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
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
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.
Hoyle state Energy (MeV) Exp. 7.65 Theor. 7.74 decay width (ev) 8.7±.7 7.7 M(0 0 ) (fm ) 5.4±0. 6.7 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
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.
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) 053060
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. 0 0.3 < 0 7.66 0 3 0 3 7.7 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) 68-7 5 0 4.44 0 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.
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.
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.
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) 0 5. -64. 5.48-7.47.40 55.6-65.8 5.55-4.65.38 4 68. -7. 5.77.74.3 0 7.6-0.9 3.58 0.6 3.47 9.84 -..80.54 5.40 0 and : smaller cancellation of kinetic and nucleon-nucleon interaction energies Both states have a similar structure.
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.
5 0 5 0 Low lying 0 levels of 6 O E (MeV) x 4 4.44 MeV 3.5.05 ( Γ=.5 kev).6 ( Γ=.6 MeV) C 7. MeV 6.06 Exp. 0 5 0 4 0 3 0 0 4.94 0.34 0.08 ( 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) 75-768 0 5 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
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)
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.)