy. Many investigations of these reactions have been published l-e) ttt, but most theoretical investigations are concerned with the calculation of the angular distribution rather than of the magnitude of the cross section.

In the case of the one-nucleon stripping reaction the magnitude of the cross section is governed by the "spectroscopic factor", and it has been shown that the study of this factor is very important for nuclear spectroscopy 7, s). We expect that the two-nucleon stripping process also is useful for nuclear spectroscopy and that in some cases it may give new information which was not obtainable from the one-nucleon stripping process. Our at tempt in this note is to study the two-nucleon stripping (or pick-up) reaction from the viewpoint of nuclear structure and to try to find its usefulness in nuclear spectroscopy. The two-nucleon spectroscopic factor is calculated using the pairing interaction model, and this is applied also to the collective states. For some types of reaction collective enhancement is predicted, which is not found in the one-nucleon stripping but which is similar to that found in y-transitions and in inelastic scattering.

If the plane wave Born approximation is assumed, and the incident particle is assumed to be small compared to the target nucleus, then the differential cross section is given by 8)

da tzttt, kt 2Jr+IF(K)• I,~A(jLSlal2)iL_,x_,, d.Q 4:~ z~,4 kt 2 J t + l SLJ (1)

"F (2zl + 1 ) <2z,÷ 1 ) (ll0Z,01L0) j o 1 (,)R*,, fL (Q')' 2d" ,

t Present address: Insti tute for Nuclear Study, University of Tokyo, Tokyo, Japan. t t This work was supported b y the Office of Naval Research. t t t Only recent papers are listed in ref. 6); in these papers, howevcr, references to previous

papers can be found.

685

Pairing correlations enhance two-nucleon transfer cross sections

Form factor for the transfer of a J=0 pair

PHYSICS REPORTS (Review Section of Physics Letters) 199, No. 1(1991)1—72. North-Holland

TWO-NUCLEON TRANSFER REACTION MECHANISMS

Masamichi IGARASHI*Department of Physics, Tokyo Medical College, Shinjuku, Sinjuku-ku, Tokyo 160, Japan

Ken-ichi KUBODepartment of Physics, Tokyo Metropolitan University. Fukazawa. Setagaya-ku, Tokyo 158, Japan

and

Kohsuke YAGIInstitute of Physics and Tandem Accelerator Center, University of Tsukuba, Tsukuba, Ibaraki 305, Japan

Editor: G.E. Brown Received April 1990

Contents:

1. Introduction 3 3.1. The measurement of (p, t) analyzing powers and cross.1. Review of previous studies 3 sections 381.2. Aim of this work 6 3.2. Experimental results for (p t) analyzing powers and

2. Formulation and method of calculation 6 cross sections 392.1. The first- and second-order distorted wave Born ap- 3.3. Experimental studies of the (p, d) and (d, t) reactions 39

proximations 6 4. Results and discussions 402.2. The triton and deuteron wave functions 9 4.1. The one-step (p—d) process 402.3. The one-step process 14 4.2. The sequential transfer (p—d—t) process 492.4. The sequential transfer (p—d—t) process 25 4.3. Some properties of one- and two-step processes 562.5, The spectroscopic amplitude for the particle transfer 4.4. Summary and conclusions 65

reactions 32 Appendix A. Hard-core correction to the one-nucleon trans-2.6. The differential cross section and analyzing power 35 fer form factor 682.7. Why analyzing power? 37 References 69

3. Experimental study of (p 1) cross sections and analyzingpowers 38

*Deceased July 1988.

Single orders for this issue

PHYSICS REPORTS (Review Section of Physics Letters) 199, No. 1 (1991)1—72.

Copies of this issue may be obtained at the price given below. All orders should be sent directly to the Publisher. Orders must beaccompanied by check.

Single issue pnce Dfi. 54.00, postage included.

0 370-1573/91/$25.20 © 1991 — Elsevier Science Publishers B.V. (North-Holland)

Review of two-neutron transfer experiments; pairing rotations and pairing vibrations

2nd order DWBA: Sequential and simultaneous contributions

Coupled channel effects Semiclassical theory

INSTITUTE OF PHYSICS PUBLISHING REPORTS ON PROGRESS IN PHYSICS

Rep. Prog. Phys. 64 (2001) 1247–1337 PII: S0034-4885(01)98206-5

Pairing correlations of nucleons and multi-nucleontransfer between heavy nuclei

W von Oertzen1,2 and A Vitturi3

1 Hahn-Meitner-Institut Berlin, Glienicker Strasse 100, D-14109 Berlin, Germany2 Fachbereich Physik, Freie Universitat Berlin, Germany3 Dipartimento di Fisica, Universita di Padova, and INFN, Padova, Italy

Received 20 April 2001Published 28 September 2001Online at stacks.iop.org/RoPP/64/1247

Abstract

This review presents the status of the field of two-nucleon and multi-nucleon transfer reactionsinduced by heavy ions. The role of the pairing interaction in nuclei as a finite quantum systemis illustrated. These reactions serve as a unique tool for the study of the dynamical aspectsof pairing correlations in nuclei in particular for the cases of nuclear superfluidity. The newpossibilities offered by the development of experimental techniques, e.g. the large gamma-ray detectors combined with charged particle detectors are presented, which allows the studyof the enhanced pair transfer between well selected states, and the problem of the expectedquenching of pairing correlation at high spin. Further, the new possibilities offered by theadvent of radioactive beam facilities are discussed.

