Particle Condensation in Nuclear systems A. Tohsaki, H. Horiuchi, G. Röpke, P. Sch. T. Yamada and Y. Funaki -condensation in matter 8 Be and Hoyle state in 12 C -condensate wave function Effective GPE for n. condensate November 29, 2005 1 Conclusions, outlook
Clusters important aspect and richness of nuclear systems due to 4 Fermions : Dimer : Quartet : E A =1 MeV Proposal : E = 7 MeV, E * = 20 MeV A 2 Trapping of 4 different species of Fermionic atoms.
Infinite matter : Pair Condensation (nn or pn) ɛ 1 = p2 1 2m Thouless criterion for T c : f 1 = 1 1 + e (ɛ 1 µ) T (2µ ɛ 1 ɛ 2 ) ψ 12 = (1 f 1 f 2 ) 1 2 v 121 2 ψ 1 2 µ chemical-potential and f 1, f 2 Fermi-Dirac at T = T c. k k 3
-Particle Condensation : G. Röpke, M. Beyer (4µ ɛ 1 ɛ 2 ɛ 3 ɛ 4 ) ψ 1234 = (1 f 1 f 2 ) 1 2 v 121 2 ψ 1 2 34 +permutations K = 0 P + permutations P 10 10 temperature T [MeV] 8 6 4 2 T t c (deuteron pairing) T 4 c ( quartetting) temperature T [MeV] 8 6 4 2 (deuteron pairing) T c 4 ( quartetting) T t c 0 20 0 20 40 60 chemical potential µ* [MeV] 0 10 4 10 3 10 2 10 1 10 0 density n 1 [fm 3 ] 4 -Condensation only at very low density!
Cooper pair n p High Density n p Cooper pairs Strongly overlapping not Bosons BCS BCS Low density : smooth transition BEC BEC gas of Deuterons ~ Bosons 5
Finite nuclei? Exact 8 Be : Density : ρ 0 3 3 rd particle Fermi gas collapse 12 C compact ground state V/ 3 6 V
Does a dilute 3 12 C state exist? Similar to 8 Be +? At T = 10 8 K helium burning thermal equilibrium + + 8 Be 12 C O + 2 7 O + 2 : dilute 3 state hypothesis!
it seems impossible to get Hoyle state from shell model calculation! 45 MeV B. Barret 8
If O + 2 in 12 C dilute state then -condensate infinite matter ρ crit ρ 0 3 Conjecture all n. nuclei possess exited n condensed state all s in s wave s p 9 Analogy with atoms in traps! ρ(r) = N φ 0 (r) 2 N = 10 6
Bosons Back to nuclei many s condensate 0 + 2 7,65 MeV 12 C proton neutron alpha 0 + 1 strong cluster phenomena in lighter nuclei 10
11
Theoretical Description Ideal Bose condensate : 0 = bb b vac 0 0 0 -particle condensate : Φ = C CC C vac In r-space : r, r,, r Φ = A Φ r, r, r, r Φ r, r, r r Φ r r r r { ( ) (, ) (,,, )} 1 2 4n C 1 2 3 4 5 6 7 8 4n 3 4n 2 4n 1 4n In comparison with pairing : r, r, BCS = A Φ r, r Φ r, r { ( ) ( ) } 1 2 1 2 3 4 Variational ansatz for Φ,,, : ( r r r r ) 1 2 3 4 Φ = 2 R 2 B ( r1, r2, r3, r4) e φ ( ri rj) 2
Center of mass : 1 R= r + r + r + r 4 ( ) 1 2 3 4 Intrinsic -wave function : φ r r = e ( ) i j {( ) 2 ( ) 2 r ( ) 2 } 4 r 1 r 4 r 2 r 4 r 3 1 + + + 2 8b Two variational parameters : B, b Two limits : B = b Φ = B b C Φ = C Slater determinant gas of independent -particles Two dimensional surface : (, ) E B b = Φ C Φ C H Φ Φ C C
-34 MeV 1.9 1.8 1.7-39 MeV -44 MeV 8 B b [fm] 1.6 1.5 1.4 1.3 1.2 1.1-54.33 MeV -54 MeV 0 1 2 3 4 5 6 7 8 9 10 β x =β y =β z [fm] -49 MeV Saddle Point -53.64 MeV -49 MeV -44 MeV -39 MeV -34 MeV 12 C 16 O 14
