Effect of effective interaction potentials used inquantum molecular dynamics onnucleon-induced reactions
D. N. Kadrev1,2 and Y. Watanabe1
1Department of Advanced Energy Engineering Science, Kyushu University, Kasuga,Fukuoka 816-8580, Japan
2
Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences,Sofia 1784, Bulgaria

A quantum molecular dynamics (QMD) model is applied to investigate the light-ion production in nucleon-induced reactions. The JQMD code is employed and the generalized evaporation
model (GEM) is used to account for statistical decay processes after the QMD stage. In order toimprove the ground state properties of the created nucleus other important effective interaction
potentials, such as Pauli, momentum-dependent and Yukawa are added to the JQMD. The effectof the newly added effective interactions on the total reaction cross sections and differential crosssections of light-ion production in nucleon-induced reactions is also investigated.

1Introduction

Various applications involving fast neutrons with energies over several tens of MeV haverecently been developed, e.g., dosimetry at commercial aircraft altitudes, radiation therapyof cancer, soft-error estimation in microelectronic devices, accelerator-driven transmutation ofnuclear waste, etc. Elementary interaction of a neutron with matter occurs via only a nuclear
reaction because it has no charge. As a result, the kinetic energy is transferred into matter bylight ions (p, d, t, 3 He and 4 He) and recoils produced by the nuclear reaction. As the incident
neutron energy increases, it is known that emission of light ions with relatively high energy inthe forward direction through direct and preequilibrium processes becomes important. Thus,
it is necessary to enhance our understanding of these mechanisms and to establish a reliable
model predicting accurately the energy and angular distributions of produced light ions fromboth viewpoints of fundamental nuclear physics and the above-mentioned applications.
To meet the needs, we have chosen the quantum molecular dynamics (QMD) model [1,2] as one of the models describing the preequilibrium light-ion production in nucleon-induced
reactions, and examined the applicability of the JAERI QMD (JQMD) code [2]. It has beenfound that the JQMD calculation can reproduce preequilibrium nucleon emission fairly well[2, 3], while it underestimates remarkably preequilibrium emission of light clusters (d, t, 3 Heand 4 He) [4, 5]. More recent work [5] has suggested that implementation of a phenomenologicalsurface coalescence model into the QMD leads to a reasonable description of preequilibrium
light-cluster production in neutron-induced reactions. For further refinement, it is important
to investigate how the model parameters used in the QMD calculations have an impact on thenucleon-induced light-cluster production.
It is well known that the usage of realistic initial ground state configuration in the QMD
model is essential for the high predictable ability of the studied reactions. It has been foundthat the JQMD model can reproduce satisfactorily ground state properties of nuclei, e.g. thebinding energy, root-mean-square radii, density and momentum distributions. The results show
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overestimation of the root-mean-square radii in comparison with the experimental data, underestimation of the density in the center of the nucleus and its increase in the surface region andcannot reproduce experimentally observed high-momentum component of the total momentum
distribution. One of the reasons for this is expected to be the simple effective interaction usedin the JQMD model. It includes an attractive 2-body and repulsive 3-body simple Skyrmeinteractions, Coulomb and symmetry terms. The total wave function is a direct product of
single-particle wave functions and is not antisymmetric. The fermionic properties of nucleons
are taken into account only by checking the Pauli principle in the ground state and by the Pauliblocking effect in the collision process. It has been demonstrated [6] that it is possible to devise
a Pauli potential so that the fermionic nature of the nucleons in a semiclassical manner could
be reproduced, and this potential could yield an effect similar to that of the antisymmetrizationof the total wave function.
In order to improve the ground state properties we modify the JQMD model by introducing
the Pauli potential and also other important interactions such as momentum-dependent andYukawa. The effects of these interactions on the total reaction cross sections and differential
cross sections of light-ion production in nucleon-induced reactions are investigated.