0034-4885/01/101247+91$90.00 © 2001 IOP Publishing Ltd Printed in the UK 1247

A. Vi?uri, Kyoto, DCEN 2011

h?p://www2.yukawa.kyoto-‐u.ac.jp/~ykis2011/dcen/slide/workshop4/vi?uri.pdf

50 years of nuclear BCS ,eds. R.A. Broglia and V. Zeelvinsky, World ScienUfic, to be published

Erratum: Calculation of the Transition from Pairing Vibrational to Pairing RotationalRegimes between Magic Nuclei 100Sn and 132Sn via Two-Nucleon Transfer Reactions

[Phys. Rev. Lett. 107, 092501 (2011)]

G. Potel, F. Barranco, F. Marini, A. Idini, E. Vigezzi, R. A. Broglia(Dated: Received 31 May 2011; published 22 August 2011)

PACS numbers: 25.40.Hs, 25.70.Hi, 74.20.Fg, 74.50.+r

Figure 1 should be substituted by

This figure is identical to the published one, except for the value of the absolute cross section associated with the108Sn(p,t)106Sn reaction (vertical number in the N = 58− 56 box), which now reads 2316 instead of 1035 (the unitsbeing, as stated in the published and correct caption, µb)

EN =1

2J (N −N0)2

Pairing rotaUonal band in superfluid Un isotopes

g.s.

p.v.

•  Static deformation of the pair field • Rotational-like spectrum formed by a sequence of ground states of even-N

systems

Pairing Vibrations (Nobel Lecture, Ben R. Mottelson, 1975)

• Near closed shell nuclei (like 208Pb) no static deformation of pair field •  Vibrational-like excitation spectrum. •  Enhanced pair-addition and pair-removal cross-sections seen in (t,p) and

(p,t) reactions (indicated by arrows).

E−

λ(N−

N0)

(MeV

)

y x 2x

2y y x

Two-‐neutron transfer is the specific probe of pairing correlaUons

α0 =< BCS|P+|BCS >=

ν>0

UνVν ∼ ∆/G

In superfluid nuclei,

In normal nuclei, α0 = 0

dσ/dΩ(A, g.s.− > A+ 2, g.s) ∼ α20

dσ/dΩ ∼ < (α− α0)2 >= [< 0|P+P |0 > + < 0|PP+|0 >]/2

A wise opinion (John Schiffer)

Simple ways of treating data can allow us to extract essential structural information, even if reaction theorists are not completely happy.

One hopes that a new generation will be able to start from what was Learned, and build on it, and not get bogged down with the fascinating sophistries of reaction theories. We need not have to rediscover all the blind alleys (of two-step processes, coupled channels … etc) that obscured simple underlying information

All experiments should be feasible, exciting and interpretable, but they most rarely enjoy more than two of the three properties, frequently failing the interpretability test (V. Telegdi)

Some recent 2-‐nucleon transfer experiment t(30Mg,32Mg)p K. Wimmer et al. ,PRL 105 (2011) 252501 p(11Li, 9Li)t I. Tanihata et al., PRL 100 (2008) 192502 p(8He,6He)t N. Keeley et al., PLB 646 (2007) 222

121Sb(p,t)119Sb P. Guazzoni et al., J.Phys.G 34 (2007)2665 120Sn(p,t)118Sn P. Guazzoni et al., PRC 78 (2008)064608 134Ba(p,t)132Ba S. Pascu et al., PRC 81 (2010)014304 65Cu(6He ,4He)67Cu A. Cha?erjee et al., PRL 101 (2008) 032701 9Be (18O,16O)11Be M. Cavallaro et al., J. Phys G Conf. Ser. 312(2011)092020 40Ca(96Zr,94Zr)42Ca L. Corradi et al. , PRC 84(2011)034603

“Standard procedure”: first order DWBA

112Sn(p,t)110Sn reaction, Ep = 26 MeV(Guazzoni et al. PRC 74 054605 (2006))with first order DWBA one obtains theangular distribution of the angulardifferential cross sectionabsolute normalization ⇒ relative crosssections

Give up absolute cross sections!!

Kyoto, October 24th, 2011 slide 10/22

SystemaUc high-‐quality A+2Sn(p,t)ASn angular distribuUons for 112 < A <124: Guazzoni et al., PRC 60 054603 (1999); 74 054605 (2006); ecc.

Fits obtained using normalizaUon factors and assuming dineutron transfer (one step); angular distribuUons are well described.

112Sn(p,t)110Sn

0+ P. GUAZZONI et al. PHYSICAL REVIEW C 74, 054605 (2006)

G.S.

2.309

2.573

2.742

3.183

104

103

102

101

102

101

102

101

102

101

100

10-10 20 6040

≈

≈

≈

≈

θc.m. (deg)

dσ/d

Ω (µ

b/sr

)

FIG. 2. Differential cross sections for the excitation of 0+ statesby the 112Sn(p, t)110Sn reaction. The dots represent the experimentaldata, the solid lines the theoretical estimates obtained with semimicro-scopic DWBA calculations. The energies attributed to the observedlevels are those given in the present work.

