Microscopic description of production cross sections including deexcitation effects

TL;DR

This paper introduces TDHF+GEMINI, a new microscopic method combining time-dependent Hartree-Fock and statistical models to predict nuclear production cross sections, including deexcitation effects, improving agreement with experimental data.

Contribution

The paper presents a novel combined microscopic and statistical approach, TDHF+GEMINI, for more accurate predictions of nuclear reaction cross sections involving deexcitation processes.

Findings

Inclusion of secondary deexcitation improves agreement with experiments.

TDHF+GEMINI outperforms GRAZING in describing multi-proton transfer channels.

Discrepancies suggest the need for beyond mean-field theories.

Abstract

Background: At the forefront of the nuclear science, production of new neutron-rich isotopes is continuously pursued at accelerator laboratories all over the world. To explore the currently-unknown territories in the nuclear chart far away from the stability, reliable theoretical predictions are inevitable. Purpose: To provide a reliable prediction of production cross sections taking into account secondary deexcitation processes, both particle evaporation and fission, a new method called TDHF+GEMINI is proposed, which combines the microscopic time-dependent Hartree-Fock (TDHF) theory with a sophisticated statistical compound-nucleus deexcitation model, GEMINI++. Results: The method is applied for multinucleon transfer processes in low-energy heavy ion reactions. It is shown that the inclusion of secondary deexcitation processes, which are dominated by neutron evaporation in the present systems, substantially improves agreement with the experimental data. The magnitude of the evaporation effects is very similar to the one observed in GRAZING calculations. TDHF+GEMINI provides better description of the absolute value of the cross sections for channels involving transfer of more than one protons, compared with the GRAZING results. However, there remain discrepancies between the measurements and the calculated cross sections, indicating a limit of the theoretical framework that works with a single mean-field potential. Possible causes of the discrepancies are discussed. Conclusions: In order to perfectly reproduce experimental cross sections for multinucleon transfer processes, one should go beyond the standard self-consistent mean-field description. Nevertheless, the proposed method will provide valuable information to optimize production mechanisms of new neutron-rich nuclei through its microscopic, non-empirical predictions. (Shortened due to the word limit)

Tables

Table 1. Table 1: A list of information of the reactions investigated.

System	$E_{c.m.}$ (MeV)	$N_P/Z_P$	$N_T/Z_T$	$Z_P{Z}_T$	Expt.
40Ca+124Sn	128.54	1.00	1.48	1000	Corradi(40Ca+124Sn)
48Ca+124Sn	125.44	1.40	1.48	1000	Corradi(48Ca+124Sn)
40Ca+208Pb	208.84	1.00	1.54	1640	Szilner(40Ca+208Pb)
58Ni+208Pb	256.79	1.07	1.54	2296	Corradi(58Ni+208Pb)
64Ni+238U	307.35	1.29	1.57	2576	Corradi(64Ni+238U)
136Xe+198Pt	644.98	1.52	1.54	4212	Watanabe(136Xe+198Pt)

Equations

$$\Psi(\bm{r}_{1}\sigma_{1}q_{1},\dots,\bm{r}_{A}\sigma_{A}q_{A},t_{\rm f})=\frac{1}{\sqrt{A!}}\det\bigl{\{}\psi_{i}^{(q_{j})}(\bm{r}_{j}\sigma_{j},t_{\rm f})\bigr{\}},$$

$$\big{|}\Psi\bigr{>}=\sum_{N,Z}\big{|}\Psi_{N,Z}\bigr{>},$$

$$\big{|}\Psi_{N,Z}\bigr{>}=\hat{P}_{N}^{(n)}\hat{P}_{Z}^{(p)}\big{|}\Psi\bigr{>},$$

$${\hat{P}}_n^{(q)}=\frac{1}{2\pi }{\int }_0^{2\pi }{e}^{\mathrm{i}(n-{\hat{N}}_V^{(q)})\theta }d\theta .$$

$${\hat{N}}_V^{(q)}={\int }_V\sum _{i=1}^{N^{(q)}}\delta (\mathbf{r}-{\hat{\mathbf{r}}}_i)d\mathbf{r}=\sum _{i=1}^{N^{(q)}}{\Theta }_V({\hat{\mathbf{r}}}_i),$$

$$\Theta_{V}(\bm{r})=\left\{\begin{array}[]{ccc}1&\;\mbox{for}&\bm{r}\in V,\\[1.42262pt] 0&\;\mbox{for}&\bm{r}\notin V.\end{array}\right.$$

$$P_{N,Z}=\bigl{<}\Psi_{N,Z}\big{|}\Psi_{N,Z}\bigr{>}=P_{N}^{(n)}P_{Z}^{(p)}.$$

$$P_n^{(q)}=\frac{1}{2\pi }{\int }_0^{2\pi }{e}^{\mathrm{i}n\theta }\det {\mathcal{B}}^{(q)}(\theta )d\theta ,$$

$$\displaystyle\bigl{(}\mathcal{B}^{(q)}(\theta)\bigr{)}_{ij}$$

$$\psi_{i}^{(q)}(\bm{r}\sigma,\theta)=\bigl{[}\Theta_{\bar{V}}({\bf r})+e^{-\mathrm{i}\hskip 0.56905pt\theta}\Theta_{V}({\bf r})\bigr{]}\psi_{i}^{(q)}(\bm{r}\sigma).$$

$${\sigma }_{N,Z}(E)=2\pi {\int }_{b_{\min }}^{b_{\mathrm{cut}}}b{P}_{N,Z}(b,E)db,$$

$${\hat{\mathbf{J}}}_V=\sum _{i=1}^A{\Theta }_V({\hat{\mathbf{r}}}_i){\hat{\mathbf{j}}}_i,$$

$$\bm{J}_{N,Z}=\frac{\bigl{<}\Psi_{N,Z}\big{|}\hat{\bm{J}}_{V}\big{|}\Psi_{N,Z}\bigr{>}}{\bigl{<}\Psi_{N,Z}\big{|}\Psi_{N,Z}\bigr{>}}=\bm{J}_{N}^{(n)}+\bm{J}_{Z}^{(p)}.$$

$$\bm{J}_{n}^{(q)}=\frac{1}{2\pi P_{n}^{(q)}}\int_{0}^{2\pi}e^{\mathrm{i}\hskip 0.56905ptn\theta}\det\mathcal{B}^{(q)}(\theta)\sum_{i=1}^{N^{(q)}}\bigl{<}\psi_{i}^{(q)}\big{|}\hat{\bm{j}}\big{|}\tilde{\psi}_{i}^{(q)}(\theta)\bigr{>}_{V}d\theta,$$

