A new dynamical mechanism of incomplete fusion in heavy-ion collision

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The incomplete fusion (ICF) of nuclei at the heavyion collision is observed in reactions of light projectiles with the intermediate-mass target nucleus.This phenomenon was first observed more than 60 years ago [1].
From the analysis of experimental data it was found that the peripheral collisions are a favorable condition for the incomplete fusion.This phenomenon is studied by observation of the α particle flying in the forward angles or other light clusters or by identification of the evaporation residue accompanied with the emitted fast light clusters.
A mean value < L >= 40 of the angular momentum distribution of the entrance channel corresponding to the incomplete fusion was observed for the 159 Tb( 14 N, αxn) 169−x Yb reaction [2], while the estimated value < L >= 30 was presented in Ref. [3] for the 16 O+ 146 Nd reaction.The cross section of the evaporation residues (ER) formed in the incomplete fusion increases gradually with the mass asymmetry of the reaction entrance channel [4].The ER presence accompanied with the emission of the α particle formed in the incomplete fusion has been established from the analysis of the total cross section for the α-particle production by the standard statistical models ALICE-91 [4] and PACE4 [5,6].The conclusion of the authors is based on the enhancing underestimation of the measured cross sections of α-emitting residues by the theoretically predicted cross sections.
The breakup fusion model [7] was suggested to describe ICF.In this model, the projectile is assumed to break up into a light cluster and a conjugate nucleus at close distances to the target nucleus.
The sum-rule model [8,9] developed by Wilczynski et al. was used to calculate the ER cross section of various projectile-like fragments formed in ICF reactions.The authors concluded that ICF reactions are localized in the angular momentum space above the critical angular momentum cr for complete fusion (CF) of projectile and target.
The breakup of the projectile was analysed in the recent paper [10] by R. V. den Bosshe and A.D. Torres by combining a classical trajectory model with stochastic breakup with the quantum-mechanical fragmentation * nasirov@jinr.rutheory [11] treatment of two-body clusterization and decay of a projectile.The angular distributions of the clusters 4 He and 8 Be produced at the breakup of projectile in the 20 Ne+ 208 Pb reactions explored and compared with the experimental data.
We should stress that the projectile breakup mechanism of the incomplete fusion was assumed in all of the above-listed theoretical methods to analyse measured data.
In this work, we consider a new mechanism of the ICF reaction as a quasifission to calculate the excitation function of the evaporation residues (ER) survived against fission, which is accompanied by the α-particle emission from the dinuclear system (DNS) (see Fig. 1).The DNS is formed at the capture(full momentum transfer) of the projectile nucleus by the target nucleus.
This mechanism is based on the DNS concept of the complete fusion which operates with such physical quantities as intrinsic fusion barrier B * fus , quasifission barrier B qf and the excitation energy E * Z of the DNS with the charge asymmetry Z.The DNS evolution by the diffusion process due to the nucleon transfer between fragments leads to the formation of the α particle in collisions with the large orbital angular momentum if capture takes place at the given beam energy.
The increase of the orbital angular momentum in the entrance channel leads to the following changes of the physical quantities causing the enhance of the incomplete fusion probability.
i) The increase of the dynamical intrinsic barrier B * fus to complete fusion at the very asymmetric charge and mass distribution corresponding to the α particle (see Figs. 2 and 3. ii) The decrease of the stability of the DNS due to decrease the depth of the potential well of the nucleusnucleus interaction.It is called the quasifission barrier B qf (see Fig. 2).
iii) The excitation energy E * Z of the DNS with the charge asymmetry Z, which is generated from the total kinetic energy loss at the capture of the projectile by the target nucleus, decreases due to increase the DNS rotational energy (see Fig. 2).Therefore, the residue nucleus formed in the incomplete fusion is less heated than the compound nucleus formed in the complete fusion.iv) The effect of the centrifugal force on the very massasymmetric DNS enhances leading to the incomplete fusion which can be considered as the DNS quasifission producing very mass-asymmetric products.