2The QMD model

The JQMD model is described in details in Ref. [2]. In this section we give only some basic
equations and ingredients which are changed or added to the JQMD.
The QMD model is a semi-classical simulation model in which each nucleon state is represented by a Gaussian wave function of width L#"(r − Ri )2i1
exp−+ r · Pi ,φi (r) =4Lh̄(2πL)3/4(1)where Ri and Pi are the centers of position and momentum of the i-th nucleon, respectively.
The total wave function is assumed to be a direct product of these wave functions.
The one-body distribution function is obtained by the Wigner transform of the wave function,
Xf (r, p) ="#(r − Ri )2 2L(p − Pi )2fi (r, p) = 8 exp −−.
2Lh̄2fi (r, p),i(2)The equation of motion of nucleons is given by the Newtonian equations
Ṙi =∂H,∂Pi

Ṗi = −∂H,∂Ri(3)and the stochastic two-body collision term.
The Hamiltonian H in the present work consists of the kinetic energy and the energy of
two-body effective interactions
H = T + Vlocal + VPauli + VMD + VYukawa ,where T =(4)Xq

m2i + P2i is the kinetic energy including the mass term. Vlocal , VPauli , VMD andi

VYukawa are the local potential, the Pauli potential, the momentum-dependent (MD) potential
and the Yukawa potential parts, respectively.

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For the local potential part we adopt the effective interaction used in the JQMD model. Itcontains the Skyrme type, the Coulomb and the symmetry terms,³√ ´XA XBe2 X
ci cjhρi i + τhρi iτ +erf |Ri − Rj |/ 4L
2ρ0 i
ρ0 (1 + τ ) i
2 i,j(6=i) |Ri − Rj |XCs+
(1 − 2|ci − cj |) ρij ,2ρ0 i,j(6=i)

Vlocal =(5)where erf denotes the error function, ci is 1 for protons and 0 for neutrons and hρi i is an overlapof density with other nucleons defined ashρi i ≡XXZ

ρij ≡j6=idrρi (r)ρj (r) =j6=iwithZρi (r) ≡X

hi(4πL)−3/2 exp −(Ri − Rj )2 /4L ,(6)j6=ihidp−3/22f(r,p)=(2πL)
exp−(r−R)/2L.
ii(2πh̄)3(7)In addition to the JQMD effective interaction we include also Pauli potential, momentumdependent term and Yukawa potential.
The Pauli potential [6, 7] is introduced for the sake of simulating fermionic properties ofnucleons in a semiclassical way. This phenomenological potential prohibits nucleons of the samespin σ and isospin τ from coming close to each other in the phase space. Here we employ theGaussian form of the Pauli potential
CPVPauli =2µh̄
q0 p0"¶3 X#(Ri − Rj )2 (Pi − Pj )2exp −δτi τj δσi σj .
−2q02
2p20i,j(6=i)(8)

The momentum-dependent term [7] is introduced as a Fock term of the Yukawa-type interaction. It is divided into two ranges so as to fit the effective mass and the energy dependenceof the real part of the optical potential, as(1)VMD =(2)Cex X
11Cex X
ρij +ρij .
22ρ0 i,j(6=i) 1 + [(Pi − Pj )/µ1 ]2ρ0 i,j(6=i) 1 + [(Pi − Pj )/µ2 ]2(9)The Yukawa potential [6, 8] has the form
½VYukawa =·|Ri − Rj |CY X exp(L/γY2 )exp(−) 1 − erf
2 i,j(6=i) 2 |Ri − Rj |γY·|Ri − Rj |− exp() 1 − erf
γYµ2L/γY + |Ri − Rj |√4L

µ2L/γY − |Ri − Rj |√4L

¶¸¾

¶¸

(10)

.