$$\tilde{\psi}_{i}^{(q)}(\bm{r}\sigma,\theta)\equiv\sum_{j=1}^{N^{(q)}}\psi_{j}^{(q)}(\bm{r}\sigma,\theta)\bigl{(}\mathcal{B}^{(q)}(\theta)\bigr{)}_{ji}^{-1}.$$

$$E_{N,Z}=\frac{\bigl{<}\Psi_{N,Z}\big{|}\hat{H}_{V}\big{|}\Psi_{N,Z}\bigr{>}}{\bigl{<}\Psi_{N,Z}\big{|}\Psi_{N,Z}\bigr{>}},$$

$$E_{N,Z}$$

$$E_{N,Z}^{\ast }=E_{N,Z}-E_{N,Z}^{g.s.},$$

$$P_{\mathrm{decay}}(E_{N,Z}^{\ast },J_{N,Z},N,Z;N^{\prime },Z^{\prime })=\frac{N_{N^{\prime },Z^{\prime }}}{N_{\mathrm{trial}}}.$$

$${\tilde{\sigma }}_{N^{\prime },Z^{\prime }}(E)$$

$${\tilde{P}}_{N^{\prime },Z^{\prime }}=\sum _{N\ge N^{\prime }}\sum _{Z\ge Z^{\prime }}{P}_{N,Z}{P}_{\mathrm{decay}}(E_{N,Z}^{\ast },J_{N,Z},N,Z;N^{\prime },Z^{\prime }).$$

$$\bar{J}=\bigl{<}\Psi\big{|}\hat{J}_{V}\big{|}\Psi\bigr{>}=\sum_{i=1}^{A}\bigl{<}\psi_{i}\big{|}\hat{j}\big{|}\psi_{i}\bigr{>}_{V},$$

$$\mathrm{TKE}=\frac{1}{2}\mu {\dot{\mathbf{R}}}^2+\frac{Z_1{Z}_2{e}^2}{|\mathbf{R}|},$$

$${\bar{E}}_{\mathrm{tot}}^{\ast }=E-\mathrm{TKE}-Q,$$

$${\bar{E}}_{N,Z}^{\ast }=\frac{N+Z}{A_1+A_2}{\bar{E}}_{\mathrm{tot}}^{\ast }.$$

Full text

At the forefront of the nuclear science, production of new neutron-rich isotopes is continuously pursued at accelerator laboratories all over the world. To explore the currently-unknown territories in the nuclear chart far away from the stability, reliable theoretical predictions are inevitable.

Purpose

To provide a reliable prediction of production cross sections taking into account secondary deexcitation processes, both particle evaporation and fission, a new method called TDHF+GEMINI is proposed, which combines the microscopic time-dependent Hartree-Fock (TDHF) theory with a sophisticated statistical compound-nucleus deexcitation model, GEMINI++.

Methods

Low-energy heavy ion reactions are described based on three-dimensional Skyrme-TDHF calculations. Using particle-number projection method, production probabilities, total angular momenta, and excitation energies of primary reaction products are extracted from the TDHF wavefunction after collision. Production cross sections for secondary reaction products are evaluated employing GEMINI++. Results are compared with available experimental data and widely-used GRAZING calculations.

Results

The method is applied to describe cross sections for multinucleon transfer processes in 40Ca+124Sn ($E_{\rm c.m.}$ $\simeq$ 128.54 MeV), 48Ca+124Sn ($E_{\rm c.m.}$ $\simeq$ 125.44 MeV), 40Ca+208Pb ($E_{\rm c.m.}$ $\simeq$ 208.84 MeV), 58Ni+208Pb ($E_{\rm c.m.}$ $\simeq$ 256.79 MeV), 64Ni+238U ($E_{\rm c.m.}$ $\simeq$ 307.35 MeV), and 136Xe+198Pt ($E_{\rm c.m.}$ $\simeq$ 644.98 MeV) reactions at energies close to the Coulomb barrier. It is shown that the inclusion of secondary deexcitation processes, which are dominated by neutron evaporation in the present systems, substantially improves agreement with the experimental data. The magnitude of the evaporation effects is very similar to the one observed in GRAZING calculations. TDHF+GEMINI provides better description of the absolute value of the cross sections for channels involving transfer of more than one protons, compared to the GRAZING results. However, there remain discrepancies between the measurements and the calculated cross sections, indicating a limit of the theoretical framework that works with a single mean-field potential. Possible causes of the discrepancies are discussed.

Conclusions

In order to perfectly reproduce experimental cross sections for multinucleon transfer processes, one should go beyond the standard self-consistent mean-field description. Nevertheless, the proposed method will provide valuable information to optimize production mechanisms of new neutron-rich nuclei through its microscopic, non-empirical predictions.

I INTRODUCTION

To expand our knowledge of the nature of the atomic nuclei, it is obviously important to produce new neutron-rich unstable isotopes that have not yet been produced to date, and study their properties both experimentally and theoretically. However, the optimal reaction condition, such as projectile-target combinations and incident energies, to produce such extremely unstable nuclei is not obvious, and reliable theoretical predictions are mandatory to guide experiments at current and future radioactive-ion beam facilities. This paper aims to provide a predictive model of production cross sections in low-energy heavy ion reactions.

To describe low-energy heavy ion reactions, various models have been developed. Semi-classical models, called GRAZING GRAZING and complex Wentzel-Kramers-Brillouin (CWKB) CWKB , have shown remarkable successes in describing multinucleon transfer (MNT) processes in peripheral collisions Corradi(review) . The GRAZING code was extended to include effects of transfer-induced fission in competition with particle evaporation GRAZING-F . A possible drawback of those models lies in insufficient description of deep-inelastic processes at small impact parameters. On the other hand, different theoretical approaches have also been developed: e.g. a dynamical model based on Langevin-type equations of motion Zagrebaev(2005) ; Zagrebaev(2007)1 ; Zagrebaev(2008)1 ; Zagrebaev(2013)IQF ; Zagrebaev(2014)light_N-rich ; Zagrebaev(2008)2 ; Zagrebaev(2007)2 ; EXP2015(136Xe+208Pb) , dinuclear system model (DNS) Antonenko(1995) ; Adamian(1997)1 ; Adamian(1997)2 ; Adamian(1998) ; Adamian(2003) ; Feng(2009) ; Adamian(2010)1 ; Adamian(2010)2 ; Adamian(2010)3 ; Mun(2014) ; Mun(2015) ; DNS(2015) ; DNS(2017) ; DNS(2017)2 ; DNS(2017)3 ; DNS(2017)4 , and improved quantum molecular dynamics model (ImQMD) ImQMD(2002) ; ImQMD(2004) ; ImQMD(2008) ; ImQMD(2013) ; ImQMD(2015)1 ; ImQMD(2015)2 ; ImQMD(2016)1 ; ImQMD(2016)2 ; ImQMD(2016)3 ; ImQMD(2017) . Although those models can describe both peripheral and damped collisions, including fusion and quasifission (QF) processes, they are to some extent empirical containing model parameters. In order to provide a reliable prediction to produce new neutron-rich isotopes, it is desirable to have less adjustable parameters as possible. In the present paper, a method is developed to predict production cross sections based on a microscopic framework of the time-dependent Hartree-Fock (TDHF) theory.