The potential energy surface (PES) presented in Fig. 2 is calculated as a sum of the nucleus-nucleus interaction V and reaction energy balance Q gg [12]: where Z and A are charge and mass numbers, respectively, of a DNS fragment, and the ones of the conjugate fragment are , and B CN are the binding energies of the interacting nuclei and CN, respectively, which are obtained from the nuclear mass tables in Refs.[13,14]; β i and α i represent deformation parameters (quadrupole and octupole) of the DNS fragments and orientation angles of the axial symmetry axis of the deformed nuclei to the beam direc- The driving potential of the DNS formed in the 16  tion, respectively.In case of the nuclei with at spherical shape at their ground state, the change in their shape due to surface vibration at the zero-point motion is considered [12].The amplitudes of vibrations are taken equal to the values of the deformation parameters of the first quadrupole 2 + and octupole 3 − collective excitations of nuclei (β + 2 ) [15] and (β − 3 ) [16].The dependence of the barrier B * fus on L is seen from the analysis Fig. 2 which shows the PES values for the very asymmetric charge asymmetry (Z → 2 and Z c → 58) increase strongly with L due to smallness of the moment of inertia DNS with the α particle.Therefore, the fusion probability decreases by the increase of L, since the intrinsic fusion barrier B * fus increases and quasifission barrier B qf decreases (see Fig. 2) by increasing L. This circumstance is a reason leading to the creation of the favorable range of the angular momentum for the incomplete fusion in the peripheral collisions at the large beam energies.The enhance of the rotational energy in the PES with L is related with the strongly decrease of moment of inertia of the very asymmetric DNS: which is used in calculation of the rotational energy: The reduced mass of DNS and moments inertia of the interacting nuclei are calculated by the expressions µ = , respectively; m is a nucleon mass; a i and b i are small and large radii of nuclei; R m is the distance corresponding to the minimum of the potential well of the nucleus-nucleus interaction; α i and β i are the orientation angle of the axial symmetry axis and the deformation parameter of the DNS fragments, respectively.
The excitation energy E * Z of DNS at the given value of the beam energy is calculated taking into account the change in the intrinsic energy of DNS at the change of nucleon numbers of fragments: where ∆Q gg (Z, and B c are binding energies of the initial ("P and "T ") and interacting fragments; is the minimum value of the potential well, and it is a function of the nuclear shape β i and orientation angles α i of the axial symmetry axis of the deformed nuclei to the beam direction [17].The probability of the α-particle formation and its yield has been estimated as a quasifission fragment by the formula where D Z (A, E * Z , L, t) is the probability of population of the DNS configuration (Z, Z CN − Z) for a given set of E * Z and L; a value of k max corresponds to the interaction time t of the DNS fragments when D Z (A, E * Z , L, t int ) < 10 −5 , i.e. the DNS has gone to complete fusion or it has broken up as quasifission products (see Fig. 4).The part of D Z going to region Z < 2 is a contribution to the complete fusion.The evolution of the charge distribution D Z is calculated by the transport master equation [18] with initial conditions D Z (A, E * Z , L, t = 0) = 1 for Z = Z P (Z T ) and A = A P (Z T ); Λ qf Z is the width of the decay through the quasifission barrier which is calculated by the expression where T Z is the effective temperature of the DNS with the charge asymmetry Z: The transition coefficients of the transport master equation depend on the energy, spin, and occupation numbers of the single-particle states of the nucleons in the DNS fragments (see Refs. [12,18] for details).The occupation numbers of nucleons in the DNS fragments depend on T Z .
Evolution of the charge distribution D Z of the DNS and yield Y Z of fragments calculated for the 16  of the quasifission process by the following expression: (6) where σ cap (E lab , L) = πλ −2 P cap (E lab , L), where λ −2 is the de Broglie wavelength corresponding to the collision energy E lab and P cap is the capture probability which is found from the calculation of the collision trajectory for the given values of E c.m. and orbital angular momentum L [12].
The increase of the probability of the mass and charge distribution at the Z = 2 and A = 4 corresponding to the α particle occurs in the collisions with L =40-60 for the 16 O+ 130 Te reaction and in the collisions with L =30-40 for the 18 O+ 93 Nb reaction.These results allow us to make theoretical analysis of the incomplete fusion mechanism by the calculation of the evaporation residues excitation function and to compare with the measured data from Refs.[6,19].
The partial cross sections of the incomplete fusion for the 16 O+ 130 Te and 18 O+ 93 Nb reactions are presented in Figs. 6 and 7 system (see Fig. 8).The important result is the formation of the wide range E * Z (E lab , , {β i , α i })=10-50 MeV of the DNS excitation energy for the angular momenta L =35-50 of DNS corresponding to the incomplete fusion.The nearly plateau of the excitation function of the evaporation residues of the 1n-5n channels for the wide of the beam energy (65-105 MeV) is related with this Comparison of the theoretical cross sections (solid curve) of the evaporation residues formed in the 18 O+ 93 Nb incomplete fusion reaction after emission of 3 neutrons with the measured experimental data (squares) presented in Ref. [6]. phenomenon.
