Semi-Classical Model of Neutron Rearrangement Using Quantum Coupled-Channel Approach

ISSN 16-8738, Bulletin of the Russian Academy of Sciences. Physics, 16, Vol. 8, No. 3, pp. 73 8. Allerton Press, Inc., 16. Original Russian Text V.A. Rachkov, A.V. Karpov, V.V. Samarin, 16, published in Izvestiya Rossiiskoi Akademii Nauk. Seriya Fizicheskaya, 16, Vol. 8, No. 3, pp. 34 313. Semi-Classical Model of Neutron Rearrangement Using Quantum Coupled-Channel Approach V. A. Rachkov, A. V. Karpov, and V. V. Samarin Flerov Laboratory of Nuclear Reactions, Joint Institute for Nuclear Research, Dubna, Moscow oblast, 14198 Russia e-mail: rachkov@jinr.ru Abstract A quantum coupled-channel approach with collective degrees of freedom (the rotation of deformed nuclei and/or their surface vibrations) is combined with an empirical coupled-channel model to add neutron rearrangement channels to vibrational and rotational excitations. The calculated fusion cross sections and the barrier distribution functions for several combinations of nuclei are in good agreement with experimental data. DOI: 1.313/S1687381634 INTRODUCTION Interest in the fusion reactions of heavy atomic nuclei is due mainly to the possibility of obtaining new superheavy elements [1] and new isotopes of elements away from the line of stability, along with the display of the collective properties of atomic nuclei during their approach to the distance of nuclear forces. Nuclear fusion at energies below the Coulomb barrier is essentially of a quantum nature and is largely determined by the laws of quantum tunneling through the potential barrier of nucleus nucleus interaction, the possibility of the deformation of nuclei as they collide, and the properties of collective and single-particle excitations in isolated and close nuclei. Experimental dependences of the cross section of fusion on the energy of colliding nuclei [ 4] are mainly reproduced with high accuracy in the calculations of quantum [5 1] and empirical [1 15] coupled channel models. Additional information on the process of nuclear fusion comes from the so-called barrier distribution function d Eσ de [5, 6]. To estimate the second derivative from the values of the fusion cross section, determined with some experimental error and the digitizing of graphs from original publications, the authors of [16 18] proposed several versions of the smoothing of cubic splines described in [19] that yield similar results, particularly a fine structure from two or three peaks for a number of reactions, e.g., 16 O + 144 Sm [6, 16], 36 S + 9 Zr [16], and 4 Ca + 9 Zr [5, 16 18, ]. Such fine structures are reproduced in calculations that include the strong coupling between the relative motion of nuclei with quadrupole and octupole vibrations of the nucleus surfaces [6, 16 18, ]. Computational schemes for solving systems of differential equations with a matrix of coupling according to the potential energy are employed in the CCFULL program [7] and in the network knowledge base on low energy nuclear physics [1]. For a number of reactions, particularly 4 Ca + 96 Zr [5, 1, 17, 1, ] and 18 O + 58 Ni [1, 3], taking into account only such a coupling is not enough to describe the fusion process, since the calculation results for the sub-barrier energies are much lower than the experimental values of the cross sections. Additional enhancement of the sub-barrier fusion cross section occurs due to the rearrangement of neutrons with positive Q values [1 5]. This effect becomes apparent when we compare the fusion cross section for two close projectile target combinations where in one of which the neutron rearrangement channels with positive Q values are possible, whereas for another all neutron transfer channels have negative Q values, e.g., 4 Ca + 96 Zr and 4 Ca + 9 Zr [1, ], and 