Microscopic Calculation of Fusion: Light to Heavy Systems
aa r X i v : . [ nu c l - t h ] O c t October 29, 2018 15:17 WSPC - Proceedings Trim Size: 9in x 6in sanibel˙umar MICROSCOPIC CALCULATION OF FUSION; LIGHT TOHEAVY SYSTEMS
A. S. UMAR ∗ and V. E. OBERACKER Physics and Astronomy, Vanderbilt University,Nashville, TN 37235, USA ∗ E-mail: [email protected]
J.A. MARUHN
Institut f¨ur Theoretische Physik, Goethe-Universit¨at,D-60438 Frankfurt am Main, Germany
R. KESER
RTE University, Science and Arts Faculty, Department of Physics, 53100,Rize, TURKEY
The density-constrained time-dependent Hartree-Fock (DC-TDHF) theory isa fully microscopic approach for calculating heavy-ion interaction potentialsand fusion cross sections below and above the fusion barrier. We discuss recentapplications of DC-TDHF method to fusion of light and heavy neutron-richsystems.
Keywords : Time-Dependent Hartree-Fock, Heavy-Ion Fusion, DC-TDHF.
1. Introduction
The investigation of internuclear potentials for heavy-ion collisions is offundamental importance for the study of fusion reactions as well as for theformation of superheavy elements and nuclei far from stability. Recently, wehave developed a new method to extract ion-ion interaction potentials di-rectly from the time-dependent Hartree-Fock (TDHF) time-evolution of thenuclear system. In the density-constrained TDHF (DC-TDHF) approachthe TDHF time-evolution takes place with no restrictions. At certain timesduring the evolution the instantaneous density is used to perform a staticHartree-Fock minimization while holding the neutron and proton densi-ties constrained to be the corresponding instantaneous TDHF densities. Inessence, this provides us with the TDHF dynamical path in relation to the ctober 29, 2018 15:17 WSPC - Proceedings Trim Size: 9in x 6in sanibel˙umar multi-dimensional static energy surface of the combined nuclear system. Inthis approach there is no need to introduce constraining operators which as-sume that the collective motion is confined to the constrained phase space.In short, we have a self-organizing system which selects its evolutionarypath by itself following the microscopic dynamics. Some of the effects nat-urally included in the DC-TDHF calculations are: neck formation, massexchange, internal excitations, deformation effects to all order, as well asthe effect of nuclear alignment for deformed systems. The DC-TDHF the-ory provides a comprehensive approach to calculating fusion barriers inthe mean-field limit. The theory has been applied to calculate fusion cross-sections for Ni+
Sn, Ni+ Ni, O+ Pb, Zn+
Pb, Ca+
U,and , Sn+ Zr systems.
In this paper we will outline the DC-TDHFmethod and give new examples of its application to the calculation of fusioncross-sections for various systems.
2. Density-Constrained TDHF Method
The concept of using density as a constraint for calculating collective statesfrom TDHF time-evolution was first introduced in Ref. 10, and used incalculating collective energy surfaces in connection with nuclear molecularresonances in Ref. 9.In this approach we assume that a collective state is characterized onlyby density ρ , and current j . This state can be constructed by solving thestatic Hartree-Fock equations < Φ ρ, j | a † h a p ˆ H | Φ ρ, j > = 0 , (1)subject to constraints on density and current < Φ ρ, j | ˆ ρ ( r ) | Φ ρ, j > = ρ ( r , t ) < Φ ρ, j | ˆ ( r ) | Φ ρ, j > = j ( r , t ) . Choosing ρ ( r , t ) and j ( r , t ) to be the instantaneous TDHF density andcurrent results in the lowest energy collective state corresponding to theinstantaneous TDHF state | Φ( t ) > , with the corresponding energy E coll ( ρ ( t ) , j ( t )) = < Φ ρ, j | ˆ H | Φ ρ, j > . (2)This collective energy differs from the conserved TDHF energy only by theamount of internal excitation present in the TDHF state, namely E ∗ ( t ) = E T DHF − E coll ( t ) . (3) ctober 29, 2018 15:17 WSPC - Proceedings Trim Size: 9in x 6in sanibel˙umar However, in practical calculations the constraint on the current is difficultto implement but we can define instead a static adiabatic collective state | Φ ρ > subject to the constraints < Φ ρ | ˆ ρ ( r ) | Φ ρ > = ρ ( r , t ) < Φ ρ | ˆ ( r ) | Φ ρ > = 0 . In terms of this state one can write the collective energy as E coll = E kin ( ρ ( t ) , j ( t )) + E