The ER cross sections of the xn channels of the incomplete fusion accompanied with the emission of the α particle have been calculated in the framework of the DNS model [12].We should note that the excitation energy of the conjugate nucleus after emission of α particle is for the given values of orientation angles α i of the DNS fragments.The evaporation residue (ER) cross section of the xn channel (x neutrons have been emitted) has an excitation energy E * (x) and its value is calculated a sum of the partial cross sections: ER ( * x , L) is the partial cross section of the ER formation as the survival cross section of the intermediate nucleus at each step x of the de-excitation cascade by the formula [12,20] (E  * x−1 , L) the survival probability of the xth intermediate nucleus against fission along each step of the de-excitation cascade.It is calculated by the statistical model implanted in KEWPIE2 [21].
Obviously σ ER (E * ICF , L) = σ ICF (E * ICF , L) which is calculated by (6).This procedure is similar to the calculation of the cross section of the evaporation residues formed after emission of neutrons from the heated and rotating compound nucleus formed at complete fusion [12,20] where the equation σ In Fig. 9, the results of calculations using Eq. ( 7) are compared with the measured cross sections of the  evaporation residues of 108 Ag and 107 Ag formed in the 18 O+ 93 Nb incomplete fusion reaction after emission of 3 and 4 neutrons [6], respectively.The agreement our results with the measured data is better than that calculated in Ref. [6] by the standard methods ALICE-91 [4] and PACE4 [5,6].In Fig. 10, the cross sections of the evaporation residues of 107 In and 108 In formed in the 18 O+ 93 Nb complete fusion reaction after emission of 4 and 3 neutrons, respectively, are compared with the measured experimental data.It is seen from Fig. 10 that the behaviours of the experimental data and theoretical curves do not have a plateau as a function of the beam energy.From this point of view, the behaviours of the excitation functions of the evaporation residues formed in the incomplete and complete fusion reactions are different.The difference is explained by increasing the rotational energy V rot (Z, L) of the DNS with the very mass-asymmetric configuration corresponding to the α-particle emission in collisions with the large beam energies: V rot (Z) takes an appreciable part of the kinetic energy of the relative motion.As a result, the residual nucleus formed after emission of αparticle in the incomplete fusion is less excited than compound nucleus formed in the complete fusion in heavy-ion collision with the same values of the orbital angular momentum and beam energy.The fission barrier B f of the heated and rotating compound nucleus decreases due to its large excitation energy E * CN and angular momentum L.
The theoretical excitation function of the ER 139 Ce formed in the incomplete fusion of the 16 O+ 130 Te reaction after emission of the α particle and 3 neutrons is compared with the measured experimental data [19] in Fig. 11.Our approach does not allow us to reach an agreement at low energies where the excitation function of the ER 140 Ce formed after emission of the α particle and 2 neutrons dominates over 3n neutron channel.
The new mechanism of the incomplete fusion has been explored as a very mass-asymmetric quasifission of a DNS formed at the capture of the projectile nucleus (full momentum transfer) by the target nucleus.The L-dependence of the charge distribution of the DNS fragments leads to formation of its configuration consisting of the α particle and a conjugate nucleus.The centrifugal force related with the rotation of a very massasymmetric DNS is strong for the L > 30.Consequently, it leads to the incomplete fusion which can be considered as the quasifission producing α particle and a conjugate heavy fragment.This phenomenon is related with the transformation of a significant part of the kinetic energy of the collision energy to the rotational energy of the DNS formed at capture of the projectile by target nucleus.As a result the conjugate fragment is less heated than the compound nucleus formed in the complete fusion.Therefore, the plateau of the excitation function of the ER of the 3n-4n channels of the incomplete fusion in the high energy range is observed for the 18 O+ 93 Nb and 16 O+ 130 Te reactions.The measured cross sections of ER formed in the incomplete fusion and complete fusion channels have been reproduced well by the DNS model and the statistical model implanted in KEWPIE2 [21].
Our exploration of the incomplete mechanism has confirmed a decisive role of the orbital angular momentum in the reaction mechanisms of the heavy-ion collisions at energies above the Coulomb barrier and below 10 MeV/nucleon.