16 O + 6 Ni and 18 O + 58 Ni [3]. The rearrangement role of neutrons with positive Q of the reaction can be explained as follows: as the colliding nuclei approach each other, the wave function of a valence neutron (initially localized in one of the nuclei) begins to penetrate into the volume of another nucleus before the nuclei overcome the Coulomb barrier [6 3]. This can occur with the rearrangement of energy between the neutron and the relative motion of the colliding nuclei. This can be demonstrated for one-dimensional quantum three-body (two nuclear core and neutron) systems by solving the nonstationary Schrödinger equation in [6, 9]. The effect of neutron rearrangement upon the transition to the underlying (with Q > ) and overlying (with Q < ) twocenter states on the cross section of nuclear fusion was studied in [18, 9, 3] using the simplified equations of perturbed stationary states proposed in [31, 3] with a matrix of coupling according to kinetic energy. Considering the coupling between the relative motion of nuclei and both the collective (vibrational) degrees of freedom and the rearrangement of nucleons is a complicated problem that has yet to be completely 73

74 RACHKOV et al. solved. In a stationary statement, the problem arises upon the decomposition of the total wave function on the excited states of the colliding nuclei and simultaneously on the states with the rearrangement of nucleons. Using a non-orthogonal and overcomplete set of basis functions in this case complicates calculations with the use of special mathematical methods. Choosing simplified models is a better alternative. In the CCFULL [7] program, which uses a phenomenological additive to the coupling matrix on the potential energy, the transition of pair of neutrons between the ground states of nuclei is considered. In the model proposed in [33], it was assumed that the properties of collective excitations of colliding nuclei change after the transfer of neutrons near the top of the Coulomb barrier, affecting the fusion cross section. An empirical coupled channel model with neutron rearrangement was proposed in [1] and successfully used in [1 15] to describe experimental data on the cross sections of fusion. The basic approach of this model is to introduce an empirical distribution function over the dynamic barriers to consider the coupling between the relative motion of colliding nuclei with collective degrees of freedom. Simplified approximations of this using the Gaussian distribution often result in visible deviations from the experimental data near the Coulomb barrier. The calculated barrier distribution function in this model always has one peak, while a complex nonmonotonic structure is observed in experiments. The ambiguous determination of the parameters of empirical distribution function over the dynamic barriers (using the liquid-drop model and/or from systematic study of the available experimental data) also limits its possibilities for microscopic descriptions of the fusion of atomic nuclei. To eliminate these disadvantages, the distribution function over dynamic barriers was determined in [4] using the quantum method of strong coupling with collective degrees of freedom. This led to good agreement between calculations of the cross section and the experimental data for the fusion reaction 3 S + 96 Zr. In this work, a semi-classical model is proposed that combines the quantum coupled channel aproach (without introducing an artificial distribution function over the dynamic barriers) and the empirical method of considering the rearrangement of neutrons using a quasi-classical approximation for the probability of neutron transfer. THEORETICAL ANALYSIS The fusion cross section can be represented in the form of partial decomposition σ ( E) = π ( l + 1 ) Tl ( E), k l = (1) where μ is the reduced mass of the colliding nuclei, l is the orbital momentum, E = ħ k µ is the energy of collision in the center-of-mass system, and T l (E) is the partial penetrability of the potential barrier. We define penetration probability T l (E) for the Hamiltonian of the colliding nuclei Hˆ = ħ Δ+ V(, r α) + Hˆ int( α), () μ where α denotes the internal (single-particle and collective) degrees of freedom, H ˆ int ( α ) is the Hamiltonian corresponding to these variables, and V(, r α) is the interaction potential of the colliding nuclei. The solution of the stationary Schrödinger equation HΨ ˆ k = EΨ k with boundary conditions corresponding to the collision of nuclei in the approximation of smallness of their spins, compared to the orbital angular momentum of relative motion, can be expanded in partial waves 1 l iσl Ψ k( r, θα ; ) = i e ( l + 1 ) χl ( r, α) Pl ( cos θ), (3) kr l= where Coulomb scattering phase σ i = argγ(l + 1 + iη), kz1z e η= is the Sommerfeld parameter, and Pl E denotes Legendre polynomials. Substituting expression (3) into the Schrödinger equation produces the system of equations for partial wave functions ll ( + 1) χ l ( r, α ) χ l ( r, α ) r r (4) μ + [ E V ( r, α) Hint ( α) ] χ ( α ) = l r,. ħ Functions χl ( r, α) can be expanded into the complete set of wave functions ϕ( α) of the internal Hamiltonian Ĥ int H ˆ int ϕ ( α) = εϕ ( α), (5) χl ( r, α) = ψl, ( r) ϕ( α). (6) The radial wave functions describing the relative motion of nuclei in channel satisfy the system of radial Schrödinger equations [7 9, 31] d ll ( + 1) μ ψ ψ + [ ε l, () r l, () r E V() r ] dr r ħ (7) μ ψl, () r Vγ() r ψ γ = l, () r ħ γ with the coupling matrix on potential energy V * γ() r = ϕγ ( α) V ( r, α) ϕ( α) dα, (8) where non-diagonal elements are responsible for the system s transition from one channel to another. For the collective degrees of freedom, potential energy of two deformable and/or rotating nuclei V ( r, α) can be BULLETIN OF THE RUSSIAN ACADEMY OF SCIENCES. PHYSICS Vol. 8 No. 3 16

SEMI-CLASSICAL MODEL OF NEUTRON REARRANGEMENT 75 presented as the sum of the Coulomb and the nuclear parts V ( r, α ) = VC( r, α ) + VN ( r, α). (9) The internal motion of nuclei is described by the Hamiltonian ħ Iˆ i Hint( α ) = + 1 + 1 C λβ (1) i iλ, λ = 1, i = λ d λ i i 1, i i which includes the kinetic energy of the nuclei rotation ( i denotes the inertia moments of nuclei, i = 1, ) and the kinetic and potential energy of small (harmonic) oscillations of their surfaces (where Ci λ is the rigidity parameter; β iλ is the amplitude of the deformation of the nuclear surface of the i-th nucleus with multipolarity λ =, 3, ; and di λ is the mass parameter). The eigenfunctions of Hamiltonian (1) for surface oscillations are expressed in terms of Hermite polynomials, and are proportional to the spherical harmonics for rotation. The boundary conditions needed to solve coupled radial Schrödinger equation (7) were formulated so that the equations would describe the fusion of atomic nuclei. It was assumed that upon contact between the surfaces of light- and medium-weight colliding nuclei, fusion occurs with a probability close to unity. The flow of probability behind the Coulomb barrier is completely absorbed (i.e., a compound nucleus is formed) and is not reflected from the inner region. This means that when r < R fus R1 + R, only convergent waves correspond to functions χl(, r α) [7, 9]. At great distances, we have the standard boundary conditions for the wave function, which are in the form of convergent (incident) wave in the entrance channel and divergent waves in all other channels: ( ) ψl, ( r ) = i hl ( η, kr) δ (11) 1 k + l ( ) Shl ( η, kr), k µ where k = E, E = E ε, ε is the excitation ħ energy of the nucleus in channel ; ε =, kzze 1 ( ± η =, h ) l ( η, k r) denote the Coulomb partial E wave functions with asymptotics exp ( ±ixl, ), where xlv, = kr η lnkr + σl, lπ and σ l, = arg Γ ( l + 1 + iη) represent the Coulomb partial scattering phases; and S indicates partial elements of the l scattering matrix. System of coupled equations (7) with the given boundary conditions is solved numerically [7, 1]. The fusion cross section taking into account collective degrees of freedom can thus be calculated using formula (1), in which the probability of passing through the barrier in channel is defined as the ratio of the probability flow reaching a certain distance R fus to the incident flow (CC) T 1 l ( E) = jl, ( E). (1) j( E) Let us consider a semi-classical method that includes the channels of rearrangement of x ( 1 x x max ) neutrons with positive and negative values of Q xn of the reaction in describing nuclear fusion within coupled channel method. When Q xn >, part of energy Q Qxn can become the kinetic energy of the relative motion of nuclei, which increases the penetrability of the barrier T l(cc) ( E + Q) T l(cc) ( E). For the complete probability of neutron rearrangement max α ( ElQ,, ) = α ( ElQ,, ). tr x x= 1 (13) The expression for total penetrability of Coulomb barrier is obtained after integration over Q: max{ } Q xn 1 T ( E) = N [ δ ( Q) +α ( E, l, Q) ] l tr tr E ( CC) (14) Tl ( E + Q) dq. The constant in expression (14) is determined from the normalization condition N tr Q xn = 1 + α ( E, l, Q) dq. E tr (15) The case α tr = and N tr = 1 corresponds to no rearrangement of neutrons. As an expression for probability α x of the subsequent rearrangement of neutrons, we use an estimate of the probability of the subsequent transfer of neutrons in the quasi-classical approximation [1, 33, 34] 1 ( ElQ,, ) N exp( Q ) α = σ exp( κ [ D( E, l) D ]). x x x x x (16) Here κ = max x κ( ε denote the = 1 i), κ( ε i i) = μnεi ħ, εi energy of separation of the i-th transferred neutron, and the normalization constant is determined by the condition Q xn (17) In expression (16), D( E, l) is the distance of the closest approach between interacting nuclei moving along the Coulomb trajectory with angular momentum l; D = R1 + R + d, R i = r A denote the ( n) ( n) ( n) ( n) 13 ( n) radii of orbits of valence (transferred) neutrons ( r and are adjustable parameters). Dispersion of dis- d x ( x) N = exp Q σ dq. x E α tr BULLETIN OF THE RUSSIAN ACADEMY OF SCIENCES. PHYSICS Vol. 8 No. 3 16

76 1. α tr Q 1n RACHKOV et al. tive values of Q makes a negligible contribution to integral (16), due to the rapid decrease in penetrability (CC) T l at sub-barrier energies. For practical application of the developed model, the lower limit of integration in the expression (14) can thus be replaced by the value max ( ). Q xn.5 1 1 Q, MeV tribution of the value Q for the probability of neutron transfer in formula (17) can be written as Q n Q 3n Q 4n Fig. 1. Typical dependence of the total probability of neutron transfer (13) on Q for Q xn = x. α tr ħ κ xb σ x =, (18) μrb where B and R B are the height and position of the Coulomb barrier for two spherical nuclei, respectively. Parameters r 1 = 1.5 fm, r = 1.5 fm, and d =.5 were the same as in [14, 15]. Slightly larger values of the parameters were obtained in [34, 36, 37] by analyzing experimental data on transfer reactions: r 1 = r = 1.4 fm. This led to lower values of α x for transfer reactions, compared to fusion reactions. The difference could be due to the probability of neutron transfer being determined after the separation of nuclei, while the probability of neutron rearrangement is determined at the time of their closest approach. A typical form of the distribution over Q α x ( ElQ,, ) is shown in Fig. 1. Each value of Q xn defines the threshold for the transmission