Sonja Orrigo Correlation effects and continuum spectroscopy in light exotic nuclei Unbound Nuclei Workshop, Pisa, Italy, November 3-5, 2008

ContentsContents Physical scenario: light exotic nuclei Correlations effects and continuum spectroscopy  DCP correlations: Fano resonances  Experimental results on 11 Be and 15 C via CEX reactions  Pairing correlations in the continuum  Transfer reactions to unbound states: 9 Li(d,p) 10 Li Summary and conclusions

Why correlation effects in light exotic nuclei? light n-rich nuclei Dissolution of shell structures Influence of correlation dynamics Peculiar conditions: Large charge asymmetry Weak binding of valence n Proximity of s.p. continuum low density (halos) isovector interaction open quantum systems

Continuum spectroscopy of light exotic nuclei Two topics: 15 C: weakly-bound S n = 1218 keV effects due to the DCP correlation dynamicsFano Resonances 10 Li: n-unbound by 25 keV effects due to pairing correlations continuum spectroscopy by one-neutron transfer s.p. excitations Dynamical Core Polarization (DCP) correlations Pairing correlations Important to investigate their effects in the low-energy continuum

ContentsContents Physical scenario: light exotic nuclei Correlations effects and continuum spectroscopy  DCP correlations: Fano resonances  Experimental results on 11 Be and 15 C via CEX reactions  Pairing correlations in the continuum  Transfer reactions to unbound states: 9 Li(d,p) 10 Li Summary and conclusions

Fano Resonances Fano Resonancesa new continuum excitation mode Fano Resonances are investigated as a new continuum excitation mode in exotic nuclei Atomic physics H.Feshbach, Ann. of Phys. 5 p. 357 (1958), Ann. of Phys. 19 p. 287 (1962), Ann. of Phys. 43 p. 410 (1967) F.H.Mies, Phys. Rev. 175 p. 164 (1968) A.F.Starace, Phys. Rev. B 5 p (1972) A.K.Bhatia and A.Temkin, Phys. Rev. A 29 p (1984) J.P.Connerade and A.M.Lane, Rep. on Progr. in Phys. 51 p (1988) Hadron physics N.E.Ligterink, PiN Newslett. 16 p. 400 nucl-th/ (2002) Solid-state physics S.Glutsch, Phys. Rev. B 66 p (2002) Nuclear physics G.Baur and H.Lenske, Nucl. Phys. A 282 p. 201 (1977) General phenomenon observed in many different areas of physics

Fano interference Originally detected in atomic spectra 1960’s, Fano: first model for atomic states excited in the inelastic scattering e - -atoms Typical for interacting many-body systems at all scales ! Fano interference quantum-mechanical interaction between discrete and continuous configurations asymmetric line shape

Fano Resonances in nuclear physics BSEC: narrow resonances in the continuum (E x > S n ) DCP model: BSEC as quasi-bound core-excited configurations Experimental signature of the DCP correlations G.Baur and H.Lenske, Nucl. Phys. A 282(1977)201; H.Lenske et al., Jour. Progr. Part. Nucl. Phys. 46(2001)187 Predicted theoretically by Mahaux and Weidenmüller (1969) C.Mahaux and H.A.Weidenmüller, Shell Model Approach to Nuclear Reactions, North-Holland, Amsterdam (1969) 1 st observed BSEC (1980): 13 C (stable), E x = MeV (J  = 3/2 + ) H.Fuchs et al., Nucl. Phys. A 343(1980)133 Bound States Embedded in the Continuum (BSEC) And in exotic nuclei?

