Coupling with two-center neutron states and two-surface collective excitations at fusion reactions in the vicinity of the Coulomb barrier

A clear physical interpretation of barrier distribution peaks and depressions is proposed. It is based on distance dependent vibration excitation energies near the barrier. Microscopic coupled channel equations with neutron rearrangement coupling were proposed and solved. They used distance dependent Q-values near the barrier. The satisfactory agreement between experimental data and calculation results is obtained for fusion cross sections in two reactions O+Ni, O+Ni in the vicinity of the Coulomb barrier.


Introduction
Two interesting aspects of fusion reactions are the fine structure of the barrier distribution function and the increase of fusion cross sections in reactions with some neutron-rich nuclei.The well known [1][2][3] coupled channel method is evolved to include these aspects in the microscopic description by solving two-center and timedependent Schrödinger equations.
Here L is the orbital momentum of partial wave, m is the reduced mass, c.m.

E E
n n = -e , ( , ) V R β is the nucleus-nucleus interaction [3], R is the internuclear distance.An approximate expression for the function ( , ) V R β with the proximity potential was obtained in [3].The well-known CCFULL code [2] (for Woods-Saxon potential) and the code included in the NRV scientific web server [4] (for Woods-Saxon and proximity potentials) are often used for the fusion cross section s calculation (Fig. 1 Here Lv j is the incoming fusion flux to the compound nucleus region in the channel n and 0 0 / j k = m ℏ .( ) The function D(E) is ambiguously determined because of experimental errors.To obtain D(E) we used the mathematically correct procedure of the two-stage spline smoothing.In the initial stage the smoothing function The barrier distribution function values ( ) and estimations of their errors were calculated on the second stage by the smoothing function It was found from the condition of the minimum of the functional analogous to (5).The function 0 ( ) g E is the result of the calculation without smoothing at the limit 0 for the reaction 40 Ca + 90 Zr and data from [5] is shown in Fig. 1 b.
Where i H l is the Hamiltonian of the independent vibration of the ith (i = 1, 2) nucleus surface, that has a multipolarity l and 0 ( ) 0 e ¥ = .In order to solve approximately the problem specified by Eq. ( 11), we use an expansion in harmonic-oscillator functions.In just the same way as in the coupled channel method [3], we take into account quadrupole and octupole vibrations of both the nuclei, with numbers 1 n and 2 n of corresponding phonons satisfying the amplitude and energy limitation.The total vibration energy of each nucleus is limited to Here max,i

De
is the upper limit for energies of vibration excitations that corresponds to their mixing with closely lying non-collective excitations.We used the estimate of the upper limit max  In the adiabatic approximation, the potential energy of the nuclear interaction for the modified vibration state with number a can be characterized by the effective potential and by the potential barrier of height B V a at B R R a = , which were studied in [7].Most populated states before the distance B R a of adiabatic potentials (13) are labelled A, B, and C in Fig. 3. Peaks of the calculated barrier distribution c.m.

( ) D E
Fig. 2) correspond to them, and there is a satisfactory agreement between the experimental and calculated distributions for the reaction 40 Ca + 90 Zr in the number of peaks, their relative heights, and the curve shape.

The neutron transfer channels coupling
Neutron transfer channels coupled equations based on the perturbed stationary states method [1] and similar to (1) were proposed in work [8,9] [ ] ( ) In the simplest approximation coupling matrix elements The equation ( 16) was solved by the method based on the series expansion of Bessel functions [7 -10].The total angular momentum projection 1 2,3 2, W = -… onto the axis connecting the centers of colliding nuclei (the internuclear axis) is the quantum number of two-center states , ( ) Energies of some two-center (molecular) states in the 18 O + 58 Ni system are plotted in Fig. 4.    y W (Fig. 8) and coefficients a a of series expansion of the time-dependent valence neutron wave function ( , ) t Y r [9,12,13] for the head-on nucleus-nucleus collision occurs for the coupling strength value 15 t F = .In the vicinity of the Coulomb barrier the transition between two-center energy levels from the 1d 5/2 (O) to 1g 9/2 (Ni) with ( ) 0 Q R > explains the essential increase in the fusion cross section in the reaction 18 O + 58 Ni as compared with the reaction 16 O + 50 Ni [11].
the deformation parameter multipolarity l ;

for the reaction 40
Ca+ 90 Zr, L=0 is shown in Fig. 2. For the energy c.m. 96 = E MeV near the first peak A in Fig. 1 b the partial probability density has one jet across the multi-It is similar to the ground deformation vibrational state.For the energy c.m. 100 = E MeV near the second peak B in Fig. 1 b the wave function 0 ( , ) R Y b has two jets across the multi-dimension potential barrier (Fig. 2 b).It is similar to the first excited deformation vibrational state.Therefore the barrier distribution may be interpreted using the energy levels ( ) R a e of the twosurface quadrupole and octupole vibrations of nuclei closely located at the distance R [7].

Figure 2 .
Figure 2. The probability density e is the neutron separation energy.Such calculations by the coupledchannel method with the proximity potential for the 40 Ca + 90 Zr fusion led to satisfactory agreement with the experimental data on the cross section c.m. two-surface vibration states EPJ Web of Conferences in the 40 Ca + 90 Zr systems are plotted in Fig. 3.In the vicinity of the Coulomb barrier (at a distance B deformations of nuclei meeting one another.The spacings De between neighbouring vibrational levels depend on R; therefore, in general, excitation energies in the barrier region differ from respective excitation energies in isolated nuclei.

Figure 3 .
Figure 3. Energies ε(R) of two-surface quadrupole and octupole vibration levels in the 40 Ca + 90 Zr system: the ground state (the curve 0), excited states going over to single-and two-phonon states of 90 Zr nuclei for R fi ¥ (curves 1 and 2, respectively).Points A, B, and C correspond to most populated states before the barriers of adiabatic potentials (14), R B is the radius of the Coulomb barrier top for spherical nuclei of radii R 1 and R 2 .
-e is the distance dependent Q-value, 0 ( ) g e = e ¥ is the energy of the initial neutron state g in the distant nucleus, ( ) R a e is the two-center (molecular) energy level and ( ) T R ¢ aa is the reduced kinetic energy coupling matrix only for states , a b appurtenant to different nuclei in the limit R fi ¥ .The coupling strength (the renormalized factor) 1 t F > is used for compensating the deletion of some complicated expressions in the exact equations of the perturbed stationary states method.Wave functions neutron with the mass m may be calculated in the two-center shell model by solving a stationary Schrödinger equation with the potential U and the spin-orbital interaction LS U

Figure 4 . 2 p
Figure 4. Energies , ( ) n R W e of two-center (molecular) states of the valence neutron with angular momentum projections onto the inter-nuclear axis Ω = 1/2 (full curves) and Ω = 3/2 (dashed curves) in the 18 O + 58 Ni system versus the nucleus-nucleus distance R; scale labels: R B is the radius of the barrier; R 1 and R 2 are radii of nuclei.The notation for states in the separated nuclei 18 O and 58 Ni is indicated.Neutron transfers from the initial state 5 / 2 1d of 18 O to unoccupied levels

Figure 7 .
Figure 7.The experimental fusion cross section