of the corresponding number of neutrons, which explains the stepwise behavior of the distribution function. The probability of neutron rearrangement at subbarrier energies is quite low, so the contribution from delta function δ(q) predominates in formula (14), and is responsible for fusion without rearrangement. The probability of neutron transfer (16) falls rapidly with an increase in the number of transferred neutrons x. The sub-barrier fusion of atomic nuclei is thus strongly affected by intermediate rearrangements of only one and/or two neutrons with Q >. In this work, rearrangements of up to four valence neutrons were considered. It should be noted that integration over nega- RESULTS AND DISCUSSION Let us consider the use of the above model for describing the fusion of nuclei that are spherical in the ground state: 3 S, 4 Ca, 6, 64 Ni, 9, 94, 96 Zr, and 1 Mo. We ignore the rotation of the nuclei, i.e., omit the corresponding term in the Hamiltonian (1). For all calculations, nucleus nucleus interaction V N took the form of Woods Saxon potential with Akyüz Winther parameterization [38]. The properties of the low-lying vibrational states of these nuclei (multipolarity λ, phonon energy E λ, and rms amplitudes of zero-point vibrations β λ ) are given in Table 1 [1, 39, 4]. Quadrupole and octupole vibrational modes were considered in all combinations of colliding nuclei, with the exception of the 4 Ca nucleus ( E = 3.9 MeV, β + = +.13), the quadrupole mode of which is rather weak with respect to the octupole. Adding it to the calculation scheme shifts the σ( E) curve slightly toward lower energies in the region of sub-barrier energies. Agreement between the calculation results and the experimental data can in this case be achieved simply by slightly varying the parameters of the potential. The barrier distribution functions DE ( ) = 1 d Eσ (19) π RB de were estimated via smoothing by splines [18]. Values Q xn for the rearrangament of x neutrons from the ground to the ground state for the considered reactions are shown in Table. For reactions 4 Ca + 9 Zr and 3 S+ 9 Zr, all values of Q xn < and calculations of the fusion cross section according to formula (14) were performed without rearrangement of neutrons: α tr =, N tr = 1. For the other reactions in Table, Q xn > and the rearrangement of neutrons must be considered. The tabulated values of dispersions σ x Qxn indicate the high probability of the rearrangement of neutrons with Q >. The experimental data and calculation results for the fusion cross section without rearrangement of neutrons in the reaction 4 Ca + 9 Zr are shown in Fig., along with the distribution function on the barriers. Both the fusion cross section and the distribution function on the barriers are satisfactorily reproduced using quantum coupled channel model taking into account the vibrational properties of colliding nuclei. BULLETIN OF THE RUSSIAN ACADEMY OF SCIENCES. PHYSICS Vol. 8 No. 3 16

SEMI-CLASSICAL MODEL OF NEUTRON REARRANGEMENT 77 Table 1. Vibrational properties of the nuclei used in our calculations with strong channel coupling. Parameter n ph is the number of phonons included in the calculations. The data were taken from [1, 39, 4] Nucleus π ( λ ) n E l, MeV b l 4 Ca (3 ) 1 3.737.411 3 S 9 Zr 94 Zr 96 Zr 6 Ni 64 Ni 1 Mo ph ( + ) 1.3.315 (3 ) 1 5.6.4 ( + ) 1.186.89 (3 ) 1.748.11 ( + ) 1.919.9 (3 ) 1.58.193 (+) 1 1.751.8 (3 ) 1 1.897.84 ( + ) 1 1.333.7 (3 ) 1 4.4.9 ( + ) 1 1.346.179 (3 ) 1 3.56.1 ( + ) 1.536.31 (3 ) 1 1.98.18 One possible reason for the formation of a structure containing at least two distinct maxima is excitation with a high probability of coupled vibrations on the surfaces of approaching nuclei [17, 18, ]. The energies of one or several phonons are in this case approximately equal to the