DC ( ρ ( r , t )) , (4)where the density-constrained energy E DC , and the collective kinetic energy E kin are defined as E DC = < Φ ρ | ˆ H | Φ ρ >E kin ≈ m X q Z d r j q ( t ) /ρ q ( t ) , where the index q is the isospin index for neutrons and protons ( q = n, p ).From Eq. 4 is is clear that the density-constrained energy E DC plays therole of a collective potential. In fact this is exactly the case except for thefact that it contains the binding energies of the two colliding nuclei. Onecan thus define the ion-ion potential as V = E DC ( ρ ( r , t )) − E A − E A , (5)where E A and E A are the binding energies of two nuclei obtained froma static Hartree-Fock calculation with the same effective interaction. Fordescribing a collision of two nuclei one can label the above potential withion-ion separation distance R ( t ) obtained during the TDHF time-evolution.This ion-ion potential V ( R ) is asymptotically correct since at large initialseparations it exactly reproduces V Coulomb ( R max ). In addition to the ion-ionpotential it is also possible to obtain coordinate dependent mass parame-ters. One can compute the “effective mass” M ( R ) using the conservationof energy M ( R ) = 2[ E c . m . − V ( R )]˙ R , (6)where the collective velocity ˙ R is directly obtained from the TDHF evolu-tion and the potential V ( R ) from the density constraint calculations. ctober 29, 2018 15:17 WSPC - Proceedings Trim Size: 9in x 6in sanibel˙umar
3. Results
In this Section we give some recent examples of DC-TDHF calculationsof heavy-ion potentials and cross-sections. Recently, we have studied thefusion of very neutron rich light nuclei that may be important to deter-mine the composition and heating of the crust of accreting neutron stars. The main focus was the O+O and C+O systems. For the O+ O systemwe have shown excellent agreement between our calculations and the lowenergy data from Refs.
We have also extended this work to higher en-ergies to see how our results compare with the available data. The reactionsof light systems at high energies (2 − Experimental findings differ considerably in this energyregime as can be seen from Fig. 1. Recent analysis of the O+ O systemsystem by H. Esbensen primarily uses the data of Tserruya et al. Weexpect the TDHF results to yield a higher fusion cross-section since manyof the breakup channels are not naturally available in TDHF. However,a close investigation of the TDHF dynamics and the microscopically cal-culated excitation energy clearly indicate that a significant portion of thecollective kinetic energy is not equilibriated. It may be plausible to consider
R, R (fm) -1001020 V ( R ) , V D C ( R ) ( M e V ) Point Coulomb O + OM(R)/ µ V DC (R) V(R) E c.m. (MeV) σ f u s i on ( m b ) O + O L = 14 16 18 20B. Fernandez, et. al. (1978)I. Tserruya, et. al. (1978)J. Thomas, et al. (1986)J. Kolata, et al. (1979)
Fig. 1. (a) Ion-Ion potential and effective mass for O+ O. (b) Corresponding fusioncross-sections. the direct influence of the excitation energy, E ∗ ( R ), on the fusion barriersby making an analogy with the coupled-channel approach and construct anew potential V ∗ ( R ) = V ( R ) + E ∗ ( R ), which has all the excitations addedto the ion-ion potential V ( R ) that should be calculated at higher energies to ctober 29, 2018 15:17 WSPC - Proceedings Trim Size: 9in x 6in sanibel˙umar minimize the nuclear rearrangements (frozen-density limit). The resultingpotentials somewhat resemble the repulsive-core coupled-channel potentialsof Ref. This approach does lead to improvements in cases where most ofthe excitation energy is in the form of collective excitations rather thanirreversible stochastic dissipation (true especially for lighter systems). Theviability of this approach requires further examination and will be studiedin the future. It is interesting to note that the gross oscillations in the cross-section at higher energies are correctly reproduced in our calculations. Thisis simply due to opening of new L -channels as we increase the collision en-ergy. Individual contributions to the cross-section from higher L vales arealso shown on the lower part of the plot.In Fig. 2a we show the DC-TDHF potential barriers for the C+O sys-tem. The higher barrier corresponds to the C+ O system and has apeak energy of 7 .