The TDHF theory allows to describe nuclear dynamics microscopically from nucleonic degrees of freedom. The theory itself was proposed in 1930 Dirac(TDHF) , and its application to nuclear systems already started about forty years ago BKN(1976) ; Negele(review) . Since then, it has been developed as an omnipotent tool, rooted with the time-dependent density functional theory (TDDFT), to study nuclear structure and dynamics in a unified way Simenel(review) ; Nakatsukasa(PTEP) ; Sky3D ; Nakatsukasa(review) . Recently, we have applied the theory to study MNT and QF processes in various systems at energies around the Coulomb barrier KS_KY_MNT ; KS_KY_PNP ; MyPhD ; Bidyut(2015) ; KS_KY_Ni-U ; KS_SH_Kazimierz . Applying particle-number projection (PNP) method Projection , transfer cross sections were evaluated based on the TDHF theory. Comparisons with measured cross sections revealed that the theory, being with no adjustable parameters, can describe transfer cross sections quite well in accuracy comparable to the existing models KS_KY_MNT . However, the calculated cross sections were of primary (excited) reaction products which are to be deexcited through particle evaporation and/or fission. Because of this fact discrepancy arises when compared with experimental data, especially for channels accompanying transfer of many nucleons. The TDHF description of production cross sections has thus been beset with the absence of deexcitation processes that limits its predictive power.

In this paper, a method, called TDHF+GEMINI, is proposed to cure the drawback of the TDHF description. Namely, secondary deexcitation processes of primary reaction products, both particle evaporation and fission, are simulated employing a state-of-the-art statistical model, GEMINI++ GEMINI++ . Those secondary processes are difficult to investigate within TDHF, because of, e.g., its much longer timescales. The inputs of statistical-model calculations, spin and excitation energy of primary reaction products, are extracted from the TDHF wavefunction after collision, using an extended PNP method KS_KY_PNP . The method is applied to 40,48Ca+124Sn, 40Ca+208Pb, 58Ni+208Pb, 64Ni+238U, and 136Xe+198Pt reactions for which measured cross sections are available. To demonstrate the accuracy of the proposed method is the main purpose of this work.

It has been appreciated that TDHF provides valuable insight into complex many-body dynamics of low-energy heavy ion reactions. However, it works with a single mean-field potential which is deterministically associated with the initial condition. In other words, fluctuations in collective space is absent in the TDHF dynamics. Indeed, it has been shown that the theory is optimized to describe the expectation value of one-body observables BV(1981) ; Simenel(review) , and fluctuations of them are known to be severely underestimated Koonin(1977) ; Davies(1978) ; Dasso(1979) ; Simenel(2011) . How and to what extent beyond-mean-field fluctuations play a role in MNT reactions is an open question. Moreover, inter-nucleon correlations are also not included in TDHF. Outcomes of MNT reactions may reflect effects of inter-nucleon correlations, because of possible transfer of a correlated-pair or a cluster of nucleons. Recently, it has become possible to pursue microscopic simulations of heavy ion reactions including the pairing correlations Scamps(2012) ; Scamps(2013) ; Ebata(2014) ; Ebata(2015) ; Hashimoto(2016) ; MSW(2017) ; SMW(2017) ; SWM(2017) . It should be noted here that the proposed method can, in principle, be extended to incorporate with the pairing correlations. In the present paper, however, we will focus on a treatment without pairing and leave further extension/application to include the pairing correlations as a future task. Nevertheless, it has to be noted here that, by extending the application of the theoretical framework based on the TDHF theory as far as possible, this work will shed light on the validity of the theoretical framework that works with a single mean-field potential, without inter-nucleon correlations.

The article is organized as follows. In Sec. II, the methodology of TDHF+GEMINI is outlined. In Sec. III, numerical results for various reactions are presented and are compared with available experimental data. In Sec. IV, a summary of the present work is given.

II TDHF+GEMINI

In this Section, we focus on the analysis of the TDHF wavefunction using the PNP method and its coupling with statistical-model calculations. For details of the TDHF theory and its application to nuclear systems, see, e.g., Refs. Negele(review) ; Simenel(review) ; Nakatsukasa(PTEP) ; Sky3D ; Nakatsukasa(review) , and references therein.

II.1 Cross sections for primary products

Let $A_{\mu}$, $Z_{\mu}$, and $N_{\mu}$, respectively, be mass, charge, and neutron numbers of a projectile ($\mu={\rm P}$) and a target ($\mu={\rm T}$). The total numbers of neutrons and protons in the system are $N^{(n)}=N_{\rm P}+N_{\rm T}$ and $N^{(p)}=Z_{\rm P}+Z_{\rm T}$, respectively. The total number of nucleons is denoted as $A=N^{(n)}+N^{(p)}$.

Suppose that we have performed a TDHF calculation for a reaction at a given energy $E$ and an impact parameter $b$, and we observed generation of binary reaction products. Now we have a many-body wavefunction at a certain time $t=t_{\rm f}$ after collision, which is given by a single Slater determinant:

[TABLE]

where $\psi_{i}^{(q)}(\bm{r}\sigma,t_{\rm f})$ is $i$th single-particle orbital at $t=t_{\rm f}$ with spatial, spin, and isospin coordinates, $\bm{r}$, $\sigma$, and $q$, respectively. Our aim is to evaluate production cross sections for secondary reaction products based on the TDHF wavefunction after collision, Eq. (1).