In exotic nuclei n-dripline nuclei: easily polarizable core BSEC at low-energy C-isotopes: presence of low-energy 2 + core states good candidates Importance of a systematic study H. Lenske, from HFB & QRPA calculations

Fano Resonances in exotic nuclei F.Cappuzzello, S.E.A. Orrigo et al., EuroPhys. Lett. 65 p. 766 (2004) S.E.A. Orrigo et al., Proceedings Varenna 122 p. 147 (2003)  8.50* 8.50  7.30  7.30*  6.77 ]  6.4  g.s.  g.s.* 0.77  lab = 14°, 55 keV/ch C Excitation energy [MeV] Counts DCP regime Single particle regime SnSn  0.77* Fano interference: BSEC – s.p. continuum  8.50*  L =8° 109 keV/ch 15 C Excitation energy [MeV] Counts  N( 7 Li, 7 Be) MeV

F.Cappuzzello, S.E.A. Orrigo et al., EuroPhys. Lett. 65 p. 766 (2004)S.E.A. Orrigo et al., Proceedings Varenna 122 p. 147 (2003); a) level density natural parity transitions C Excitation energy [MeV] d QRPA (  ) [MeV  1 ] b) level density unnatural parity transitions C Excitation energy [MeV] d QRPA (  ) [MeV  1 ] s. p. SnSn SnSn Results of microscopic QRPA calculations E x [MeV]  [keV] 0.00    0.06 <  0.06 <  0.06 < 140  8.50* 8.50  7.30  7.30*  6.77 ]  6.4  g.s.  g.s.* 0.77 DCP regime Single particle regime SnSn  0.77* C Excitation energy [MeV] Counts  lab = 14° 55 keV/ch Strength well reproduced for single particle transitions (1/2 + g.s., 5/2 + state at 0.77 MeV) Observed fragmentation for E x > 2 MeV not reproduced 15 C: Fano Resonances Strong competition of mean-field and correlation dynamics mean-field approaches are no longer appropriate Enhanced correlation effects (Dynamical Core Polarization DCP) new excitation modes involving core-excited configurations (BSEC)

11 B( 7 Li, 7 Be) MeV a) level density natural parity transitions Be Excitation energy [MeV] d QRPA (  ) [MeV  1 ] b) level density unnatural parity transitions Be Excitation energy [MeV] d QRPA (  ) [MeV  1 ] s. p. QRPA calculations F.Cappuzzello, H.Lenske et al., Phys. Lett B 516(2001)21 7 Be detected with the IPN-Orsay Split-Pole magnetic spectrometer DCP regime Single particle Be Excitation energy [MeV] SnSn Counts Strength well reproduced for single particle transitions (1/2 + g.s., 1/2 - state at 0.32 MeV and 5/2 + state at 1.77 MeV) Observed fragmentation for E x > 2 MeV not reproduced

The QPC model H.Lenske, J. Phys. G: Nucl. Part. Phys. 24 (1998) 1429 H.Lenske, C.M.Keil, N.Tsoneva, Progr. in Part. and Nucl. Phys. 53 (2004) 153 Quasiparticle-core coupling (QPC) model (Bohr & Mottelson) DCP correlations described by coupling 1QP to the core-excited configurations 1QP 3QP Core excitations: 2QP exc. given by QRPA Strength fragmentation not reproduced by QRPA DCP effects QPC Hamiltonian of the odd-mass system: 3QP states 1QP states coupled by V 13

To study resonances in the low-energy continuum and their line shapes Theoretical model QPC eigenstates: S.E.A.Orrigo, H.Lenske et al., Phys. Lett. B 633 (2006) 469 s.p. mixing 1QP 3QP E < 0 bound states E  > 0 continuum BSEC (E C ) Bound core-excited states (E – E C < 0)

Theoretical model Coupling of a single particle elastic channel to closed core-excited channels By projecting the Schrödinger equation onto the 1-QP and 3-QP components N coupled equations 1QP Channel 1 3QP Channels i = 2, …, N Ch. 11QP continuum Ch. i = 2, …, N3QP states S.E.A.Orrigo, H.Lenske et al., Phys. Lett. B 633 (2006) 469

Numerical methods The coupled channels problem is solved in coordinate space N coupled equations for the radial wave functions 1QP Channel 1 (open) 3QP Channels i = 2, …, N (closed) r < R A r >> R A Potentials U i from HFB calculations Transition form factors from QRPA calculations & data S.E.A.Orrigo, H.Lenske et al., Phys. Lett. B 633 (2006) 469

