Continuum States in Drip-line Oxygen isotopes

Continuum States in Drip-line Oxygen isotopes EFES-NSCL WORKSHOP, Feb. 4-6, 2010 @ MSU Department of Physics The University of Tokyo Koshiroh Tsukiyama *Collaborators : Takaharu Otsuka (Tokyo), Rintaro Fujimoto (Hitachi, Japan), Morten Hjorth-Jensen (Oslo) and Gaute Hagen (ORNL) 1

Motivation The study of nuclei far from stability. Understanding of the stability of matter itself. Potential implications for the synthesis of elements. How magic numbers and shells appear and evolve with increasing N/Z. => understand nuclear force at the extreme. Treatment of continuum => description of open quantum system requires the explicit coupling to the scattering continuum. 2

Why Oxygen isotopes The heaviest isotopes for which the drip line is well established. A doubly magic nature is seen for 22 O and 24 O The isotopes 25-28 O are all believed to be unstable toward neutron emission. ( 28 O is a doubly magic nucleus within the standard picture) Difficulty in many-body description within two-body Hamiltonian in sd-shell. Normal shell model in sd-shell fails to reproduce the drip line without tuning of effective Hamiltonian. Without fitting, typically MBPT, gives us bound 28 O. 3

Overview of my talk Description of open quantum system Real energy approach Complex energy approach CDCC Continuum Coupled Shell Model Gamow Shell Model Continuum Shell Model Complex Scaling Method empirical NN empirical NN starting from realistic NN K.T., T. Otsuka, R. Fujimoto, arxiv: 1001.0729 K.T., M. H-Jensen, G. Hagen Phys.Rev.C80:051301(R), (2009) 4

Continuum-Coupled Shell Model (CCSM) Extend normal shell model Continuum states are discretized in a spherical box. Effective interaction is determined keeping close relation to the normal shell model. Continuum spectra are obtained by taking the overlap with the doorway states of the reaction assumed. 6

Continuum-Coupled Shell Model (CCSM) Define effective Hamiltonian for CCSM in (1s1/2, id3/2) space Continuum states are generated as eigen states of H 0 r Given by hand, results are insensitive to the mu s CCSM Hamiltonian is defined so as to reproduce, in the filling configuration, the spectra of O isotopes obtained by SDPF-M (Utsuno et al., sucsessful Heff for sd and p3/2 f7/2 region ). No fit to experimental data. 7

Simulation of the knockout reaction @MSU (2009) 9 Be( 26 F, 24 O)X C. Hoffman, M. Thoennessen et al. 16 O less probable <== large s 1/2 -d 3/2 neutron gap 16 O -p -n 16 O 16 O bound nucleus doorway state excited states in 24 O 16 O ground state 1s 1/2 is well bound. Kanungo et al. (2009) 8

Low-lying States in 24 O 9 Be ( 26 F, 24 O)X Doorway state ==> excited states in 24 O Bound approximation: Normal shell model with CCSM Hamiltonian : NO continuum effect CCSM : With continuum effect No int. : With continuum effect but no residual interaction. Energy of emitted neutron Continuum effect is about 1 MeV or larger. No bound excited state. 1 + -2 + splitting by 2b interaction The splitting is in good agreement with experiments. 9

Low-lying states in exotic O isotopes Continuum effects Smaller paring gap SDPF-M bound approx. ==> Justification of CCSM Hamiltonian. Continuum effect is larger than 1 MeV. USD[ab] has continuum effect in its H eff by fit. 10

Radial density (w.f.) of continuum states in 24O 1 + 2 + Notable difference between 1 + and 2 + stares. The peak states in CCSM reproduce the behavior of far distance. 11

Effective phase shift and one-body reduction Can many-body resonance be described by effective one-body problem? CCSM Effective phase shift 22 O 22 O CCSM: continuum spectra are obtained by taking the overlap between the doorway state and CCSM eigen states. 23 O We define effective phase shift by introducing 1b reduction of CCSM wave function. Then one gets phase shift which is used obtain the cross section. 12