Because of possible nucleon transfer processes, the TDHF wavefunction after collision is, in general, not an eigenstate of a number operator in a subspace $V$ that contains one of the reaction products, but a superposition of states with different particle-number distributions:

[TABLE]

where $N$ and $Z$ specify neutron and proton numbers of a reaction product inside the spatial region $V$, respectively. Here and henceforth, brackets, such as \big{|}\Psi\bigr{>} and \big{|}\psi_{i}^{(q)}\bigr{>}, are often used omitting indexes to simplify notations. \big{|}\Psi_{N,Z}\bigr{>} can be expressed as

[TABLE]

where $\hat{P}_{n}^{(q)}$ is the PNP operator for neutrons ($q=n$) or protons ($q=p$),

[TABLE]

$\hat{N}_{V}^{(q)}$ is the number operator for neutrons ($q=n$) or protons ($q=p$) in the spatial region $V$,

[TABLE]

where

[TABLE]

The probability that a reaction product composed of $N$ neutrons and $Z$ protons is produced, $P_{N,Z}$, is given by

[TABLE]

Note that $P_{N,Z}$ is a product of probabilities for neutrons $P_{N}^{(n)}$ and for protons $P_{Z}^{(p)}$ in TDHF. These probabilities can be expressed in terms of the single-particle orbitals as

[TABLE]

where

[TABLE]

with

[TABLE]

By repeating TDHF calculations for various impact parameters $b$ at a given incident energy $E$, we obtain $P_{N,Z}(b,E)$. The production cross section for a primary reaction product composed of $N$ neutrons and $Z$ protons before secondary deexcitation is then given by

[TABLE]

where $b_{\rm min}$ is the minimum impact parameter for binary reactions, inside which fusion reactions take place; $b_{\rm cut}$ is a cutoff impact parameter for the numerical integration. Note that if $b_{\rm cut}$ is chosen large enough it merely affects magnitude of the cross section for the elastic scattering.

II.2 Total angular momentum

In Ref. KS_KY_PNP , the PNP method has been extended to calculate the expectation value of operators. The method allows to evaluate the total angular momentum, $J$, of a reaction product in each transfer channel, which is an input of a statistical-model calculation.

The idea KS_KY_PNP was to introduce operators for a reaction product inside the spatial region $V$. In the case of the total angular momentum operator, it can be expressed as

[TABLE]

where $\hat{\bm{j}}_{i}=(\hat{\bm{r}}_{i}-\bm{R}_{\rm c.m.})\times\hat{\bm{p}}_{i}+\hat{\bm{s}}_{i}$. $\bm{R}_{\rm c.m.}$ is the center-of-mass position of the reaction product, $\hat{\bm{p}}_{i}$ and $\hat{\bm{s}}_{i}$ are the momentum and the spin operators, respectively,

The expectation value of the total angular momentum of a reaction product composed of $N$ neutrons and $Z$ protons is then defined as

[TABLE]

It satisfies an identity, \bigl{<}\Psi\big{|}\hat{\bm{J}}_{V}\big{|}\Psi\bigr{>}=\sum_{N,Z}P_{N,Z}\bm{J}_{N,Z}. The contribution from neutrons ($q=n$) or protons ($q=p$) is given by

[TABLE]

where

[TABLE]

Note that $\{\tilde{\psi}_{i}(\theta)\}$ are biorthonormal to $\{\psi_{i}\}$, i.e., \bigl{<}\psi_{i}\big{|}\tilde{\psi}_{j}(\theta)\bigr{>}=\delta_{ij}. The subscript $V$ of the bracket in Eq. (14) indicates that the spatial integration is taken only over the spatial region $V$. In practice, the total angular momentum perpendicular to the reaction plane will be used as an input for statistical-model calculations. It will be denoted simply as $J_{N,Z}$.

II.3 Excitation energy

Applying the PNP method, we can also evaluate the excitation energy, $E^{*}$, of a reaction product in each transfer channel KS_KY_PNP . The energy expectation value of a reaction product composed of $N$ neutrons and $Z$ protons is defined as

[TABLE]

where $\hat{H}_{V}$ is a Hamiltonian for a reaction product inside the spatial region $V$. There also follows an identity, \bigl{<}\Psi\big{|}\hat{H}_{V}\big{|}\Psi\bigr{>}=\sum_{N,Z}P_{N,Z}E_{N,Z}.

In practice, one may work with an energy density functional (EDF). In such a case, $E_{N,Z}$ is given by

[TABLE]

where $\det\mathcal{B}(\theta,\varphi)=\det\mathcal{B}^{(n)}(\theta)\det\mathcal{B}^{(p)}(\varphi)$. $\mathcal{E}(\bm{r},\theta,\varphi)$ denotes an EDF kernel which has the same form as the EDF used, but is composed of complex mixed densities, e.g., $\rho_{q}(\bm{r},\theta)=\sum_{i,\sigma}\psi_{i}^{(q)*}(\bm{r}\sigma)\tilde{\psi}_{i}^{(q)}(\bm{r}\sigma,\theta)$, etc.

For our purpose, Eq. (16) or Eq. (17) should be evaluated in the rest frame of the reaction product in order to remove kinetic energy associated with center-of-mass translational motion. Regarding it as internal energy of the reaction product, excitation energy can be evaluated as

[TABLE]

where $E_{N,Z}^{\rm g.s.}$ denotes energy of a nucleus specified by $N$ neutrons and $Z$ protons in its Hartree-Fock ground state.

II.4 Cross sections for secondary products

Having $P_{N,Z}$, $J_{N,Z}$, and $E_{N,Z}^{*}$ at hand, we can apply a statistical model to obtain production cross sections for secondary reaction products. In this paper, GEMINI++ GEMINI++ is employed. For a given set of ($N$, $Z$, $E^{*}$, $J$), GEMINI++ provides a sequence of statistically-selected binary decays, including both particle evaporation and fission, until it becomes energetically forbidden or improbable due to competing $\gamma$-ray emission.

Because of its statistical nature, a different sequence of binary decays can be obtained with the same set of ($N$, $Z$, $E^{*}$, $J$). To evaluate decay probabilities, one may repeat computations of a binary-decay sequence, let’s say, $N_{\rm trial}$ times. Among the obtained $N_{\rm trial}$ decay sequences, one can count the number of processes in which a sequence of deexcitation processes of a primary reaction product composed of $N$ neutrons and $Z$ protons ends up with a nucleus specified by $N^{\prime}$ neutrons and $Z^{\prime}$ protons. Then, denoting it as $N_{N^{\prime},Z^{\prime}}$, the decay probability for the process ($N,Z$) $\rightarrow$ ($N^{\prime},Z^{\prime}$) can be defined by

[TABLE]

The number $N_{\rm trial}$ sets a lower limit on the decay probabilities, since processes with $P_{\rm decay}$ $\lesssim 1/N_{\rm trial}$ will not be taken into account.

The cross section for a secondary reaction product after evaporation and/or fission processes is then evaluated as

[TABLE]

where $\widetilde{P}_{N^{\prime},Z^{\prime}}$ denotes the probability that a reaction product composed of $N^{\prime}$ neutrons and $Z^{\prime}$ protons is produced after secondary deexcitation processes:

[TABLE]

Note that the input quantities, $P_{N,Z}$, $J_{N,Z}$, and $E_{N,Z}^{*}$, are dependent on the incident energy $E$ and the impact parameter $b$.