Numerical methods r < R A i = 1, …, N 1) Internal w.f. by solving the NxN eigenvalue problem r >> R A 2) Asymptotic w.f. i = 1, …, N Matching2N equations with complex coefficients b m, C 1i i = 1, …, N S.E.A.Orrigo, H.Lenske et al., Phys. Lett. B 633 (2006) 469

Results for 15 C Elastic scattering cross section Analytic calculation for 2 ch. A single excited state of the 14 C core: E C = MeV U i from HFB V 13 is the only free parameter S.E.A.Orrigo, H.Lenske et al., Phys. Lett. B 633 (2006) s-wave p- d- V 13  0  11 [mb] 15 C excitation energy [MeV] Fano interference s-, p-, d-waves

Results for 15 C 15 C theo. (  11 d-wave) V 13 is the only free parameter 15 C exp. from ( 7 Li, 7 Be) Full 5-channels calculation 4 14 C states: E C (J  ) = 6.094(1 – ), 6.728(3 – ), 7.012(2 + ), 8.317(2 + ) MeV U i from HFB V 13 weighted by  (i) of 14 C( ,  ’) JCJC Qualitative comparison: E th. = 6.67, 7.36, 7.70, 8.92 MeV  th. = 66, 80, 141, 85 keV E exp. = (6.77, 7.30, 8.50)  0.06 MeV  exp. ≤ 160, 70, 140 keV   V 13 affects  of the resonances (here V 13 = 5 MeV) S.E.A.Orrigo, H.Lenske et al., Phys. Lett. B 633 (2006) 469

Results for 17 C and 19 C State parameters by QRPA V 13 is the only free parameter (here 5 MeV) 18 C states: E C (J  ) = 1.620(2 + ), 2.967(4 + ), 3.313(2 + ), 5.502(1 – ) MeV 16 C states: E C (J  ) = 1.766(2 + ), 3.986(2 + ), 4.142(4 + ) MeV d-wave  11 [mb] 17 C excitation energy [MeV] 17 C S n = 0.73 MeV d-wave  11 [mb] 19 C excitation energy [MeV] 19 C S n = 0.16 MeV BSEC structures move towards lower energies with increasing the neutron excess Increased effect of the correlations Systematic study of the evolution of the phenomenon when going towards more n-rich nuclei S.E.A. Orrigo, H.Lenske et al., Proceedings INPC07, Tokyo

The ( 7 Li, 7 Be) CEX reaction Structural properties: Single particle isovector excitations BSEC and Fano resonances in the continuum Reaction dynamics: One-step / two-step contributions Spin transfer probabilities N = 1 7 He N = 2 11 Be N = 3 15 C N = 4 19 O N = 5 23 Ne N = 6 27 Mg … N  + 3 n MAGNEX IPN-Orsay References: S.E.A. Orrigo et al., Core excited Fano-resonances in exotic nuclei, Phis.Lett. B 633(2006)469 F.Cappuzzello, S.E.A. Orrigo et al., Excited states of 15 C, EuroPhys.Lett. 65(2004)766 F.Cappuzzello et al., Analysis of the 11 B( 7 Li, 7 Be) 11 Be reaction at 57 MeV in a Microscopic Approach, Nucl.Phys. A 739(2004)30 S.E.A. Orrigo et al., Spectroscopy of 15 C by ( 7 Li, 7 Be) Charge Exchange Reaction, Proc. “10 th Int. Conf. on Nuclear Reaction Mechanisms”, Varenna, Italy, 122(2003)147 C.Nociforo et al., Investigation of light neutron-rich nuclei via the ( 7 Li, 7 Be) reaction, Acta Physica Polonica, B 34(2003)2387 F.Cappuzzello et al., Excited states of 11 Be, Phys.Lett B 516(2001)21

Maximum magnetic rigidity1.8 T m Solid angle 51 msr E max /E min 1.7 Total energy resolution (target 1 mm 2, 90% of full acceptance)  1000 Mass resolution 250 A.Cunsolo et al., NIMA 481 (2002) 48 A.Cunsolo et al., NIMA 484 (2002) 56 E < 30 AMeV 2 < A < 40 E < 25 AMeV 40 < A < 93 Upper bent limits