3.059 MeV. The adopted level scheme [14] gives a 0+ levelat 3060 keV, on the basis of the (p, t) reaction measuredby Blankert [8]. This attribution is incompatible with ourfindings because it would imply the typical L = 0 patternin the differential cross section angular distribution of a verysteeply rising cross section at very small reaction angles andsharp minimum at the detector angle of about 20. This is inconflict with the typical L = 4 transfer shape of the angulardistribution that we observed. The J π value assigned in thepresent work is 4+.

3.083 MeV. No level is given at this energy by the adoptedlevel scheme [14]. The angular distribution is quite accurately

1.212

2.545

2.857

2.965

3.083

3.421

3.751

3.812

103

102

101

101

101

100

101

100

101

101

100

101

100

101

100

10-10 20 40 60

≈

≈

≈

≈

≈

≈

θ c.m. (deg)

dσ/d

Ω (µ

b/sr

)

FIG. 3. Differential cross sections for the excitation of 2+ statesby the 112Sn(p, t)110Sn reaction. The dots represent the experimentaldata, the solid lines the theoretical estimates obtained with semimicro-scopic DWBA calculations. The energies attributed to the observedlevels are those given in the present work.

reproduced by considering an L = 2 transfer. The presentassignment is J π = 2+.

3.183 MeV. The adopted level scheme [14] reports a levelat an energy of 3182.8 keV with tentative spin and parityassignments (2+, 3+, 4+, 5+), deduced from 110Sb β+ decay[2]. In our measurement, the angular distribution displaysa typical L = 0 shape, well reproduced by the DWBAcalculation. We assign J π = 0+ to this level, which mostprobably does not coincide with the level reported in NDS [14]at 3182.8 keV.

3.421 MeV. At this energy no level is reported in the adoptedlevel scheme [14]. In our experiment this level is weakly

054605-6

2+

Note change of scale of cross secUons and of angular distribuUons between 0+ and 2+

2-‐body spect. ampl.

1-‐body spect. ampl.

Ujf Vjf

Ujf x Vjf

Both successive and simultaneous terms are coherent

T =

jijfBjf (B,A)Bji(a, b)T

(1)(jf , ji) +Cjf (B,F )Cjf (F,A) Cji(a, f)Cji(f, b)

× [T (2)succ(jf , ji)− T (2)

NO(jf , ji)]

120 U Gotz ci al., Reaction mechanism of two.nucleon transfer between heavy ions

‘ // \ -/ /

4 r© ~/ /\ r2,5I / I

rib I / r2f /

/ r2b ~/ r?a

/ r~AB

// / rf~/ /

I / F/ I /

r~ r0~

b r5~

Fig. I . Set of coordinates. The labels a, A, f, F, b. B refer to the centers-off-mass of the respective systems.

Assuming that the intrinsic wavefunctions I~are known, we derive the equations that should

be satisfied by the wavefunctions u~of relative motion. We multiply the SchrOdinger equation

(H—E)’I’=0 (2.13)from the left by one of the intrinsic wavefunctions ~ ~ ~ 21 and integrate over all argu-ments by fixing the coordinate raA, rhB, rf1F2 and rf2F~,respectively. We then obtain the set ofbasic coupled equations for the wavefunctions uci(raA), uO(rbB), u.1.~2(rf1F2) and u~2(rf2F1): *

+ ~1aA~c* — E(c~))u(r~~)

= ~ ‘1’~A U~(TaA)_~(~sIHl5~JUp) _- \/‘) ~< lH_EI-~.~2u.1~~(2.14a)

(ThB +(VbB)O — E(13))l1O(rbB)

= --- ~ (~l~IcI~’)UO’(rbB) -- ~ (rT~I[I_El4~u0>-- 2~(~I~lH—El~.©2u~12),(2.14b)

ci

(Tf1F2 + ~VfF — E(y)) u.~ (rfIF2)

~ <~i2 V~1.l ~i2 (rt~F2) + ~ ~i2 H--El72i ~2i

~‘i2 ~~I2 ~2t

— ~2 H—El ~0u0)— ~ IH—El ~0u~). (2. l4c)

* The coupled equations for rearrangement collisions are derived in ref. [12] from a variational principle. An alternative derivationis given by Mittleman [13] and by Coz [14], based on a projection operator method.

CalculaUon of absolute two-‐nucleon transfer cross secUon by finite-‐range DWBA calculaUon

B.F. Bayman and J. Chen, Phys. Rev. C 26 (1982) 1509 G. Potel et al., arXiv:0906.4298

Two particle transfer in second order DWBA

Some details of the calculation of the differential cross section for the

two–neutron transfer (1H(

11Li,

9Li)

3H) reaction

Simultaneous transfer

T (1)(ji , jf ) = 2

σ1σ2

drfFdrb1drA2[Ψ

jf (rA1, σ1)Ψjf (rA2, σ2)]

0∗0 χ(−)∗

bB (rbB)

× v(rb1)[Ψji (rb1, σ1)Ψ

ji (rb2, σ2)]00χ

(+)aA (raA)