II.5 Computational details

For the TDHF calculation and the PNP analysis, our own computational code has been extended and applied MyPhD . In the code, single-particle orbitals are represented on three-dimensional Cartesian coordinates (without any symmetry restrictions) with isolated boundary conditions. The lattice spacing is set to 0.8 fm. Spatial derivatives are computed with 11-point finite-difference formula. For time evolution, 4th-order Taylor expansion method is used with a single predictor-corrector step. The time step is set to 0.2 fm/$c$. Coulomb potential is computed using Fourier transformations. For the EDF, Skyrme SLy5 functional Chabanat is employed.

Note that results of the PNP analysis with Eqs. (7), (13), and (16) are invariant under unitary transformations for the single-particle orbitals, $\psi_{i}^{\prime}(\bm{r}\sigma)=\sum_{j}\mathcal{U}_{ij}\psi_{j}(\bm{r}\sigma)$, where $\mathcal{U}$ is a unitary matrix: it only changes the phase of the many-body wavefunction, i.e., $\Psi^{\prime}=\det\mathcal{U}\,\Psi$. Thus, the quantities are well-defined and the results may merely depend on the accuracy of the numerical integration and the choice of the spatial region $V$. The latter is expected to be small in the case where reaction products are well separated spatially. In the present analysis, $V$ is taken as a sphere with a radius $R_{V}$ around the center-of-mass of a reaction product. Indeed, it has been confirmed, by varying $R_{V}$ from 11 fm to 15 fm for the 40Ca+124Sn system, that $J_{N,Z}$ and $E_{N,Z}$ of lighter fragments are affected only less than $0.01\,\hbar$ and a few tens of keV, respectively.

The numerical integration over the gauge angle $[0,2\pi]$ is performed using the trapezoidal rule with an $M$-point uniform mesh. $P_{N,Z}$ and $J_{N,Z}$ can be stably computed for all transfer channels and do not depend on $M$, if it is taken larger than $M\approx 100$–200 for the systems analyzed in the present paper. The accuracy of $E_{N,Z}$, on the other hand, is somewhat worse, which depends on the probability $P_{N,Z}$. For example, for different values of $M$ (200–500), $E_{N,Z}$ varies roughly several keV for a main process with $P_{N,Z}\approx 10^{-1}$, whereas for a process with smaller probabilities, $P_{N,Z}\approx 10^{-5}$, difference can be several hundreds of keV. Moreover, absolute value of $E_{N,Z}$ becomes unphysically large for processes with tiny probabilities, $P_{N,Z}\lesssim 10^{-5}$, as was observed for a lighter system KS_KY_PNP . In the present paper, $M=300$ and $R_{V}=15$ fm are utilized. It ensures at least 1-MeV accuracy of $E_{N,Z}$, which would be sufficient for the present purpose of quantifying effects of secondary deexcitation processes on production cross sections.

The ground-state energy of nuclei, $E_{N,Z}^{\rm g.s.}$ in Eq. (18), is computed employing a cubic box with $20.8$ fm on each side. The mesh spacing is set to 0.8 fm, as in the TDHF calculations. To find the energy minimum solution, avoiding those in local minima, static Hartree-Fock calculations are first performed with constraints on various deformation parameters ($\beta=0$, and $\beta=0.1,0.2$ with $\gamma=0^{\circ},30^{\circ},60^{\circ}$). After getting a moderately convergent solution, the constraints on $\beta$ and $\gamma$ are released, and the energy is re-minimized. The state with the lowest energy among the seven candidates is regarded as the Hartree-Fock ground state. The ground-state energies of various isotopes with $Z\leq 35$ are prepared for the evaluation of excitation energies of respective transfer products. Equal-filling treatment is adopted for odd number of nucleons, where the last nucleon resides in a time-reversal pair of orbitals with half occupation.

To describe secondary processes, the statistical compound-nucleus deexcitation model, GEMINI++ GEMINI++ , is employed. GEMINI++ is an improved version of a statistical model, GEMINI, developed by R.J. Charity GEMINI . It takes into account not only evaporation of light particles, i.e., $n$, $p$, $d$, $t$, 3He, $\alpha$, 6He, 6-8Li, and 7-10Be, with the Hauser-Feshbach formalism Hauser-Feshbach , but also emission of heavier fragments with a binary-decay formalism of L.G. Moretto Moretto1975 , as well as fission based on the Bohr-Wheeler formalism Bohr-Wheeler . The default setting of the code is used for all reactions analyzed in the present paper. The ingredients of the statistical model have been parametrized and determined so as to allow a good systematic description of the evaporation spectra for the entire mass region. Detailed discussions on various modifications and fine-tuning of the model parameters that were implemented in the GEMINI++ code can be found in Refs. Charity2010 ; Mancusi2010 . It has been tested, for all the systems under study (see Table 1), that $N_{\rm trial}=100$, 1000, and 10000 provide almost identical cross sections. (Even $N_{\rm trial}=10$ gives very similar cross sections, because of neutron evaporation dominance in those reactions.) In the following, results obtained with $N_{\rm trial}=1000$ are presented.

III RESULTS AND DISCUSSION

In the present paper, TDHF+GEMINI is applied to six reaction systems listed in Table 1, for which precise experimental data are available. One should note that the reactions investigated cover a range of systems which are expected to show different properties of the reaction dynamics. In reactions of projectile and target nuclei with different $N/Z$ ratios, the charge equilibration process takes place Freiesleben(1984) , where neutrons and protons are transferred toward the opposite directions, to reduce the $N/Z$ imbalance of the colliding nuclei. Fusion reaction in a system with a large charge product exceeding a critical value, $Z_{\rm P}Z_{\rm T}\gtrsim 1600$, is known to be substantially hindered Sahm(1984) . Those characteristic quantities are also listed in Table 1.

A detailed analysis of the TDHF results for MNT processes in the 40,48Ca+124Sn, 40Ca+208Pb, and 58Ni+208Pb reactions was carried out in Ref. KS_KY_MNT . Recently, TDHF calculations for MNT and QF processes in the 64Ni+238U reaction were also performed. Comprehensive discussions on interplay between orientations of deformed 238U and quantum shells in the reaction dynamics can be found in Ref. KS_KY_Ni-U . Here, the same TDHF wavefunctions are re-analyzed applying TDHF+GEMINI.

