Entrance-channel potentials in the synthesis of the heaviest nuclei

Transcription

1 Entrance-channel potentials in the synthesis of the heaviest nuclei Vitali Yu. DENISOV 1,2 and Wolfgang Nörenberg 1,3 1 Gesellschaft für Schwerionenforschung, Darmstadt, Germany 2 Institute for Nuclear Research, Kiev, Ukraine 3 Institut für Kernphysik, Technische Universität Darmstadt, Darmstadt, Germany Plan Capture is the first decisive step for the fusion Definition of a semi-microscopic potential (SMP) in the entrance channel SMP for cold-fusion systems SMP for hot-fusion systems SMP for warm-fusion systems Conclusion

2 Definition of a semi-microscopic potential (SMP) in the entrance channel The interaction potential V (R, ϑ) V (R, ϑ) =E 12 (R, ϑ) E 1 E 2. In the frozen-densityapproximation these binding energies are determinated bythe energydensityfunctional E[ρ p (r),ρ n (r)], i.e. E 12 (R, ϑ) = E[ρ 1p (r)+ρ 2p (R, ϑ, r),ρ 1n (r)+ρ 2n (R, ϑ, r)] dr, E 1 = E[ρ 1p (r),ρ 1n (r)] dr, E 2 = E[ρ 2p (r),ρ 2n (r)] dr, where ρ 1p, ρ 2p, ρ 1n and ρ 2n are the frozen proton and neutron densities of the spherical nucleus (index 1) and the deformed nucleus (index 2), respectively. Energy-density functional: E[ρ p (r),ρ n (r)] = 2 2m [τ p(r)+τ n (r)] + V Skyrme (r)+v Coul (r). ρ 1p (r),ρ 2p (R, ϑ, r),ρ 1n (r),ρ 2n (R, ϑ, r) Hartree-Fock-Bogoliubov (HFB) with Skyrme forces.

3 The kinetic parts for the protons (i = p) and neutrons (i = n) τ i (r) = 3 5 (3π2 ) 2/3 ρ 5/3 i + 1 ( ρ i ) ρ i f i + ρ i f i 6 f i + 1 ( 2m 2 ρ W 0 i 2 2 ρ i 1 12 ρ i ρ i ( fi f i (ρ + ρ i ) f i ) 2 ) 2, where W 0 - the strength of the Skyrme spin-orbit interaction, ρ = ρ p + ρ n, f i (r) =1+ 2m ( 3t1 +5t 2 + t ) 2x 2 ρ 2 i (r) The potential part V sk, Skyrme interaction, V Skyrme (r) = t 0 2 [( x 0)ρ 2 (x )(ρ2 p + ρ 2 n)] t 3ρ α [( x 3)ρ 2 (x )(ρ2 p + ρ 2 n)] [t 1( x 1)+t 2 ( x 2)]τρ [t 2(x ) t 1(x )](τ pρ p + τ n ρ n ) [3t 1( x 1) t 2 ( x 2)]( ρ) [3t 1(x )+t 2(x )]( ρ n) 2 +( ρ p ) 2 ) W 0 2 [ 2m ρp (2 ρ 4 2 p + ρ n ) 2 + ρ ] n (2 ρ n + ρ p ) 2, f p f n where t 0, t 1, t 2, x 0, x 1, x 2, α and W 0 are Skyrme force parameters. The Coulomb energydensity V Coul (r) = e2 2 ρ ρ p (r ) p(r) r r dr 3e2 4 ( ) 1/3 3 (ρ p (r)) 4/3. π

4 Entrance channel dynamics The nuclear interaction time τ coll (collision time) τ coll π [ ] 1/2 ma 1 A 2 = π s. ω pocket (A 1 + A 2 )V (R pocket ) The relaxation of the intrinsic nuclear state due to nucleon-nucleon interactions τ relax (G.F. Bertsch) τ relax ɛ F 3.2σv F ρ 0 E E s s. τ relax >> τ coll. Conclusion: Frozen-densities of nucleons in nuclei can be applied for the evaluation of the nucleus-nucleus potential.