Effective phase shift and one-body reduction CCSM(doorway state approach ) and effective phase shift approach give very similar results. ==> 1b nature of decay The difference appears for the width of 2 + in 24 O. ==>doorway state or not. Unit : MeV 23 O 24 O states 3/2+ 1+ 2+ CCSM E 0.92 1.35 0.61 CCSM Γ 0.11 0.28 0.06 Phase shift E 0.92 1.36 0.61 Phase shiftγ 0.11 0.28 0.04 13

Convergence with respect to boundary condition r The peak energies discussed as a function of L. The results do not change so much if L is taken to be sufficiently large. Even usual values of L 50 fm are not stable. 14

Summary of CCSM We introduced Continuum-Coupled Shell Model (CCSM). CCSM treats continuum coupling, taking NN interaction into account in the level of filling configuration. Note: NN correlation is treated by fitting to SDPF-M. ==> What is the relation to the realistic NN? The effect of continuum coupling for the excited states is more than 1 MeV. CCSM assure the correct asymptotic behavior of wave functions. ==> Are all the results specific for CCSM? General consequence? 15

Overview of my talk Description of open quantum system 1st part Real energy approach Complex energy approach 2nd part CDCC Continuum Coupled Shell Model Gamow Shell Model Continuum Shell Model Complex Scaling Method empirical NN empirical NN starting from realistic NN K.T., T. Otsuka, R. Fujimoto, arxiv: 1001.0729 K.T., M. H-Jensen, G. Hagen Phys.Rev.C80:051301(R), (2009) 16

Complex energy approach In quantum mechanics, resonance states and scattering states have exponentially diverging wave function and then norm. They cannot be expressed in Hilbert space and have to be in Rigged Hilbert space. Continuum Shell model (CSM) Feshbah s approach ==> effective Hamiltonian in P space spanned by discrete states includes effects of continuum space (=Q space). Energy dependent, non-hermitian. Gamow Shell Model (GSM) Bound and scattering degrees of freedom are treated equally Most importantly allows for an exact treatment of the anti-symmetry of the wave functions No limitation on the number of particles in the continuum. 17

Gamow shell model The completeness relation of Newton Resonance states are embedded in the continuum. can be extended to construct a bi-orthogonal basis Resonance states are extracted and described as discrete states. Im(k) Bound Boundary condition for one particle is automatically satisfied Resonance states are defined as poles of S-matrix. Capturing states Anti-bound Re(k) Decaying states 18

Discretized representation of Gamow basis WS The contour integral along C is limited to the finite range and then discretized, WS potential A resonance pole is independent of the choice of contour or discretization. 21

Quantity in Gamow basis It is practically difficult to calculate the matrix elements in lab system by Gamow basis (delta and Heaviside function in vector brakets transformation). CCSM effective interaction One can represent an operator in HO basis ==> transform into Gamow basis. Gamow states, H.O. states Hamiltonian: complex and symmetric 27

Contour dependence and convergence n Gamow integration points for two different contours. L 1 L 2 n Gamow 24 O 1 + 24 O 2 + L 1 L 2 L 1 L 2 25 1.35-0.130i 1.36-0.128i 0.596-0.018i 0.624-0.001i 30 1.35-0.130i 1.35-0.129i 0.596-0.018i 0.604-0.008i 35 1.35-0.130i 1.35-0.130i 0.602-0.019i 0.602-0.019i 40 1.35-0.130i 1.35-0.130i 0.602-0.019i 0.602-0.019i 45 1.35-0.130i 1.35-0.130i 0.602-0.019i 0.602-0.019i 100 1.35-0.130i 1.35-0.130i 0.602-0.019i 0.602-0.019i Results are independent on the contour Converge wrt the number of integration points. 28

Results Note that we solved the same problem by two different method; Continuum-Coupled Shell Model and Gamow Shell Model. 23 O 24 O 25 O 26 O States 3/2 + 1 + 2 + 3/2 + 0 + 2 + Full sd 2.22 2.77 1.74 2.15 2.19 4.19 Bound app. 2.22 2.82 1.82 2.17 2.33 4.15 CCSM E 0.92 1.35 0.61 0.86 0.53 1.66 CCSM Γ 0.11 0.28 0.06 0.11 0.24 0.33 Phase shift E 0.92 1.36 0.61 - - - Phase shift Γ 0.11 0.28 0.04 - - - GSM E 0.92 1.35 0.60 0.85 - - GSM Γ 0.11 0.26 0.04 0.08 - - EXP E 1.26 1.24(7) 0.63(4) 0.77(2) - - EXP Γ 0.2 - - 0.17 - - Good agreement for the peak energy and certain difference for the width. Doorway state or not 29