FIG. 1 .
FIG. 1.The sketch of the incomplete fusion mechanism as very asymmetric quasifission in the case of the 18 O+ 93 Nb reaction.

FIG. 2 .
FIG. 2. Potential energy surface calculated for the DNS formed in the 16 O+ 130 Te reaction at the collisions with the values of orbital angular momentum L = 40 as a function of the fragment charge numbers (Z) and relative distance (R) between centres-of-mass fragments.The DNS excitation energy E * DNS (ZP ), the intrinsic fusion B * fus barrier is shown for Z = 4 and quasifission B qf barrier is shown for Z = 2 by the corresponding arrows.
FIG. 3. The driving potential of the DNS formed in the 16 O+ 130 Te reaction calculated for the orbital angular momentum L = 10, 25, and40 .the intrinsic fusion B * fus and quasifission B qf barriers of the entrance channel ZP are shown by the corresponding arrows.

FIG. 4 .
FIG. 4.Evolution of the charge distribution for the projectile-like fragments for the16 O+ 130 Te reaction at Ec.m. = 78.4MeV and L = 40 .The results have been obtained for the orientation angles α1 = 45 o andα2 = 30 o .
FIG. 6.The partial cross section of the incomplete fusion σICF(E lab , L) as a function of the angular momentum L for the set of collision energy values E lab for the 16 O+ 130 Te reaction.

FIG. 8 .
FIG. 7. The partial cross section of the incomplete fusion σICF(E lab , L) as a function of the angular momentum L for the set of collision energy values E lab for the 18 O+ 93 Nb reaction.
) Here, σ x−1 ER (E * x−1 , L) is the partial cross section of the intermediate excited nucleus formation at the (x − 1)th step, and W (x) sur (E * CN , L) = σ fus (E * CN , L) was used.

FIG. 10 .
FIG.10.Comparison of the theoretical cross sections (dashed and solid curves) of the evaporation residues formed in the18 O+ 93 Nb complete fusion reaction after emission of 3 (triangles) and 4 (squares) neutrons with the measured experimental data presented in Ref.[6].

FIG. 11 .
FIG. 11.Comparison of theoretical cross sections (solid curve) of the evaporation residues formed in the16 O+ 130 Te complete fusion reaction after emission of 3 neutrons with the measured experimental data (triangles) presented in Ref.[19].