Berkeley, August 9, 2010 slide 4/8


‘ // \ -/ /

4 r© ~/ /\ r2,5I / I

rib I / r2f /

/ r2b ~/ r?a

/ r~AB

// / rf~/ /

I / F/ I /

r~ r0~

b r5~





+ ~1aA~c* — E(c~))u(r~~)




ci

(Tf1F2 + ~VfF — E(y)) u.~ (rfIF2)

~ <~i2 V~1.l ~i2 (rt~F2) + ~ ~i2 H--El72i ~2i

~‘i2 ~~I2 ~2t

— ~2 H—El ~0u0)— ~ IH—El ~0u~). (2. l4c)


Simultaneous transfer


‘ // \ -/ /

4 r© ~/ /\ r2,5I / I

rib I / r2f /

/ r2b ~/ r?a

/ r~AB

// / rf~/ /

I / F/ I /

r~ r0~

b r5~





+ ~1aA~c* — E(c~))u(r~~)




ci

(Tf1F2 + ~VfF — E(y)) u.~ (rfIF2)

~ <~i2 V~1.l ~i2 (rt~F2) + ~ ~i2 H--El72i ~2i

~‘i2 ~~I2 ~2t

— ~2 H—El ~0u0)— ~ IH—El ~0u~). (2. l4c)


Successive transfer




11Li,

9Li)

3H) reaction

Successive transfer

T (2)succ(ji , jf ) = 2

K ,M

σ1σ2σ

1σ2

drfFdrb1drA2[Ψ


0∗0

× χ(−)∗bB (rbB)v(rb1)[Ψ

jf (rA2, σ2)Ψji (rb1, σ1)]

KM

×

drfFdrb1drA2G (rfF , rfF )[Ψjf (rA2, σ2)Ψ

ji (rb1, σ1)]

KM

× 2µfF

2v(rf 2)[Ψ

ji (rA2, σ2)Ψ

ji (rb1, σ1)]

00χ

(+)aA (raA)



‘ // \ -/ /

4 r© ~/ /\ r2,5I / I

rib I / r2f /

/ r2b ~/ r?a

/ r~AB

// / rf~/ /

I / F/ I /

r~ r0~

b r5~





+ ~1aA~c* — E(c~))u(r~~)




ci

(Tf1F2 + ~VfF — E(y)) u.~ (rfIF2)

~ <~i2 V~1.l ~i2 (rt~F2) + ~ ~i2 H--El72i ~2i

~‘i2 ~~I2 ~2t

— ~2 H—El ~0u0)— ~ IH—El ~0u~). (2. l4c)


Non orthogonal contribution




11Li,

9Li)

3H) reaction

Non–orthogonality term

T (2)NO

(ji , jf ) = 2

K ,M

σ1σ2σ

1σ2

drfFdrb1drA2[Ψ


0∗0

× χ(−)∗bB

(rbB)v(rb1)[Ψjf (rA2, σ2)Ψ

ji (rb1, σ1)]KM

×

drb1drA2[Ψjf (rA2, σ

2)Ψ

ji (rb1, σ1)]

KM

× [Ψji (rA2, σ2)Ψ

ji (rb1, σ1)]

00χ

(+)aA

(raA)


Ingredients of the calculation

Structure input for, e.g., the 112Sn(p,t)110Sn reaction:

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15−50

−45

−40

−35

−30

−25

−20

−15

−10

−5

0

r (fm)

E (M

eV)

real partimaginary part

optical potential

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

r (fm)

2d5/23s1/21g7/22d3/21h11/2

single particle wave functions

0 0.5 1 1.5 2 2.5 3−300

−250

−200

−150

−100

−50

0

r (fm)

E (M

eV)

proton−neutron potential

plus the Bj spectroscopic amplitudes needed to define the two–neutronwavefunction:

Φ(r1, σ1, r2, σ2) =

j

Bjψj(r1, σ1)ψ

j(r2, σ2)0

0


Elements of the calculaUon for the 112Sn(p,t)110Sn reacUon

Two neutron spectroscopic amplitude:

Bj = (j + 1/2)1/2U (A−2)j V (A)

j

Bj =< Ψ(A−2)|Pj |ΨA >

T =

j

Bj

T (1)(j) + T (2)

succ(j) + T (2)NO

(j)

BCS

122Sn(p,t)120Sn Elab= 26 MeV

Successive

Simult.+Non orth.

Non orthogonal

Simultaneous

ASn(p,t)A−2Sn, results

0 10 20 30 40 50 60 70

102

103

104

θCM

dσ/d

Ω(µ

b/sr

)

102

103

104

dσ/d

Ω(µ

b/sr

)

0 10 20 30 40 50 60 70θCM

0 10 20 30 40 50 60 70θCM

102

103

104

dσ/d

Ω(µ

b/sr

)

101

0 10 20 30 40 50 60 70θCM

102

103

104

dσ/d

Ω(µ

b/sr

)0 10 20 30 40 50 60 70

θCM

101

102

103

104

dσ/d

Ω(µ

b/sr

)

0 20 40 60 80θCM

102

103

104

dσ/d

Ω(µ

b/sr

)

105

101

Comparison with the experimentaldata available so far for superfluid tinisotopesPotel et al., PRL 107, 092501 (2011)


G. Potel et al., PRL107 (2011) 092501

•  BCS wavefuncUons reproducing experimental pairing gaps

•  Tang-‐Herndon wavefuncUons for the triton

•  OpUcal potenUal fi?ed by Guazzoni et al

4 E. Vigezzi, F. Barranco, R.A. Broglia, A. Idini and G. Potel

0 10 20 30 40 50 60 70100

101

102

103

104

!CM

d"/d#

(µ b

/sr)

experimentBCSShell Modelpure (d5/2)2 configuration

112Sn(p,t)110Sn, Elab=26 MeV112Sn(p,t)110Sn!