distance between the peaks. This results in splitting of the probability flow upon crossing the multidimensional potential barrier, depending in this case on internuclear distance r and parameters β λ of the dynamic deformation of nuclei, and most of all on the parameter of the dynamic octupole deformation of the 9 Zr nucleus (Fig. 3) [18]. The experimental data and calculation results for fusion cross sections with and without neutron rearrangament in the reactions of 4 Ca + 94, 96 Zr are shown in Fig. 4, along with the distribution functions on the barriers. We can see that considering only the collective degrees of freedom greatly underestimates the fusion cross section for combinations of the 4 Ca + 94, 96 Zr reactions (dashed curves in Fig. 4) for which the channels of neutron transfer are open from zirconium to calcium with positive Q xn (Table ). At the same time, there is good agreement between the experimental data and theoretical curves, calculated taking into account the rearrangement of neutrons in the model described above. A similar situation is observed for the fusion reactions 3 S + 9, 94, 96 Zr (Fig. 5). As before, additional enhancement of the sub-barrier fusion cross section due to the rearrangement of neutrons is observed for Table. Values of Q xn for transferred neutrons and the values of σ x. All values are given in MeV Reaction Q 1n σ 1 Q n σ Q 3n σ 3 Q 4n σ 4 4 Ca + 9 Zr 3.61 1.44 5.86 4.18 4 Ca + 94 Zr +.14 4. +4.89 5.79 +4.19 7.19 +8.13 8.6 4 Ca + 96 Zr +.51 4.13 +5.53 5.69 +5.4 7.6 +9.64 8.1 3 S + 9 Zr 3.61 1.44 5.86 4.18 3 S + 94 Zr +.4 4.18 +5.11 5.75 +3.46 7.15 +6.15 8.1 3 S + 96 Zr +.79 4.1 +5.74 5.65 +4.51 7. +7.66 8.5 64 Ni + 1 Mo.19 +.83 5.48. 1.3 6 Ni + 1 Mo.47 +4. 5.65 +.39 7.4 +5.3 8.8 BULLETIN OF THE RUSSIAN ACADEMY OF SCIENCES. PHYSICS Vol. 8 No. 3 16

78 RACHKOV et al. σ, mb (a) β 3 (a).4 1. 1 1..4 1 (b) 85 9 95 1 15 11 D, MeV 1 (b).3.4.. 4 Ca + 9 Zr..4.1 (c).4 85 9 95 1 15 11. Fig.. (a) Fusion cross section σ and (b) distribution function D on the barriers for the reaction 4 Ca + 9 Zr. Calculations with and without taking into account dynamic deformations are denoted by the dashed and dotted lines, respectively. Experimental data on the fusion cross sections (dots) were taken from [1]. The estimate of function D(E) from [] (solid symbols) and one obtained as described in [18] (empty symbols) are shown. the combination of 3 S + 94, 96 Zr, for which the values of Q xn are positive (Table ). It should be noted that despite the good agreement between the calculations and experimental data, the cross section for the reaction 3 S + 96 Zr (the solid line in Fig. 5c) is slightly overestimated at deep sub-barrier energies and underestimated for the fusion reaction of 3 S + 94 Zr (the solid curve in Fig. 5d). Note that the cross section of 4 Cа + 94 Zr is slightly underestimated at deep sub-barrier energies. These differences are easily eliminated by, e.g., slightly varying the potential parameters of nucleus nucleus interactions or making minor changes to neutron radius R () n. In this work, we did not try to adjust the parameters of the model so as to achieve the best description of the experimental data...4 (d).4...4 8 1 1 14 R, fm Fig. 3. Probability density χ(, r β3) intersecting twodimensional potential barrier V(, rβ for the fusion reaction 4 Ca + 9 3) Zr with l = and E cm = (a) 94, (b) 96, (c) 98, and (d) 1 MeV; r is the distance between the centers of nuclei; β 3 is the diameter of dynamic octupole deformation the 9 Zr nucleus. The values of V from [18] are shown. BULLETIN OF THE RUSSIAN ACADEMY OF SCIENCES. PHYSICS Vol. 8 No. 3 16