TDHF calculations for the 136Xe+198Pt reaction have been newly performed. The ground state of 136Xe and 198Pt turned out to be slightly deformed in a triaxial shape with deformation parameters, ($\beta\simeq 0.06$, $\gamma\simeq 29^{\circ}$) and ($\beta\simeq 0.12$, $\gamma\simeq 33^{\circ}$), respectively. Those nuclei were placed in such a way that the axis around which \bigl{|}Q_{22}\bigr{|} takes the smallest value is set perpendicular to the reaction plane. TDHF calculations were performed for various impact parameters in a range of $[0,\,12]$ (fm). No fusion reaction was observed for all impact parameters. Reaction mechanisms including incident energy dependence will be investigated in the forthcoming paper. In this paper, we will focus on the production cross sections for the $E_{\rm c.m.}\simeq 644.98$ MeV case.

It would be useful to first digest success and failure of the TDHF description. Because of the microscopic nature of the TDHF theory, it was not known, before our study KS_KY_MNT , that to what extent TDHF can quantitatively describe cross sections for MNT processes. By performing a number of TDHF calculations for various impact parameters $b$ and by plugging $P_{N,Z}(b)$ into Eq. (11), one can evaluate production cross sections for primary reaction products, based on the TDHF calculations. It should be noted here that no empirical parameters are introduced that can be adjusted to reproduce experimental data.

The calculations were first performed for the 40,48Ca+124Sn, 40Ca+208Pb, and 58Ni+208Pb reactions KS_KY_MNT . In Figs. 1–4, production cross sections for lighter (projectile-like) fragments are shown. Cross sections are classified according to the number of transferred protons $x$, indicated by ($\pm x$p; X), where X stands for the corresponding element. The plus sign corresponds to proton transfer from the target to the projectile (pickup), while the minus sign corresponds to the opposite (stripping). The horizontal axis is the neutron number of the fragments. Experimental data are shown by red filled circles. The cross sections for primary reaction products obtained from the TDHF calculations KS_KY_MNT are shown by red filled areas.

It is evident from the figures that TDHF nicely captures the main features of the reaction dynamics. In the reactions with a large $N/Z$ asymmetry (40Ca+124Sn, 40Ca+208Pb, and 58Ni+208Pb), neutron-pickup and proton-stripping are favored, because of the charge equilibration process (Figs. 1, 3, 4). On the other hand, in the 48Ca+124Sn reaction where $N/Z$ ratios of the projectile and the target are already very close to each other, the number of transferred nucleons becomes very small on average, resulting in transfer of neutrons and protons in both directions (Fig. 2).

From a careful look at the figures, one can see that TDHF quantitatively reproduces the measured cross sections for few-nucleon transfer processes: e.g. processes with pickup of a few neutrons in ($0p$), ($\pm 1p$), and ($-2p$) proton-transfer channels. The agreements are noteworthy, since no adjustable parameters are included in the calculations. However, there exist discrepancies between the TDHF results and the experimental data. Namely, the measurements show that substantial cross sections for fragments with a smaller number of neutrons emerge when more than one protons are transferred. TDHF fails to reproduce the observed tendency. Moreover, cross sections for multi-proton stripping processes are considerably underestimated by the TDHF calculations. On the other hand, in the 40Ca+208Pb and 58Ni+208Pb reactions, TDHF overestimates cross sections for pickup of many neutrons in ($0p$), ($-1p$), and ($-2p$) channels. The latter originated from trajectories at small impact parameters accompanying large energy losses, where dynamics of a thick-neck formation and its breaking is responsible for the amount of nucleon transfer (see Ref. KS_KY_MNT for a detailed discussion).

One can expect that the discrepancies may be remedied by including secondary deexcitation processes in the TDHF description. After transfer of many nucleons reaction products can be highly excited, and what were measured experimentally must be the cross sections for secondary products after deexcitation processes. For example, if neutrons were evaporated from primary reaction products, the resulting cross sections would shift toward less neutron-number side, the left direction in the figures; if protons were emitted to the continuum, then the cross sections would be re-distributed to channels classified as a process with a larger number of stripping of protons. It is now feasible to disentangle the possible origins of the discrepancies owing to TDHF+GEMINI.

In Figs. 1–4, the cross sections for secondary reaction products obtained with TDHF+GEMINI are represented by blue solid lines. In the evaluation of the cross sections, the PNP calculations for $J_{N,Z}$ and $E_{N,Z}^{*}$ presented in Secs. II.2 and II.3, respectively, were fully performed for all impact parameters investigated. For comparison, online GRAZING calculations GRAZING-online which include neutron-evaporation effects were also performed, and the results are shown by green shaded histograms.

From the figures, one can see that the inclusion of secondary deexcitation processes improves the description. For instance, TDHF+GEMINI quantitatively reproduces the experimental data, mainly for neutron-pickup processes, accompanying stripping of several protons. The description of the overestimated cross sections of neutron-pickup processes for ($0p$), ($-1p$), and ($-2p$) channels in the 40Ca+208Pb and 58Ni+208Pb reactions is also improved. However, the effects of secondary deexcitation processes are not large enough to remedy all the discrepancies between the TDHF results and the experimental data. The peak positions of the cross sections still locate at the larger neutron-number side compared to the experimental data. Note that the magnitude of evaporation effects is very similar to the one observed in the GRAZING calculations as well as our prior attempts KS_KY_FUSION14 ; KS_KY_Maruhn with a simple evaporation model developed by I. Dostrovsky et al. Dostrovsky(1959) .

It is worth emphasizing here that, although the peak positions are different compared to the experimental cross sections, its absolute value is well reproduced by TDHF+GEMINI up to around ($-4p$) channel. The widely-used GRAZING calculations provide few orders of magnitude smaller cross sections, when more than one protons are transferred. In Refs. Szilner(40Ca+208Pb) ; Corradi(58Ni+208Pb) , the 40Ca+208Pb and 58Ni+208Pb reactions were analyzed by the CWKB semi-classical model. It provides quantitatively very similar results as the GRAZING calculations, i.e., the absolute values of the cross sections for ($-xp$) ($x\geq 2$) channels were substantially underestimated. The authors of Refs. Szilner(40Ca+208Pb) ; Corradi(58Ni+208Pb) phenomenologically introduced pair-transfer modes to explain the experimental data. Although they could obtain better description of the cross sections by adjusting an additional macroscopic form factor to reproduce the experimental cross section for pure two-proton stripping process without neutron transfer, i.e., ($0n,-2p$) channel, the validity of the hypothetical pair-transfer modes was unclear. The present results indicate that the pair-transfer processes may play a minor role, at least for the above-barrier MNT processes, as the absolute value of the cross sections for ($-2p$) channel is well reproduced without pairing by TDHF+GEMINI.