Two sampling methods are employed in the JQMD model for creation of the ground state,
random packing and frictional cooling. The random packing method accepts initial configurations which binding energy per nucleon lies within ±0.5 MeV interval of the liquid-drop modelvalue. The frictional cooling method cool or heat the system to adjust the binding energy to theliquid-drop model value. We also use the Metropolis sampling method [9], which give possibility
to obtain a ground state at certain temperature, and so to examine the thermostatic propertiesof nuclei.
The QMD simulations of nucleon-induced reactions are performed up to certain time (100–150 fm/c in our case) and then at the end of dynamical stage we switch to statistical decay
calculations. The generalized evaporation model (GEM) [10] is used to account for the decay
processes.
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3Results and Discussion

The modified QMD model has been used to explore the effect of the different effective interactions on the ground state properties of nuclei, reaction cross sections and light-ion production
in nucleon-induced reactions. Calculations have been done for several nuclei of interest, such as
12 C, 16 O, 27 Al, and 28 Si.
We use two parameter sets, which take into account different effective interactions. They arelisted in Table 1 along with the JQMD one. The first of them (denoted as P&MD) in addition
to the JQMD interaction includes Pauli and momentum-dependent potentials and is taken fromRef. [11], the other one (Yukawa) with Pauli and Yukawa potentials is from [12]. We use theTable 1: Parameters of the model for the different interactions.

A (MeV)B (MeV)L (fm2 )τCP (MeV)
q0 (fm)p0 (MeV)(1)
Cex (MeV)(1)
Cex (MeV)µ1 (MeV)µ2 (MeV)CY (MeV fm)
γY

JQMD-219.4
165.3
2.0
1.33

P&MD
-127.86204.28
1.75
1.33
140
1.644
120
-258.54-375.60
2.35
0.4

Yukawa-163.0125.95
1.0
1.67
30
5.81
400

-0.498
1.4

JQMD values for the saturation density ρ0 = 0.168 fm−3 and the symmetry energy parameter
Cs = 25 MeV.
The ground state in the JQMD and with the P&MD parameter set is generated by thefrictional cooling method. In order to investigate the thermostatic properties of the ground state
and to check the effect of the temperature on the reaction cross section, the ground state for
the P&MD parameter set is created also using the Metropolis sampling method at temperature
T = 3 MeV. For the Yukawa parameter set we use random packing because of difficulties in thecreation of stable ground state by means of the frictional cooling and adjustment of the binding
energy.
In Fig. 1 we show the density ρ(r) and momentum n(p) distributions of the ground stateof 12 C and 28 Si nuclei calculated with different parameter sets. The corresponding root-meansquare radii are listed in Table 2. The QMD simulations are performed up to 150 fm/c andthe averaged quantities over 1000 events are plotted. The results are compared with the empirical density distribution of Negele [13] and with “experimental” total momentum distribution
deduces by y-scaling analysis [14] of inclusive electron-scattering data. The results of both parameter sets show improvement of the ground state properties in comparison with the JQMDones. We have higher density in the center of nucleus and not so wide surface shape, root-meansquare radii are smaller and closer to the experimental ones, and the momentum distributions
develop a high-momentum component. The inclusion of the Pauli and momentum-dependent
potentials shows, for the 12 C case, fair agreement of the density distribution and root-mean4
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square radius with the empirical ones. Increasing of the temperature of the system leads torearrangement of the nucleons from the center to the periphery. The momentum distribution of
12 C calculated with Yukawa parameter set is in better agreement with the experimental data.
0.2
100

12C0.15exp
JQMD
P&MDP&MD T=3
Yukawa10-2

0.05

n(p) [fm3]ρ(r) [fm-3]

C

10-1

Negele
JQMD
P&MD
P&MD T=3
Yukawa

0.1

120
0.2
28

10-3
10-4
10028Si

Si

0.15

10-1

0.1

10-2

0.05

10-3

001

2

3

45

6

7

10-4

00.5

1

1.5

2

2.5

3-1r [fm]p [fm ]

Figure 1: Density (left) and momentum (right) distributions of the ground state of 12 C and 28 Si
obtained with different parameter sets.