SEMI-CLASSICAL MODEL OF NEUTRON REARRANGEMENT 79 σ, mb D, MeV 1 (a) (c).4 1 4 Ca + 94 Zr.3 4 Ca + 94 Zr 1 1. 1.1 1 1 1 (b).4 (d) 1 1 1 1 1 1 1 3.3 4 Ca + 96 Zr 4 Ca + 96 Zr..1 1 4 85 9 95 1 15 11 85 9 95 1 15 11 Fig. 4. Cross section of fusion σ (a, b) and the barrier distribution function D (c, d) for the reactions of 4 Ca + 94, 96 Zr. The dashed and dotted curves denote calculations with and without taking into account dynamic deformations, respectively. The solid curves represent calculations using model taking into account the rearrangement of neutrons. Experimental data on the cross sections 4 Ca + 94 Zr were taken from []. The data on the 4 Ca + 96 Zr reaction are from [1] (empty symbols) and [] (solid symbols). The estimate of function D(E) uses the same symbols as in Fig.. The obtained good agreement between the calculations and experimental data using the standard parameters of the models show that the above approach can be used not only to describe existing data, but also to predict new experimental results. For fusion reactions with the rearrangement of neutrons ( 4 Ca, 3 S + 94, 96 Zr), the distribution functions on the barriers, calculated taking into account the collective degrees of freedom only (the dashed curves in Figs. 4, 5) are much narrower when compared to the experimental values. Adding the channels of neutron rearrangement broadens the distribution function on the barriers without noticeable change in its structure (solid curves). There is good agreement between the calculated distribution functions on the barriers in the proposed model and the experimental data over the entire range of energies, including the low-energy region. This cannot be achieved by considering collective degrees of freedom only. In [14, 15], it was concluded that the contributions to the enhancement of sub-barrier cross section resulting from the rearrangement of neutrons and the excitation of collective degrees of freedom and/or rotation were not additive. This means that the greatest effect due to the rearrangement of neutrons with Q > is observed for the combinations of the most rigid colliding nuclei, i.e. for those whose coupling with collective degrees of freedom is slight. For medium-mass and heavy nuclei, the slightest effect on coupling with collective degrees of freedom is achieved for spherical nuclei with the high energy of the first vibrational state (magic nuclei or similar to them). All these conclusions are confirmed in this model. Let us illustrate this using the example of fusion reaction 6, 64 Ni + 1 Mo [41, 4]. In the fusion of 64 Ni and the 1 Mo nucleus, BULLETIN OF THE RUSSIAN ACADEMY OF SCIENCES. PHYSICS Vol. 8 No. 3 16

8 RACHKOV et al. σ, mb 1 1 1 (a) D, MeV 1.4 3 S + 9 Zr.3 3 S + 9 Zr (d) 1. 1 1.1 1 1 (b).4 (e) 1 1 3.3 S + 94 Zr 3 S + 94 Zr 1. 1 1.1 1 1 1 1 (c).4 3 S + 96 Zr.3 3 S + 96 Zr (f) 1. 1 1.1 1 7 75 8 85 9 95 7 75 8 85 9 Fig. 5. Cross section of fusion σ (a c) and the barrier distribution function D (d f) for reactions 3 S + 9, 94, 96 Zr. The designations of the curves are same as in Fig. 4. The experimental data on the cross sections and the estimates of function D(E) for reactions 3 S + 9, 96 Zr were taken from [4]; the estimates for 3 S + 94 Zr are from [5]. the channels of neutron rearrangement are energetically unfavorable (Q xn < ), and it is necessary to take into account only the coupling between the relative motion of colliding nuclei and the collective degrees of freedom. In the fusion reaction of 6 Ni + 1 Mo, the values of Q xn are positive and close to those of the reaction 4 Ca + 96 Zr (Table ). According to experimental observations, the differences between the fusion cross sections of reactions 6, 64 Ni + 1 Mo are minor (Fig. 6). This is explained by our calculations using this approach and is associated with different vibrational properties of the nuclei involved in the reactions. The 1 Mo nucleus is softer than the more rigid 96 Zr nucleus, so the amplitude of vibrations on the nuclear surface of 1 Mo is greater. This results in greater enhancement of the fusion due to coupling BULLETIN OF THE RUSSIAN ACADEMY OF SCIENCES. PHYSICS Vol. 8 No. 3 16