0 10 20 30 40 50 60 70100

101

102

103

104

!CM

d"/d#

(µ b

/sr)

totalsuccsimnonsim+non

108Sn(p,t)106Sn, Elab=26 MeV

Fig. 2. (left) The absolute differential cross section for the reaction 112Sn (p,t)110Sn, calculatedmaking use of the BCS amplitudes (solid curve), already shown in Fig. 1, is compared to theresult obtained making use of shell-model amplitudes.18) Also shown is the result calculatedassuming a pure (d5/2)

2 configuration. (right) The theoretical differential cross section forthe reaction 108Sn (p,t)106Sn is decomposed into successive, simultaneous and non orthogonalcontributions. Also shown is the sum of non ortohogonal and successive amplitudes.

102 106 110 114 118 122 126 130500

1000

1500

2000

2500

3000

3500

4000

4500

5000

A

!(µ

b)

10 12 14 16 18 20 22 24 26 28 30

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

Elab (MeV)

b)a)

100 110 120 13010

20

30

40

A

E max

Fig. 3. (a) The squares show the experimental values of the cross section integrated in the range0o < θ < 80o for the two-neutron transfer reaction ASn (p,t)A−2Sn, (cf. Fig. 1), compared withthe theoretical value (triangles). The solid curve shows the integrated cross sections along the Snisotopic chain, calculated at the matching (optimum) bombarding energy where it reaches themaximum value for a given mass number A. (b) Dependence of the integrated cross section ofthe matching reaction 112Sn (p,t)110Sn on the laboratory bombarding energy in the laboratory.Also shown in the inset is the optimum energy as a function of mass number.

4 E. Vigezzi, F. Barranco, R.A. Broglia, A. Idini and G. Potel

0 10 20 30 40 50 60 70100

101

102

103

104

!CM

d"/d#

(µ b

/sr)

experimentBCSShell Modelpure (d5/2)2 configuration

112Sn(p,t)110Sn, Elab=26 MeV112Sn(p,t)110Sn!

0 10 20 30 40 50 60 70100

101

102

103

104

!CM

d"/d#

(µ b

/sr)

totalsuccsimnonsim+non

108Sn(p,t)106Sn, Elab=26 MeV

Fig. 2. (left) The absolute differential cross section for the reaction 112Sn (p,t)110Sn, calculatedmaking use of the BCS amplitudes (solid curve), already shown in Fig. 1, is compared to theresult obtained making use of shell-model amplitudes.18) Also shown is the result calculatedassuming a pure (d5/2)

2 configuration. (right) The theoretical differential cross section forthe reaction 108Sn (p,t)106Sn is decomposed into successive, simultaneous and non orthogonalcontributions. Also shown is the sum of non ortohogonal and successive amplitudes.

102 106 110 114 118 122 126 130500

1000

1500

2000

2500

3000

3500

4000

4500

5000

A

!(µ

b)

10 12 14 16 18 20 22 24 26 28 30

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

Elab (MeV)

b)a)

100 110 120 13010

20

30

40

A

E max

Fig. 3. (a) The squares show the experimental values of the cross section integrated in the range0o < θ < 80o for the two-neutron transfer reaction ASn (p,t)A−2Sn, (cf. Fig. 1), compared withthe theoretical value (triangles). The solid curve shows the integrated cross sections along the Snisotopic chain, calculated at the matching (optimum) bombarding energy where it reaches themaximum value for a given mass number A. (b) Dependence of the integrated cross section ofthe matching reaction 112Sn (p,t)110Sn on the laboratory bombarding energy in the laboratory.Also shown in the inset is the optimum energy as a function of mass number.

The ground state is a coherent state well described by BCS calculations constrained to reproduce the phenomenological pairing gap

-10 -8 -6E (MeV)

0

0.5

1

B(jj;

0)

SHELL MODELBCS

d5/2 d3/2s1/2g7/2 h11/2s1/2g7/2d5/2 d3/2

h11/2h11/2

112Sn -> 110Sn

Shell model : P. Guazzoni et al., PRC 74 054605 (2006

Some challenges: Check pair vibrational scheme around new magic nuclei Transfer from heavy, weakly bound nuclei Effects of core excitation Transfer from halo nuclei Continuum and coupled channel effects Pair transfer as probe of shape coexistence

Pair vibraUons around 100Sn and 132Sn in the harmonic approximaUon

parametrized according to [17], its strength adjusted so asto reproduce the intermediate channel deuteron bindingenergy.