Table 2: Root-mean-square radii (in fm) of 12 C and 28 Si obtained with different parameter sets.
Nucleus12 C28 Si

JQMD
3.082
3.488

P&MD
2.771
3.413

P&MD T = 3 MeV
2.987
3.447

Yukawa2.555
3.265

Negele
2.707
3.165

A comparison between QMD calculations of the total reaction cross section for the protoninduced reaction on 12 C is shown in Fig. 2. The results are averaged quantities over time
evolution up to 100 fm/c and over 104 events. The experimental data are taken from Ref. [15].
The JQMD model describes satisfactorily the reaction cross section, especially at high energies.
The QMD calculations with the Pauli and momentum-dependent parameter set underestimate
the experimental cross section for whole energy spectrum, but reproduce very well the energy
dependence. We can improve the results by increasing the temperature of the ground state. The
calculations with the Yukawa parameter set show good agreement with the experimental data.
Similar results have been obtained for the 27 Al nucleus.
The results with the P&MD parameter set for angle-integrated energy spectra of light-ion
production for 28 Si in neutron-induced reaction at incident energy 96 MeV are presented in Fig. 3.
The QMD simulations are performed with 106 events and up to 100 fm/c. It can be seen that
the JQMD results reproduce very well the proton production, but show large underestimation
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600

p + 12C500

exp
JQMD
P&MD
P&MD T=3 MeVYukawa

σR [mb]400300200

1000101102103

Ep [MeV]Figure 2: Total reaction cross section of 12 C for incident proton.
for other light clusters. The introduction of Pauli and momentum-dependent potentials withpresent parameters doesn’t change significantly the description of light-ion emission. The protoncross sections are slightly reduced, while for the cross sections of other light clusters we havesmall enhancement. The results for the double-differential cross sections are similar. Thus, thepresent work indicates that the effective interaction potentials used in QMD calculations havelittle influence on preequilibrium light-cluster production in neutron-induced reactions. It should
be noted that agreement with the experimental data is fairly improved by implementation of aphenomenological coalescence model into the QMD calculation as shown in our recent work [5].

4Conclusions

The JQMD model is used to explore the preequilibrium light-cluster production in nucleoninduced reactions at intermediate energies. In order to improve ground state properties of nuclei,
new effective interactions, for instance Pauli, momentum-dependent and Yukawa, are added tothe nuclear Hamiltonian. Two parameter sets are used, the first one includes the Pauli andmomentum-dependent potentials, the second one – the Pauli and Yukawa potentials.
The inclusion of the Pauli potential and momentum-dependent interaction leads to enhancement of the density distribution and the root-mean-square radii of nuclei. Calculations show
underestimation of the proton-induced total reaction cross section but the energy dependence
is reproduced rather well. Increasing of the temperature of the system leads to enhancement ofthe results.
The introduction of the Yukawa potential improves the nucleon momentum distribution, it
develops a high-momentum component in accordance to the experimental data. The protoninduced total reaction cross section is reproduced rather well.
Newly added interactions with present parameter sets don’t affect significantly the JQMD
description of the light-cluster emission in nucleon-induced reactions.

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28Si(n,x)X @ 96 MeVproton

deuterontriton110-1

d /dE (mb/MeV)

10exp
JQMD
P&MD-31020
31104HeHe406080

100

E (MeV)-1

10-310020406080

020406080

100

E (MeV)

Figure 3: Angle-integrated energy spestra of light ions produced from neutron-induced reaction
on 28 Si at 96 MeV. The experimental data are taken from [16].

Acknowledgments
The authors would like to thank Drs. K. Niita and S. Chiba for valuable comments on
this work. D.N.K. is grateful to the Japan Society for the Promotion of Science (JSPS) for a
postdoctoral fellowship. This work is partly supported by the Grants-in-Aid (No.17-05374) for
Scientific Research from JSPS.

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