σ, mb 1 (a) SEMI-CLASSICAL MODEL OF NEUTRON REARRANGEMENT 81 ACKNOWLEDGMENTS This work was supported by the Russian Foundation for Basic Research, project no. 15-7-7673-a. 1 1 1 4 1 1 1 (b) 1 4 1 13 14 15 16 with the collective states, and the effect of neutron rearrangement becomes less noticeable. CONCLUSIONS 64 Ni + 1 Mo 6 Ni + 1 Mo Fig. 6. Cross section of fusion σ for the reactions 64, 6 Ni + 1 Mo. Designations are the same as in Fig. 4. Experimental data for the reactions 64 Ni + 1 Mo were taken from [41]. The data for 6 Ni + 1 Mo are from [4]. The proposed new method of treatment of the rearrangement of neutrons in the quantum coupled channel approach is based on the quasi-classical expression for the probability of neutron transfer and quantum treatment of the coupling of relative motion and the collective degrees of freedom. For the considered fusion reactions calculations performed with the proposed model resulted in good agreement with the experimental data for both the fusion cross section and the distribution functions on the barriers without varying the parameters of the model. The proposed model can be also applied to the fusion of both statically deformed nuclei and a statically deformed nucleus and a spherical nucleus. REFERENCES 1. Oganessian, Yu.Ts., Utyonkov, V.K., Lobanov, Yu.V., et al., Nucl. Phys. A, 4, vol. 734, p. 19.. Wei, J.X., Leigh, J.R., Hinde, D.J., et al., Phys. Rev. Lett., 1991, vol. 67, p. 3368. 3. Stefanini, A.M., Ackermann, D., Corradi, L., et al., Phys. Rev. Lett., 1995, vol. 74, p. 864. 4. Trotta, M., Stefanini, A.M., Corradi, L., et al., Phys. Rev. C, 1, vol. 65, p. 1161. 5. Rowley, N., Nucl. Phys. A, 1998, vol. 63, p. 67. 6. Dasgupta, M., Hinde, D.J., Leigh, J.R., et al., Nucl. Phys. A, 1998, vol. 63, p. 78. 7. Hagino, K., Rowley, N., and Kruppa, A.T., Comput. Phys. Commun., 1999, vol. 13, p. 143. 8. Samarin, V.V. and Zagrebaev, V.I., Nucl. Phys. A, 4, vol. 734, p. E9. 9. Zagrebaev, V.I. and Samarin, V.V., Phys. At. Nucl., 4, vol. 67, p. 146. 1. Fusion Code of the NRV. http://nrv.jinr.ru/nrv/ 11. Zagrebaev, V.I., Phys. Rev. C, 1, vol. 64, p. 3466. 1. Zagrebaev, V.I., Phys. Rev. C, 3, vol. 67, p. 6161. 13. Adel, A., Rachkov, V.A., Karpov, A.V., et al., Nucl. Phys. A, 1, vol. 876, p. 119. 14. Rachkov, V.A., Karpov, A.V., Denikin, A.S., and Zagrebaev, V.I., Phys. Rev. C, 14, vol. 9, p. 14614. 15. Rachkov, V.A., Karpov, A.V., Denikin, A.S., et al., Bull. Russ. Acad. Sci.: Phys., 14, vol. 78, p. 1117. 16. Samarin, V.V., Bull. Russ. Acad. Sci.: Phys., 5, vol. 69, p. 176. 17. Samarin, V.V. and Balyasnikova, N.S., Bull. Russ. Acad. Sci.: Phys., 5, vol. 69, p. 38. 18. Samarin, V.V., EPJ Web Conf., 15, vol. 86, p. 39. 19. Marchuk, G.I., Metody vychislitel noi matematiki (Methods of Calculus Mathematics), Moscow: Nauka, 198.. Samarin, V.V., Phys. At. Nucl., 9, vol. 7, p. 168. 1. Timmers, H., Ackermann, D., Beghini, S., et al., Nucl. Phys. A, 1998, vol. 633, p. 41.. Stefanini, A.M., Behera, B.R., Beghini, S., et al., Phys. Rev. C, 7, vol. 76, p. 1461. 3. Borges, A.M., Silva, C.P., Pereira, D., et al., Phys. Rev. C, 199, vol. 46, p. 36. 4. Zhang, H.Q., Lin, C.J., Yang, F., et al., Phys. Rev. C, 1, vol. 8, p. 5469. 5. Jia, H.M., Lin, C.J., Yang, F., et al., Phys. Rev. C, 14, vol. 89, p. 6465. 6. Zagrebaev, V.I., Samarin, V.V., and Greiner, W., Phys. Rev. C, 7, vol. 75, p. 3589. 7. Umar, A.S. and Oberacker, V.E., Phys. Rev. C, 8, vol. 77, p. 6465. 8. Simenel, C., Wakhle, A., and Avez, B., J. Phys.: Conf. Ser., 13, vol. 4, p. 1118. BULLETIN OF THE RUSSIAN ACADEMY OF SCIENCES. PHYSICS Vol. 8 No. 3 16

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