With the help of a Saxon-Wood potential ([18] p. 239)and a !GPyP pairing interaction single-particle levelswere calculated and BCS occupation parameters U!V!

were obtained adjusting G so as to reproduce the (fourpoint) odd-even mass difference for the nuclei of massnumber A and A! 2. Making use of the associatedtwo-particle transfer spectroscopic amplitudes B! ¼ðj! þ 1=2Þ1=2U!ðA! 2ÞV!ðAÞ [10], and standard opticalparameters [11–15], the absolute differential cross sectionsassociated with the reactions ASnðp; tÞA!2Snðg:s:Þ (102 &A & 130) were calculated. In all cases, successive, simul-taneous, and nonorthogonality contributions (postrepre-sentation) to the cross section were considered (see[19–21] and references therein, see also [22]). We displayin Fig. 2 the theoretical predictions in comparison withthe experimental data for all of the six mass numbers(A ¼ 124, 122, 120, 118, 116, and 112) for which obser-vations have been carried out ([11–15]). Theory provides,without any free parameters, an account of the absolute

value of all measured differential cross sections withinlimits well below the (estimated) 15% (systematic) experi-mental error, and almost within statistical errors.In keeping with the results displayed in this figure, we

present below predictions concerning the expected pairingvibrational spectrum of the closed shell nuclei 100

50 Sn50,13250 Sn82, and the associated absolute differential cross sec-tions. Within the harmonic approximation [8,10], the two-phonon pairing vibrational 0þ states of 132Sn and 100Sn arepredicted at an excitation energy of 6.6 and 7.1 MeV,respectively [see Figs. 3 (I) and (II)]. At variance withthe superfluid (pairing rotational) case (see Fig. 1), theseexcited 0þ state are predicted to be populated with a crosssection comparable to or larger than that associated withthe g:s: ! g:s: transition, a direct consequence of the cleardistinction which can be operated between occupied(V2 ' 1, U2 ' 0) and empty (V2 ' 0, U2 ' 1) states

FIG. 2. Absolute calculated cross section predictions in com-parison with the experimental results of [11–15].

FIG. 3. (I) The solid bold lines represent the values of theexpression E ¼ Bð132SnÞ ! BðASnNÞ ! 4:75ð82! NÞ MeV (seealso caption to Fig. 1) corresponding to the pair addition (a), pairremoval (r), and two-phonon pairing vibrational state (E ¼6:6 MeV) of 132Sn. The absolute differential cross sectionsassociated with the reaction 134Snðp; tÞ132Snðg:s:Þ (pair addition,a) and 132Snðp; tÞ130Snðg:s:Þ (pair removal, r) at Ec:m: ¼ 20 MeVand 26 MeV, respectively, are also displayed. At the bottom weshow the ratio of cross section associated with the reactions134Snðp; tÞ132Snðpv; 6:6 MeVÞ and 134Snðp; tÞ132Snðg:s:Þ, i.e., therelative cross section of the 132Sn, 0þ two-phonon pairing vibra-tional state. (II) The same as above but for 100Sn. In this caseE ¼ Bð100SnÞ ! BðASnNÞ ! 14:5ð50! NÞ MeV.

PRL 107, 092501 (2011) P HY S I CA L R EV I EW LE T T E R Sweek ending

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092501-3

parametrized according to [17], its strength adjusted so asto reproduce the intermediate channel deuteron bindingenergy.

With the help of a Saxon-Wood potential ([18] p. 239)and a !GPyP pairing interaction single-particle levelswere calculated and BCS occupation parameters U!V!

were obtained adjusting G so as to reproduce the (fourpoint) odd-even mass difference for the nuclei of massnumber A and A! 2. Making use of the associatedtwo-particle transfer spectroscopic amplitudes B! ¼ðj! þ 1=2Þ1=2U!ðA! 2ÞV!ðAÞ [10], and standard opticalparameters [11–15], the absolute differential cross sectionsassociated with the reactions ASnðp; tÞA!2Snðg:s:Þ (102 &A & 130) were calculated. In all cases, successive, simul-taneous, and nonorthogonality contributions (postrepre-sentation) to the cross section were considered (see[19–21] and references therein, see also [22]). We displayin Fig. 2 the theoretical predictions in comparison withthe experimental data for all of the six mass numbers(A ¼ 124, 122, 120, 118, 116, and 112) for which obser-vations have been carried out ([11–15]). Theory provides,without any free parameters, an account of the absolute

value of all measured differential cross sections withinlimits well below the (estimated) 15% (systematic) experi-mental error, and almost within statistical errors.In keeping with the results displayed in this figure, we

present below predictions concerning the expected pairingvibrational spectrum of the closed shell nuclei 100

50 Sn50,13250 Sn82, and the associated absolute differential cross sec-tions. Within the harmonic approximation [8,10], the two-phonon pairing vibrational 0þ states of 132Sn and 100Sn arepredicted at an excitation energy of 6.6 and 7.1 MeV,respectively [see Figs. 3 (I) and (II)]. At variance withthe superfluid (pairing rotational) case (see Fig. 1), theseexcited 0þ state are predicted to be populated with a crosssection comparable to or larger than that associated withthe g:s: ! g:s: transition, a direct consequence of the cleardistinction which can be operated between occupied(V2 ' 1, U2 ' 0) and empty (V2 ' 0, U2 ' 1) states

FIG. 2. Absolute calculated cross section predictions in com-parison with the experimental results of [11–15].