19 F( 7 Li, 7 Be) MeV X foc [m] Counts  = 50 msr Energy byte = ± 27% PRELIMINARY g.s. 96 keV 19 O E/  E ~ 1000  lab = 7° ° 19.8 keV/ch S n = 3.9 MeV

ContentsContents Physical scenario: light exotic nuclei Correlations effects and continuum spectroscopy  DCP correlations: Fano resonances  Experimental results on 11 Be and 15 C via CEX reactions  Pairing correlations in the continuum  Transfer reactions to unbound states: 9 Li(d,p) 10 Li Summary and conclusions Single particle dynamics: the relevant energy scale is S n   Stable nuclei: S n ~10MeV → static MF in the p-h channel + paring for the p-p correlations Weakly-bound n-rich nuclei: S n ~few keV-MeV → pairing correlations in the p-h channel are also important

Theory of Pairing in the Continuum Extended MF approach for pairing in weakly-bound or unbound nuclei 2x2 coupled channel problem described by the Gorkov equations H.Lenske, F.Hofmann, C.M.Keil, J. Progr. Part. Nucl. Phys. 46(2001)187 Particle channel (open) Hole channel (closed) Similarity between the Pairing and DCP approaches Particle-stable system: q <0 e  <0, hole w.f. v  decaying exponentially for r»R A In the continuum e  + >0, particle w.f. u  like a scattering wave:

Theory of Pairing in the Continuum The same type of effects is produced by any types of correlations (e.g., DCP → Fano) Partial wave elastic scattering cross section: Resonances in resonances in  ℓ j Observables involving the states u  will show a characteristic energy dependence (e.g., transfer cross sections through S  (E)) Relationship scattering observables – pairing strength Continuum level density: spectral distribution of particle strength per energy E = Density of states S.E.A. Orrigo and H. Lenske, submitted to PLB (2008)

Neutron s.p. spectral functions in 9 Li Effects of the dynamical correlations (particle-hole coupling) due to the pairing field The widths of the hole distributions (E<0) are due to the bound-continuum coupling The deeper-lying s-wave levels are coupled more efficiently to the particle continuum The 5/2 + d-wave strength is lowered into the bound state sector (intruder component) A small amount of 1/2 + and 3/2 + strengths is above the p½ peak Dramatic change in dynamics at the n-dripline: the level ordering is not determined by simple MF E<0 hole sector E>0 particle sector S.E.A. Orrigo and H. Lenske, submitted to PLB (2008)

Partial wave cross sections for elastic scattering 9 Li+n Comparison: full HFB Gorkov-pairing – bare MF calculations Pairing gives an attractive self-energy in the p-wave channels → 1/2 – and 3/2 – resonances at very low energy (E<<3MeV ~ threshold for DCP correlations) Slight attraction in the 1/2 + channel and repulsion for the d-waves The structure results are used as input for transfer reaction calculations S.E.A. Orrigo and H. Lenske, submitted to PLB (2008)

Continuum spectroscopy of 10 Li by transfer reactions Transfer reactions well established tool for structural studies of bound and exotic nuclei Weakly-bound final state: prevalence of small momentum components the cross section maximum is at much lower incident energies H. Lenske and G. Schrieder, Eur. Phys. J 2 (1998) 41 Single-nucleon transfer reactions as a tool for continuum spectroscopy in exotic nuclei Study of the low-energy s.p. resonances in unbound systems also Method based on a DWBA approach. Main innovations: to treat the case of unbound final states and to calculate the double differential cross section for one-neutron transfer The model is applied to the 9 Li(d,p) 10 Li reaction to explore the structure of 10 Li 10 Li is neutron-unbound by 25 ± 15 keV