The remained discrepancies between the results of TDHF+GEMINI and the experimental data would rather suggest a limit of the theoretical framework. For example, although we could obtain cross sections for various transfer channels using the PNP method, they merely come from distribution around the average trajectory described by a single Slater determinant. Because of this fact, the peak positions are strongly correlated with the average number of transfered nucleons. Since we do not have a mean-field potential for, e.g., ($-6p$) channel, the underestimation of multi-proton stripping processes may be an artifact of the usage of the single mean-field potential. In reality, the potential should be transfer-channel dependent. When many protons are removed from the projectile, for instance, the potential felt by neutrons inside the proton-removed (proton-added) nucleus would become shallower (deeper) that may suppress/enhance neutron pickup/stripping processes. This transfer-channel dependent potential may explain the observed discrepancies. Part of the effects may be seen as beyond-mean-field fluctuations and correlations. One should note, however, that possible underestimation of evaporation effects has not been excluded yet. For instance, $E^{*}_{N,Z}$ and $J_{N,Z}$ that were used for the statistical-model calculations are the expectation values of energy and angular momentum of a reaction product in each transfer channel and, thus, are averaged over all possible quantal states populated by the reaction. However, these quantities should have certain distributions, and substantial part of evaporation processes might originate from components which are not in the vicinity of the mean values. Those distributions may not adequately be described within the TDHF theory, since for evaluation of the variance of them proper description of two- and four-body observables is necessary. In addition, e.g., the neglected prompt (pre-equilibrium) nucleon emissions before compound-nucleus formation and any underestimation of negative $Q$-value effect would enhance the particle evaporation. In any case, the present results by TDHF+GEMINI suggest that, in order to fully reproduce experimental cross sections for MNT processes, one should extend the theoretical framework that goes beyond the TDHF theory: e.g. a variational method of Balian and Vénéroni BV(1981) ; Simenel(2011) , stochastic mean-field theory (SMF) Ayik(2008) ; Tanimura(2017) , time-dependent density matrix theory (TDDM) Shun-jin(1985) ; Tohyama(1985) ; Gong(1990) ; Tohyama(2016) , time-dependent generator coordinate method (TDGCM) Berger(1984) ; Simenel(2014) , multiconfiguration TDHF (MCTDHF) Kato(2004) ; Caillat(2005) , and so forth.

It should be emphasized here that, although there still remain certain discrepancies as discussed above, TDHF+GEMINI will be a powerful tool, especially for systems for which no experimental data exist, to figure out optimal conditions to produce objective neutron-rich isotopes, as a microscopic model without empirical parameters that has better accuracy than the widely-used semi-classical models. In the following, let us thus consider further applications of TDHF+GEMINI.

Although it is possible to perform the PNP calculations for $J_{N,Z}$ and $E_{N,Z}^{*}$ as shown above, it requires quite large computational effort. For reactions involving heavy nuclei, for instance, the number of discreet points for the numerical integration over the gauge angles $\theta$ and $\varphi$ is typically a few hundreds. Then, to evaluate Eqs. (14) and (17), one needs to compute derivatives of 3D complex spatial functions many times, which could be as time-consuming as a TDHF calculation. Moreover, to compute the excitation energies with Eq. (18), one has to compute the energy of nuclei in their Hartree-Fock ground state in a wide mass region. It would also be problematic if one wants to change the working EDF, then the ground-state energies have to be re-computed for the evaluation of $E_{N,Z}^{*}$. One may consider to use experimental masses for ground-state energies, $E_{N,Z}^{\rm g.s.}$, however, then the result would be dependent on quality of the EDF itself.

Since it is not desirable to pay much effort for the computation of the input parameters of the statistical-model calculations, let us consider a simpler evaluation. Namely, one may replace $E_{N,Z}^{*}$ and $J_{N,Z}$ in Eq. (19) with average quantities that can be easily obtained from the TDHF wavefunction after collision. At the time $t=t_{\rm f}$ after collision, one can compute the average total angular momentum as

[TABLE]

where operators $\hat{J}_{V}$ and $\hat{j}$ are those for the component perpendicular to the reaction plane. One can also compute such quantities as average mass and charge numbers, $A_{i}$ and $Z_{i}$ ($i=1,2$), the relative vector connecting the center-of-mass positions of the reaction products, $\bm{R}$, and its time derivative, $\dot{\bm{R}}$. Then, the average total kinetic energy (TKE) of the outgoing fragments reads:

[TABLE]

where $\mu=m_{n}A_{1}A_{2}/(A_{1}+A_{2})$ is the reduced mass, being $m_{n}$ the nucleon mass. From this quantity, average total excitation energy shared by the two fragments may be evaluated as

[TABLE]

where $E$ is the incident relative energy and $Q$ denotes the $Q$ value of the reaction. By putting the actual $Q$ value for each transfer channel, $E_{\rm tot}^{*}$ becomes effectively transfer-channel dependent. To evaluate the $Q$ value for each transfer channel, experimental masses from the latest atomic mass evaluation, AME2016 AME2016(I) ; AME2016(II) , are utilized, whenever available; for nuclei whose mass has not been measured experimentally, theoretical values from the newest version of the finite-range droplet model, FRDM(2012) FRDM(2012) , are adopted. One may distribute the total excitation energy to respective reaction products in such a way that it is proportional to their mass:

[TABLE]

It is equivalent to assume that the thermal equilibrium is realized before formation of the fragments. The assumption may not always be correct especially in a transitional regime from quasielastic to deep-inelastic reactions. However, since in such a regime energy loss is not that large and secondary processes play less important role, genuineness of the assumption may not be significant.

The results of TDHF+GEMINI calculations with the average quantities, $\bar{E}_{N,Z}^{*}$ and $\bar{J}$, are presented in Figs. 5–8 by blue shaded histograms. As seen in the figures, this treatment provides quantitatively similar results as those obtained with the elaborated PNP calculations (shown in Figs. 1–4). For some channels, evaporation effects are less pronounced (e.g. for the 40Ca+124Sn reaction shown in Fig. 5), which might be due to the assumption that the excitation energy is shared as it is proportional to fragment masses. On the other hand, the angular momentum effects on neutron evaporation processes turned out to be negligibly small in the reactions investigated. It may play a certain role in fission processes of heavier fragments, and the validity of using $\bar{J}$ should be re-examined for such cases. Nevertheless, the results shown in Figs. 1–8 may support the usage of the average values for the evaluation of secondary deexcitation processes, significantly reducing the computational cost. In the following, TDHF+GEMINI is applied to the 64Ni+238U and 136Xe+198Pt reactions employing the simpler treatment with the average values, $\bar{E}_{N,Z}^{*}$ and $\bar{J}$, for statistical-model calculations (We will keep the usage of Eq. (25) as a simple, conservative choice).