FIG. 3. (I) The solid bold lines represent the values of theexpression E ¼ Bð132SnÞ ! BðASnNÞ ! 4:75ð82! NÞ MeV (seealso caption to Fig. 1) corresponding to the pair addition (a), pairremoval (r), and two-phonon pairing vibrational state (E ¼6:6 MeV) of 132Sn. The absolute differential cross sectionsassociated with the reaction 134Snðp; tÞ132Snðg:s:Þ (pair addition,a) and 132Snðp; tÞ130Snðg:s:Þ (pair removal, r) at Ec:m: ¼ 20 MeVand 26 MeV, respectively, are also displayed. At the bottom weshow the ratio of cross section associated with the reactions134Snðp; tÞ132Snðpv; 6:6 MeVÞ and 134Snðp; tÞ132Snðg:s:Þ, i.e., therelative cross section of the 132Sn, 0þ two-phonon pairing vibra-tional state. (II) The same as above but for 100Sn. In this caseE ¼ Bð100SnÞ ! BðASnNÞ ! 14:5ð50! NÞ MeV.

PRL 107, 092501 (2011) P HY S I CA L R EV I EW LE T T E R Sweek ending

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092501-3

G. Potel et al., PRL107 (2011) 092501

E. PLLUMBI et al. PHYSICAL REVIEW C 83, 034613 (2011)

0 20 40 60 80 100 120θc.m. (deg)

10-8

10-7

10-6

10-5

10-4

diff

. cro

ss s

ectio

n (a

rb. u

nits

)

x=1, Ep=30 MeV

x=1, Ep=35 MeV

x=0.35, Ep=30 MeV

x=0.35, Ep=35 MeV

20 40 60 80 100 120θc.m. (deg)

x=1, Ep=15 MeV

x=0.35, Ep=15 MeV

gs0

+2

136Sn(p,t)

134Sn

FIG. 4. (Color online) Left: Cross sections for the reaction136Sn(p, t)134Sn obtained in the cases of a pure surface interaction(x = 1) and a mixed interaction (x = 0.35). The transition to theground state of the final nucleus is considered and two valuesof the incident proton energy are selected, Ep = 30 and 35 MeV.Right: Cross sections for the reaction 136Sn(p, t)134Sn obtained in thecases of a pure surface interaction (x = 1) and a mixed interaction(x = 0.35). The transition to the first excited 0+ state of the finalnucleus is considered and the value of 15 MeV is selected for theincident proton energy.

thus expected to be typically more important. Owing to this, wehave considered also the pair-transfer reaction 136Sn(p, t)134Snwhere a more neutron-rich nucleus is involved. With thenext-generation facilities, it is expected that beams of veryneutron-rich tin isotopes such as 134,136Sn will be producedwith sufficiently high intensity for performing two-nucleontransfer experiments. We have performed the same kind ofcalculations as for the case 124Sn(p, t)122Sn (Ep ranging from15 to 35 MeV). In this case, we have checked the sensitivityof the calculations with respect to the choice of the opticalpotential in the entrance channel. We have found similar resultsusing the optical potential of Ref. [34].

For some values of the proton incident energy, the locationof the diffraction minima is not the same when different pairinginteractions are used. We show in Fig. 4 only the relevantcases, i.e., those corresponding to the incident energies forwhich some discrepancies have been found in the angularprofiles associated to different pairing interactions. The twocases of mixed interactions (x = 0.35 and x = 0.65) donot actually significantly differ one from the other whilediscrepancies are found between the pure surface case andthe mixed cases. As an illustration, we show for the mixedinteraction only the case x = 0.35. We compare in Fig. 4 theresults obtained for x = 1 and x = 0.35. In the left panel,the gs transition is described for the two values of incidentenergy Ep = 30 and 35 MeV. For Ep = 30 MeV, the curvesto compare are the solid black (x = 1) and the dashed green(x = 0.35). For Ep = 35 MeV, the curves to compare arethe solid red (x = 1) and the dashed blue (x = 0.35). In theright panel, the 0+

2 transition is described for Ep = 15 MeV.One observes in both panels that the profiles of the cross

15 20 25 30 35Ep(MeV)

1

2

4

8

CS gs

/CS 0+ 2

x=1x=0.35x=0.65

136Sn(p,t)

134Sn

FIG. 5. (Color online) Ratios of the cross sections associated tothe g.s. and to the 0+

2 transitions at different proton energies for thereaction 136Sn(p, t)134Sn.

sections corresponding to x = 1 and x = 0.35 differ at largeangles (!c.m. > 70 degrees). We are aware that measurementsare more difficult at large angles because the correspondingcross sections are very low. Nevertheless, this result indicatesthat very neutron-rich Sn isotopes may be interesting cases toanalyze. On the basis of this first indication, we continue ourinvestigation for 136Sn and we show in Fig. 5 the ratios of thecross sections associated to the g.s. and to the 0+

2 transitionsat different proton energies. Even if absolute cross sectionscannot be calculated within the present reaction model, theratios of the cross sections related to the g.s. and the 0+