Methodological relation between CCSM and GSM Resonance state A point where phase shift goes through pi/2 The point where cross section gets its peak Asymptotic radial w.f. is a standing wave. Pole of S-matrix (Green s function) The radial w.f. has purely outgoing boundary condition. 30

Methodological relation between CCSM and GSM near k=k 0 one can approximate the energy dependence of scattering cross section. pole of S-matrix which is analytically continued to the complex k-plane. Difference between CCSM and GSM can be explained by the definition of resonance, then different boundary condition. doorway state 31

Gamow shell model starting from realistic NN interaction We want understand the structure of exotic nuclei from microscopic points of view. Recent development for Oxygen drip line. Coupled-cluster calculations G. Hagen et al.phys. Rev. C 80, 021306(R) (2009). ==>* chiral NN interactions cannot rule out the existence of 28 O * It was concluded that 3NFs will eventually decide the matter. Shell model within the sd shell ==>* inclusion of effective 3NFs gives added repulsion in the heavier oxygen isotopes and results in an unstable 28 O T. Otsuka et al., arxiv:0908.2607. Disentangle the effects coming from many-body correlations and the proximity of the continuum How different parts of the underlying NN interaction affect the structure of nuclei close to the drip line. 33

GSM K.T., M. H-Jensen, G. Hagen, Phys.Rev.C80:051301(R), (2009) Chiral NN potential N 3 LO (500 MeV) Not so hard as conventional models, but still hard to be non perturbative. Renormalized to lower momentum interaction Keeping low-energy NN observable unchanged. Transform into Gamow basis representation in lab frame Many-body perturbation theory Continuum coupling is explicitly taken into account. Effective Hamiltonian in s1-d3 space Complex and symmetric 34

Continuum coupling in many-body theory In many-body perturbation theory can be continuum states Valence line include resonance and scattering states. ==>explicit continuum coupling for effective interaction. Effective vertices for bound orbits also include virtual excitation to continuum states. 35

Results Energies measured from the ground state of 24 O For excited states There is significant continuum effect. For ground state No big difference between GSM and normal one. excited states ground state GSM widths 1 + 0.44 MeV (Exp. 0.03 +0.12-0.03) 2 + 0.26 MeV,(Exp. 0.05 +0.21 0.05 ) 3/2 + 0.5 MeV (Exp 0.17) poor description due to filling configuration. 25 O is particle unstable wrt 24 O irrespective to HO or Gamow, as long as 22 O is taken as an inert core. 36

3NF d3 s1 d5 16 O core 22 O core Strongly attractive V d3d5 Only s1 and d3 Repulsive correction (stemming from 3NF) Implicitly included by renormalization of SPE s For Shell model in (s1-d3) space Missing many-body physics is not crucial to determine the drip line of Oxygen. Continuum coupling is crucial to explain the properties of excited states. 37

Paring correlation to resonance Which parts of the NN interaction may play an important role in our understanding of the low-lying states in 24 O? we single out the 1 S 0 partial wave component of the staring NN interaction and vary its strength. The ground state gains additional binding when α increases. With no paring, 1+ 2+ are interchanged and degenerate. 38

Summary We consider continuum effects on shell model (CI method) by Continuum- Coupled Shell Model (CCSM) and Gamow Shell Model (GSM) Continuum effects are more than 1 MeV. CCSM can reproduce the asymptotic behavior of resonance wave functions and has clear connection to what is measured in experiments. Methodological relation was discussed in detail. Similarity and difference in CCSM and GSM are clearly seen. GSM starting from realistic NN interaction was implemented. Continuum effects is crucial for the description of excited states. Many-body (3N) correlation is dominant for binding-energy systematics Paring channel in the bare NN is important for the understanding of drip-line O isotopes. Treatment of NN correlation is still crucial in the exotic O isotopes. CCSM and GSM are the natural extension of normal shell model. More systematic calculations are important. 39