5 Main features of SMP in light systems: Deep pocket inside the barrier Light ions easilyfuse after tunneling through or passing over the barrier The barrier height and the potential pocket are well above the ground-state energy The potential surface exhibits large gradients in the fusion direction driving the system into the compound-nucleus shape The barriers obtained with the help Bass-74, Bass-80, Proximity-77 and Krappe-Nix-Sierk (KNS) potentials are spread over a wide interval

6 The Bass-74, -80, Prox-77 and KNS interaction potentials are spread over even larger intervals for heavier systems as compared to light system The potential pockets are much shallower than for lighter systems and tend to vanish with increasing size of the projectile We attribute the observed reduction of the SHE formation with increasing size of the projectile, at least partially, to decreasing pocket depth The observed fusion windows lie about 5 to 10 MeV below SMP barriers. There is a correlation between the width of fusion window and the depth of potential pocket (cases 50 Ti+ 208 Pb, 58 Fe+ 208 Pb and 64 Ni+ 208 Pb)

7 The difference between the barrier position and the ground-state Q-value for fusion decreases with increasing charge of the projectile

8 Symmetric systems The capture process is suppressed bythe shallowness of the potential pocket The shape of the system at capture is less compact, and hence a longer shape evolution is needed to reach the compound-nucleus shape. the formation probabilityof compound nucleus is reduced due to the larger competition of other decays

9 Large distances between spherical and prolate nuclei ϑ =90 due to the Coulomb interaction (ϑ =90 side position) The time for the rotating the deformed nucleus by90 τ rot π 2ω rot = s, where ω rot 50 kev. Typical collision times on the approaching part of the Coulomb trajectoryare order s. Strong orientation effect on the barrier and pocket, stronglydeformed plolate target High excitation energyof compound nucleus Fusion relates with side orientation (ϑ 90 ) Fusion suppressed for tip position (ϑ 0 ) The height of the barrier reduces with increasing neutron number

10 Warm-fusion systems 198 Pt - oblate -β 2 = 0.10 Recent GSI experiment: 40 Ar, 50 Ti+ 198 Pt. The cross sections for reaction 50 Ti+ 198 Pt is comparable with the one for coldfusion reaction 40 Ar+ 208 Pb. Large distances between spherical and oblate nuclei ϑ =0 due to the Coulomb interaction (ϑ =0 tip position)

11 Conclusion Rules for the determination of the best candidates for the synthesis of SHEs The SMP barrier should lie about 5 to 15 MeV above the 1n fusion threshold, but not above the 2n fusion threshold to avoid the reduction of the fusion cross-section byan additional factor Γ n /Γ f The deeper the pocket the larger the capture window better the chance of synthesis It is best to have a most compact capture configuration

12 The synthesis of 118 with hot-, cold- and warm-fusion systems The cold-fusion system 86 Kr+ 208 Pb has its capture window below the 1nfusion channel and shallow pocket, and hence is not expected to be a good candidate The symmetric system 144 Ce+ 150 Nd has no pocket and hence no capture window at all The hot-fusion system 48 Ca+ 252 Cf has nice capture properties, however needs to emit about 3 to 4 neutrons, which reduce the survival probability byseveral orders due to factor Γ n /Γ f << 1 The hot-fusion system 40 Ca+ 252 Cf has less attractive capture properties (as compared to the 48 Ca case) and needs to emit even 5 to 6 neutrons The system 58 Fe+ 238 U has onlya tinypocket and needs to emit about 3-4 neutrons the warm-fusion system 96 Zr+ 198 Pt has also a tinytip-positioned pocket but needs to emit only1n The most attractive projectile-target are: 48 Ca+ 252 Cf at E coll 206 MeV 96 Zr+ 198 Pt at E coll 330 MeV. While 48 Ca+ 252 Cf is more compact, 96 Zr+ 198 Ptneedstoemitonly1neutron. It is hard to judge which of these features are more important

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