77 MeV. The barrier for the C+ O system occurs ata slightly larger R value with a barrier peak of 6 .
64 MeV. Figure 2b showsthe corresponding cross sections for the two reactions. Also shown are theexperimental data from Refs. 19–21. The DC-TDHF potential reproducesthe experimental cross-sections quite well for the C+ O system, and thecross section for the neutron rich C+ O is predicted to be larger thanthat for C+ O. R (fm) -30-20-1001020 V ( R ) ( M e V ) Point Coulomb C + O C + O E c.m. (MeV) -5 -4 -3 -2 -1 σ f u s i on ( m b ) C + O C + OExperiment
Fig. 2. (a) Ion-Ion potential for various isotopes of the C+O system. (b) Correspondingcross-sections.
Figures 3a and 3b show the corresponding potentials and cross-sectionsfor the Ca+Ca system, which was the subject of recent experimental stud-ies. The observed trend for sub-barrier energies is typical for DC-TDHFcalculations when the underlying microscopic interaction gives a good rep- ctober 29, 2018 15:17 WSPC - Proceedings Trim Size: 9in x 6in sanibel˙umar resentation of the participating nuclei. Namely, the potential barrier corre-sponding to the lowest collision energy gives the best fit to the sub-barriercross-sections since this is the one that allows for more rearrangements totake place and grows the inner part of the barrier. Considering the fact thathistorically the low-energy sub-barrier cross-sections of the Ca+ Ca sys-tem have been the ones not reproduced well by the standard models, theDC-TDHF results are quite satisfactory, indicating that the dynamical evo-lution of the nuclear density in TDHF gives a good overall description ofthe collision process. The shift of the cross-section curve with increasingcollision energy is typical. In principle one could perform a DC-TDHF cal-culation at each energy above the barrier and use that cross-section forthat energy. However, this would make the computations extremely timeconsuming and may not provide much more insight. The trend at higherenergies for the Ca+ Ca system is atypical. The calculated cross-sectionsare larger than the experimental ones by about a factor of two. Such low-ering of fusion cross-sections with increasing collision energy is commonlyseen in lighter systems where various inelastic channels, clustering, andmolecular formations are believed to be the contributing factors.
R (fm) V ( R ) ( M e V ) Point Coulomb Ca + Ca Ca + Ca Ca + Ca
45 50 55 60 65 E c.m. (MeV) -4 -3 -2 -1 σ f ( m b ) Ca + Ca Ca + Ca Ca + Ca Fig. 3. (a) Ion-Ion potential for various isotopes of the Ca+Ca system. (b) Correspond-ing cross-sections.
Recently, we have also provided extensive studies of the neutron-richsystems
Sn+ Ca and
Sn+ Ca. Such systems typically have en-ergy dependent potentials. Figure 4a shows our DC-TDHF calculationscompared with the experimental data for the Sn+ Ca system. If onecompares the measured fusion cross sections for both systems at low ener-gies, one finds the surprising result that fusion of
Sn with Ca yields ctober 29, 2018 15:17 WSPC - Proceedings Trim Size: 9in x 6in sanibel˙umar a larger cross section than with Ca. For example, at E c . m . = 110 MeVwe find an experimental cross section of ≈ Sn+ Ca as com-pared to 0 . Sn+ Ca. This be-havior can be understood by examining the DC-TDHF heavy-ion poten-tials, in Fig. 4b, which have been calculated at the same center-of-massenergy E TDHF = 120 MeV. We observe that while the barrier heights andpositions for both systems are approximately the same, the width of theDC-TDHF potential barrier for
Sn+ Ca is substantially smaller thanfor
Sn+ Ca, resulting in enhanced sub-barrier fusion at low energy.
110 120 130 140 150 E c.m. (MeV) σ ( m b ) Sn+ CaDC-TDHFexp. (HRIBF) V ( R ) ( M e V ) Sn + Ca E TDHF = 120 MeVDC-TDHF potentials
Sn + Ca Fig. 4. (a) DC-TDHF results for
Sn+ Ca system compared to data. (b) Ion-ionpotentials for the
Sn+ , Ca systems.
4. Conclusions
We have provided some of the recent result for fusion cross-sections ob-tained by the microscopic DC-TDHF method. DC-TDHF method is shownto be a powerful method for such calculations and can be readily general-ized to other dynamical microscopic theories. Considering the fact that thefitting of the Skyrme force parameters contains no dynamical informationthe result are very promising.
Acknowledgments
This work has been supported by the U.S. Department of Energy undergrant No. DE-FG02-96ER40975 with Vanderbilt University. ctober 29, 2018 15:17 WSPC - Proceedings Trim Size: 9in x 6in sanibel˙umar References
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