Hamiltonian : H = T + V N N + V C + V N N N Kin. energy Gaussien Coulomb Gaussian Quantization of energy surface E(B, b) : Force : A. Tohsaki 1990 no adjustable parameters! Hill-Wheeler : ψ = B f B Φ C (B) Without adjustable parameters : 12 C : (E O + 2 E 3 ) = Theory + 0.50 MeV Exp. + 0.38 MeV 16 O : (E O + 5 E 4 ) = Theory 0.70 MeV Exp. 0.44 MeV 15
r.m.s. of O 2 + in 12 C 3.83 fm groundstate 2.40 fm V O + 2 V O + 1 3 4 0.2 Density : r 2 (r) [fm -1 ] 0.1 0.0 0 2 4 6 8 10 12 r [fm] 16 Exp. Cal. M(O 2 + O+ 1 ) 5.4 ± 0.2fm2 6.7 R rms (O 1 + ) 2.43fm 2.40 R rms (O 2 + ) 3.47
Radial behavior of S-wave orbit vs. R rms R rms =2.43 fm 4.84 fm (ρ/ρ 0 =1.1 0.14) R rms =2.43 fm (ρ/ρ 0 =1.1) R rms =2.70 fm (ρ/ρ 0 =0.83) R rms =3.11 fm (ρ/ρ 0 =0.53) 33% 49% 63% R rms =3.76 fm (ρ/ρ 0 =0.30) R rms =4.84 fm (ρ/ρ 0 =0.14) 79% Increasing R rms, we see smooth change 90% from 2S to 0S orbit. Yamada & Schuck, EPJA 2005
R : r.m.s. radius β1 : = ( βx = βy, βz) = (5.27 fm, 1.37 fm)
Some more numbers : 12 C : Theory Exp. O 1 + 89.52 92.16 O 2 + 81.79 84.51 7.73 7.65 18 Threshold states E E thresh. 8 Be 12 C 16 O 20 Ne 2 3 4 5 O 1 + O 2 + O 3 + O 4 + 0.17 0.50 0.7 1.8 O 1 + O 2 + O 5 +? 0.09 0.38 0.44? ] theory ] experimental
Spectrum of 8 Be : 4 + ~11.4 MeV Fully reproduced 2 + 2.9 MeV O + ~ 12 C : Second excited 2 + : 2 + 2 It has been discovered recently by Itoh et al. 2.6 MeV above 3 rhreshold Width 1 MeV : resonance in continuum 19
Theory : We start with deformed condensate state : Φ n A n i=1 { exp 2 X2 ix Bx 2 2 X2 iy B 2 y 2 } X2 iz Bz 2 Φ i Then projection on good angular momentum Then Hill Wheeler or GCM For width : ACCC method Position Width Experiment : 2.6 ± 0.3 MeV 1.0 ± 0.3 MeV Theory : 2.1 MeV 0.64 MeV With in error bars! halo! RMS : 4.43 fm V 2 + 2 V 8!! O + 1 20
Internal structure : S D Extremely dilute 3 state Suggests a pure Boson picture Hartree Fock (Gross Pitaevsky eq) for ideal bosons ( s) : φ 0 = b + 0 b+ 0 b+ 0... vac [ 2m + N d 3 r v( r r ) φ 0 ( r ) 2 ] φ 0 ( r) = ɛ 0 φ 0 ( r) effective + Coulomb T. Yamada 21
Estimate for maximum number 22 N limit 10 40 Ca
BUT Some neutrons can stabilise 8 Be unbound, 9 Be bound (2.5 MeV) 10 Be strongly bound! May be 20-30 s possible! 8 Be 4 fm 10 Be 3 fm 23
Boson occupancy : -particle density matrix : ρ ( R, R ), R : c.m. of Diagonalization : 12 C : O + 2 70% S-wave occupancy 24
Conclusions Strong indications that extended -condensates exist, - condensate-halos Adding some neutron glue, condensates may become very large : ten s of s! Highly excited but long life time! Indra : 36 Ar + 36 Ni 9 (M.F. Rivet, B. Borderie) -energies wavefcts Chimera Why, not 16 O, 40 Ca very inert and compact, almost ideal Boson, 20 MeV first excitation May be 48 Cr 3 16 O 25 Ultimate question : How to prove -particle superfluidity?
Eventually important in nuclear astrophysics : collapsing massive stars Nuclei immersed in a gas of neutrons + some protons at finitet Can lead to gas of -particles may be -particle condensate change of equation of state change of neutrino absorption influence on collapse scenario 26
27