Why 10 Li? Crucial for the comprehension of the structure of 11 Li as a three body system ( 11 Li is a Borromean 2-n-halo nucleus) Information on the n- 9 Li interaction, important for the theoretical models of 11 Li Interest in the structure of 10 Li itself: low-lying states are not yet well known Ground state: p-wave or s 1/2 virtual state at the n+ 9 Li threshold? 4 states are expected at low energy with J  = 1 −, 2 − and 1 +, 2 + (neutron in 2s 1/2 or 1p 1/2  unpaired 1p 3/2 proton of the 9 Li core) Two resonances seen at E x ~ 250 and 500 keV; several resonances at higher E x D.R. Tilley et al., Nucl. Phys. A 745 (2004) 155 and refs. therein Interest in the 9 Li(d,p) 10 Li reaction in inverse kinematics: experiments at 20 AMeV ( P. Santi at al., Phys. Rev. C 67 (2003) ) resonance at ~ 350 keV with  ~ 300 keV experiments at 2.36 AMeV ( H.B. Jeppesen et al., Phys. Lett. B 642 (2006) 449 ) resonance at ~ 380 keV with  ~ 200 keV

Theory for transfer reactions Transfer reaction A(a,b)B (a = b+x, B = A+x) in a DWBA approach [Satchler] Optical model Hamiltonians and DW Schroedinger equations: H  = H A + H a + K  + U  + V  (  = a+A) ; H  = H B + H b + K  + U  + V   = b+B) (K  + U  – E  )   (±) (r , k  ) = 0 (  =  ) The optical model wave functions   (±) (r , k  ) describe the elastic scattering determined by the optical potentials U  at the channel energies E  = E – e A – e a Hinterberger Menet d-potentials p-potentials NPA111(1968) PRC4(1971) Residual interaction V  (post) chosen according to effective self-energy (full HFB Gorkov- pairing), it reproduces B.E., r M, r C of 9 Li

for a fixed energy In the post representation for a stripping reaction in which x is transferred from a to B: F  =  J B M B s b m b | V | J A M A s a m a  = =  jl (S jl ) ½ R jl (r xA )(l s m  – m | j  )(s b s m b m a – m b | s a m a )(J A j M A M B – M A | J B M B )D(r xb ) Y l m* ( ) spectroscopic amplitude radial wave function for the transferred particle x projectile internal function times x-b interaction potential Zero-Range Approximation: D(r xb ) = D 0  (r x – r b ) T  = D 0 (S  ) ½   (–) | R(r x )Y * ( )  (r x – r b ) |   (+)  which contains dynamics and structure information xA r ˆ Theory for transfer reactions First order DWBA transition amplitude: T  =   (–) | F  |   (+)  F  =  bB|V|aA  form factor

Double differential cross section for one-nucleon transfer to unbound final states Momentum distribution (Dynamics: Fourier transform of the wave function) Spectral function (Structure: probability per energy for finding the particle in state ℓ j at energy E) Theory for transfer reactions to unbound states Transfer into the continuum: The B = A+x final states are unbound against the reemission of the nucleon x (e x < 0) the overlap form factor oscillates at large distances the DWBA radial integrals converge very slowly Vincent and Fortune: powerful method of contour integration in the complex radius plane to overcome the convergence problem C.M. Vincent and H.T. Fortune, PRC2(1970); PRC7(1973); PRC8(1973) S.E.A. Orrigo and H.Lenske, submitted to PLB (2008)

p 1/2 resonance at E R = 400 keVMainly a potential resonance p 3/2 resonance at E R = 850 keVCoupled-channels pairing resonance New feature essential to describe data obtained by adding a polarization repulsive surface potential (acting in  E~250keV around E R ) to reproduce the full HFB Gorkov-pairing results The theoretical results include the experimental energy resolution  FWHM ~250keV Good agreement with data (shape and resonances position) Spectroscopy of 10 Li = 9 Li+n at the continuum threshold E [MeV] d( 9 Li, 10 Li)p  cm = [98°,134°] Total data 1/2+ 3/2- 1/2- 5/2+ 3/2+ d   [mb/MeV] dE Angle-integrated cross section d( 9 Li, AMeV S.E.A. Orrigo and H. Lenske, submitted to PLB (2008) H.B. Jeppesen et al., PLB 642(2006)449 E R ~ 380 keV,  ~ 200 keV