Figure 9 shows production cross sections for the 64Ni+238U reaction at $E_{\rm c.m.}\simeq 307.35$ MeV. Although 238U is largely deformed in a prolate shape, it has been shown that MNT processes in peripheral collisions do not depend much on the nuclear orientations KS_KY_Ni-U . Here, the results of side collisions ($z$-direction case in Ref. KS_KY_Ni-U ), where the symmetry axis of 238U is always set perpendicular to the reaction plane, are shown for better visibility. (Other cases resulted in quantitatively similar cross sections.) Since the 64Ni+238U reaction has a relatively large $N/Z$ asymmetry, neutron-pickup and proton-stripping processes dominate the nucleon transfer. The agreement and disagreement are indeed similar to the 40Ca+124Sn, 40Ca+208Pb, and 58Ni+208Pb cases shown in Figs. 1, 3, 4 (and 5, 7, 8). A characteristic feature specific to the 64Ni+238U reaction is that cross sections were measured also for proton-pickup processes, ($+1p$) and ($+2p$). While GRAZING substantially underestimates those cross sections, especially for the ($+2p$) channel, TDHF+GEMINI provides significant cross sections, even greater than the experimental data. In TDHF, the latter originated from a transitional regime from quasielastic to more complicated reactions, like deep-inelastic and QF processes. Indeed, if we exclude contributions from small impact parameters ($b\lesssim 4$ fm), where energy loss is already saturated KS_KY_Ni-U , the overestimation of cross sections for proton-pickup processes can be removed, leaving cross sections for proton-stripping processes almost unaffected. Similarly, the “shoulders” due to the overestimation of neutron-pickup processes seen in Figs. 3 and 4 for the 40Ca+208Pb and 58Ni+208Pb reactions can be removed if one excludes contributions from small impact parameters, where the onset of mass equilibration process through a thick-neck formation and its breaking is observed KS_KY_MNT . However, all figures in the present paper show cross sections obtained by integration over the full impact-parameter range, avoiding any ad hoc manipulation of the obtained results.

The primary goal of this work is to predict optimal conditions to produce new neutron-rich unstable isotopes. Aiming at production of neutron-rich nuclei around the neutron magic number $N=126$, whose properties are crucial to understand the detailed scenario of the $r$-process nucleosynthesis, an experiment was recently carried out for the 136Xe+198Pt reaction Watanabe(136Xe+198Pt) . In Ref. Watanabe(136Xe+198Pt) , production cross sections for heavier (target-like) fragments were deduced from detected outcomes with respect to lighter (projectile-like) fragments, obtaining promising results. Thus, the comparison with the experimental data for this system and TDHF+GEMINI is expected to be a benchmark to check the accuracy and usefulness of the proposed method.

In Fig. 10, production cross sections for the 136Xe+198Pt reaction at $E_{\rm c.m.}\simeq 644.98$ MeV are shown. It should be noted here that the projectile and the target have very similar $N/Z$ ratios (cf. Table 1), and one expects transfer of neutrons and protons toward both directions (similar to the 48Ca+124Sn reaction shown in Figs. 2 and 6). By comparing the cross sections for primary (red filled areas) and secondary (blue solid lines) reaction products, one can see significant effects of deexcitation processes. For proton-stripping channels ($-xp$), TDHF+GEMINI reproduces the measurements surprisingly well, both the magnitude and the centroid of the cross sections for the secondary products. On the other hand, too large deexcitation effects are observed for proton-pickup channels ($+xp$). This type of disagreement is peculiar to the 136Xe+198Pt reaction. The experimental data indicate that reaction products in proton-pickup channels might be less excited compared to those in proton-stripping channels. Note that GRAZING also provides similar magnitude of evaporation effects, although the absolute value of the cross sections are substantially underestimated.

Lastly, it is to be reminded that TDHF+GEMINI also allows to evaluate production cross sections for heavier fragments, where transfer-induced fission is expected to play an important role. Detailed investigation of production mechanisms of heavy neutron-rich nuclei which are survived against transfer-induced fission as well as particle evaporation is the next step of this work.

IV SUMMARY

In this paper, a method, called TDHF+GEMINI, has been proposed, which enables us to evaluate production cross sections for secondary products in low-energy heavy ion reactions. In the method, the reaction dynamics, on the timescale of $10^{-21}$–$10^{-20}$ sec, is described microscopically based on the time-dependent Hartree-Fock (TDHF) theory. Production probabilities, total angular momenta, and excitation energies of primary reaction products are extracted from the TDHF wavefunction after collision, using the particle-number projection method. Based on those quantities derived from TDHF, secondary deexcitation processes of primary reaction products, including both particle evaporation and fission, are described employing the GEMINI++ compound-nucleus deexcitation model.

The method was applied to 40,48Ca+124Sn, 40Ca+208Pb, 58Ni+208Pb, 64Ni+238U, and 136Xe+198Pt reactions for which precisely-measured experimental cross sections are available. The inclusion of deexcitation effects, which are dominated by neutron evaporation in the present cases, changes the cross sections toward the direction consistent with the experimental data. However, there still remain discrepancies between the measured cross sections and the TDHF+GEMINI results, especially for multi-proton transfer processes. It may indicate the importance of description going beyond the standard self-consistent mean-field theory to correctly describe multinucleon transfer processes in low-energy heavy ion reactions.

In conclusion, it has been demonstrated that, even though some discrepancy still remains, the combination of TDHF and a statistical model offers an excellent starting point toward a complete modeling of low-energy heavy ion reactions. It is important to stress that, in the proposed method, there is no room to adjust the model parameters specific to each reaction: energy density functional is determined so as to reproduce known properties of finite nuclei and nuclear matter Chabanat ; GEMINI++ GEMINI++ and its ongoing developments Charity2010 ; Mancusi2010 allow a systematic reproduction of a large body of data for the entire mass region. Therefore, it will be a promising tool that can predict, in a non-empirical way, optimal reaction mechanisms to produce new neutron-rich isotopes that have not yet been produced to date.

Acknowledgements.

The author wish to thank Kazuhiro Yabana (University of Tsukuba) for useful comments on this article. The author is grateful to Lorenzo Corradi (INFN-LNL) and Yutaka Watanabe (KEK-Japan) for providing the experimental data. Lu Guo (University of Chinese Academy of Sciences) is also thanked for useful information on GEMINI++. The author acknowledges support of Polish National Science Centre (NCN) Grant, decision No. DEC-2013/08/A/ST3/00708. This work is indebted for continuous usage of computational resources of the HPCI system (HITACHI SR16000/M1) provided by Information Initiative Center (IIC), Hokkaido University, through the HPCI System Research Projects (Project IDs: hp120204, hp140010, 150081, hp160062, and hp170007).