2transitions are meaningful quantities to analyze. These ratiosare proportional to the ratios of the transition probabilitiesassociated to the two transitions and that the proportionalityfactor is the same independently of the pairing interaction.The comparison of the ratios obtained with different pairinginteractions can thus provide interesting predictions about thesensibility of the cross sections to the choice of the pairinginteraction. It can be seen that differences exist among thethree sets of results and they are more important at thelowest energy of 15 MeV, that represents the case wherethe reaction takes place mostly in the surface region of thenucleus. Hence, we suggest very neutron-rich Sn isotopesand proton energies around 15 MeV as favorable cases forfuture (p, t) or (t, p) pair-transfer experiments that can providea deeper insight into the surface/volume character of thepairing interaction. Performing (t, p) reaction measurementsin inverse kinematics is quite more challenging than (p, t), butsuch reactions would allow one to populate different states ofSn isotopes that may also represent very favorable cases forpairing studies.

IV. SUMMARY AND PERSPECTIVES

We have evaluated (p, t) two-neutron transfer-reactioncross sections in the zero-range DWBA approximation for

034613-4

Pllumbi et al., PRC 83 (2010) 034613

Two –neutron transfer cross secUons for neutron-‐rich Sn isotopes (zero-‐range approximaUon)

BCS wavefunctions are enough to calculate 2n transfer between ground states of superfluid nuclei. BUT 2n transfer reactions to collective surface vibrational states may reveal the existence of shape fluctuations in the condensate

A A

A-‐2 A-‐2

λ λ

aj1aj2)λ

aj1aj2)λ

+ Ground state correlaUons

R.A. Broglia, C. Riedel, T. Udagawa, NPA169 (1971) 225

r

Probing 11Li halo-‐neutrons correla4ons via (p,t) reac4on

Introduction

We will try to draw information about the halo structure of11

Li from the

reactions1H(

11Li,

9Li)

3H and

1H(

11Li,

9Li

∗(2.69 MeV))

3H (I. Tanihata et

al., Phys. Rev. Lett. 100, 192502 (2008))

Schematic depiction of11

Li First excited state of9Li


F. Barranco et al., EPJ A11 (2001) 385

Vnp Vnp

p

11Li 9Li 10Li

d t

3 3 3

n1,n2

an1,n2 [ψn1(r1)ψn2(r2)]00

G. Potel et al., PRL 105 (2010) 172502

11Li(p,t)9Li

g.s. 3/2-‐

First exc. 1/2-‐

InterpretaUon of direct reacUons: ex of 8He+p @ 15.6 MeV/nucleon

E405s –GANIL-‐MUST 8He +p @ 15.6 MeV/n

CRC: PLB 646(’07)

8He(p,t)6He (0+)

8He(p,t)6He (2+)

8He(p,d)

8He(p,p)

Coupled reacUon channel (CRC) calculaUons needed: Cf 8He+p Analysis à N. Keeley, SPhN [now: univ of Varsaw] F. Skaza et al., PLB 619. 82 (’05) ; PRC 73, 044301 (’06) N. Keeley et al., PLB 646, 222(’07)

Spectroscopic factors C2S from (dσ/dΩ)theo % (dσ/dΩ)exp

The transferred angular momentum Lt indicates Jπ

V. Lapoux IRFU/SPhN November 17th 2011 [email protected]

Two neutron transfer reaction to 32Mg

coexistence of spherical and deformed states

deformed 2p−2h configuration becomes ground state in32

Mg

where is the excited 0+

state?

predictions for the 0+2

state in32

Mg:

between 1.5 and 3 MeV

E. Caurier et al., NPA 693 374, T. Otsuka, EPJA 20 69,

R. Rodriguez-Guzmán et al., NPA 709 201

this state has not been observed so far

similar particle-hole structure:

populate the excited 0+

state by a two neutron transfer reaction

large overlap of wavefunctions

large spectroscopic factor for transfer

→ t(30

Mg,p)32

Mg

Kathrin Wimmer ECT* workshop, Trento

The radioactive tritium target

Challenge: radioactive target

first experiment with a Tritium target and a radioactive heavy ion beam

use of Tritium loaded Titanium foil

0.5 mg/cm2

Ti foil

atomic ratio3H/Ti ≈ 1.5

corresponds to 40 µg/cm2

Tritium

activity 10 GBq

Problem:

Fusion reactions with the carrier material Ti produce protons

→ lower beam energy, below Fusion barrier


(TREX at ISOLDE)

The excitation energy of the 0+2

state

E. Caurier et al., NPA 693 374, T. Otsuka, EPJA 20 69, R. Rodriguez-Guzmán et al., NPA 709 201

0+2

state lower than predicted

long lifetime several ns

similar cross sections

simple picture

ground state 2p−2hexcited state 0p−0h


Ground state configurationGround state:

two particle - two hole configuration

simple assumption: neutron pair in 1f7/2

Occupation numbers for the ground state

within the Monte Carlo SM

SDPF-M effective interaction

T. Otsuka et al., Prog. Part. Nucl. Phys. 47 319

J. R. Terry et al., Phys. Rev. C 77 014316

simple assumption (1f7/2)2amplitude

can not reproduce the data


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