Alpha particle condensation in nuclear systems

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1 Alpha particle condensation in nuclear systems Contents Introduction ncondensate wave function 3system (0 and states ) 4system (05 state) Yasuro Funaki (Kyoto Univ.) Peter Schuck (IPN, Orsay) Akihiro Tohsaki (Suzuki Corp.) Hisashi Horiuchi (Kyoto Univ.) Taiichi Yamada (Kanto Gakuin) Gerd Röpke (Rostock Univ.) Summary

2 Energy region of cluster gas states appearing Energy 00 MeV Nucleon Nucleon gas C cluster Cluster gas Appearing near the 3threshold ~0 MeV Condensed into the lowest orbit 0 MeV luquid Lowest energy state

3 Typical mysterious 0 state in nuclear structure problem 0 state of C(Hoyle state) Important for C synthesis in stars Ab initio non-core shell model calculation 0 state : missing (excitation energy ~44 MeV) One of the typical excited states which resist a shell model description P. Navratil et al., Phys. Rev. Lett. 84 (000), 578.

4 Hoyle state Energy (MeV) Exp Theor decay width (ev) 8.7± M(0 0 ) (fm ) 5.4± B(E; 0 ) (e fm 4 ) 3±4 5.6 Many experimental data exist. Three-body (3) problem is fully solved a quarter century ago, and the obtained w. f. well reproduces almost all experimental data. (Kamimura et al. (RGM), Uegaki et al. (GCM)) Resonating Group Method(RGM) φ φ φ χ, φ φ( ) φ = 0 ( ) ( ) ( ) ( H E) A ( s r ) ( ) ( ) { } 3 3 A s r Traditional cluster model in 3 system Fully solved without any model assumption between clusters

5 ncondensed wave function n ( ) ( ), b exp X ( ) Φ = A β φ n B i i i= A. Tohsaki, H. Horiuchi, P. Schuck, and B = b β We don t explicitly treat inter-cluster distance. G. Ropke, Phys. Rev. Lett. 87 (00) 950 b : width parameter of the internal wave function of particle φ() (size of cluster) X: center-of-mass coordinate of particle A: anti-symmetrizing operator acting on all of constituent nucleons β Large : Bose condensed state C. M. motion of nclusters occupy the same S- orbit exp(-x /B ), forming a gas-like structure. β 0 : (when normalized)shell model w. f.

6 Ψ= { χ ( s, r ) φ ( ) φ( ) φ ( )} 3 A s r RGM Extremely reliable solution was obtained for the Hoyle state As for the Hoyle state, both are almost equivalent (~90 %). 3 Ψ= A i= ( ) X φ( ) exp i i B 3cluster condensate model A 3 clusters occupy the same S orbit Y. Funaki, A. Tohsaki, H. Horiuchi, P. Schuck, and G. Ropke, Phys. Rev. C 67 (003)

7 5 Observed levels of C E x (MeV) 4. 4 Hoyle state : It seemed that we have understood this state. However, our w. f. revealed essential physics which has been hidden in the full solution, 3gas picture < MeV state : E=9.9±0.3 MeV Γ=.0±0.3 MeV C ( ) Recently observed M. Itoh et al., Nucl. Phys. A 738 (004) Possible excitation mode based on condensate particles:s-orbit particle:d-orbit Y. Funaki, A. Tohsaki, H. Horiuchi, P. Schuck and G. Ropke, Euro. Phys. Jour. A, 4 (005), 3.

8 state of C (.6 MeV above the 3threshold) state of C (.6 MeV above the 3threshold) cond. w. f.accc method E=9.38 MeV Γ=0.64 MeV Volkov No. force is adopted. Experimental values for state of C E=9.9±0.3 MeV C( ) Γ=.0±0.3 MeV M. Itoh et al., Nucl. Phys. A 738 (004) 68-7 Energy and width are well reproduced. Single 3condensate w. f. projected onto J= with optimum parameter value, (Bx=By, Bz) (6 fm, 0 fm) 3 ˆ ˆ P P exp ( ) J= A Xix Xiy Xiz φ i i= Bx By B z state can also be expressed by the single 3condensate w. f.. (more than 90%) (no need to superpose various w. f. via GCM.) We can show that the above w.f. has a dominant configuration, where one of the 3 s jump into D-orbit and s stay S-orbit.

9 Density distribution of 0,, 4 states(shell model structure) and 0, states (gas-like structure) 0,,4 : R r.m.s. ~.4 fm 0 : R r.m.s. =3.83 fm (~ρ 0 /4) : R r.m.s. =5.4 fm (~ρ 0 /0) ``-halo state Form factor ρ ( a) = δ ( r ) i XG a i= Density operator Volkov No. force is adopted.

10 Expectation values of energies for the 0,,4 and 0, states Volkov No. force, Unit is in MeV J π <T-T G > <V N > <V C > E-E 3 th R r.m.s (fm) and : smaller cancellation of kinetic and nucleon-nucleon interaction energies Both states have a similar structure.

11 Calculation by Orthogonality Condition Model (OCM) Single -orbits in C(0 ) N : Occupation number (normalized to 3) 0 (g.s.): R rns =.44 fm S-orbit (N =.05) 0 : Rrms =4.3 fm S-orbit (N =.6=70% 3) G-orbit (N =0.8) D-orbit (N =.06) ref.: r N0( a) exp( ar ) Large oscillation : strong Pauli blocking effect Compact structure SU(3) model: [f](λµ)=[444](04)-like structure Small oscillation: weak Pauli blocking effect Long tail: dilute structure Radial behavior: Gaussian form with a= 0.04 fm - T. Yamada and P. Schuck, Euro. Phys. Jour. A, 6 (005), 85.

12 Low lying 0 levels of 6 O E (MeV) x MeV ( Γ=.5 kev).6 ( Γ=.6 MeV) C 7. MeV 6.06 Exp ( 0 3 ) theor ( 0 ) theor ( 0 ) 4cond. w.f. theor C OCM 0 : C(0 ) 04 : C( ) Y. Suzuki, Prog. Theor. Phys. Vol. 55, No. 6, (976) state: A candidate of 4condensate E=3.5MeV Γ=0.8MeV 6 O( ) Wakasa etal. (0 3 ) theor : 4condensed state E=4.9 MeV Γ=.5 MeV (based on R-matrix theory) A. Tohsaki, H. Horiuchi, P. Schuck, and G. Ropke, Nucl. Phys. A 738 (004) 59-63

13 Cross section of inelastic scattering to 4condensed state (3.5 MeV)(preliminary) Both agree well, and reasonably 05 state can be assigned to the 4 condensate Calculated by M. Takashina(RIKEN)

14 Summary 0 w. f. of C (and the ground state of 8 Be) which is obtained by fully solving three-body problem without any model assumption about inter-cluster motions can be simply expressed by the single 3condensate w. f. We theoretically find the decisive evidence that 0 state of Cis considered to be the 3 Bose-condensed state. state of C belongs to the family of 3Bose-condensation. (: D state, : S state) As for 6 O, the recently observed 0 5 state (3.5MeV) can be assigned to the 0 3 state (4condensate). (decay width ( )cross section(preliminary)) Possibility that cluster gas states exist more widely For example, (3/)3 - state of B (triton gas) (Kawabata and En yo et al.)

Citation PHYSICAL REVIEW C (2006), 74(5) RightCopyright 2006 American Physical So

Title alphac-12 in angular distri 12(O-2()) Author(s) Takashina, M; Sakuragi, Y Citation PHYSICAL REVIEW C (2006), 74(5) Issue Date 2006-11 URL http://hdl.handle.net/2433/50458 RightCopyright 2006 American