Angular distributions Good agreement with data (no scaling) The p 1/2 -wave is dominant As expected, transfer is favoured at low incident energies: 20 AMeV (MSU exp.) →  transfer at E R (p 1/2 ) is lowered by a factor of 26 The measurement of angular distributions is important to identify the 10 Li states (ℓ  values) Angular distributions d( 9 Li, AMeV S.E.A. Orrigo and H. Lenske, submitted to PLB (2008) H.B. Jeppesen et al., PLB 642(2006)449 E R ~ 380 keV,  ~ 200 keV Total data 1/2+ 3/2- 1/2- 5/2+ 3/2+ d( 9 Li, 10 Li)p  cm [deg.] d  [mb/sr] d 

Transfer 9 Li(d,p) AMeV, before folding,  cm =[98°,134°] Transfer 9 Li(d,p) AMeV, before folding,  cm =[98°,134°] Elastic scattering n+ 9 Li (p-wave) Elastic scattering n+ 9 Li (p-wave) S.E.A. Orrigo and H. Lenske, submitted to PLB (2008) p1/2 p3/2 d  R  [mb/MeV] dE E [MeV] p1/2 p3/2 E [MeV] d  E  [mb/MeV] dE Same structure for  elastic and  transfer : a physical resonance appears in both  ℓj (E) [°] → S ℓj (E) Access to the spectroscopic information by transfer However, in  transfer there can be not physical resonances also, due only to the reaction dynamics part E [MeV]

Spectral distribution ↔ properties of n- 9 Li interaction S.E.A. Orrigo and H. Lenske, submitted to PLB (2008) Variations by ±10% of the potential diffuseness c and radius R I Vol = (AMeV)·fm 3 = constant p 1/2 -wave: E R,  strongly sensitive to c (asymptotic shape of potential, halo tail) Scattering length a s = 1.69 fm (c+10%)=1.68 fm, (c-10%)=1.72 fm (R+10%)=2.21 fm, (R-10%)=1.06 fm s 1/2 -wave: larger sensitivity to R (no centrifugal barrier) Information on the residual n- 9 Li interaction c, R c + 10% c – 10% R + 10% R – 10% E [MeV] Elastic scattering n+ 9 Li (p 1/2 -wave) Elastic scattering n+ 9 Li (p 1/2 -wave) d  E  [mb/MeV] dE

SummarySummary Fano resonances can be expected to be of particular importance for the continuum dynamics of exotic nuclei The coupled channels model extends the QPC into the continuum: the interference between open 1-QP and closed 3-QP ch. gives sharp and asymmetric resonances (  →V 13 ) The calculations performed for 15 C, 17 C, 19 C show increased effects of correlations Exp. evidence of DCP correlations in the 15 C spectra, qualitatively reproduced by theoretical calculations, and in the 11 Be and 19 O spectra Transfer reactions are a powerful tool to do continuum spectroscopy in exotic nuclei Innovations of the DWBA approach: to treat unbound final states and to calculate d 2  /d  dE Calculations performed for the d( 9 Li, 10 Li)p reaction at E Li = 2.36 and 20 AMeV 10 Li continuum: p 1/2 -resonance at ~400 keV and p 3/2 -pairing resonance at ~850 keV in very good agreement with experimental data Same behaviour of  elastic and  transfer : same structure S  (E) Correlation: spectral distributions ↔ n- 9 Li interaction (sensitivity to the halo tail)

Analogy:Configuration Mixing due to 15 C continuum = n+ 14 C, BSEC = n+ 14 C*core polarization (DCP) 10 Li continuum = n+ 9 Li unbound, (particle-hole)pairing correlations (MF-level) ☺ Thank you for your attention ☺ Pairing in unbound nuclear states explored in terms of an extended MF approach Paring effects may introduce pronounced structures and shifts in the low-energy continuum of all the channels Configuration mixing acts at the MF level, but mechanisms similar to the mixing due to dynamical correlations SummarySummary